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What were your impressions of “The Myth of the Mathematical Gifted Child”?

What strategies can you use to actively overcome inequalities in your class?

Is Jo saying that with the right attitude and work ethic, we could all be Einstein?  What do you think?

Jo has some strong feelings about homework and its place in the math classroom.  What are your thoughts/feelings about homework?

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Why Math Education in the U.S. Doesn’t Add Up

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Research shows that an emphasis on memorization, rote procedures and speed impairs learning and achievement 

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Math fact fluency........something that has been a struggle for so many of my students in the past. I am not one of those who believes that math fact fluency is the most important math skill and spends countless hours using flashcards with my students (I do ask them to practice their math facts at home using either archaic flashcards or a fun app). I prefer to spend our time together problem solving and applying mathematics in real wold situations. However, I do realize that students need to be able to compute math facts fairly quickly because, to some degree, fluency does interfere with problem solving.

In the past I have done what many teachers do: give my students timed practice tests. This works great for most of my students. Their math fact fluency improves. However, for a handful of my other students, their math fact fluency never improves. Those kids who get stressed out by a timer, who begin sweating the moment you give them their test, whose hearts are racing by the time you say, "Flip your paper over and begin", who are defeated when they see a page full of problems, and who get more and more jammed up with every completed paper they hear flipping over, this simply does.not.work. I have even had some students whose scores decreased over time. This doesn't mean they knew fewer facts, it meant that they could not handle the anxiety of timed math tests. For adults, we probably think of a timed math test as no big deal. But to a child, a timed math test could be the fine line between having a great day at school or feeling like a complete failure.

I have also given timed math facts because usually it is a skill that is included on a grading rubric for the report card. "Students will correctly solve 30 addition facts in 3 minutes." Or something like that. How else can you assess this skill without giving a timed math test?

With all that said, I still give my anxiety-ridden students timed math tests, but who says the timer has to be counting down? Not even does the grading rubric state that the time has to be winding down on the student. I don't remember who gave me this idea, but the moment I heard it I was like, "How ingenious! This is perfect!"

Backward timing. Instead of the timer counting down to a buzzer (which may be the most dreaded sound in the world to some kids...or whatever sound you like to use to signal time's up. I used my ActivBoard to count down and at the end it would play "Rocky Top" instead of a buzzer...teaching in Georgia this probably WAS the SINGLE worst noise some of my students could have heard!), the timer begins at "0" and counts up until students are done. Once my students figured out there would be no "BUZZZZZZZZZZZ!" or, in my case, "Oh, Rocky Top, you'll always be.....", their shoulders relaxed, sweat beads disappeared, and over time they began improving. So how does it work?
**Let me preface by saying that I only use backward timing with my students who are stressed out by the traditional timed test and those who are not showing improvement because I like to use this as a strategy for those kids.**

First, you have to decide how many correct facts they need to get correct. I had it easy because it was on the grading rubric. Then you need to discuss this with the student and set a SMART goal with the student. Students need to be let in on goal setting, especially since it involves them! In case you aren't sure what a SMART goal is, it is a goal that is Specific, Measurable, Attainable, Realistic, and Timely. You could set the goal to be what is expected for the end of the grading period, but for some kids this is so far off from their current performance, you need to lower it so they feel accomplished by reaching smaller goals and gain confidence.If you aren't sure what a reasonable goal would be, talk to other teachers first to see what they think a reasonable goal BEFORE you meet with the students.

So, as an example, let's say the expectation for the end of the quarter is that students will answer 30 addition facts correctly in 3 minutes. Let's say the student currently answers 7 correct in 3 minutes because you will need to do a traditional timed-test to get that baseline. It is the beginning of the quarter. Setting a goal of 30 correct in 3 minutes by the end of the quarter is specific, but is certainly isn't timely. That is 9 weeks away. A better goal might be, "Johnny will answer 15 addition facts correctly in 3 minutes" with a goal date of 3 weeks away. Then in 3 weeks, or before if he has met this goal, you can increase the goal to, "Johnny will answer 25 addition facts correctly in 3 minutes" with another goal date of 3 weeks away. If Johnny did not answer 15 correctly in 3 minutes at the end of 3 weeks, you will need to see how far away he is from that goal. If he is at 12, 13, or 14, clearly he is improving and you may want to extend the date by a week.

Second, once you have your baseline and goal set, you need to decide how many problems are you going to put on one page. If your goal is for students to get 15 problems correct in 3 minutes, then you don't want to slap 50 problems on a page. Remember, some kids get overwhelmed at the sheer sight of a page full of problems because they know they struggle and this makes them feel defeated before they even begin. I would put about 20 problems on that page for them. Or you can always put the goal on the page, and in this case it would be 15 problems on the page. The problem with this, though, is if they miss just 1, they will never meet their goal. Accuracy is important (which is why I record how many they got correct out of how many they attempted), but even we as adults will make errors every now and again. We can't expect our students to be perfect every time. You may ask, "Well, what if they are taking 3 minutes and 15 seconds to solve all 20 correctly? How am I supposed to know if they can solve 15 correctly in 3 minutes?" You may want to tell students to stop for a moment while you stop the timer at 3 minutes, and ask them to circle the last problem they completed, then resume the timer and let students finish. This way you can see where they were at at the 3 minute mark, and they are able to complete the page.

Third, explain to students how their new timed tests will work. Let them know that they will have as much time as they need to finish the page. If they need 3 minutes, 4 minutes, or 5 minutes, let them know they will not hear a buzzzzzzzzzzzz. When they are finished, you will stop the timer that is counting up, and you will record that time (or if you have several students who are using backward timing, you may want them to raise their hand when they are finished so you can write the time down, or you may want them to take the test in small groups if you have several to make it more manageable). Let me add, that if students are taking longer than 5 minutes to complete the page, you have too many on the page for them. I would scale back the number on the page. If your goal is 15 and you have 20 on the page, I would probably lower the goal and lower the number of problems on the page. Remember, you want your students to feel successful, so if that means setting a goal that is more easily attained at first, then do that.

Fourth, after students have finished the test and you have graded the tests, fill in a graph WITH the students. I like to record how many they got correct out of how many they attempted because this helps me see if their accuracy is improving. If you don't have one to use, check out the FREEBIE I made below! Graphing the results allows the students to see their progress. It gives them motivation to do better next time, and students soon realize they are no longer really competing with a timer but with themselves, trying to do better with each test they take. Intrinsic motivation can be a powerful thing!

