Category: headaches

Total 14 Posts

Don’t Teach Math the “Smart Way”

Smartness and mathematics have an unhealthy relationship.

If you have been successful in math, by public consensus, you must be smart. If you have been successful in the humanities, you may also be smart but we cannot really be sure about that now can we, says public consensus.

In a world where our finest mathematical minds ruined the global economy and perpetuate unequal social outcomes, outcomes most ably critiqued by people trained in the humanities, public consensus is wrong.

A worksheet that asks students to use the 'smart way' to tell time.

This worksheet is worse.

This worksheet associates smartness with a certain way of doing math, diminishing other ways your students might develop to do the same math. Because there are lots of possible ways to tell time — some new, some old, and some not-yet-invented!

Worse, this worksheet associates smartness with a certain way of doing math that is culturally defined, diminishing entire cultures. For example, depending on your location in the world, “2/5/19” and “5/2/19” can refer to the same calendar date. Neither of those ways are “smart” or “dumb.” They work for communication or they don’t.

Try This Instead

If I’d like students to learn a certain way of doing math — whether that’s adding numbers a certain way or solving equations a certain way — I need to understand the reasons why we invented those ways of doing math and put students in a position to experience those reasons. I also need to be excited — thrilled even! — if students create or adapt their own ways of doing math when they’re having those experiences. Anything less is to diminish their creativity.

If I want students to learn how to communicate mathematically, I need to ask them to communicate.

So in this Desmos activity, one student will choose a clock and another student will ask questions to narrow 16 clocks down to 1.

I have no idea what ways students will use, create, or adapt in order to tell time. I will be excited about all of them.

I will also be excited to share with them the ways that lots of cultures use to tell time. When I share those ways, I will be honest that those ways aren’t “smart” any more than they are “moral.” They are merely what one group of people agreed upon to help them get through their day.

So I’d also offer students this Desmos activity, which tells students the time using several different cultural conventions, including the one the worksheet calls “smart” above.

Students set the clock and then they see how easy or hard it was for the class to come to consensus using that convention.

Later, we invite students to set the clock themselves and name the time using three different conventions. They make two of them true, one of them a lie, and submit the whole package to the Class Gallery where their classmates try to determine the lie.

The words we use matter. “Real world” matters. “Mistakes” matter. “Smart” matters. Those words have the power to shape student experiences, to extend or withdraw opportunities to learn, to denigrate or elevate students, their cultures, and the ideas they bring to our classes.

Defining smartness narrowly is to define “dumbness” broadly. Instead, we should seek to find smartness as often as possible in as many students as possible.

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Recipes for Surprising Mathematics

What does it take to ask students a question like this?


A poker face? A bit of malice? Nitsa Movshovits-Hadar argues [pdf] that it requires only the ability to trick yourself into forgetting that you know every triangle has the same interior angle sum. “Suppose we do not know it,” she writes, which is easier said than done.

The premise of her article is that “… all school theorems, except possibly a very small number of them, possess a built-in surprise, and that by exploiting this surprise potential their learning can become an exciting experience of intellectual enterprise to the students.”

This is such a delightful paper — extremely readable and eminently practical. Without knowing me, Movshovits-Hadar took several lessons that I love, but which seemed to me totally disparate, and showed me how they connect, and how to replicate them. I’m pretty sure I was grinning like an idiot the whole way through this piece.

[via Danny Brown]

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Not easy for math teachers to do!

Kent Haines:

What if you asked two questions: which triangle has the longest perimeter and which triangle has the largest angle sum? It might clarify what can change in a triangle and what cannot. Also it hides the surprise better. If you teach via surprise consistently, kids start looking for the punchline.

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Elementary may actually have an advantage here! We play these games all the time. Some favorites:

Draw me a two-sided quadrilateral
Draw me a triangle with three right angles (or three obtuse angles)
(We have a manipulative that consist of little plastic sticks that snap together to build things)–Build me a triangle with the red stick (6″), the purple stick (1″) and the green stick (2″ )

Once the whole class is convinced they can’t you can get at why and then writing a rule for it. There is nothing an 8 year old likes better than proving the teacher wrong.

