## “All the time.”

Yesterday, a student gave me step-by-step directions to solve a Rubik’s Cube. I finished it, but had no idea what I was doing. At times, I just watched what he did and copied his moves without even looking at the cube in my hands.

When we were finished, I exclaimed, “I did it!”, received a high-five from the student and some even applauded. For a moment, I felt like I had accomplished something. That feeling didn’t last long. I asked the class how often they experience what I just did.

They said, “All the time.”

Featured Comment

Is there an argument to be made that sometimes the conceptual understanding comes from repeating a procedure, then reflecting on it? Discovering/noticing patterns through repetition?

Great question. I wrote a comment in response.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### Jian Li

November 21, 2016 - 4:12 pm -

Hi, Dan. I know you use this to symbolize some learning process. It is true sometimes. And it’s hard to totally understand something immediately after one process of doing it. Repetition and deep reflection is needed.

Back to the Rubik Cube manipulation, the last couple of operations are hard to imagine or understand in 3-D. Most players are just remember the procedure. Those who can restore it from any position may have get the true essence of it.

They can do it very fast as they repeate many many times.

Math learning is bit like manipulating Rubik Cube.

I really enjoy what you mentioned here. We do need work hard to get students understand what behind the process and explore the learning further themselves.

2. #### Lauren Beitel

November 21, 2016 - 4:23 pm -

Is there an argument to be made that sometimes the conceptual understanding comes from repeating a procedure, then reflecting on it? Discovering/noticing patterns through repetition?

• #### Dan Meyer

November 21, 2016 - 9:42 pm -

Thanks for the comments, Lauren and Jian. I’m sure you’re both right. Don’t let me overreach and suggest that it’s impossible for students to learn anything from procedural repetition. But we should base our instruction on what probably will result in learning, not what possibly might. My concern is that students who aren’t making sense of their procedural repetition aren’t likely to learn much from it.

I found an example of that phenomenon in a study this week. A summary of the design:

In sum, the key features that by design differed between the AR [Algorithmic Reasoning] condition and the CMR [Creative Mathematical Reasoning] condition were: (a) the AR condition was given a solution method while the CMR condition was not, (b) the AR condition was given five instances to practice using the formula while the CMR condition was given two instances to construct a solution method and one instance to formalize it into a mathematical function, and (c) in the CMR condition participants had 10 min at their disposal for each instance while participants in the AR condition had 5 min.

The AR group had more repetition than the CMR group. To your point, they didn’t learn nothing. Their post-test scores weren’t zero. But they learned significantly less from their repetition than the CMR group did from trying to make sense of the concept.

So I’m happy that if students find themselves in a position where they are repeating procedures without understanding their conceptual underpinnings, hope isn’t lost. It’s still possible for them to learn the concepts. It just isn’t as probable as emphasizing the concepts first.

• #### Kyle Pearce

November 22, 2016 - 11:50 am -

Great points here, Lauren, Jian and Dan.

Reflecting on my personal mathematics journey, I was one who mostly memorized procedures and *sometimes* made connections along the way. I personally believe that there is a group of students, like myself, who can make it out just fine this way. However, there is another group of students who fail to make those connections, even when the teacher attempts to explain the “concept” behind the procedure.

During Chapter 4 in “Adding It Up”, the section that summarizes the importance of procedural fluency mentions a few points that I will paraphrase below:

1) Learning procedures without conceptual understanding presents the possibility of a student learning the procedure incorrectly and this can be difficult to reverse – especially since there is little or no conceptual understanding to help guide them.

2) Students may struggle to determine whether an answer is realistic/reasonable.

3) Learning a procedure without understanding requires extensive practice in order to remember the steps; much more practice than is required when students understand the inner workings of the concept and can “reconstruct” their understanding if difficulty should arise. The reasoning behind why it can be much more difficult to remember without conceptually understanding is that each step is stored as an individual piece of information as opposed to one “chunk” when the concept is connected to prior knowledge.

Something that also resonated with me was the line “once students have learned procedures without understanding, it can be difficult to get them to engage in activities to help them understand the reasons underlying the procedure.” I saw this in my own classroom when I first began my shift from teaching more procedurally to a more conceptual approach via inquiry. Once students “knew” the formula, there was an unwillingness (or maybe fear) to attempt unpacking the “why”.

My big takeaway is that math is an experience and if we want that experience to be memorable (and useful), we need to help students by seeing math class as differentiable experiences. When I think back to teaching via procedures primarily, I see the same experience. When I think back to teaching via 3 act math and inquiry approaches, I see separate chapters of the learning unfolding. Hopefully my students do as well.

• #### Dimitra Georganas

November 22, 2016 - 12:20 pm -

Awesome response, Kyle! I’m teaching a Math Methods graduate course and I have been relaying the same message to my students. It’s so gratifying to witness the amount of eagerness they have to learn the conceptual approach to teaching as they are not accustomed to it. It amazes me how much they have not seen or learned yet (which is not their fault). We need a LOT more training taking place for these soon-to-be teachers as one course is simply not enough in my eyes. It’s almost as if you need a topic (i.e., number sense, fractions, etc.) covered per college course to really be prepared to teach these young learners who are experiencing the critical stages of building a strong foundation of number sense. A great emphasis needs to be placed at the laying of the foundation which is engulfed with conceptual knowledge.

• #### Kyle Pearce

November 22, 2016 - 12:42 pm -

Dimitra,

I completely agree about the necessity to help teacher candidates learn about mathematics concepts conceptually. I am amazed at how much I learn every day that I was unaware of when I learned as a student and taught as a teacher. I am still ignorant to many pieces of the mathematics puzzle, but I think as the math community continues to dig and share, we will make significant progress.

In our district, we are using a continuum model to attempt starting with a concept from the beginning and work with it as it develops into bigger ideas. Seems as though there is so much that we will never “get there”, however I think the more we do, the more others will continue exploring in other areas and share back to the community.

I’m convinced that for at least grades K-10 in Ontario, that there is a way we can make the learning concrete and visual for students. Sometimes we might not know what a specific topic “looks like”, but that just means we need to dig deeper.

• #### Laurie Hailer

November 22, 2016 - 1:55 pm -

I’m wondering about some areas where repetition may make sense – like recognizing patterns with familiar numbers. Maybe for factoring. I just learned how much less my students know about how to factor than I thought they knew. It may be quiz stress, it may be not enough practice, it may be too many types of problems mixed together, but I was surprised that, given the formula for sum/difference of cubes, they still really struggled with using it for something like 27x^3+8. I don’t do many drill style problem sets. But, maybe 10 problems on this alone (not mixed with other factoring problems) would help. I would love suggestions or ideas.