Here is what the graph might look like. You may want to alternate colors each time so you can more easily see the different days, or you may want to keep the same color.

Repeat, as needed. Decide how often you will give students this timed test. Twice a week? Three times a week? I don't recommend every day, but that's just me. I like 3 times a week. But choose what is best for you and your students. Once students reach their goal, celebrate it with them, then create a new one with them and start a new graph. Repeat the process until they are meeting expectations. And, honestly, when they are meeting expectations, I may not set goals with them anymore because they are where they need to be, but I do continue to use the backward timing test with them. They do well with it, and I am not going to use the traditional timed test anymore.

How can you use the backward timing test for grading purposes? Well, you are writing down the time that the students finish, so if they finish in under 3 minutes (or whatever the given time is), and they correctly answer 30 (or whatever the minimum correct is), you have evidence of them meeting expectations. Let's say they don't finish in under 3 minutes, and the grading rubric says they are "in progress" of meeting expectations if they correctly answer 20-29 in 3 minutes. You are going to have to tell students to stop at the 3 minute mark, you will stop the timer, ask students to circle the last problem they finished, then resume the timer and let students continue working until they are done. You can go back to using the traditional timed test for report card assessments, but I feel as though I would be setting up my students for failure because I know they get anxiety and we haven't been using the traditional method. The grading rubric doesn't say it has to be a countdown timer.

If you want to use the FREEBIE I made click here!

I hope I have given you an alternative way to help increase your students' fact fluency in lieu of using traditional timed tests. Backward timing does take a little more effort because you have to set goals and decide how many to put on a page and keep track of when students finish, but I have seen such great growth out of my most struggling students, that I feel it is worth it.

I would love to hear your thoughts and comments about your experiences with backward timing. Thanks for reading!

You can join my email list by using the form below. I will send you an exclusive freebie for joining, and you will stay updated with the latest happenings at Brainy Apples 🙂

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Improving Math Fluency Using Backward Timing - Brainy Apples

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Math fact fluency........something that has been a struggle for so many of my students in the past. I am not one of those who believes that math fact fluency is the most important math skill and spends countless hours using flashcards with my students (I do ask them to practice their math facts at home ...

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“ “Math is a universal language.” This is a common belief, and there is some truth to it. Educators need to understand that not all cultures use the same numerical symbols and function signs”

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PreK-12 math teachers often rely upon worksheets and timed drills to develop math fact fluency in students. While research on the effectiveness of worksheets and timed drills varies, some research indicates that such approaches to developing fact fluency - particularly drilling for speed prematurely - can negatively affect students' feelings toward math.

While drill, both timed and untimed, can be appropriate in certain circumstances, other approaches can foster more positive attitudes towards mathematics while also developing math fact fluency. Learning science research can help us identify those strategies and integrate them into math instruction. By harnessing what we know about how learning happens, we can design learning experiences that are not only effective in promoting positive outcomes, but also enjoyable for students.

Fluency requires students to notice relationships and use strategies. It's about the speed at which a student produces an answer, but it's also about the methods they used to produce that answer and the flexibility they have in their thinking.

However, fluency should not require rote memorization. Instead, students should either have a fact meaningfully memorized or be able to produce that fact through a highly efficient, automatically executed, strategy. A student has mastered a math fact if they can produce an answer within 3 seconds, through either recall or a highly efficient strategy application (Kling & Bay-Williams 2015).

First, it's important to understand the definition and characteristics of math fact fluency. How do we know if a student is truly fluent in math facts? Gina Kling, mathematics education researcher and author, points us to two core aspects of fluency:

According to Kling, fluency is developed when students have the opportunity to deliberately and explicitly move through three developmental phases by building reasoning strategies. In general, children begin solving math facts through counting (Phase 1), progress to using reasoning strategies to derive unknown facts (Phase 2), and finally, develop mastery with their facts (Phase 3). If students simply memorize math facts as rote facts (in other words, skip Phase 2), they might fail to develop important conceptual understandings, which puts them at a disadvantage when attempting to engage in more advanced math work (Kling & Bay-Williams 2015). "Fluency allows us to retrieve and use math facts with a certain automaticity, having internalized them from prior experience," according to Learning Scientist Claire Cook. "It's not about memorization per se - just as a master chef doesn't go about selecting the right ingredients in the right amounts because he's memorized recipes, but rather because he knows what he's doing at that level without thinking about it too hard or too explicitly."

Premature drill for speed and other practices that rely heavily on rote memorization should be replaced or supplemented with practices that promote reasoning - and, in many cases, replace math anxiety with math positivity. Learning science research has enabled us to identify a variety of practices that encourage students to develop conceptual understandings, reasoning skills, and fact fluency - all in a math-positive learning environment. We have deliberately integrated such practices intoEveryday Mathematics, a program built on decades of research about how learning happens. Here are a few of the research-based practices Everyday Mathematics employs to develop fact fluency and foster math positivity:

Interleaving

We know that developing fact fluency requires practice. But what kind of practice is most effective? While timed worksheets with many similar facts can sometimes foster math anxiety, particularly when they are used too early in the learning process or when they are tied to high-stakes outcomes, certain kinds of worksheets do have their place in math instruction - particularly when they employ a technique called interleaving. Interleaving essentially involves practicing skills in a rotating or variable order rather than starting with one skill and mastering it completely before moving on to another. For example, a math facts worksheet or activity might include both addition and subtraction problems, including some set within story contexts. Or a facts game might include practice with all four operations (addition, subtraction, multiplication, and division). Research shows that this kind of practice is more effective for mastering skills in a variety of domains (sports, even music), including math. For more on interleaving and the research mentioned above, see:

Games

Returning to the phases of fact fluency development: it's critical that students do not skip phases that give them the opportunity to develop reasoning skills. Explicitly teaching reasoning strategies doesn't mean simply teaching a specific strategy and then asking students to use it to solve a problem - in that case, students may use the strategy in a rote way, without understanding why it works. Instead, explicitly teaching reasoning should involve encouraging students to develop and share efficient strategies they can use and allowing them to practice. Games provide students with the opportunity to practice these reasoning strategies and move through the developmental stages of mastering math facts at their own pace (Bay-Williams & Kling 2014). And because games are fun, students play them willingly and even on their own, thus getting much more practice than they would from worksheets.

