I’ve always felt conflicted about repetitive practice.

On the one hand, I see how vital practice is. Musicians repeat the same piece again and again. Soccer players run drills. Chefs hone their chopping motion. Shouldn’t math students do the same: rehearse the skills that matter?

But sometimes, I backtrack. “This is just going to bore them,” I fret, scanning a textbook exercise. “I’m emphasizing the rote aspects of math at the expense of the creative ones. They’re going to forget this skill anyway, and be left only with the insidious impression that math is a jackhammer subject of tooth-grinding repetition.”

(Then I assign the exercise anyway, because class starts in five minutes and— despite my repeated petitions—the administration has denied me access to a time turner.)

These two trains of thought suffer daily collisions in my mind: repetition is dull, but repetition is necessary. This inner conflict takes for granted the idea that repetitive practice is a separate endeavor, a distinct stage of the learning process. First, you learn the concept. Second, you practice it. In this view, practice is like cleaning up after a picnic: absolutely essential, but not much fun.

But this summer, a very wise teacher showed me a path forward, a way to reconciliation.

I’m referring, of course, to a two-year-old named Leo.

Toddlers have always fascinated me. They’re clumsy little people who laugh spontaneously, nap at random, and feast on Cheerios—basically, my heroes. And I love imagining a complex inner life for them, full of discovery and improvisation. I think they experience a far more voluminous river of thought than can trickle through the narrow spout of their language skills.

I spent an afternoon with Leo and his parents. When we got to the playground, Leo wanted to go down the slide.

Then, Leo wanted to go down the slide.

Then, Leo wanted to go down the slide.

Then, Leo wanted to go down the slide.

Then, Leo wanted to go down the slide.

Watching Leo’s joyful, oblivious repetition of the same activity, over and over and over, I could see why babies get a bad rap as theoreticians and intellectuals. I couldn’t help thinking—as Leo ascended the stairs and descended the slide for the seventeenth consecutive time—that maybe there just isn’t much going on inside a two-year-old’s thoughts.

But then, something changed: Leo wanted to go down the slide.

Okay, fine, no change there. What surprised me wasn’t the down, but the *up*. Instead of the usual steps, Leo sought out a ladder on the opposite side of the structure. From there, he could reach the slide only by crawling through a sort of cage-tunnel, full of wide gaps, through which his whole body could easily fall.

“Oh, lovely,” his dad groaned. “Now you want to do the danger path.”

By Leo’s fourth journey through the cage-tunnel, his father was getting a little worn down from the heightened vigilance of supporting him. So I offered to step in as Leo’s spotter, and began watching more closely.

Every moment brought a tentative new movement. Probing forward with his feet. Shifting his balance slightly. Releasing and re-gripping the bars above. From a distance, he must have looked merely squirmy. But up close, I could see him challenging his kinetic abilities.

It dawned on me that he’d chosen this path specifically because it taxed and excited him. Because he desired a new frontier.

Soon, we came to a particularly tricky step. Leo had exhausted his repertoire of wiggles. Up to this point, he’d paid me no notice, but now—without looking up—he grabbed by hand and pressed it against his chest. He needed help. This was his way of asking for it.

It struck me in that moment: Leo’s repetition on the structure isn’t mindless. It’s a deliberate path to mastery.

Leo is practicing.

No one needs to command or force Leo onto the climbing structure. It’s instinctive. He wakes up every day with four funny limbs attached to a silly little body, and—naturally—he wants to learn how to move the whole apparatus around in a coordinated, effective way. He wants mastery.

Sometimes, that means repetition.

Sometimes, that means trying a “danger path,” something hard and novel.

Sometimes, that means pausing and reflecting.

Sometimes, that means asking for help.

Repetitive practice doesn’t belong in quarantine. It’s not some separate chapter in the book of learning, to be consulted only at one precise spot in the sequence. Instead, practice functions best as part of an integrated and organic whole, enmeshed and woven in with other aspects of learning.

Practice is the best way to hone and solidify a skill. When students want the skill, practice comes naturally.

I ought to be striving for that same ideal in teaching mathematics.

Now, that’s not necessarily easy. Whereas Leo can easily gaze upon the entire play-structure, math is less tangible. Often, students possess only a hazy, uninspected idea of what they’re building towards. They may see their own purposes as obscure, opaque—unlike Leo, who knew exactly what he wanted.

