The Math Teachers of Tomorrow will be the Students Themselves

It’s always seemed like a no-brainer: put 25 students in a classroom with a knowledgeable math teacher, let the teacher teach, and the 25 students will learn the math the teacher teaches. It’s such a simple strategy. What could possibly go wrong?


This student is functioning at a 2nd grade math level. That one is ready for algebra. This one has a crush on the one over there and can’t think of anything else. That one didn’t sleep last night. This one developed a severe math phobia in elementary school. That one can’t read and can only guess on the word problems. This one has a medical condition that requires frequent absences. That one keeps getting pulled out of class for band lessons. And on and on. And on.

The one-math-teacher-to-many-students method seems like it should work.

There are countless strategies designed to make it work.

Society needs it to work.

But it doesn’t work.

Because kids have vastly different amounts of background knowledge. Because one teacher can’t give 25 kids the individualized and immediate attention that learning math requires. Because kids work at different rates. Because kids’ minds - like all minds - wander, even when they don’t want them to. Because kids are human beings, not factory items that can be processed in large batches.

At long last, let’s admit it. The problem isn’t the teachers, and it isn’t the parents, and it isn’t society, and it most certainly isn’t the students themselves; the problem is the situation. We can throw all the time and money we have at trying to improve teaching practices or student participation, but the results will never change. Until we change the situation - until we reduce the ratio from one-to-many to something closer to one-to-one - the needle won’t budge.

That last part shouldn’t even be controversial: educational research has consistently found one-to-one tutoring to be the most effective form of math instruction there is; there isn’t even a close second. But still, the idea of one-to-one math instruction for all students is ridiculous; the teacher/student ratio these days is heading in the direction of one-to-more, not one-to-less. Individualized tutoring for all students is the absolute height of wishful thinking.

Unless, that is, there was a way for the students to tutor themselves

Okay, so right now (literally as I type this) I’m on my lunch prep in an all-purpose room. A Spanish teacher just asked me to help her explain some algebra problems to two students she’s working with who don’t speak English. I’ve been helping them off and on for the past few days, using the silent teaching strategy I’ve written about previously - partly because I don’t speak Spanish! - and they’ve been thrilled to discover that they can fully understand the math involved in my explanations without their teacher having to translate. (This shouldn’t be surprising: math is its own language, and an international language to boot. It’s one area of the curriculum where spoken language shouldn’t be much of a barrier at all.)

But that’s not the most amazing part. Just now, instead of writing out the explanatory equations and steps as they watch (in other words, instead of doing silent teaching), I took a whiteboard off to the side and wrote them down first, out of their eyesight. Without saying a word or even making eye contact, I then set the whiteboard on the table in front of them. After scanning what I had written, their eyes suddenly lit up and they both gave me a smile and a thumbs up. They’re now completing similar problems as I type this, with no difficulty whatsoever. 

In other words, they didn’t really need me; they just needed to see the math.

Why is that?

Because math is self-explanatory.

I know that sounds nuts, but hear me out.

When it's broken down into bite-sized chunks, when it's sensibly sequenced, when it’s represented with simple visuals, when its patterns are revealed, when its components are connected - in other words, when it’s organized - math makes all the sense in the world. And anything that makes all the sense in the world has the potential to teach itself.

Now I know what you’re thinking: am I really suggesting that students - all of them - can be relied upon to teach themselves math, of all things? The answer is yes - I’ve seen it countless times over the course of my teaching career - but even if you disagree, what choice do we have? We can’t control what students actually learn; we can only ever put them in the position to learn something and hope they learn it. And they can only ever be in that position if the “something” lies within their individual “Goldilocks zones” (or zones of proximal development in educational jargon): not too hard, not too easy, just right. But then again, they all have different “Goldilocks zones,” with their different amounts of background knowledge, and different levels of maturity, and different learning rates, and…

See the problem? And see why, until now, one-to-one tutoring has been the only real game in town? Instruction - and especially math instruction with all of its details - must be individualized to have even a chance of working.

And the only way to have genuinely individualized math instruction - especially in the current educational and economic climate - is to turn control of the process over to the students themselves.

Turns out we’ve had it upside-down all these years. Students are perfectly capable of teaching themselves math, all of them, so long as the math materials meet them halfway - with examples, examples, examples instead of just the same old problems, problems, problems.

But how is this even possible?

When I wrote out the math beforehand for my Spanish-speaking friends a couple of paragraphs ago, I was using the most powerful instructional strategy I’ve ever encountered: pre-worked examples. (Want to know why math education in the past had such a dismal success rate? Think back to all of those math textbooks just loaded with problems, and not loaded with examples. For crying out loud, humans learn by example!)

Instructional materials based on pre-worked examples will be the future of math education (and, I predict, many other areas of education too). Because they focus attention. Because they’re self-explanatory. Because they can deliver completely individualized instruction. Because they can be produced inexpensively. Because they can promote actual mastery of real-world skills. Because they can be made available to all. Because they can be sequential. Because they’re always there to come back to if a student’s learning gets interrupted. Because they can be self-paced, self-leveling, and self-checking. Because they make immediate sense to teachers, parents, and tutors - in addition to students. Because they can consistently deliver instruction within the “Goldilocks zone.” Because they can reduce teachers’ workloads. Because they inspire students to take responsibility for their own learning. Because they can be “low floor, high ceiling.” Because they promote higher-level thinking. Because they can provide instant review. Because they permit affordable tutoring for all - and self-tutoring at that. Because they permit affordable self-remediation for students of any age. Because they can put a permanent end to “summer slide.” Because they can be used for test prep. Because they can make classroom math instruction far more fun (think real-world projects). Because they can make classroom math instruction far more relevant (think real-world projects).

In short, because they flat-out work.

Turns out we’ve had it upside-down all these years. Students are perfectly capable of teaching themselves math, all of them, so long as the math materials meet them halfway - with examples, examples, examples instead of just the same old problems, problems, problems.

Steve Hare

Math-Whisperer. Humanist.
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