• #### joe vignolini

November 22, 2016 - 3:59 pm -

Laurie,

I wonder if the question we need to ask first is “why do they need to knwo the differences of cubes? Early in my 25 year career I made that a big deal. I have come to realize all the drills and practice meant nothing long term as they did not know why they needed to factor. I failed to connect factoring deeply enough with prime factors and with other things. I also failed to accept when a student did long division to reduce (x^3-1)/(x-1) and other examples, including using synthetic division… Tools I also “taught without context”. I would suggest that factoring without context is useless. It is like learning vocabulary words and not grammar. I learned a new way to show a connection my kids never seemed to make. The Fund theorem of Algebra – factors make polynomials. – But I now show this by having students draw lines, create a table of values and find products and plot that. Ideally knowing that the factors make a polynomial, as the first step. Investigating and playing they see a lot. SO MAYBE that is the “repetition” needed? One sees patterns by REPEATED PLAY and not repeated algorithmic practice. We use this model in our JK-5 math and try to move it forward through the rest of the grades to 12.
“Conceptual Understanding can come from repeated Play”

• #### Jian Li

November 23, 2016 - 11:10 am -

First,it nice to see Nic our NZ fellow. It is sooo good to read your ideas. Together we open our minds to find the best way of teaching. And Joe I like your comment. Mic’s also remind me of my experience. Repetition of procedure without understanding works for some cases or some individuals.
When i was in Primary in China,like Y6 or 7 in NZ, we were learning a rule “knowing a part to find the whole using division” it involves division of fractions. As we only have experience division make number smaller e.g. integers divided by integers. I cannot figure out why division get a larger number then. I kept asking why,why why? And my teacher explained quite patiently I can remember. It confused my long time maybe two or three years until I forgot to argue why! But I remember the rule whenever that kind of word questions that doesn’t make sense to me I use that rule and get a tick right. In my mind I still can’t figure out why! It works for me. As I at least have enough literacy to know the situation and know when to use the rules. And I am sure my teacher was right,though she did not find the right context suit me .

There a lot of students they only accept the maths rules when they totally understand. Sometimes it’s hard to understand immediately. Factorising is a case like that especially factorising polynomials. Students can do expanding quite well but confused by factorising even the same expanding reverses back. It is really hard to connect to daily life with polynomials. Some students remember rules some don’t.

One of my learning strategies is to use it first or anyway if I can’t fully understand it works for maths. And it does not work for everyone.

Any ideas or advice for making sense of factorising polynomials?

• #### Arlene Farray

March 21, 2017 - 8:09 am -

Sometimes people need modeling to initiate transfer, and this is particularly important in math. I know that I have practiced many procedures to automaticity, and after seeing how it is applied in practice, then I gained a complete understanding of it and could use it generally. I work with students receiving special education, and I am privy to many math lessons, and one of the things I noticed is that after one or two lessons, the students will be given problems to solve using the concepts they have just learned. They usually have problems, because the concepts are not yet cemented in their minds, and having not seen worked examples illustrating its practical use, they are unable to apply them to the problems.
I recently did an online course in trigonometry, and I kept noting all the topics one needed to know to do trig proofs using the different identities: you had to be fluent in the basic math operations-addition, multiplication and subtraction; you had to know exponents, radicals, working with complex fractions, factorization and the use of conjugates to rationalize square roots and some more. It involved a lot of memory to know which techniques were relevant. Knowing concepts are just one part of math, you need to remember them and know when they are to be used. The only way you can remember so many things is through practice, and when deliberate practice is removed from strategies to gain competence in math in favor of knowing concepts alone there is a problem. I am not a researcher, but I know from reflection on my past actions what strategies are necessary for me to gain competence in a subject.

• #### smitch5511

November 22, 2016 - 9:22 am -

I appreciate the described scenarios and the advantage of the creative approach. The reality has to include time constraints however, since a classroom includes a group of students and a set amount of time to complete the curriculum. I do agree that covering less curriculum can be a richer experience (“less is more”), however even within this parameter there will be students with different levels of comprehension. Some may need to resort to pattern recognition and then recall the algorithm at a later opportunity.

• #### Kyle Pearce

November 22, 2016 - 6:41 pm -

Hi Laurie,

I don’t think anyone is suggesting that repetition is not useful, appropriate or even necessary. Repetition is definitely important in many aspects of life, including mathematics. However, in mathematics in particular, I think we often try to “skip” the whole understanding part and hope it will come at some point from repetition. For some students it will, but for others it may not. We are all bound by restrictions like curriculum expectations, bells and school interruptions which make it difficult for us to always feel like we can do our very best.

Some ideas to consider might be interleaving and/or spiralling content in an attempt to build conceptual understanding of multiple concepts over time rather than digging straight down from day one to day 10 of a big idea in one go. This also allows for spaced practice rather than the typical mass practice we usually see outlined in textbooks. There is a body of research that suggests students appear to understand content more quickly when we use mass practice (giving one topic lots of repetition immediately after learning), but retention is poor and when questions are later mixed, students struggle. Whereas interleaving (spaced practice with multiple concepts mixed) is more difficult for students initially, however they retain more knowledge for a longer period of time.

Just some ideas you might want to explore based on your struggles mentioned in your comment.

3. #### Rene Grothmanm

November 21, 2016 - 10:19 pm -

When I was young I solved the cube all by myself. Took me quite some time and stubbornness, giving up and trying again. My solution was by far not the shortest path, of course, but it was mine.

That attitude must be the goal. The desire to solve a problem is the key. If you let them find satisfaction by repeating a solution until mastership like an ape you are on the wrong track. Instead, they should feel the urge to go for new things. Repetitions are boring and for computers.

I know I am dreaming. In fact, we all find satisfaction by stupid repetitions. But solving only one new problem on your own is so much better.

• #### Rachel

November 22, 2016 - 12:50 pm -

Rene, as a product of an experiential math program, I can identify with your joy at having reached the solution all by yourself (or in a small group…). However, I also know that there is a difference between solving the problem once and understanding it well enough to solve it more efficiently next time and recognize similar problems in the future. It’s important that students get the conceptual foundation; the connection between the concept and the algorithm(s) needs to be made just as carefully. It’s a good thing to have a “eureka!” moment when you discover that this triangle is half of a parallelogram, but it’s a longer, deeper process to get to the point where you are really comfortable with 1) the idea that this relationship is always true (even for that weird scalene one) and 2) the fact that a=bh/2 describes what you discovered. It takes meaningful repetition combined with sophisticated reflection to get to that point (and remember that the proof is always most immediately convincing to the person who already knows the right answer, teacher).
As a teacher, you have to decide, too, whether your goal is for students to solve problems like these quickly and automatically in the future or whether it’s ok for them to have to think a bit and recreate the next logical step. Is this groundwork or mastery time? They are different.