Formative Assessment

Interviews, where teachers have open discussions with students about the strategies they use to solve a problem and utilize tools to chart student progress

Observations, where teachers might watch students employing strategies during math game play and record their observations on structured tools

Journaling, where students have the opportunity to create a written record of their reasoning

Finally, consider how formative assessment strategies contribute to math positivity and support instructional iteration. Timed tests can limit an educator's ability to understand what strategies a student uses to solve a problem. To conduct formative assessment in a way that advances math fluency without promoting math anxiety, Gina Kling recommends a few practices:

According to Kling, quizzes can have their place in a math-positive, productive formative assessment strategy - as long as they focus on efficiency and strategy use (Kling & Bay-Williams 2014).

For a deeper understanding of the way learning science serves as a foundational research basis for Everyday Mathematics, see:

References

Bay-Williams, Jennifer M., and Gina Kling. "Enriching Addition and Subtraction Fact Mastery through Games." Teaching Children Mathematics, vol. 21, no. 4, 2014, p. 238., doi:10.5951/teacchilmath.21.4.0238.

Bay-Williams, Jennifer M., and Gina Kling. "Math Fact Fluency: 60+ Games and Assessment Tools to Support Learning and Retention." ASCD and NCTM: 2019.

Kling, Gina, and Jennifer M. Bay-Williams. "Assessing Basic Fact Fluency." Teaching Children Mathematics, vol. 20, no. 8, Apr. 2014, p. 488., doi:10.5951/teacchilmath.20.8.0488.

Kling, Gina, and Jennifer M. Bay-Williams. "Three Steps to Mastering Multiplication Facts." Teaching Children Mathematics, vol. 21, no. 9, 2015, p. 548., doi:10.5951/teacchilmath.21.9.0548.

Pan, Steven C. "The Interleaving Effect: Mixing It Up Boosts Learning." Scientific American, Springer Nature America, Inc., 4 Aug. 2015, www.scientificamerican.com/article/the-interleaving-effect-mixing-it-up-boosts-learning/.

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A Research-Based Approach to Math Fact Fluency (That Also Promotes A Love of Mathematics!)

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PreK-12 math teachers often rely upon worksheets and timed drills to develop math fact fluency in students. While research on the effectiveness of worksheets and timed drills varies, some research...

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Research from the Cognitive Science, Explicit Instruction and Cognitive Load Theory sections of this page stress the importance of students having key mathematical facts ready to be retrieved from long term memory so they do not take up valuable space in working memory

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Research from the Cognitive Science, Explicit Instruction and Cognitive Load Theory sections of this page stress the importance of students having key mathematical facts ready to be retrieved from long term memory so they do not take up valuable space in working memory

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arrow_back Back to Research

Fluency with Maths Facts

Research from the Cognitive Science, Explicit Instruction and Cognitive Load Theory sections of this page stress the importance of students having key mathematical facts ready to be retrieved from long term memory so they do not take up valuable space in working memory. For want of a better expression, I will refer to this as developing fluency with maths facts. Over the last few years, I have tried to get students to develop this fluency via number talks, and more recently through drills/rote learning. This section is my attempt to find research-based evidence on the merits of these, as well as to address the issue of whether fluency is important when students have calculators and mobile phones!

For an overview of how Number Talks work, I would recommendthis article, together with this video from Jo Boaler.

For an overview of drilling, I would recommend reading the following two blog posts by teachers at Michaela Community School:Dani Quinn and Hin-Tai Ting.

Research Paper Title: Assisting Students Struggling with Mathematics
Author(s): Institute of Education Sciences
My Takeaway:
The second appearance of this brilliant paper which is full of research-based evidence and practical strategies for assisting students who struggle with mathematics. This time we turn to a recommendation that is directly relevant to this section:
Recommendation 6 - Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.
The paper then goes on to suggest three practical strategies to aid with this:
1. Provide about 10 minutes per session of instruction to build quick retrieval of basic arithmetic facts. Consider using technology, flash cards, and other materials for extensive practice to facilitate automatic retrieval.
2. For students in kindergarten through grade 2, explicitly teach strategies for efficient counting to improve the retrieval of mathematics facts.
3. Teach students in grades 2 through 8 how to use their knowledge of properties, such as commutative, associative, and distributive law, to derive facts in their heads.
For me, strategies 1 and 2 are all about drilling, whereas strategy 3 lends itself well to the concepts of Number Talks which will be covered later on in this session. The overall conclusion is clear - a knowledge of basic mathematical facts is necessary in order for students to achieve success in mathematics, and as teachers we should help students gain this by making a focus on fluency a regular part of each lesson.
My favourite quote:
These studies reveal a series of small but positive effects on measures of fact fluency128 and procedural knowledge for diverse student populations in the elementary grades.In some cases, fact fluency instruction was one of several components in the intervention, and it is difficult to judge the impact of the fact fluency component alone. However, because numerous research teams independently produced similar findings, we consider this practice worthy of serious consideration. Although the research is limited to the elementary school grades, in the panel's view, building fact fluency is also important for middle school students when used appropriately.

