# data[].attributes.body
Video

# data[].attributes.attachment_link.readable_text
Are you the driver in the lot who parks in the first spot you see? Or do you circle around and around looking for a spot by the door? Physicists Paul Krapivsky of Boston University and Sidney Redner of the Santa Fe Institute explored the mathematics of parking. The research required different equations and simulations to model the benefits of the various parking approaches. From EurkeAlert!:

In their paper, Krapivsky and Redner map three simple parking strategies onto an idealized, single row parking lot. Drivers who grab the first space available follow what the authors call a "meek" strategy. They "waste no time looking for a parking spot," leaving spots near the entrance unfilled. Those who gamble on finding a space right next to the entrance are "optimistic." They drive all the way to the entrance, then backtrack to the closest vacancy. "Prudent" drivers take the middle path. They drive past the first available space, betting on the availability of at least one other space further in. When they find the closest space between cars, they take it. If no spaces exist between the furthest parked car and the entrance, prudent drivers backtrack to the space a meek driver would have claimed straightaway.

So which strategy is best? As the name suggests, the prudent strategy. Overall, it costs drivers the least amount of time, followed closely by the optimistic strategy. The meek strategy was "risibly inefficient," to quote the paper, as the many spaces it left empty created a lengthy walk to the entrance.

This DIY hovercraft styled like the Doc Brown's DeLorean in the Back to the Future films is up for auction on Bring A Trailer. Current bid is $22,500, half of what maker Matt Riese was asking when I posted last year that the DeLorean was listed on eBay. Apparently he's done quite a bit of [...]

# data[].attributes.attachment_link.title
Video: the mathematics of where to park your car

# data[].attributes.attachment_link.description
https://www.youtube.com/watch?v=6W7UrAKcvQw Are you the driver in the lot who parks in the first spot you see? Or do you circle around and around looking for a spot by the door? Physicists Paul Krapivsky of Boston University and Sidney Redner of the Santa Fe Institute explored the mathematics of parking.

# data[].attributes.body
Article

# data[].attributes.attachment_link.readable_text
Indeed, in the eyes of Euclid, mathematics was so important because it is purely truth-oriented, and it has the beauty of art and value of abstract thought. His mathematical approach still stands as a perfect model of reasoning. He did something special that had not been done before. We may say his works were the beginning of the consideration of mathematics as an analytic deductive subject. He taught us a valuable lesson; that when we feel intuitively something is true, we need to prove that it is always true for everyone. He showed us the power of proof and a logical path that help us find universal truths. He turned mathematics into a subject that proves things with a hundred percent certainty and can be applied in diverse situations.

When you read Euclid's , you will notice that Euclid's mathematical approach is unique and simple. He starts with basic assumptions such as if this is true, then this must be true or if this is wrong, then the opposite of it has to be true. Then he either proves or disproves his assumptions, and concludes by writing the results as a theorem. What is important here is that Euclid chose to go for universality. He didn't find temporary solutions for specific issues, he made a remarkable change and made the solutions universally applicable.

Let's take a look at Euclid's proof about prime numbers. There's nothing particularly striking about prime numbers other than the fact that there are infinitely many of them. We are not 100% sure but have reason to believe that Euclid was the first human to prove there are infinitely many primes. His proof was also, most probably, the first proof ever in mathematics. However, it is important to mention that Euclid never explicitly wrote "there are infinitely many primes"; instead, he wrote: " prime numbers are more than any assigned multitude of prime numbers" (Clark University Mathematics, " Proposition 20")The reason for this strange wording was due to the fact that the idea of infinity was different than it was today and was a developing concept. [Translate from book IX]

I'm sure you've all come across prime numbers. Before giving Euclid's unique proof, we should talk about prime numbers a little bit because the definitions are important parts of understanding mathematics. So, what is a prime number?

Definition: A prime number is a whole number bigger than one that's divisible only by 1 and itself.

The number 1 is an exception to this rule. Although 1 satisfies all the conditions to be prime, we don't assume that it is a prime number for good reason. It is because mathematicians need to make practical definitions. If 1 is considered a prime number, then when you do a prime factorization for any number, you encounter a problem. For instance, if you do prime factorization for 15, you need to write: 18=2x3x3x1x1x1x1x1x1x1x1x1x1x1x1x1x1x1.... This is a good enough reason to say 1 is not prime.

When Euclid discovered these intriguing numbers, he researched them and decided to reveal some mysteries. To begin with, he wrote the first few prime numbers on a parchment paper. For the sake of example, let's say he wrote up to 100. Then he started looking for patterns because all mathematicians are interested in patterns. For instance, he saw that 2 is the only even prime number because all other even numbers larger than 2 were divisible by 2. Likewise, 3 was a prime number, but multiples of 3 couldn't be prime because he could divide them by 3.

Then he asked himself: "If I continue writing, will I ever stop?" He was hoping to have infinitely many prime numbers because otherwise, life would be tedious for him or any other mathematician after him. It was a very challenging question because he had to painstakingly check each number to see whether it was prime or not. There were no computers to do the calculations for him. Yes, it is quite easy to calculate whether 13 is prime or not, but after a certain point, each digit would take days or weeks... Even if we had the chance to donate the most powerful computer in the world to Euclid's institution, it still wouldn't be enough to quell his curiosity. The computer could find a very large prime number, but we still wouldn't know whether that's the biggest prime number. So, asking a computer to find big primes for us is never going to resolve the question of the biggest prime number.

Euclid had found numerous truths just by using the mathematics of his time. Since believing something was not enough to convince people, he had to look for perfect certainty one more time. Only then could he could approach a mathematical way to do it. He just needed a clever idea and end up with an elegant proof. That's why he defined a theorem at first.

Theorem: Prime numbers are more than any assigned multitude of prime numbers.

The proof was of great significance to Euclid because his theorem needed to be sound. His plan was to use a thought experiment, which is a mathematical technique called proof by contradiction. First, he imagined that he lived in a parallel universe where there are finitely many prime numbers. Thus, he could write them on a list. It might be a very long list, but prime numbers are finite in his universe. He didn't know the largest prime exactly, so he called it "p". He got a very large piece of paper and wrote all the primes in the world. His list started from 2, 3, 5, 7 to all the way up to "p" which is [theoretically] the largest prime number. Then, Euclid came up with a fantastic idea: "I am going to multiply all those numbers together and add 1".

He didn't know what that number was, but it was a product of all the primes in his new world plus 1. He already knew that that number had to have a prime factor because every number bigger than 1 has to have a prime factor. [The Fundamental Theorem of Arithmetic] There remains the possibility that this number was itself prime.

The Fundamental Theorem of Arithmetic: Every integer greater than 1 can be expressed as a product of primes in an essentially unique way. In other words, they are the building blocks from which all numbers are made. Teachers like to say, "Primes are the atoms of mathematics."

So, Euclid needed to check. Could that prime factor be 2? The answer was no because that number was 2 times some other number plus 1. So it had to leave the remainder 1. Could that prime factor be 3? The answer was no again because that number is 3 times some other numbers plus 1 and it leaves a remainder of 1. Could that prime factor be 5? No, because that number is 5 times some other numbers plus 1 and it leaves a remainder of 1. For each prime, the same thing was going to happen.

Euclid's act of genius made his new number always indivisible by any of the primes on his list. Euclid's number always left a remainder of 1 when he divided by any of the prime numbers he found. However, that number had to have a prime factor, as mentioned earlier. Thus, his logical argument reached an absurdity that this cannot happen. So there was a contradiction in his parallel universe which cannot exist- there must be infinitely many primes.

What Euclid did a long ago was so beautiful because our finite mind could reach infinity with this approach. He expanded our horizons of knowledge. As I said above earlier, the view from Euclid's window was a revolution that would be extended into space. For us, the possibilities of expanding the mathematical frontier should be thrilling.

# data[].attributes.attachment_link.description
Indeed, in the eyes of Euclid, mathematics was so important because it is purely truth-oriented, and it has the beauty of art and value of abstract thought. His mathematical approach still stands as a perfect model of reasoning. He did something special that had not been done before.

# data[].attributes.headline
A selection of apps supporting Japanese problem solving approach

# data[].attributes.subject
A selection of apps supporting Japanese problem solving approach

# data[].attributes.body
Website links

# data[].attributes.attachment_link.title
【東京書籍】数学|Dマークコンテンツ|平成28年度用

# data[].attributes.attachment_link.description
東京書籍が発行する，平成28年度用 中学校教科書についてのご紹介ページです。ゲーム等のサンプルを実際にご覧になれます。

# data[].attributes.body
Research article

# data[].attributes.attachment_link.readable_text
Toshiakira Fujii

Original Article

While the history of Lesson Study in Japan spans more than a century (Makinae, 2010), for Japanese educators, Lesson Study is like air, part of everyday school life. This situation possibly explains why Lesson Study is regarded as being under-theorised (e.g. Elliott, 2012). Educators outside Japan however, having had to learn about Lesson Study less naturally, may sometimes lose some important aspects of Lesson Study.

Lesson Study came to the attention of educators outside of Japan primarily through the publication of The Teaching Gap (Stigler and Hiebert, 1999), which described findings from the TIMSS video study focussing on the eighth grade mathematics lessons in USA, Germany, and Japan. Chapter seven in particular, titled "Japan's approach to the improvement of classroom teaching", which is based on Yoshida's ( 1999) doctoral dissertation, now available in book form (Fernandez and Yoshida, 2004), provoked enormous interest, not only in Lesson Study, but also in the typical structure of Japanese mathematics lessons. Independently, some educators such as Lewis also noticed the significance of Japanese Lesson Study (Lewis and Tsuchida, 1998).

Since then many mathematics teachers and teacher educators around the world have been involved in Lesson Study, and many books and research papers have been written on various aspects of Lesson Study (Lewis, 2002; Lewis et al., Lewis and R, Perry., & J. Hurd, 2009; Hart, Alston and Murata, 2011; Doig and Groves, 2011; Department for Children, Schools and Families, 2008; White and Lim, 2008; Ono and Ferreira, 2010). However, some aspects of Lesson Study, that may be taken for granted by Japanese teachers, seem not to be well understood outside Japan.

This paper aims to clarify the role and function of lesson planning in the Lesson Study process, based on case studies conducted in three schools in Tokyo.

2.1 The Lesson Study process

Lesson Study is an approach to teacher professional development that differs sharply from the professional development practices common in other countries. Liptak (cited in Lewis, 2002, p. 12) contrasted Lesson Study with traditional professional development as practised in the United States, as shown in Table 1.

Lesson Study begins with a question, not with an answer prepared by someone else. Identifying this question, which becomes the research theme for Lesson Study, is the first step in the process (see Fig. 1).

The research theme is developed through consideration of the reality of students' current state vis- à- vis educational or long-term goals for their learning and development.

The second step of Lesson Study is to develop a plan to address the research theme through lessons. This means making an instructional plan for a selected unit and a detailed plan for one of the lessons in that unit in which the planning team puts forth their ideas about how to address the research theme while teaching specific academic content. That lesson is called the research lesson.

The third and fourth steps in Fig. 1, conducting the research lesson and having a detailed discussion about the lesson, occur in one day-a big event day for the school. Typically, it is done in a half day; one class of students stays for the research lesson while the other classes are dismissed so that every teacher can come to observe the research lesson (even the school nurse and school nutritionist usually attend). At the end of the post-lesson discussion, usually there will be final comments lasting 30 min or more by a "knowledgeable other" from outside the school, who has been invited for this purpose.

The fifth step is to reflect on the process and consolidate and carry forward the learnings from it. Teachers will usually write their reflections and publish records of Lesson Study activities in the school bulletin.

Because they are the most visible aspects of Lesson Study, some people think of the research lesson and post-lesson discussion as the most important parts of Lesson Study, or even use "Lesson Study" to refer to the research lesson alone. However, these are just two of the five components of Lesson Study.

The Lesson Study cycle, with its five steps as illustrated in Fig. 1, contrasts with similar diagrams in other publications that have four steps (e.g. Lewis, 2002; Lewis and Hurd, 2011). These five steps, while overlapping with the four steps in the other diagrams, more accurately portray the reality of Japanese teachers' Lesson Study activity by having a closer correspondence between the titles of the steps and the activities undertaken by teachers.

Borrowing from Lewis' ( 2002) and Lewis and Hurd's ( 2011) descriptions, each step can be summarized as follows:

4.

Post- lesson discussion In a formal lesson colloquium, observers share data from the lesson to illuminate student learning, disciplinary content, lesson and unit design, and broader issues in teaching and learning.

There are three types of Lesson Study in Japan: School-based, District-based, and National-level Lesson Study. According to Takahashi ( 2006), participants' motivations or interests are different in these types of Lesson Study, but the cycle itself is basically the same. The difference is in the range, or scope, of students to be considered: school-based Lesson Study is concerned with students in the school; District-based Lesson Study is concerned with students in the district; and National-level Lesson Study is concerned with the reality of students across the country, and has a research theme with a nationwide view. Lesson Study is sometimes introduced as an open lesson by a veteran teacher "jumping in" to another teacher's classroom (Takahashi, 2013, p. 84). A "jumping in" lesson is just a demonstration unless the veteran teacher has a clear goal for the lesson as in Step 1, and proposes a new idea or content to be teachable, or he or she wants to demonstrate students' potential to be greater than ordenary teachers believe, so that he, or she, plans the lesson carefully as in Step 2. This kind of Lesson Study exists in Japan and in this case the collaboration among teachers is not a critical part of Lesson Study. In any case, each step in the Lesson Study cycle is closely related to the others, with the third and fourth steps particularly related to the first and second.

In school-based Lesson Study, which is the focus of this paper, the typical Lesson Study cycle begins at the end of an academic year-i.e. in February or March in Japan-when the faculty decides upon a research theme for the next school year, which starts in April. Several research lessons are scheduled from, say, May to November. Each research lesson and its post-lesson discussion occupy only one day, but the teachers reflect on what they learned at the research lessons and usually write a booklet or long summary report by the end of school year.

While the importance of a lesson plan as a product of Lesson Study is certainly understood, compared to the research lesson, of which there are many public examples, the collaborative work of Japanese teachers in creating a lesson plan is generally mysterious, because it is difficult to observe. According to Lee and Takahashi ( 2011) "Lesson plans are central resources for these teachers in that they constantly refer to, problematize and act on them during the entire cycle of the [Lesson Study] procedure" (p. 210).

Japanese teachers spend a lot of energy and time crafting a lesson plan. Although the details vary from school to school and even from teacher to teacher, Lewis ( 2002, pp.127-130) notes that a typical template for a lesson plan for a research lesson in Japan consists of the following:

The Japanese term for the document created for a research lesson is gakushushido- an (学習指導案), which is usually translated as "lesson plan". In this paper we will use that common translation, although we prefer the phrase "lesson proposal", because the document is much larger and broader in scope than what is usually meant by "lesson plan". Also the word "plan" may imply a fixed script, but in Japanese Lesson Study the teacher is expected to use his or her judgment if students respond in unanticipated ways. As Lee andTakahashi (2011) argue, researchers have taken for granted that using lesson plans, no matter how well devised, always involves judgment and interpretation, as teachers and their students face the contingencies of the lesson in the classoom. Their empirical study, in the context of Lesson Study, provided analytic descriptions of the interactive processes through which lesson plans are realized, leading to the conclusion that "classroom teachers use lesson plans as communicative resources to identify problems, specify assumptions about their teaching, and act on the evolving contingency of classroom interaction" (p. 209). However, Lee and Takahashi ( 2011) did not describe details of planning the lesson, including how teachers adapted or designed the task for the lesson, or how many hours they spent on planning.

In the context of Lesson Study, Lewis, Perry and Hurd (2009) focussed on one US lesson study group, of six teachers from five different schools, that conducted a research lesson in a 2-week summer workshop. This is an experimental situation, which is different from the Japanese traditional school-based Lesson Study setting. However it is worth considering in terms of the lesson planning activity. They documented that the group spent a total of six hours planning the lesson: "select research lesson, do task and share solutions, anticipate student thinking, write instructional plan using template" (Lewis et al., 2009, p. 290). However they have not offered descriptions of how they designed or adapted the task for the lesson.

On the other hand, Fernandez and Yoshida ( 2004) described in detail the process of planning lessons in the context of Lesson Study. This ethnographic study, focussed on a local elementary school in Hiroshima, vividly shows Japanese teachers' activities. However, the Lesson Study described there has the rather unique feature in that, following the research lesson being taught by a young inexperienced teacher, observed by the whole school and discussed by only the lower grade group of teachers and the principal, the lesson was revised by these teachers and then re-taught by a veteran teacher, with the whole school and an ouside advisor observing the lesson and taking part in the post-lesson discussion. The notion of Re- Teaching is extremely problematic and sensitive. In fact, the need to revise and re-teach a lesson is one of the misconceptions identified in foreign countries implementing Japanese Lesson Study (Fujii, 2014b). Whether Re- Teaching exists or not in the Lesson Study process affects the nature of the planning and the discussion of the lesson.

2.2 Structured problem solving

The structure of Japanese mathematics lessons is often regarded as unique by non-Japanese eyes, with researchers from outside Japan having noted patterns in Japanese mathematics lessons. For example, Becker et al. ( 1990) identified eight components in a typical Japanese mathematics lesson, while Stigler and Hiebert ( 1999) identified five components and labelled these lessons as structured problem solving. But their points of view are those of observers, while Japanese teachers usually do not think about the structure of their lessons in the same way. For instance, the first component of Stigler and Hiebert ( 1999), reviewing the previous lesson, is not an important activity from a Japanese teacher's point of view. Instead Japanese teachers typically consider a mathematics lesson as problem solving in terms of the four phases shown in Table 2 (see, for example, Shimizu, 1999).

This type of lesson imposes certain demands on how to interpret the lesson plan. Phase 1, presenting the problem, means helping students understand the context of the problem or task and what it will mean to solve the task-but it specifically excludes any exposition by the teacher about how to solve the task. Instead, students are expected to work independently on the task for 10-20 min (phase 2). Therefore teachers need to discuss the appropriateness of the task described in the lesson plan. The third phase, called neriage in Japanese, assumes that students will arrive at different solution methods and focusses on a comparison and discussion of those different solution methods. Therefore teachers need to discuss the plausibilty of the anticipated student solutions listed in the lesson plan. In the fourth phase, matome, the teacher may say something about which strategy may be the most sophisticated and why, but it should go beyond that to include comments by the teacher concerning the mathematical and educational values of the task and lesson (Fujii et al., 1998). Therefore teachers need to discuss the resonableness of the matome by the teacher as foreshadowed in the lesson plan. For a lesson to work in this way, the task should be understandable by the students with minimal teacher intervention; it should be solvable by at least some students (but not too quickly), and it should lend itself to multiple strategies.

This paper focusses on the second, planning step in the Lesson Study cycle, and aims to illuminate the nature of the collaborative work among teachers, based on three case studies where re-teaching was not part of the Lesson Study process, with particular emphasis on planning for these four phases of the research lessons.

This research took place in three local public elementary schools in Tokyo, which will be referred to as schools M, S and T. These schools were participating in the International Math- teacher Professionalization Using Lesson Study project (IMPULS), a recently established project funded by the Ministry of Education, Culture, Sports, Science and Technology of Japan, located at Tokyo Gakugei University, Tokyo. The purpose of this project is two-fold. First, as an international centre of Lesson Study in mathematics, Tokyo Gakugei University and its network of laboratory schools help teacher professionals learn about authentic Japanese Lesson Study, and thereby prepare them to create Lesson Study systems in their own countries for long-term, independent, educational improvement in mathematics teaching. Second, the project conducts research projects examining the mechanism of Japanese Lesson Study in order to maximize its impact on schools in Japan.

Although several research lessons were scheduled for each year, this study focusses on just one research lesson at each of these schools, and the planning meetings for those research lessons-that is, just one lesson study cycle in each school.

The author observed each lesson-planning meeting and took fieldnotes. In addition, each lesson-planning meeting was video-recorded and later transcribed; and all lesson plans and revised versions were collected and analyzed with respect to their evolution.

This paper provides a descriptive analysis of the planning process undertaken by these groups of teachers in preparation for the research lessons. In a similar vein to the research carried out by Lee and Takahashi ( 2011), discourse-in-interaction analysis (Sacks et al., 1974) was used to examine "the methods and procedures by which participants carry out ordinary tasks of classroom teaching and collaboration among teachers" (Lee and Takahashi, 2011, p. 215). The analysis began with unmotivated looking (Sacks, 1992) during the observations of the planning meetings in order to identify key discussions that eventually led to consensus regarding the lesson plans.

Through this overview of the lesson planning processes, the author came to realize that the discussions were based on the flow of the lesson. In particular, it seemed that teachers could imagine or visualize clearly what would happen at the research lesson through reading the lesson plan. Therefore it was clear that this study could focus on analyzing the planning of the flow of the research lesson.

Based on the flow of Japanese problem-solving lessons, thematic content analysis (see, for example, Fereday and Muir-Cochrane, 2006; Braun and Clarke, 2006) was carried out on transcripts of the lesson planning discussions. Using the framework of the four phases of problem-solving lessons (Table 2), participants' comments were coded with appropriate keywords to track their views of the lessons. These comments were examined with respect to the role of the lesson plan and planning meetings, in order to make visible an important part of Lesson Study-namely the planning process.

The following section is organized according to the main results obtained through the inductive process of examining the trajectory of revising lesson plans, transcribed records of planning meetings, research lesson, and post-lesson discussion, and field notes.

The results of this study are presented in three sections. First, we report on the lesson planning meetings overall-e.g. the number of meetings and participants, and the duration of meetings. Second, we examine the major component of the meetings. Finally, we identify major concerns at the meetings, such as the appropreateness of the task for the lesson, anticipated student solutions, and how to organize the comparison and discussion phase in the lesson.

4.1 The lesson planning process overall

The dates of the research lessons held at school M, S and T, together with the dates of the planning meetings are shown in Table 3. The planning meetings began between 4 and 6 weeks before the research lessons. Two schools, M and S, had four planning meetings and school T had just two meetings.

It should be noted that there was no rehearsal or trial implementation of a tentative lesson plan between planning meetings. It should be noted also that this schedule fails to reveal the amount of time that the teachers may have spent thinking about their research lesson beforehand, since the grade, unit, and lesson may have been selected at the end of the previous academic year in March.

Table 4 shows the number of participants at each of the planning meetings.

In the case of school M, the regular members of planning meetings were: the leader of the research steering committee, who also chaired the meeting and was the lead teacher for mathematics in the school; three Grade 3 teachers, one of whom taught the research lesson; and four Grade 4 teachers-a total of eight participants. The first planning meeting, held in the principal's office, was rather informal, since the knowledgeable other, who had given a talk at a research lesson that day, joined the meeting, together with the principal of the school. Beside these two participants, three Grade 3 teachers and two Grade 4 teachers attended. But at later meetings, in the school conference room, the only participants were the eight regular members.

At school S, which is a small school with only one class at each grade, the first meeting included five regular members: two classroom teachers for Grades 5 and 6, the music teacher, the art teacher, and the teacher for mathematics. The Grade 6 teacher was the leader of the school research steering committee and taught the research lesson. In Tokyo, in the case of mathematics only, if a school wants to divide classes into two or three groups for teaching mathematics, in order to help cater for individual differences, the school gets an extra teacher-in this case this teacher. The music teacher and the art teacher were teaching Grade 5 and 6 students, therefor the regular members were the upper year level team. At the second meeting, the principal joined them; at the third meeting, the knowledgeable other also joined; but the fourth meeting included only the Grade 1 teacher and the Grade 6 teacher, the music teacher, and a special needs teacher-these four constituted the school research steering committee. The venue was always a meeting room in the school.

At school T, regular members were the leader of the research steering committee, three Grade 3 teachers and three Grade 4 teachers, and the principal of the school, who attended the planning meetings-so the total number was 8. One of the Grade 4 teachers taught the research lesson. There were only two meetings, both of which were held in the principal's office.

School M, S, and T each organized a research steering committee. According to Takahashi and McDougal ( 2014), a research steering committee in Japan consists of representatives of each grade level and, in the case of the Lesson Study focussing on mathematics, the lead teacher for mathematics. In addition, representatives of special subject teams, such as music, science and home economics may join. The research steering committee leads the school's efforts and maintains the cohesion of ideas across the grades. Takahashi and McDougal ( 2014, p. 16) list roles and functions of research steering committees as follows (parenthesis added by author):

Developing a master plan for the school research;

Scheduling and leading monthly meetings to find strategies to address the school's research theme based on the ideas of the teachers;

Publishing a monthly (not always the case) internal newsletter to record the findings from each research lesson;

Planning, editing, and publishing the school research reports, including those for the research open house; and

Arranging for knowledgeable others to present lectures, teach demonstration lessons (not always the case), and give final comments at research lessons.

As shown in Table 5, the duration of the planning meetings ranged from a minimum of 30 min to a maximum of 128 min.

The chairperson of the school research steering committee led most of the meetings at schools M, S, and T. As these schools were conducting Lesson Study focussing on mathematics, the lead teacher for mathematics tended to also be in charge of the school research steering committee. Besides regular members from the school, the knowledgeable other, who had given comments on a research lesson that day, attended the first meeting at school M and the third meeting at school S. Involving a knowledgeable other in this way is common; after a research lesson and discussion ends, the team responsible for the next research lesson will meet with the knowledgeable other for further discussion and to get advice for their lesson.

As both of the 30-min meetings were with the knowledgeable other, these could be regarded as atypical. The average duration was 72 min, with the average duration excluding the 30-min meetings being 83 min.

One reason that may account for the differences in the duration of planning meetings between schools could be that the principals of schools S and T attended and participated actively in these meetings, with teachers in both schools appearing to have great confidence in them. When teachers asked, these principals gave suggestions to help break deadlocks. As a result, the duration could become shorter. In the case of school M, some of the regular members of planning meetings were young and inexperienced. Therefore, the leader of the research steering committee, who was also the lead teacher for mathematics, sometimes needed to explain the position of the lesson in the scope and sequence of the Japanese course of study, and the mathematical value of the task for use in the lesson. These factors may have had an effect on the longer duration of the meetings.

4.2 Major components and structure of the planning meetings

The first meetings held at school M and S were unusual in that the teachers discussed ideas about the research lesson in depth without a written lesson plan. At all other meetings, the discussion was based on a draft lesson plan, which had been written, either with or without the support of colleagues, by the teacher who would be teaching the lesson. Furthermore, the flow of the planning meetings followed the flow of the lesson plan. Other issues, such as the logistics of the research lesson or post-lesson discussion, were not discussed.

The format of the first draft of the lesson plan for schools M, S, and T was basically the same as Lewis' ( 2002) template as described earlier in this paper. In the case of school M, component 5 in Lewis' ( 2002) template, Learning plan for the unit, was missing at the beginning, but was added later.

Among the seven components in Lewis' ( 2002) template, component 6, Plan for the research lesson-which we will refer to here as Planning the flow of the research lesson in order to distinguish it from the overall lesson plan-is the most prominent in terms of both quantity and quality. At school T, the draft lesson plan had already been prepared for the first meeting, written by the teacher who was to teach the research lesson. The items discussed at the first meeting were as follows:

These items were exactly the items written in the draft lesson plan.

In both meetings at school T, discussion relating to planning the flow of the research lesson occupied the majority of the time: 51 min (65 %) of the first meeting as shown above, and 87 min (78 %) of the second meeting.

At school S, the first meeting was held without a written lesson plan. At this stage, teachers had not yet decided exactly which unit or content to teach for the research lesson and how. From the second meeting onwards, the teachers' discussions were based on the lesson plan drafted by the teacher who was to teach the research lesson. The knowledgeable other attended the third meeting. Excluding the first and third meetings, the proportion of time spent on planning the flow of the research lesson was 74 %, while when all four meetings are included, 52 % of the time was spent on planning the flow of the research lesson.

At school M, the first meeting was also held without the written lesson plan. From the second meeting onwards, the discussion was based on the draft lesson plan which had been written mainly by the teacher who was to teach the research lesson, but as a team, with support from the third grade teachers. In the second, third and fourth meetings, the proportion of time spent planning the flow of the research lesson was 74 %, while if the first meeting is included the proportion was 66 %. Across the three schools, omitting meetings without the lesson plan, the average proportion of time spent on planning the flow of the research lesson was 72 %; while if all meetings are included the proportion was 63 %.

Thus we have two findings: one, that the planning meetings followed the structure of the lesson plan; and two, that the discussion among teachers was particularly focussed on planning the flow of the research lesson.

The discussions specific to the flow of the research lesson during the planning meetings at the three schools could be aligned with the four phases of a problem-solving lesson (see Table 2). For example, at the second meeting at school S, a discussion on how students might grasp the given task (15 min) was related to phase 1, Presenting the problem for the day; discussion about likely student responses (14 min) was related to phase 2, Problem solving by the students; discussion about how to organize the comparison and discussion period (15 min) was obviously related to phase 3, Comparing and discussing; and discussion about how to conclude the lesson (5 min) was related to phase 4, Summing up by the teacher. Of the 49 min focussed on the flow of the research lesson, the proportions of time related to these four phases was approximately 31, 29, 31 and 10 %. The other two schools showed a similar pattern.

In the next section we will present, in more detail, what the teachers talked about regarding each phase of their lessons.

4.3 Major concerns when planning the flow of the research lesson

Discussions by the teachers, while planning the flow of the research lesson, were classified into three key categories: Appropriateness of the task, Plausibility of the anticipated student solutions, and Quality of the comparison and discussion ( neriage) phase.

4.3.1 Appropriateness of the task

Discussions about the task for the research lesson can be classified into two types. One is discussion about the task and unit from an advanced mathematical perspective, where teachers clarify the scope and sequence of relevant topics, or relationships within and expansion of the content. The second is to discuss the appropriateness of the task to the goal of the lesson, including detailed consideration of the numbers in the task, the context of the task, and so on.

When teachers talked about the position of the unit within the curriculum, they carefully referred to the National Course of Study ( 2008) published by the Ministry of Education, Culture, Sports, Science and Technology. According to Lewis' ( 2002) typical lesson plan template, this discussion is related to "connections to standards and prior and subsequent learning", which is included in the fifth componentof the template, Learning plan for the unit, where related units in former and later grades are explained and shown by using a diagram. In fact, teachers at school M used their own diagram as they discussed why the unit was important and as they traced the students' learning path leading to the unit. In the case of school S, at the second meeting where teachers talked about sequence of units, they recalled an old version of the National Course of Study (1998) in which "speed" was placed in fifth grade. "Speed" was now in sixth grade in the National Course of Study ( 2008). In fact, one teacher said "At fourth grade we teach multiplication and division of decimal numbers, and in fifth grade we teach the size of per-unit quantities.1 The closest content to speed is size of per-unit quantities.... We used to teach speed in fifth grade, together with the size of per-unit quantities".

Teachers also talked a lot about the task itself. The tasks in all three cases were not directly from textbooks; they were newly created, or modified from tasks in the textbook. Teachers discussed why they selected the particular tasks; what roles the tasks were expected to play in the unit; what benefits students might gain from solving the tasks: whether it helped to develop a new concept, a new way of thinking, or some important procedure.

The discussion of the curriculum was closely related to the solution of the task, because related content in the curriculum was expected to be a resource for students to solve the task. For example, in the second meeting in School S, there was the following exchange:

This kind of detailed and concrete consideration of previously-learned content was observed in all three schools.

Teachers also engaged in detailed discussions about the task itself, including which numbers to use and why. This aspect of Lesson Study was noted by Stigler and Hiebert ( 1999), who reported that teachers would talk about the "problem with which the lesson would begin, including such details as the exact wording and numbers to be used" (p. 117). However, the selection of numbers is not always from a purely mathematical point of view.

For example, in the case of School S, teachers thought about numbers both in terms of their students' reality and also from a procedural or calculation point of view. The teacher who would teach the research lesson said:

Child A in the problem can run 40 metres in only 6 s. In my class there is no such fast runner. However I decided to use these numbers, because these numbers are easier for children to calculate.

Time and distance data for the first three people in the problem (A, B, C) were not changed, but data for two people (E, F) were changed from E (42 metres in 6.7 s), F (28 metres in 4.9 s) to E (45 metres in 6.5 s), F (50 metres in 8 s), in order to provide some faster speeds. Numbers for D, E, and F were considered hard for students to calculate and the teachers also worried about having decimal fractions as the result of calculations. However, they decided to keep the numbers and let students use calculators if they wanted.

In the case of school M, the task was to contrast partitive and quotitive division problems obtained from one mathematical sentence. The teachers chose to use 8 ÷ 2 after also discussing 12 ÷ 3, 18 ÷ 6, 6 ÷ 2, and 10 ÷ 2 as possible candidates. They considered the numbers 8, 2, and 4 as the most easily distinguishable for students, so that students would not confuse them in using, or explaining, their ideas.

In the case of school T, the research lesson was on learning about quadrilaterals and the task was to classify quadrilaterals. The teachers changed the plan from asking students to draw figures freely on dot paper to giving students figures already drawn by the teacher. The teacher worried that students might not construct certain figures that the teacher particularly wanted to discuss in the lesson. The teachers also discussed what would be a suitable number and what types of quadrilaterals to give. If the number of figures was too small, students would not be interested in classifying them, or they would not feel any necessity to make groups. Eventually the teachers decided on nine figures: a square, a rectangle, two parallelograms, two rhombi, an isosceles trapezium, a general trapezium, and a general quadrilateral. The team decided not to include a trapezium with a right angle. As part of their discussion, teachers simulated individual students solving the problem to get an idea of the time required. Further, they considered the quality of the problem-solving activity in terms of the appropriateness of the task and the goal of the lesson.

At all three schools, the teachers discussed the unit in reference to the curriculum, as well as discussing the main task in terms of its appropriateness within the unit, its value for clarifying mathematical ideas, and its appropriateness for accomplishing the goal of the lesson. In terms of the appropriateness of the task for the goals of the lesson, teachers considered what solutions or ideas the students would be likely to bring up. This is the topic of the next section.

4.3.2 Anticipated student solutions

In all three schools, teachers spent time discussing likely student responses to the main task in the research lesson. These discussions usually began by considering what was most likely from the class as a whole. They then went on to consider likely responses from students who were rather slow learners and from students who were fast learners.

In the case of school S, teachers pretended to be students in order to solve the speed task, Who is faster? (see Table 6), from the students' point of view. Through this activity, teachers confirmed the plausibility of the four anticipated solutions already written in the lesson plan: (1) finding a common multiple of distance to compare; (2) finding a common multiple of time to compare; (3) finding the amount of time per metre to compare; and (4) finding the distance per second to compare.

In the case of school T, one teacher was asked to pretend to be a student to solve the task, and the other teachers watched his activity. In the case of school M, teachers wondered whether students would be able to create two kinds of division stories or just one story. The team leader asked the other teachers if they felt uneasy partly because of their own experiences. Teachers made explicit reference to their own experiences as they tried to anticipate students' responses to the task.

