Math = Structured Problem-Solving

Can you solve this?

Sketching out our liberal-arts-major thoughts on what math in a new-kind-of-STEM school could look like, we mentioned something we dubbed "The Japanese Method" of teaching math — structured group problem-solving. 

Pics or it didn't happen!

That, at least, was the response of one reader. What does "structured group problem-solving" actually look like?

Today: one of the most glorious ideas alive today in math instruction. (Note: all this is stolen shamelessly from the book The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom, by James W. Stigler & James Hiebert. If you haven't a copy, buy one — it's that eye-opening! 

In short: great mathematics is a culture, not just a method — but methods can help carry cultures. 

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The gist: before showing students how to do a type of problem, just present them the problem, and let them struggle with it. 

Start by letting them struggle by themselves. Then (often, though not always) let them struggle in small groups.

As they struggle, stroll around the classroom, noting how different groups have solved the problem in different ways. Make notes of a few different methods, and invite the creators of those methods to present them on the board.

Start with confusion. Spin understanding out of chaos. 

An example, you say? Let's!

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Try this problem on your own. Find x:

This problem, by the way, comes from The Teaching Gap.

If you're bamboozled, good! I'm a math teacher, and I was stumped when I saw this.

If it helps, here are a few basic geometry rules the kids would be familiar with:

• There are 180º in a half turn (or "in a line") — so the fat angle above 'B' would be 150º, and the fat angle below 'A' would be 130º.
• There are also 180º in a triangle (or "three-pointed-three-liney-thing").

Let's pretend we gave this problem to a class of students.

First, we'd let them puzzle it over on their own. A minute or two might suffice.

Then, we might ask them to form small groups, and share what happened. What did they try? What did they find? What questions do they have?

As they chat, we (the teacher) would stroll about the room, noting different methods kids have come up with.

Let's assume we witness three different methods. After a few minutes, we ask a few students (let's call them Josh, Dana, and Sanket) to come to the board and share what they figured out.

Josh's Method

What did Josh do? He stretched out a line, and made a triangle — yay, triangles! Triangles are simple, and as previously mentioned, we know stuff about them — like that their inside angles add up to 180º. 

Josh has labelled that top-left angle "30º", because it matches the 30º angle in the bottom right. (This is a cool previous pattern that they kids would come into this class already knowing.)

How big is that remaining angle? Well, we have the two other angles in that upper-triangle: 50º and 30º. And 50 + 30 = 80. So the remaining angle has to be 100º.

And our blessed x is smack up against that 100º angle, so it has to be 80º.

Well, done, Josh.

Now, much American math instruction would stop here: we've figured out x, after all! But is Josh's method the only way to solve the problem? Heck no!

A standard Japanese math lesson would progress to Dana's explanation. 

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Dana's Method

What did Dana do? She decided to drop a line straight down, and make two triangles — right triangles. If triangles are magic, then right triangles are even... magicaler? 

Dana can quickly figure out the missing angles in these new triangles, because she recalls that the angles inside a triangle always add to 180º. 

Using this, she finds that one angle is 40º, and the other is 60º.

Now she has a half turn (aka a "line") — 40º, xº, and 60º. Together, they have to equal 180º — so x must be 80.

Well done, Dana!

Even a quite progressive American-style math teacher might stop here. They've done their job: they've demonstrated to their students that math is creative, and that there isn't just one solution method.

Not our Japanese math teacher. Demonstrating a plurality of methods — student-originated methods — is the norm. So let's move onto Sanket.

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Sanket's Method

Whoa: weird.

What's Sanket up to here? Well, like Dana, he decided to just drop a line down from the top to the bottom. But unlike Dana, he didn't make two triangles — he made one four-sided shape. 

It's an interesting move. Four-sided shapes ("quadrilaterals", but aren't our lives already complicated enough?) are more complex then triangles. On the other hand, he just has one of them, rather than Dana's two triangles.

Then he gets to labeling: one angle is 130º, and another is 90º. A third angle is 60º. And the final angle is, of course, x.

Sanket recalls one thing about four-sided shapes: their angles add up to 360º. And so he adds 130 + 90 + 60 + x = 360, and finds (drumroll!) that x = 80!

Well done, Sanket!

