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As an educator, I would absolutely love to just explore math each day with my students. Lockhart's theory on Math gives a magical insight to organic learning and exploration that children love. I believe that we need like-minded educators that set the expectations, make the laws and requirements for teachers following some of his suggestions. I say some, because all would crash the public school system like an earthquake. We are set up with TIME in mind for everything. It seems that in reading this book, time is not a factor. Discovery is on one's own time, therefore, not only would curriculum be useless, but also grade levels. What a great read, and insightfulness to ignite the curiosity of exploration in a classroom!

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I love the art of discovery in this chapter. "First of all-forget the symbols- they don't matter". This is so hard to make myself do after being trained to "what they are, they are" mentality. He compares the quantity to entities.  Numbers have nicknames and they do what they do - join together to become another number with a nickname - 12. It's an interesting chapter and concept that would definitely take a sufficient amount of time in a classroom that is not there. Concepts would need months to explore instead of days, weeks that are in place now.

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I agree wholeheartedly with Lockhart about Geometry. I not only went back to my high school experience while reading this book, but also in my reflection sided with him. Something so simple, was presented difficult and blew my mind. I loved how he said letting students explore on their own, with their own questions and representations became - "the student's own idea", which is what we spend our whole career trying to get them to do.

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I love how he stays in a comparison state with Math and Art - " Real mathematics doesn't come in a can", "Problems lead you to where they take you", "Art is not a race". I agree that the curriculum idea stifles learning organically and binds educators to teaching by topics and curriculum.

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My favorite quote from this chapter was in the section titled "The Standard School Mathematics Curriculum - Lower School Math (p83) - " The indoctrination begins.  Students learn that mathematics is not something you do, but something that is done to you."
In the paragraphs that follow the author discusses contrived and artificial 'word problems' making math drudgery.  Inspiration for change for sure.

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I love when Lockhart wrote, "Let me make a couple important points.  First of all, notice that once we know why something is true, then in particular we know that is it true.  Another thing I want you to appreciate is the finality of mathematical proof."  This makes me think of my High School math class.  We were taught the math behind the math.  The patterns that make something true - every time.  That's the thing about math that I like- the finality of mathematical proof.

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In reading this book, I have to step back and take a look at my classroom. I, like many, fall prey to too strong a focus on state assessments and even local testing data pressures. This is a reminder that I am teaching children not tests, and that should be my primary focus.

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It is a tricky thing as a math teacher to wander too far from the given curriculum.  Lockhard said, "In place of discovery and exploration, we have rules and regulations."  That's how I feel as a math teacher sometimes - the need to follow rules and regulations.  A co-worker once said her favorite lesson in math did not seem like an official lesson.  Her teacher had a watermelon seed spitting competition.  The kids (3rd or 4th grade) was to measure the distances and compare different students distance.
Problem solving in practice and fun. No official lesson and one she rememberd many years later.

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I am struck by this sentence:  "Teaching is not about information.  It's about having an honest intellectual relationship with your students.  It requires no method, no tools, and no training.  Just the ability to be REAL."
I love using Number Talks in my classroom. There is a posed problem, and the students discuss their thinking - how they solved the problem in their head. It is wonderful to hear the different approaches to solve the same problem. Then see other students apply the different strategies to see what works for them. It has. nothing to do with tests just how we think. I am guilty of no being intentional and making time daily for this activity.

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Math teachers glide along a slippery slope at times dealing with standardized testing. We live in an era of accountability, which weighs heavy on teachers and administrators. Lockhart noted that we teach a list of topics in a particular order (required curriculum). the slippery part is not letting the 'list' control the exploration, creativity and individual thinking of mathematics.

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The analogy of music and art being taught in such a rudimentary way - much like math. The sarcasm made this book easy to read and easy to see the problem with math class today. It actually made me sad to think of math class being boring and children not being able to explore and find new ways of thinking.

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I love that Devlin is so excited about the content of Lockhart's book. It gets me excited to read his thoughts on mathematics and see what I can apply to my teaching.