Ethan Hall:

Theorems and formulae in textbooks should be marked with a “spoiler alert”.

Math: Improve the Product Not the Poster


Danny Brown has expressed an interest in teaching mathematics that is relevant to students, relevant in important, sociological ways especially. This puts him in a particular bind with mathematics like Thales’ Theorem, which seems neither important nor relevant.


Danny Brown:

Here is Thales’ theorem. Every student in the UK must learn this theorem as part of the Maths GCSE. You are explaining Thales’ theorem, when one of the students in your class asks, “When will we ever need this in real life?” How might you respond?

He proceeds to offer several possible responses and then, with admirable empathy for teenagers, rebut them. Brown finds none of our best posters for math particularly compelling. You know the ones.

  • Math is everywhere.
  • Math develops problem solving skills.
  • Math is beautiful.
  • Etc.

So instead of fixing our posters, let’s fix the product itself.

Brown’s premise is that students are listening to him “explaining Thales’ theorem.” Let’s question that premise for a moment. Is that the only or best way to introduce students to that proof? [2016 Jun 3. Brown has informed me that explanation is not his preferred pedagogy around proof and I have no reason not to take him at his word. So feel free to swap out “Brown” in the rest of this post with your recollection of nearly every university math professor you’ve ever had.]

Among other purposes, every proof is the answer to a question. Every proof is the rejection of doubt. It isn’t clear to me that Brown has developed the question or planted the doubt such that the answer and the explanation seem necessary to students.

So instead of starting with the explanation of an answer, let’s develop the question instead.

Let’s ask students to create three right triangles, each with the same hypotenuse. Thales knows what our students might not: that a circle will pass through all of those vertices.


Let’s ask them to predict what they think it will look like when we lay all of our triangles on top of each other.

Let’s reveal what several hundred people’s triangles look like and ask students to wonder about them.


My hypothesis is that we’ll have provoked students to wonder more here than if we simply ask students to listen to our explanation of why it works.


To test that hypothesis, I ran an experiment that uses Twitter and the Desmos Activity Builder and is pretty shot through with methodological flaws, but which is suggestive nonetheless, and which is also way more than you oughtta expect from a quickie blog post.

I asked teachers to send their students to a link. That link randomly sends students to one of two activities. In the control activity, students click slide by slide through an explanation of Thales’ theorem. In the experimental activity, students create and predict like I’ve described above.

At the end of both treatments, I asked students “What questions do you have?” and I coded the resulting questions for any relevance to mathematics.

77 students responded to that final prompt in the experimental condition next to 47 students in the control condition. 47% of students in the experimental group asked a question next to 30% of students in the control group. (See the data.)

This suggests that interest in Thales’ theorem doesn’t depend strictly on its social relevance. (Both treatments lack social relevance.) Here we find that interest depends on what students do with that theorem, and in the experimental condition they had more interesting options than simply listening to us explain it.

So let’s invite students to stand in Thales’ shoes, however briefly, and experience similar questions that led Thales to sit down and wonder “why.” In doing so, we honor our students as sensemakers and we honor math as a discipline with a history and a purpose.

BTW. For another example of this pedagogical approach to proof, check out Sam Shah’s “blermions” lesson.

BTW. Okay, study limitations. (1) I have no idea who my participants are. Some are probably teachers. Luckily they were randomized between treatments. (2) I realize I’m testing the converse of Thales’ theorem and not Thales’ theorem itself. I figured that seeing a circle emerge from right triangles would be a bit more fascinating than seeing right triangles emerge from a circle. You can imagine a parallel study, though. (3) I tried to write the explanation of Thales’ theorem in conversational prose. If I wrote it as it appears in many textbooks, I’m not sure anybody would have completed the control condition. Some will still say that interest would improve enormously with the addition of call and response questions throughout, asking students to repeat steps in the proof, etc. Okay. Maybe.

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Danny Brown responds in the comments.