In all three schools, teachers considered how to deal with slow learners. In the case of school S, the teacher had already decided to provide hints to students who wanted them during the individual problem-solving period. The team discussed specifically what should be on these hint cards. While a hint card suggesting using common multiples was reasonable from the teachers' initial point of view, they no longer thought this might be the case when they imagined, or visualized, the lesson. They thought this strategy would eventually be rejected in favour of a better strategy: finding the distance per second. One teacher said, "Students might ask the teacher, ' Why did you not give me the best hint, if you knew?'" The other teachers agreed that was likely to happen. So they discussed how to let students notice the per second strategy. Finally teachers thought of using 30 metres and 5 s as the data. "It divides beautifully". "If the teacher asks a question such as, ' Five seconds to go (30 m), so if it were one second how far could you go?, students may be able to notice the idea of per second '". "It will work," one teacher said, "it looks fine". Eventually the teacher decided to suggest using the "per second method" to solve the task using the data of 30 metres and 5 s.

In all three schools, teachers also considered how to deal with fast learners in the lesson. For instance, at school M, a teacher said, "Students who have finished solving the task, I would ask them to write mathematical sentences, possibly like 4 × 2 = 8 or 2 × 4 = 8, showing the process to get the answer".

4.3.3 The comparison and discussion (neriage) phase

The comparision and discussion ( neriage) phase follows the problem solving by the students. This phase in the structured problem-solving lesson is the most difficult for teachers to deal with. Each correct solution has equal value in terms of getting an answer. However, the ideas involved may not have equal value. The neriage phase is when the teacher elicits these ideas and discusses the value of each solution. The teacher at school S clearly stated, "Although each strategy is sure to get the correct answer, we should not end there ... I want the students to know that getting the answer is not the final goal".

In the case of school M, teachers wanted students to compare two word problems, for partitive and quotitive division, through the use of multiplication sentences to model situations. (See the " Appendix " for the actual task.) The lead teacher of the research steering committee posed the question, "What should we ask to elicit a multiplication maths sentence?" For the next 17 min the teachers discussed what the question should be, including its exact wording.

At school T, teachers talked about which point or theme for discussion would be best: the number of groups of quadrilaterals, where the teacher might say "this student made two groups and the other student made three groups, what made these difference? What were the thoughts behind these categorizations?" or how to characterize each group, for example "This student made two groups. Can you see the common characteristics of the quadrilaterals in each group?" One teacher asked, "Which is the higher level of thinking?" to which another teacher responded, "Probably the number of groups is higher. This point is proposed in the lesson plan". So they decided to ask students to discuss how many groups there were and reasons behind this in the neriage phase.

The teams at all three schools discussed how to elevate students' mathematical thinking by comparing individual students' solutions.

It is well known that Japanese teachers get together before a research lesson to discuss the lesson. What do teachers discuss? This study reveals that their discussions followed the lesson plan, which had been drafted or created before meetings, and they devoted approximately two thirds of the time to discussing the flow of the research lesson. Within that time, teachers focussed on the appropriateness of the task, anticipated student solutions, and the plan for comparing and discussing those student solutions. The teachers also referred to the Japanese National Course of Study and its guide for teachers.

5.1 The role of the Japanese National Course of Study in designing and adapting the task for the research lesson

At planning meetings, teachers frequently referred to the National Course of Study when they needed to confirm the role of the unit, or focus lesson, within the entire curriculum. Teachers at school S talked about the placement of speed in the previous National Course of Study. This is a more difficult conversation to have in countries lacking a clear curriculum. Lewis and Tsuchida ( 1998) argued that having a frugal, shared curriculum was necessary for implementing Lesson Study. With a clear curriculum sequence, teachers could identify the value of the research lesson and the unit within the curriculum: by identifying closely related content in former and later grades, teachers can understand why the research lesson is important for later learning. And, identifying similar units or content in earlier grades helps teachers infer what students might do to solve the task, based on their previous learning. All three teams of teachers identified the position of the research lesson in the curriculum in order to clarify students' learning trajectory.

5.2 The value of discussing anticipated solutions

Data from the three schools revealed that teachers tried hard to anticipate student solutions in detail; and what they anticipated influenced the design of the lesson. For example, it influenced the design of the task, such as in the case of school T where the decision whether to include a trapezium with a right angle was made through considering students' anticipated solutions. Anticipating student responses also influenced how teachers decided to pose the problem. For instance, teachers at school S considered how students would react to the question of which person is faster when only times were given. Also teachers tried to predict students' difficulties, and discussed how to reduce students' confusion in comparing three speeds. They eventually decided to erase the slowest person's data in order to focus on only two people.

Based on their experience, Japanese teachers know that the conditions, or characteristics, of the task influence students' thinking processes and solution methods. In the case of school T, the teachers thought that the right angle might cause students to go in a direction that was not consistent with the goal of the lesson. Anticipating student solutions at planning meetings is therefore important in Lesson Study, and this unique activity is a characteristic of task design in Lesson Study (Fujii, 2015).

Teachers also think carefully about the numbers used in a task because this can strongly influence students' ways of solving the task. In the case of school S, teachers deliberately chose awkward numbers for the additional speed data, of persons D, E, F. The teacher explained, "I want students to say that it is awkward to calculate common multiples among them". She deliberately chose numbers that would push students to calculate distance divided by time. On the other hand, the numbers for B and C were (30, 6) and (30, 5) respectively, with these chosen because the numbers "divide beautifully". The teacher clearly anticipated that some students would calculate 30 ÷ 6 and 30 ÷ 5 to get the distance per second.

Close attention to the specific numbers does not mean that teachers are sticking to a concrete level of thinking or encouraging students to think concretely. On the contrary, teachers consider the general aspect of the numbers-their quasi-variable aspects. A quasi-variable is a number deliberately used in a general way, so that it serves as a representative of many numbers, just as a variable would (Fujii and Stephens, 2001, 2008; Fujii, 2008, 2010). Numbers are often chosen based on their quasi-variable power, or how well they can demonstrate a general truth-a general truth that is brought out during whole class discussion.

A structured problem-solving lesson includes a neriage-comparison and discussion-phase for students to compare or experience their friends' methods and discuss similarities and differences between strategies as a whole class. When designing the task, there needs to be consideration of whether the task will elicit the alternative approaches needed for an effective neriage. Therefore teachers carefully discuss and choose appropriate numbers for the task.

Discussing students' anticipated solutions while considering the specific numbers in the task clarifies the mathematical value of the task. In their book The Teaching Gap Stigler and Hiebert ( 1999, p. 118) have another example: teachers discuss appropriate number sentences to use in the context of teaching subtraction across 10. Subtraction across 10 can be solved by subtraction-addition (e.g. 12-9 = 10-9 + 2), subtraction-subtraction (e.g. 12-9 = 12-2-7), counting down, and counting up. In this example, the teachers believed that the subtraction-addition strategy was the most valuable for students to learn, so they examined the potential of different choices of numbers to lead to that strategy. For the same reason, almost all textbooks in Japan choose 13-9 or 12-9 to elicit the subtraction-addition strategy (Doig, Groves, and Fujii, 2011). In the case of school S, numbers were chosen to lead students to calculate distance divided by time. In the case of school T, teachers chose geometrical figures which could lead students to classify them in terms of characteristics related to their parallel or perpendicular sides. Anticipating student solutions in Lesson Study helps clarify the mathematical value of the task, and helps teachers make sure that the goal of the lesson is reached.

5.3 The value of designing the neriage phase of the lesson

The comparison and discussion of multiple student solutions needs to be more than "show and tell" (Takahashi, 2008). This neriage phase of a lesson should be an actualization of Vygotsky's zone of proximal development (Ohtani, 2014), and the role of the teacher is critical. Teachers at the three schools, M, S, and T, discussed at length how to deepen students' ideas in the neriage phase. A teacher at school S said, "Although each strategy is sure to get the correct answer, we should not end there". This comment shows teachers' deliberate efforts to elevate all students' ways of thinking.

During the planning meetings, the focus of designing the neriage phase of the lesson was on deepening students' understanding and ways of thinking. From the point of view of mathematical value, the lesson should clarify the relative value of the different solutions, generally by contrasting these. The lesson is less likely, obviously, to do this without sufficiently rich and diverse solutions to compare. Therefore, teachers carefully examine anticipated student solutions in detail in order to make sure valuable solutions are likely to appear in the comparison and discussion phase. The value of designing the neriage phase of the lesson lies in its potential to elucidate or expose ways to highlight different solutions, and how to compare them in order to reach the goal of the lesson.

5.4 Designing and adapting tasks in lesson planning goes with lesson evaluation

As we have seen, teachers give much thought to the selection and design of the task during the planning phase of Lesson Study. The task is later evaluated during the post-lesson discussion. This is another distinguishing aspect of Lesson Study. The task is not judged based on some abstract determination of whether it is good for teaching a certain skill or concept, but based on concrete evidence from the research lesson of how the students responded to it. In the case of school S, three pairs of data points were added for students to compare, but at the post-lesson discussion teachers argued about whether these additional data were useful or not. The arguments were based on how students actually responded to the task in the lesson. Similar arguments occurred at the other two schools.

In the case of school S, the arguments progressed from evaluating the task to modifying the task. In fact, the final commentator, the knowledgeable other, suggested more direct ways to manipulate numbers to identify faster speed without calculating six pairs of numbers. He gave the example shown in Table 7 of two pairs of numbers in the context of population density:

The final commentator suggested using these numbers instead the six pairs of numbers that were used in the research lesson, as some students struggled to carry out the calculations in the time available, and then missed the educational value of the task, and the whole-class discussion. The post-lesson discussion provided a context for revising the task used at the research lesson, since points missed in planning meetings were revealed in the post-lesson discussion. This shows that the planning meetings of the Lesson Study cycle are closely related to the research lesson itself, and to the post-lesson discussion.

The post-lesson discussion provided a context for revising the task used at the research lesson. However, this does not imply that re-teaching is necssarily part of Japanese Lesson Study. Based on their experience, Japanese teachers know that if students are different then their reactions will be different. They understand that a lesson is itself an organic system, it is not like a machine. A non-organic system, such as a car, is composed of parts that may be easily replaced. However, in organic systems, like a lesson, each part supports the whole ecology. In the case of school S, important ideas missed in planning meetings were revealed in the post-lesson discussion. Teachers then regretted that their kyozai- kenkyu (study or research on teaching materials- see, for example, Watanabe, Takahashi, and Yoshida, 2008) was not profound enough and broad enough to cover the idea. In other words, Japanese teachers' attitude towards research lessons and lesson plans is that their best lesson plan should be implemented at a research lesson, and that a research lesson is the proving ground for teachers (c.f. Lewis and Tsuchida, 1998).

It is widely understood that a lesson plan is an important product of Lesson Study, but despite much research into Lesson Study, the process of creating a lesson plan, as a collaborative effort by teachers, is largely invisible to non-Japanese adopters of Lesson Study. This paper tries to clarify the process of lesson planning and the role and function of the lesson plan, based on case studies of Lesson Study in three Japanese schools.

In each of these case studies, we see that the planning meetings began with a lesson plan already written by the teachers and most of the time was spent discussing the flow of the research lesson. While discussing the flow of the research lesson, teachers spent time designing and adapting the task for the lesson, during which time they typically did the following: consulted the National Course of Study to clarify the position of the task in curriculum, as well as for guidelines in designing and adapting tasks; verified the mathematical value of the task by anticipating student solutions; carefully designed the comparison and discussion ( neriage) phase of the lesson to ensure that the goal of the lesson was reached.

In addition, teachers evaluated the task during the post-lesson discussion in light of the actual student responses in the research lesson, and they also explored how the task might be revised based on this discussion.

Some potentially interesting aspects of lesson planning were not addressed in this paper: the author did not consider the relationship between the quality of the lesson planning and the quality of the research lesson. This paper did not look at the impact of lesson planning on teachers' mathematical and pedagogical knowledge (Lee and Takahashi 2011, Lewis 2009). And the paper did not look at how the lesson planning process exposes teachers' beliefs. The author hopes, however, that by making aspects of the planning phase of Lesson Study visible, this paper will contribute to helping educators outside Japan appreciate the full richness of Lesson Study, and better understand how it can improve teaching and learning.

# data[].attributes.attachment_link.title
Designing and adapting tasks in lesson planning: a critical process of Lesson Study

# data[].attributes.attachment_link.description
Toshiakira Fujii Original Article While the history of Lesson Study in Japan spans more than a century (Makinae, 2010), for Japanese educators, Lesson Study is like air, part of everyday school life. This situation possibly explains why Lesson Study is regarded as being under-theorised (e.g. Elliott, 2012).

# data[].attributes.body
Article

# data[].attributes.attachment_link.readable_text
Mathematics is the science that deals with the logic of shape, quantity, and arrangement. Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports. It represents a high level of abstraction attained by the human mind.

10 Indian Inventions on Mathematics

In India, mathematics has its roots in the Vedic literature which is nearly 4000 years old. Between 1000 B.C. and 1000 A.D. various concepts on mathematics were given by Indian mathematicians in which were set forth for the first time, the concept of zero, the techniques of algebra and algorithm, square root and cube root. There are various examples of mathematics from ancient India that are applied even today.

If it weren't for Indian mathematician-astronomer Aryabhata, there wouldn't have been a number zero.

Though people have always understood the concept of nothing or having nothing, the concept of zero is relatively new; it fully developed in India around the fifth century A.D. Before then, mathematicians struggled to perform the simplest arithmetic calculations. Today, zero - both as a symbol (or numeral) and a concept meaning the absence of any quantity - allows us to perform calculus, do complicated equations, and to have invented computers.

"The zero is widely seen as one of the greatest innovations in human history, is the cornerstone of modern mathematics and physics, plus the spin-off technology," said Peter Gobets, secretary of the ZerOrigIndia Foundation, or the Zero Project.

As much as the students hate it today, India has had big contributions in the field of Algebra in ancient times.

In ancient India, conventional mathematics termed Ganitam was known before the development of algebra. This is borne out by the name - Bijaganitam, which was given to the algebraic form of computation. Bijaganitam means 'the other mathematics' (Bija means 'another' or 'second' and Ganitam means mathematics).

In India, around the 5th century A.D., a system of mathematics that made astronomical calculations easy was developed. In those times its application was limited to astronomy as its pioneers were Astronomers. As astronomical calculations are complex and involve many variables that go into the derivation of unknown quantities. Algebra is a short-hand method of calculation and by this feature, it scores over conventional arithmetic.

3. Trigonometry

Though Trigonometry goes back to the Greek period, the character of the subject started to resemble modern form only after the time of Aryabhata. From here it went to Europe through the Arabs and went into several modifications to reach its present form. In ancient times Trigonometry was considered a part of astronomy. Three functions were introduced: jya, kojya and ukramajya.

The first one is r sina where r is the radius of the circle and sina is the angle subtended at the center. The second one is r cosa and the third one is r (1 - cosa). By taking the radius of the circle to be 1, we get the modern trigonometric functions. Various relationships between the sine of an arc and its integral and fractional multiples were used to construct sine tables for different arcs lying between 0 and 90°.

4. Decimal System and Quadratic formula

In the classical period of Indian mathematics (400 CE to 1600 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji. The decimal number system in worldwide use today was first recorded in Indian mathematics.

It was in 7th century CE when Brahmagupta found the first general formula for solving quadratic equations. The decimal system (or the Hindu number system), which was a precursor of the Arabic numeric system, was developed in India between the 1st and 6th centuries CE.

5. Fibonacci Numbers

The Fibonacci sequence is a series of numbers where a number is found by adding up the two numbers before it. Starting with 0 and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth. It was first described by Virahanka, Gopala, and Hemachandra as an outgrowth of earlier writings by Pingala.

Rulers are believed to have been used by the Indus Valley Civilization prior to 1500 BCE. Made of ivory, the rulers found during excavation, reveal the amazing accuracy of decimal subdivisions on it.

The people of the Indus Civilization achieved great accuracy in measuring length, mass, and time. They were among the first to develop a system of uniform weights and measures. A comparison of available objects indicates large scale variation across the Indus territories. Their smallest division, which is marked on an ivory scale found in Lothal in Gujarat, was approximately 1.704 mm, the smallest division ever recorded on a scale of the Bronze Age.

The history of measurement systems in India begins in early Indus Valley Civilisation with the earliest surviving samples dated to the 5th millennium BCE. Since early times the adoption of standard weights and measures has reflected in the country's architectural, folk, and metallurgical artifacts. A complex system of weights and measures was adopted by the Maurya empire (322-185 BCE), which also formulated regulations for the usage of this system. Later, the Mughal Empire (1526-1857) used standard measures to determine land holdings and collect the land tax as a part of Mughal land reforms.

A total of 558 weights were excavated from Mohenjodaro, Harappa, and Chanhu-Daro, not including defective weights. They did not find statistically significant differences between weights that were excavated from five different layers, each measuring about 1.5 m in depth. This was evidence that strong control existed for at least a 500-year period. The 13.7-g weight seems to be one of the units used in the Indus valley. The notation was based on the binary and decimal systems. 83% of the weights which were excavated from the above three cities were cubic, and 68% were made of chert.

Indian mathematicians had their contribution even in the area of Geometry. There was an area of mathematical applications called Rekha Ganita (Line Computation). The Sulva Sutras, which literally mean 'Rule of the Chord' give geometrical methods of constructing altars and temples. The temples layouts were called Mandalas. Some of the important works in this field are by Apastamba, Baudhayana, Hiranyakesin, Manava, Varaha, and Vadhula.

9. Infinite Series

Kerala mathematicians produced rules for second-order interpolation to calculate intermediate sine values. The Kerala mathematician Madhava may have discovered the sine and cosine series about three hundred years before Newton. In this sense, we may consider Madhava to have been the founder of mathematical analysis. Madhava (circa 1340 - 1425 A.D.) was the first to take the decisive step from the finite procedures of ancient Indian mathematics to treat their limit-passage to infinity.

His contributions include infinite-series expansions of circular and trigonometric functions and finite-series approximations. His power series for p and for sine and cosine functions is referred to reverentially by later writers.

10. Binary Code

Binary numbers form the basis for the operation of computers. Binary numbers were discovered in the west by German mathematician Gottfried Leibniz in 1695. However, new evidence proves that binary numbers were used in India prior to the 2nd century A.D., more than 1500 years before their discovery in the west.

The source of this discovery is a text of music by Pingala named "Chhandahshastra" meaning science of meters. This text falls under the category of "Sutra" or aphorismic statements. Detailed discussions of these short but profound statements are found in later commentaries. "Chhandahshastra" can be conservatively dated to 2nd century A.D. The main commentaries on "Chhandahshastra" are "Vrittaratnakara" by Kedara in probably 8th century, "Tatparyatika" by Trivikrama in the 12th century and "Mritasanjivani" by Halayudha in the 13th century.

# data[].attributes.attachment_link.title
10 Mathematical Inventions In Ancient India That Changed The World

# data[].attributes.attachment_link.description
Mathematics is the science that deals with the logic of shape, quantity, and arrangement. Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports.

# data[].attributes.body
Website

# data[].attributes.attachment_link.readable_text
We are a team of teacher educators from the Faculty of Education , University of Malta:

Dr James Calleja

Dr Michelle Attard Tonna (team member)

Dr Michael Buhagiar (team member)

The aim of the website is to showcase the various initiatives undertaken by different schools in Malta to initiate lesson study. This website is also intended to offer support by providing a range of resource materials that include articles, websites, videos, sample lesson plans, observation sheets and lesson study reports.

A learning together approach to CPD is also driven by the notion of teachers supporting teachers . Teachers benefit from support they receive from their colleagues particularly because they are in the best position to understand the situations faced in the day-to-day classroom life.

Lesson study is a diffused example of ongoing teacher learning originating from Japan. Put simply, lesson study involves teacher-led research in which a group of teachers work together to identify an area for development in student learning. Through lesson study teachers collaboratively engage in ongoing cycles to plan, observe and evaluate lessons.

Stigler, J. W., & Hiebert, J. (2009). The teaching gap: Best ideas from the world's teachers for improving education in the classroom. New York: Free Press.

Source:

This image is taken from this paper :

Source: Lesson Study Alliance

Available on YouTube at:

Resources

Contact

# data[].attributes.attachment_link.title
Supporting Schools | Collaborative Lesson Study Malta | Malta

# data[].attributes.attachment_link.description
This page provides information about the CLeStuM team of teacher educators, from the Faculty of Education at the University of Malta, and their work on lesson study as a continuing professional development model for teachers. Our aim is supporting schools to build lesson study communities.

# data[].attributes.body
Article

# data[].attributes.attachment_link.readable_text
Fractals and Dimensions

Fractals are some of the most beautiful and most bizarre geometric shapes. They look the same at various different scales - you can take a small extract of the shape and it looks the same as the entire shape. This curious property is called self-similarity.

To create a fractal, you can start with a simple pattern and repeat it at smaller scales, again and again, forever. In real life, of course, it is impossible to draw fractals with "infinitely small" patterns. However we can draw shapes which look just like fractals. Using mathematics, we can think about the properties a real fractal would have - and these are very surprising.

Here you can see, step by step, how to create two famous fractals: the Sierpinski Gasket and the von Koch Snowflake.

The name "fractals" is derived from the fact that fractals don't have a whole number dimension - they have a fractional dimension. Initially this may seem impossible - what do you mean by a dimension like 2.5 - but it becomes clear when we compare fractals with other shapes.

Calculating the Dimension of Fractals

Fractals are very popular in mathematical visualisation, because they look very beautiful even though they can be created using simple patterns like the ones above. You can zoom into a fractal, and the patterns and shapes will continue repeating, forever.

The Sierpinski Gasket

One of the examples above was the Sierpinski Gasket consisting of countless triangles. We created it by repeatedly cutting out a triangle in the centre of all the other triangles. However there are many other methods for creating this shape - and here are just a few.

The Mandelbrot Set

One of the most famous and most intriguing fractals is the Mandelbrot Set, named after the French mathematician Benoît Mandelbrot (1924 - 2010). When rotated by 90°, it looks a bit like a person, with head, body and two arms. Here is how you can create it:

We start with the plane of complex numbers (you can think about it like the two dimensional coordinate system). Every point on the plane is represented by a different number c, and we repeat the following steps for every single point:

First we create an infinite sequence of numbers according to the following pattern: We start with 0. Every new number is the previous number squared, plus c. In mathematical notation, we have a sequence (), where z = ² + c.

If this sequence of numbers always increases and tends to infinity (it diverges), we colour the point white. However if the sequence does not increase beyond a certain limit (if it is bounded), we colour the point black.

We repeat this process for every point in the coordinate system. The collection of all the black points is the Mandelbrot set. Move the blue pin below to explore what happens at various points:

For the point c = 1 we create the following sequence:

This sequence will always increase and tends to infinity, so 1 is not part of the Mandelbrot set. The point is coloured white.

A computer can do these computations very quickly for millions of numbers c - like all pixels on a screen. The code required is simple, but the resulting fractal is unbelievable complex.

When Benoît Mandelbrot studied the fractal in the early 1920s, there weren't any computers to visualise it - in fact he didn't know exactly what it looked like. But using mathematics he was able to predict its complexity. The first computer generated image of the Mandelbrot set was produced by an IBM supercomputers in 1980; today everybody can do the same calculations on a normal laptop.

Black points in the image below are part of the Mandelbrot set. Coloured areas are not in the Mandelbrot set, and the colour indicates the speed with which the respective sequence of complex numbers diverges (tends to infinity).

Fractals in Nature and Technology

Fractals clearly can't appear in Nature - if you would zoom in further and further, you would eventually arrive at molecules and atoms and the pattern has to stop. However, there are many shapes in nature which are very similar to fractals:

These shapes appear to be completely random, but - as with fractals - there is an underlying pattern that determines how the shapes are formed and what they will look like. Mathematics can help us understand the shapes better, and thus has applications in medicine, biology, geology and meteorology.

At the beginning of this article we created a very realistic snowflake using a fractal. Similar processes can be applied to all kinds of computer generated graphics. The water, mountains and clouds in this image are generated entirely by a computer using fractals. These methods can be used, for example, when creating textures for computer games.

The process can also be reversed and used for image compression. Usually, pictures on a computer are saved by remembering the colour of every individual pixel in the image. For large images this can take up a lot of disk space and we want to reduce its size: we want to compress it.

However some parts of the picture may look similar to fractals. Instead of saving the pixels individually, a computer could try to find certain patterns in the picture and only save these patterns - this could save a lot of space. Methods of fractal image compression were developed by Michael Barnsley and Alan Sloan in the 1980s.

Fractals, Contraction Mapping and Compression

Fractals in 3-Dimensional Space

So far we have only looked at "flat" fractals like the Sierpinski Gasket or the Mandelbrot set. However there is no restriction regarding the number of dimensions a fractal may live in. Here you can see some fractals in 3-dimensional space:

# data[].attributes.attachment_link.description
Fractals are some of the most beautiful and most bizarre geometric shapes. They look the same at various different scales - you can take a small extract of the shape and it looks the same as the entire shape. This curious property is called self-similarity.

# data[].attributes.body
Article

# data[].attributes.attachment_link.description
Artists around the world are finding inspiration in the curves, equations and patterns of mathematics. Here are some of their stories.

# data[].attributes.body
Research summary

# data[].attributes.body
Article

# data[].attributes.attachment_link.readable_text
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ...

This is the Fibonacci Sequence. It goes on infinitely and is made up of the series of numbers starting with 0, followed by 1, where each subsequent number is the sum of the previous two numbers.

November 23rd is Fibonacci day because when written in mm/dd format as 11/23, these four numbers form a Fibonacci sequence.

Fibonacci is probably not new to you

If you're a programmer, there's a good chance you've implemented a function to compute the nth number of the Fibonacci sequence at some point. It's often used to introduce new programmers to the concept of recursion,¹ and is a common interview question.² There is also an algorithm for searching sorted arrays that leverages Fibonacci numbers, called Fibonacci search. Furthermore, Fibonacci is a popular choice for the scale used by Scrum teams for estimating work.

The context of Fibonacci goes far beyond programming though, the sequence draws origins from as early as 200BC, and can be found in many aspects of nature. There are also lots of peculiar qualities and patterns related to the numbers themselves. So in honor of Fibonacci Day, here are some interesting Fibonacci facts:

1. About Fibonacci the Mathematician

Leonardo of Pisa, more commonly known as Fibonacci, is the medieval Italian mathematician the Fibonacci sequence is named for, and the man most often credited for its invention. However, Leonardo of Pisa did not invent the sequence. It is unclear who actually first discovered it, but Leonardo was born around 1170AD, while references to the sequence have been found to date back to well over a thousand years before his time.

What Leonardo of Pisa did do though, is publish the most influential text on mathematics in Europe at the time, his book Liber Abaci helped spread the usage of Hindu-Arabic numerals (our now used 0, 1, 2, 3, 4, ...) in place of Roman numerals (I, II, III, IV, ...). The Fibonacci sequence of numbers also appeared in this book, as a solution to the following problem:

"A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair [produces] a new pair which from the second month on becomes productive?"

The solution:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

Fibonacci's rabbit problem is a good illustration of the Fibonacci sequence, but it is not very realistic for obvious reasons. A more real life, found in nature example is the ancestral lineage of the Honeybee.

In a Honeybee colony there are three types of bees:

Queen Bee - A single Female that produces eggs
Worker Bees - Females that do work and produce no eggs
Drone Bees - Males that do no work

♀ Female Honeybees are produced from the Queen's fertilized eggs, so they have two parents, one Female and one Male

♂ Male Drone Honeybees are produced from the Queen's unfertilized eggs, so they only have one parent, a Female

When you trace back the ancestral tree of any Honeybee (Male or Female), you'll see that the number of bees in each prior generation matches the numbers in the Fibonacci sequence.

3. In the decimal expansion of 1/89

The decimal expansion of 1/89 is approximately:

0.01123595505618...

Notice how the first six numbers after the decimal point are also the first six numbers of the Fibonacci sequence: 0, 1, 1, 2, 3, and 5

Actually, in a way, it's not just the first six numbers, it's all of the numbers. The catch is when the number spans multiple digits, it will overlap with the previous numbers.

0.0
0.01
0.001
0.0002
0.00003
0.000005
0.0000008
0.00000013
0.000000021
0.0000000034
0.00000000055
0.000000000089
0.0000000000144
...etc...
__________________________
0.01123595505618... = 1/89

So you can think of 1/89 as the value of the infinite series, where each term is a decimal fraction with the tens place digit of the nth Fibonacci number in the n th decimal place.

4. Adjacent pairs approach The Golden Ratio

When you divide a line segment into two parts, such that the ratio between the longer part and shorter part is equal to the ratio between the entire line segment and the longer part, this ratio is The Golden Ratio, commonly denoted by the Greek letter "phi" φ.

The Golden Ratio is an irrational number, approximately 1.6180339887...

Although the ratio itself has existed indefinitely in our universe and in nature, it's hard to pinpoint when it was first referenced by mankind. The Great Pyramids (2589-2504 BC) of the early Egyptians have dimensions very close to The Golden Ratio, Plato (~428/427-348/347 BC) considered the ratio to be the most universally binding of all mathematical relationships and the key to unlocking the physics of the cosmos, and Leonardo Da Vinci (1452-1519AD) made extensive use of the proportion in his works of art.

It is unclear whether Fibonacci himself ever made the connection between his Fibonacci numbers and it's relationship to The Golden Ratio, but here's what happens when we create a sequence by dividing each Fibonacci number by the number before it:

1 / 1 = 1
2 / 1 = 2
3 / 2 = 1.5
5 / 3 = 1.667
8 / 5 = 1.6
13 / 8 = 1.625
21 / 13 = 1.615
34 / 21 = 1.619
55 / 34 = 1.618

We can see that each subsequent number in this sequence is approaching, or getting closer and closer to The Golden Ratio.

Interestingly enough, this also works with any Fibonacci-like sequence of numbers. We can start the sequence with any two whole numbers, as long as every next number still follows the rule of being the sum of the previous two numbers.

Starting with 123 and 8 for example, we have:

123, 8, 131, 139, 270, 409, 679, 1088, 1767, 2855, 4622, 7477, ...

And here are the ratios, once again approaching The Golden Ratio:

8 / 123 = 0.065
131 / 8 = 16.375
139 / 131 = 1.061
270 / 139 = 1.942
409 / 270 = 1.514
679 / 409 = 1.660
1088 / 679 = 1.602
1767 / 1088 = 1.624
2855 / 1767 = 1.616
4622 / 2855 = 1.619
7477 / 4622 = 1.618

5. Also known as "Nature's Secret Code"

Not all petals on flowers or leaves on plants correspond to Fibonacci numbers, but there are enough that do to suggest that these numbers do bear some sort of significance in nature. And the most common form in which we see these numbers, is through spiral arrangements.

Leaves on branches, for example, often grow in a helix pattern, spiraling around the branch. If we start with a leaf at the bottom of the branch and follow the spiral of the other leaves up the branch, counting the number of leaves as well as the number of turns around the branch until we come to a leaf in the same position directly above the leaf we started with, we'll have a leaf count as well as a turn count. This leaf count and turn count is what often corresponds to Fibonacci numbers in many tree species.

Pinecones will typically have 8 spirals going one way, and 13 going the other, with smaller varieties having 5 and 8. A pineapple shell, built from hexagons, will have three kinds of spirals, and when counted, these are usually 8 to the right, 13 to the left, and 21 vertical. Also, the number of spirals in a sunflower core is very typically 55 going one way, and 89 going the other way, or in some varieties 34:55 or 89:144.

One plausible explanation for these numbers popping up all over nature, is that it probably has more to do with The Golden Ratio, and that it is because of nature's tendency towards The Golden Ratio, that the Fibonacci numbers show up often. But why does nature tend towards The Golden Ratio? One particularly Darwinian explanation would be that over time nature has figured out that by utilizing this ratio to create angles of rotation, seeds can be placed to maximize space, leaves arranged to optimize exposure to sunlight and other nutrients for each leaf, and so on...

¹ The recursive approach is actually the worst approach for doing this, since the cost of each calculation increases exponentially with every subsequent number, and you'll probably end up with a stack overflow in most languages.

² I was asked to implement this function during my technical interview at Vincit 3 years ago.

Boom, newsletter sign up form.

# data[].attributes.attachment_link.description
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ... This is the Fibonacci Sequence. It goes on infinitely and is made up of the series of numbers starting with 0, followed by 1, where each subsequent number is the sum of the previous two numbers.

# data[].attributes.body
Article

# data[].attributes.attachment_link.readable_text
Different Way To Solve Quadratics That Everyone Is Talking About

I was curious after reading MIT Technology Review's headline about a new way to solve quadratic equations. I was even more interested when I saw Po-Shen Loh devised the method.

For those that don't know, here is Loh's background:

Caltech BS Mathematics
Cambridge MASt Mathematics
Princeton PhD Mathematics
Faculty at Carnegie Mellon University

Professor Loh is also the coach of the USA Math Olympiad team, and he coached for the team's first ever back-to-back #1 rank victories in 2015 and 2016. And for good measure, the team won back-to-back again in 2018 and 2019.

The results speak for themselves. When Professor Loh suggests a problem solving method, it is certainly worth learning!

After Professor Loh devised the method, he researched the literature to see if anyone was teaching it. Mostly it is an unknown method. But after he wrote an article, someone pointed out a similar method written by John Savage 30 years ago, and it is quite possible other people have written about the idea. In any case, the method is an alternative to the standard quadratic formula taught in classrooms, and we can all benefit from learning a different approach. Here's a video explaining and illustrating the method.
"New" Way To Solve Quadratic Equations That Everyone Is Talking About
(Pretty much all posts are transcribed quickly after I make the videos for them-please let me know if there are any typos/errors and I will correct them, thanks).

For the most accuracy, please see Professor Loh's articles. He has carefully chosen the words to meet the standards of mathematical correctness. He also provides historical context for which parts of his method are new versus old. Generally all the steps of the method have been known for hundreds of years. But putting them together is a newer concept he had not seen taught popularly (though some people have mentioned the idea before).

Po-Shen Loh website articles on quadratic equations
https://www.poshenloh.com/quadraticdetail
https://www.poshenloh.com/quadratic

Po-Shen Loh paper on quadratic equations
https://arxiv.org/abs/1910.06709

Po-Shen Loh video
https://www.youtube.com/watch?v=ZBalWWHYFQc&feature=youtu.be

Po-Shen Loh video with examples
https://www.youtube.com/watch?v=XKBX0r3J-9Y
My explanation
My explanation below is a summary of how to calculate the answer. I have done my best to follow the idea of Professor Loh's articles, but I have focused more on the calculation part.

Start with the general quadratic equation.

Then divide both sides by a.

What would happen if a quadratic equation has two roots that are R and S? (We should not assume there are two roots for strict mathematical correctness). But if there were two roots, we could factor the quadratic equation to get:

B would be the sum of the roots
C would be the product of the roots

This gives us an idea of how we could solve a quadratic equation. We can try to find two numbers that have the appropriate sum and product. If we can find such numbers, then we could factor the equation and solve it. So go back to the quadratic equation we are trying to solve.

When - B is the sum of two roots, we would have - B/2 is the average of the roots. So one root would be the average plus a deviation z, and another would be the average minus a deviation z.

We would also want the product of these to be equal to C. The neat thing is the product is a difference of squares so it is easy to compute.

We can solve for z by equating the last two lines. (This value always exists in complex numbers.)