Research Paper Title: Improving Basic Multiplication Fact Recall for Primary School Students
Author(s): Monica Wong and David Evans
My Takeaway:
This is a great overview into the importance of times tables, and the strategies students employ. There are a number of key points addressed:
1) Whilst conceptual understanding of the times tables is important, it is not sufficient. Automaticity must be achieved in orer to free up space in working memory to solve more complex problems (I love this quote: The importance of automaticity becomes apparent when it is absent. Lessons may stall as students look up facts they should recall from memory. Thus conceptual understanding is necessary, but insufficient for mathematical proficiency)
2) Basic multiplication facts are considered to be foundational for further advancement in mathematics. They form the basis for learning multi-digit multiplication, fractions, ratios, division, and decimals Many tasks across all domains of mathematics and across many subject areas call upon the recall of basic multiplication facts as a lower-order component of the overall task. To enable students to focus on more sophisticated tasks such as problem solving, proficiency in basic facts and skills is an advantage. Without procedural fluency and the ability to recall facts from memory, the student's focus during problem solving will be on basic skills rather than the task at hand, thus drawing attention away from the learning objectives of the task. If the student cannot perform these basic calculations without the need to use calculators or other aids, higher-order processing in problem solving will be impeded
3) When answers are predominantly recalled from memory, the student should be able to answer approximately 40 basic mathematics questions correctly in one minute
4) The order in which facts are introduced and sequenced can assist students to become proficient in learning and recalling basic multiplication facts. Facts that can be learned easily should be presented first during practice (e.g., 0, 1, 10, 2, 5, 9), then they should be followed by the more demanding multiplication sequences (e.g., 4, 7, 3, 8, and 6). Students should also be proficient at counting from 1 to 100 and be able to skip count.
5) To improve speed of fact recall, students should be given a specific time to respond to a question or a constant time delay, typically starting at five seconds and gradually reducing to one and a half seconds. Reducing the response time forces the student to abandon inefficient counting strategies and attempt to retrieve the answer from memory
6) Practice on computers is said to afford some advantages over more traditional delivery modes. Students can progress at their own rate and practise using varying representations (horizontal or vertical). Feedback is immediate and scoring systems automatically monitor progress. Students who used computers as part of their usual instruction generally learn more in less time and retain the information for longer
7) A constant time delay approach is effective for developing fluency in students with learning difficulties. for The students were then taught 15 unknown multiplication facts using computer-assisted instruction based on a five second constant time delay procedure. The results indicated that the constant time delay procedure was an effective method of teaching multiplication facts to those students.
8) Interspersion of known and unknown facts in each practice session increases the speed at which facts are committed to, maintained in, and retrieved from long- term memory. It also assists in the remediation of errors from previous sessions and improves the speed of retrieval of known facts from long-term memory.
9) Procedural and conceptual misunderstandings need addressing. Some students consistently calculated the A x 0 multiplication fact incorrectly by writing the value of the A, whereas its commuted counterpart, 0 x A was answered as 0. The revision of concepts and procedures needs to be included in the practice sessions to ensure students possess the necessary pre-skills to answer questions accurately. This can be achieved through building conceptual knowledge though the use of manipulatives, through building 'generalisable' rules, and through providing practice in the use of virtual manipulatives. Virtual manipulatives allow the introduction of concepts, and provide practice and remediation
10) While proficiency in multiplication facts is important, there are also other basic facts that require practice to maintain ongoing development of mathematical proficiency, such as addition and subtraction facts. Therefore practising other basic arithmetic skills could be included in practice sessions like those used in this study. Initially, multiplication facts may be practised separately to promote proficiency; later they could be mixed with other facts to allow students to become more proficient in selecting from and discriminating between operations. Although it is a more
My favourite quote:
In summary, the belief that the development of basic multiplication fact recall is enhanced by practice has been supported. Results generalised to the study group have shown that a systematic practice of basic multiplication facts by interspersing known and unknown facts improved students' recall of these facts for all but a few. For these few students, revisiting multiplication concepts may be necessary. Without this improved recall of basic multiplication facts, working memory is consumed by the most fundamental of problems. Releasing working memory capacity allows students to tackle more difficult tasks such as multi-step problems or questions demanding higher-order thinking.

Research Paper Title: Developing Multiplication Fact Fluency
Author(s): Jonathan Brendefur, S. Strother, K. Thiede and S. Appleton
My Takeaway:
This paper investigates if presenting times tables with different representations can help students develop fluency in them more effectively than drilling. The researchers used third, fourth and fifth grade students, splitting them up into two groups. the Strategy group received instruction based on cognitive and social-interactional framework for fluency development; whereas, the Drill group received fluency instruction for basic multiplication facts using an approach emphasizing memorization and rehearsal techniques typically practiced in schools. The techniques used in the Strategy group were Based around Bruner's theory of Modes of Representation, and used the following procedure:
1. Strategy group students began by building arrays with physical models (e.g. tiles) and finding arrays in pictures as well as the surrounding environment. Students then drew diagrams of the arrays on either grid paper (to structure the drawings) or freehand.
2. Students then transitioned from arrays to using a 12x12 blank grid as a multiplication table as both a way to list facts they knew but also as an example of an array. Students overlaid their derived facts strategies (e.g. 8x5 + 8x1 to recall 8x6) on these 12x12 multiplication grids.
3. Eventually, students' materials were removed and they were engaged in what was called "fluency-talks". Students sat on the floor with no writing materials or manipulatives available and were presented with various facts. They had to discuss as a
class how they might use related facts to solve the unknown facts presented on the board.
4. To culminate the strategy group's fact development, pairs of students created sets of strategy cards, which were essentially multiplication flash cards (with a fact on the front of the card), but strategy cards included derived facts strategies the pair preferred written on the back of the card. The pairs would alternate describing two or three facts strategies for each card's fact.
The results were that sstudents receiving instruction through drill and rehearsal gained on average 0.79 facts per minute over the five weeks, with positive gains of 2 facts per minute in fourth grade 2.36 facts per minute in fifth grade. Students receiving instruction grounded in the social-interactional approach demonstrated an average gain of 6.08 facts per minute, with the highest gains in fourth grade of 6.65 facts per minute. This suggests a different approach is needed to introduce students to times tables at a younger age, or possibly to help those older students who do not have a firm grasp of times tables develop the fluency they need.
My favourite quote:
The evidence from this study also demonstrates that students receiving instruction grounded in a framework built upon Bruner's Modes of Representation combined with social- interactional elements significantly outperform students who receive instruction grounded in a behavioristic theory of learning. These instructional activities designed for the Strategy group emphasized strategic thinking and mathematical relationships between multiplication facts and created greater and more consistent gains in fact fluency than activities emphasizing memorization and repetition. Although Brownell (1935) had similar conjectures and findings, more recently Russell (2000) found that students build an understanding of multiplication facts through problem solving and, then, sharing and examining their own strategies. One of the implications to this finding involves memory: one might presume that students will remember any piece of information (including multiplication facts) more accurately if the information is connected to already easily-remembered information (Hiebert & Carpenter, 1992). In the case of multiplication facts, this would mean that students who know 8x5 would ideally spend instructional time learning to use that knowledge to solve 8x4, 8x6, and 8x7 and so forth. Time spent building flexibility with facts would in turn produce fluency with facts as students would have related strategies to refer to should memory fail them. This would contrast with a similar amount of time being spent by students trying to commit 8x4, 8x6, 8x7, etc. to memory.