All this helps define my job. I need to help supply a vision of what mathematics is, a sense of the powers they can acquire. I need to paint them a compelling picture of the play-structure.

A goal-motivated human is a powerful, capable thing.

Even if that human naps at random and feasts on Cheerios.

No, not “even if”—*especially* if.

If you can make learning and practise a kind of play, you win everything. The internets, the kudos, the stage at teaching conferences. And the prize of changing the lives of children who would otherwise be switched off.

There are a great many children who dislike STEM subjects for just those reasons, and unfortunately it means that only those children who like STEM *despite* the experience get to the professional world. More creative children, female children, personable children find other pursuits, and it really undermines the potential of those that do make it.

Seek ye out the holy grail.

I agree with you wholeheartedly about making math a kind of play. I have always seen it that way and that is why I have always loved math. Math is just a huge set of puzzles, which is a game. I love the fact that there are definite answers and I just have to fit the pieces together in the right way to make the picture.

However, the stereotype that female children do not like STEM subjects really riles me up. I detest this stereotype. When I got to college and was in education classes, I was warned not to perpetuate the stereotype that girls were not good at math. I was dumbfounded. Where did this nonsense come from? Before that, it never even once came close to entering my mind that girls were not BETTER at math than boys. That had been my experience. As a female, and former female child, I loved math and everything about it (although I was soon to have the rude awakening that real math isn’t really about the numbers, although I came to love that too). When we would have math races at the board in elementary school, they usually pitted the girls against the boys and the girls won more often than not. I had exactly one male math teacher in all of my elementary and high school education, and that was in 6th grade. In my high school, there wasn’t a male teacher in the entirety of the math department. I couldn’t comprehend a world where math was a male subject. After that, I started looking around and did notice more guys in the upper level math classes, but never would have noticed if it hadn’t been pointed out.

math usually give me a head ache

I have wondered about Malcom Gladwell’s 10,000 hours rule relative to learning mathematical concepts and skills, but, like your playground comparison, hours of work that are not practicing the same idea, but fitting ideas together. The playground comparison is helpful to me!

The main problem is that math is treated too formally, too abstractly, and without the real world experiences that would give it some purpose. The rush to symbols doesn’t help. Doing puzzles, making things, taking things apart, being creative. All of this needs to be experienced before and while any attempt at formalisation and abstraction is embarked on. “But they are not doing math!” is the inevitable cry.

I found this site the day I planned to drop out of calculus and change my major to something else. I am not sure what because everything else is just so… less. It is a struggle because, on one side, it is so hard and I think I must have made a mistake because, surely if this was right for me, it would not be so. But, at the same time, I really cannot express how much I love it – especially because it does not come so easily. I am like a small child, doing the same problems over and over and I never get sick of it. I need to know “why”, not just learn the patterns so I can pass the tests. Everything here resonates so well with me, at this immediate point in my life. I just wanted to say I am grateful.

You are a true mathematician. Upper level math is not about the patterns but about the why. Sure, recognizing patterns is a huge part of it, but even recognizing patterns is figuring out why the next number is what it is. This is why I think so many students have a hard time with math…they think it’s this random decision on why things behave as they do, but the real math is the underlying logic behind why those steps are taken. Stick with it!

Hmm. I wonder why Keith Devlin of Stanford and various colleagues in the world of higher mathematics call mathematics “the science of patterns”?

I think you kinda missed her point. Consider proof by induction, arguably one of the most explicit examples of using patterns in math, requires something more than observing patterns – it takes the ability to define and create structure and understand why those things satisfy the pattern. To claim math is simply the science of patterns is to claim that words are letter combinatorics.

Because the ‘science of ‘ implies the how and why of patterns?

Calculus is one of those challenging transitions at first for everybody…. mathematicians included.

Never ever give up! 🙂

Mathematics is hard, even for mathematicians. Struggling is normal, it shouldn’t deter you. Plus, you are allowed to help yourself: draw a picture, search for a similar problem, play the abstract thing through with an example or ten.

Ben: one of your best posts in a cascade of excellent, insightful contributions to mathematics teaching and learning. Keep up the outstanding work.

Thanks so much! I’ve had a traffic bump of late, which means I’ve been bad about replying to comments, but I really appreciate your reading and adding to the discussion.