Sum this quantity with the average - B/2 and we get two numbers with the desired sum and product. Thus we can factor the quadratic equation with these values, and they are the two roots to the original equation.

One could memorize these equations-and that is the method taught in Germany. However, the entire point of this method is you don't have to memorize equations. You just need to understand and apply the method. Let's see how it works in a few examples.
Example 1
We will start off easy with the following problem:

This is easily factorable by guessing, but let's practice the method. The coefficient on x is -8, so if the equation could be factored into two roots, the sum of the roots would be 8, and their average would be 4. We then consider 4 + z and 4 - z to have a product equal to the constant term 15. So we would have:

Then we have 4&pm;1:

These values 5 and 3 have the sum (4) and product (15), so they are roots of the original equation.

Not too bad! Let's try another example.
Example 2
Solve:

The coefficient on x is -4, so calculate -(-4/2) = 2 and consider the values 2&pm; z whose product we want to be the constant term 6. So we have:

Then consider 2&pm; i √2 which are the terms:

These have the desired sum (4) and product (6), so these values are roots to the original equation.

This worked out well. Now let's do one final example.
Example 3
Solve:

First we will divide by 2 so we get a coefficient of 1 on the x ² term. This won't affect the value of the roots to the equation.

The coefficient on x is -2, so consider -(-2/2) = 1. We then consider values 1&pm; z, whose product will be the constant term 5. So we have:

Then add these to the value of 1 to get:

These have the desired sum (2) and product (5), so these values are roots to the original equation.
Conclusion
Let's recap the method. Start with a general quadratic equation:

First divide by a so we get the equation:

Next, consider two values of the form:

Then you can solve for z by setting the product, which will be a difference of squares, equal to C.

Once you have z, you can add - B/2 and that gives two values with the desired sum - B and product C. Thus - B/2 &pm; z will be the two roots to the original equation.

This is an interesting way to solve quadratic equations. The method may be useful for students who have trouble memorizing the standard quadratic formula. And the method also related to concepts like the sum of roots and the product of roots, which are useful in advanced mathematics (such as the elementary symmetric polynomials, which are mentioned in this university level problem: solve if you are a genius.)

So give this new method to solve quadratic equations a try. After all, it never hurts to know another way to solve quadratic equations!

Po-Shen Loh website articles on quadratic equations
https://www.poshenloh.com/quadraticdetail
https://www.poshenloh.com/quadratic

Po-Shen Loh paper on quadratic equations
https://arxiv.org/abs/1910.06709

Po-Shen Loh video
https://www.youtube.com/watch?v=ZBalWWHYFQc&feature=youtu.be

Po-Shen Loh video with examples
https://www.youtube.com/watch?v=XKBX0r3J-9Y

Po-Shen Loh tweet about method
https://twitter.com/PoShenLoh/status/1202765493192548352

MIT Technology review article
https://www.technologyreview.com/s/614775/a-new-way-to-make-quadratic-equations-easy/

Popular Mechanics article
https://www.popularmechanics.com/science/math/a30152083/solve-quadratic-equations/

Girls Angle Bulletin Oct/Nov 2017 (See Quadratic Formula Revisited)
http://girlsangle.org/page/bulletin-archive/GABv11n01E.pdf

Similar method taught in Germany
https://math.stackexchange.com/questions/1314373/how-do-pupils-solve-2nd-degree-equations-in-germany-different-from-spain

MY BOOKS

If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.

Mind Your Decisions is a compilation of 5 books:

(1) The Joy of Game Theory: An Introduction to Strategic Thinking
(2) 40 Paradoxes in Logic, Probability, and Game Theory
(3) The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias
(4) The Best Mental Math Tricks
(5) Multiply Numbers By Drawing Lines

The Joy of Game Theory shows how you can use math to out-think your competition. (rated 4.2/5 stars on 48 reviews)

40 Paradoxes in Logic, Probability, and Game Theory contains thought-provoking and counter-intuitive results. (rated 4.2/5 stars on 15 reviews)

The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 3.5/5 stars on 4 reviews)

The Best Mental Math Tricks teaches how you can look like a math genius by solving problems in your head (rated 4.8/5 stars on 10 reviews)

Multiply Numbers By Drawing Lines This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 4.4/5 stars on 7 reviews)

Mind Your Puzzles is a collection of the three "Math Puzzles" books, volumes 1, 2, and 3. The puzzles topics include the mathematical subjects including geometry, probability, logic, and game theory.

Math Puzzles Volume 1 features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 16 reviews.

Math Puzzles Volume 2 is a sequel book with more great problems. (rated 4.5/5 stars on 6 reviews)

Math Puzzles Volume 3 is the third in the series. (rated 4.1/5 stars on 7 reviews)

Prime Reading

I am thrilled to announce my ebook The Joy of Game Theory was selected for the Amazon.com Prime Reading program! Right now Prime members can read the book for free. Go to the Prime Reading catalog and find my book "The Joy of Game Theory" and select the option to borrow for free.

Be sure to read it soon! The promotion is valid for about 90 days, starting December 18.

Here is a link to the book's main page:
The Joy of Game Theory: An Introduction to Strategic Thinking
You don't need a Kindle to read the ebook. You can install the Kindle app on virtually any computer/smart phone/tablet/etc.

KINDLE UNLIMITED

Teachers and students around the world often email me about the books. Since education can have such a huge impact, I try to make the ebooks available as widely as possible at as low a price as possible.

Currently you can read most of my ebooks through Amazon's "Kindle Unlimited" program. Included in the subscription you will get access to millions of ebooks. You don't need a Kindle device: you can install the Kindle app on any smartphone/tablet/computer/etc. I have compiled links to programs in some countries below. Please check your local Amazon website for availability and program terms.

MERCHANDISE

Grab a mug, tshirt, and more at the official site for merchandise: Mind Your Decisions at Teespring.

# data[].attributes.attachment_link.title
The New Way To Solve Quadratic Equations Everyone Is Talking About

# data[].attributes.attachment_link.description
Different Way To Solve Quadratics That Everyone Is Talking About I was curious after reading MIT Technology Review's headline about a new way to solve quadratic equations. I was even more interested when I saw Po-Shen Loh devised the method.

# data[].attributes.body
Article

# data[].attributes.attachment_link.readable_text
There are some great uses. Shutterstock

Despite the fact that mathematics is often described as the underpinning science, it is often not given enough credit when scientific discoveries are presented. But the contribution of mathematics and statistics is essential and has transformed entire areas of research - many discoveries would not have been possible without it. In fact, as a mathematician, I have contributed to scientific discoveries and provided solutions to problems that biology was yet to solve.

Seven years ago, I attended a lecture on some biological research that was taking place at Heriot-Watt University. My colleagues had an unsolved problem which related to the movement of bag-like structures called vesicles which move hormones and neurotransmitters such as insulin or serotonin around cells and the body.

Their problem lay in that vesicles were known to follow specific tracks along the cell skeleton which lead to special molecules which then caused the vesicle to release its contents into the cell. However, when the biologists themselves tried to find these tracks, they were not in the expected places.

OpenStax, CC BY

It is important to understand how vesicles behave, or in fact misbehave, as they have been linked to conditions such diabetes and neurological disorders. The biologists were struggling to find a way to understand the vesicles - but I had a solution in my mathematical toolkit.

Maths can beat biology

After two years of collaboration I told my colleagues: "my model and computer experiments are better than your microscope!"

What I meant by this rather confident statement was that by using mathematics to model how molecules move in a cell we could predict and run multiple experiments on a computer at a smaller scale and faster rate than a microscope. It could allow us to uncover things that the biologist's resources could not, and might even point us in the direction of target molecules for future treatments of diabetes and neurological disorders.

The mathematical model allowed us to recognise that the movement of vesicles requires energy - and the maths models it through an energy landscape. It imagined a vesicle to be like a cyclist riding a bicycle - the landscape may have easy level sections but also hills that require more energy input to get over them, and so we wanted to test whether they actually avoided these hills.

After seven years of working in partnership with the biologists, my colleagues and I proved our hypothesis was correct. Vesicles do follow lower energy "valleys" in the landscape, avoiding molecules which create the high energy hills in the energy landscape - taking the easiest path. The overall result is just the same as the biologists had found - the vesicles end up in the same end location and they reuse similar routes over and over again. But the difference lies in the way in which they do it, and it was not by following the cell skeleton as biologists had first believed - they take an easier route. It really shows the power of maths and how it can change the way we see things.

Mathematical models allow you to capture many gigabytes of raw data in a compact form in a way that is impossible for a biologist with a microscope. You can make modifications to the model easily and show how vesicle behaviour may change during disease, when they are disrupted or mutated. It could then reveal which molecules to target in future treatment studies - and lay the groundwork for larger and more thorough modelling of complex biological processes.

Shutterstock

The integration of cutting-edge microscopy with cell biology and mathematical modelling could be applied to many other problems in bio-medicine and will accelerate discovery in the years to come. The movement of molecules and other cell components is just one example of where the power of mathematics is unrivalled, but it is by no means its limit.

Useful is an understatement

Maths is often criticised by the public for lacking in "real-world" applications, but it is being applied to many real-world problems all the time. Groundwater contamination, financial and economic forecasting, plume heights in volcanic eruptions, the modelling of biological processes and drug delivery are just a few places where maths is making a huge difference.

I'm proud to say that I co-authored a paper with my biology colleagues, and I hope to see more mathematicians coming to the fore for science research in the future. Mathematics plays a central role in so many of the world's scientific breakthroughs and deserves a headline role in more academic publications. Power to the mathematician - they're behind more discoveries than you think.

# data[].attributes.attachment_link.title
Maths: why many great discoveries would be impossible without it

# data[].attributes.attachment_link.description
Despite the fact that mathematics is often described as the underpinning science, it is often not given enough credit when scientific discoveries are presented. But the contribution of mathematics and statistics is essential and has transformed entire areas of research - many discoveries would not have been possible without it.

# data[].attributes.body
Teaching resource

# data[].attributes.body
Teaching resource

# data[].attributes.body
Magazine article

# data[].attributes.attachment_link.readable_text
When I wore a Diarrablu dress to celebrate Thanksgiving, everyone the beautiful wrap design, dramatic sleeves, and bold print made quite a splash.

When I told my friends and family the pattern was generated by a math equation and designed by a Senegalese woman who lost her memory and nearly died on her path to becoming a designer, they wanted to know the whole story, naturally.

I'm guessing you're intrigued now, too, so voila: Diarra Bousso's evolution from math nerd to Wall Street bond trader to creative math teacher and fashion designer-and how she nearly lost her life along the way.

And no, she doesn't plan on choosing between the math and fashion any time soon.

"I almost feel like it's too good to be true, that I can do both things," Bousso told me. "Not just for me-the fashion brand is uplifting entire communities and artisans and providing jobs. And my work in education and research is getting kids to care and learn and be successful."

Her arrival at this poised position didn't come easy. The fulcrum of Bousso's story is a life-threatening accident in 2012 when she was 23 that left her paralyzed and with no memory. The event was precipitated by a bout of severe depression, and in Senegalese culture at the time, Bousso says mental health was considered embarrassing and not discussed openly. Hence, she kept much of her story private.

But with nearly eight years of recovery and perspective behind her, plus her current success as both a high-school math teacher and a designer (she's currently a designer-in-residence at Fashion Incubator San Francisco)-she's ready to share.

"It took being a teacher to realize the importance of mental health and the need to share it openly and ask for help when needed," Bousso said. "In my case, I fell into depression while working on Wall Street because I had chosen the wrong path and was miserable and felt no purpose. I didn't know what depression was, in my culture we never talked about such a thing."

Bousso is private about the specifics of the accident itself, although she's certain her depression caused it. Her art, fashion, and, surprisingly, Tumblr, all played a role in her journey to recovery.

As a teen, Bousso was a bright student who wanted to be rich. She also wanted to be a cool, respected artist and fashion maven, inspired by her stylish mother who designed most of her own wardrobe using fabrics collected from travels around the world. But no one in Bousso's family believed art was a viable career path. It was a fun hobby, and she was a gifted painter inspired by Picasso and cubism. But art wouldn't make her the earner she impatiently dreamt of becoming.

At 16, she declared to her parents that she was bored with school in Senegal and wanted a challenge. She informed them she'd be moving to Norway as one of just three students accepted by United World Colleges (she'd applied without her parents' knowledge) to study in the Nordic country. They said she was too young-she could go abroad for college in France after graduating high school, the natural progression for a smart Senegalese student.

She went to Norway anyway. She spoke no English or Spanish like her classmates, but learned both in a matter of months. The challenge she sought was granted in spades.

"I went from a tropical country to snow, spicy food to bland Norwegian food, French and Wolof to English and Spanish," Bousso recalls. "Everything was so hard."

But she soon found her groove, made friends, and graduated from high school after two years abroad. The program funneled students to various locations around the world, and she wound up at Macalester College in Minnesota. "I was used to the cold by then," she said. She studied economics, math, and squeezed in art classes when it worked with her schedule. After graduation, she went straight to Wall Street, taking a job as a bond trader and fulfilling her dream of wealth.

Bousso was making six figures, lived in a luxury high rise, and wore all the luxury designers as she commuted to work each day in a limousine. But she soon found her new life in New York City was not what she'd imagined.

"My whole life I was chasing being the best at everything, and all of a sudden I'm there and it feels like there's nothing else to do," Bousso said. "What's next after this? I thought, there's no purpose here. I don't think I like this. I don't enjoy it. I miss my family."

It was a devastating realization, and the depression that followed was so debilitating that she went home to Senegal on medical leave. In July 2012, a few days before her scheduled return to New York, she suffered the accident that nearly killed her and left her in a coma for two weeks. When she finally woke up, she had no memory, 23 missing teeth, a ruptured knee cap, a broken femur, multiple lacerations on her face, chin, and body, and she couldn't walk.

When she finally opened her eyes, Bousso recognized her tearful father. She remembered her two sisters and her brother. But her memory of her career and much of her identity were gone. Details of the previous day were erased with each new morning. A psychologist recommended writing and drawing in a journal to help her remember what had happened the previous day.

"I started drawing and drawing and drawing," Bousso said. "I would also write, write write. I became so prolific, so I started a Tumblr page. I had nothing better to do."

Her mother was terrified that if Bousso started remembering Wall Street, she would fall back into depression. So she concocted a plan. She hung up Bousso's childhood paintings on the white walls of the hospital room and told her daughter a story she believed would motivate her to recover: She was a successful artist, and when she could walk again, there would be a grand exhibition of her work. She made up her daughter's face, did her hair in beautiful head wraps, and took photos, all with the hope of lifting her daughter's spirit.

Bousso didn't know any better, so she went along with the fictitious persona on Tumblr, making up stories about a glamorous life of art and fashion-something she'd always dreamed of now appeared to be a reality, at least on social media. Between exhibitions and runway shows along with the glamorous photos, fictional Bousso was the toast of the town on Tumblr, where she quickly acquired 19,000 followers.

But by November, the story began to unravel. She saw flashes of her New York life and discovered the exhibition was a hoax.

"I was furious and told (my parents) I would quit my job on Wall Street as soon as I could walk." She vowed to transform her fabricated life into a reality.

Her parents promised if she could somehow prove that art could become a viable career they'd support her and not insist that she return to her Wall Street job, though they didn't believe the idea would lead anywhere.

"I'm the type of person that if you tell me I can't do something, I'll do it," she said.

Bousso email blasted every public relations agency in Senegal announcing that she was an artist and designer with a robust Tumblr following. One of them soon took her on, and not much later she was featured in the Street Style section of The New York Times, a Florida print magazine, Vogue Italia (which featured a portrait of her father-she had also taken up photography).

She printed out the articles, arranged them in a folder, and handed it to her father.

"That's you, by the way," she told him regarding the portrait. "He was like, 'is this the real New York Times or some Tumblr New York Times?' He was just so mind blown and confused: 'You can't even walk, you haven't left the house, how is all this happening?'" she recalls him asking.

Her parents were becoming convinced, but they held out one more requirement before they OK'd her ditching Wall Street: "You have to walk first," they said.

True to form, she soon did just that.

Bousso flew back to NYC to quit her job in person, and while she was there, volunteered at New York Fashion Week with the goal of discovering how it all works in real life. She learned where the editors sit, where the press sits, and most importantly what attracts photographers.

She once again took to blasting press representatives, this time at fashion brands. The following season she was invited to Jeremy Scott, BCBG, Herve Leger among other shows.

"I would go dressed up with this big head wrap, massive necklaces, and be very dramatic," she said.

The photographers flocked, and everyone wanted to know who she was. No one noticed her slight limp. By the end of Fashion Week, she had appeared in street style roundups in Vogue Italia, Glamour, The New York Times and more.

Meanwhile with the help and inspiration of her stylish mother, she began collecting fabric from local markets and employing local artisans to create clothing she designed.

"My dad was just like, I can't with her, what is she doing?"

She had the PR down, but she wanted to prove herself as a designer and artist. So with the help of her new agency and her life savings, she produced "African Voyage," a daylong event that took guests from a private island to the city to an artist's private home. Throughout the day, Bousso showed her paintings and and incredible 300 garments, which included spring, summer, fall, winter, resort, bridal, shoes, and bags. The traveling show was accompanied by live drummers, poets, and models who theatrically played out an allegory of Diarra's accident and recovery. She didn't want any success to be based on pity, so she disguised her story as a lost princess finding her identity.

"The idea was that my art and my designs are too cool to come to you, you had to walk to them. It was the most obnoxious thing," Bousso recalls. "It was also super expensive and the biggest waste of money. But I was totally free-(my family) was like, 'She survived, let's be nice to her.'"

The event created the recognition she desired. She was the talk of the town (in real life!), and buyers placed orders. The problem was, she had no business plan or production facilities in place. She returned to the markets where she had bought fabric, and it was all gone.

"That's when I realized it was not going to be just parties all the time," she said. "You're going to have to work."

At the same time, she was feeling the itch to get back to mathematics and she applied to graduate schools. She was accepted in 2015 to Stanford Graduate School of Business with a $150,000 scholarship. But after some soul searching, she turned it down. She knew she needed to honor her creative side. The following year, she applied and was accepted to Stanford's Graduate School of Education with the idea of becoming a teacher focusing on creative mathematics.

One day in class with revolutionary Math Education Professor Jo Boaler, the teacher passed out sugar cubes and paint to illustrate complex equations.

"I was like 'Wow! I want to do this with fashion,'" Bousso said.

She was elated at the prospect of combining math and art. It gave her a sense of "happiness, purpose and freedom in general," and it meant that she was finally on the right path. Whereas she once felt she had a foot in each world and wasn't sure if she could be taken seriously in either, she now realized her hybrid interests were her strength, and something to be celebrated.

The realization gave her the courage to move forward with Diarrablu. This time she had a business plan and production facilities, managed by her mother back home in Senegal.

Earlier this year, she landed a meeting with editors at Vogue.

"Virginia Smith, the U.S. fashion director, literally left her meeting to welcome me," Bousso said. She kept her entire rack of clothing, and Kendall Jenner appeared in a Diarrablu swimsuit in the June issue.

You might think that was enough for Bousso to ditch everything else for fashion. Instead her experience with Vogue solidified her belief that the multidisciplinary approach was her super power.

"Both worlds thrive on the fact that I have the other one. That multidimensional approach to being is very rich," she said. "And I think for students it's very empowering because (it says) 'I can be many things,' but also from a design standpoint, it makes the process a lot more enriching. It just seems limitless."

She currently teaches high school math in Northern California where she's showing kids that math is more than boring, abstract concepts by joining it with art.

"When I'm thinking of algebra and graphing, instead of asking for a particular graph, I ask them to design shapes and find the equation that will make you have (for example) a star. Kids who were borderline D/F students got A's in my class. When your identity as a learner is boosted, it changes your whole experience."

She was recently accepted to Fashion Incubator San Francisco's Designer-in-Residence program, which provides a space at Bloomingdale's, mentorship, and pop-up opportunities, including a recent " Math Lab " that attracted both fashion and math mavens from around the Bay Area. Math teachers sent their students for extra credit.

Several educators asked Bousso to share her math-art lesson plans, and Bousso plans to do just that. In collaboration with her former Stanford professors Jo Boaler and Nancy Lobell, she's creating an interdisciplinary program to share with educators across the country.

"I never imagined all my doodling and nerding out on math and art could be so well-received," Bousso said.

She's especially thrilled with the impact her approach has on female students: "A lot of my girl teenagers are getting more excited about STEM when the fashion is involved and I am super excited to work on it more."

It makes girls feel like they belong in math class, which is vital, Bousso said.

"If you feel that way, you're not going to fail because you believe that you should get it. You should understand. You should be there. You should belong. Changing that mindset took about nine months (for one of my students), but she's like a different person. If I can have that impact on students and community, it's amazing. They can grow up and do anything."

At this point, Bousso is feeling like she can do just about anything, too. How does she possibly make time for two jobs? Each day, she teaches until three, goes home and naps for two hours, then wakes up to work on Diarrablu. And even though it seems like she has two jobs, they're inextricably entwined. When she's writing algorithms for her designs, she's also lesson planning for class.

"Sometimes I can't tell if I'm doing design work or teaching work," she said, "because they are the same."

In other words, she has found her flow-and perhaps more importantly, an optimistic and holistic life path.

"For me, it's not about money or success," Bousso said. "It's about survival. It's what made me survive a very tragic thing."

# data[].attributes.attachment_link.title
How Fashion Saved This High School Math Teacher's Life

# data[].attributes.attachment_link.description
When I wore a Diarrablu dress to celebrate Thanksgiving, everyone the beautiful wrap design, dramatic sleeves, and bold print made quite a splash. When I told my friends and family the pattern was generated by a math equation and designed by a Senegalese woman who lost her memory and nearly died on her path to becoming a designer, they wanted to know the whole story, naturally.

# data[].attributes.body
Journal

# data[].attributes.attachment_link.readable_text
Welcome to PAUL ERNEST'S page

Based at School of Education, University of Exeter, United Kingdom.

Philosophy of Mathematics Education Journal

Number 38 December 2021

Number 37 August 2021 (Dedicated to Ubi D.Ambrosio)

Number 36 December 2020

Number 35 December 2019

Number 34 December 2018

Number 33 January 2018

Number 32 November 2017 (Special issue on Mathematics Education and the Living World: Responses to Ecological Crisis)

Number 31 November 2016

Number 30 October 2016

Number 29 July 2015 (25th Anniversary Issue)

Number 28 October 2014

Number 27 April 2013

Number 26 December 2011

Number 25 October 2010 (Special Issue on Critical Mathematics Education)

Number 24 December 2009 (Special Issue on Mathematics and Art)

Number 23 October 2008

Number 22 November 2007

Number 21 September 2007 (Special Issue on Social Justice, Part 2)

Number 20 June 2007 (Special Issue on Social Justice, Part 1)

Number 19 December 2006

Number 18 October 2004

Number 17 May 2003

Number 16 July 2002

Number 15 March 2002

Number 14 May 2001

Number 13 September 2000

Number 12 November 1999

Number 11 March 1999

Number 10 November 1997

Number 9 November 1996

Number 8, May 1995

Number 8, May 1995

Number 7, February 1994

Number 6, December 1992

Number 4/5, February 1992

Number 2, November 1990

Number 1, July 1990

Advanced Study in Mathematics Education at Exeter

Papers by Paul Ernest

Download Paul Ernest CV

To contact Paul Ernest please mail to p.ernest@exeter.ac.uk

Copyright Notice.
All material accessed through this page may be consulted and copied freely for non-profit purposes provided full acknowledgment is given.

© P. Ernest 2006

There have been visitors to this page since 1996.

# data[].attributes.attachment_link.title
Philosophy of Mathematics Education Journal

# data[].attributes.attachment_link.description
Based at School of Education, University of Exeter, United Kingdom.

# data[].attributes.body
A presentation

# data[].attributes.attachment_link.readable_text
Meet masters of the microscope and see their tools of the trade in this tale of discovery of new worlds in miniature.

Hooke's sumptuously illustrated volume went on to become a scientific bestseller, capturing the imagination of a wide audience.

Micrographia was the second book ever to be published by the Royal Society.

Its vivid plate illustrations helped to make it Hooke's most influential work.

Hooke also studied the forms of ice crystals in frozen water, in snow, and even in frozen urine.

These hand-drawn illustrations from the Royal Society collection were later reproduced in Micrographia and are some of the few surviving originals from the book.

Hailed by Samuel Pepys (1633-1703) as 'the most ingenious book that I ever read', Micrographia inspired a generation of natural philosophers. It included fellow microscopists Jan Swammerdam (1637-1680), Antoni van Leeuwenhoek (1632-1723), Nehemiah Grew (1641-1712) and Marcello Malpighi (1628-1694).

In this study of berries under the microscope for his Anatomy of plants, Grew illustrates the complex structures within fruits.

Malpighi produced these fine drawings for his Anatomes plantarum (1675-1679) published by the Royal Society.

In this appendix to the Philosophia Britannica (1759), the instrument maker Benjamin Martin (1704-1782) illustrates the latest improvements in compound and solar microscopes.

He described the former as 'universal' and boasted of an additional 'between-lens' for improved resolution.

William Cary (1759-1825), a London-based cartographer and instrument maker produced and marketed this boxed naturalist's microscope.

The accompanying slides include, inevitably, a flea, in homage to Hooke's famous Micrographia illustration.

As well as experimenting with new instruments, scientists used different methods for preparing samples to observe reactions.

These samples include lung sections taken from goats during compressed air experiments by Commander Guybon Chesney Castell Damant (1881-1963) and the physiologist John Scott Haldane (1860-1936).

Damant, Haldane and Arthur Edwin Boycott (1877-1938) collaborated on a paper that made the case for staged decompression of divers and modern dive tables.

All rights reserved © The Royal Society 2019

For more information about the Royal Society Library and Archive please visit our website: https://royalsociety.org/collections/

The story featured may in some cases have been created by an independent third party and may not always represent the views of the institutions, listed below, who have supplied the content.

# data[].attributes.attachment_link.title
Seeing closer: the birth of microscopy - The Royal Society - Google Arts & Culture

# data[].attributes.attachment_link.description
Meet masters of the microscope and see their tools of the trade in this tale of discovery of new worlds in miniature. Hooke's sumptuously illustrated volume went on to become a scientific bestseller, capturing the imagination of a wide audience. Micrographia was the second book ever to be published by the Royal Society.

# data[].attributes.body
Article

# data[].attributes.attachment_link.readable_text
Toshiakira Fujii

Original Article

While the history of Lesson Study in Japan spans more than a century (Makinae, 2010), for Japanese educators, Lesson Study is like air, part of everyday school life. This situation possibly explains why Lesson Study is regarded as being under-theorised (e.g. Elliott, 2012). Educators outside Japan however, having had to learn about Lesson Study less naturally, may sometimes lose some important aspects of Lesson Study.

Lesson Study came to the attention of educators outside of Japan primarily through the publication of The Teaching Gap (Stigler and Hiebert, 1999), which described findings from the TIMSS video study focussing on the eighth grade mathematics lessons in USA, Germany, and Japan. Chapter seven in particular, titled "Japan's approach to the improvement of classroom teaching", which is based on Yoshida's ( 1999) doctoral dissertation, now available in book form (Fernandez and Yoshida, 2004), provoked enormous interest, not only in Lesson Study, but also in the typical structure of Japanese mathematics lessons. Independently, some educators such as Lewis also noticed the significance of Japanese Lesson Study (Lewis and Tsuchida, 1998).

Since then many mathematics teachers and teacher educators around the world have been involved in Lesson Study, and many books and research papers have been written on various aspects of Lesson Study (Lewis, 2002; Lewis et al., Lewis and R, Perry., & J. Hurd, 2009; Hart, Alston and Murata, 2011; Doig and Groves, 2011; Department for Children, Schools and Families, 2008; White and Lim, 2008; Ono and Ferreira, 2010). However, some aspects of Lesson Study, that may be taken for granted by Japanese teachers, seem not to be well understood outside Japan.

This paper aims to clarify the role and function of lesson planning in the Lesson Study process, based on case studies conducted in three schools in Tokyo.

2.1 The Lesson Study process

Lesson Study is an approach to teacher professional development that differs sharply from the professional development practices common in other countries. Liptak (cited in Lewis, 2002, p. 12) contrasted Lesson Study with traditional professional development as practised in the United States, as shown in Table 1.

Lesson Study begins with a question, not with an answer prepared by someone else. Identifying this question, which becomes the research theme for Lesson Study, is the first step in the process (see Fig. 1).

The research theme is developed through consideration of the reality of students' current state vis- à- vis educational or long-term goals for their learning and development.

The second step of Lesson Study is to develop a plan to address the research theme through lessons. This means making an instructional plan for a selected unit and a detailed plan for one of the lessons in that unit in which the planning team puts forth their ideas about how to address the research theme while teaching specific academic content. That lesson is called the research lesson.

The third and fourth steps in Fig. 1, conducting the research lesson and having a detailed discussion about the lesson, occur in one day-a big event day for the school. Typically, it is done in a half day; one class of students stays for the research lesson while the other classes are dismissed so that every teacher can come to observe the research lesson (even the school nurse and school nutritionist usually attend). At the end of the post-lesson discussion, usually there will be final comments lasting 30 min or more by a "knowledgeable other" from outside the school, who has been invited for this purpose.

The fifth step is to reflect on the process and consolidate and carry forward the learnings from it. Teachers will usually write their reflections and publish records of Lesson Study activities in the school bulletin.

Because they are the most visible aspects of Lesson Study, some people think of the research lesson and post-lesson discussion as the most important parts of Lesson Study, or even use "Lesson Study" to refer to the research lesson alone. However, these are just two of the five components of Lesson Study.

The Lesson Study cycle, with its five steps as illustrated in Fig. 1, contrasts with similar diagrams in other publications that have four steps (e.g. Lewis, 2002; Lewis and Hurd, 2011). These five steps, while overlapping with the four steps in the other diagrams, more accurately portray the reality of Japanese teachers' Lesson Study activity by having a closer correspondence between the titles of the steps and the activities undertaken by teachers.

Borrowing from Lewis' ( 2002) and Lewis and Hurd's ( 2011) descriptions, each step can be summarized as follows:

4.

Post- lesson discussion In a formal lesson colloquium, observers share data from the lesson to illuminate student learning, disciplinary content, lesson and unit design, and broader issues in teaching and learning.

There are three types of Lesson Study in Japan: School-based, District-based, and National-level Lesson Study. According to Takahashi ( 2006), participants' motivations or interests are different in these types of Lesson Study, but the cycle itself is basically the same. The difference is in the range, or scope, of students to be considered: school-based Lesson Study is concerned with students in the school; District-based Lesson Study is concerned with students in the district; and National-level Lesson Study is concerned with the reality of students across the country, and has a research theme with a nationwide view. Lesson Study is sometimes introduced as an open lesson by a veteran teacher "jumping in" to another teacher's classroom (Takahashi, 2013, p. 84). A "jumping in" lesson is just a demonstration unless the veteran teacher has a clear goal for the lesson as in Step 1, and proposes a new idea or content to be teachable, or he or she wants to demonstrate students' potential to be greater than ordenary teachers believe, so that he, or she, plans the lesson carefully as in Step 2. This kind of Lesson Study exists in Japan and in this case the collaboration among teachers is not a critical part of Lesson Study. In any case, each step in the Lesson Study cycle is closely related to the others, with the third and fourth steps particularly related to the first and second.

In school-based Lesson Study, which is the focus of this paper, the typical Lesson Study cycle begins at the end of an academic year-i.e. in February or March in Japan-when the faculty decides upon a research theme for the next school year, which starts in April. Several research lessons are scheduled from, say, May to November. Each research lesson and its post-lesson discussion occupy only one day, but the teachers reflect on what they learned at the research lessons and usually write a booklet or long summary report by the end of school year.

While the importance of a lesson plan as a product of Lesson Study is certainly understood, compared to the research lesson, of which there are many public examples, the collaborative work of Japanese teachers in creating a lesson plan is generally mysterious, because it is difficult to observe. According to Lee and Takahashi ( 2011) "Lesson plans are central resources for these teachers in that they constantly refer to, problematize and act on them during the entire cycle of the [Lesson Study] procedure" (p. 210).

Japanese teachers spend a lot of energy and time crafting a lesson plan. Although the details vary from school to school and even from teacher to teacher, Lewis ( 2002, pp.127-130) notes that a typical template for a lesson plan for a research lesson in Japan consists of the following:

The Japanese term for the document created for a research lesson is gakushushido- an (学習指導案), which is usually translated as "lesson plan". In this paper we will use that common translation, although we prefer the phrase "lesson proposal", because the document is much larger and broader in scope than what is usually meant by "lesson plan". Also the word "plan" may imply a fixed script, but in Japanese Lesson Study the teacher is expected to use his or her judgment if students respond in unanticipated ways. As Lee andTakahashi (2011) argue, researchers have taken for granted that using lesson plans, no matter how well devised, always involves judgment and interpretation, as teachers and their students face the contingencies of the lesson in the classoom. Their empirical study, in the context of Lesson Study, provided analytic descriptions of the interactive processes through which lesson plans are realized, leading to the conclusion that "classroom teachers use lesson plans as communicative resources to identify problems, specify assumptions about their teaching, and act on the evolving contingency of classroom interaction" (p. 209). However, Lee and Takahashi ( 2011) did not describe details of planning the lesson, including how teachers adapted or designed the task for the lesson, or how many hours they spent on planning.

In the context of Lesson Study, Lewis, Perry and Hurd (2009) focussed on one US lesson study group, of six teachers from five different schools, that conducted a research lesson in a 2-week summer workshop. This is an experimental situation, which is different from the Japanese traditional school-based Lesson Study setting. However it is worth considering in terms of the lesson planning activity. They documented that the group spent a total of six hours planning the lesson: "select research lesson, do task and share solutions, anticipate student thinking, write instructional plan using template" (Lewis et al., 2009, p. 290). However they have not offered descriptions of how they designed or adapted the task for the lesson.

On the other hand, Fernandez and Yoshida ( 2004) described in detail the process of planning lessons in the context of Lesson Study. This ethnographic study, focussed on a local elementary school in Hiroshima, vividly shows Japanese teachers' activities. However, the Lesson Study described there has the rather unique feature in that, following the research lesson being taught by a young inexperienced teacher, observed by the whole school and discussed by only the lower grade group of teachers and the principal, the lesson was revised by these teachers and then re-taught by a veteran teacher, with the whole school and an ouside advisor observing the lesson and taking part in the post-lesson discussion. The notion of Re- Teaching is extremely problematic and sensitive. In fact, the need to revise and re-teach a lesson is one of the misconceptions identified in foreign countries implementing Japanese Lesson Study (Fujii, 2014b). Whether Re- Teaching exists or not in the Lesson Study process affects the nature of the planning and the discussion of the lesson.

2.2 Structured problem solving

The structure of Japanese mathematics lessons is often regarded as unique by non-Japanese eyes, with researchers from outside Japan having noted patterns in Japanese mathematics lessons. For example, Becker et al. ( 1990) identified eight components in a typical Japanese mathematics lesson, while Stigler and Hiebert ( 1999) identified five components and labelled these lessons as structured problem solving. But their points of view are those of observers, while Japanese teachers usually do not think about the structure of their lessons in the same way. For instance, the first component of Stigler and Hiebert ( 1999), reviewing the previous lesson, is not an important activity from a Japanese teacher's point of view. Instead Japanese teachers typically consider a mathematics lesson as problem solving in terms of the four phases shown in Table 2 (see, for example, Shimizu, 1999).