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To sum up:

This "structured group problem-solving method" starts with hard questions. (American math classes, by contrast, typically start by explaining "what to do".) 

This method forces students to think for themselves, and then to think with peers. (American math classes typically start by forcing students to follow the teacher and book.)

And this method demonstrates that math is a creative, flexible pursuit — an art as well as a science. (American math classes typically demonstrate that math is about following set procedures — like filling out taxes.)

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Is this method dangerous?

There is, I think, a very sensible apprehension that many of us might have to this method — that while students may learn the creative possibilities of math, they won't learn what works best.

There may be, for example, three different ways to solve the problem above, but students will benefit from learning the most efficient way. 

This is an even more potent objection when we're teaching foundational processes — like adding, subtracting, multiplying, or dividing.

In our next post, I'd like to address this concern head on.

Faith-based math

Our school shall have no faith-based math. Before I set off an Internet flame war (or is it too late already?!): I'm not talking about religion right now. Except maybe I am?

The Calvin & Hobbes strip above really nails the experience of many students in math class. Doing well in math amounts to taking things (formulas, for instance) on the authority of the textbook. Students who do well in math class are those who can best memorize these bits of dogma.

Obviously, this has nothing to do with actual mathematical understanding.

I know that this idea sounds incontestable — and, well, it is. Of course students should understand what they're doing in math!

Yet this principle is broken in nearly every textbook, in nearly every class.

I'm reminded of this today as I prepare my economics lesson for the afternoon. We're reading a popular book on economics — I won't mention the title — and are trying to understand how supply and demand curves shift when products are taxed.

The students are struggling to understand it. They're model students: reading carefully, testing their comprehension. But they're frustrated. I should be able to help them, because I should have a full understanding of the topic at hand.

The thing is: I don't. And the book is no help.

The book — at least this portion of the book — is, in effect, faith-based. It doesn't explain taxation the way it claims to. It doesn't matter how hard the reader works: they're stuck in faith-based math (or, in this case, faith-based economics). They're forced to kowtow to the author, and simply assume the theory makes sense.

Ack. Uck.

I'll see what I can do for the class — I may need to bring in an outside economist to help us make sense of this. I'll certainly own up to my own non-understanding, and help the students explicate the gaps in their understanding.

That is, I'll help them see what they don't see.

And that's useful, in the short-term. But here's a long-term promise we can make for our school:

When studying any analytical, reasoning-based subject, students will never be expected to take anything on faith. We'll inculcate them in the truth that, if some idea (a math formula, an economic concept, a chemistry… chemically-thing!) has been understood by someone else's mind, it can be understood by their mind.

And we'll rear them in the conviction that achieving this understanding — capturing its complexity in their own head — is one of the most beautiful experiences available to us humans.

A School for Difficult, Exhilarating Math

The problem:

Math is more than following someone else's recipe. Math is about prolonged puzzled, creative daring, and brilliant insights.

In my last two posts, I argued that we need to make math as simple as possible. If we're going to be risky, let it be in making mastery too easy.

That sounds snarky, but I mean it seriously. It's our duty to students to all but guarantee that they'll succeed in coming to a full understanding of math.

But that's not enough. It's not enough that all our students excel at math. Our job is also to lead them to love math.

How can we do that? How can we lead kids to mathematical infatuation?

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Well, there are a number of ways we'll be pursuing this, but one major route:

Bring in creative puzzles. Puzzles that are challenging. Puzzles that can't be unraveled right away — that need to be put away and considered hours or days later. Puzzles that offer multiple solution methods. Puzzles that require creative daring — trying something that, on the face of it, might seem strange or stupid. Puzzles that, week by week and month by month, grow creativity.

Some of these puzzles will incorporate old ideas — the concepts that the kids have learned in previous years, only shown in an unfamiliar form. Others of these puzzles will preview new ideas — the concepts that the kids will be learning in the following months and years.

What matters is that the puzzles not just be technically difficult, but conceptually clever — that they be about ideas.

As a class, the goal isn't merely to get the right answer, though that's (of course) very important. The goal is also to explore diverse routes for finding that answer.

Good, complex math puzzles typically can be solved using multiple methods. For example, consider a classic math puzzle: What's the sum of all the whole numbers from 1 to 100?

There's a straightforward way to solve this: just add away! 1 + 2 + 3 + 4… + 99 + 100 = 5,050.