• #### Arlene Farray

March 21, 2017 - 8:39 am -

I agree with Kyle Pearce on the practice of interleaving -the use mixed sets as a way to boost understanding. However, that is not what is practiced. With the focus on assessment, and what I view as piecemeal assessment, students are tested after every topic; thus they only encounter mass practice. This leads to the inability of the students to discriminate where things are similar but a small difference requires a different treatment. When a varied set of problems are presented, they find it difficult to integrate different skills.I know the benefits of mixed or interleaved practice, but who is to institute it where the syllabus is a mile long and an inch deep, so everything receives a cursory hit. I was schooled on interleaved practice, since our end of year test covered all the topics we had learned during the school year. I also studied accounting for which interleaved practice is par for the course.

4. #### Lane Walker

November 22, 2016 - 6:07 am -

I would also argue that routinely teaching concept first, followed by “step-by-step,” will likely train students to tune out until they are being told what to do. Better: coach students toward understanding the concept. Next practice applying the concept. When students see the pattern, they can use the pattern; but if they forget the pattern, go back to coaching concept. Coach next concepts based on previous understandings.

5. #### Dimitra

November 22, 2016 - 7:01 am -

I agree 100% with Dan and those who responded. Thoughts on dividing fractions? The conceptual is taught first followed by “Keep, Change, Flip”. I would like to request a list of topics that teachers should and should not spend a lot of time on (i.e. rounding vs. estimating). How much paper and pencil computation should be taking place (i.e. long division) vs. calculator usage as a tool?

• #### Arlene Farray

March 21, 2017 - 8:53 am -

I agree with you about one thing spending reams of time on rounding and estimating is wasting valuable instruction time, and students seem to have a difficult time understanding rounding. However, on the use of pen and paper math, I think wunderlic testing which many employers use as a means of elimination. in wundelic participants are not allowed to use a calculator, and they have 50 questions to complete in 12 minutes and they need to make between 20 and 25 to pass.
This is one of the questions four men entered into a partnership contributing capital in the ratio of 50, 15, 15 and 20 percent. They agree to share the profits equally. In the first year they make 120 000. How much more the partner contributing the highest amount of capital would have received if profit was shared in proportion to the capital contributed. This must be done in 14.4 seconds without a calculator. You may think that students need to know pen and paper math, but they do need to know it, because employers know that they may not be fluent in it and use it as a means of elimination. This test also contain a lot of vocabulary and interpretation of proverbs. Given the dismal state of language arts it could be a chalennge in all departments.

6. #### Angela

November 22, 2016 - 7:26 am -

But if I’m Mr. Cox, what did I want that whole time? Did I want to understand how to solve a Rubik’s cube? Or did I want to 1) please my teacher by getting it right 2) impress those watching by getting it right and 3) feel accomplished by getting it right? To me, the question isn’t whether I learned to solve the Rubik’s cube, it’s whether I really wanted to do it in the first place. Maybe that’s too philosophical.

• #### Trevor

November 22, 2016 - 7:34 am -

I think that is an important point and something Ts need to pay attention to. What is a student trying to accomplish in our class? Are they setting their sights too low? How can we expand their vision both of what they are capable and what is possible in a math class?

7. #### Erik Von Burg

November 22, 2016 - 7:47 am -

The analogy I often use to describe the procedural/conceptual split is the difference between a person with a recipe book and a chef. Given a collection of recipes, a person with minimal training should be able to proficiently prepare most of those recipes and appear to most people be a good cook. However, if you placed this person on the show, Iron Chef, their success would be predicated on the off-chance that the given ingredients would fortuitously match one of their recipes in their collection. Absent this situation, the recipe book chef would mostly likely be either blindly experimenting or attempting to generalize one of the prescribed procedures in hopes that it works. The chef would recognize the different variables at play (of which I cannot even fathom to describe) and apply their understanding to create with the different ingredients.

Assuming the recipe book cook only robotically follows the recipes, they practice move them any closer to being more chef-like. If they deviated from the recipes a times, they would at best be left to their own devices to develop and intuit the knowledge, concepts, and connections that a chef relies upon. Naturally talented culinary artists would be able to to proficiently make the jump. Others, not so much. Regardless, hoping that practice would move a cook to a chef remands the more important, interesting, and difficult understandings to the realm of unfacilitated self-discovery.

It’s not the practice itself that’s important; it’s the nature and point of the practice.

• #### Nic Petty

November 22, 2016 - 9:13 am -

I really like this analogy, and it also helps us to understand and express how mistakes help us. If the person with the recipe book only ever follows the recipe without mistake, they never gain understanding of the roles of the ingredients. However if they leave out an ingredient, they can learn more about the underlying principles of cooking. They do need to value mistakes, though, rather than quickly tipping them in the bin without working out what caused the result.

• #### John Schnatterly

November 23, 2016 - 9:40 am -

I like the analogy too, Erik. I believe being this “chef” is a worthy destination. But what about the chef’s journey?

I find that in most cases the only chef is the one standing in front of the room. So in this case, the question is whether it’s possible to train chefs by letting them work with recipes. Do we build sous-chefs first and assuming so, how do we get there? Frequently, there are instances when we need to just practice with a knife for a while. Should we develop these skills with the occasional recipe? Of course there are occasions for this; but you’re right, this alone will never help us grow into chefs.

Some of my classroom’s most successful days are more like the TV show “Chopped”: Here are a few ingredients that are required, the kitchen is stocked with utensils you might need, let’s explore this question, GO! But then there are days when we need to work on some basics. Today, my PreCalculus students were asked to draw their own graphs of functions with different types of discontinuities at given points. After seeing repeated examples of graphs that weren’t even functions, I knew we had some knife practice to do. Of course this means that the first time around with functions, they likely didn’t cook their own soup but it’s often impractical to start from scratch.

So we have a messy kitchen, and we need more thyme.

• #### Erik Von Burg

November 23, 2016 - 4:54 pm -

John, I agree with you. There are skills that require practice, and these practices are a piece of the puzzle in becoming a chef; and, these practices are far from the sum total of skills/understandings required to become a successful chef.

In my career, I have tried to figure out ways to couch these practices in something larger, i.e. the idea that some necessity instigates the need for the practice not prescribing the practice and explaining that they will need to be proficient to help with some situation in the future. Sometimes I have been successful; other times not so much. Creating that necessity–without artificiality (and by this I don’t mean necessarily outside the realm of “real world”)–is so much tougher than just teaching prescribed lessons, but it is so much more rewarding for both myself and my students.