Research Paper Title: Automaticity in Computation and Student Success in Introductory Physical Science Courses
Author(s): JudithAnn R. Hartman, Eric A. Nelson
My Takeaway:
Students (and parents!) often say to me something along the lines of: "why do I need to work that out when I can just bang it into my calculator/smart phone?". To be honest, my answers over the years have not been great. Fortunately, with the findings of this fascinating paper, I now have some evidence up my sleeve. This paper looks at the effect the move away from the practice of key mathematical skills in US high schools has had on students taking science degrees. The authors find that between 1984 and 2011, the percentage of US bachelor's degrees awarded in physics declined by 25%, in chemistry declined by 33%, and overall in physical sciences and engineering fell 40%. Data suggest that these declines are correlated to a K-12 (kindergarten to the end of high school) de-emphasis in most states of practicing computation skills in mathematics. The authors cite recent studies in cognitive science that have found that to solve well-structured problems in the sciences, students must first memorize fundamental facts and procedures in mathematics and science until they can be recalled "with automaticity," then practice applying those skills in a variety of distinctive contexts. Even with access to a calculator, students working memories can become overloaded, which can prevent them being able to solve more complex problem and hence inhibit learning. To explain this further, I can do no better than to quote from the paper itself:
My favourite quote:
As one example, if as part of a calculation "8 times 7" cannot be recalled, the calculator answer of 56 must be stored in working memory so that it can be transferred to where the calculation is being written. On a problem of any complexity, that storage may bump out of working memory an element that is needed to solve the problem. An answer from a calculator takes up limited working memory space; an answer recalled from long term memory does not.

If arithmetic and algebraic fundamentals are automated, when examples are based on simple ratios or equations, room is available in novel WM for the context that builds conceptual understanding, and problem solving builds an intuitive, fluent understanding of when to apply facts and procedures (Willingham 2006). Conversely, if a student lacks "mental math" automaticity, conceptual explanations based on proportional reasoning or "simple whole-number-mole ratios" will likely not be simple. If a student must slowly reason their way through steps of algebra that could be performed quickly if automated, the "30 seconds or less" limit on holding the goal, steps, and data elements of the problem in working memory ticks away.

In recent years, the internet has facilitated the finding of facts and procedures, but new information occupies the limited space in novel WM that is needed to process the unique elements of a problem. Unless new information is moved into long term memory by repeated practice at recall, during future problem solving that new information will again need to be sought, and when found, it will again restrict cognitive processing (Willingham 2004, 2006).Fluency without Fear
Author(s): Jo Bolaer
My Takeaway:
Jo Boaler agrees on the importance of knowing mathematical facts, but argues that the memorisation of math facts through times table repetition, practice and timed testing is unnecessary and damaging. Boaler goes further to argue that when teachers emphasise the memorisation of facts, and give tests to measure number facts, students suffer in two important ways. Firstly, for about one third of students the onset of timed testing is the beginning of math anxiety, which can block working memory and prevent learning taking place. Secondly, they can put students off mathematics for life. Boaler advocates a move away from seeped and memorisation, and towards encouraging students to work with, explore and discuss numbers. This will allow them to commit important facts to memory, but in a fun and engaging context. She then goers on to describe Number Talks, as well as some other strategies for developing a sense of number. My concern with this is founded in personal experience of using and observing Number Talks extensively over the last three years - whilst it is supposed to be the lowest achieving students who gain the most from them, I have found they are often held back from fully participating by their lack of knowledge of key maths facts. It ends up being the teacher suggesting strategies, students copying them down and seemingly understanding then, only for them to be unable to transfer them to a new calculation in the next Number Talk. Contrast this to higher achieving students, who can happily break apart and put back together numbers in wonderfully efficient ways, and really seem to get a lot out of the Number Talks, sharing and discussing each others' approaches. I believe all students can (and should) get to this level, but I don't think this can be done through Number Talks alone. Students cannot develop fluency with numbers without these facts in long term memory. I have found that the more comfortable students get with their times tables and number bonds, the more readily they can think of, and successfully carry out the kind of efficient strategies that Number Talks promote. I draw the analogy with problem solving - you cannot teach problem solving by just showing students how to solve problems. Likewise, I believe you cannot teach fluency with numbers simply by showing students how to be fluent.
My favourite quote:
High achieving students use number sense and it is critical that lower achieving students, instead of working on drill and memorization, also learn to use numbers flexibly and conceptually. Memorization and timed testing stand in the way of number sense, giving students the impression that sense making is not important. We need to urgently reorient our teaching of early number and number sense in our mathematics teaching in the UK and the US. If we do not, then failure and drop out rates - already at record highs in both countries - will escalate.

Research Paper Title: Developing Fluency with Basic Number Facts: Intervention for Students with Learning Disabilities
Author(s): Katherine Garnett
My Takeaway:
This paper provides a fascinating insight into attempts to develop fluency with students with learning difficulties. The authors cite that on timed assessments, 5th grade students with learning disabilities completed only one-third as many multiplication fact problems as their non disabled counterparts. Interestingly, the students with learning disabilities were very much slower, but not significantly less accurate, than their non-disabled peers. Additionally, they demonstrated basic conceptual understanding of the basic maths operations. Thus, many students with learning disabilities establish basic understanding of the number relations involved in basic facts, but continue using circuitous strategies long after their non-disabled peers have developed fluent performance. And as we have seen from Cognitive Load Theory, it is the inability to recall facts from long term memory that will hinder such students in solving more complex problems as their working memories will become overloaded. However, the author points out that becoming fluent is not simply a case of remembering a load of facts - it is about forming a well-developed network of number relationships, easily activated counting and linking strategies, and well-practiced navigational rules for when to apply which maneuver". The author goes on to argue that the only way to develop these skills is though several years of frequent and varied number experiences and practice, and drilling is not enough. The author recommends presenting Challenge Problems and discussing strategies, much in the same way as Number Talks, and a really useful collection of prompt questions is provided. I am still not convinced that strategy comes before knowledge, but I am convinced that simply knowing facts is not enough given the immeasurable number of combinations of facts that would be required to answer every single maths problem!
My favourite quote:
In investigating the effects of challenge problems, many of the guidelines offered here would be useful, especially the emphasis on interactive, oral work. Regularly including challenge problems in student/teacher interactive math work could well promote the "mental math" prowess needed by so many students with learning disabilities who cling to number lines and paper-pencil routines.