There's a clever way to solve this: Look for similar pairs. 1 + 100 = 101 2 + 99 = 101 3 + 98 = 101 … 50 + 51 = 101.

There are 50 pairs. Each pair adds to 101. 50 * 101 = 5,050.

There's a weird way to solve this: Add sets of 10s, and spot the pattern. Sum of 1–10: 55. Sum of 11–20: 155 Sum of 21–30: 255 Sum of 31–40: 355 … Sum of 91–10: 955 And then add all those together: 5,050.

On their own, students will come up with all these methods — and more besides! Our teachers need only give them the encouragement to do so. (According to the excellent book The Teaching Gap, this is actually pretty close to how math is taught in Japan, and to a lesser extent Germany.)

One of these super-challenging problems can be given each week. At the week's end, our students can present their methods. And teachers can lead the class in exploring how, ultimately, each method is exactly the same thing.

This is deep mathematical understanding.

Diverse routes lead to fuller understanding.

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But the results could be even better than that. Devising (and valuing) diverse routes to solving puzzles changes the nature of math. No longer is math something out there to be obeyed — it's something in you to be explored.

Diverse routes make math personal.

As a class, we might award a prize each week to the method that is the most clever, and to the method that is easiest to perform in your head, and to the method that is the weirdest!

It's not that there's a single right method, and many wrong methods. It's that there are many methods, each beautiful or ugly or useful or pointless in its own way. 

Math is an expression of humanity.  It's a human thing, not a robot thing.

To best appreciate these puzzles, we might collect them (and our favorite methods) in binders, and encourage students to re-visit them from time to time.

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The focus of this post is to talk about love, not mastery — but mastery is exactly what will result as students slowly internalize these puzzles and their methods. As students write these ideas in their long-term memory, they will become more and more brilliant at mathematical problem-solving.

The SAT and ACT are made up (nearly exclusively!) of these sorts of puzzles. Bizarrely, a curriculum of creative math, of loving math, will end up being the best standardized-test-prep curriculum imaginable.

Not that we're putting much weight on that.

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In brief:

Alongside a micro-scaffolded curriculum of tiny mathematical discoveries (based on JUMP Math), our school will also have a curriculum of unguided math puzzles. We might have 1 super-challenging problem per week. Students can work on the puzzles by themselves, or in teams. At the week's end, students will present their methods, and the teacher will help the class explore why each method works.

Our hope is that this won't just help raise kids who are adept at math — but kids who truly enjoy it.

A School for Complete Mathematical Understanding (2 of 2)

Complete Math Understanding and Social Justice

In my last post, I identified a huge problem with traditional schools: they don't reliably bring all students up to a complete understanding of math.

This was a problem in the middle of the 20th century. This is a disaster at the beginning of the 21st. 

If I can interject a bit of social justice: the inequalities in contemporary American society as numerous as they are complicated — but there is a strong correlation between economic success and mathematical understanding. This holds through many inequalities.

There's a gap between the outcomes of males and females, but when you filter out differential math abilities, the gap gets smaller. There's a gap between the outcomes of white students and students of color, but when you filter out differential math abilities, the gap gets smaller.

Obviously — obviously! — these disparities are not reducible to math performance. There is sexism, and it matters. There is racism, and it matters.

But there's good evidence to say that if we provide a way for all students to excel at math, we will make a significant stride toward reducing inequality in American society. This is something worth fighting for.

All right. So: how can we accomplish this?

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Step #1:  JUMP Math

We'll start by using the gold standard for curricula that achieve full comprehension: JUMP Math.

JUMP is published by a non-profit organization from Canada, the brainchild of John Mighton: an actor-turned-playwright-turned-math-tutor-turned-Math-Ph.D.-turned revolutionary-curriculum-designer. (Y'know, one of those people).

(Not that it particularly matters, but you might have seen Mighton before — he played the inspirational teacher in Good Will Hunting.)

The heart of JUMP Math is the insight that each math concept — even the very most complex ones — can be broken down into smaller and smaller chunks, until they're small enough for students to understand in mere seconds. Students come to understand (not merely perform) each chunk quickly, and then jump onto the next micro-concept.

Emphasis on quickly. In JUMP, students move from insight to insight, with only a small bit of struggle in between. There's little of the floundering that makes many (many) students feel that they're just spinning their wheels, that they'll never understand math.