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I love the statement he made about how “art is not a collection of nouns, it’s a verb”. This sums up very easily the point that Lockhart has been making throughout the entire book. Math is something we do, not simply memorizing formulas, steps and acronyms in order to pass a test.

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This year I’m wanting to start using Flipgrid and a variety of jamboards to help get students expressing their thoughts in a variety of ways.

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This book really made me think about instruction in math. Often times educators are trained and encouraged to dissect our data from assessments to drive our instruction and fill gaps. Ultimately our end goal is to maximize our students performance and our data as a whole. We as educators feel success in this because we’ve been trained to feel like our students performance on an assessment equals how successful we are as an educator.
I do think the amount of stress placed on state testing does hinder how much time we are able to designate to exploratory thinking in our classrooms. We do need to allow students to explore because I feel like it has greater meaning when they have the ability to discover for themselves instead of simply being told what steps need to be done.

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I’ll be honest, when I got to the proof on page 71 I skipped right over it! It instantly brought back memories of high school geometry for me and I was not mentally prepared to revisit that. I had always felt such success in math until I took that class.
I do agree that math needs to be kept simple but with the required material that teachers are told to teach, it’s easy to see how a teacher could fall into the trap of “training” our students in processes instead of allowing for exploration.

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Reading the words “training methods” instead of teaching methods hit me like a ton of bricks. Teachers are using data to drive their instruction, studying the teks, and dissecting released tests in order to perfect instruction for the students to pass a TEST. So even though we may have an engaging and interesting classroom, we really are “training” our students in math.

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“Include mathematics as a major component of standardized tests and you virtually guarantee the education system will suck the life out of it!”  I absolutely love this passage in this section. The amount of pressure placed on students and teachers trying to prepare for a test is tremendous and exhausting.
Students need to be given the opportunity to discover concepts and express those discoveries to their peers. One thing I try to do in my class is have students elaborate on their answers instead of simply giving me the one word answer. This helps show their true understanding of the concept instead of simply reciting the steps.

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This section was spot on for me. Growing up I memorized proofs, formulas and steps without truly understanding why in the world I needed to do them.
I feel that allowing students to have an opportunity to explore a topic freely or with a small group is so beneficial. This allows them some struggle time which is helpful in teaching them that things will not always be instantaneous or necessarily easy to accomplish.
The section about “the realm of pure idea being fascinating, fun and free”  reminds me of how number talks can look in a classroom. These talks allow students to think freely and express their opinions and ideas without fear of being wrong. Often times these talks give spot on explanations in student language so that it can be received easier than if I’d explained it to them.

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I loved reading the section about artistry with paint. The nightmare of children being taught “art” as if they were robots, memorizing colors, strokes, brushes,etc.  School curriculum is constantly being sped up and pushed sooner. Students are coming to class with a set curriculum that they are being taught systematically with limited time for exploration and “play” with the concepts. I feel that this places such a stress on the students which drives a wedge between them and math instead of fostering a passion for it.

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I love that Devlin has such a passion for education in math. So much so that he felt compelled to share the opinions of Lockhart to not only people in the educational field, but parents of school aged children as well.
I’m very excited to read this book to hear Lockhart’s perspective on mathematics education.

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Hello! I’m Michelle Hall and I’m starting my 6th year in education. I’m a 3rd grade teacher with Mabank and am excited to read the opinions expressed in this book.

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I like how he says that he doesn't blame the teachers for following the curriculum, that he blames the culture  that produces such teachers.
We do know in our hearts that some of the information is, well, useless or inappropriate for the child. Therefore, they cannot understand what is being asked of them. It is not only frustrating to the student, but the teacher as well.
Unfortunately,  It has always been a push to have them ready for the next year.

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This is also a great resource for educators. It allows students to learn through the feedback of their peers, share ideas and discussions for new explorations of a problem.

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This is a great resource that explains the WHAT & WHY for kiddos and allows natural problem solving skills. A mentor teacher gave me hers and I reference it quite a bit.