Research Paper Title: Mastering Maths Facts: Research and Results
Author(s): Otter Creek Institute
My Takeaway:
This authors provide a really good summary of the key findings from this paper: " Learning math facts proceeds through three stages: I) procedural knowledge of figuring out facts; 2) strategies for remembering facts based on relationships; 3) automaticity in maths facts-declarative knowledge. Students achieve automaticity with math facts when they can directly retrieve the correct answer, without any intervening thought process. The development of automaticity is critical so students can concentrate on higher order thinking in maths. Students who are automatic with math facts answer in less than one second, or write between 40 to 60 answers per minute, if they can write that quickly. Research shows that math facts practice that effectively moves students towards automaticity proceeds with small sets of no more than 2 -4 facts at a time. During practice, the answers must be remembered rather than derived. Practice must limit response times and give correct answers immediately if response time is slow. Automaticity must be achieved with each small set of facts, and maintained with the facts previously mastered, before more facts are introduced. Suggestions for doing this with flashcards or with worksheets are offered." So, this paper stresses the importance of the automaticity of maths facts, and advocates the use of timed drills to achieve this. I found the suggested order to be interesting: strategies comes before automaticity. This did not seem to fit in with my experiences described in my Takeaway on the Boaler article above. However, digging a little deeper, the authors suggest that strategy comes first for addition and subtraction facts, but memorisation is needed for multiplication facts. As most Number Talks require an element of times table knowledge, this certainly fits in with my experiences. The paper also makes one point that I feel is of paramount importance: if students are relying on a counting strategy to solve basic maths facts (such as counting on fingers for multiplication), then no amount of drilling will help them transfer these facts to long term memory. For students relying on these strategies, drills can become a painful process. Students need to be moved away from inefficient strategies as soon as possible, and the authors suggest that timed drills might be a way to achieve this. Of course, we need to bear in mind Boaler's important point about the dangers of maths anxiety (see Anxiety section for more) - but there are plenty of ways to make this kind of drilling fun and non-threatening, as the Michaela blogs demonstrate.
My favourite quote:
What is required for students to develop automaticity is a particular kind of practice focused on small sets of facts, practiced under limited response times, where the focus is on remembering the answer quickly rather than figuring it out. The introduction of additional new facts should be withheld until students can demonstrate automaticity with all previously introduced facts. Under these circumstances students are successful and enjoy graphing their progress on regular timed tests. Using an efficient method for bringing math facts to automaticity has the added value of freeing up more class time to spend in higher level mathematical thinking.

Research Paper Title: Mental calculation methods used by 11-year-olds in different attainment bands
Author(s): Derek Foxman
My Takeaway:
This is fascinating. A sample of 247 eleven year old children was divided into three bands of attainment as measured independently by their scores on a written test of concepts and skills.They were then given a series of mental arithmetic questions to answer during one-ton-one interviews, and crucially asked to explain how they arrived at their answer. An example of one of the questions children were given is "I buy fish and chips for £1.46. How much change should I get from £5 ?". Their responses were classified as either being Complete (e.g. £5 − £1 − 46p), or Split (e.g. £5 − £1; £1 − 46p). There were three main findings:
1) Complete number methods were far more successful than Split number methods, even more so in the two lower attainment bands than in the top band.
2) For all three questions, Complete number strategy use declined from the Top to the Bottom attainment band, while Split number strategy use increased from Top to Bottom.
3) Complete number strategies were used far more frequently than either Split methods or the Algorithm for working out the in context questions.
I was surprised by these results - for me the Split strategy seems more efficient, and is how I approach the problems. But, then again, I can see how it is prone to error. Relating this to Cognitive Load Theory - if a students does not have facts and processes stored in long term memory, then think of the cognitive demands placed on working memory when trying to process a problem, split it up, work out the individual components, and then put it back together again. Without such facts and procedures in long term memory, the complete strategy was always going to be the most successful.
My favourite quote:
The main significance of these findings is that the two mental computation strategies represent different attitudes towards numbers. The Split strategies suggests that numbers up to 100 are viewed as consisting of tens and units and children using them attempt to deal with these values separately. Such strategies can frequently lead to the sort of errors that occur when using the written standard algorithm. By contrast, Complete number strategies treat numbers as wholes. Furthermore, the calculation steps are sequential so that subtotals are operated on as they occur and do not have to be stored separately in memory.

Research Paper Title: Developing Automaticity in Multiplication Facts: Integrating Strategy Instruction with Timed Practice Drills
Author(s): John Woodward
My Takeaway:
This study seeks to establish whether it is better to use a strategy of timed drills to teach multiplication facts, or a combination of timed drills with activities that promote the use of strategies to teach these facts. The strategies involved were those similar found in Number Talks, for example the multiplication fact 6 x 7 was shown, through discussion, to be equivalent to 6 x 6 + 6. Groups of students were taught using either Drills or and Integrated approach for 4 weeks. They were then given three types of test: Computation, Extended Facts and Approximations, followed by an Attitude Towards Maths survey. Both groups performed equally well on Computations, with the Integrated group performing better on the Extended Facts and Approximations tests. Both groups reported the same level of happiness in the attitudes survey, which may surprise those who fear the Drill and Kill strategy. My only reservation about recommending an integrated approach based upon these findings is that we do not have longitudinal data to see the levels of retention. The students in the Drill approach got through far more computations, and hence there is a chance that their degree of retention will be higher than the Integrated group.This study supports my view that an integrated approach can work - but the key knowledge and facts must be in students' long term memories before attempting to develop and discuss these strategies with numbers, and drilling may be the best way to achieve that. Then, hopefully, you can end up with the best of both worlds.
My favourite quote:
Results from this study indicate that an integrated approach and timed practice drills are comparable in their effectiveness at helping students move toward automaticity in basic facts. If educators were only considering facts as a foundation for traditional algorithm proficiency, either method would probably suffice. Yet, the educationally significant differences between groups found on the extended facts and approximations tests should encourage special educators to consider how strategy instruction can benefit students' development of number sense.

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Fluency with Maths Facts on Mr Barton Maths

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arrow_back Back to Research Fluency with Maths Facts Research from the Cognitive Science, Explicit Instruction and Cognitive Load Theory sections of this page stress the importance of students having key mathematical facts ready to be retrieved from long term memory so they do not take up valuable space in working memory.

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We provide resources related to effective math instruction. Our goal is to ensure that all students, regardless of background or status, have equitable access to high-quality math instruction. To guide the Science of Math, we rely on well-researched instructional strategies and research about how st

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Should we teach children to memorize the multiplication tables?

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A new article argues that mathematical reasoning and complex problem-solving benefits from a foundation of well-memorized math facts

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The Biggest Misconception about Math Fact Fluency

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Let students share out their different strategies and thinking. Working in partner pairs is very helpful to engage students in discussion about math. 

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Let students share out their different strategies and thinking. Working in partner pairs is very helpful to engage students in discussion about math. 