This type of lesson imposes certain demands on how to interpret the lesson plan. Phase 1, presenting the problem, means helping students understand the context of the problem or task and what it will mean to solve the task-but it specifically excludes any exposition by the teacher about how to solve the task. Instead, students are expected to work independently on the task for 10-20 min (phase 2). Therefore teachers need to discuss the appropriateness of the task described in the lesson plan. The third phase, called neriage in Japanese, assumes that students will arrive at different solution methods and focusses on a comparison and discussion of those different solution methods. Therefore teachers need to discuss the plausibilty of the anticipated student solutions listed in the lesson plan. In the fourth phase, matome, the teacher may say something about which strategy may be the most sophisticated and why, but it should go beyond that to include comments by the teacher concerning the mathematical and educational values of the task and lesson (Fujii et al., 1998). Therefore teachers need to discuss the resonableness of the matome by the teacher as foreshadowed in the lesson plan. For a lesson to work in this way, the task should be understandable by the students with minimal teacher intervention; it should be solvable by at least some students (but not too quickly), and it should lend itself to multiple strategies.

This paper focusses on the second, planning step in the Lesson Study cycle, and aims to illuminate the nature of the collaborative work among teachers, based on three case studies where re-teaching was not part of the Lesson Study process, with particular emphasis on planning for these four phases of the research lessons.

This research took place in three local public elementary schools in Tokyo, which will be referred to as schools M, S and T. These schools were participating in the International Math- teacher Professionalization Using Lesson Study project (IMPULS), a recently established project funded by the Ministry of Education, Culture, Sports, Science and Technology of Japan, located at Tokyo Gakugei University, Tokyo. The purpose of this project is two-fold. First, as an international centre of Lesson Study in mathematics, Tokyo Gakugei University and its network of laboratory schools help teacher professionals learn about authentic Japanese Lesson Study, and thereby prepare them to create Lesson Study systems in their own countries for long-term, independent, educational improvement in mathematics teaching. Second, the project conducts research projects examining the mechanism of Japanese Lesson Study in order to maximize its impact on schools in Japan.

Although several research lessons were scheduled for each year, this study focusses on just one research lesson at each of these schools, and the planning meetings for those research lessons-that is, just one lesson study cycle in each school.

The author observed each lesson-planning meeting and took fieldnotes. In addition, each lesson-planning meeting was video-recorded and later transcribed; and all lesson plans and revised versions were collected and analyzed with respect to their evolution.

This paper provides a descriptive analysis of the planning process undertaken by these groups of teachers in preparation for the research lessons. In a similar vein to the research carried out by Lee and Takahashi ( 2011), discourse-in-interaction analysis (Sacks et al., 1974) was used to examine "the methods and procedures by which participants carry out ordinary tasks of classroom teaching and collaboration among teachers" (Lee and Takahashi, 2011, p. 215). The analysis began with unmotivated looking (Sacks, 1992) during the observations of the planning meetings in order to identify key discussions that eventually led to consensus regarding the lesson plans.

Through this overview of the lesson planning processes, the author came to realize that the discussions were based on the flow of the lesson. In particular, it seemed that teachers could imagine or visualize clearly what would happen at the research lesson through reading the lesson plan. Therefore it was clear that this study could focus on analyzing the planning of the flow of the research lesson.

Based on the flow of Japanese problem-solving lessons, thematic content analysis (see, for example, Fereday and Muir-Cochrane, 2006; Braun and Clarke, 2006) was carried out on transcripts of the lesson planning discussions. Using the framework of the four phases of problem-solving lessons (Table 2), participants' comments were coded with appropriate keywords to track their views of the lessons. These comments were examined with respect to the role of the lesson plan and planning meetings, in order to make visible an important part of Lesson Study-namely the planning process.

The following section is organized according to the main results obtained through the inductive process of examining the trajectory of revising lesson plans, transcribed records of planning meetings, research lesson, and post-lesson discussion, and field notes.

The results of this study are presented in three sections. First, we report on the lesson planning meetings overall-e.g. the number of meetings and participants, and the duration of meetings. Second, we examine the major component of the meetings. Finally, we identify major concerns at the meetings, such as the appropreateness of the task for the lesson, anticipated student solutions, and how to organize the comparison and discussion phase in the lesson.

4.1 The lesson planning process overall

The dates of the research lessons held at school M, S and T, together with the dates of the planning meetings are shown in Table 3. The planning meetings began between 4 and 6 weeks before the research lessons. Two schools, M and S, had four planning meetings and school T had just two meetings.

It should be noted that there was no rehearsal or trial implementation of a tentative lesson plan between planning meetings. It should be noted also that this schedule fails to reveal the amount of time that the teachers may have spent thinking about their research lesson beforehand, since the grade, unit, and lesson may have been selected at the end of the previous academic year in March.

Table 4 shows the number of participants at each of the planning meetings.

In the case of school M, the regular members of planning meetings were: the leader of the research steering committee, who also chaired the meeting and was the lead teacher for mathematics in the school; three Grade 3 teachers, one of whom taught the research lesson; and four Grade 4 teachers-a total of eight participants. The first planning meeting, held in the principal's office, was rather informal, since the knowledgeable other, who had given a talk at a research lesson that day, joined the meeting, together with the principal of the school. Beside these two participants, three Grade 3 teachers and two Grade 4 teachers attended. But at later meetings, in the school conference room, the only participants were the eight regular members.

At school S, which is a small school with only one class at each grade, the first meeting included five regular members: two classroom teachers for Grades 5 and 6, the music teacher, the art teacher, and the teacher for mathematics. The Grade 6 teacher was the leader of the school research steering committee and taught the research lesson. In Tokyo, in the case of mathematics only, if a school wants to divide classes into two or three groups for teaching mathematics, in order to help cater for individual differences, the school gets an extra teacher-in this case this teacher. The music teacher and the art teacher were teaching Grade 5 and 6 students, therefor the regular members were the upper year level team. At the second meeting, the principal joined them; at the third meeting, the knowledgeable other also joined; but the fourth meeting included only the Grade 1 teacher and the Grade 6 teacher, the music teacher, and a special needs teacher-these four constituted the school research steering committee. The venue was always a meeting room in the school.

At school T, regular members were the leader of the research steering committee, three Grade 3 teachers and three Grade 4 teachers, and the principal of the school, who attended the planning meetings-so the total number was 8. One of the Grade 4 teachers taught the research lesson. There were only two meetings, both of which were held in the principal's office.

School M, S, and T each organized a research steering committee. According to Takahashi and McDougal ( 2014), a research steering committee in Japan consists of representatives of each grade level and, in the case of the Lesson Study focussing on mathematics, the lead teacher for mathematics. In addition, representatives of special subject teams, such as music, science and home economics may join. The research steering committee leads the school's efforts and maintains the cohesion of ideas across the grades. Takahashi and McDougal ( 2014, p. 16) list roles and functions of research steering committees as follows (parenthesis added by author):

Developing a master plan for the school research;

Scheduling and leading monthly meetings to find strategies to address the school's research theme based on the ideas of the teachers;

Publishing a monthly (not always the case) internal newsletter to record the findings from each research lesson;

Planning, editing, and publishing the school research reports, including those for the research open house; and

Arranging for knowledgeable others to present lectures, teach demonstration lessons (not always the case), and give final comments at research lessons.

As shown in Table 5, the duration of the planning meetings ranged from a minimum of 30 min to a maximum of 128 min.

The chairperson of the school research steering committee led most of the meetings at schools M, S, and T. As these schools were conducting Lesson Study focussing on mathematics, the lead teacher for mathematics tended to also be in charge of the school research steering committee. Besides regular members from the school, the knowledgeable other, who had given comments on a research lesson that day, attended the first meeting at school M and the third meeting at school S. Involving a knowledgeable other in this way is common; after a research lesson and discussion ends, the team responsible for the next research lesson will meet with the knowledgeable other for further discussion and to get advice for their lesson.

As both of the 30-min meetings were with the knowledgeable other, these could be regarded as atypical. The average duration was 72 min, with the average duration excluding the 30-min meetings being 83 min.

One reason that may account for the differences in the duration of planning meetings between schools could be that the principals of schools S and T attended and participated actively in these meetings, with teachers in both schools appearing to have great confidence in them. When teachers asked, these principals gave suggestions to help break deadlocks. As a result, the duration could become shorter. In the case of school M, some of the regular members of planning meetings were young and inexperienced. Therefore, the leader of the research steering committee, who was also the lead teacher for mathematics, sometimes needed to explain the position of the lesson in the scope and sequence of the Japanese course of study, and the mathematical value of the task for use in the lesson. These factors may have had an effect on the longer duration of the meetings.

4.2 Major components and structure of the planning meetings

The first meetings held at school M and S were unusual in that the teachers discussed ideas about the research lesson in depth without a written lesson plan. At all other meetings, the discussion was based on a draft lesson plan, which had been written, either with or without the support of colleagues, by the teacher who would be teaching the lesson. Furthermore, the flow of the planning meetings followed the flow of the lesson plan. Other issues, such as the logistics of the research lesson or post-lesson discussion, were not discussed.

The format of the first draft of the lesson plan for schools M, S, and T was basically the same as Lewis' ( 2002) template as described earlier in this paper. In the case of school M, component 5 in Lewis' ( 2002) template, Learning plan for the unit, was missing at the beginning, but was added later.

Among the seven components in Lewis' ( 2002) template, component 6, Plan for the research lesson-which we will refer to here as Planning the flow of the research lesson in order to distinguish it from the overall lesson plan-is the most prominent in terms of both quantity and quality. At school T, the draft lesson plan had already been prepared for the first meeting, written by the teacher who was to teach the research lesson. The items discussed at the first meeting were as follows:

These items were exactly the items written in the draft lesson plan.

In both meetings at school T, discussion relating to planning the flow of the research lesson occupied the majority of the time: 51 min (65 %) of the first meeting as shown above, and 87 min (78 %) of the second meeting.

At school S, the first meeting was held without a written lesson plan. At this stage, teachers had not yet decided exactly which unit or content to teach for the research lesson and how. From the second meeting onwards, the teachers' discussions were based on the lesson plan drafted by the teacher who was to teach the research lesson. The knowledgeable other attended the third meeting. Excluding the first and third meetings, the proportion of time spent on planning the flow of the research lesson was 74 %, while when all four meetings are included, 52 % of the time was spent on planning the flow of the research lesson.

At school M, the first meeting was also held without the written lesson plan. From the second meeting onwards, the discussion was based on the draft lesson plan which had been written mainly by the teacher who was to teach the research lesson, but as a team, with support from the third grade teachers. In the second, third and fourth meetings, the proportion of time spent planning the flow of the research lesson was 74 %, while if the first meeting is included the proportion was 66 %. Across the three schools, omitting meetings without the lesson plan, the average proportion of time spent on planning the flow of the research lesson was 72 %; while if all meetings are included the proportion was 63 %.

Thus we have two findings: one, that the planning meetings followed the structure of the lesson plan; and two, that the discussion among teachers was particularly focussed on planning the flow of the research lesson.

The discussions specific to the flow of the research lesson during the planning meetings at the three schools could be aligned with the four phases of a problem-solving lesson (see Table 2). For example, at the second meeting at school S, a discussion on how students might grasp the given task (15 min) was related to phase 1, Presenting the problem for the day; discussion about likely student responses (14 min) was related to phase 2, Problem solving by the students; discussion about how to organize the comparison and discussion period (15 min) was obviously related to phase 3, Comparing and discussing; and discussion about how to conclude the lesson (5 min) was related to phase 4, Summing up by the teacher. Of the 49 min focussed on the flow of the research lesson, the proportions of time related to these four phases was approximately 31, 29, 31 and 10 %. The other two schools showed a similar pattern.

In the next section we will present, in more detail, what the teachers talked about regarding each phase of their lessons.

4.3 Major concerns when planning the flow of the research lesson

Discussions by the teachers, while planning the flow of the research lesson, were classified into three key categories: Appropriateness of the task, Plausibility of the anticipated student solutions, and Quality of the comparison and discussion ( neriage) phase.

4.3.1 Appropriateness of the task

Discussions about the task for the research lesson can be classified into two types. One is discussion about the task and unit from an advanced mathematical perspective, where teachers clarify the scope and sequence of relevant topics, or relationships within and expansion of the content. The second is to discuss the appropriateness of the task to the goal of the lesson, including detailed consideration of the numbers in the task, the context of the task, and so on.

When teachers talked about the position of the unit within the curriculum, they carefully referred to the National Course of Study ( 2008) published by the Ministry of Education, Culture, Sports, Science and Technology. According to Lewis' ( 2002) typical lesson plan template, this discussion is related to "connections to standards and prior and subsequent learning", which is included in the fifth componentof the template, Learning plan for the unit, where related units in former and later grades are explained and shown by using a diagram. In fact, teachers at school M used their own diagram as they discussed why the unit was important and as they traced the students' learning path leading to the unit. In the case of school S, at the second meeting where teachers talked about sequence of units, they recalled an old version of the National Course of Study (1998) in which "speed" was placed in fifth grade. "Speed" was now in sixth grade in the National Course of Study ( 2008). In fact, one teacher said "At fourth grade we teach multiplication and division of decimal numbers, and in fifth grade we teach the size of per-unit quantities.1 The closest content to speed is size of per-unit quantities.... We used to teach speed in fifth grade, together with the size of per-unit quantities".

Teachers also talked a lot about the task itself. The tasks in all three cases were not directly from textbooks; they were newly created, or modified from tasks in the textbook. Teachers discussed why they selected the particular tasks; what roles the tasks were expected to play in the unit; what benefits students might gain from solving the tasks: whether it helped to develop a new concept, a new way of thinking, or some important procedure.

The discussion of the curriculum was closely related to the solution of the task, because related content in the curriculum was expected to be a resource for students to solve the task. For example, in the second meeting in School S, there was the following exchange:

This kind of detailed and concrete consideration of previously-learned content was observed in all three schools.

Teachers also engaged in detailed discussions about the task itself, including which numbers to use and why. This aspect of Lesson Study was noted by Stigler and Hiebert ( 1999), who reported that teachers would talk about the "problem with which the lesson would begin, including such details as the exact wording and numbers to be used" (p. 117). However, the selection of numbers is not always from a purely mathematical point of view.

For example, in the case of School S, teachers thought about numbers both in terms of their students' reality and also from a procedural or calculation point of view. The teacher who would teach the research lesson said:

Child A in the problem can run 40 metres in only 6 s. In my class there is no such fast runner. However I decided to use these numbers, because these numbers are easier for children to calculate.

Time and distance data for the first three people in the problem (A, B, C) were not changed, but data for two people (E, F) were changed from E (42 metres in 6.7 s), F (28 metres in 4.9 s) to E (45 metres in 6.5 s), F (50 metres in 8 s), in order to provide some faster speeds. Numbers for D, E, and F were considered hard for students to calculate and the teachers also worried about having decimal fractions as the result of calculations. However, they decided to keep the numbers and let students use calculators if they wanted.

In the case of school M, the task was to contrast partitive and quotitive division problems obtained from one mathematical sentence. The teachers chose to use 8 ÷ 2 after also discussing 12 ÷ 3, 18 ÷ 6, 6 ÷ 2, and 10 ÷ 2 as possible candidates. They considered the numbers 8, 2, and 4 as the most easily distinguishable for students, so that students would not confuse them in using, or explaining, their ideas.

In the case of school T, the research lesson was on learning about quadrilaterals and the task was to classify quadrilaterals. The teachers changed the plan from asking students to draw figures freely on dot paper to giving students figures already drawn by the teacher. The teacher worried that students might not construct certain figures that the teacher particularly wanted to discuss in the lesson. The teachers also discussed what would be a suitable number and what types of quadrilaterals to give. If the number of figures was too small, students would not be interested in classifying them, or they would not feel any necessity to make groups. Eventually the teachers decided on nine figures: a square, a rectangle, two parallelograms, two rhombi, an isosceles trapezium, a general trapezium, and a general quadrilateral. The team decided not to include a trapezium with a right angle. As part of their discussion, teachers simulated individual students solving the problem to get an idea of the time required. Further, they considered the quality of the problem-solving activity in terms of the appropriateness of the task and the goal of the lesson.

At all three schools, the teachers discussed the unit in reference to the curriculum, as well as discussing the main task in terms of its appropriateness within the unit, its value for clarifying mathematical ideas, and its appropriateness for accomplishing the goal of the lesson. In terms of the appropriateness of the task for the goals of the lesson, teachers considered what solutions or ideas the students would be likely to bring up. This is the topic of the next section.

4.3.2 Anticipated student solutions

In all three schools, teachers spent time discussing likely student responses to the main task in the research lesson. These discussions usually began by considering what was most likely from the class as a whole. They then went on to consider likely responses from students who were rather slow learners and from students who were fast learners.

In the case of school S, teachers pretended to be students in order to solve the speed task, Who is faster? (see Table 6), from the students' point of view. Through this activity, teachers confirmed the plausibility of the four anticipated solutions already written in the lesson plan: (1) finding a common multiple of distance to compare; (2) finding a common multiple of time to compare; (3) finding the amount of time per metre to compare; and (4) finding the distance per second to compare.

In the case of school T, one teacher was asked to pretend to be a student to solve the task, and the other teachers watched his activity. In the case of school M, teachers wondered whether students would be able to create two kinds of division stories or just one story. The team leader asked the other teachers if they felt uneasy partly because of their own experiences. Teachers made explicit reference to their own experiences as they tried to anticipate students' responses to the task.

In all three schools, teachers considered how to deal with slow learners. In the case of school S, the teacher had already decided to provide hints to students who wanted them during the individual problem-solving period. The team discussed specifically what should be on these hint cards. While a hint card suggesting using common multiples was reasonable from the teachers' initial point of view, they no longer thought this might be the case when they imagined, or visualized, the lesson. They thought this strategy would eventually be rejected in favour of a better strategy: finding the distance per second. One teacher said, "Students might ask the teacher, ' Why did you not give me the best hint, if you knew?'" The other teachers agreed that was likely to happen. So they discussed how to let students notice the per second strategy. Finally teachers thought of using 30 metres and 5 s as the data. "It divides beautifully". "If the teacher asks a question such as, ' Five seconds to go (30 m), so if it were one second how far could you go?, students may be able to notice the idea of per second '". "It will work," one teacher said, "it looks fine". Eventually the teacher decided to suggest using the "per second method" to solve the task using the data of 30 metres and 5 s.

In all three schools, teachers also considered how to deal with fast learners in the lesson. For instance, at school M, a teacher said, "Students who have finished solving the task, I would ask them to write mathematical sentences, possibly like 4 × 2 = 8 or 2 × 4 = 8, showing the process to get the answer".

4.3.3 The comparison and discussion (neriage) phase

The comparision and discussion ( neriage) phase follows the problem solving by the students. This phase in the structured problem-solving lesson is the most difficult for teachers to deal with. Each correct solution has equal value in terms of getting an answer. However, the ideas involved may not have equal value. The neriage phase is when the teacher elicits these ideas and discusses the value of each solution. The teacher at school S clearly stated, "Although each strategy is sure to get the correct answer, we should not end there ... I want the students to know that getting the answer is not the final goal".

In the case of school M, teachers wanted students to compare two word problems, for partitive and quotitive division, through the use of multiplication sentences to model situations. (See the " Appendix " for the actual task.) The lead teacher of the research steering committee posed the question, "What should we ask to elicit a multiplication maths sentence?" For the next 17 min the teachers discussed what the question should be, including its exact wording.

At school T, teachers talked about which point or theme for discussion would be best: the number of groups of quadrilaterals, where the teacher might say "this student made two groups and the other student made three groups, what made these difference? What were the thoughts behind these categorizations?" or how to characterize each group, for example "This student made two groups. Can you see the common characteristics of the quadrilaterals in each group?" One teacher asked, "Which is the higher level of thinking?" to which another teacher responded, "Probably the number of groups is higher. This point is proposed in the lesson plan". So they decided to ask students to discuss how many groups there were and reasons behind this in the neriage phase.

The teams at all three schools discussed how to elevate students' mathematical thinking by comparing individual students' solutions.

It is well known that Japanese teachers get together before a research lesson to discuss the lesson. What do teachers discuss? This study reveals that their discussions followed the lesson plan, which had been drafted or created before meetings, and they devoted approximately two thirds of the time to discussing the flow of the research lesson. Within that time, teachers focussed on the appropriateness of the task, anticipated student solutions, and the plan for comparing and discussing those student solutions. The teachers also referred to the Japanese National Course of Study and its guide for teachers.

5.1 The role of the Japanese National Course of Study in designing and adapting the task for the research lesson

At planning meetings, teachers frequently referred to the National Course of Study when they needed to confirm the role of the unit, or focus lesson, within the entire curriculum. Teachers at school S talked about the placement of speed in the previous National Course of Study. This is a more difficult conversation to have in countries lacking a clear curriculum. Lewis and Tsuchida ( 1998) argued that having a frugal, shared curriculum was necessary for implementing Lesson Study. With a clear curriculum sequence, teachers could identify the value of the research lesson and the unit within the curriculum: by identifying closely related content in former and later grades, teachers can understand why the research lesson is important for later learning. And, identifying similar units or content in earlier grades helps teachers infer what students might do to solve the task, based on their previous learning. All three teams of teachers identified the position of the research lesson in the curriculum in order to clarify students' learning trajectory.

5.2 The value of discussing anticipated solutions

Data from the three schools revealed that teachers tried hard to anticipate student solutions in detail; and what they anticipated influenced the design of the lesson. For example, it influenced the design of the task, such as in the case of school T where the decision whether to include a trapezium with a right angle was made through considering students' anticipated solutions. Anticipating student responses also influenced how teachers decided to pose the problem. For instance, teachers at school S considered how students would react to the question of which person is faster when only times were given. Also teachers tried to predict students' difficulties, and discussed how to reduce students' confusion in comparing three speeds. They eventually decided to erase the slowest person's data in order to focus on only two people.

Based on their experience, Japanese teachers know that the conditions, or characteristics, of the task influence students' thinking processes and solution methods. In the case of school T, the teachers thought that the right angle might cause students to go in a direction that was not consistent with the goal of the lesson. Anticipating student solutions at planning meetings is therefore important in Lesson Study, and this unique activity is a characteristic of task design in Lesson Study (Fujii, 2015).

Teachers also think carefully about the numbers used in a task because this can strongly influence students' ways of solving the task. In the case of school S, teachers deliberately chose awkward numbers for the additional speed data, of persons D, E, F. The teacher explained, "I want students to say that it is awkward to calculate common multiples among them". She deliberately chose numbers that would push students to calculate distance divided by time. On the other hand, the numbers for B and C were (30, 6) and (30, 5) respectively, with these chosen because the numbers "divide beautifully". The teacher clearly anticipated that some students would calculate 30 ÷ 6 and 30 ÷ 5 to get the distance per second.

Close attention to the specific numbers does not mean that teachers are sticking to a concrete level of thinking or encouraging students to think concretely. On the contrary, teachers consider the general aspect of the numbers-their quasi-variable aspects. A quasi-variable is a number deliberately used in a general way, so that it serves as a representative of many numbers, just as a variable would (Fujii and Stephens, 2001, 2008; Fujii, 2008, 2010). Numbers are often chosen based on their quasi-variable power, or how well they can demonstrate a general truth-a general truth that is brought out during whole class discussion.

A structured problem-solving lesson includes a neriage-comparison and discussion-phase for students to compare or experience their friends' methods and discuss similarities and differences between strategies as a whole class. When designing the task, there needs to be consideration of whether the task will elicit the alternative approaches needed for an effective neriage. Therefore teachers carefully discuss and choose appropriate numbers for the task.

Discussing students' anticipated solutions while considering the specific numbers in the task clarifies the mathematical value of the task. In their book The Teaching Gap Stigler and Hiebert ( 1999, p. 118) have another example: teachers discuss appropriate number sentences to use in the context of teaching subtraction across 10. Subtraction across 10 can be solved by subtraction-addition (e.g. 12-9 = 10-9 + 2), subtraction-subtraction (e.g. 12-9 = 12-2-7), counting down, and counting up. In this example, the teachers believed that the subtraction-addition strategy was the most valuable for students to learn, so they examined the potential of different choices of numbers to lead to that strategy. For the same reason, almost all textbooks in Japan choose 13-9 or 12-9 to elicit the subtraction-addition strategy (Doig, Groves, and Fujii, 2011). In the case of school S, numbers were chosen to lead students to calculate distance divided by time. In the case of school T, teachers chose geometrical figures which could lead students to classify them in terms of characteristics related to their parallel or perpendicular sides. Anticipating student solutions in Lesson Study helps clarify the mathematical value of the task, and helps teachers make sure that the goal of the lesson is reached.

5.3 The value of designing the neriage phase of the lesson

The comparison and discussion of multiple student solutions needs to be more than "show and tell" (Takahashi, 2008). This neriage phase of a lesson should be an actualization of Vygotsky's zone of proximal development (Ohtani, 2014), and the role of the teacher is critical. Teachers at the three schools, M, S, and T, discussed at length how to deepen students' ideas in the neriage phase. A teacher at school S said, "Although each strategy is sure to get the correct answer, we should not end there". This comment shows teachers' deliberate efforts to elevate all students' ways of thinking.

During the planning meetings, the focus of designing the neriage phase of the lesson was on deepening students' understanding and ways of thinking. From the point of view of mathematical value, the lesson should clarify the relative value of the different solutions, generally by contrasting these. The lesson is less likely, obviously, to do this without sufficiently rich and diverse solutions to compare. Therefore, teachers carefully examine anticipated student solutions in detail in order to make sure valuable solutions are likely to appear in the comparison and discussion phase. The value of designing the neriage phase of the lesson lies in its potential to elucidate or expose ways to highlight different solutions, and how to compare them in order to reach the goal of the lesson.

5.4 Designing and adapting tasks in lesson planning goes with lesson evaluation

As we have seen, teachers give much thought to the selection and design of the task during the planning phase of Lesson Study. The task is later evaluated during the post-lesson discussion. This is another distinguishing aspect of Lesson Study. The task is not judged based on some abstract determination of whether it is good for teaching a certain skill or concept, but based on concrete evidence from the research lesson of how the students responded to it. In the case of school S, three pairs of data points were added for students to compare, but at the post-lesson discussion teachers argued about whether these additional data were useful or not. The arguments were based on how students actually responded to the task in the lesson. Similar arguments occurred at the other two schools.

In the case of school S, the arguments progressed from evaluating the task to modifying the task. In fact, the final commentator, the knowledgeable other, suggested more direct ways to manipulate numbers to identify faster speed without calculating six pairs of numbers. He gave the example shown in Table 7 of two pairs of numbers in the context of population density:

The final commentator suggested using these numbers instead the six pairs of numbers that were used in the research lesson, as some students struggled to carry out the calculations in the time available, and then missed the educational value of the task, and the whole-class discussion. The post-lesson discussion provided a context for revising the task used at the research lesson, since points missed in planning meetings were revealed in the post-lesson discussion. This shows that the planning meetings of the Lesson Study cycle are closely related to the research lesson itself, and to the post-lesson discussion.

The post-lesson discussion provided a context for revising the task used at the research lesson. However, this does not imply that re-teaching is necssarily part of Japanese Lesson Study. Based on their experience, Japanese teachers know that if students are different then their reactions will be different. They understand that a lesson is itself an organic system, it is not like a machine. A non-organic system, such as a car, is composed of parts that may be easily replaced. However, in organic systems, like a lesson, each part supports the whole ecology. In the case of school S, important ideas missed in planning meetings were revealed in the post-lesson discussion. Teachers then regretted that their kyozai- kenkyu (study or research on teaching materials- see, for example, Watanabe, Takahashi, and Yoshida, 2008) was not profound enough and broad enough to cover the idea. In other words, Japanese teachers' attitude towards research lessons and lesson plans is that their best lesson plan should be implemented at a research lesson, and that a research lesson is the proving ground for teachers (c.f. Lewis and Tsuchida, 1998).

It is widely understood that a lesson plan is an important product of Lesson Study, but despite much research into Lesson Study, the process of creating a lesson plan, as a collaborative effort by teachers, is largely invisible to non-Japanese adopters of Lesson Study. This paper tries to clarify the process of lesson planning and the role and function of the lesson plan, based on case studies of Lesson Study in three Japanese schools.

In each of these case studies, we see that the planning meetings began with a lesson plan already written by the teachers and most of the time was spent discussing the flow of the research lesson. While discussing the flow of the research lesson, teachers spent time designing and adapting the task for the lesson, during which time they typically did the following: consulted the National Course of Study to clarify the position of the task in curriculum, as well as for guidelines in designing and adapting tasks; verified the mathematical value of the task by anticipating student solutions; carefully designed the comparison and discussion ( neriage) phase of the lesson to ensure that the goal of the lesson was reached.

In addition, teachers evaluated the task during the post-lesson discussion in light of the actual student responses in the research lesson, and they also explored how the task might be revised based on this discussion.

Some potentially interesting aspects of lesson planning were not addressed in this paper: the author did not consider the relationship between the quality of the lesson planning and the quality of the research lesson. This paper did not look at the impact of lesson planning on teachers' mathematical and pedagogical knowledge (Lee and Takahashi 2011, Lewis 2009). And the paper did not look at how the lesson planning process exposes teachers' beliefs. The author hopes, however, that by making aspects of the planning phase of Lesson Study visible, this paper will contribute to helping educators outside Japan appreciate the full richness of Lesson Study, and better understand how it can improve teaching and learning.

# data[].attributes.attachment_link.title
Designing and adapting tasks in lesson planning: a critical process of Lesson Study

# data[].attributes.attachment_link.description
Toshiakira Fujii Original Article While the history of Lesson Study in Japan spans more than a century (Makinae, 2010), for Japanese educators, Lesson Study is like air, part of everyday school life. This situation possibly explains why Lesson Study is regarded as being under-theorised (e.g. Elliott, 2012).

# data[].attributes.body
Interactive article

# data[].attributes.attachment_link.readable_text
Please click or tap in the gray square below. It will fill with a jolly tableau of colored disks-first big blue ones, then somewhat smaller purple ones, and eventually lots of tiny red dots.

Sorry. My program and your browser are not getting along. None of the interactive elements of this page will work. Could you try a different browser? Current versions of Chrome, Firefox, and Safari seem to work.

The disks are scattered randomly, except that no disk is allowed to overlap another disk or extend beyond the boundary of the square. Once a disk has been placed, it never moves, so each later disk has to find a home somewhere in the nooks and crannies between the earlier arrivals. Can this go on forever?

The search for a vacant spot would seem to grow harder as the square gets more crowded, so you might expect the process to get stuck at some point, with no open site large enough to fit the next disk. On the other hand, because the disks get progressively smaller, later ones can squeeze into tighter quarters. In the specific filling protocol shown here, these two trends are in perfect balance. The process of adding disks, one after another, never seems to stall. Yet as the number of disks goes to infinity, they completely fill the box provided for them. There's a place for every last dot, but there's no blank space left over.

Or at least that's the mathematical ideal. The computer program that fills the square above never attains this condition of perfect plenitude. It shuts down after placing just 5,000 disks, which cover about 94 percent of the square's area. This early exit is a concession to the limits of computer precision and human patience, but we can still dream of how it would work in a world without such tiresome constraints.

This scheme for filling space with randomly placed objects is the invention of John Shier, a physicist who worked for many years in the semiconductor industry and who has also taught at Normandale Community College near Minneapolis. He explains the method and the mathematics behind it in a recent book, Fractalize That! A Visual Essay on Statistical Geometry. (For bibliographic details see the links and references at the end of this essay.) I learned of Shier's work from my friend Barry Cipra.

Shier hints at the strangeness of these doings by imagining a set of 100 round tiles in graduated sizes, with a total area approaching one square meter. He would give the tiles to a craftsman with these instructions:

"Mark off an area of one square meter, either a circle or a square. Start with the largest tile, and attach it permanently anywhere you wish in the marked-off area. Continue to attach the tiles anywhere you wish, proceeding always from larger to smaller. There will always be a place for every tile regardless of how you choose to place them. " How many experienced tile setters would believe this?

Shier's own creations go way beyond squares and circles filled with simple shapes such as disks. He has shown that the algorithm also works with an assortment of more elaborate designs, including nonconvex figures and even objects composed of multiple disconnected pieces. We get snowflakes, nested rings, stars, butterflies, fish eating lesser fish, faces, letters of the alphabet, and visual salads bringing together multiple ingredients. Shier's interest in these patterns is aesthetic as well as mathematical, and several of his works have appeared in art exhibits; one of them won a best-of-show award at the 2017 Joint Mathematics Meeting.

Shier and his colleagues have also shown that the algorithm can be made to work in three-dimensional space. The book's cover is adorned with a jumble of randomly placed toruses filling the volume of a transparent cube. If you look closely, you'll notice that some of the rings are linked; they cannot be disentangled without breaking at least one ring. (The 3D illustration was created by Paul Bourke, who has more examples online, including 3D-printed models.)

After reading Shier's account of his adventures, and admiring the pictures, I had to try it for myself. The experiments I'm presenting in this essay have no high artistic ambitions. I stick with plain-vanilla circular disks in a square frame, all rendered with the same banal blue-to-red color scheme. My motive is merely to satisfy my curiosity-or perhaps to overcome my skepticism. When I first read the details of how these graphics are created, I couldn't quite believe it would work. Writing my own programs and seeing them in action has helped persuade me. So has a proof by Christopher Ennis, which I'll return to below.

Filling a region of the plane with disks is not in itself such a remarkable trick. One well-known way of doing it goes by the name Apollonian circles. Start with three disks that are all tangent to one another, leaving a spiky three-pointed vacancy between them. Draw a new disk in the empty patch, tangent to all three of the original disks; this is the largest disk that can possibly fit in the space. Adding the new disk creates three smaller triangular voids, where you can draw three more triply tangent disks. There's nothing to stop you from going on in this way indefinitely, approaching a limiting configuration where the entire area is filled.

There are randomized versions of the Apollonian model. For example, you might place zero-diameter seed disks at random unoccupied positions and then allow them to grow until they touch one (or more) of their neighbors. This process, too, is space-filling in the limit. And it can never fail: Because the disks are custom-fitted to the space available, you can never get stuck with a disk that can't find a home.

Shier's algorithm is different. You are given disks one at a time in a predetermined order, starting with the largest, then the second-largest, and so on. To place a disk in the square, you choose a point at random and test to see if the disk will fit at that location without bumping into its neighbors or poking beyond the boundaries of the square. If the tests fail, you pick another random point and try again. It's not obvious that this haphazard search will always succeed-and indeed it works only if the successive disks get smaller according to a specific mathematical rule. But if you follow that rule, you can keep adding disks forever. Furthermore, as the number of disks goes to infinity, the fraction of the area covered approaches \(1\). It's convenient to have a name for series of disks that meet these two criteria; I have taken to calling them fulfilling series.