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As an educator now, I look back and think, man, that would have been greatness to be asked - "Jodye, you want to be a teacher - what do you love to learn about yourself and what could you passionately teach someone?"
What a powerful tool for someone. Being reflective before you ever pursue something.
Teacher's are given a set of requirements based on those that do not even teach.

I loved that he says that questions are asked of third graders that are not authentic to a third grader. It is so true. He states that we should not worry about notation or technique. Play games! This would be an overhaul for sure. He also says that you learn by doing. Agreed! Teachers place this throughout their daily routine to help, but until we as a society place authentic learning as a priority, we will not ensure everyone LOVES learning and creates new knowledge /skills independently . It is amazing what we have stifled. How great would it be to ask a child entering school for the first time or even each year. What are you interested in learning?

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"A Mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with IDEAS!" Paul Lockhart

I was definitely taught that Art was Art and to be enjoyed as a past time or hobby and Math was more of a life skill that you need to be a successful adult. As an educator, collaborative learning has been the closest form of engagement that teachers can distribute among their students. Most students are intimidated by math because of those culturized stigmas. I agree with Lockhart's perception of Mathematics as an art form.

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I loved his opening argument with a musician's dream. The freedom to create being stolen. He explains that we regurgitate what we want a student to know, say and think. That explorations is the key to true creativity. It is also humorous that he repeatedly states that creativity is only presented in the higher grades. That students in elementary are not ready to create, play or explore.

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Thank goodness for Keith Devlin sharing this fascinating information to educators. Math as an art! I whole heartedly agree.

"I would have loved to have Paul Lockhart as my school mathematics teacher'" Keith Devlin

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Hey friends, my name is Jodye and I am a 3rd grade teacher at Central Elementary. I have taught Kinder, 1st, 2nd & 3rd with 3rd being the favorite! I look forward to reading this material and being able to reflect and collaborate with each of you.

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hello!  I am Cindy Clayburn.  I currently teach second grade math, but have taught 4th and 1st as well.  This is my 23rd year teaching and 10th year in second grade.  

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hello!  I am Cindy Clayburn.  I currently teach second grade math, but have taught 4th and 1st as well.  This is my 23rd year teaching and 10th year in second grade.  

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Number Rock videos
Quizz
Kahoot
Giant game boards

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This book was extremely insightful to me on how mathematics has been viewed for centuries.  It brought so much awareness to the methodology that I have been taught.  That mathematics is a set of rules and formulas to be taught and memorized to solve problems.  He expanded my thinking that it is a true art form, a way to creative express our imagination through an adventure of discovery.

Even through my education, I loved Math and was always drawn to it. I feel that I fell into the trap he talks about, because I was good at memorizing and following instructions; therefore, I felt I was always "good" at Math. Now that I am a Math teacher, I can try and change my thought pattern to reflect one of curiosity of the endless adventure mathematics can not only take me, but an adventure my students and I can travel on together. The opportunities are endless.

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Moving further along the lines of simplify when viewing mathematics is where the author elaborates his argument.  He metaphorically compares viewing a set of numbers to explorations of animals and their respective habitat in the wild.  He states that numbers can be viewed by their behavior and not just a set of rules to perform.

Because "Mathematical Reality" is an imaginary place, the depths to which ideas can be discovered is endless.   It's not that the set of numbers change, but the way they are seen and interpreted that is the real adventure.  To take a set of numbers and through observation and insight develop a proof that can be expressed, not just simply, but "simply beautiful."  Mathematics isn't something you do, it is something we can be.  It can shape our mind in ways that we simply cannot imagine if we don't understand that it is a true art form, a world of discovery waiting to be uncovered.

He advises us to leave the idea that Math is just a subject to be taught, and pick up the mentality that is a world to be discovered.

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The author argues that the curriculum fails the most at stunting the creativity of the students when Geometry is first introduced.  The rigorous work involved to prove an obvious observation is taking away the in the depth insight a student can achieve if all of the unnecessary work to prove the obvious was eliminated.

The idea is that we are overcomplicating the simple and obvious observations and not allowing the student room to expand their intuitive thought process.  There is a time and and place to prove your answer, but it isn't when the solution is obvious.