People don't like floundering. People don't like struggle without hope. People love to struggle and achieve.

Video game makers understand this. In the last few decades, they've mastered the psychology of struggle and reward, and have made video games into feedback systems so well-suited for human brains that they are nearly addictive.

JUMP stokes the ego. JUMP (metaphorically) turns math into video games.

Learning anything — feeling the change from not-knowing to knowing well — feels fulfilling. Learning quickly feels especially fulfilling.

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a (Crucial) Side Note

Crucial side note: small struggles are not enough. To be psychologically healthy, humans also need big struggles — we need to take on enormous projects that we're not confident we'll be able to solve.

In our school, our math curriculum will also have another component — baffling puzzles that students will need hours and weeks to unravel; puzzles that will allow for creativity and individualized solutions.

Our school's math curriculum will be both/and: students will fully learn the core K-12 math curriculum through a micro-scaffolded JUMP Math curriculum, and they will cultivate their creative brilliance through non-scaffolded puzzles.

I'll be blogging later on the second half of this.

End of side note.

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What about Struggling Students?

Some students, of course, have more difficulty learning math. (Again: people are not blank slates.) That doesn't mean their mathematical understanding has a ceiling. 

JUMP Math works wonderfully for them, too. Teachers simply break the micro-concepts down into still-smaller chunks — however small the student needs to quickly and fully understand the concept.

Every student can learn one more concept. Every student can learn another concept after that. There are no ceilings in math.

This psychological insight is perhaps the most revolutionary piece of JUMP. My students who use JUMP report having new faith in their abilities to learn. JUMP teaches that anything is possible in learning.

Learning to teach the JUMP Math way is an art: one of the most joy-inducing skills I've honed as a teacher.

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How is This Different?

Traditional math books have two phases: they introduce the concept (the first couple pages of each chapter, replete with 2-3 sample problems), and then they ask students to apply the concept (the next few pages, featuring about 20-30 problems).

JUMP Math doesn't do that — it teaches new concepts through the very problems it presents.

Every question enlightens. Students learn constantly. No problem is wasted.

I recall, when I was in high school, staring blankly at my math book, reading the sample problems a third, fourth, and fifth time, wondering what I wasn't getting.

(I also remember stabbing my book in frustration. Lost a good pen that way!)

JUMP, again, makes learning math easy. It makes achieving a fundamental skill of the 21st century simple, something everyone can do.

This seems, to me, a fundamental human right.

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But Wait, There's More!

Any curriculum that did all the above would be excellent, but JUMP Math goes an extra step.

Instead of asking students to merely perform math, JUMP leads them into the messy guts of understanding.

JUMP helps all students clearly understand somewhat-obtuse concepts that I recall merely memorizing.

Students understand why order matters in subtraction and division. Students understand why order doesn't matter in addition and multiplication. Students understand why you can't divide by zero. (Hint: it has nothing to do with blowing up the universe.) And so on.

Again, I'm pretty "good" at math: I got a perfect score on my GRE Quantitative, for example. But I regularly learn new things when I teach with JUMP. Big things. Things I never thought to ask about. Things that make me aghast I didn't know them before.

JUMP Math makes it simple for every student to develop full mathematical understanding. We'll ground our curriculum in it — and move beyond it, too.

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I listed, in the last post, four things we should be able to promise students vis-à-vis math. By explaining JUMP, I think I've handled the first two of them — complete (2) understanding and (3) solving of the K-12 curriculum.

I haven't touched on (4) remembering everything that students learn, and (4) allowing students to be active learners, rather than passive receivers.

But I suspect I'm pushing the upper bounds of how long a blog post ought be already. I'll look forward to addressing those in future posts!

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In Brief:

Understanding math is (and will continue to be) crucial in the 21st century. Yet our brains aren't built for it. What's needed — and what our school will set itself to delivering — is a math curriculum that takes seriously how difficult and unnatural math learning is, and then helps students master it entirely. To do this, we will start with the JUMP Math curriculum, and build from there.

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For Further Reading:

John Mighton has written two books — The Myth of Ability and The End of Ignorance. Both are excellent, though start with the first. For a quicker overview of JUMP, however, take a peek at these two excellent posts in the New York Times Opinionator column — "A Better Way to Teach Math," and "A Better Way to Teach Math, Part 2."