And, I think there is the questions about what causes the intrinsic desire to learn something. Would a person be inspired to be a chef by learning and following different rote recipes? I suppose so, and I would imagine that this could be true for many people. But, I would argue that most likely people who launch into their path to chef-hood from learning some recipes probably were already inclined to do so without any assistance from the teacher. In this case, the teacher simply provided the opportunity. What I am interested in is–how might I inspire someone to explore chef-hood or question like chef who did not already have that inherent proclivity? Can I make people act more chef-like even if they are not really interested to go down that path. And, can I show them that chef-like thinking is truly universal, and it’s really only the content/context that changes. I don’t know the answers, but I am working hard to explore them.

• #### joe

November 23, 2016 - 5:08 pm -

hmm.. we know one can “practice music” and play the notes as they are listed – but the “feel of the music” is very different. Procedure is the process/notes/recipe. But great musicians and chefs do not repeat – they envision, they inspire, they own it.
To get students to own it, means we need to tap into their needs and dreams. 10,000 repeats does not lead to mastery, it leats to repeats.
Students (and all people) need DEEP practice – not MORE practice. In The “talent code” this is discussed. Deep practice needs to be more than a procedure.

• #### Erik Von Burg

November 24, 2016 - 7:04 am -

I vehemently agree, Joe. But, I think the question is: what place and purpose does repeated procedural practice inhabit in the world of learning? My guess is that a budding musician at some time repeatedly performed procedural practice to their benefit. When I dabbled in guitar, I know that I played scales and practiced with the intent of strengthening my fingers, build callouses, and improve my accuracy with the pick. These skills do not constitute the totality of a musician by a long shot; there are all the things you describe. You could easily imagine a person with strong calloused fingers and superb plucking accuracy who falls far short of being a true original musician.

The question I tangle with is then–where does this practice most benefit budding learners while not conflating the mastery of procedural practice with true mastery of the larger skill? Procedural practice has to be part albeit small and well couched of the growth of learners.

• #### Arlene Farray

March 21, 2017 - 9:30 am -

Well said Erik,
You articulated well my thoughts. I can cook fairly well and I know that feeling. But there are some times that you need to follow recipes. I am a good artisan baker, and I am always researching how to make the perfect bread. One day a friend saw my well-leavened golden loaf, and decided to come for some lessons on making bread, but after seeing the rigmarole that was required, she said it was too difficult. But becoming proficient in math is something like that chef, but the self discovery is really self realization. individuals must realize that they know all the techniques required for problem solving. Sometimes they know it without knowing that they know it. I have looked at some of Salman Khan’s videos, and I see the same problems that most teachers make -they assume that everyone could follow their applications. The videos are good for me, but not for my daughter. Sometimes you know something, but you do not know how to use it. And it would help if someone would remind you when solving a problem why they do what they do.
I think the worse situation is sometimes you get it and sometimes you do not. I remember sometimes getting all my factorization correct, and sometimes getting it wrong. Then, I realized I was patterning without understanding- although I always wondered why I sometimes seemed fluent and other time not. When I got it correct was when I regurgitated something I did before (standard recipe), when I saw a problem that I had never done I was stuck, but all along I had the requisite knowledge, but did not know the technique. The moment I realized how to work out the middle term ‘bingo’. I knew my tables very well, but I missed the explanation of the technique-find two factors of the last term that will return the middle term. The chef has flexibility, and a person who can only pattern has no flexibility. A chef has flexibility because he can bring to bear all his knowledge of cooking to create new dishes, our artisan cook may not have that flexibility. The chef easily transfers his skills to create new recipes. Students need to have skills, and to know how to transfer those skills, but to transfer those skills, they must have them. So it is in cooking so it is in math.

8. #### Micahel C

November 22, 2016 - 8:09 am -

If we assume not all students learn at the same rate, if we assume all the suggestions above are worth while, how do we change state testing, course sequencing, and even our own assignments and grading policies to encourage and to support “Creative Mathematical Reasoning” as the norm rather than the factory model of Math education that we have right now? It seems that Math education is due for a bigger paradigm shift than what Common Core has attempted to do. The standards and focus on habits of mind may have changed but the framework of Math education still seems to be a one size fits all model (as evident in the requirment to “meet or exceed the standard” in the new Common Core state tests); and this frustrates me on trying to find a happy compromise of two seemingly disjointed models of Math education.

• #### Merryl Polak

November 22, 2016 - 9:00 am -

Micahel C, I agree with you. Living in two opposing math worlds is quite a challenge. The paradigm shift needs momentum, and quickly…

• #### Arlene Farray

March 21, 2017 - 9:38 am -

Most state testing is not consistent with the curriculum; that is why it is difficult and futile. The people creating the test make assumptions about what students should know without actually knowing what they know. I came from a system where we did exit exams a form of standard testing. But the testing was based on our syllabus. Standard testing are designed by people who pay no attention to what students are actually studying. So it is reduced to the teacher teaching to the test and neglecting the syllabus, or following the syllabus and neglecting the testing. Testing needs practice on what is to be tested.

9. #### Dick Fuller

November 22, 2016 - 9:19 am -

How do students come to understand the meaning and power of understanding? How do they come to expect to understand, and to insist upon it? How do they stop the assembly line until they understand?

• #### Arlene Farray

March 21, 2017 - 9:49 am -

That is exactly what needs to happen. But remember how it goes-a student does not understand something, and asks for help. The teacher asks what don’t you understand. The student cannot articulate. As I am older, and I revisit some math that i did in school without the pressure to perform; I realize that an effective teacher should recognize a student’s problem. I learned this by sometimes looking at a demonstration, and thinking I do not understand it. I would usually reflect on it , and then an ‘aha’ moment arrives, and I would ask myself why my teacher did not recognize my problem when I was at school.