He states that the monotony of the current set of standards for K-12 Mathematics is damaging innocent students. and not allowing them to experience the fun that the freedom of exploration in regards to Math can bring.

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The rigidity of the mathematics curriculum is argued to be destroying the curiosity and creativity that math can bring to learners.  Other subjects, such as English and History has adopted the idea that instead of memorizing dates and rule, giving the meaning behind those dates and rules allow for a more permanent understanding versus and forced learning style.

He asks the question, "Why is mathematics stuck?" As a teacher myself, I often question the rigor that we are putting these students through. We have our hand forced to get information to the students so they can perform. Are they really learning? The goal is to get them to use their creativity to discover patterns and meaningful explanations vs just following a standard set curriculum.

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Mathematics, according to this author, has lost its enthusiasm due to the idea that it needed to become a mandate in the school system.  The argument posed by the author is the mathematics curriculum doesn't need to be reformed, but instead scrapped all together in order to preserve its real meaning...."to engage in an act of discovery...., intuition and inspiration."

He argues to be a successful Math teacher, vulnerability and realness from the teacher are a must.  We must share our enthusiasm with the students, and if we don't have passion and interest in mathematics, then maybe we are in the wrong subject area.  "Teaching is not about information.  It's about having an honest intellectual relationship with your students."
Teaching requires an openness and honesty, and an ability to share with them our excitement about learning.

The standardized set of rules and problems that we are giving students to solve is taking away their ability to be creative in their own problem solving. He argues that a young child needs to play games that cause them to be creative in their mathematical thought process versus making them memorize a specific set of rules and formulas to follow. I honestly can say that this is opening my eyes to the beauty of learning vs mandate of test scores.

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To bring the idea that Math in its truest form should be viewed as a form of creative expression much like Art is an idea that seems so profound to me.  The notion that our students just roll through, from grade to grade, memorizing a set of formulas to calculate the answer to a problem, but never understanding the why behind it.  As a whole, we have eliminating the curiosity of what Math really is and why it is important.

The points brought up in this chapter were so valid as to the decline in individuals who are interested in a career as a mathematician, much less respected.  I have honestly never viewed Math as an art form, and I teach it.  The idea that the fun that could be obtained when learning Math is more about understanding that it is a creative expression, versus just creating engaging lessons.  The understanding that we have been viewing Math in the wrong context of what it essentially is.....a unrecognized art form.

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The picture painted by the author regarding the current state of our mathematics curriculum is astounding and eye opening.   The visual that he created with his words of a fallen system geared to train children for adulthood through a series of steps is so accurate.

Are we just here to teach a subject or are we here to ignite a passion and an interest in learning and growing. Not just allowing children to follow a set of steps and levels before being allowed to "move on" to the next level. It begs to ask ourselves, are we training mathematical robots or are we inspiring our future. I feel that this picture allows the truth to be seen in a world blinded by statistics and scores.

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I love how he thought so highly of the written works of Paul Lockhard, that he felt compelled to seek him out to gain his permission to publish his works.  He believed so strongly that this information was so necessary for all mathematicians to read in order to become better educators.

I also love how attention was brought to "much debated" issues in the mathematics world, in order to get necessary conversations began and hopefully, change and inspiration to follow.

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I love how he thought so highly of the written works of Paul Lockhard, that he felt compelled to seek him out to gain his permission to publish his works.  He believed so strongly that this information was so necessary for all mathematicians to read in order to become better educators.

I also love how attention was brought to a "much debated" issues in the mathematics world, in order to get necessary conversations to begin and change to hopefully follow.

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I am a 4th grade Teacher in Mabank. This is my second year at a 4th grade teacher. I taught 2nd grade Math before I moved to 4th.

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I'm a first grade teacher at Mabank. Next year will be my 5th year in education and my 4th year teaching first grade.

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Wow, this really opened my eyes to more of the things I had already in my own mind had questions about before reading.