In exploring these ideas computationally, it makes sense to start with the simplest case: disks that are all the same size. This version of the process clearly cannot be fulfilling. No matter how the disks are arranged, their aggregate area will eventually exceed that of any finite container. Click in the gray square below to start filling it with equal-size disks. The square box has area \(A_{\square} = 4\). The slider in the control panel determines the area of the individual disks \(A_k\), in a range from \(0.0001\) to \(1.0\).

Program 1: Fixed-Size Disks

Sorry, the program will not run in this browser.

If you play with this program for a while, you'll find that the dots bloom quickly at first, but the process invariably slows down and eventually ends in a state labeled "Jammed," indicating that the program has been unable The program gives up after trying 10 million random locations. to find a place where one more disk will fit. If you move the slider to the right, specifying larger disks, this impasse is reached very quickly, sometimes after placing just one or two disks. If you select very small disks, the program may churn away for five or ten minutes and fill the square with more than 20,000 disks before running out of options. Nevertheless, for any disk size greater than zero, a jammed outcome is inescapable.

The densest possible packing of equal-size disks places the centers on a triangular lattice with spacing equal to the disk diameter. The resulting density (for an infinite number of disks on an infinite plane) is \(\pi \sqrt{3}\, /\, 6 \approx 0.9069\), which means more than 90 percent of the area is covered. A random filling in a finite square is much looser. My first few trials all halted with a filling fraction fairly close to one-half, and so I wondered if that nice round number might be the expectation value of the probabilistic process. Further experiments suggested otherwise. Over a broad range of disk sizes, from \(0.0001\) up to about \(0.01\), the area covered varied from one run to the next, but the average was definitely above one-half-perhaps \(0.54\). After some rummaging through the voluminous literature on circle packing, I think I may have a clue to the exact expectation value: \(\pi / (3 + 2 \sqrt{2}) \approx 0.539012\). Where does that weird number come from? The answer has nothing to do with Shier's algorithm, but I think it's worth a digression.

Consider an adversarial process: Alice is filling a unit square with \(n\) equal-size disks and wants to cover as much of the area as possible. Bob, who wants to minimize the area covered, gets to choose \(n\). If Bob chooses \(n = 1\), Alice can produce a single disk that just fits inside the square and covers about \(79\) percent of the space. Can Bob do better? Yes, if Bob specifies \(n = 2\), Alice's best option is to squeeze the two disks into diagonally opposite corners of the square as shown in the diagram at right. These disks are bounded by right isosceles triangles, which makes it easy to calculate their radii as \(r = 1 / (2 + \sqrt{2}) \approx 0.2929\). Their combined area works out to that peculiar number \(\pi / (3 + 2 \sqrt{2}) \approx 0.54\).

If two disks are better than one (from Bob's point of view), could three be better still? Or four, or some larger number? Apparently not. In 2010, Erik Demaine, Sándor Fekete and Robert Lang conjectured that the two-disk configuration shown above represents the worst case for any number of equal-size disks. In 2017 Fekete, Sebastian Morr, and Christian Scheffer proved this result.

Is it just a coincidence that the worst-case density for packing disks into a square also appears to be the expected density when equal-size disks are placed randomly until no more will fit? Wish I knew.

Let us return to the questions raised in Shier's Fractalize That! If we want to fit infinitely many disks into a finite square, our only hope is to work with disks that get smaller and smaller as the process goes on. The disk areas must come from some sequence of ever-diminishing numbers. Among such sequences, the one that first comes to mind is \(\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\) These fractions have been known since antiquity as the harmonic numbers. (They are the wavelengths of the overtones of a plucked string.)

To see what happens when successive disks are sized according to the harmonic sequence, click in the square below.

Program 2: Disk Sizes from Harmonic Series

Sorry, the program will not run in this browser.

Again, the process halts when no open space is large enough to accommodate the next disk in the sequence. If you move the slider all the way to the right, you'll see a sequence of disks with areas drawn from the start of the full harmonic sequence, \(\frac{1}{1} , \frac{1}{2}, \frac{1}{3}, \dots\); at this setting, you'll seldom get beyond eight or nine disks. Moving the slider to the left omits the largest disks at the beginning of the sequence, leaving the infinite tail of smaller disks. For example, setting the slider to \(1/20\) skips all the disks from \(\frac{1}{1}\) through \(\frac{1}{19}\) and begins filling the square with disks of area \(\frac{1}{20}, \frac{1}{21}, \frac{1}{22}, \dots\) Such truncated series go on longer, but eventually they also end in a jammed configuration.

The slider goes no further than 1/50, but even if you omitted the first 500 disks, or the first 5 million, the result would be the same. This is a consequence of the most famous property of the harmonic numbers: Although the individual terms \(1/k\) dwindle away to zero as \(k\) goes to infinity, the sum of all the terms,

\[\sum_{k = 1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots,\]

does not converge to a finite value. As long as you keep adding terms, the sum will keep growing, though ever more slowly. This curious fact was proved in the 14th century by the French bishop and scholar Nicole Oresme. The proof is simple but ingenious. Oresme pointed out that the harmonic sequence

\[\frac{1}{1} + \frac{1}{2} + \left(\frac{1}{3} + \frac{1}{4}\right) + \left(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\right) + \cdots\]

is greater than

\[\frac{1}{1} + \frac{1}{2} + \left(\frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\right) + \cdots\]

The latter series is equivalent to \(1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} \cdots\), and so it is clearly divergent. Since the grouped terms of the harmonic series are even greater, they too must exceed any finite bound.

The divergence of the harmonic series implies that disks whose areas are generated by the series will eventually overflow any enclosing container. Dropping a finite prefix of the sequence, such as the first 50 disks, does not change this fact.

Let me note in passing that just as the filling fraction for fixed-size disks seems to converge to a specific constant, 0.5390, disks in harmonic series also seem to have a favored filling fraction, roughly 0.71. Can this be explained by some simple geometric argument? Again, I wish I knew.

Evidently we need to make the disks shrink faster than the harmonic numbers do. Here's an idea: Square each element of the harmonic series, yielding this:

\[\sum_{k = 1}^{\infty}\frac{1}{k^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots.\]

Click below (or press the Start button) to see how this one turns out, again in a square of area 4.

Program 3: Disk Sizes from Zeta(2)

Sorry, the program will not run in this browser.

At last we have a process that won't get stuck in a situation where there's no place to put another disk. It could run forever, but of course it doesn't. It quits when the area of the next disk shrinks down to about a tenth of the size of a single pixel on a computer display. The stopped state is labeled "Exhausted" rather than "Jammed."This is an algorithm that could truly run forever. And yet the result is still not quite what we were hoping for-it's not fulfilling. The disks are scattered sparsely in the square, leaving vast open spaces unoccupied. The configuration reminds me of deep-sky images made by large telescopes.

Why does this outcome look so different from the others? Unlike the harmonic numbers, the infinite series \(1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots\) converges to a finite sum. In the 18th century the task of establishing this fact (and determining the exact sum) was known as the Basel Problem, after the hometown of the Bernoulli family, who put much effort into the problem but never solved it. The answer came in 1735 from Leonhard Euler (another native of Basel, though he was working in St. Petersburg), who showed that the sum is equal to \(\pi^2 / 6\). This works out to about \(1.645\); since the area of the square we want to fill is \(4\), even an infinite series of disks would cover only about \(41\) percent of the territory.

Given that the numbers \(\frac{1}{1^1}, \frac{1}{2^1}, \frac{1}{3^1}, \dots\) diminish too slowly, whereas \(\frac{1}{1^2}, \frac{1}{2^2}, \frac{1}{3^2}, \dots\) shrink too fast, it makes sense to try an exponent somewhere between \(1\) and \(2\) in the hope of finding a Goldilocks solution. The computation performed below in Program 4 is meant to facilitate the search for such a happy medium. Here the disk sizes are elements of the sequence \(\frac{1}{1^s}, \frac{1}{2^s}, \frac{1}{3^s}, \dots\), where the value of the exponent \(s\) is determined by the setting of the slider, with a range of \(1 \lt s \le 2\). We already know what happens at the extremes of this range. What is the behavior in the middle?

Program 4: Disk Sizes from Zeta(s)

Sorry, the program will not run in this browser.

If you try the default setting of \(s = 1.5\), you'll find you are still in the regime where the disks dwindle away so quickly that the box never fills up; if you're willing to wait long enough, the program will end in an exhausted state rather than a jammed one. Reducing the exponent to \(s = 1.25\) puts you on the other side of the balance point, where the disks remain too large and at some point one of them will not fit into any available space. By continuing to shuttle the slider back and forth, you could carry out a binary search, closing in, step by step, on the "just right" value of \(s\). This strategy can succeed, but it's not quick. As you get closer to the critical value, the program will run longer and longer before halting. (After all, running forever is the behavior we're seeking.) To save you some tedium, I offer a spoiler: the optimum setting is between 1.29 and 1.30.

At this point we have wandered into deeper mathematical waters. A rule of the form \(A_k = 1/k^s\) is called a power law, since each \(k\) is raised to the same power. And series of the form \(\sum 1/k^s\) are known as zeta functions, denoted \(\zeta(s)\). Zeta functions have quite a storied place in mathematics. The harmonic numbers correspond to \(\zeta(1) = \sum 1/k^1\), which does not converge. If \(\zeta(1)\) grows without limit whereas \(\zeta(2)\) converges, you might well wonder where the boundary lies between these two behaviors. As it happens, \(\zeta(1 + \epsilon)\) converges for any \(\epsilon \gt 0\). As mentioned above, Euler found that \(\zeta(2) = \pi^2 / 6\), and he went on to show that \(\zeta(s)\) also converges for all integer values of \(s\) greater than \(1\). Later, Pafnuty Chebyshev extended the domain of the function beyond the integers to all real numbers \(s\) greater than \(1\). And then Bernhard Riemann went further still: He devised a smoke-and-mirrors trick for defining the zeta function over the entire plane of complex numbers, with the single exception of \(s = 1\).

Today, Riemann's version of the zeta function is the engine (or enigma!) driving a major mathematical industry. Shier's use of this apparatus in making fractal art is far removed from that heavy-duty research enterprise-but no less fascinating. Think of it as the zeta function on vacation.

If a collection of disks are to fill a square exactly, their aggregate area must equal the area of the square. This is a necessary condition though not a sufficient one. In all the examples I've presented so far, the containing square has an area of 4, so what's needed is to find a value of \(s\) that satisfies the equation:

\[\zeta(s) = \sum_{k = 1}^{\infty}\frac{1}{k^s} = 4\]

Except for isolated values of \(s\), Those isolated values are the negative integers and the even positive integers.there is no known method for solving this equation exactly. But numerical approximation works well enough for a computer program that draws pictures. The binary search described above is a crude and cumbersome version of this numerical method. SageMath presumably does something more sophisticated. When I ask it to find a root of the equation \(zeta(s) = 4\), it returns the result \(1.2939615055572438\).

Having this result in hand solves one part of the square-filling problem. It tells us how to construct an infinite set of disks whose total area is just enough to cover a square of area \(4\), with adequate precision for graphical purposes. We assign each disk \(k\) (starting at \(k = 1\)) an area of \(1/k^{1.2939615}.\) This sequence begins 1.000, 0.408, 0.241, 0.166, 0.125, 0.098,...

In the graph above, the maroon curve with \(s = 1.29396\) converges to a sum very close to 4. Admittedly, the rate of convergence is not quick. More than 3 million terms are needed to get within 1 percent of the target.

Our off-label use of the zeta function defines an infinite sequence of disks whose aggregate area is equal to \(4\). The disks in this unique collection will exactly fill our square box (assuming they can be properly arranged). It's satisfying to have a way of reliably achieving this result, after our various earlier failures. On the other hand, there's something irksome about that number \(4\) appearing in the equation. It's so arbitrary! I don't dispute that \(4\) is a perfectly fine and foursquare number, but there are many other sizes of squares we might want to fill with dots. Why give all our attention to the \(2 \times 2\) variety?

This is all my fault. When I set out to write some square-filling programs, I knew I couldn't use the unit square-which seems like the obvious default choice-because of the awkward fact that \(\zeta(s) = 1\) has no finite solution. The unit square is also troublesome in the case of the harmonic numbers; the first disk, with area \(A_1 = 1\), is too large to fit. So I picked the next squared integer for the box size in those first programs. Having made my choice, I stuck with it, but now I feel hemmed in by that decision made with too little forethought.

We have all the tools we need to fill squares of other sizes (as long as the size is greater than \(1\)). Given a square of area \(A_{\square}\), we just solve for \(s\) in \(\zeta(s) = A_{\square}\). A square of area 8 can be covered by disks sized according to the rule \(A_k = 1/k^s\) with \(s = \zeta(8) \approx 1.1349\). For \(A_{\square} = 100\), the corresponding value of \(s\) is \(\zeta(100) \approx 1.0101\). For any \(A_{\square} \gt 1\) there is an \(s\) that yields a fulfilling set of disks, and vice versa for any \(s \gt 1\).

This relation between the exponent \(s\) and the box area \(A_{\square}\) suggests a neat way to evade the whole bother of choosing a specific container size. We can just scale the disks to fit the box, or else scale the box to accommodate the disks. Shier adopts the former method. Each disk in the infinite set is assigned an area of

\[A_k = \frac{A_{\square}}{\zeta(s)} \frac{1}{k^s},\]

where the first factor is a scaling constant that adjusts the disk sizes to fit the container. In my first experiments with these programs I followed the same approach. Later, however, when I began writing this essay, it seemed easier to think about the scaling-and explain it-if I transformed the size of the box rather than the sizes of the disks. In this scheme, the area of disk \(k\) is simply \(1 / k^s\), and the area of the container is \(A_{\square} = \zeta(s)\). (The two scaling procedures are mathematically equivalent; it's only the ratio of disk size to container size that matters.)

Program 5 offers an opportunity to play with such scaled zeta functions. I'm not actually changing the physical size of the box-the number of pixels it occupies on-screen remains the same. I'm scaling the units of measure. No matter where you move the \(s\) slider, the area of the square container will adjust to match the total area of the infinite set of disks. As the ratio of disk size to container size shifts, so does the overall texture or appearance of the pattern. At a setting of \(s = 1.35\), the largest disk fills almost a third of the square; at \(s = 1.08\), the first disk occupies only about \(8\) percent of box area. At those lower settings it takes a very long time to reach a high filling percentage, but if you have true faith in mathematical certainties, your patience will be rewarded.

Program 5: Disk Sizes from Scaled Zeta(s)

Sorry, the program will not run in this browser.

At the other end of the scale, if you push the value of \(s\) up beyond about \(1.40\), you'll discover something else: The program more often than not halts after placing just a few disks. At \(s = 1.50\) or higher, it seldom gets beyond the first disk. This failure is similar to what we saw with the harmonic numbers, but more interesting. In the case of the harmonic numbers, the total area of the disks is unbounded, making an overflow inevitable. With this new scaled version of the zeta function, the total area of the disks is always equal to that of the enclosing square. In principle, all the disks could all be made to fit, if you could find the right arrangement. I'll return below to the question of why that doesn't happen.

In Fractalize That! Shier introduces another device for taming space-filling sets. He not only scales the object sizes so that their total area matches the space available; he also adopts a variant zeta function that has two adjustable parameters rather than just one:

A note on notation: Shier writes the Hurwitz zeta function as \(\zeta(c, N)\), whereas most of the mathematical literature seems to favor \(\zeta(s, a)\). I'm going with the majority.\[\zeta(s, a) = \sum_{k=0}^{\infty} \frac{1}{(a + k)^s}\]

This is the Hurwitz zeta function, named for the German mathematician Adolf Hurwitz (1859-1919). Before looking into the details of the function, let's play with the program and see what happens. Try a few settings of the \(s\) and \(a\) controls:

Program 6: Disk Sizes from Scaled Hurwitz Zeta Function

Sorry, the program will not run in this browser.

Different combinations of \(s\) and \(a\) produce populations of disks with different size distributions. The separate contributions of the two parameters are not always easy to disentangle, but in general decreasing \(s\) or increasing \(a\) leads to a pattern dominated by smaller disks. Here are snapshots of four outcomes:

Within the parameter range shown in these four panels, the filling process always continues to exhaustion, but at higher values of \(s\) it can jam, just as it does with the scaled Riemann zeta function.

Adolf Hurwitz in the 1880s. Photo from Wikipedia. Until I began this project, I knew nothing of Adolf Hurwitz or his work, although he is hardly an obscure figure in the history of mathematics. He earned his Ph.D. under Felix Klein and also studied with Karl Weierstrass, Ernst Eduard Kummer, and Leopold Kronecker-key figures in the founding of modern analysis and number theory. Among his own pupils (and close friends) were David Hilbert and Hermann Minkowski. Albert Einstein was another of his students, although apparently he seldom went to class.

Hurwitz wrote just one paper on the zeta function. It was published in 1882, when he was still quite young and just beginning his first academic appointment, at the University of Göttingen. (The paper is available from the Göttinger Digitalisierungszentrum; see pp. 86-101.)

Hurwitz modified the Riemann zeta function in two ways. First, the constant \(a\) is added to each term, turning \(1/k^s\) into \(1/(a + k)^s\). Second, the summation begins with \(k = 0\) rather than \(k = 1\). By letting \(a\) take on any value in the range \(0 \lt a \le 1\) we gain access to a continuum of zeta functions. The elements of the series are no longer just reciprocals of integers but reciprocals of real numbers. Suppose \(a = \frac{1}{3}\). Then \(\zeta(s, a)\) becomes:

\[\frac{1}{\left(\frac{1}{3} + 0\right)^s} + \frac{1}{\left(\frac{1}{3} + 1\right)^s} + \frac{1}{\left(\frac{1}{3} + 2\right)^s} + \cdots\ = \left(\frac{3}{1}\right)^s + \left(\frac{3}{4}\right)^s + \left(\frac{3}{7}\right)^s + \cdots\]

The Riemann zeta function and the Hurwitz zeta function differ substantially only for small values of \(k\) or large values of \(a\). When \(k\) is large, adding a small \(a\) to it makes little difference in the value of the function. Thus as \(k\) grows toward infinity, the two functions are asymptotically equal, as suggested in the graph at right. When the Hurwitz function is put to work packing disks into a square, a rule with \(a > 1\) causes the first several disks to be smaller than they would be with the Riemann rule. A value of \(a\) between \(0\) and \(1\) enlarges the early disks. In either case, the later disks in the sequence are hardly affected at all.

If \(a\) is a positive integer, the interpretation of \(\zeta(s, a)\) is even simpler. The case \(a = 1\) corresponds to the Riemann zeta sum. When \(a\) is a larger integer, the effect is to omit the first \(a - 1\) entries, leaving only the tail of the series. For example,

\[\zeta(s, 5) = \frac{1}{5^s} + \frac{1}{6^s} + \frac{1}{7^s} + \cdots.\]

In his fractal artworks, Shier chooses various values of \(a\) as a way of controlling the size distribution of the placed objects, and thereby fine-tuning the appearance of the patterns. Having this adjustment knob available is very convenient, but in the interests of simplicity, I am going to revert to the Riemann function in the rest of this essay.

Before going on, however, I also have to confess that I don't really understand the place of the Hurwitz zeta function in modern mathematical research, or what Hurwitz himself had in mind when he formulated it. Zeta functions have been an indispensable tool in the long struggle to understand how the prime numbers are sprinkled among the integers. The connection between these two realms was made by Euler, with his remarkable equation linking a sum of powers of integers with a product of powers of primes:

Euler's other famous equation, \(e^{i\pi} + 1 = 0\), has a bigger fan club, but this is the one that revs my motor.\[\sum_{k = 1}^{\infty} \frac{1}{k^s} = \prod_{p \text{ prime}} \frac{1}{1 - \frac{1}{p^s}}.\]

Riemann went further, showing that everything we might want to know about the distribution of primes is encoded in the undulations of the zeta function over the complex plane. Indeed, if we could simply pin down all the complex values of \(s\) for which \(\zeta(s) = 0\), we would have a master key to the primes. Hurwitz, in his 1882 paper, was clearly hoping to make some progress toward this goal, but I have not been able to figure out how his work fits into the larger story. The Hurwitz zeta function gets almost no attention in standard histories and reference works (in contrast to the Riemann version, which is everywhere). Wikipedia notes: "At rational arguments the Hurwitz zeta function may be expressed as a linear combination of Dirichlet L-functions and vice versa"-which sounds interesting, but I don't know if it's useful or important. A recent article by Nicola Oswald and Jörn Steuding puts Hurwitz's work in historical context, but it does not answer these questions-at least not in a way I'm able to understand.

But again I digress. Back to dots in boxes.

If a set of circular disks and a square container have the same total area, can you always arrange the disks so that they completely fill the square without overflowing? Certainly not! Suppose the set consists of a single disk with area equal to that of the square; the disk's diameter is greater than the side length of the square, so it will bulge through the sides while leaving the corners unfilled. A set of two disks won't work either, no matter how you apportion the area between them. Indeed, when you are putting round pegs in a square hole, no finite set of disks can ever fill all the crevices.

Only an infinite set-a set with no smallest disk-can possibly fill the square completely. But even with an endless supply of ever-smaller disks, it seems like quite a delicate task to find just the right arrangement, so that every gap is filled and every disk has a place to call home. It's all the more remarkable, then, that simply plunking down the disks at random locations seems to produce exactly the desired result. This behavior is what intrigued and troubled me when I first saw Shier's pictures and read about his method for generating them. If a random arrangement works, it's only a small step to the proposition that any arrangement works. Could that possibly be true?

Computational experiments offer strong hints on this point, but they can never be conclusive. What we need is a proof. Ennis's proof was published in Math Horizons, a publication of the Mathematical Association of America, which keeps it behind a paywall. If you have no library access and won't pay the $50 ransom, I can recommend a video of Ennis explaining his proof in a talk at St. Olaf College.And so I turn to the work of Christopher Ennis, a mathematician at Normandale Community College, who met Shier when they were both teaching there.

As a warm-up exercise, Ennis proves a one-dimensional version of the area-filling conjecture, where the geometry is simpler and some of the constraints are easier to satisfy. In one dimension a disk is merely a line segment; its area is its length, and its radius is half that length. As in the two-dimensional model, disks are placed in descending order of size at random positions, with the usual proviso that no disk can overlap another disk or extend beyond the end points of the containing interval. In Program 7 you can play with this scheme.

Program 7: One-Dimensional Disks

Sorry, the program will not run in this browser.

I have given the line segment some vertical thickness to make it visible. The resulting pattern of stripes may look like a supermarket barcode or an atomic spectrum, but please imagine it as one-dimensional.

If you adjust the slider in this program, you'll notice a difference from the two-dimensional system. In 2D, the algorithm is fulfilling only if the exponent \(s\) is less than a critical value, somewhere in the neighborhood of 1.4. In one dimension, the process continues without impediment for all values of \(s\) throughout the range \(1 \lt s \lt 2\). Try as you might, you won't find a setting that produces a jammed state. (In practice, the program halts after placing no more than 10,000 disks, but the reason is exhaustion rather than jamming.)

Ennis titles his Math Horizons article "(Always) room for one more." He proves this assertion by keeping track of the set of points where the center of a new disk can legally be placed, and showing the set is never empty. Suppose \(n - 1\) disks have already been randomly scattered in the container. The next disk to be placed, disk \(n\), will have an area (or length) of \(A_n = 1 / n^s\). Since the geometry is one-dimensional, the corresponding disk radius is simply \(r_n = A_n / 2\). The center of this new disk cannot lie any closer than \(r_n\) to the perimeter of another disk. It must also be at a distance of at least \(r_n\) from the boundary of the containing segment. We can visualize these constraints by adding bumpers, or buffers, of thickness \(r_n\) to the outside of each existing disk and to the inner edges of the containing segment. A few stages of the process are illustrated below.

Placed disks are blue, the excluded buffer areas are orange, and open areas-the set of all points where the center of the next disk could be placed-are black. In the top line, before any disks have been placed, the entire containing segment is open except for the two buffers at the ends. Each of these buffers has a length equal to \(r_1\), the radius of the first disk to be placed; the center of that disk cannot lie in the orange regions because the disk would then overhang the end of the containing segment. After the first disk has been placed (second line), the extent of the open area is reduced by the area of the disk itself and its appended buffers. On the other hand, all of the buffers have also shrunk; each buffer is now equal to the radius of disk \(2\), which is smaller than disk \(1\). The pattern continues as subsequent disks are added. Note that although the blue disks cannot overlap, the orange buffers can.

For another view of how this process evolves, click on the Next button in Program 8. Each click inserts one more disk into the array and adjusts the buffer and open areas accordingly.

Program 8: One-Dimensional with Buffers

Sorry, the program will not run in this browser.

Because the blue disks are never allowed to overlap, the total blue area must increase monotonically as disks are added. It follows that the orange and black areas, taken together, must steadily decrease. But there's nothing steady about the process when you keep an eye on the separate area measures for the orange and black regions. Changes in the amount of buffer overlap cause erratic, seesawing tradeoffs between the two subtotals. If you keep clicking the Next button (especially with \(s\) set to a high value), you may see the black area falling below \(1\) percent. Can we be sure it will never vanish entirely, leaving no opening at all for the next disk?

Ennis answers this question through worst-case analysis. He considers only configurations in which no buffers overlap, thereby squeezing the black area to its smallest possible extent. If the black area is always positive under these conditions, it cannot be smaller when buffer overlaps are allowed.

The basic idea of the proof I have altered some of the notation and certain details of presentation to conform with choices I made elsewhere in this exposition. is to start with \(n - 1\) buffered disks already in place, arranged so that none of the orange buffer areas intersect. The total area of the blue disks is \(\sum_{k=1}^{k=n-1} 1/k^s\). Each buffer zone has a width equal to \(r_{n}\), the radius of the next disk to be added to the tableau; since each disk has two buffers, the total area of the orange buffers is \(2(n-1)r_{n}\). The black area is whatever's left over. In other words,

\[A_{\square} = \zeta(s), \quad A_{\color{blue}{\mathrm{blue}}} = \sum_{k=1}^{k = n - 1} \frac{1}{k^s}, \quad A_{\color{orange}{\mathrm{orange}}} = 2(n-1)r_{n}.\]

Then we need to prove that

\[A_{\square} - (A_{\color{blue}{\mathrm{blue}}} + A_{\color{orange}{\mathrm{orange}}}) \gt 0.\]

A direct proof of this statement would require an exact, closed-form expression for \(\zeta(s)\), which we already know is problematic. Ennis evades this difficulty by turning to calculus. He needs to evaluate the remaining tail of the zeta series, \(\sum_{k = n}^\infty 1/k^s\), but this discrete sum is intractable. On the other hand, by shifting from a sum to an integral, the problem becomes an exercise in undergraduate calculus. Exchanging the discrete variable \(k\) for a continuous variable \(x\), we want to find the area under the curve \(1/x^s\) in the interval from \(n\) to infinity; this will provide a lower bound on the corresponding discrete sum. Evaluating the integral yields:

\[\int_{x = n}^{\infty} \frac{1}{x^{s}} d x = \frac{1}{(s-1) n^{s-1}}.\]

Some further manipulation reveals that the area of the black regions is never smaller than

\[\frac{2 - s}{(s - 1)n^{s - 1}}.\]

If \(s\) lies strictly between \(1\) and \(2\), this expression must be greater than zero, since both the numerator and the denominator will be positive. Thus for all \(n\) there is at least one black point where the center of a new disk can be placed.

Ennis's proof is a stronger one than I expected. When I first learned there was a proof, I guessed that it would take a probabilistic approach, showing that although a jammed configuration may exist, it has probability zero of turning up in a random placement of the disks. Instead, Ennis shows that no such arrangement exists at all. Even if you replaced the randomized algorithm with an adversarial one that tries its best to block every disk, the process would still run to fulfillment.

The proof for a two-dimensional system follows the same basic line of argument, but it gets more complicated for geometric reasons. In one dimension, as the successive disk areas get smaller, the disk radii diminish in simple proportion: \(r_k = A_k / 2\). In two dimensions, disk radius falls off only as the square root of the disk area: \(r_k = \sqrt{A_k / \pi}\). As a result, the buffer zone surrounding a disk excludes neighbors at a greater distance in two dimensions than it would in one dimension. There is still a range of \(s\) values where the process is provably unstoppable, but it does not extend across the full interval from \(s \gt 1\) to \(s \lt 2\).

Program 9, running in the panel below, is one I find very helpful in gaining intuition into the behavior of Shier's algorithm. As in the one-dimensional model of Program 8, each press of the Next button adds a single disk to the containing square, and shows the forbidden buffer zones surrounding the disks.

Program 9: Two-Dimensional Disks with Buffers

Sorry, the program will not run in this browser.

Move the \(s\) slider to a position somewhere near 1.40. In the control panel for this program, the registers showing fill percentages for blue, orange, and black areas are not entirely trustworthy. In the one-dimensional case, it's easy to calculate the areas to high precision. That's much harder in two dimensions, where the overlapping regions of buffers can have complex shapes. I have resorted to counting pixels, a procedure that has limited resolution and is subject to errors caused by fuzzy boundaries.At this setting, there's a fair chance (maybe 10 or 20 percent) that the very first disk and its orange buffer zone will entirely cover the open black region, creating a jammed state. On other runs you might get as far as two or three or 10 disks before you get stuck. If you make it beyond that point, however, you are likely to continue unimpeded for as long as you have the patience to keep pressing Next. Shier describes this phenomenon as "infant mortality": If the placement process survives the high-risk early period, it is all but immortal.

There's a certain whack-a-mole dynamic to the behavior of this system. Maybe the first disk covers all but one small corner of the black zone. It looks like the next disk will completely obliterate that open area. And so it does-but at the same time the shrinking of the orange buffer rings opens up another wedge of black elsewhere. The third disk blots out that spot, but again the narrowing of the buffers allows a black patch to peek out from still another corner. Later on, when there are dozens of disks, there are also dozens of tiny black spots where there's room for another disk. You can often guess which of the openings will be filled next, because the random search process is likely to land in the largest of them. Again, however, as these biggest targets are buried, many smaller ones are born.

Ennis's two-dimensional proof addresses the case of circular disks inside a circular boundary, rather than a square one. (The higher symmetry and the absence of corners streamlines certain calculations.) The proof strategy, again, is to show that after \(n - 1\) disks have been placed, there is still room for the \(n\)th disk, for any value of \(n \ge 1\). The argument follows the same logic as in one dimension, relying on an integral to provide a lower bound for the sum of a zeta series. But because of the \(\pi r^2\) area relation, the calculation now includes quadratic as well as linear terms. As a result, the proof covers only a part of the range of \(s\) values. The black area is provably nonempty if \(s\) is greater than \(1\) but less than roughly \(1.1\); outside that interval, the proof has nothing to say.

As mentioned above, Ennis's proof applies only to circular disks in a circular enclosure. Nevertheless, in what follows I am going to assume the same ideas carry over to disks in a square frame, although the location of the boundary will doubtless be somewhat different. I have recently learned that Ennis has written a further paper on the subject, expected to be published in the American Mathematical Monthly. Perhaps he addresses this question there.

With Program 9, we can explore the entire spectrum of behavior for packing disks into a square. The possibilities are summarized in the candybar graph below.

The leftmost band, in darker green, is the interval for which Ennis's proof might hold. The question mark at the upper boundary line signifies that we don't really know where it lies.

In the lighter green region no proof is known, but in Shier's extensive experiments the system never jams there.

The transition zone sees the probability of jamming rise from \(0\) to \(1\) as \(s\) goes from about \(1.3\) to about \(1.5\).

Beyond \(s \approx 1.5\), experiments suggest that the system always halts in a jammed configuration.

At \(s \approx 1.6\) we enter a regime where the buffer zone surrounding the first disk invariably blocks the entire black region, leaving nowhere to place a second disk. Thus we have a simple proof that the system always jams.

Still another barrier arises at \(s \approx 2.7\). Beyond this point, not even one disk will fit. The diameter of a disk with area \(1\) is greater than the side length of the enclosing square.

Can we pin down the exact locations of the various threshold points in the diagram above? This problem is tractable in those situations where the placement of the very first disk determines the outcome. At high values of \(s\) (and thus low values of \(\zeta(s)\), the first disk can obliterate the black zone and thereby preclude placement of a second disk. What is the lowest value of \(s\) for which this can happen? As in the image at right, the disk must lie at the center of the square box, and the orange buffer zone surrounding it must extend just far enough out to cover the corners of the inner black square, which defines the locus of all points that could accommodate the center of the second disk. Finding the value of \(s\) that satisfies this condition is a messy but straightforward bit of geometry and algebra. With the help of SageMath I get the answer \(s = 1.282915\). This value-let's call it \(\overline{s}\)-is an upper bound on the "never jammed" region. Above this limit there is always a nonzero probability that the filling process will end after placing a single disk.

The value of \(\overline{s}\) lies quite close to the experimentally observed boundary between the never-jammed range and the transition zone, where jamming first appears. Is it possible that \(\overline{s}\) actually marks the edge of the transition zone-that below this value of \(s\) the program can never fail? To prove that conjecture, you would have to show that when the first disk is successfully placed, the process never stalls on a subsequent disk. That's certainly not true in higher ranges of \(s\). Yet the empirical evidence near the threshold is suggestive. In my experiments I have yet to see a jammed outcome at \(s \lt \overline{s}\), not even in a million trials just below the threshold, at \(s = 0.999 \overline{s}\). In contrast, at \(s = 1.001 \overline{s}\), a million trials produced 53 jammed results-all of them occuring immediately after the first disk was placed.

The same kind of analysis leads to a lower bound on the region where every run ends after the first disk (medium pink in the diagram above). In this case the critical situation puts the first disk as close as possible to a corner of the square frame, rather than in the middle. If the disk and its orange penumbra are large enough to block the second disk in this extreme configuration, then they will also block it in any other position. Putting a number on this bound again requires some fiddly equation wrangling; the answer I get is \(\underline{s} = 1.593782\). No process with higher \(s\) can possibly live forever, since it will die with the second disk. In analogy with the lower-bound conjecture, one might propose that the probability of being jammed remains below \(1\) until \(s\) reaches \(\underline{s}\). If both conjectures were true, the transition region would extend from \(\overline{s}\) to \(\underline{s}\).

The final landmark, way out at \(s \approx 2.7\), marks the point where the first disk threatens to burst the bounds of the enclosing square. In this case the game is over before it begins. In program 9, if you push the slider far to the right, you'll find that the black square in the middle of the orange field shrinks away and eventually winks out of existence. This extinction event comes when the diameter of the disk equals the side length of the square. Given a disk of area \(1\), and thus radius \(1/\sqrt{\pi}\), we want to find the value of \(s\) that satisfies the equation

\[\frac{2}{\sqrt{\pi}} = \sqrt{\zeta(s)}.\]

Experiments with Program 9 show that the value is just a tad more than 2.7. That's an interesting numerical neighborhood, no? A famous number lives nearby. Do you suppose?

Another intriguing set of questions concerns the phenomenon that Shier calls infant mortality. If you scroll back up to Program 5 and set the slider to \(s = 1.45\), you'll find that roughly half the trials jam. The vast majority of these failures come early in the process, after no more than a dozen disks have been placed. At \(s = 1.50\) death at an early age is even more common; three-fourths of all the trials end with the very first disk. On the other hand, if a sequence of disks does manage to dodge all the hazards of early childhood, it may well live on for a very long time-perhaps forever.