10. #### Michael Whalen

November 22, 2016 - 1:22 pm -

What an intuitive thought to ask his students if they ever felt like that. To hear their response “All the time.” must have made that teacher cringe. Is it a wonder that students have trouble retaining their teachings over time if they don’t fully understand what it is they are doing or why they are doing it.
One of the main goals in my class is to teach for understanding, however, that is not a reality for every student. Material is covered at a fast pace and even though the what they learn is reinforced in applying their math & science to their Trade, learning and retaining key concepts, principles and skills is a difficult task for many. At some point you start giving weaker students ‘cheats’ or shortcuts so that they can have a method for solving problems and doing the work. (For example I might show them how to use the AB/C key on the calculator to help them do fractions or the ‘magic circle’, for a wide range of topics, because they can never seem to be able to solve an equation when the unknown is in the denominator).
For some core topics I will give students problem after problem with the thought that they will come to understand the concept and the process through repetition. It is somewhat saddening to know that some students understand what they are doing, and therefore can reason and apply in new and different circumstances, while others simply recognize a type of problem and recall the pattern of steps needed to solve. For a number of students this will be the highest level of understanding they will achieve.
We have to provide students with skills, techniques and strategies that will help them be successful in their jobs. Teaching for understanding is best but we have to give something to those who will not get there over the timeframe of our course.
Apologies for the long comment. Thank you for this post. It definitely struck a chord and I will share it with my staff.

11. #### Richard

November 23, 2016 - 9:35 pm -

This is an awesome analogy on mathematics practice today. While there are times where we need “skill and drill” practices, we also can’t ignore the conceptual understanding behind it. Too often I’ve seen where we teach the students the procedures first and have them apply it to other problems without ever asking or understanding why. It’s true that some students can maybe understand the concept through many repetitions, but there’s no guarantee they will. They might just be satisfied without knowing why because they can still get good results on their test scores. We need to be able to teach the students through problem solving so that the students themselves can conjure the mathematical idea behind it, not teach the procedure so they can solve for problem solving and never get to understand why.

12. #### Dick Fuller

November 25, 2016 - 8:35 am -

I’ve learned a lot, and understood some too, I hope. I am still confused about the role of repetition in understanding. Familiarity I understand, and repetition does result in a kind of familiarity, generally a shallow one; presumably you are being asked to do a lot of problems in order see they can be done without taking time for reflection. And this takes us back to the earlier analogy made with performance art.
I think the distinction between understanding and craft is apparent in the earliest school mathematics: counting-number addition is inherent in real counting of real objects in collections of them. If you have a good feeling for what you are doing, you understand whole number addition. Retrieving a number from memory in response to reading an addition expression does not lead to understanding addition.
My working understanding of understanding: it is my mental video of a process that I made and can manipulate. Most importantly different videos combine to make wholes greater than their parts.

• #### Arlene Farray

March 21, 2017 - 2:29 pm -

In response to Dick. Repetition is a part of the learning process. As someone who had learned math by the old method of repetition, I swear by it because, I do not think that concepts help children very much in understanding. Recently, I witnessed a lesson where children were shown the concept of area through counting squares, and the concept of the perimeter through counting the squares on the edge of the figure. They were then taught how to calculate area and perimeter using dimensions, and asked if they knew what they were doing with the expectation that they would associate this new way of computing area and perimeter with the concrete way they had previously learned. Unfortunately they did not.

I will now talk about my daughter who was educated in North America. She recently had to do a standardized test to obtain a job. The math questions were easier than the language questions, but she was unable to work them out mentally. she had to rely on the language problems. Let us hope they were enough to gain a pass. The questions were simple, any student with fair arithmetic skills should have been able to do it. But her problem is that the way she was taught in Canada, she never did repetition, she also never did interleaved practice, and interleaved practice is not new. i am 62, and that is what I was exposed to in my little third world country. I do not blame her for her situation, although she should not really be in the position she is now because I had always predicted that the way they were being taught in school, only the very best will be competent in math. I offered to teach her arithmetic, algebra and some geometry, but since that idid not correspond with the school syllabus it seemed like a different kind of math. What they did at school is work out problems without being taught the methods that will allow them to generalize or transfer their knowledge. They were not taught changing the subject of a formula. They applied it in solving an equation, but were not aware that if you have one unknown in any equation, you can solve it by changing the subject of the formula. Thy were not taught a general method for calculating the Lowest Common Multiple, so she is unable to do fractions fluently. They were taught to find out by doing iterations until you get the same product. I can name many more instances of what should have been taught but never was. She is now trying do undo the damage. Recently, she told me that she realized that you must know your tables to master factorization. I think that it is unfair to students when the authorities who control what they learn indulge in educational innovations for which there is no robust evidence that they will work. At the end of the day, the bells a whistles never stay, and what should have been learned to form the foundation for higher math was never learned. My daughter is a victim of the education system like so many other children are victims. I have done many math courses that involved application of math, and learning by repetition never hampered my conceptual understanding instead it fostered it. It provided me with a good foundation for higher math. By the way repetition alone is not responsible for making one remember multiplication tables; using them all the time is.

13. #### Josh Hornick

November 25, 2016 - 5:33 pm -

One size does not fit all.
Two ideas
1) Back in the ’60s and ’70s, when I was a K-12 student, our curriculum was all about lecture, practice, lecture, practice. I was in a “higher track” and those of us in that track always made sure that we understood the “why” as well as the “how.” Figuring out the why was a good intellectual challenge. That sort of instruction served us just fine. There were many who it did not serve so well.
I use inquiry methods now to teach my untracked 11th and 12th grade math classes. Some of my students figure out the math fast. They often become frustrated during discovery lessons where I ask groups to figure it out together. Sometimes I wonder if these students wouldn’t learn more better and be more challenged with “old school” drilling, developing strong rote skills and applying their learning to a larger array of variations.

2) Many of my students need some drilling.
I tested my 11th grade on basic multiplication, just digits 2-9. It was crazy, how many of them can’t multiply 7×8 in their heads. I wish that these students had more rote practice. Not being aware intuitively that 7 is a factor of 35 makes much of what we do in Algebra 2 much more difficult for them. This is a real problem, and some drilling might be just the ticket here. I’m starting to experiment. (In fact, if anyone can recommend resources/approaches to teaching 4-7th grade math to 11th graders in the context of Algebra 2, I’d love to hear from you.)

• #### Joe v

November 25, 2016 - 6:05 pm -

I have dyslexia and ADHD… and a masters in math, was on my way to a phd…. before I loved teaching high school math….. but I do not have that in my head… instead I think 7×4 =28 x2…. takes me a few extra seconds… but not memorized… visualized…

I think strategies beats memorizing….

I would suggest as a person who works as A jk-12 math chair… use the manipulative and multiple approaches… seeing 7×8 is believing….

• #### Arlene Farray

March 21, 2017 - 2:35 pm -

Good skills in addition, multiplication and division is the key. nothing more nothing less. it is skirted in the earlier grades, but when students encounter higher level math, the absence of the basic foundation becomes a serious impediment to learning.

14. #### Chris Heddles

November 26, 2016 - 3:25 am -

I’m with Josh Hornick on this – one size does not fit all so there is no one “best way” to teach maths. Different students and different classes need different approaches at different times. This is the human endeavour of teaching.