I wonder if the slide is a reward for practising the climbing or something to practise in itself. Or both.

My guess would be more the former, but probably a bit of both!

I think the mastering of climbing is a reward in itself for that toddler.

The sliding down is also an ‘excercise’. Something like that baby that drops his toys on the floor over and over again. He discovered gravity and experiments with it over and over again. They are grinding a path in their brains that helps them predict the world.

Who said exercising can’t be fun? 😉

Most children stop going ‘compulsivly’ on that slide when they get older. A few times because it’s fun, but than they look for something more fun/challenging.

I’ve had some success with getting students to practice by using online resources like Quizlet and IXL.com. Being timed or scored seems to get kids fired up a bit. Like knowing they have to get 30 correct on IXL.com. Also, you get the huge learning advantage of immediate feedback. The tech is out there now to make “grinding” a little more palatable (and educational).

Good to hear from you Bryan! And yeah, I think this is a good role for computers. There’s a lot that teachers do that computers can’t; but structured practice is one where they’re a big help.

No, I just asked a question.

Making it into a game, well, you will need creativity for that. But, you already have it. 🙂 The above mentioned immediate feedback and getting scores thing is something worth to think about, because it is really effective. I am an adult and played with addition at Khan Academy for hours. Ridiculous.

Incredible!!!!! is the one word i could write and there series of appreciation rolling up and down in my mind

I really like math and do unterstand it very well,because I really like to have a clear solution 😅That’s kinda embaressing.Every one in school is like ew math ,I can’t understand it and I’m like well…it’s pretty easy…..and they look like WTF?! What I like about math are the solution and the way you get the solutions like finding the right formular (formel idk it in english) .It’s so funny and I like it so much its so weird.Last day my sience teacher came and told me that I solved the task very well but he hasn’t thought some one could solve that from our class.I was shocked.Cause it was easy.Well what i wanted to say was , you are totally right!

Btw: I love your blog! ♡

Math finally made sense when I got into physics in high school. Ahh! Numbers, letters, formulas, they were necessary and DID something. I wish I’d had that kind of math instead if abstract whatever to solve. I liked being able to see it in action.

This was a beautifully written and insightful piece – thank you! I’m a big proponent of 20% time in the class (student initiated learning/passion driven learning) and it’s got me re-thinking how to bring the mastery to THAT process through repetition. I also shared it on my FB page https://www.facebook.com/20procenttijdnl, hope that’s okay!

Definitely – thanks for reading!

A terrific piece – but it leaves out a few important details:

First, what you observed in your 2YO is how 2YO’s – well, kids in pretty much most of the Early Childhood window! – are hard-wired to learn: exploration and repetition and then extension/refinement. HE chose the activity, HE was driven to repeat it, and then to explore other ways to get on, and over time he will likely get bored with it but eventually come back to sliding with a new approach entirely (going UP the slide, for example, with the requisite repetition, or transferring sliding to other slides or to things that look like slides (the surfaces the security guards at the mall were always chasing my own kids off of LOL)). It’s also worth noting that the activity involved not only mental but physical practice; his whole being was involved here.

However, and this is a BIG “however,” while repetition is still important for older kids, they are not driven to it the way younger children naturally are. As a music teacher who has to impress this on my private music students, I share with them what we know about learning, about muscle memory, about myelination, and WHY they need to do some things repetitively (incidentally, 3 times in a row w/o mistakes is their goal, because something about that third time helps “seal the deal.”) – but by 7-8YO it no longer comes as naturally to most of them as it did as toddlers. A change in their neurodevelopment has taken place. It’s not only a matter of Making It Fun, but also taking those changes into account and working with them.

Once we get to school, kids are by and large no longer choosing their own activities, or how long to spend with each before moving on, so their own process of learning through repetition is disrupted by the necessities of scheduling and large-group paced instruction; the difference between a Kindergartener in a public school and one in a Montessori setting, where s/he can choose the activities and how long to stay with each, is very telling. When we assume that Making It Fun will create similar conditions by “Motivating” students, we don’t allow for the other factors that make it natural to kids: self-selection of activity, and self-direction in terms of how long to practice it and which aspects to practice.

So – a good start, but without a deeper understanding of What’s Going On In There (to borrow Lise Eliot’s book title), it’s only part of the story.