Should we be surprised by this behavior? I am. As Shier points out, the patterns formed by our graduated disks are fractals, and one of their characteristic properties is self-similarity, or scale invariance. If you had a fully populated square-one filled with infinitely many disks-you could zoom in on any region to any magnification, and the arrangement of disks would look the same as it does in the full-size square. By "look the same" I don't mean the disks would be in the same positions, but they would have the same size distribution and the same average number of neighbors at the same distances. This is a statistical concept of identity. And since the pattern looks the same and has the same statistics, you would think that the challenge of finding a place for a new disk would also be the same at any scale. Slipping in a tiny disk late in the filling operation would be no different from plopping down a large disk early on. The probability of jamming ought to be constant from start to finish.

But there's a rejoinder to this argument: Scale invariance is broken by the presence of the enclosing square. The largest disks are strongly constrained by the boundaries, whereas most of the smaller disks are nowhere near the edges and are little influenced by them. The experimental data offer some support for this view. The graph below summarizes the outcomes of \(20{,}000\) trials at \(s = 1.50\). The red bars show the absolute numbers of trials ending after placing \(n\) disks, for each \(n\) from \(0\) through \(35\). The blue lollipops indicate the proportion of trials reaching disk \(n\) that halted after placing disk \(n\). This ratio can be interpreted (if you're a frequentist!) as the probability of stopping at \(n\).

It certainly looks like there's something odd happening on the left side of this graph. More than three fourths of the trials end after a single disk, but none at all jam at the second or third disks, and very few (a total of \(23\)) at disks \(4\) and \(5\). Then, suddenly, \(1{,}400\) more fall by the wayside at disk \(6\), and serious attrition continues through disk \(11\).

Geometry can explain some of this weirdness. It has to do with the squareness of the container; other shapes would produce different results.

At \(s = 1.50\) we are between \(\overline{s}\) and \(\underline{s}\), in a regime where the first disk is large enough to block off the entire black zone but not so large that it must do so. This is enough to explain the tall red bar at \(n = 1\): When you place the first disk randomly, roughly \(75\) percent of the time it will block the entire black region, ending the parade of disks. If the first disk doesn't foreclose all further action, it must be tucked into one of the four corners of the square, leaving enough room for a second disk in the diagonally opposite corner. The sequence of images below (made with Program 9) tells the rest of the story.

The placement of the second disk blocks off the open area in that corner, but the narrowing of the orange buffers also creates two tiny openings in the cross-diagonal corners. The third and fourth disks occupy these positions, and simultaneously allow the black background to peek through in two other spots. Finally the fifth and sixth disks close off the last black pixels, and the system jams.

This stereotyped sequence of disk placements accounts for the near absence of mortality at ages \(n = 2\) through \(n = 5\), and the sudden upsurge at age \(6\). The elevated levels at \(n = 7\) through \(11\) are part of the same pattern; depending on the exact positioning of the disks, it may take a few more to expunge the last remnants of black background.

At still higher values of \(n\)-for the small subset of trials that get there-the system seems to shift to a different mode of behavior. Although numerical noise makes it hard to draw firm conclusions, it doesn't appear that any of the \(n\) values beyond \(n = 12\) are more likely jamming points than others. Indeed, the data are consistent with the idea that the probability of jamming remains constant as each additional disk is added to the array, just as scale invariance would suggest.

A much larger data set would be needed to test this conjecture, and collecting such data is painfully slow. Furthermore, when it comes to rare events, I don't have much trust in the empirical data. During one series of experiments, I noticed a program run that stalled after \(290\) disks-unusually late. The 290-disk configuration, produced at \(s = 1.47\), is shown at left below.
This pattern, compared with those produced at lower values of \(s\), has a strongly Apollonian texture. Many of the disks are nestled tightly among their neighbors, and they form the recursive triangular motifs characteristic of Apollonian circles.
I wondered if it was truly jammed. My program gives up on finding a place for a disk after \(10^7\) random attempts. Perhaps if I had simply persisted, it would have gone on. So I reset the limit on random attempts to \(10^9\), and sat back to wait. After some minutes the program discovered a place where disk \(291\) would fit, and then another for disk \(292\), and kept going as far as 300 disks. The program had an afterlife! Could I revive it again? Upping the limit to \(10^{10}\) allowed another \(14\) disks to squueze in. The final configuration is shown at right above (with the original \(290\) disks faded, in order to make the \(24\) posthumous additions more conspicuous).

Is it really finished now, or is there still room for one more? I have no reliable way to answer that question. Checking \(10\) billion random locations sounds like a lot, but it is still a very sparse sampling of the space inside the square box. Using 64-bit floating-point numbers to define the coordinate system allows for more than \(10^{30}\) distinguishable points. And to settle the question mathematically, we would need unbounded precision.

We know from Ennis's proof that at values of \(s\) not too far above \(1.0\), the filling process can always go on forever. And we know that beyond \(s \approx 1.6\), every attempt to fill the square is doomed. There must be some kind of transition between these two conditions, but the details are murky. The experimental evidence gathered so far suggests a smooth transition along a sigmoid curve, with the probability of jamming gradually increasing from \(0\) to \(1\). As far as I can tell, however, nothing we know for certain rules out a single hard threshold, below which all disk sequences are immortal and above which all of them die. Thus the phase diagram would be reduced to this simple form:

The softer transition observed in computational experiments would be an artifact of our inability to perform infinite random searches or place infinite sequences of disks.

Here's a different approach to understanding the random dots-in-a-box phenomenon. It calls for a mental reversal of figure and ground. Instead of placing disks on a square surface, we drill holes in a square metal plate. And the focus of attention is not the array of disks or holes but rather the spaces between them. Shier has a name for the perforated plate: the gasket.

Program 10 allows you to observe a gasket as it evolves from a solid black square to a delicate lace doily with less than 1 percent of its original substance.

Program 10: The Gasket

Sorry, the program will not run in this browser.

The gasket is quite a remarkable object. When the number of holes becomes infinite, the gasket must disappear entirely; its area falls to zero. Up until that very moment, however, it retains its structural integrity. This statement about the connectedness of the gasket requires a more careful consideration of what it means for two disks or holes to overlap. Are they allowed to touch, or in other words to be tangent, sharing a single point? The answer makes no difference to the calculation of areas, but it does matter for connectivity. Allowing tangency (as in Apollonian circles) would shatter the gasket, leaving tiny shards. To preserve connectivity, a computer program must test for overlaps with "\(\gt\)" rather than "\(\ge\)". Although it may be reduced to a fine filigree, the perforated square never tears apart into multiple pieces; it remains a single, connected component, a network with multiple paths linking any two points you might choose.

As the gasket is etched away, can we measure the average thickness of the surviving wisps and tendrils? I can think of several methods that involve elaborate sampling schemes. Shier has a much simpler and more ingenious proposal: To find the average thickness of the gasket, divide its area by its perimeter. It was not immediately obvious to me why this number would serve as an appropriate measure of the width, but at least the units come out right: We are dividing a length squared by a length and so we get a length. And the operation does make basic sense: The area of the gasket represents the amount of substance in it, and the perimeter is the distance over which it is stretched. (The widths calculated in Program 10 differ slightly from those reported by Shier. The reason, I think, is that I include the outer boundary of the square in the perimeter, and he does not.)

Calculating the area and perimeter of a complicated shape such as a many-holed gasket looks like a formidable task, but it's easy if we just keep track of these quantities as we go along. Initially (before any holes are drilled), the gasket area \(A_0^g\) is the area of the full square, \(A_\square\). The initial gasket perimeter \(P_0^g\) is four times the side length of the square, which is \(\sqrt{A_\square}\). Thereafter, as each hole is drilled, we subtract the new hole's area from \(A^g\) and add its perimeter to \(P^g\). The quotient of these quantities is our measure of the average gasket width after drilling hole \(k\): \(\widehat{W}_k^g\). Since the gasket area is shrinking while the perimeter is growing, \(\widehat{W}_k^g\) must dwindle away as \(k\) increases.

The importance of \(\widehat{W}_k^g\) is that it provides a clue to how large a vacant space we're likely to find for the next disk or hole. If we take the idea of "average" seriously, there must always be at least one spot in the gasket with a width equal to or greater than \(\widehat{W}_k^g\). From this observation Shier makes the leap to a whole new space-filling algorithm. Instead of choosing disk diameters according to a power law and then measuring the resulting average gasket width, he determines the radius of the next disk from the observed \(\widehat{W}_k^g\):

\[r_{k+1} = \gamma \widehat{W}_k^g = \gamma \frac{A_k^g}{P_k^g}.\]

Here \(\gamma\) is a fixed constant of proportionality that determines how tightly the new disks or holes fit into the available openings.

The area-perimeter algorithm has a recursive structure, in which each disk's radius depends on the state produced by the previous disks. This raises the question of how to get started: What is the size of the first disk? Shier has found that it doesn't matter very much. Initial disks in a fairly wide range of sizes yield jam-proof and aesthetically pleasing results.

Graphics produced by the original power-law algorithm and by the new recursive one look very similar. One way to understand why is to rearrange the equation of the recursion:

Perhaps this equation would be a little easier to interpret if the average width were defined in terms of hole diameters rather than hole perimeters. Then the denominator of the right hand side would be the sum of the first \(k\) diameters scaled by the \((k+1)\)st diameter.\[\frac{1}{2 \gamma} = \frac{A_k^g}{2 r_{k+1} P_k^g}.\]

On the right side of this equation we are dividing the average gasket width by the diameter of the next disk to be placed. The result is a dimensionless number-dividing a length by a length cancels the units. More important, the quotient is a constant, unchanging for all \(k\). If we calculate this same dimensionless gasket width when using the power-law algorithm, it also turns out to be nearly constant in the limit of karge \(k\), showing that the two methods yield sequences with similar statistics.

Setting aside Shier's recursive algorithm, all of the patterns we've been looking at are generated by a power law (or zeta function), with the crucial requirement that the series must converge to a finite sum. The world of mathematics offers many other convergent series in addition to power laws. Could some of them also create fulfilling patterns? The question is one that Ennis discusses briefly in his talk at St. Olaf and that Shier also mentions.

Among the obvious candidates are geometric series such as \(\frac{1}{1}, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots\) A geometric series is a close cousin of a power law, defined in a similar way but exchanging the roles of \(s\) and \(k\). That is, a geometric series is the sum:

\[\sum_{k=0}^{\infty} \frac{1}{s^k} = \frac{1}{s^0} + \frac{1}{s^1} + \frac{1}{s^2} + \frac{1}{s^3} + \cdots\]

For any \(s > 1\), the infinite geiometric series has a finite sum, namely \(\frac{s}{s - 1}\). Thus our task is to construct an infinite set of disks with individual areas \(1/s^k\) that we can pack into a square of area \(\frac{s}{s - 1}\). Can we find a range of \(s\) for which the series is fulfilling? As it happens, this is where Shier began his adventures; his first attempts were not with power laws but with geometric series. They didn't turn out well. You are welcome to try your own hand in Program 11.

Program 11: Disk Sizes from Geometric Series

Sorry, the program will not run in this browser.

There's a curious pattern to the failures you'll see in this program. No matter what value you assign to \(s\) (within the available range \(1 \lt s \le 2\)), the system jams when the number of disks reaches the neighborhood of \(A_\square = \frac{s}{s-1}\). For example, at \(s = 1.01\), \(\frac{s}{s - 1}\) is 101 and the program typically gets stuck somewhere between \(k = 95\) and \(k = 100\). At \(s = 1.001\), \(\frac{s}{s - 1}\) is \(1{,}001\) and there's seldom progress beyond about \(k = 1,000\).

For a clue to what's going wrong here, consider the graph at right, plotting the values of \(1 / k^s\) (red) and \(1 / s^k\) (blue) for \(s = 1.01\). These two series converge on nearly the same sum (roughly \(100\)), but they take very different trajectories in getting there. On this log-log plot, the power-law series \(1 / s^k\) is a straight line. The geometric series \(1 / s^k\) falls off much more slowly at first, but there's a knee in the curve at about \(k = 100\) (dashed mauve line), where it steepens dramatically. If only we could get beyond this turning point, it looks like the rest of the filling process would be smooth sledding, but in fact we never get there. Whereas the first \(100\) disks of the power-law series fill up only about \(5\) percent of the available area, they occuy 63 percent in the geometric case. This is where the filling process stalls.

Even in one dimension, the geometric series quickly succumbs. (This is in sharp contrast to the one-dimensional power-law model, where any \(s\) between \(1\) and \(2\) yields a provably infinite progression of disks.)

Program 12: Disk Sizes from Geometric Series in One Dimension

Sorry, the program will not run in this browser.

And just in case you think I'm pulling a fast one here, let me demonstrate that those same one-dimensional disks will indeed fit in the available space, if packed efficiently. In Program 13 they are placed in order of size from left to right.

Program 13: Deterministic One-Dimensional Geometric Series

Sorry, the program will not run in this browser.

I have made casual attempts to find fulfillment with a few other convergent series, such as the reciprocals of the Fibonacci numbers (which converge to about \(3.36\)) and the reciprocals of the factorials (whose sum is \(e \approx 2.718\)). Both series jam after the first disk. There are plenty of other convergent series one might try, but I doubt this is a fruitful line of inquiry.

All the variations discussed above leave one important factor unchanged: The objects being fitted together are all circular. Exploring the wider universe of shapes has been a major theme of Shier's work. He asks: What properties of a shape make it suitable for forming a statistical fractal pattern? And what shapes (if any) refuse to cooperate with this treatment? (The images in this section were created by John Shier and are reproduced here with his permission.)

Shier's first experiments were with circular disks and axis-parallel squares; the filling algorithm worked splendidly in both cases. He also succeeded with axis-parallel rectangles of various aspect ratios, even when he mixed vertical and horizontal orientations in the same tableau. In collaboration with Paul Bourke he tried randomizing the orientation of squares as well as their positions. Again the outcome was positive, as the illustration above left shows.

Equilateral triangles were less cooperative, and at first Shier believed the algorithm would consistently fail with this shape. The triangles tended to form orderly arrays with the sharp point of one triangle pressed close against the broad side of another, leaving little "wiggle room." Further efforts showed that the algorithm was not truly getting stuck but merely slowing down. With an appropriate choice of parameters in the Hurwitz zeta function, and with enough patience, the triangles did come together in boundlessly extendable space-filling patterns.

The casual exploration of diverse shapes eventually became a deliberate quest to stake out the limits of the space-filling process. Surely there must be some geometric forms that the algorithm would balk at, failing to pack an infinite number of objects into a finite area. Perhaps nonconvex shapes such as stars and snowflakes and flowers would expose a limitation-but no, the algorithm worked just fine with these figures, fitting smaller stars into the crevices between the points of larger stars. The next obvious test was "hollow" objects, such as annular rings, where an internal void is not part of the object and is therefore available to be filled with smaller copies. The image at right is my favorite example of this phenomenon. The bowls of the larger nines have smaller nines within them. It's nines all the way down. When we let the process continue indefinitely, we have a whimsical visual proof of the proposition that \(.999\dots = 1\).

These successes with nonconvex forms and objects with holes led to an Aha moment, as Shier describes it. The search for a shape that would break the algorithm gave way to a suspicion that no such shape would be found, and then the suspicion gradually evolved into a conviction that any "reasonably compact" object is suitable for the Fractalize That! treatment. The phrase "reasonably compact" would presumably exclude shapes that are in fact dispersed sets of points, such as Cantor dust. But Shier has shown that shapes formed of disconnected pieces, such as the words in the pair of images below, present no special difficulty.

Fractalize That! is not all geometry and number theory. Shier is eager to explain the mathematics behind these curious patterns, but he also presents the algorithm as a tool for self-expression. MATH and ART both have their place.

Finally, I offer some notes on what's needed to turn these algorithms into computer programs. Shier's book includes a chapter for do-it-yourselfers that explains his strategy and provides some crucial snippets of code (written in C). My own source code (in JavaScript) is available on GitHub. And if you'd like to play with the programs without all the surrounding verbiage, try the GitHub Pages version.

The inner loop of a typical program looks something like this:

let attempt = 1; while (attempt <= maxAttempts) { disk.x = randomCoord(); disk.y = randomCoord(); if (isNotOverlapping(disk)) { return disk; } attempt++; } return false;

We generate a pair of random \(x\) and \(y\) coordinates, which mark the center point of the new disk, and check for overlaps with other disks already in place. If no overlaps are discovered, the disk stays put and the program moves on. Otherwise the disk is discarded and we jump back to the top of the loop to try a new \(xy\) pair.

The main computational challenge lies in testing for overlaps. For any two specific disks, the test is easy enough: They overlap if the sum of their radii is greater than the distance between their centers. The problem is that the test might have to be repeated many millions of times. My program makes \(10\) million attempts to place a disk before giving up. If it has to test for overlap with \(100{,}000\) other disks on each attempt, that's a trillion tests. A trillion is too many for an interactive program where someone is staring at the screen waiting for things to happen. To speed things up a little I divide the square into a \(32 \times 32\) grid of smaller squares. The largest disks-those whose diameter is greater than the width of a grid cell-are set aside in a special list, and all new candidate disks are checked for overlap with them. Below this size threshold, each disk is allocated to the grid cell in which its center lies. A new candidate is checked against the disks in its own cell and in that cell's eight neighbors. The net result is an improvement by two orders of magnitude-lowering the worst-case total from \(10^{12}\) overlap tests to about \(10{10}\).

All of this works smoothly with circular disks. Devising overlap tests for the variety of shapes that Shier has been working with is much harder.

From a theoretical point of view, the whole rigmarole of overlap testing is hideously wasteful and unnecessary. If the box is already 90 percent full, then we know that 90 percent of the random probes will fail. A smarter strategy would be to generate random points only in the "black zone" where new disks can legally be placed. If you could do that, you would never need to generate more than one point per disk, and there'd be no need to check for overlaps. But keeping track of the points that comprise the black zone-scattered throughout multiple, oddly shaped, transient regions-would be a serious exercise in computational geometry.

For the actual drawing of the disks, Shier relies on the technology known as SVG, or scalable vector graphics. As the name suggests, these drawings retain full resolution at any size, and they are definitely the right choice if you want to create works of art. They are less suitable for the interactive programs embedded in this document, mainly because they consume too much memory. The images you see here rely on the HTML canvas element, which is simply a fixed-size pixel array.

Another point of possible interest is the evaluation of the zeta function. If we want to scale the disk sizes to match the box size (or vice versa), we need to compute a good approximation of the Riemann function \(\zeta(s)\) or the Hurwitz function \(\zeta(s, a)\). I didn't know how to do that, and most of the methods I read about seemed overwhelming. Before I could get to zeta, I'd have to hack my way through thickets of polygamma functions and Stieltjes constants. For the Riemann zeta function I found a somewhat simpler algorithm published by Peter Borwein in 1995. It's based on a polynomial approximation that yields ample precision and runs in less than a millisecond. For the Hurwitz zeta function I stayed with a straightforward translation of Shier's code, which takes more of a brute-force approach. (There are alternatives for Hurwitz too, but I couldn't understand them well enough to make them work.)

The JavaScript file in the GitHub repository has more discussion of implementation details.

Bibliography: Works by John Shier and colleagues

Shier, John. 2018. Fractalize That! A Visual Essay on Statistical Geometry. Singapore: World Scientific. Publisher's website.

Shier, John. Website: http://www.john-art.com/

Shier, John. 2011. The dimensionless gasket width \(b(c,n)\) in statistical geometry. http://www.john-art.com/gasket_width.pdf

Shier, John. 2012. Random fractal filling of a line segment. http://www.john-art.com/gasket_width.pdf

Dunham, Douglas, and John Shier. 2014. The art of random fractals. In Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture pp. 79-86. PDF.

Shier, John. 2015. A new recursion for space-filling geometric fractals. http://www.john-art.com/gasket_width.pdf

Dunham, Douglas, and John Shier. 2015. An algorithm for creating aesthetic random fractal patterns. Talk delivered at the Joint Mathematics Meetings January 2015, San Antonio, Texas.

Dunham, Douglas, and John Shier. 2018. A property of area and perimeter. In ICGG 2018: Proceedings of the 18th International Conference on Geometry and Graphics, Milano, August 2018, pp. 228-237.

Dunham, Douglas, and John Shier. 2017. New kinds of fractal patterns. In Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture,
pp. 111-116. Preprint.

Shier, John, and Paul Bourke. 2013. An algorithm for random fractal filling of space. Computer Graphics Forum 32(8):89-97. PDF. Preprint.

Proof

Ennis, Christopher. 2016. (Always) room for one more. Math Horizons 23(3):8-12. PDF (paywalled).

Apollonian circles

Dodds, Peter Sheridan, and Joshua S. Weitz. 2002. Packing-limited growth. Physical Review E 65: 056108.

Lagarias, Jeffrey C., Colin L. Mallows, and Allan R. Wilks. 2001. Beyond the Descartes circle theorem. https://arxiv.org/abs/math/0101066. (Also published in American Mathematical Monthly, 2002, 109:338-361.)

Mackenzie, Dana. 2010. A tisket, a tasket, an Apollonian gasket. American Scientist 98:10-14. https://www.americanscientist.org/article/a-tisket-a-tasket-an-apollonian-gasket.

Manna, S. S. 1992. Space filling tiling by random packing of discs. Physica A 187:373-377.

Zeta functions

Bailey, David H., and Jonathan M. Borwein. 2015. Crandall's computation of the incomplete Gamma function and the Hurwitz zeta function, with applications to Dirichlet L-series. Applied Mathematics and Computation, 268, 462-477.

Borwein, Peter. 1995. An efficient algorithm for the Riemann zeta function. http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf

Coffey, Mark W. 2009. An efficient algorithm for the Hurwitz zeta and related functions. Journal of Computational and Applied Mathematics 225:338-346.

Hurwitz, Adolf. 1882. Einige Eigenschaften der Dirichletschen Funktionen \(F(s) = \sum \left(\frac{D}{n} \frac{1}{n^s}\right)\), die bei der Bestimmung der Klassenzahlen binärer quadratischer Formen auftreten. Zeitschrift für Mathematik und Physik 27:86-101. https://gdz.sub.uni-goettingen.de/id/PPN599415665_0027.

Oswald, Nicola, and Jörn Steuding. 2015. Aspects of zeta-function theory in the mathematical works of Adolf Hurwitz. https://arxiv.org/abs/1506.00856.

Xu, Andy. 2018. Approximating the Hurwitz zeta function. PDF.

# data[].attributes.attachment_link.description
Please click or tap in the gray square below. It will fill with a jolly tableau of colored disks-first big blue ones, then somewhat smaller purple ones, and eventually lots of tiny red dots. Sorry. My program and your browser are not getting along. None of the interactive elements of this page will work.

# data[].attributes.body
Blog

# data[].attributes.attachment_link.readable_text
Breaking away from the traditional-style, gradual release classroom was the best move I ever made for my teaching craft and the mathematicians in my classroom. The gradual release model, I do-We do-You do, has its place in the classroom, and it's a valid way to deliver your lesson and content to young mathematicians. I definitely have a handful of these lessons throughout the school year. Teaching long division comes to mind. Making a change to a higher level of active learning can open the door toward deeper learning and understanding.

One way to promote active learning in your classroom is to break away from the gradual release model when possible. A majority of the time, I use a rapid release of responsibility model. I'm currently teaching the Open Up Resources 6-8 Math curriculum authored by Illustrative Mathematics , so the lessons I deliver to my students are intentionally designed by experts of the Standards around the principle of a rapid release of responsibility to these young mathematicians. So, instead of modeling, practicing together, and independent work time, we flip the approach to You do, We do, I do.

Regardless of how you structure lessons, it's important for all students to be able to access the math content. Using Low Floor High Ceiling tasks lend a hand in opening up the math to all learners and can also be extended. So these tasks reach quite a range of learners that can be found in a heterogeneous classroom. I mean, check out this range of responses! The Launch.

Routines like Notice and Wonder, Which One Doesn't Belong?, and Think Pair Share, along with presenting open-ended tasks, allow students to share their ideas about visuals or extend to more abstract ideas. All students can engage. All students can access the math. Students are playing an active role in their learning and understanding. This builds students' confidence and sparks thinking before diving deeply into the Standards-based objectives for the day.

You do. Nearly every math activity begins with my students jumping into a task independently. They are given quiet time to think or work through a carefully-selected task. Some students are able to really attack the problem and justify their thinking in the process. But more often than not, learners engage in productive struggle. As a teacher, it's a challenge to watch my students grapple with a task I've presented them without trying to dive in and save them with a lifeguard rescue floating device, or my pencil. A deeper understanding of math and engagement comes with challenges; this process of learning while wrestling with difficult content is important and should be valued.

We do. After the students engage in the math independently, they get a chance to collaborate and engage with their peers with the grade-level content. The "We do" part of the lesson lends to a deeper understanding for a couple of reasons. First, the students get a chance to articulate their thinking to peers. Learners also get the opportunity to listen to the thinking of their peers. I typically set a goal for this conversation to guide the discussion in the right direction. Maybe the students will focus on certain parts of the task that the just completed, or other times I pose a question that will build toward a generalization or a deeper learning. I might say something like, "Focus your team time around the diagram and how it connects to the scenario." Allowing learners to talk, hear, and reflect about the content leads better understanding. Making a habit to revise and refine thinking along the way is an added bonus.

Another "We do" routine is using vertical non-permanent surfaces (#VNPS). I have big whiteboards hanging around my classroom made from giant sheets of shower board material from a home store. We sometimes use these during team time, where one teammate gets a marker and records the thinking of their peers. When teams hit a roadblock, they can glance around the room to find a strategy to get started. When teams finish early, they typically check around the room that the work is on the right track. During sharing time, there's no easier way to compare and connect students' ideas! It's all up and displayed for us to dissect and find relationships between concrete or abstract ideas.

"I do. " After students have engaged in productive struggle independently and refined their thinking with peers, it's time to solidify that thinking. I'm not sure it's accurate to even call the next step "I do" in my classroom. Let's just call it Synthesis! Enough about semantics. This is the point in the activity where we discuss, share, and synthesize the learning as a whole class. Sometimes I'll select and sequence the strategies and work of students to share and make connections. As opposed to the traditionally-taught math lesson, this is where the Big Picture comes into play. We notice and wonder. We connect. We find efficiencies. We generalize. And we check if we've made our learning goals.

Routines at the Core. Effective instructional routines are practiced and learned throughout the school year, so students' concentration can be on the math at hand. The routines should be purposeful and intentional to move learning forward. With a student-centered approach, learners can engage in active learning and collaboration. Number Talks, Notice and Wonder, Poll the Class, Card Sorts, and Which One Doesn't Belong?, are great examples of instructional routines that promote active engagement and learning in math class.

Mathematical Language Routines, spelled out , provide opportunities for all students to read, speak, listen, and write during math class. Students communicate in their learning to build toward understanding and mastery of grade-level content. The learners really begin to own their learning and understanding of the content. They're given multiple passes at accessing and articulating their understanding of the math content, while also developing their language and soft skills as they communicate their ideas with their peers. As the teacher, I record ideas and discussions on my whiteboard. I think-aloud and help my students make connections between the strategies and ideas shared.

The kinds of opportunities presented to students can promote active learning. All students should be given the chance to access the math at hand. Find the strength to allow your students to experience productive struggle and collaborate with their peers during challenging tasks. Then engage them in purposeful sharing and connecting. One of my students recently articulated this well. He said, "In Math class, I have to really push my brain. Like, it's hard, but I know I can do it." Active learning wins every time!

#msmathchat #openupmath #learnwithim #activelearning #mtbos #coreadvocates #coreadvocates

Previous Blog: "Our Community Wasn't Built in a Day..."

# data[].attributes.attachment_link.description
Breaking away from the traditional-style, gradual release classroom was the best move I ever made for my teaching craft and the mathematicians in my classroom. The gradual release model, I do-We do-You do, has its place in the classroom, and it's a valid way to deliver your lesson and content to young mathematicians.

# data[].attributes.body
Short read

# data[].attributes.attachment_link.readable_text
This article was adapted from Takahashi, A., & McDougal, T. (2016). Collaborative lesson research: maximizing the impact of lesson study. ZDM, 1-14. doi:10.1007/s11858-015-0752-x, which may be accessed for free by clicking on the link.
"Lesson Study" is a translation of the Japanese phrase jugyou kenkyuu, which refers to a set of practices that have been used in Japan to improve teaching and learning for over 100 years. Lesson Study is credited with enabling profound changes in math and science instruction in Japan in recent decades, but Japanese teachers use Lesson Study to hone instruction in all content areas, including p.e. and foreign language.

Many educators outside of Japan have tried to use Lesson Study in the hope of effecting similar changes in their locales. Lacking first-hand experience with Japanese Lesson Study, and lacking some of the structural supports enjoyed by Japanese teachers, they have implemented lesson study in ways that sometimes varied significantly from how it is practiced in Japan. It should not be surprising that the impact of lesson study on teaching and learning in these circumstances has been uneven.

Dr. Akihiko Takahashi has 19 years of personal experience with Lesson Study as a teacher in Japan. Tom McDougal, who has been working with Dr. Takahashi since 2008, has traveled twice to Japan to observe Lesson Study there first-hand. But we, too, have had a lot to learn about how to effectively translate the practice into the U.S. context. This article summarizes what we have learned.

Critical aspects of effective Lesson Study

Based on our experience with jugyou kenkyuu in Japan and our experience with various adaptations of it in the U.S. and other countries, we believe that certain aspects of Japanese jugyou kenkyuu are critical for improving teaching and learning. We use the phrase Collaborative Lesson Research (CLR) to refer to Lesson Study that includes those critical aspects.

The characteristic components of Collaborative Lesson Research are:

A clear research purpose

Kyouzai kenkyuu (see below)

A written research proposal

A live research lesson and discussion

Contributions by knowledgeable others

A process for sharing results.

A clear research purpose: What problem are you trying to solve?

Sometimes teachers want to use a favorite lesson for a research lesson - one that will favorably impress the observers. Sometimes teachers will conduct multiple iterations of a research lesson, in order to perfect it. Both of these approaches miss out on the purpose of Lesson Study, which is to conduct research: to seek a solution to a teaching/learning problem.

CLR usually has two research objectives. One objective relates to the teaching of specific content: How can we design a lesson so that students learn such-and-such concept or skill better than they have in the past? Thus the topic of the research lesson should usually be one which students or teachers have previously struggled with. The second objective involves a broad teaching/learning goal that is shared by the CLR community (e.g. all teachers of mathematics at the school), and that goes beyond any particular topic or grade level and may even be cross-disciplinary. This second research objective is referred to as the research theme.

Ideally, a research theme describes (a) a desired outcome for students, and (b) an entry point for achieving that outcome. At Prieto Math and Science Academy, one of our partner schools in Chicago, teachers seek to improve their students' ability to give a viable argument and to critique the reasoning of others; their entry point is teaching students to use journals to record their own ideas and the ideas of others, and using the whiteboard strategically to support student journal writing and classroom discussion. Almost every research lesson at Prieto since 2013 has sought to address this theme.

Kyouzai kenkyuu

Kyouzai kenkyuu translates roughly as "the study (research) of materials for teaching." It is a routine part of lesson planning for Japanese teachers, but in Lesson Study it is done with an extra intensity, analogous to a literature review in scientific research. Kyouzai kenkyuu involves an investigation of the intended learning trajectory related to the topic, from prior to later grades, through a review of the standards and curriculum, and research into teaching and learning issues such as typical misunderstandings around the topic. Kyouzai kenkyuu also includes consideration of possible tools, manipulatives, or materials that may be used, and possible tasks that may be presented to students. Thorough helps avoid "reinventing the wheel", making it more likely that CLR will contribute new knowledge to the education community. This is also where teachers frequently deepen or strengthen their content knowlege.

One obstacle for teachers in the U.S. is the quality of available materials to support kyouzai kenkyuu, especially compared to the materials available to Japanese teachers. This problem has been especially acute with the early implementation of the Common Core, as no curricula were immediately available that aligned well with the new standards. Many of our partner schools have relied on a translated version of a popular Japanese textbook series, since the CCSSM were largely inspired by the Japanese curricula, and since the Japanese textbooks support teaching through problem solving.

A written research proposal

A CLR planning team - usually 3 to 6 teachers - creates a written document, called the lesson research proposal, to communicate what the team learned from their kyouzai kenkyuu, and to explain their instructional thinking. In addition to a detailed teaching/learning plan for one particular lesson (the research lesson), the research proposal includes an overview of the unit in which the research lesson will occur, the rationales for the design of the unit and research lesson, and a clear statement of how the research lesson aims to address the research theme and the learning goals of the lesson. In our experience, a thorough lesson research proposal may be 9 pages long! We developed (and continue to refine) a template document to guide CLR teams in organizing their Lesson Study work and in writing their proposal.

A live research lesson and post-lesson discussion

Based on the lesson plan in the research lesson proposal, one member of the team teaches the research lesson, observed by the entire planning team and by additional members of the CLR community. Observers are responsible for collecting data on how the lesson impacts the students, relative to the research theme and the learning goals. A video recording of the lesson can be useful for some purposes, but CLR requires observations from multiple viewpoints, so video cannot substitute for live observation.

(Lesson Study Alliance developed an iPad application called LessonNote to help observers collect valuable data during a research lesson. The app is available for free.)

As soon as practical after the research lesson, observers share data and discuss implications, especially with respect to the learning goals of the lesson and the research theme. The primary goal of the discussion is to gain insights into teaching and learning and to inform the design of future lessons, not to revise the lesson plan. These discussions generally benefit from a moderator, someone not on the planning team, who helps focus the discussion on important issues and keeps the conversation grounded in data.

Some groups refer to this activity as a "debriefing", but we avoid that word because, in its strict sense, "debriefing" means the extracting of information from someone, such as a spy, rather than a collaborative discussion about a shared experience. Thus we prefer the phrase "post-lesson discussion." Our guide for facilitating a post-lesson discussion can be found here.

Contributions by knowledgeable others

A knowledgeable other is someone from outside of the planning team with deep expertise in the content, often deep expertise in teaching, and much experience with Lesson Study. A knowledgeable other helps the CLR team go beyond what they know. Ideally a CLR community needs two knowledgeable others: one for supporting development of the research proposal and another for providing the final comments at the end of the post-lesson discussion.

During planning, a knowledgeable other may help the team identify instructional examples to review, valuable resources in the form of articles or results from other CLR work, and may give feedback on the proposal. Besides having extensive knowledge of the subject matter and the topic, this knowledgeable other should be familiar with the school's curriculum and students. An experienced teacher or a content coach who often works at the school may play the role of this kind of knowledgeable other.

Dr. Tad Watanabe gives final comments on a lesson.

Another knowledgeable other is needed at the research lesson. In final comments at the end of the post-lesson discussion, he or she is expected to share observations from and analysis of the lesson, highlight important events from the research lesson that were not discussed, and make connections between the lesson and new knowledge from research and standards. The knowledgeable other also provides suggestions to the CLR community of possible steps they could take toward accomplishing their research theme. We often refer to this person as the "final commentator."