Like many maths teachers, I found mathematics easy and fun at school. Rote learning bored me and I needed to understand a technique before it would stick. Once I understood how it fit my existing framework of understanding, I remembered it easily and could use it effectively. Crucially, I had the confidence and interest to tussle extensively with complex, challenging problems and puzzles, many of which had no answer (or at least none available with the tools at my disposal.)

When I started teaching, I taught this way as well, assuming that my students would also be receptive to learning this way. Wow, was I wrong. Most arrived in my (senior secondary) classroom with huge gaps in prior understanding and very low confidence in their ability to learn mathematics. They lacked the tools required to tussle meaningfully with complex problems that were relevant to their assigned content so they freaked out when asked to do so. There were exceptions of course, but the majority of my students didn’t respond well to such openness in early lessons.

Since then, I have found much more success in using procedural teaching to build basic manipulation skills and confidence in mathematical learning, even though it pained me at first to do this. This allows students to then begin applying those skills to more complex problems in order to gain an understanding of how the skills work and some of the deeper connections. Procedural learning is a remedial intervention and an incredibly powerful one for many students.

I use a range of analogies with my students to help them understand the connection between learning procedures and “doing maths”. Many of my current students are dancers so they can think of procedures as barre work; it doesn’t make them great on stage but not doing it makes the on-stage work nearly impossible. For musicians, it’s like the scales and arpeggios. It’s not music but making great music is really tough without a solid base of technical skills. Learning times tables won’t make you a mathematician but it certainly frees up brain space to concentrate on higher-order learning.

To avoid a false dichotomy, I always try to follow up the teaching of procedures with puzzling challenges that require students to use a range of tools, often with combination and modification, to solve. Starting with this type of problem in order to give purpose to the rote skills is an appealing idea but takes more confidence than most of my students have when they arrive in my classroom.

• #### Jian

November 26, 2016 - 1:37 pm -

Agree with you Chris. Most students didn’t realize how important the procedure and repetition are. We don’t want too much repetition but certain amount is necessary to make them confident to apply to various problems especially during assessments. Just like your analogy of memory of time tables. Students sometimes just see the short sighted their learning. As you mentioned mist students come to class with lot of gaps and they even believe themselves to know everything what they need is just manipulation of mathematics basic. And when we try to make up the gaps they start to complain. Even for those who have deep understanding and logical connection are still need some procedure repetition especially true if they want excellent results to participate in some maths competitions. Speed comes from practice and repetition like sports.

15. #### Dan Meyer

November 26, 2016 - 9:45 pm -

Quite a fascinating conversation. I’ll stir the pot in three ways.

One, I don’t think anyone here (certainly not me) is discouraging practice, repetition, or procedures. To argue that “practice is important” is to counter a point that nobody here has made. (Here’s a conversation about practice, FWIW.)

The point I’m trying to make with David’s anecdote (I won’t try to read his mind) is that it’s very possible for students to become fluent with procedures and to feel happy after achieving that fluency, all without understanding much math beyond the procedure itself. Those students can’t transfer the procedure across contexts. They can’t undo the operation. They can’t work backwards from an answer to the question. They don’t know under what circumstances to use the operation unless the chapter heading or problem preface clearly signals it.

Two, different students have different needs, sure, but the consequence of that fact can’t be “anything goes.” Different medical patients have different needs, even those with the same illness, but we don’t think well of the doctor who prescribes leeches.

Three, the opposite of mindless, procedural fluency isn’t “inquiry-based learning through open and rich tasks.” It’s having a fallback for when you forget the procedure. It’s knowing the informal siblings of those formal procedures.

At primary, it’s knowing that you can always add up eight 7’s if you forget the number fact 8 x 7. Or it’s knowing you just need to double 4 x 7.

At secondary, it’s knowing that when you graph the equation of a line, you’re graphing all the pairs of numbers that make that equation true. And if you ever forget all the many procedures we have for graphing all the different forms of lines, just find a couple of pairs that work. Any two pairs. You don’t have to be stuck because you forgot the procedure. This isn’t saying anything about some experiential, project-based rich task.

Having tried to set aside some straw men, I’ll now say I’m very interested in two items:

One, studies that show drilling and practice are superior to some alternative. Because that result depends very much on the alternative, I have no doubt such studies exist. I’m curious what the researchers studied as the alternative.

Two, more ideas for helping students become conceptually and procedurally fluent. Given some of the frustration I’m reading in this thread, I’m sure I’m not alone here.

• #### Chris Heddles

November 27, 2016 - 2:10 am -

Dan,

I don’t think anybody is advocating purely for mindles procedures or rich, open tasks. Both are important for mathematical ability. The question is more about which should come first.

Position One. Procedural practice should only come once the purpose for the procedure is known and the student understands how any why the procedure works. Practice of procedures that are not understood is pointless or even harmful.

Position Two. Procedural practice can usefully be practiced before being understood. Once some fluency is gained, students have the confidence and tools required to engage with the rich, open tasks that build connected understanding.

Ideally, we could all take position one but our context often forces us into position two. Our students often arrive with significant gaps in their prior learning. These would be relatively unimportant if they weren’t also completely convinced of their inability to learn maths in a context that makes the subject compulsory.

The point we all seem to be missing or downplaying about David’s anecdote is the feeling of accomplishment that he had. For many of our students, that’s the closest they have felt to success in a maths class – ever. This feeling of accomplishment is no small thing and can be the spark that reengages students with the idea that they can learn maths and that their effort yields progress.

Imagine that teaching someone how to solve a rubiks cube was the actual goal. By starting with a rote procedure, David is given a starting point that this is possible for him. This is a critical factor for many of our students, who believe maths to be beyond their ability to learn at all. From here, the teacher could lead him to break apart each step of the procedure to understand how it works and how it might be modified to solve a slightly different situation.

Let’s consider a different approach to teaching someone the rubiks cube. Each step must be properly understood before moving on to the next. By the time most students have worked out how the steps work to solve the bottom 4 corners, they have mostly lost interest and decided that they will never get it because progress is painfully slow and solving a whole cube seems so far away.

Now imagine a closer analogy to our secondary maths classes. Students arrive to learn the final few steps on the assumption that they already have a solid understanding of how the prior steps work. Of course, most don’t and are convinced that they aren’t “rubiks cube types”. Despite this, they must successfully show that they can solve a rubiks cube to pass school. In this situation, a taste of success and progress through rote procedures can be an essential requirement for engagement.