A School for Complete Mathematical Understanding

The thing is, this is actually true — for all genders.

The problem:

The 21st century rewards those who can think mathematically, but the human brain isn't built to do that.

A new kind of school can train students — all students — to understand math, all the way from 4+5=9 to statistics and calculus. Our goal is nothing less than that. And there's good evidence that this isn't just a utopian wish.

But before we continue with the optimistic thinking, we have to make something perfectly clear:

Math is unnatural. Math is hard.

Sure, there are a few rare humans who apprehend new mathematical ideas as readily as the majority of us apprehend the plots of, say, Michael Bay movies. Those people face entirely different problems in math class — we'll ignore them for right now.

For the grand majority of us, learning math is difficult. Vexing. Boring.

We shouldn't be surprised by this — the human mind wasn't designed to do complex, abstract math. Being able to solve for x wasn't of particular use on the African savannah.

In contrast, the human mind was designed for learning a different abstract, rule-based, and knowledge-heavy system: language. Each human language has a syntax and vocabulary that is far more complex than the sum total of everything we ask students to learn in K-12 math.

Yet every virtually every 6-year-old speaks their local dialect flawlessly, while few 18-year-olds have mastered math.

We're built for language. We're not built for math.

Maybe there's some species out there who can do complex math automatically — but it ain't us.

We need to face this reality — until we do, we'll underestimate the difficulty of math learning, and sell kids short. From this dismal starting point comes the outline of a plan:

We need to make learning math easy.

Once we understand how alien math learning is, and yet how crucial it is to success in the 21st century, we're left with the resolve to re-invent math learning so every student can succeed.

This is already being done. We can piggyback on it, and make it even better.

By understanding how human cognition works, we can lead all students to learn math, and learn it well. We can help people succeed. We can have a better world.

We should be able to promise a few things about math to every student who comes into our school:

1. They will be active learners, not passive receivers. 2. They will be led to fully understand every concept in the curriculum. 3. They will be adept at solving every mathematical problem they encounter. 4. They will easily remember everything they learn, all the way up until they graduate.

In the next post, I'll sketch out how.

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(A note: I'm worried about two things in this post. First, that what I've just sketched out will seem too optimistic. Is it really true that all kids — even those who aren't predisposed to math — can master calculus? There's excellent evidence that, excepting students with interesting neurological difficulties, the answer is yes. Second, I'm terrified that I've wrongly conveyed the sense that math can't be enjoyable. This would be horrible, as I love math: love teaching it and learning it for myself. The joy of fully grokking a math concept is one of the sweetest pleasures I know. Hopefully I'll be able to explain how we'll bring the love of math into our school in a future post.)

Can every kid pwn calculus? (or: where does mathematical brilliance come from, and can we make it boringly normal?)

Can every student succeed brilliantly at math? Where does talent come from, anyhow? As we imagine what our school might look like, and what it might aspire to pull off, we should keep this fundamental question of expertise always in view.

Let’s overstate this a little: if we can’t guarantee that every student who wishes to, regardless of IQ, can actually succeed at a subject as famously vexing as math, why start a school in the first place? What right do we have to claim to teach the “higher level” skills (creativity, empathy, &c.) if we can’t iron out something as straightforward as math?

To be perfectly clear: I don’t doubt that some people’s brains are better attuned than others’ to learn math. (Years ago, I did — I held, almost as a matter of dogma, that people were born as blank slates, cognitively equal. Then I started [1] reading summaries of twin studies, and [2] actually teaching math.) Genes matter. But do they limit?

We usually approach this sort of question through the old “nature/nurture” dichotomy. But as a chorus of cognitive scientists have been pointing out, this framework isn’t helpful — mental traits (e.g. mathematical brilliance, or mathematical stupidity) arise through the dynamic interplay of genes and culture. As David Shenk writes in The Genius in All of Us:

Contrary to what we’ve been taught, genes do not determine physical and character traits on their own. Rather, they interact with the environment in a dynamic, ongoing process that produces and continually refines an individual. (p. 13)

So: can we build a school that makes it easy for all kids to understand everything in the math curriculum? Can we make it boringly normal for even the kids who stink at math to brilliantly succeed?