I know that children and adults tend to learn more when they are able to practice hands on with the skill they are trying to master. I love how in this book he talked about "proof" and that proof was just the explanation of why.

As a 1st grade teacher, it is very difficult for students to remember all of the terms we use in mathematics. Giving students time to do their own discovery on the concepts that we are teaching is so vital to their understanding (mastery) of that particular skill.

I will definitely be more mindful after reading this book about allowing students time to explore and deepen their understanding.

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"...the goal of the mathematician: to explain in the simplest;.."

I loved this particular part in the text. It encouraged me, instead of making me second guess the techniques and the ways I have been teaching.

Teaching students to take the basic knowledge that they have about a topic to formulate a "proof" of "why" it works in a way that it does. Not necessarily right and wrong answers, but does my proof explain my wondering?

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At this point in the book, I almost feel like he is "beating a dead horse." I get it you do not agree with anything that the public school has to offer when instructing students about mathematics.

I understand that the way we approach different "topics" within or course work sometimes if not always depends on how passionate that we are about that topic. Not saying that it is correct or incorrect, but I do feel that it all comes back to our experiences and what we value as important.

I remember absolutely hating Geometry in high school. As my teacher was teaching, it was like she was speaking in a foreign language to us all. She was very passionate about Geometry. From time to time, she would began writing on the chalkboard and get so lost in her own fascinations that she would forget that she had a classroom full sitting behind her waiting for a reasonable explanation.

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When reading I had thought of why we don't teach in the exploration of learning?
Do we hold our students back from exploring because our knowledge is just surface level and we don't really have an understanding for what we are teaching?

As a teacher, I see how year after year we can get stuck in a rut or routine of teaching. Students have to be taught this and that so that they can be successful. What we really mean is they need to know these things to be successful on the state or district tests. I feel like this is very true in ALL areas, not just mathematics.

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After reading, I couldn't help think about all the different types of word problems we teach even in 1st grade. We try to relate it to the students to make it more "engaging."

"You see, kids, if you know algebra then you can figure out how old Maria is if we know that she is two years older than twice her age seven years ago?" Something that I laughed completely out loud when I read.

I am guilty of just about everything that he has shared thus far. I believe in what he is expressing, but how can we let students explore in their learning and still cover all the MUSTS in our course?

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This section made me really think about the ways I have personally taught math. He is right. We teach math as a bunch of formulas and steps with absolutely no explanation as to why that even is. I can remember hearing a teacher explain to a student that asked why. I remember the response of that is just the way it is. So I can agree with what was stated in this section about the knowledge that math teachers actual have about the subject that they are teaching.

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During the reading, I had to stop a few times and think to myself. Why do we as educators fall into this trap? Is it is because of all the testing that we have these days?

This was very well expressed and is accurate. Children in most classrooms aren't given an opportunity to explore in their own learning. If we gave them an opportunity to do so, how much more would they learn about the things that we are teaching them? How much more would they actually learn instead of memorize?

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I loved reading about the passion that the mathematician had. Tracking down the author and publishing the article so that others would have a chance to read is very impressive.

I absolutely loved the quote. It is definitely something that we as teachers hope that our students will capture. I love watching my littles love for their favorite subject grow and often times the ones that have to work a little harder develop a passion for digging deeper into learning.

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I'm a 1st grade teacher in Mabank. Next year will be my 10th year in education and my 4th year teaching 1st grade.

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I’m so glad he showed some examples of “proofs.”  This is so different to how I was taught, but even so, I can remember “understanding” a concept at its most basic level and feeling excited.  I love finding patterns and connections but I’m also usually looking as to how I can make these patterns and connections useful.  Lockhart’s explanation that there doesn’t have to be a purpose beyond the joy of discovering them is freeing

If we could stop setting our curriculum at a pace for “honors” students and instead have time for discovery, then our “mathematicians” would naturally develop just like painters, sculptors, musicians, etc. develop.  We could let them soar in their imaginary math reality and allow others to develop concepts that have meaning to them.