To me, it’s not about “which method is better” as both are important. Instead, it’s about deciding when to ask students to do procedural practice and why. How then do we ensure that they connect it to genuine understanding. The destination we seek is the same but some students need different paths to get there.

• #### Arlene Farray

March 23, 2017 - 11:54 am -

The use of visuals does not advance understanding in any way because most times the children to not transfer from the concrete to the abstract. And the problem that most students face in math is working in the abstract. I have received most of my education outside of North America and I am no researcher but early in my learning there was rote, and now I understand because of all the procedures that I have learned to automaticity. I remember before we were taught fractions we were taught to find the LCM of numbers by factoring out primes, and I was wondering why we were spending so much time learning that. When we started to do fractions, it was like drinking milk because I realized we were creating equivalent fractions all with a common denominator when we computed an LCM.
The point about this debate over rote or discovery, I view as mathematical innovation-consistent with views of Andrew Nikiforuk author of the book” If Learning so Natural, Why Am I Going to School”. Nikiforuk in his book, talks about Innovations versus Reforms” He defines innovations as ideas that administrators implement in the hope that they may do some good. He goes on to say that ” innovations are implemented with no foreknowledge or subsequent measurable proof of their effectiveness”. He posits that genuine educational reforms yields measurable improvement in student learning. There is some research by Jennifer Kaminski and Vladimir Sloutsky that says that manipulative or visuals impedes the ability of students to work in the abstract. The proponents of discovery math sell the idea that if people can visualize procedures they will understand them. I have a daughter who is the victim of discovery math. Every time I show her how to change the subject formula to solve an equation she is flummoxed. That is what you get from learning by problem solving rather than learning a defined technique that can be generalized. I remember reading that the late mathematician Herb Wilf said that after examining a slew of math research that they are not robust enough to be relied on. He described some of them as being simplistic.

16. #### Dan Meyer

November 27, 2016 - 8:21 am -

Thanks for the follow-up, Chris.

I don’t think anybody is advocating purely for mindless procedures or rich, open tasks. Both are important for mathematical ability. The question is more about which should come first.

FWIW, this isn’t my question. The question you pose is comparable in my mind to, “Which is better for us – gluttony or veganism?” There are other options besides mindless procedural fluency and rich, open tasks. Namely, mindful procedural fluency, fluency that starts with intuition, that asks students to derive early procedures, that helps students learn formal procedures by building on those early procedures.

Procedural practice can usefully be practiced before being understood. Once some fluency is gained, students have the confidence and tools required to engage with the rich, open tasks that build connected understanding.

As I said earlier, I have no doubt that some students make the leap from mindless procedural fluency to something better. But I’m interested in the techniques that will probably help students, rather than the techniques that might help students. That’s why I’m asking for research. Given how easy it is to stack a deck in favor of an intervention, there must be papers out there that conclude, “We gave students lots of practice with solutions they didn’t fully understand yet. Then we did a thing and they understood them.” I’m curious what that thing is.

• #### joe v

November 27, 2016 - 8:41 am -

Research is difficult to gather – without bias… I have looked at a lot of brain research and I am still confused. But I think it has a lot to do with how we “wire” our brains and that changes as kids mature. SO we have looked to ST Math (mindresearch.org) as a spatial temporal tool to enhance learning and “Investigations series” games as another tool when the students first learn.

The question to me is simple – why do I want to know anything? – facts or ideas or higher thoughts? My answer is – because I find a purpose in it. I do not know the capitals of the states, or the square numbers to 20, or the formulas I need for some math problems. Why? I can find those on a webs search easily, and I never “used them”. I do know the powers of 2 (computer programming) and the velocity formula for parabolic motion (as I can use calculus to get position/acceleration.) Yet we ask kids to “learn” things by doing them over and over again. This is not learning.

Now if we look at early child development – kids do things in their own ways – and can be correct. We make a MISTAKE when we say “do it this algorithmic way, because you will see it is easier in the future.” (essentially like watching a movie and being told the ending just when you got interested. WHY?) kids like to create – let them create an idea of addition, and multiplication etc… Then gradually share all strategies kids did. Finally show the algorithm.

It all starts with “trusting the kids”, “giving them time to learn” and “making it meaningful”. Hard to do with constraints we have, but worth the result.

• #### Chris Heddles

November 28, 2016 - 5:45 pm -

I think we’re talking past each other here so maybe an example would help:

If the current learning is Pythagoras’ theorem then students need skills in basic algebraic rearrangement (including square and square root) to even begin playing with the new content. Many of my students arrived at this topic without those prior skills.

The rote procedures that I would push upon students at this point are the rearrangement and use of square and square root. They won’t gain a deep understanding of how these work but we need to prioritise their available time (and boost confidence in generally “doing maths”) so it’s the “least bad” approach to the situation.

Once students have those prior procedures, they can start to play with Pythagoras theorem using any of the discovery activities that we would normally use. With the prerequisite algebraic skills, the students are then able to engage in the exploration of the new ideas.

What I’m not proposing is mindless practice of Pythagoras’ theorem in the hope that they somehow gain insight into its workings.

Basically, the “uncomprehending skill and drill” is reserved for prerequisite learning that students need for the current content. They will also need practice for the current content but this should be understood when carried out.

• #### Kyle Pearce

November 28, 2016 - 6:45 pm -

Hi Chris,

I see where you’re coming from here. However, I think that it is not accurate that students could not even begin to play with the new concept. I think tackling it from a visual approach using square tiles and/or grid paper to introduce the relationship would be completely possible (and useful) without having to jump straight to the algebra.

If one approaches each concept directly from a symbolic approach using algebraic representations, then I think your argument holds up. However, if you were to begin from a concrete representation, move to a visual representation and then finally make connections to the symbolic (see “Concreteness Fading”), then I think the game changes significantly.

Is it possible that through some tinkering and exploration with concrete / visual representations that more deep connections could be made algebraically? Imagine how awesome it would be if a student saw that an array of 3 groups of 3 actually made a square and thus that is why 3 to the exponent 2 is called “squaring”. Once students “see” the Pythagorean relationship, students could then solve problems based on simple composing and decomposing skills they learned in primary/junior grades prior to reaching the abstract. However, none of this would be possible if we rush to the symbolic especially when we are well aware that many students are not ready for it.

• #### Chris Heddles

November 28, 2016 - 9:03 pm -

Hi Kyle,

Thanks for your thoughts. Overstated my case a little. Students can begin to explore something about almost any topic with no prior learning but they are significantly limited in doing so.