Michael Ruppel responds to the charge that Thales theorem isn’t important mathematics:

As to the previous commenter, Thales’ theorem is not a particularly important piece of content in and of itself, but it’s one of my favorite proofs for students to build. It requires careful attention to definitions and previously-learned theorems as well as a bit of creativity. (Drawing that auxiliary line.) Personally, my favorite part of the proof is that students don’t solve for a or b, and in fact have no knowledge of what a and b are. but they prove that a+b=90. The proof is a different flavor than they are used to.

Creating a Need for Coordinate Parentheses & Combining Like Terms

Our first approach in preparing a new lesson is often to ask, “Where does this skill apply in the world of work or in the world outside the classroom?” There may well be a great answer for some skills, but this strategy generalizes very poorly to lots of mathematics. So instead, I try first to ask myself, “Why did we invent this skill? How does this skill resolve the limits of older skills? If this skill is aspirin, then what is the headache and how do I create it?”

Two examples from my recent past.

Combining Like Terms

Why did we invent the skill of combining like terms in an expression? Why not leave the terms uncombined? Maybe the terms are fine! Why disturb the terms?

One reason to combine like terms is that it’s easier to perform operations on the terms when they’re combined. So let’s put students in a place to experience that use:

Evaluate for x = -5:

3x + 5 + 2x2 – 7 + 8x – 5x2 – 11x + 4 – 5x + 3x2 + 4 + 3x – 6 + 2x + x2

Put it on an opener. The expression simplifies to x2, giving students an enormous incentive to learn to combine like terms before evaluating.

[I’m grateful to Annie Forest for bringing the example to mind. She also adds a context, if that’s what you’re into.]


When students first learn to graph points, the parentheses are the first convention they throw out the window. And it’s hard to blame them. If I told you to graph the point 2, 5, would you need the parentheses to know the point I’m talking about? No.

So why did mathematicians invent parentheses? What purpose do they serve, assuming that purpose isn’t “tormenting middle school students thousands of years in the future.”

It turns out that, while it’s very easy to graph a single point with or without parentheses, graphing lots of points becomes very difficult without the parentheses. So let’s put students in a place to experience that need:

Graph the coordinates:

-2, 3, 5, -2, 8, 1, -4, 0, -10, 4, -7, -3, -2, 7, 2, -5, -3

You can’t even easily tell if there are an even number of numbers!

[My thanks to various workshop participants for helping me understand this.]


The need for combining like terms is Harel’s need for computation and the need for parentheses is Harel’s need for communication. I can’t recommend his paper enough in which he outlines five needs for all of mathematics.

My point isn’t that we should avoid real-world or job-world applications of mathematics. My point is that for some mathematics that is actually impossible. But that doesn’t mean the mathematics was invented arbitrarily or for no reason or for malicious reasons. There was a need.

Math sometimes feels purposeless to students, a bunch of rules invented by people who wanted to make children miserable thousands of years in the future. We can put students in a place to experience those purposes instead.


We explored these ideas in a summer series.

Real World, But Unnecessary


There are lots of great reasons to use this task from NCTM’s Illuminations site, which asks students to derive an algebraic function from a problem situation. But one of those reasons isn’t “to show students why they should derive algebraic functions.”

It’s a real world problem, by most definitions of the term. But let’s not let that fact satisfy us. It’s possible for math to be real-world, but also unnecessary. For example, I can ask students to use trigonometry to calculate the height of a file cabinet. But that math isn’t necessary when a measuring tape would suffice.

The same is true here. I can find Stages 1 through 5 by multiplying by three successively. So why did we invent algebraic representations? Life would be so much easier for both the student and the teacher if we relaxed that condition.

But if we added the question, “How long would it take the entire world to experience a good deed?” we will have both a) identified the need for algebraic functions — to calculate outputs given any input, even distant inputs — and b) put students in a position to experience that need.


That’s a two-step process. With the line, “Describe a function that would model the Pay It Forward process at any stage,” the author satisfies the first step. He understands the value of algebraic functions, himself. But without our added question, that’s privileged knowledge and we’re hoping students infer it. Instead, let’s put them directly in the path of that knowledge.

Real world, and also necessary.