Yeah, I think that’s all fair.

(Interestingly and impressively, you totally called Leo’s next move: he ended his practice session by going up the slide, rather than down it.)

There’s a few threads in what you said that I’d tease apart:

1. Kids are less interested in repetitive practice when it’s an activity they didn’t choose. I agree.

2. “Making it fun” can’t totally replace that sense of purpose you get from choosing the activity. (i.e., “Fun” isn’t enough to guarantee deep engagement with practice.) I agree.

3. “A change in their neurodevelopment” takes place such that older kids are inherently less inclined to conduct repetitive practice. I disagree with this bit. I don’t think there’s anything biological about older kids’ disinclination to practice; rather, I think it’s precisely the cause you identify in #1 and #2, which is to say, they don’t particularly WANT this skill, so they’re not that jazzed about practice.

In short (and this is a blindingly dull conclusion for such verbosity on my part): kids need to want the skill. Part of the purpose of math class needs to be guiding them to want these skills.

A fair point about Point #3. I may have worded it less clearly than I should have, but there *is* less of a tendency to engage in so much repetitive practice because once a child reached a point where he can reason out What Will Happen If, rather than having to do it by physical trial-and-error and *actual* experience. As kids realize what they’ll likely be capable of, the drive for that kind of toddler-like repetition does diminish somewhat for most kids.

I agree that kids should *want* the skill. My point, I think, is that we need to also know about neurodevelopment – and about what’s inborn and how it develops and when/how – in order to best take advantage of it. We can’t necessarily equate Toddler Learning with, say, Middle School learning. Some similarities, but many other new and exciting things take precedence then which mean there has to be a different approach, an acknowledgement that no, 13YO’s don’t learn the same way toddlers do. 🙂

To hopefully clarify a little further, now that I’ve had a bit more time to mull on this: As a music teacher, there are things that most kids find deadly dull as music students – except, for some reason, guitarists and drummer, who will practice till their fingers are blistered and bleeding (and a few crazy pianists. LOL). Scales and etudes are meant to be skill-builders, and some kids actually enjoy the repetition of scales and will work on them with a kind of Zen approach, while for other kids, the only reason they bother taking the instrument out of the case or sitting down at the piano is because the parents paying for the lesson insists that they do so. I do try to find ways to make these less drudgery, but the fact remains that building the muscle memory for those discrete skills helps them in the Big Picture. I found a poster (at ToneDeaf Comics – LOVE those things!) that shows the progression of progress as part of a bigger picture: “If you practice, you get better. If you get better, you play with better players. If you play with better players, you play better music. If you play better music, you have more fun. If you have more fun, you want to practice more. If you practice more, you get better.” It’s a feedback loop – similar to the one Leo was caught up in, but in this case there’s an element of delayed gratification that toddlers don’t come with, and so their repeat and they refine in ever-so-slight degrees because they’re driven to Do It Now, and the End Game changes. In the case of practice for older kids and adults, we get defined objectives (standards), the SWBAT on the board at the beginning of class or at the start of a new unit, and students are more or less led thru the learning. The advantage is that at this age, they CAN be led at all – I had a colleague once describe teaching Kindergarten as “taking chickens to the opera.” LOL

Anyway, for me, I’m learning to adjust my music teaching to try to make the etudes less stultifying: I harmonize etudes/turn them into duets, or suggest that students “tweak” them to make them more interesting (change up the rhythm, mess with the dynamics or tempo, even adjust the melody with trills or other ornaments – as long as they can demonstrate that they can also play as written), and with some students I hold off on scales while for others I’ll introduce them early (some kids love the sense of accomplishment they get from mastering them, others hate the things). I have the luxury of teaching my private students one at a time, so I can make these adjustments on the fly; this would be harder in, say, a band class where by definition we all play together at the same time.