For the Pythagoras example, my preferred opening activity (modified from something I think I saw here from Dan) is to draw an accurately-measured right triangle on the whiteboard and the three squares from the sides. While sitting in their seats, I ask them to decide which has more area, the big square or the two smaller ones (sometimes with a story about kings and sheets of gold). This is concrete and informal. Once they have all chosen a side, I invite them to measure my drawing to decide who “won” in their guesses. So far this only uses primary school skills of length measurement and area of a square.

Where the additional skills come in is for the next part of this activity. Obviously the two sides have equal area and are disappointed that I “set up the situation to be a tie”. I then pass out sheets of paper and invite students to rule off corners to create right triangles that would make their guess correct. In order to rapidly iterate their triangles, they need some basic symbolic manipulation skills and many of my students lack this when they arrive in my class. Cutting out grid paper could be an alternative but would take considerable additional time in class.

Of course it’s not just Pythagoras. Most senior secondary topics require some prior symbolic manipulation skills in order to do anything with a concrete exploration. This is where some “mindless repetition” can be useful, even though it would be better (if we had the time) to go back and help students properly understand the learning that they missed from prior years.

17. #### joe

November 28, 2016 - 6:24 pm -

Question… IF students did “wrote practice” when they were in elementary school and now in HS they do not understand multiplication and other concepts…. they why go back to that “well” to get them to the “next level”? Isn’t the definition of insanity “doing what did not work the first time over and over again.”?

and isn’t the key point about the cube comment – that the KIDS did not feel the sense of accomplishment when they just followed rules?

“For a moment, I felt like I had accomplished something. That feeling didn’t last long. I asked the class how often they experience what I just did.

They said, “All the time.”….. that last line is the most depressing……. since what we do should be for the kids… and not for us…. the teachers…

Imagine – every day going to teach feeling disappointed….

• #### Arlene Farray

March 23, 2017 - 11:52 am -

18. #### Courtney Rindgen

November 30, 2016 - 6:52 pm -

Too much to comment on! However, I see factoring as completely useless, and contrived. Our Math curricula need a complete overhaul. We have computers, calculators and Sophisticated machines for calculating. Employers need problem solvers! Think about it like this: we all use computers, but do we need to know how to code? I think there is a need for s significant shift in perspective in what exactly mathematics curriculum needs to look like. Out with the old!

• #### Kyle Pearce

November 30, 2016 - 7:05 pm -

Hi Courtney,

There was a time very recently when I would have agreed that factoring should be out of the curriculum, but my thinking is shifting. If we are teaching students how to factor without making any connections to their prior knowledge (i.e.: factoring quadratics is much like dividing quantities with base ten blocks in younger grades), then definitely yank it. However, if these connections can be made and students can visualize what is really happening when base ten blocks become “base x blocks” – aka algebra tiles – then I think there is a lot of power packed into those concepts.

• #### Arlene Farray

March 23, 2017 - 10:43 am -

Michael C,

19. #### education realist

December 1, 2016 - 10:49 am -

I’ve been reading this conversation as it developed, and the recent comments remind me of the Zombie question: https://educationrealist.wordpress.com/2015/11/25/understanding-math-and-the-zombie-problem/ — that is, while it’s legitimate to ask if elementary school kids should learn rote arithmetic before conceptual understanding, most research done at high school level suggests the opposite. Which, of course, doesn’t mean that there isn’t place for practice.

I don’t understand why anyone would say factoring is useless and contrived. The answer to the question “Why do we factor?” is an essential point of understanding. We factor because multiplication offers essential advantages that addition doesn’t. We can’t simplify rational expressions, and we can’t use the Zero Product Property, if we can’t factor. I don’t see how that’s not an important idea to communicate.

• #### Dan Meyer

December 1, 2016 - 11:16 am -

I don’t see how that’s not an important idea to communicate.

Agreed. And I don’t want to parse “communicate” too much here, but I’d like students to experience the power of the Zero Product Property before their teacher talks about it.

• #### joe vignolini

December 1, 2016 - 12:09 pm -

Dan-
Agree 100% with this

” but I’d like students to experience the power of the Zero Product Property before their teacher talks about it.”

• #### Saoshyant

December 1, 2016 - 12:07 pm -

“while it’s legitimate to ask if elementary school kids should learn rote arithmetic before conceptual understanding, most research done at high school level suggests the opposite.”

Most research at the elementary level agrees with what you say the High school research is… conceptual before wrote….

• #### education realist

December 1, 2016 - 6:43 pm -

Yeah, I mean “communicate” broadly. But I’m teaching pre-calc right now, and at some point I casually listed why we factor, and why I counted it an essential skill. The kids were literally agog, and very appreciative. They had no real idea why they bothered factoring until that point (well, except the kids who had me in algebra 2–and there I do let them discover some of the properties).

• #### Kyle Pearce

December 4, 2016 - 7:15 pm -

I had the pleasure of hearing Roger Antonsen speak when I was in Oslo last fall. It was a truly inspirational talk!

• #### Dick Fuller

December 5, 2016 - 8:10 am -

Your’s is a good perspective. Experiencing a part of mathematics from different perspectives provides the opportunity to make the connections with other parts, but you still have to make the connections. Experiencing the result of them is not sufficient. I experience understanding as an active component of experience, at the same time generated and generating.
This is a long and fascinating thread, but I still don’t think I would know how how it feels to understand. Seems to me this is the most valuable gift education can bestow. Here is my take: when I come to an understanding/insight now, I can look back and see more reason and more structure, and I can look forward to greater success with less stress.

20. #### Paul Latiolais

December 11, 2016 - 8:55 am -

I used the 5 overlapping circles diagram as a problem stem (removing the question) as a warm-up exercise in a PD lesson study with high school teachers. It worked very well. The question of what is the area of each circle came up almost immediately (I used the I notice, I wonder protocols). Another question came up that was very interesting. What is the length of the diagonal? As a mathematician, I could not resist working on it. I am still working on it, but am now convinced it can be answered.

21. #### Arlene Farray

March 24, 2017 - 6:42 am -

One thing no one is asking is; why do children have problems with high school math. They have never developed proper foundations at the elementary level. I know from my experience, how concepts work in math do not help the average student. They have labored away learning concepts in elementary school and no or very little time time was spent on deliberate practice to get better at things. Then they reach to high school and nothing is cemented in their minds. Only the very best will survive. Discovery math wastes time on concepts that never help students in the long run, and it is especially disadvantageous to students with weak numeracy skills.
People like Jo Boaler and Constance Kamii have poisoned the well for the math strategies that were used in the past to help students. They with their questionable research and limited models that do not work all the time have left the educationalists disparaging tried and tested methods. As for those who promote the idea that computers can do everything; knowledge helps brain development. This is a fact that is neglected when some practice are deemed archaic.