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We can share during building teacher meetings

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80% of school math is performing calculations by hand, but computation as a valued skill has dropped from 2nd to 12th.

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How do we as teachers train ourselves to not jump in right away and help?

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Read Chapter 4 & Fluency without Fear for October. Add any ideas you may have for sharing with the staff to the Padlet.

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The posts in yellow in this column are all the resources that were listed on page 93 that incorporate the features from Chapter 5.

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Illuminations Home - National Council of Teachers of Mathematics

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Welcome! Explore these patterns with your students and watch their natural tendencies to see patterns morph into powerful algebraic thinking and reasoning. It's an ideal routine to foster mathematical practice #7 - look for and make use of structure.

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Explore these patterns with your students and watch their natural tendencies to see patterns morph into powerful algebraic thinking and reasoning. It's an ideal routine to foster mathematical practice #7 - look for and make use of structure.

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How do I get started? 1 USE THE FREE MATH LESSONS Estimation 180 Tasks 3-Act Tasks 2 SUBSCRIBE Email Newsletter Podcast

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All Grade Levels Challenging Math Problems

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All Grade Levels Challenging Math Problems

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Home

CHALLENGING MATH PROBLEMS WORTH SOLVING

DOWNLOAD OUR FAVORITE PROBLEMS FROM EVERY GRADE LEVEL

WANT GOOGLE SLIDE VERSIONS OF ALL PROBLEMS?

HERE'S OUR GROWING COLLECTION

WANT TO SHARE OPEN MIDDLE WITH OTHERS?

CHECK OUT THESE FREE WEBINARS TO HELP TEACHERS RETHINK CLASSWORK

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CHALLENGING MATH PROBLEMS WORTH SOLVING DOWNLOAD OUR FAVORITE PROBLEMS FROM EVERY GRADE LEVEL Get Our Favorite Problems Take The Online Workshop WANT GOOGLE SLIDE VERSIONS OF ALL PROBLEMS? HERE’S OUR GROWING COLLECTION Get Google Slide Versions WANT TO SHARE OPEN MIDDLE WITH OTHERS? CHECK OUT THESE FREE WEBINARS TO HELP TEACHERS RETHINK CLASSWORK Elementary Version

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search

Enriching mathematics for all learners

A free-to-use collection of mathematics activities, lessons and problems designed to nurture curious, resourceful and confident learners of mathematics. Produced by The University of Cambridge.

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The Nrich Maths Project Cambridge,England. Mathematics resources for children,parents and teachers to enrich learning. Problems,children's solutions,interactivities,games,articles.

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Classroom Resources - National Council of Teachers of Mathematics

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Please Note: Balanced Assessment printed materials are available from this site, except for those indicated below. Please use the order form and follow the ordering directions carefully as they have changed. Current prices can be found on our order form.

Please also note that the Balanced Assessment Primary & Elementary Tasks have been published by Corwin Press. The Balanced Assessment Transition & Middle School Tasks have been published by Teachers' College Press. These tasks may still be viewed in.pdf format on this website but they may not be copied or printed.

About the program

From 1993 to 2003, the Balanced Assessment in Mathematics Program existed at the Harvard Graduate School of Education. The project group developed a large collection of innovative mathematics assessment tasks for grades K to 12, and trained teachers to use these assessments in their classrooms.

The library of over 300 mathematics assessment tasks developed during the project remains freely available through this web site. Teachers may use these materials in their own classrooms at no cost.

This site also offers reports on our approach to mathematics assessment and our distinctive objects x actions scoring system. A more detailed overview of the project is also available.

One of the most satisfying results of this ten-year project was the enthusiastic response of teachers to our assessment tasks and publications. We hope that you will continue to find the Balanced Assessment tasks to be a valuable resource.

Ordering Balanced Assessment publications

It is possible to purchase printed Balanced Assessment publications using our order form.

Our library of assessment tasks

If you are looking for a specific task, you can find it by name or by number.

If you're just getting to know our materials, we'd like to suggest a few of our favorite tasks:

Our approach to assessment and scoring

These reports and materials were developed while the Balanced Assessment program operated at the Harvard Graduate School of Education, between 1993 and 2003.

Assessing Mathematical Understanding and Skills Effectively (AMUSE)
Also available as a Word document.
This 1995 report describes our project group's approach to mathematics assessment.

Mathematics Content and Process Scoring System (MCAPS)
Also available as a Word document.
This report describes our objects x actions approach to scoring assessment tasks.

MCAPS Scoring Tool
Requires Microsoft Excel or compatible spreadsheet.
This spreadsheet implementation of the MCAPS scoring system enables teachers to efficiently use the system in their classes.

Judah L. Schwartz,
Principal Investigator/Director
Joan Kenney,
Project Coordinator/Director
Susan Ilias
Kevin Kelly
Teresa Sienkiewicz
Yesha Sivan
Victor Steinbok
Amy Warren
Michal Yerushalmy

For information on using BA tasks in the classroom, professional development for teachers and administrators, and issues regarding orders, please contact:
Joan Kenney
Joan_Kenney@post.harvard.edu

For technical web site issues, contact:
dan@mathshell.org

The materials on this web site were developed under National Science Foundation grants MDR-9252902 and ESI-9726403 (subcontracts from the University of California at Berkeley and Michigan State University), with additional funding from the American Academy of Arts and Sciences, the Boston Public Schools, the Cambridge Public Schools, and the Noyce Foundation..

All documents are Copyright © 1995-2003, President and Fellows of Harvard College.With the exceptions listed above in the heading to this page you are hereby granted the right to duplicate or print these materials for use in your classroom or school. Otherwise, these materials are protected by copyright.

This web site was originally designed by Kevin Kelly.

Last Update: 10/17/2015
Copyright © 2023, The Concord Consortium. All rights reserved.

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Procedural Fluency in Mathematics - National Council of Teachers of Mathematics

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Procedural fluency is a critical component mathematical proficiency and is more than memorizing facts and procedures.

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Math Fact Fluency: Automaticity and the Brain

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Wednesday, September 27 4-5 Centennial Elem Media Center

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Wednesday, September 27 4-5 Centennial Elem Media Center

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National Louis University

Three years ago, National Louis University launched its Pathways program to help more Chicago-area high school graduates earn college degrees. In addition to providing supports such as financial assistance, academic advising and peer mentoring, NLU works hard to help its students develop a growth mindset about their ability to persist and succeed in college.