In addition to Dr. Takahashi and Tom McDougal, a few Chicago teachers with several years of Lesson Study experience have served as discussion moderators and knowledgeable others. Please contact us if you want our support.

A process for sharing results

Lesson Study can be powerful not only for the team of teachers who planned the research lesson but for others as well. Thus CLR requires a structure or process for disseminating what is learned from each research lesson to a larger community.

A simple way to enable others to learn from the research is to invite people from outside of the planning team to observe and discuss the research lesson; meanwhile, the team can benefit from the additional eyes and expertise that the additional observers bring. All of our partner schools find ways to enable teachers to observe each others' research lessons and to observe research lessons at other schools.

In addition, the team may distribute their research lesson proposal, which encapsulates the team's research and their instructional ideas, and can be useful to other educators. This document is made more powerful through the addition of a written reflection by the team, completed within a few days after the research lesson, which describes what they learned from the live lesson observation and post-lesson discussion about their research hypothesis, mathematics, student thinking, teaching, etc. In Japan, these documents are sometimes published in collections of research lessons organized by grade level or topic, and can be found in major bookstores!

Conclusion

Collaborative Lesson Research is an intense process; one CLR cycle typically spans 6 to 10 weeks. Some U.S. educators, accustomed to thinking about professional development in terms of 2-hour workshops, will object: "How can you spend so much time on just one lesson?" But CLR is not about just one lesson; a good research theme ensures that what is learned from CLR applies to many lessons, perhaps to all lessons. Teachers who have gone through CLR with us often say that the process leads them to think differently in planning their everyday lessons, and we have seen significant changes in teacher practice resulting from CLR.

At the same time, we acknowledge that multiple CLR cycles are almost always required, because most worthwhile research themes are difficult to accomplish. Lessons learned from one CLR cycle lead to revised theories about how to address the theme, or adjustments in the theme itself, which lead to another CLR cycle involving a different topic perhaps at a different grade level.

We chose the six defining characteristics of Collaborative Lesson Research listed above based on findings from research on Lesson Study outside of Japan and jugyou kenkyuu in Japan, and our own experience working with schools. As defining characteristics, we consider them required elements of CLR: if any of them is missing, then the activity cannot be called CLR. And, we believe, all six elements are critical to ensuring that Lesson Study is as effective as it should be.

As Knowledgeable Other, Dr. Takahashi gives final comments on a kindergarten lesson at the Chicago Lesson Study Conference at Prieto in 2013.

# data[].attributes.attachment_link.description
This article was adapted from Takahashi, A., & McDougal, T. (2016). Collaborative lesson research: maximizing the impact of lesson study. ZDM, 1-14. doi:10.1007/s11858-015-0752-x, which may be accessed for free by clicking on the link.

# data[].attributes.body
Framework

# data[].attributes.attachment_link.readable_text
Explore the main sections below, click on the interactive framework components, or download the full PISA 2021 Mathematics Framework Draft in PDF format.

The PISA 2021 mathematics framework defines the theoretical underpinnings of the PISA mathematics assessment based on the fundamental concept of mathematical literacy, relating mathematical reasoning and three processes of the problem-solving (mathematical modelling) cycle. The framework describes how mathematical content knowledge is organised into four content categories. It also describes four categories of contexts in which students will face mathematical challenges.

The PISA assessment measures how effectively countries are preparing students to use mathematics in every aspect of their personal, civic and professional lives, as part of their constructive, engaged and reflective 21st Century citizenship.

What is Mathematical Literacy?

Mathematical literacy is an individual's capacity to reason mathematically and to formulate, employ and interpret mathematics to solve problems in a variety of real-world contexts. It includes concepts, procedures, facts and tools to describe, explain and predict phenomena. It helps individuals know the role that mathematics plays in the world and make the well-founded judgments and decisions needed by constructive, engaged and reflective 21st Century citizens.

What's new in PISA 2021

PISA 2021 aims to consider mathematics in a rapidly changing world driven by new technologies and trends in which citizens are creative and engaged, making non-routine judgments for themselves and the society in which they live. This brings into focus the ability to reason mathematically, which has always been a part of the PISA framework. This technology change is also creating the need for students to understand those computational thinking concepts that are part of mathematical literacy. Finally, the framework recognises that improved computer-based assessment is available to most students within PISA.

The ability to reason logically and present arguments in honest and convincing ways is a skill that is becoming increasingly important in today's world. Mathematics is a science about well-defined objects and notions that can be analysed and transformed in different ways using "mathematical reasoning" to obtain certain and timeless conclusions.

In mathematics, students learn that, with proper reasoning and assumptions, they can arrive at results that they can fully trust to be true in a wide variety of real-life contexts. It is also important that these conclusions are impartial, without any need for validation by an external authority.

At least six key understandings provide structure and support to mathematical reasoning. These key understandings include

understanding quantity, number systems and their algebraic properties;

appreciating the power of abstraction and symbolic representation;

seeing mathematical structures and their regularities;

recognising functional relationships between quantities;

using mathematical modelling as a lens onto the real world (e.g. those arising in the physical, biological, social, economic and behavioural sciences); and

understanding variation as the heart of statistics.

Use the arrows below to review the key understandings in-depth:

QUANTITY, NUMBER SYSTEMS AND THEIR ALGEBRAIC PROPERTIES

MATHEMATICS AS A SYSTEM BASED ON ABSTRACTION AND SYMBOLIC REPRESENTATION

MATHEMATICAL STRUCTURES AND THEIR REGULARITIES

FUNCTIONAL RELATIONSHIPS BETWEEN QUANTITIES

MATHEMATICAL MODELLING AS A LENS ONTO THE REAL WORLD

VARIATION AT THE HEART OF STATISTICS

The word formulate in the mathematical literacy definition refers to the ability of individuals to recognise and identify opportunities to use mathematics and then provide mathematical structure to a problem presented in some contextualised form. In the process of formulating situations mathematically, individuals determine where they can extract the essential mathematics to analyse, set up and solve the problem. They translate from a real-world setting to the domain of mathematics and provide the realworld problem with mathematical structure, representations and specificity. They reason about and make sense of constraints and assumptions in the problem. Specifically, this process of formulating situations mathematically includes activities such as the following:

selecting an appropriate model from a list; **

identifying the mathematical aspects of a problem situated in a real-life context and identifying the significant variables;

recognising mathematical structure (including regularities, relationships and patterns) in problems or situations;

simplifying a situation or problem in order to make it amenable to mathematical analysis;

identifying constraints and assumptions behind any mathematical modelling and simplifications gleaned from the context;

representing a situation mathematically, using appropriate variables, symbols, diagrams and standard models;

representing a problem in a different way, including organising it according to mathematical concepts and making appropriate assumptions;

understanding and explaining the relationships between the context-specific language of a problem and the symbolic and formal language needed to represent it mathematically;

translating a problem into mathematical language or a representation;

recognising aspects of a problem that correspond with known problems or mathematical concepts, facts or procedures;

using technology (such as a spreadsheet or the list facility on a graphing calculator) to portray a mathematical relationship inherent in a contextualised problem; and

creating an ordered series of (step-by-step) instructions for solving problems.

** This activity is included in the list to foreground the need for the test-item developers to include items that are accessible to students at the lower end of the performance scale.

The word employ in the mathematical literacy definition refers to the ability of individuals to apply mathematical concepts, facts, procedures and reasoning to solve mathematically formulated problems to obtain mathematical conclusions. In the process of employing mathematical concepts, facts, procedures and reasoning to solve problems, individuals perform the mathematical procedures needed to derive results and find a mathematical solution. They work on a model of the problem situation, establish regularities, identify connections between mathematical entities and create mathematical arguments. Specifically, this process of employing mathematical concepts, facts, procedures and reasoning includes activities such as

performing a simple calculation; **

drawing a simple conclusion; **

selecting an appropriate strategy from a list; **

devising and implementing strategies for finding mathematical solutions;

using mathematical tools, including technology, to help find exact or approximate solutions;

applying mathematical facts, rules, algorithms and structures when finding solutions;

manipulating numbers, graphical and statistical data and information, algebraic expressions and equations, and geometric representations;

making mathematical diagrams, graphs and constructions and extracting mathematical information from them;

using and switching between different representations in the process of finding solutions;

making generalisations based on the results of applying mathematical procedures to find solutions;

reflecting on mathematical arguments, and explaining and justifying mathematical results; and

evaluating the significance of observed (or proposed) patterns and regularities in data.

** These activities are included in the list to foreground the need for the test-item developers to include items that are accessible to students at the lower end of the performance scale.

The word interpret (and evaluate) used in the mathematical literacy definition focuses on the ability of individuals to reflect upon mathematical solutions, results or conclusions and interpret them in the context of the real-life problem that initiated the process. This involves translating mathematical solutions or reasoning back into the context of the problem and determining whether the results are reasonable and make sense in the context of the problem.

Specifically, this process of interpreting, applying and evaluating mathematical outcomes includes activities such as the following:

interpreting information presented in graphical form and/or diagrams; **

evaluating a mathematical outcome in terms of the context; **

interpreting a mathematical result back into the real-world context;

evaluating the reasonableness of a mathematical solution in the context of a real-world problem;

understanding how the real world impacts the outcomes and calculations of a mathematical procedure or model in order to make contextual judgments about how the results should be adjusted or applied;

explaining why a mathematical result or conclusion does or does not make sense given the context of a problem;

understanding the extent and limits of mathematical concepts and mathematical solutions;

critiquing and identifying the limits of the model used to solve a problem; and

using mathematical thinking and computational thinking to make predictions, to provide evidence for arguments, and to test and compare proposed solutions.

** This activity is included in the list to foreground the need for the test-item developers to include items that are accessible to students at the lower end of the performance scale.

An understanding of mathematical content - and the ability to apply that knowledge to solving meaningful contextualised problems - is important for citizens in the modern world. That is, to reason mathematically and to solve problems and interpret situations in personal, occupational, societal and scientific contexts, individuals need to draw upon certain mathematical knowledge and understanding.

The following content categories used in PISA since 2012 are again used in PISA 2021 to reflect the mathematical phenomena that underlie broad classes of problems, the general structure of mathematics and the major strands of typical school curricula:

Four topics have been identified for special emphasis in the PISA 2021 assessment. These topics are not new to the mathematics content categories. Instead, these are topics that deserve special emphasis:

The notion of quantity may be the most pervasive and essential mathematical aspect of engaging with and functioning in our world. It incorporates the quantification of attributes of objects, relationships, situations and entities in the world; understanding various representations of those quantifications; and judging interpretations and arguments based on quantity. To engage with the quantification of the world involves understanding measurements, counts, magnitudes, units, indicators, relative size, and numerical trends and patterns.

Quantification is a primary method for describing and measuring a vast set of attributes of aspects of the world. It allows for the modelling of situations, for the examination of change and relationships, for the description and manipulation of space and shape, for organising and interpreting data, and for the measurement and assessment of uncertainty.

Computer simulations

Both mathematics and statistics involve problems that are not so easily addressed because the required mathematics is complex or involves a large number of factors all operating in the same system. Increasingly in today's world, such problems are being approached using computer simulations driven by algorithmic mathematics.

Identifying computer simulations as a focal point of the quantity content category signals that, in the context of the computer-based assessment of mathematics, there is a broad category of complex problems. For example, students can use computer simulations to analyse budgeting and planning as part of the test item.

In science, technology and everyday life, uncertainty is a given. Uncertainty is therefore a phenomenon at the heart of the mathematical analysis of many problem situations, and the theory of probability and statistics as well as techniques of data representation and description have been established to deal with it. The uncertainty and data content category includes recognising the place of variation in processes, having a sense of the quantification of that variation, acknowledging uncertainty and error in measurement, and knowing about chance. It also includes forming, interpreting and evaluating conclusions drawn in situations where uncertainty is central. Quantification is a primary method for describing and measuring a vast set of attributes of aspects of the world.

Conditional decision-making

Identifying conditional decision-making as a focal point of the uncertainty and data content category signals that students should be expected to appreciate how the assumptions made in setting up a model affect the conclusions that can be drawn and that different assumptions/relationships may well result in a different conclusion.

The natural and designed worlds display a multitude of temporary and permanent relationships among objects and circumstances, where changes occur within systems of interrelated objects or in circumstances where the elements influence one another. In many cases, these changes occur over time. In other cases, changes in one object or quantity are related to changes in another. Some of these situations involve discrete change; others involve continuous change. Some relationships are of a permanent, or invariant, nature. Being more literate about change and relationships involves understanding fundamental types of change and recognising when they occur in order to use suitable mathematical models to describe and predict change. Mathematically, this means modelling the change and the relationships with appropriate functions and equations, as well as creating, interpreting and translating among symbolic and graphical representations of relationships.

Growth phenomena

Understanding the dangers of flu pandemics and bacterial outbreaks, as well as the threat of climate change, demands that people not only think in terms of linear relationships but recognise that such phenomena need non-linear models reflecting a very rapid growth. Linear relationships are common and easy to recognise and understand, but to assume linearity can sometimes be dangerous.

Identifying growth phenomena as a focal point of the change and relationships content category does not signal an expectation that participating students should have studied the exponential function, and certainly the items will not require knowledge of the exponential function. Instead, the expectation is that there will be items that expect students to recognise (a) that not all growth is linear and (b) that nonlinear growth has profound implications on how we understand certain situations.

Space and shape encompass a wide range of phenomena that are encountered everywhere in our visual and physical world: patterns, properties of objects, positions and orientations, representations of objects, decoding and encoding of visual information, and navigation and dynamic interaction with real shapes as well as with representations. Geometry serves as an essential foundation for space and shape, but the category extends beyond traditional geometry in content, meaning and method, drawing on elements of other mathematical areas such as spatial visualisation, measurement and algebra.

Geometric approximation

Today's world is full of shapes that do not follow typical patterns of evenness or symmetry. Because simple formulas do not deal with irregularity, it has become more difficult to understand what we see and to find the area or volume of the resulting structures.

Identifying geometric approximations as a focal point of the space and shape content category signals the need for students to be able use their understanding of traditional space and shape phenomena in a range of a typical situations.

An important aspect of mathematical literacy is that mathematics is used to solve a problem set in a context. The context is the aspect of an individual's world in which the problems are placed. The choice of appropriate mathematical strategies and representations is often dependent on the context in which a problem arises. For PISA, it is important that a wide variety of contexts are used.

Personal

Problems classified in the personal context category focus on activities of one's self, one's family or one's peer group. Personal contexts include (but are not limited to) those involving food preparation, shopping, games, personal health, personal transportation, sports, travel, personal scheduling and personal finance.

Occupational

Problems classified in the occupational context category are centred on the world of work. Items categorised as occupational may involve (but are not limited to) such things as measuring, costing and ordering materials for building, payroll/accounting, quality control, scheduling/inventory, design/architecture, and job-related decision-making. Occupational contexts may relate to any level of the workforce, from unskilled work to the highest levels of professional work, although items in the PISA survey must be accessible to 15-year-old students.

Societal

Problems classified in the societal context category focus on one's community (whether local, national or global). They may involve (but are not limited to) such things as voting systems, public transport, government, public policies, demographics, advertising, national statistics and economics. Although individuals are involved in all of these things in a personal way, in the societal context category, the focus of problems is on the community perspective.

Scientific

Problems classified in the scientific category relate to the application of mathematics to the natural world and issues and topics related to science and technology. Particular contexts might include (but are not limited to) such areas as weather or climate, ecology, medicine, space science, genetics, measurement and the world of mathematics itself. Items that are intra-mathematical, where all the elements involved belong in the world of mathematics, fall within the scientific context.

There is increased interest worldwide in what are called 21st Century skills and their possible inclusion in educational systems. The OECD has put out a publication that focuses on such skills and has sponsored a research project entitled The Future of Education and Skills: Education 2030. Some 25 countries are involved in this cross-national study of curriculum including the incorporation of such skills. The project has as its central focus what the curriculum might look like in the future, focusing initially on mathematics. Some of the key 21st Century skills are:

critical thinking;

creativity;

research and inquiry;

self-direction, initiative and persistence;

information use;

systems thinking;

communication; and

reflection.

Although test-item developers recognise these 21st Century skills, the mathematics items in PISA 2021 are not specifically developed according to these skills.

Below are some example exercises from the PISA 2021 Mathematics assessment. Each button below opens an overlay that shows an example experience from the application.

# data[].attributes.attachment_link.description
Explore the main sections below, click on the interactive framework components, or download the full PISA 2021 Mathematics Framework Draft in PDF format. The PISA 2021 mathematics framework defines the theoretical underpinnings of the PISA mathematics assessment based on the fundamental concept of mathematical literacy, relating mathematical reasoning and three processes of the problem-solving (mathematical modelling) cycle.

# data[].attributes.body
Article

# data[].attributes.attachment_link.description
Japanese Temple Read more about sangaku, geometry, tablets, mathematics, temple and fukagawa.

# data[].attributes.body
Dynamic geometry

# data[].attributes.attachment_link.readable_text
De petits sangaku, niveau collège Normalement pas trop durs, enfin... il vaut mieux prendre un peu l'habitude des constructions et des raisonnements, très spécifiques. Certains sont de moi, inspirés des sangaku classiques. La plupart sont extraits de ce bouquin indispensable: Sacred Mathematics Japanese Temple Geometry Fukagawa Hidetoshi & Tony Rothman Princeton University Press, 2008

# data[].attributes.attachment_link.description
De petits sangaku, niveau collège Normalement pas trop durs, enfin... il vaut mieux prendre un peu l'habitude des constructions et des raisonnements, très spécifiques. Certains sont de moi, inspirés des sangaku classiques.

# data[].attributes.body
GeoGebra file

# data[].attributes.body
Article

# data[].attributes.body
Video and article

# data[].attributes.attachment_link.readable_text
Are you the driver in the lot who parks in the first spot you see? Or do you circle around and around looking for a spot by the door? Physicists Paul Krapivsky of Boston University and Sidney Redner of the Santa Fe Institute explored the mathematics of parking. The research required different equations and simulations to model the benefits of the various parking approaches. From EurkeAlert!:

In their paper, Krapivsky and Redner map three simple parking strategies onto an idealized, single row parking lot. Drivers who grab the first space available follow what the authors call a "meek" strategy. They "waste no time looking for a parking spot," leaving spots near the entrance unfilled. Those who gamble on finding a space right next to the entrance are "optimistic." They drive all the way to the entrance, then backtrack to the closest vacancy. "Prudent" drivers take the middle path. They drive past the first available space, betting on the availability of at least one other space further in. When they find the closest space between cars, they take it. If no spaces exist between the furthest parked car and the entrance, prudent drivers backtrack to the space a meek driver would have claimed straightaway.

So which strategy is best? As the name suggests, the prudent strategy. Overall, it costs drivers the least amount of time, followed closely by the optimistic strategy. The meek strategy was "risibly inefficient," to quote the paper, as the many spaces it left empty created a lengthy walk to the entrance.

This DIY hovercraft styled like the Doc Brown's DeLorean in the Back to the Future films is up for auction on Bring A Trailer. Current bid is $22,500, half of what maker Matt Riese was asking when I posted last year that the DeLorean was listed on eBay. Apparently he's done quite a bit of [...]

# data[].attributes.attachment_link.title
Video: the mathematics of where to park your car

# data[].attributes.attachment_link.description
https://www.youtube.com/watch?v=6W7UrAKcvQw Are you the driver in the lot who parks in the first spot you see? Or do you circle around and around looking for a spot by the door? Physicists Paul Krapivsky of Boston University and Sidney Redner of the Santa Fe Institute explored the mathematics of parking.

# data[].attributes.body
News article

# data[].attributes.attachment_link.title
Opinion | The Rich Really Do Pay Lower Taxes Than You (Published 2019)

# data[].attributes.attachment_link.description
The 400 wealthiest Americans now pay a lower rate than the middle class.

# data[].attributes.body
Blog

# data[].attributes.attachment_link.readable_text
At a time when decolonisation, part of which involves changing the content of what's taught, is dominating debate at many universities, the discipline of mathematics presents an interesting case.

But it's not obvious how mathematics can be decolonised at the level of content. This means that those within the discipline must consider other aspects: curriculum processes, such as critical thinking and problem solving; pedagogy - how the subject is taught and, as a number of people have argued, addressing the issue of identity.

Students' mathematical identities - how they see themselves as learners of mathematics and the extent to which mathematics is meaningful to them - are important when thinking about teaching and learning in mathematics.

In his book Leading for change, South African educationist Jonathan Jansen suggests that transforming university campuses into deracialised spaces requires attention to both the academic and the human project. I take the human project to mean how students see themselves. What might this mean for mathematics?

So what is mathematics?

For starters, it's important to explore what mathematics actually is.

Mathematician and academic Jo Boaler points out that mathematics is the only subject where students and mathematicians give very different answers to this question.

Mathematicians view the subject as an exciting, creative endeavour in which problem solving, curiosity, excitement, intuition and perseverance play important roles - albeit in relation to abstract objects of study.

For school and even undergraduate mathematics students, these aspects of mathematics are often not experienced and remain opaque. Students tend to believe that mathematics is a set of procedures to be followed. They think only particularly gifted people can do and understand these procedures. This suggests that the way mathematics is usually taught doesn't provide opportunities for accessing mathematical knowledge. It doesn't allow students to identify with mathematics, nor make them aspire to become mathematicians.

As a result, mathematics has a problem with diversity. All over the world, black and women mathematicians remain rare. They simply don't take mathematics at higher academic levels as much as their white and male peers.

One reason for this is given by a study in the US, which showed that the more a field attributes success to giftedness rather than effort, the fewer female and black academics are in that field. This is because the field perpetuates stereotypes about who belongs in the field. The same study found that mathematics professors hold the most fixed ideas about giftedness.

But this view of giftedness versus effort is not borne out by research. A number of scholars have argued that all people are capable of learning mathematics, to high levels.

This suggests that a lot of the "bad press" around mathematics as a subject and discipline lies with how it is taught and learned.

What is learning?

When scholars theorise learning, the thinking always happens in two directions: to the past, and to the future.

Some see learning as building on current knowledge in a step-wise linear way. Some see it as working in a spiral -- coming back to old ideas in new ways. Still others view learning as disrupting or transforming current knowledge.

For teachers, working with current knowledge means finding ways to ascertain, predict, anticipate and think about students' ideas - and finding ways to engage with these. An important part of students' ideas about mathematics is how they see themselves in relation to mathematics. Research in schools has shown that one of the key factors in students' mathematics achievement is a teacher who believes that they can do mathematics.

The future is important because universities must produce future thinkers, leaders, professionals and citizens. These institutions are the bridge between the past and the future.

Educational theorist Etienne Wenger argues that learning is fundamentally about becoming a certain kind of person. At universities, students are inducted into disciplines, fields and professions that require them to be certain kinds of people with certain orientations to the world, to knowledge, to other people and to practice.

Traditionally universities have focused on knowledge and hoped that identity will follow. This hasn't been entirely unsuccessful. But to genuinely transform the academic project, universities must do explicit identity work with their students. Academics must engage in the human project, thinking about who their students are and what their previous experiences of mathematics and of learning mathematics have been.

Towards genuine change

There have been attempts to transform the content of school mathematics curricula. These include ethnomathematics, which excavates the mathematics in cultural objects, artefacts and practices; and critical mathematics, where mathematics is used to critique aspects of society and where students critique mathematics, for example, how algorithms structure our lives in ways which reproduce inequality.

However, not all of mathematics can be accessed in these ways. For true epistemological access to mathematics, students need to study it systematically, as a body of knowledge in and of itself. This can be both empowering or disempowering.

Much, though certainly not all, of mathematics was created by dead white men. But maths should and does belong to everybody. Everybody deserves access to its beauty and its power - and everybody should be able to push back when the discipline is used to destroy and oppress.

To transform mathematics teaching and learning in ways that empower students, universities need to give students the theoretical grounding they need to access the subject and support them to identify with it -- to want to learn it, to become the mathematicians of the future, to enjoy and critique mathematics and its applications.

This means that as teachers, my colleagues and I need to believe - to know - that all students can do mathematics. This knowledge must be transmitted to them. They must be shown that mathematics is a human enterprise: it belongs to all, and it can be taken forward to transform society.

# data[].attributes.attachment_link.title
Yes, mathematics can be decolonised. Here's how to begin

# data[].attributes.attachment_link.description
Some have suggested that deracialising the academy requires all researchers, teachers and students to link knowledge and identity. What might this mean for mathematics?

# data[].attributes.body
Resources

# data[].attributes.body
Newspaper (NYTimes) article

# data[].attributes.attachment_link.readable_text
Counting will get you nowhere. Try a little combinatorics instead.

Image

A puzzle by Po-Shen Loh, a mathematician at Carnegie Mellon University. Six lines in a plane, none parallel, and no points where more than two lines intersect. How many triangles are created? Credit Credit Po-Shen Loh

A couple of weeks ago, I wrote about a viral math problem - 8÷2(2+2) = ? - that drew the disdain of many mathematicians because it contained little actual math. In its place I described a more interesting math problem, offered by Po-Shen Loh of Carnegie Mellon University.

Unlike the viral "problem," this puzzle did not hinge on misleading notation. In fact, there's no notation at all, just a diagram and a question: How many triangles are formed by these six intersecting lines? (The caveat, evident in looking the diagram, is that no two lines are parallel, and there are no points where more than two lines intersect.) The challenge, ideally, was to figure out a procedure to calculate the answer, not to count carefully.

Image

Credit Po-Shen Loh

I did not reveal the answer, although I did provide a major hint from Dr. Loh: "A triangle is made of three lines. So if I have six lines, then the whole question becomes how can I pick three lines out of those six lines?"

On Twitter and in the comments on The Times's website, it was clear that some readers knew immediately how to solve the problem because they had learned combinatorics, an area of math that figures out the number of ways things can be shuffled. Other readers just tried to count, an almost hopeless task. "I don't think I've ever met anybody who's counted right," Dr. Loh said.
[Like the Science Times page on Facebook.| Sign up for the Science Times newsletter.]
To solve the problem, start with this observation: Any three lines in the diagram define one and only one triangle. It follows that the total number of triangles will be equal to the number of three-line combinations that can be chosen from a group of six lines.

How do you calculate that? Choose a line, any line. Since there are six lines, there are six choices. Next, choose a line to be the second side of the triangle. At first, you might think there are again six choices, but one line has already been chosen as a line for the first side, so only five choices are left. Likewise, for the third side of the triangle, there are four lines left to choose from.

Thus, the total number of ways you can choose the sides of the triangle equals 6 ×5 × 4, or 120. Clearly, there are not 120 triangles in the diagram. That's because all of those combinations are being counted more than once.

For clarity, number the lines from 1 to 6, and look at the triangle defined by lines 1, 2 and 3. It's the same triangle whether you choose line 1, then line 2, then line 3; or line 1, then line 3, then line 2.

Indeed, there are as many ways to create that triangle as there are ways to choose lines 1, 2 and 3. Jumble the digits every which way: 123, 132, 213, 231, 312, 321. There are six possibilities; likewise, any triangle in the diagram could be created six possible ways.

So now, divide out the redundancy. The total number of triangles created by those six lines is (6 ×5 × 4)/6, or 20. That's the answer.

Here's where math becomes powerful. The same procedure works for any number of lines. How many triangles are created by seven nonparallel lines? That's (7 ×6 ×5)/6, or 35. What about 23 lines? (23 ×22 ×21)/6, or 1,771. How about 2,300 lines? That's (2300 ×2299 × 2298)/6, which is a big number: 2,025,189,100.

The same calculation applies no matter how many lines there are. Compare that approach to brute-force counting, which is not only laborious and error-prone but provides no way to check the answer. Math produces the solution and the rationale for it.

It also reveals that other problems are, at heart, identical. Put balls of six different colors into a bag. Pull out three. How many different possible color combinations are there? 20, of course.

That's combinatorics, and it's useful for solving problems of this type. It comes with its own notation, to simplify the process of calculating, and involves a lot of exclamation points. The expression n! - " n factorial," when said aloud - describes the product of multiplying all the integers from 1 to n. So 1! equals 1; 2! equals 2 ×1, or 2; 3! equals 3 ×2 × 1, or 6. And so on.

In the problem by Dr. Loh, the calculation for the number of triangles can be rewritten like this: 6!/(3!3!).

It can be written as C(6,3), which is read as "6 choose 3." More broadly, it's mathspeak for the number of ways to choose 3 items out of 6. It is generalized into this form:

C( n, r) = n!/(( n- r)! r!)

That's the equation that students memorize, the useful shortcut. Give it a closer look. The first part - n!/( n- r)! - is what captures the 6 ×5 ×4 in the triangle calculation. The r! is what eliminates the redundancies.

Which is to say, if you forgot the formula, you can reconstruct it by remembering what it's supposed to solve. Therein lies the beauty of math, or part of it. A good equation is more than a recitation of rules; it expresses why those rules exist, reveals patterns and points to new problems to tackle.

Earlier reporting on mathematics

Correction:

An earlier version of a picture caption with this story misidentified the affiliation of Po-Shen Loh. He is a mathematician at Carnegie Mellon University, not Cornell University.

Kenneth Chang has been at The Times since 2000, writing about physics, geology, chemistry, and the planets. Before becoming a science writer, he was a graduate student whose research involved the control of chaos. @kchangnyt

A version of this article appears in print on , Section D, Page 3 of the New York edition with the headline: How Many Triangles? Count on Math to Help. Order Reprints | Today's Paper | Subscribe

# data[].attributes.attachment_link.title
How Many Triangles Are There? Here's How to Solve the Puzzle

# data[].attributes.attachment_link.description
Counting will get you nowhere. Try a little combinatorics instead. A couple of weeks ago, I wrote about a viral math problem - 8÷2(2+2) = ? - that drew the disdain of many mathematicians because it contained little actual math. In its place I described a more interesting math problem, offered by Po-Shen Loh of Carnegie Mellon University.

# data[].attributes.body
Research article

# data[].attributes.attachment_link.readable_text
Abstract

The purpose of this article is to describe how, within the Dutch approach to mathematics education, called Realistic Mathematics Education (RME), models are used to elicit students' growth in understanding of mathematics. First some background information is given about the characteristics of RME related to the role of models in this approach. Then the focus is on the use of the bar model within a longitudinal trajectory on percentage that has been designed for Mathematics in Context, a curriculum for the U.S. middle school. The power of this model is that it develops alongside both the teaching and the students: from a drawing that represents a context related to percentage to a strip for estimation and reasoning to an abstract tool that supports the use of percentage as an operator.

contexts curriculum design mathematics education models percentage primary school shift in levels of understanding

This revised version was published online in August 2006 with corrections to the Cover Date.

This is a preview of subscription content, log in to check access.

# data[].attributes.attachment_link.title
The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage

# data[].attributes.attachment_link.description
The purpose of this article is to describe how, within the Dutch approach to mathematics education, called Realistic Mathematics Education (RME), models are used to elicit students' growth in understanding of mathematics. First some background information is given about the characteristics of RME related to the role of models in this approach.

# data[].attributes.body
Research article

# data[].attributes.attachment_link.title
Teaching mathematics to lower attainers: dilemmas and discourses

# data[].attributes.attachment_link.description
ABSTRACT This article draws on Foucault's concepts of power and discourse to explore the issues of teaching mathematics to low attainers in primary schools in England. We analyse a data set of interviews, from a larger study, with the mathematics teachers of one child across three years, showing how accountability practices, discourses of ability and inclusion policies interrelate to regulate both teachers and student.

# data[].attributes.body
Magazine article

# data[].attributes.attachment_link.readable_text
Picture a mathematician. What do you see?

I'll conjure up a popular image. It's late at night. A figure is hunched over a desk scribbling into a notebook. The room smells like graphite dust, eraser shavings and body odor. Numbers and symbols gleam in the lamplight, and a calculator sits idle nearby towering textbooks.

It's not an entirely unjust stereotype. Math attracts its fair share of cave dwellers. (I should know, since I had lecture with them.) So when people discover that I studied applied mathematics in college, I get why they might raise an eyebrow like I'm from a different species.

The truth is far less exotic: I chose to study math because I was good at it and thought quantitative skills would bolster my career prospects. As it turns out, I wasn't very good at it. I got my ass kicked by Euler and eigenvalues alike. And although familiarity with statistics has proved marginally useful, I've never used 99 percent of the methods, proofs and theorems I painstakingly studied in my college courses.

There were so many moments that I wished I had picked a different major. Something easier. A little less painful. At times, I hated math. And yet, six years later, I'm so grateful that I studied it. The reasons have nothing to do with numbers and everything to do with life.

As a kid, I thought I was pretty hot shit, mathematically speaking. If my answer didn't match the key in the back of the book, I'd blame the book. "Must be a typo," I'd reassure myself. It was never a typo. I was just wrong.

As a college student, I discovered that the first incorrect answer was a rite of passage - a necessary misstep on the path to truth. This became so ingrained in me that if I got the correct answer on my first attempt, I'd think, "Must be luck."

It might sound pessimistic, but I think it's pragmatic. I was rarely discouraged, because I never expected a quick win. And if I was correct on the first stab, I was pleasantly surprised. I became well-rehearsed in failed attempts, and so much more patient as a result.

Writer's block has nothing on a tough math problem, and I've suffered through both. Writer's block usually boils down to you thinking you're not good enough. With math, it feels like the universe is mocking your ineptitude.

I've torn my hair out while multiplying matrices, watched it flutter onto thrice-erased sheets of paper. It brought me no closer to the solution, and so I couldn't help but think, "Well, what was the point of that?"

My linear algebra teaching assistant once said, "Math is pain." I agree. Studying math is masochistic. The trick is to welcome the pain. Invite it inside for Triscuits and gouda. I learned to find fun in frustration, treating it like a game where I would be the eventual victor.

As a result, my tolerance for frustration is so much higher. I'm convinced that the seeds of patience and resilience were planted and sprouted in those math notebooks.

When I was a kid, my dad had a massive red toolbox filled with wrenches, screwdrivers and various gizmos. It was so big that I needed two hands to hold it.

Studying math was like maintaining a toolbox. Each time I learned something new, into the big red box that newfound knowledge went. Who knew when it would be useful? Long-buried methods could be just the socket wrench I needed later on.

The science of math is grounded in wielding the right tool in the right step. The art lies in knowing which tool to grab. Thing is, when you're constantly encountering mysterious machinery, it's difficult to determine which tool fits.

I became adept at trying all sorts of tactics, accustomed to chipping away at a problem from different angles on the promise of a breakthrough. And that simple joy of solving a problem propels me, for there are few better feelings than when the key finally clicks and the lock opens.

Knowing how to solve problems is Act One. Act Two is not fooling yourself into thinking you know more than you do.

The arrogant mathematician is a sloppy one. True mathematicians are skeptics, wary of their own instincts. Even if I knew how to find the solution, I still proceeded with caution. One dropped decimal could blow it all.

Even once I'd solved it, I checked each step again, trying to suss out faulty reasoning or careless mistakes. Sometimes I solved the problem a different way, ensuring that I arrived at the same result.

Math reminded me that I'm fallible - prone to all sorts of mishaps and missteps. I shouldn't cut corners, because walking with practiced care was better than flailing at a full sprint. And just because I arrived at an answer, didn't immediately make it the right one.