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Gimkit
Kahoot!
Bingo games
Number rock videos
Teacher made

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Another book I read stated that our special population students most often need explicit teaching rather than discovery.  I’ve actually based my goal this year on that premise because it was something I had noticed while teaching resource math.  Now I wonder if this group just needs more time to discover than is available in our curriculum and if it could make a difference for their future understanding of math.

I’m teaching 6th grade and exponents are so abstract for them.  I’m going to let them play with counters and see if they can “see” the square and the maybe the “L

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Wow, he is passionate in his dislike of our current system. The course descriptions are rather humorous in that I have heard some of those thoughts over the years from students and adults. I’m glad he ended this chapter with a promise for help otherwise why keep reading just to hear more of his tirade.

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I agree that it is better to learn in context like reading and history. I also think that if instead of memorizing, let’s use grouping for example, we gave young students an opportunity to “need” to group, they would understand it better. Maybe set up a discover opportunity where they are trying to keep tract of an actual item and there are so many, it’s easier if they group them in bundles. Not all kids will discover it, but some will.

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I laughed when I read, “The mathematics curriculum doesn’t need to be reformed, it needs to be scrapped.”  I completely agree with most of his statements. I know I’m not truly a mathematician, but I like his idea that I can still help students “discover” math if given time and a release from high-stakes standardized testing.  It’s a horrible feeling when students do start asking those What if questions and I have to shut them down too soon or we will get too far behind.

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To think of mathematics as ‘the art of explanation” is intriguing and to teach math more like art would be a new paradigm.  I would like to believe his idea that the “mundane useful aspects would follow naturally as a trivial by-product.”

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Such a great analogy that goes right to the heart of the issue. I think many people would think it is extreme but that’s because we have been trained to think of math as absolutes with no room for creativity.

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I like the fact that another mathematician was impressed enough with the essay to track down the author and publish it on his own site.

The quote is one I wish public education would embrace. Most of us want to teach our students “to long for the endless immensity of the sea” (or the equivalent for our subject) but standardized test scores can’t measure this concept.

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I’m a 5th & 6th grade resource math teacher in Mabank. I’ve been teaching since the 80s and have been ‘lamenting’ for several years, so this study caught my eye.

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As you go through this book study I encourage you to keep this quote in mind.  You may not agree with everything said, the goal of this book is to get you to THINK about math instruction differently.  Feel free to agree or disagree and reflect honestly.

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This section made me think... What would happen if MATH was not one of the required “CORE” classes? Would more students like it?  Would more people be “good” at it?  What if we taught Math like we teach art? Are they right, is Math class stupid and boring? I’d love to take some of these thoughts back to my classroom.

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There were a lot of quotes that spoke to me in this section.  This is one of my favorites and something to remember when helping students to learn math.

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“JUST TALK TO THEM, AND MORE IMPORTANT, LISTEN TO THEM!”  P 47

I wish that more educators and educational leaders would listen to the students.  They are SMART, they know what they are doing.  WHY???

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"A proof should be an epiphany from the gods, not a coded message from the Pentagon."

I love this quote as it reminds me of having to do proofs in math classes. I was wanting them to be more like logic puzzles with explanations, but I was quicked redireced to ensure I was usting the "code" that necessary to be a proof. :(

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I HATED to read this quote.  "School has never been about thinking and creating.  School is about training, children to perform so that they can be sorted."  That broke my heart, but the more I think about it, no matter how hard I worked to make sure that my students were learning and creating their own knowledge in my classroom, ultimately they had to learn things on the district/state timeline not when they were ready to lear.

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This book and the thinking align somewhat with the XQ Super Schools Live Broadcast.  Start at (7:00 and you can stop at 8:35) Let's think about how we can change things to benefit the future of our students!

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Not a favorite resource, but a resource mentioned in the book.
www.maa.org

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With 29 geographically distinct community groups, MAA recognizes sections as a vital component of the MAA. Special Interest Groups Special Interest Groups, or SIGMAAs, provide MAA members who share common mathematical interests with opportunities to organize and interact as professionals.Committees MAA Committees govern and lead the association.