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National Louis University Three years ago, National Louis University launched its Pathways program to help more Chicago-area high school graduates earn college degrees. In addition to providing supports such as financial assistance, academic advising and peer mentoring, NLU works hard to help its students develop a growth mindset about their ability to persist and succeed in college.

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The more aware students are of their individual fingers, the higher they score on math tests.

Uber-Images/Shutterstock If you were like most kids, your mother told you there were three no-no's when it came to your fingers: Don't put them in an electrical outlet, don't stick them up your nose (at least not in public), and don't use them when you're counting. The first two laws of finger dynamics are as true as ever. But experts in education and cognition now believe that using your fingers to do math is not only a perfectly good idea but may even help children become superior students. (Is the American school system damaging our kids?)

It certainly makes sense. When children count on their fingers, they take an abstract concept-mathematics-and translate it into the most basic, tangible form. In fact, neurobiologists believe that the brain is hardwired to "see" a representation of our fingers even when we aren't literally counting on them.

There is a section of the brain, called the somatosensory finger area, that activates when we respond to heat, pressure, pain, or the use of a given finger. Studying brain scans, researchers discovered that when students ages 8 to 13 work on subtraction equations, the somatosensory region "lights up" on the scans, even if the students aren't using their fingers. The more complex the subtraction problem, the more activity is detected in the somatosensory region. The researchers theorize that when the brain is called on to subtract, it automatically marshals its finger-counting ability to get the job done, regardless of whether any actual fingers are doing the counting. Try these other activities that boost your brain power.

The connection between finger use and math ability has been demonstrated on old-fashioned math tests as well. With their eyes closed, first graders were asked to identify which of their fingers a researcher was touching, along with other finger-related exercises. A year later, the students who scored highest on the finger-ID questions consistently scored higher on a math test. When college students were given the same finger quiz, the highest scorers once again performed best on calculation tests.

So what does all this mean? For one thing, parents and teachers shouldn't discourage children from counting on their fingers. "Telling students not to use their fingers to count or represent quantities is akin to halting their mathematical development," says Jo Boaler, PhD, a professor of math education at Stanford University. (This discouragement may be more rampant than you think-a 2014 mailing from Kumon, a nationwide tutoring service, carried an article titled "Why Finger Counting Is a No-No!")

Researchers at Stanford also stress that some students simply learn better using visual tools rather than by memorizing the multiplication tables and reciting them as quickly as possible. "When I introduce math problems to my Stanford students, I say, 'I don't care about speed; in fact, I am unimpressed by those who finish quickly. That shows you are not thinking deeply,'" says Boaler. "Instead I would like to see interesting and creative representations of ideas."

Boaler believes that anyone-at any age-looking to improve his or her math ability would do well to improve basic finger dexterity. That means not only counting on your digits but also sharpening general finger "perception."

That may sound simplistic, but the researchers at Stanford offer an interesting anecdotal hypothesis: "The need for and importance of finger perception could even be the reason that pianists and other musicians often display higher mathematical understanding than people who don't learn a musical instrument."

To that end, Stanford's Youcubed center, which develops resources for math teachers, has devised a series of activities to strengthen students' perception of their fingers. We've included one of them here (see below) for you to try at home. For keyboard templates and more activities, go to youcubed.org.

© 2016 youcubed, courtesy Stanford University

You'll need a series of paper "keyboards" like the one above, each with the colors in different spots. Put a colored dot on each finger, as shown. Starting with the key on the far left, touch the corresponding finger to each colored "piano key" and hold for a few seconds. Work through all five keys. Switch hands, then add new keyboards.
The Piano Exercise

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Kids Who Count with Their Fingers Are Smarter, Researchers Say

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The more aware students are of their individual fingers, the higher they score on math tests. Uber-Images/Shutterstock If you were like most kids, your mother told you there were three no-no's when it came to your fingers: Don't put them in an electrical outlet, don't stick them up your nose (at least not in public), and don't use them when you're counting.

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Developmentally ready is outdated. Instead, "sequential skill" learning helps parents and guardians understand that all learners have a mathematical brain. 

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Developmentally ready is outdated. Instead, "sequential skill" learning helps parents and guardians understand that all learners have a mathematical brain. 

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Chapter 3. The Creativity & Beauty in Math

What messages do we send about Math when we give an assessment on the first days? How else could we gather diagnostic information about our students? (p 25)

“Most students think of mathematics as a series of answers - answers to questions that nobody has asked.” What sorts of changes can we make in our classrooms to help our students own the questions? (p 31)

How does Jo respond to parents that ask “What is the point of my child explaining their work if they can get the answer right?”  (p 33) What are your thoughts on this idea?

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Chapter 1: The Brain & Mathematics Learning

What are some common misconceptions about math ability, and how can a growth mindset help to challenge these misconceptions?

How can we help parents avoid language that promotes a fixed math mindset (“I was never any good at math either.”) and instead support the development of a growth mindset in math at home? What role does praise play in this? (p 9)

What does Jo believe is wrong with the idea of being “‘developmentally ready” for topics in math? (p 9)

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"We have a real problem with math education right now," is how Conrad Wolfram starts his TEDGlobal 2010 talk in Oxford, in which he reasons through what's wrong, why, and how we can fix it.

Central to Conrad's argument is the role of calculating-that for the mainstream subject it's not an end in itself, but a means to an end, and therefore should be wholeheartedly computer based. As he puts it, "Math ≠ Calculating, Math >> Calculating".

He's optimistic about what's possible. "We have a unique opportunity to make math both more practical and more conceptual simultaneously," and to get people to "really feel math".

Couldn't agree more? Dramatically disagree? Let us know.

PS: If you would like to get involved, check out and join computerbasedmath.org.

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technical engineering mathematics finance financial education science scientific graphics software

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Conrad Wolfram's TED Talk: "Stop Teaching Calculating, Start Teaching Math"-Wolfram Blog

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"We have a real problem with math education right now," is how Conrad Wolfram starts his TEDGlobal 2010 talk in Oxford, in which he reasons through what's wrong, why, and how we can fix it.

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Read Chapters 1-3 and prepare to discuss by answering the Reflection Questions.

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Chapter 2: The Power of Mistakes & Struggle

Why is it important for students to make mistakes and experience productive struggle in their math learning, and how can teachers facilitate this process?

Jo argues that changing the message around mistakes will be “a big change for some teachers who currently plan the tasks given in mathematics classrooms to ensure student success.”  (p 22)  What are your thoughts on this idea?