Perhaps you've heard the stereotype that Asians are good at math. Let's just say I have a large sample set and, sure, some Asian math students are very smart. Or at least smarter than me. So is there a gene that imbues Asian people with math talent? I doubt it very much .

Could it be that some Asian students were raised to be more persistent than Americans? Maybe.

In a famous study, classrooms of American and Japanese first graders were given an impossible math problem. The American students gave up after less than 30 seconds, while the Japanese students persisted for an hour before the proctors stopped them to confess that the problem was unsolvable. (How cruel!). The grand difference came down to Asian cultures prioritizing practice and persistence in education, along with embracing that struggle is a large part of the learning process. In the Western world, the idea of inherent intelligence is pushed and prized, which undercuts the importance of educational struggle. I buy that American children are a bunch of cupcakes. Give an American kid a math problem, and chances are they'll squirm, whine, complain and hit you with that tired refrain: "When am I ever going to use this in the real world?"

Answering that question directly is a mistake. When are you going to need to factor a polynomial in the "real world"? Maybe never, kid. Especially not with that attitude.

But when are you going to face a problem that requires focusing for more than 30 seconds? All the goddamn time.

I'm so fortunate that math forced me to practice persistence. It paid massive dividends later, even if I was too immature to recognize it in the moment.

My biggest lesson from math was the most counterintuitive: It wasn't for me. Although I studied applied mathematics as a 19-year-old, I grew disillusioned with the actual application of it. I wanted to go out into the world and do things, not pore over problem sets.

So I tuned out. I mailed in midterms. I submitted half-finished homework, if I turned it in at all. The only time I saw an "A" was at the start of my first name.

I was perfectly content being mired in mathematical mediocrity.

And that scared the shit out of me. Because I couldn't ignore the voice deep inside that screamed: This isn't what you were meant to do.

The truth is, math didn't move me. My stomach dropped whenever I walked by a classroom and a chalkboard. After each final, with mathematical methods rattling in my head like a tambourine in a dryer, I couldn't help but think, "So... what do I do with this now?"

As graduation grew closer, I set one career goal: not dreading going to work in the morning like I did math lecture. Money could never compensate for that feeling.

When I graduated, I closed my math textbooks for good. I discovered new passions, learned new skills and grew in new ways. Six years into my career, I can say that being comfortable with numbers and data has been useful, but what has proved invaluable are the qualities that math imbued in me - patience, attention to detail, humility and persistence. That was the true reward.

After all that pain, angst and frustration, it may have been cathartic to douse my math textbooks in kerosene and light them on fire. But I kept them instead. They're on the bottom shelf. Even though I never take them out, I like having them around.

For a long time, I thought this was just a collector's impulse. But I just moved recently and I put my bookshelf back together from top to bottom: first the novels, then the narrative nonfiction and memoirs in the middle. I could've sworn I heard the wood creak and the shelves slide forward. It was too top heavy.

I needed that bottom row of math textbooks. They were my anchor. The bedrock. The foundation for much of what I've learned, and a sturdy base for everything still to come.

# data[].attributes.attachment_link.title
Lessons from My Math Degree That Have Nothing to Do with Math

# data[].attributes.attachment_link.description
Picture a mathematician. What do you see? I'll conjure up a popular image. It's late at night. A figure is hunched over a desk scribbling into a notebook. The room smells like graphite dust, eraser shavings and body odor. Numbers and symbols gleam in the lamplight, and a calculator sits idle nearby towering textbooks.

# data[].attributes.body
Interactive

# data[].attributes.attachment_link.title
Percent - Bar Modeling * Activity Builder by Desmos

# data[].attributes.body
Variation Theory - book chapter....

# data[].attributes.body
Variation Theory

# data[].attributes.attachment_link.title
Variation theory in mathematics education

# data[].attributes.attachment_link.description
In this Espresso, Lucy Rycroft-Smith explores the use of variation theory in mathematics teaching and learning.

# data[].attributes.body
Blog

# data[].attributes.attachment_link.title
Taking the inquiry out of Inquiry Maths: A reply to Naveen Rizvi - Inquiry Maths

# data[].attributes.attachment_link.description
Inquiry maths is a model of teaching that encourages students to regulate their own activity while exploring a mathematical statement (called a prompt). The website has resources for secondary school teachers.

# data[].attributes.body
Notebook: codex Forster II

# data[].attributes.attachment_link.title
V&A · Explore Leonardo da Vinci's notebooks: Codex Forster II

# data[].attributes.attachment_link.description
Famous worldwide as the painter of masterpieces such as the Mona Lisa, Leonardo da Vinci is also renowned for his notebooks in which he recorded his thoughts.

# data[].attributes.body
EEF Guidance Report

# data[].attributes.body
A think piece.....

# data[].attributes.attachment_link.readable_text
Julian Williams

HAVE YOU heard of "Mastery maths"? You might think it's the latest fad, the latest government imposition, or the latest resource for learning mathematics. Maybe it's all these things? You can go online and check out what is being sold: you might like some of the ideas yet also object to others. But this is where the money is; the government has decided the CPD budget for maths is to be spent this way, so this is 'it'.

The NCETM hubs seem heavily involved in shaping it in practice on the ground, but they are constrained by the programme's national commitments which have been shaped by Shanghai and Singaporean influences. I've heard that the programme gets its support because the ministers who supported it thought it involved children sitting in rows being taught together through 'Direct Instruction', being properly disciplined, and learning by rote. From what I see, however, there seem to be some good things in this system: in particular some texts and approaches are inspired by Skemp and Bruner and their aficionados in the Association of Teachers of Mathematics, adapting many of these ideas from the 1970s. There are also Western influences such as the concept of 'variation' developed by Ference Marton and others that now boomerang back to us from Shanghai: well and good! The Singapore programmes emphasise what Skemp refers to as 'relational understanding' (real understanding of mathematical concepts and how they relate) to underpin the instrumental numeracy - even the recall of times tables - that Conservative education ministers value so highly .

Not so long ago a teacher asked me if I thought 'apparatus' was still 'in' for good maths teaching: their school was very much 'into' their whiteboards and so on. I was shocked. Well, manipulating manipulatives are certainly 'in' again in Singaporean maths.

The notion of the whole class moving together through the essential curriculum also feels right by equitable criteria: a true 'no child left behind' policy approach. Then, who would complain about there being text books to support the teaching? Surely materials that are well designed and crafted by informed praxis should be welcomed as resources even for the most independently minded and resourceful teachers.

Even more positive - the fact that the Pacific rim teachers tend to have a culture of research and lesson study in their professional practice is not to be sneered at... if only this kind of professional development was significantly resourced here (clearly this is not going to happen under the present government).

So, in this case, I think there is good reason to support work on 'fluency' that is understood to be emergent from experiences that are relational. Skemp never meant, I think, to suggest that instrumental mathematics does not have a role, but I think he saw that it was limited if it did not engage with relational understanding, like 'having procedures' without 'knowing why' the procedures work and sometimes might go wrong. Consider the 17 times table, not one perhaps that we have learnt by heart/rote - but perhaps it would be good to experience trying to do so? So 8 x 17 = ... er? On the other hand, I know that 17×10 = 170... and 17×2 = 34, so 17×8 has to be 34 less than 170, or 136. This is 'obvious' knowledge that flows from relational understanding to support instrumental working; surely pedagogy should encourage it. This is a 'knowing by heart' that is genuinely mathematical, in contrast to learning by rote that is quite otherwise because it has no basis in mathematical reasoning and so undermines mathematical 'sense'; perhaps we should call it mastery but not mystery.

'The notion of the whole class moving together through the essential curriculum also feels right by equitable criteria: a true 'no child left behind' policy approach.'

So what should we really worry about in "Mastery mathematics", given that many of the ideas it promotes seem to be arguably well founded, and persuasive? For myself, what I worry most about is that 'mastery' is not only mastery of mathematics per se but mastery of the teaching-learning partnership, or relationship. "Mastery" sold as good practice seems to be yet another means of telling teachers and learners how they should do things, because this is believed or has been shown to be 'effective' (in somebody's hands). In other words, what is wrong with "Mastery" for me is what is wrong with the whole policy discourse about "what works" and "best practice", which is often propagated in a regime that actually has very slim evidence or authority to make these claims. The policy becomes the master of us all, and we are obliged to suspend our critical faculties and comply.

I think what is wrong with "Mastery maths" is close to the concept of the dialectic of Master and Slave: both are alienated from their labour as well as each other. In Mastery Mathematics who is Master and who is being "mastered", who is the 'slave'? It worries me that the people who are absolutely central to learning-teaching are not given agency, not given the control, are not invited to be critical of this whole new narrative of 'best practice' which marginalises their judgment. The teachers and learners are not the masters here; are they then the slaves to their masters, working for their institutions, as Ian Stronach once put it "Doing their sums for their country" in PISA, TIMSS, and National tests?

The direction of policy, at least in recent decades (and this has been true of the Blair governments as much as of its expansion in the years since), has been to treat the profession, the teachers and classroom practice, as the object of change, rather than its agent. Barrow wrote "Giving teaching back to the teachers", in which he appealed to researchers not to steal teaching and judgment from the teachers. Everything said there applies even more emphatically now, with the 'researchers' being replaced by the what-works-technicians, managers, and policy.

We also need to give learning back to the students, because the control of teaching from outside robs learners of agency or mastery quite as much as it has done teachers. Becoming the ever-more compliant 'deliverers' of a given best practice pedagogy and national curriculum, driven by tests, the freedom of the teacher disappears. So does the freedom of the student to learn; motivation becomes a technical job for the teacher, an instrument to harass the learner into learning what we claim they need to know. We are no longer allowed as teachers to conceive of the diverse ways we might teach, and the many ways our students might learn, because best practice tells us how it is normatively effective, and anyone who does not comply- well that's fine but they have to have a convincing reason to press back against the weight of the world and its 'evidence'.

And yet - the evidence is remarkably slim. Arguably, the most important factor in effective learning-and-teaching is a powerful (and relatively 'unalienated' - if that's possible) relationship between the teacher and learner: some call this love. If the intrusion of Mastery into an effective teacher-learner relationship serves to undermine this, then I am sure it will prove ineffective - though I wonder if this will be detected by the kinds of evaluation and measurement widely employed to evaluate effects.

In conclusion: what I am arguing is not really 'against' the current trends for Mastery Maths particularly, but FOR a renewal of a different kind of professional development, where the agency of learners and teachers is more obviously respected as the vital agents of real learning. I urge the critical examination of the imposition of Mastery Maths and other such slogans and programmes insofar as they purport to have the ultimate answers to developing practice, leaving the teachers with a purely technical task of 'delivery', and presumably reducing learning activity to 'package opening' functions worthy of a future menial workforce and compliant citizenry.

# data[].attributes.attachment_link.title
Mastery mathematics but who is the slave? - education for tomorrow

# data[].attributes.attachment_link.description
Julian Williams HAVE YOU heard of "Mastery maths"? You might think it's the latest fad, the latest government imposition, or the latest resource for learning mathematics. Maybe it's all these things? You can go online and check out what is being sold: you might like some of the ideas yet also object to others.

# data[].attributes.body
Some maths

# data[].attributes.attachment_link.readable_text
Icy patches sit in wait and bare trees shiver in anticipation but spring is still dawdling, likely enjoying a final cuppa before going back to work. What are we to do while days are dreary and the wind rummages through empty branches? Why, stay inside and play with geometry software, of course! Yet winter was long and snowy, and after one too many hours indoors the treasure trove of maths is perhaps running low.

What to do once ancient theorems are re-proved and the excitement of a parametric plot has wilted? Well, there is still a nugget of algebra that could flourish into hours of exploratory fun. So grab your Geometer's Sketchpads, your Geogebra, or simply some transparencies and coloured markers, and let's play billiards with polynomials.

In the vast sea of beautiful mathematics it is quite easy to drown, finding more and more interesting papers on a subject, eventually getting lost in the plethora" of nifty theorems and charming proofs. It's best to restrict ourselves early on in the game if we are to make an afternoon of it. Let's say for today that we consider only the polynomials of degree 3. Armed with our restriction-cubics only!-let us dive into the deep end of adventure.

Build a billiards table

To play a good long game of billiards, you must first build a pool table (as Carl Sagan might have it). Look at your cubic polynomial, for now remaining mysteriously general: $a_3 x^3+a_2 x^2+a_1 x+a_0$. First, from the origin in the positive $x$-direction, draw a line segment $a_3$ units long. At the end of it, turn 90° anticlockwise and draw a line segment of length $a_2$. Turn 90° anticlockwise again to plot $a_1$, and do the same for $a_0$. Call the final point you arrive at $\Omega$, where you build a corner pocket. If ever a coefficient is negative, you 'backtrack', plotting out a segment backwards but still remembering the direction in which you face for the next turn. Now for the playing part: the segments constructed, and the lines extending them, have become walls. Our objective is to launch a ball from the origin and hit three walls, sinking it into $\Omega$, much like in three-cushion billiards. In the words of our indices, "3, 2, 1, sc0re"!

But on this special table, the physics is peculiar: our ball makes a 90°-turn each time it bounces off a wall.

Here is a path on $x^3-5x^2+17x-13$. We try different launch angles $\alpha_3$ from the origin (3 is for cubics), bouncing against walls (hitting the corners if necessary), until we find the winning path to $\Omega$. Once a path is found, $-\tan\alpha_3$ is the real root of the polynomial! The path in the picture corresponds to the root $-\tan\alpha_3= - \tan(-\frac{\pi}{2})=1$. A visual way to factor polynomials through a geometric game.

You can handle the truth

Perhaps it is far more rewarding to know why this method works than it is to practise it. If you want the magic and mystery to remain, pause now until you're ready. Otherwise, let's dive deep into geometry and fish out some nuggets of truth.

With the choice for each coefficient of whether to go forwards ($+$), stay and turn (coefficient of $0$) or to backtrack ($-$) along each segment, we need casework to do the proof properly. But with long cold nights in our near future, we will talk business here and leave the details to settle into place in due time with some well-chosen parameters.

Let's consider first the ideal case: all coefficients positive, $a_3 x^3 + a_2 x^2 + a_1 x + a_0$, our billiard ball makes two right angles to land perfectly at $\Omega$, the endpoint of $a_0$. In our 90°-bounce universe, the billiard ball path forms a series of 3 similar right-angled triangles. In each triangle, call $s_k$ the side opposite angle $\alpha_3$, starting with $s_3$ for the triangle at the origin. Let's trace out the trigonometry.

We start with $s_3 = a_3 \tan\alpha_3$. In the next triangle:
$$s_2 = (\tan\alpha_3)(a_2 - a_3 \tan\alpha_3)=a_2 \tan\alpha_3-a_3 \tan^{2} \alpha_3,$$
and $$s_1 = a_0 = (\tan\alpha_3)(a_1 - s_2)=a_1 \tan\alpha_3 - a_2\tan^{2}\alpha_3+a_3 \tan^{3} \alpha_3.$$

Replacing $x=-\tan\alpha_3$, we get exactly our polynomial, $a_0 + a_1 x + a_2 x^2+ a_3 x^3=0$, so $x=-\tan\alpha_3$ is indeed a root of the original polynomial. Simple enough.

Ever catch a side length of zero? A length, you will agree, unacceptable for a stable pool table (or a pool). And yet the maths works out! If an $a_k=0$, we generalise: draw a line where segment $a_k$ would have been: a ball hitting a line-wall outside the segment $a_k$ will go right through the wall, still making a 90° angle with the path coming in. Not much of a table in this case and not much is left of billiards but the procedure still
delivers a visual way to find real roots. Three examples with zero coefficients are showcased below.

\begin{align*}
s_3 &= a_3 \tan \alpha \\
s_2 &= s_3 \tan \alpha \\
&= a_3 \tan^2 \alpha \\
a_0 &= (a_3 \tan^2 \alpha + a_1) \tan \alpha \\
&= a_3 \tan^3 \alpha + a_1 \tan \alpha
\end{align*}

\begin{align*}
s_3 &= a_3 \tan \alpha \\
s_2 &= (s_3 - a_2) \tan \alpha \\
a_0 &= s_2 \tan \alpha \\
&= a_3 \tan^3 \alpha - a_2 \tan^2 \alpha
\end{align*}

$$a_0 = a_3 \tan^3 \alpha$$

How the tables have turned

Good news for the completionists in the audience: should you chance upon a cubic with three real roots, you can extend the process, building the walls of a new billiard table on the path of the ball's old trajectory, continuing onwards: the ball's path you have drawn becomes the walls for a new polynomial.

In the diagram above, I have done just that with $(x+3)(x+1)(x+2)$, better known as $x^3+6x^2+11x+6$. The original rectangle of a pool table can be played on via the red path. Its angle $\alpha_3$ (measured in the usual manner: anticlockwise about the origin from $a_3$) respects $\tan\alpha_3=3$, one of our roots. Build a second billiards table! For the three similar right-angled triangles with hypotenuses on the red path, we find the relationship of hypotenuses to be $AF:FG:G\ \Omega=1:3:2$. The path $AF, FG, \Omega G$ itself represents a polynomial ($x^2+3x+2$)! Well, we couldn't quite tell that it wouldn't have been $x^{\hspace{1pt}3}+3x^{\hspace{1pt}2}+2x$, or $x^{300}+3x^{299}+2x^{298}$, or, for that matter, $9x^{\hspace{1pt}2}+27x+18$. In fact, it is $\sqrt{10}x^{\hspace{1pt}2}+3\sqrt{10}x+2\sqrt{10}$, where $\sqrt{10}$ is $\sqrt{1+\tan^{2}\alpha_3}$, or $a_3/\cos\alpha_3$. You can find this from the diagram, or directly from the polynomial with a bit of work.

Now, we jolly good shots have found an orange path to sink the ball: notice that $-\tan\alpha_{2}=-FH/AF=-1$: another one of our roots! Triangles $AFG$ and $FGH$ are once again similar, with the latter twice the size of the former, so $AH/H\mathit{\Omega}=\frac{1}{2}$: the yellow path describes the path of $x+2$ (up to multiplication by $cx^{\hspace{1pt}n}$, $c \in \mathbb{R}$, $n \in \mathbb{Z}$). We end by firing off a shot down the $x$-axis from A, to E, to victory, with $-\tan\alpha_{1}=-2$. Our roots are indeed, $-3,-1,-2$.

For our friend from earlier, $x^{\hspace{1pt}3}-5x^{\hspace{1pt}2}+17x-13,$ there are no other paths to victory. This is a method for real roots, you see: there just aren't triangles around whose opp/adj ratio is $2\pm 3i$, for $i=\sqrt{-1}$. But there are plenty of polynomials in the maths sea! The method works for polynomials of any degree, you will agree: nothing in our proofs was specific to cubics.

One can factor continuously, perching a precarious sequence of billiards games one on top of the other in a whirlwind of a billiard tournament, until the real roots are all encoded by triangles for a straight and easy parting shot. The billiard game is an inefficient method, albeit a fun one, but with practice one gains enough intuition to seriously improve one's game, and this is a valuable skill of visual factorisation: give it a try!

More to explore

This is the pastime which Austrian engineer Eduard Lill gifted the maths world in 1867. Felix Klein publicised the cubic version in his 1926 book, Elementarmathematik vom höheren Standpunkte aus, II: Geometrie and for a while it seemed to feed its own craze, but eventually became forgotten. Interest was revived again in 2011 by Thomas Hull. His article tells also of Margherita Beloch, the Italian mathematician with a fascinating pastime, and of origami, and forgotten manuscripts, and much intricate maths besides. Within that paper, you'll read that $x^3+2$ (pictured right) produced a Beloch square, which is itself constructible from a Beloch fold, itself one of three mutual tangents of two parabolas... But hark! Here there be quadratics.

We have committed to a cubics-only rule in our dive and so we must emerge, nifty pastime in hand. We have yet to exhaust the topic and if you wish to dive deeper, there are beautiful and obscure results stretching far and wide. A topic for another deep dive, I'm sure.

# data[].attributes.attachment_link.title
Playing billiards with cue-bics - Chalkdust

# data[].attributes.attachment_link.description
Icy patches sit in wait and bare trees shiver in anticipation but spring is still dawdling, likely enjoying a final cuppa before going back to work. What are we to do while days are dreary and the wind rummages through empty branches? Why, stay inside and play with geometry software, of course!

# data[].attributes.body
Blog

# data[].attributes.attachment_link.readable_text
There are several ways teachers can replace student fretfulness over math with a sense of appreciation.

Be clear about why understanding math concepts matters. Kids who believe that they must simply endure algebra and calculus until they're through with school-and that the actual learning is pointless because they'll never use it again-should be reminded why understanding mathematical concepts is valuable. Most importantly, being able to comprehend a "symbolic, representative system," Immordino-Yang says, teaches the brain how to think theoretically and logically. "Learning how to think abstractly is a useful ability in all aspects of life," Immordino-Yang said. In fact, people who have studied complex math in high school tend to have better life outcomes, she said. Teachers who share this information may persuade reluctant math-learners to stay engaged.

Assign projects that help kids see math's usefulness. Students are more apt to participate if they see a practical application to their studies. "This goes beyond learning how to balance a checkbook," Immordino-Yang adds. By studying how fast and far vegetable oil spreads on tissue paper, for example, students can learn not only about the math concept of direct variation, but also about how oil spills are measured. Sharing stories from the news where math understanding is featured in the narrative-in a story about price fixing, say, or one on climbing interest rates-also can help students see its usefulness in the real world. Learner.org, a free educational resource from the Annenberg Center, provides such practical lesson plans for math at all levels, including the oil spill example.

Discuss mathematical role models, and share how their ideas have changed the world. Because math is the foundation of so many other fields-physics, engineering, finance, astronomy, among others-history teems with mathematical virtuosos whose creativity and curiosity shaped the modern world. In addition to the usual suspects of math icons, including Pythagoras, Rene Descartes, and Ptolemy, more contemporary role models might spark student appreciation for the subject. There are many: Alan Turing, who's code-breaking during World War II helped defeat the Axis powers; Ada Lovelace, who created the Analytical Engine, which presaged modern programming; even Nate Silver, a popular mathematician who uses statistical forecasting to predict outcomes in Major League Baseball and political elections. The humbler discoveries made by annual recipients of the Presidential Early Career Awards for Science and Engineering might also inspire.

Strive to minimize the sources of fear. Math anxiety stifles clear thinking. At those moments when students most need to marshal their intellectual resources-during a test, say, or when called up before class to work a problem-those who fear the subject are apt to panic and shut down. Sian Beilock, a professor of psychology at the University of Chicago, and author of Choke: What the Secrets of the Brain Reveal About Getting it Right When You Have To, describes the anxious over-reaction to high pressure situations as a "malfunction of the prefrontal cortex." For nervous students who feel pressure to perform well on a test, that worry causes them to execute beneath their skill level. "Anxiety is robbing you of working memory," Immordino-Yang explained. "You're wasting your thought powers," she added. Math phobic kids need help from teachers to lessen their fear.

Keeping the classroom "kid-centric," Immordino-Yang said, can help. Teachers who act as facilitators, or resident experts, rather than omniscient instructors, invite students to explore without fear of messing up in front of an authority. Freeing up fellow students to explain problems also allows for more personalized instruction. During Jacqui Young's happy year studying pre-calc, she worked with peers who struggled to keep up with the teacher's pace. "I think it was better because I'd be working one-on-one or two-on-one with my classmates," she said. Another way to tamp down dread is to set up class in a roundtable and encourage student-led give-and-take. Known formally as the Harkness Method of teaching, this collaborative approach to learning may be especially useful in math subjects.

# data[].attributes.attachment_link.title
How to Make Math More Emotionally Engaging For Students

# data[].attributes.attachment_link.description
There are several ways teachers can replace student fretfulness over math with a sense of appreciation. Be clear about why understanding math concepts matters. Kids who believe that they must simply endure algebra and calculus until they're through with school-and that the actual learning is pointless because they'll never use it again-should be reminded why understanding mathematical concepts is valuable.

# data[].attributes.body
Mathematical Skills Working Group Report

# data[].attributes.body
A report

# data[].attributes.body
Research review

# data[].attributes.headline
Maths of knocking over absurdly large dominoes 

# data[].attributes.subject
Maths of knocking over absurdly large dominoes 

# data[].attributes.body
Maths

# data[].attributes.attachment_link.readable_text
Everyone has probably played with dominoes. At some point in your childhood, chances are that you tried this game: Line up a bunch of dominoes in a row and then knock one over. Soon the rest of the dominoes topple in a chain reaction. I'm not sure why this is fun, but it is.

But how about something even cooler? What if you have a small domino knocking over a bigger domino? This is indeed possible. But is it possible to ramp up the domino size so much that a huge domino could tip over and crush a car? This was the goal in a recent episode of MythBusters Jr., where I am the science consultant.

Physics research has shown that a domino can knock over another one that is 1.5 times taller. OK, this is what makes physics so cool. We don't just study protons and black holes. Physicists also write papers about dominoes. Check out this paper on arXiv.org. Also, there is this much older paper (from 1983) in which someone described this taller domino crash as a way to demonstrate a nuclear chain reaction (without actually blowing stuff up).

Also, there is this very nice video showing a domino chain reaction starting with a super, super tiny domino and leading to a 13th domino that is 1 meter tall.

Notice the claim that the 29th domino would be as tall as the Empire State Building (381 meters tall). Is that true? Let's look at the math of increasing sizes. Suppose we start with a domino that is 1 meter tall. How big will the next one be? It will be 1.5 multiplied by 1 meter to give a height of 1.5 meters. What about the next one? Again, we multiply by 1.5 to get a height of 2.25 meters. This phenomenon is called a geometric sequence, where each successive number in a list is found by multiplying by a constant.

Since multiplication can be tedious after a while, let's make a computer program to calculate the height of the 29th domino assuming the starting domino is 5 millimeters tall. Wait! Don't panic. Just because I said "computer program" doesn't mean this is going to be crazy complicated. Computers are our friends (for now), and you should learn to use them. Now is a great time to start.

Check out this code for a geometric series. You can click the Play button to run the code and then click the "pencil" to go back and edit the code. Yes, you can edit the code if you like (and you should).

From the output of this program you can see that the 29th domino would be more than 400 meters tall-even taller than the Empire State Building. Yup. Things can get pretty big fairly quickly with a geometric series. Let me go over the important parts of this very short program.

In lines 4, 7, and 10 I am just creating some variables and assigning them to some value. The actually name doesn't matter, you could call them Bob, Jane, and Rhett and it would still work as long as you used these names consistently.

The green lines that start with a number sign (3, 6, 9) are comments. They are just there for the humans. The computer ignores these.

Line 12 is the most important. This is the start of a "while loop." Everything below this line that is indented will be repeated until the value of N is no longer less than 30.

Line 14 and 15 are where the calculations happen. By adding 1 to N and setting it to N, the counter value increases. The other line calculates the new height.

Now for a test. See if you can modify the code to see how tall the 29th domino would be if the multiplicative factor was 1.4 instead of 1.5. What if you want to calculate the size of the 100th domino? Try it. If you change the code and run it-congratulations. You are now a computer programmer.

What about the dominoes in MythBusters Jr.? There are two major differences. First, they started with an actual domino. According to this site on dominoes, the standard domino is 1 and 7/8th inches tall (15/16" wide and 1/4" thick). The second difference is that all the measurements are in Imperial units (inches and pounds and stuff) instead of metric units. It's not a big deal, but I just like using metric units (like most of the world). OK, we will do it both ways.

Here is a similar program that calculates the height of dominoes needed to get one big enough to possibly crush a car. This code is a bit more complicated, but you can look at it if you like. From this, I get the following graph of domino height vs. domino number.

Here you can see that the 12th domino would have a height of about 4.12 meters. But what about the mass (and the weight-yes, weight is different than mass)? Let's assume that each domino has the same density, where density is defined as the mass per unit volume of a material. Since the 12th domino is 86 times taller than the original domino, wouldn't the mass also be 86 times greater? Nope. It's even more massive than that.

Check it. If I increase the height of a domino by a factor of 1.5, it does indeed get taller. However, if I want to keep the same shape for this bigger domino, I also have to increase the width by 1.5 and increase the thickness by 1.5. This means the volume of the new domino will increase by a factor of 1.5 x 1.5 x 1.5, or 3.375. If the density is constant, then the mass (and yes the weight) will increase by 3.375 for each successive domino.

Let's just plot this mass of the 12 dominoes. I am going to use metric-but don't worry, I will convert to pounds for you too.

That's over 3,200 kilograms for that last domino. If you convert that to pounds, it's more than 7,000 pounds. Bam.

But wait! If you're thinking this domino chain reaction is as simple as a few lines of Python code, you are forgetting something. Someone has to actually build these things. Oh sure, I could probably make the first 5 dominoes-but what about those last few? Can you just make them out of wood? That's possible, but it would have to be wood with the same density as the original domino. The other option is to build a metal frame and then fill the domino with stuff to make sure the mass is correct. But be careful, you also need to keep the center of mass in the center of the domino. It's a tough build-that's for sure.

OK, let me leave you with one homework question: How many chain-reaction dominoes would you need such that the last domino could fall over and crush an aircraft carrier? How much space would you need to make this happen? The good news is, I doubt anyone will ever build that domino.

More Great WIRED Stories

# data[].attributes.attachment_link.title
The Sensible Math of Knocking Over Absurdly Large Dominoes

# data[].attributes.attachment_link.description
Everyone has probably played with dominoes. At some point in your childhood, chances are that you tried this game: Line up a bunch of dominoes in a row and then knock one over. Soon the rest of the dominoes topple in a chain reaction. I'm not sure why this is fun, but it is.

# data[].attributes.headline
European traditions in Didactics of Mathematics

# data[].attributes.subject
European traditions in Didactics of Mathematics

# data[].attributes.body
A free book

# data[].attributes.body
CoreMaths

# data[].attributes.attachment_link.readable_text
Chris is a current MSc Computer Science student at the University of Warwick, UK. He is also the co-founder of Sustain Investing. Before that, Chris worked at Citi Ventures and at Citi Markets.
It started with an excuse

Hi, I'm Chris. While building Sustain with my cofounders Andre, Nick Foden and Sylwia Zieba I've been studying for my Masters in Computer Science. For a data analytics module I took, we were tasked with finding any dataset(s) in the world and applying the techniques we learnt in class.

As a self-confessed running addict, this was an opportunity to dive into some juicy running data. And which race to analyse? Only the best race in the world would do: the London Marathon. 💪

I ran London Marathon last April with the aim of getting 2:59:59. I failed. My training for the race had been through the long cold winter but on race day itself temperatures reached a balmy 23C. My pace was on target until just over the 25km mark, and then it plummeted. I finished in 3:16:38.

This left me with big questions: does the weather affect everyone? How much does it affect them? Ultimately I wanted to know... Is it a valid excuse for my performance? 😂

Where to find the data? 🕵️
Now for some maths
While there is no public dataset for London Marathon results, the official website has searchable results for all runners for the last 5 years. To retrieve these results I built a scraping tool in the programming language python. I used popular modules requests and Beautiful Soup to call the results page of each participant for the last 5 years. I saved these results to a csv file.

Overall there were 193,439 unique runners, slightly lower than the number of participants in the races because I had no elite runner data.

The info I had about each runner was somewhat limited: Name, Runner Number, Year of Race, Club, Age Category, Place, Gender and Nationality. It would have been cool to have exact age, city of residence, charity they contributed to, fancy dress outfit and running experience (eg number of marathons completed). Oh well, I'll take what I can get.

For each runner I also had a bunch of info on how the runner's time was looking at each 5km mark. For example, at 10km in this is how I was looking:

So far, so good. Right. Let's start with some basic statistics on the runners. You can see between 2014 and 2018, the average runner finished in around 4 and a half hours. The fastest 25% of runners finished in under 3:50 and the slowest 25% finished in over 5:00. Interesting.
Let's look at the weather
Right, so let's take a look at pacing strategies. I think it's fair to say that a runner's pace at the 5km or 10km mark is pretty much the pace they wanted to hold for the whole race. As you can see below, the average runner slowed down every single year. Not so surprising I guess, but look at the purple line. In the 2018 race, the average runner slowed down significantly more than in any other year. Looks like I wasn't alone...

Alright, warning. There is a little bit of maths coming up. I promise you it will be over soon.

So the graph above gave me an idea. What if I could calculate a 'slowdown factor' for each runner? Basically I want to know how much did a given runner slow down between the 5km mark and the 40km mark. The formula below gives this number as a percentage.

Anyway, even if you ignore this formula then just know that the slowdown factor is a number that represents how much a runner's pace slowed between the beginning and the end of the race.

The above chart shows how much runners slowed down over the years. I've split the results into different finishing times (runners who finished in under 3 hours, under 4 hours etc) and then by which year they ran the race. You can see that fast runners didn't slow down that much (maybe around 10%) which makes a lot of sense. These runners are often experienced and know how to pace themselves. The slower runners experienced a much larger drop in pace however.

Also interesting that 2018 was the worst year for slowing down, no matter the speed of the runner.

So when runners complain about the weather, they normally mean its either cold (below 5C), hot (above 18C) or humid. Running in the rain sucks too, but that's more of a psychological thing. Here, I just looked at the temperatures on each race day.

The above chart is called a box-and-whisker plot. It shoes the range of temperatures for the race day each year. The box shows where the temperature was for 50% of the time. The whiskers show where the temperature was the rest of the race.

It jumps right out at you - 2018 was not only hotter, but the temperature changed a huge amount during the race. Much more than any other year!

Here's a chart of different temperatures during the 2015 race vs finish times for each runner. It's pretty clear that the hotter it got, the slower the runners were. The chart even gives the regression equation of the relationship.

More maths: every year seems to have a similar relationship. The last column in the above table shows the correlation coefficient. There is definitely a correlation. Except in 2018.

Hmmm that's strange. Didn't we just show that 2018 was the hottest year? And didn't we show that this was also the year where runners slowed down the most? A potential explanation is that 2018 was so hot from the start that it didn't matter how the temperature changed during the day.

After regressions, I built a number of classifiers for the data. If you're curious about the results there then let's chat, but I'll keep it off this page to spare everyone else.

Always pre-prepare your excuse 😏

So it appears that my defence is reasonably well founded. 2018 was hotter than every other year. It was also the year when runners slowed down the most. And even in other years, the hotter the temperature the slower the runners were.

Case closed, right?

Well, perhaps not. I don't doubt that 2018 was the hottest year (even the Daily Express agrees, and they are never wrong). It's also pretty clear from the data that runners slowed more in 2018 than in any other year before that. But I'm skeptical about the evidence that in a given race, runners who finished in hotter temperatures were slower. If they took 5 hours to complete the race then finished later in the day, and the temperature is always higher then - right?

Anyway. Let me know what you think. Do I have a good excuse or should I just admit I wasn't fit enough? This April 2019 I get another chance to break the 3 hour barrier in Brighton Marathon. Praying for cool air! 🙏🏼

# data[].attributes.attachment_link.title
How to Pace the London Marathon: Fuelled by Data 🏃 🇬🇧

# data[].attributes.attachment_link.description
Chris is a current MSc Computer Science student at the University of Warwick, UK. He is also the co-founder of Sustain Investing . Before that, Chris worked at Citi Ventures and at Citi Markets . It started with an excuse Hi, I'm Chris.

# data[].attributes.body
A free book