It’s a process for us all, students and teachers, and I think it’s neat that your observations of Leo are informing your understanding of learning. I found and continue to find, having two non-neurotypical kids (and having been one myself), that Learning About Learning and seeing how differently THEY learn compared to their more typical peers makes me a far different (and hopefully better) teacher than I would have been without that understanding. 🙂

I’d almost forgotten I’d written this, and I promise this isn’t intended as self-promotion, but this came up in response to someone on a parenting forum today and it resonated with me here, the parts about repeating and learning and growing – and persistence – something else we short-circuit when we stop kids from playing and make them learn in lockstep:

http://crunchyprogressivemusicmama.blogspot.com/2013/09/non-academic-skills-can-we-teach-those.html

My two cents…..I found a math program called Math Facts Pro. I began using it with my students last year. The student has to get 50 correct problems, and then it shows the student what facts they know and what facts aren’t fast enough and what facts are wrong. I challenged the kids to get the grid filled in. If they did what they were required for the grade, they got to join the Donut Club. I bought them a doughnut and gave it to the student at lunch. They want to practice. They want to learn. There are those students who don’t want to be bothered, but we can’t do it all. I know what’s ahead of them. I know algebra requires mastery of all four operations. Most students will take basic algebra. I want to make their lives easier. I get great enjoyment when students make progress. I also enjoy when the student is almost there. You get too see determination to complete the operation.

Check out some works by Maria Montessori. She explains the child’s innate desire to practice until satisfaction is reached. Also, when people actively create their own mathematical knowledge in the context of real world situations, they are going to understand and “know” it. It is not true that the only paradigm is “learn (ie be taught) the concepts and then repeat them.” You can repeat, experiments, manipulate, and in that way discover and learn the concepts for yourself…. the same way mathematical formulas and theories were discovered in the first place. This takes a lot more work on the teacher’s part, to set up the experiments and the activities with the right amount of guidance to reach the desired conclusions. There are some curriculum that do a better job of this than others, too. It is also generally not how we’re “taught” to teach mathematics in our education prep courses.

Yes yes yes!! Better yet, see if you can arrange a visit to a Montessori school with a Montessori teacher to talk you thru what you’re observing. (Read some of Montessori’s own work first, though; that knowledge base will come in handy!)

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I love this post! Offering repetitive practice in math instruction can sometimes be seen as a four-letter word. I, however, do see a lot of value in it. In my teaching experiences I have had a lot of low-level learners and I feel they have benefitted most from repetitive practice. Much like Leo on the slide, my students have treated equations (for example) as their “jungle gym”. The first time they look at a problem they think they can’t do it and really don’t want to try. The more they practice, the more they explore what strategies to get rid of and works best for them. In addition, when they get to a point where they are stuck, they “reach out” for help. Not to mention, giving them repetitive practice on the topic gives them a comfortability with what they are learning. In those classes with lower level learners, that comfortability means the difference between a student coming to math class dreading the next hour or one who comes in looking forward to the next challenge I am going to give them for the next hour. But you are also absolutely right in your opinion that it is just part of a well rounded math classroom. Too much repetitive practice and we miss the opportunity to offer inquiry and enrichment. Probably why teachers are part artist…..painting the perfect lesson plans tailored to their students!

Reblogged this on Schools for Humans and commented:

Oh, heavens, what a wonderful explanation of WHY and WHEN repetition is good in learning! The only thing that I’d add is the magic of spaced repetition — see my earlier posts on how that gloriousness can transform our K–12 educational praxis!

Now, I shall say what a typical teenager with a mathematically bright yet too occupied with teaching philosophical theories to math students teacher will say: please, teach my math teacher how to mathematically engage his students and teach the theories of math.

I really enjoyed this post and the discussion in the comments! I am one of those folks always asking, “Why?” and continuing to practice until I feel I’ve mastered a topic. So, studying mathematics has been very rewarding for me in that way. There is (almost, if not) always a why, and there is usually an easy way to test mastery — do the hardest problem I can get my hands on, or even try teaching it to someone else.

I don’t know if anyone else has experienced this phenomenon, but I often think about practicing things until I reach a “critical mass” of experience. There is some tipping point where I can look at a problem and say, “I bet if I treat it like [a past similar problem], but tweaked to account for this specific problem’s unique features . . .” and then I’m off to the races. Perhaps that’s just pattern recognition kicking in; by identifying the similarities I at least begin to think about what past approaches might prove useful in the current situation.

Perhaps ironically, one of the topics I struggled mightily with was proving things by induction! The only thing that seemed to help was grinding out many base cases, and then finally attempting it with n+1. Anyway, now that I’m more comfortable with them, I appreciate the power and beauty of being able to prove something with just a few specially selected cases.

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