Total 18 Posts

## Teach the Controversy

Here is how your unit on linear equations might look:

1. Writing linear equations.
2. Solving linear equations.
3. Applying linear equations.
4. Graphing linear equations.
5. Special linear equations.
6. Systems of linear equations.
7. Etc.

On the one hand, this looks totally normal. The study of the linear functions unit should be all about linear functions.

But a few recent posts have reminded me that the linear functions unit needs also to teach not linear functions, that good instruction in [x] means helping students differentiate [x] from not [x].

Ben Orlin offers a useful analogy here:

If I were trying to teach you about animals, I might start with cats and dogs. They’re simple, furry, familiar, and lots of people have them lying around the house. But I’d have to show you some other animals first. Otherwise, the first time you meet an alligator, you’re gonna be like, “That wet green dog is so ugly I want to hate it.”

Michael Pershan then offers some fantastic prompts for helping students disentangle rules, machines, formulas, and functions, all of which seem totally interchangeable if you blur your eyes even a little.

Not all rules that we commonly talk about are functions; not all functions are rules; not all formulas have rules; not all rules have machines. Pick two: not all of one is like the other. A major goal of my functions unit to help kids separate these ideas. So the very first thing I do is poke at it.

And then I was grateful to Suzanne von Oy for tweeting the question, “Is this a line?” a question that is both rare to see in a linear functions unit (where everything is a line!) and important. Looking at not lines helps students understand lines.

So I took von Oy’s question and made this Desmos activity where students see three graphs that look linear-ish. The point here is that not everything that glitters is gold and not everything that looks straight is linear. Students first make their predictions.

Then they see the graphs again with two points that display their coordinates. Now we have a reason to check slopes to see if they’re the same on different intervals.

Finally, we zoom out to check a larger interval on the graph.

I’m sure I will need this reminder tomorrow and the next day and the next: teach the controversy.

BTW. In addition to being good for learning, controversy is also good for curiosity.

Bonus. Last week’s conversation about calculators eventually cumulated in the question:

“Calculators can perform rote calculations therefore rote calculations have no place on tests.” Yay or nay?

I’ve summarized some of the best responses – both yay and nay – at this page. (I’m a strong “nay,” FWIW.)

## The Explanation Difference

Brett Gilland coined the term “mathematical zombies” in a comment on this blog:

Students who can reproduce all the steps of a problem while failing to evidence any understanding of why or how their procedures work.

When I think about mathematical zombies, I think about z-scores – how easy it is to calculate them relative to how difficult it is to explain those calculations.

Check it out. Here is the formula for a z-score:

In words:

1. You subtract the mean from your sample.
2. You divide that by the standard deviation.

Subtraction and division. Operations simple enough for a elementary schooler. But the explanation of those operations – why they result in a z-score, what a z-score is, and when you should use a z-score – is so challenging it eludes many graduates of high school statistics. Think about how easily you could solve these exercises without knowing what you’re doing.

That difference brings this chart to mind and helps me understand all of the times I’m tempted to just tell students, here’s how you do it already so now just do it. That’s where the operational shortcuts are most tempting.

All of this is preface to a lesson plan on hypothesis testing by Jeremy Strayer and Amber Matuszewski, which is one of the best I’ve read all year.

Hypothesis testing is, again, one of those skills that’s far easier to do than to understand. As you read the lesson plan, please keep in mind that difference. Also notice how capably the teachers develop the question, disclosing the mathematics progressively, and resisting the temptation to shortcut their way to operational fluency.

It’s spectacular. I’m struck every time by a moment where Strayer and Matuszewski ask students to model an experiment with playing cards, only to model the exact same experiment with a computer later. They didn’t just jump straight to the computer simulation!

Here is a video of an airline pilot landing an Airbus A380 in a crosswind. This is that for teachers.

Featured Comment

Amy:

I always think of z-scores as a set of transformations from one plain-vanilla normal curve to the hot-fudge-sundae Standard Normal curve. Maybe once you see it this way, you can’t unsee it. To me, that helps make sense of the “why” you would bother standardizing and the “how” it’s done.

I’m not sure I agree that z-score is so conceptually difficult as to be worth the shortcut. Though I suppose it requires understanding of standard deviation, which is kind of hard. But if you think of standard deviation as “typical weirdness distance” then z-score as the idea of “how many times the typical weirdness is this point” becomes pretty straightforward. A z-score magnitude of 1 becomes average weirdness, less than 1 becomes less weird than average, etc. The bigger the magnitude of the z-score, the weirder the point.

In introductory stats courses, much of what we do simply comes down to separating “Is it possible?” from “Is it plausible?”. We have seen a wonderful growth in the number of free, online applets which allow teachers and students to perform simulations designed to assess these subtly different questions.

## Testify

a/k/a Oh Come On, A Pokémon Go #3Act, Are You Kidding Me With This?

Karim Ani, the founder of Mathalicious, hassles me because I design problems about water tanks while Mathalicious tackles issues of greater sociological importance. Traditionalists like Barry Garelick see my 3-Act Math project as superficial multimedia whizbangery and wonder why we don’t just stick with thirty spiraled practice problems every night when that’s worked pretty well for the world so far. Basically everybody I follow on Twitter cast a disapproving eye at posts trying to turn Pokémon Go into the future of education, posts which no one will admit to having written in three months, once Pokémon Go has fallen farther out of the public eye than Angry Birds.

So this 3-Act math task is bound to disappoint everybody above. It’s a trivial question about a piece of pop culture ephemera wrapped up in multimedia whizbangery.

But I had to testify. That’s what this has always been – a testimonial – where by “this” I mean this blog, these tasks, and my career in math education to date.

I don’t care about Pokémon Go. I don’t care about multimedia. I don’t care about the sociological importance of a question.

I care about math’s power to puzzle a person and then help that person unpuzzle herself. I want my work always to testify to that power.

So when I read this article about how people were tricking their smartphones into thinking they were walking (for the sake of achievements in Pokémon Go), I was puzzled. I was curious about other objects that spin, and then about ceiling fans, and then I wondered how long a ceiling fan would have to spin before it had “walked” a necessary number of kilometers. I couldn’t resist the question.

That doesn’t mean you’ll find the question irresistible, or that I think you should. But I feel an enormous burden to testify to my curiosity. That isn’t simple.

“Math is fun,” argues mathematics professor Robert Craigen. “It takes effort to make it otherwise.” But nothing is actually like that – intrinsically interesting or uninteresting. Every last thing – pure math, applied math, your favorite movie, everything – requires humans like ourselves to testify on its behalf.

In one kind of testimonial, I’d stand in front of a class and read the article word-for-word. Then I’d work out all of this math in front of students on the board. I would circle the answer and step back.

But everything I’ve read and experienced has taught me that this would be a lousy testimonial. My curiosity wouldn’t become anybody else’s.

Meanwhile, multimedia allows me to develop a question with students as I experienced it, to postpone helpful tools, information, and resources until they’re necessary, and to show the resolution of that question as it exists in the world itself.

I don’t care about the multimedia. I care about the testimonial. Curiosity is my project. Multimedia lets me testify on its behalf.

So why are you here? What is your project? I care much less about the specifics of your project than I care how you testify on its behalf.

I care about Talking Points much less than Elizabeth Statmore. I care about math mistakes much less than Michael Pershan. I care about elementary math education much less than Tracy Zager and Joe Schwartz. I care about equity much less than Danny Brown and identity much less than Ilana Horn. I care about pure mathematics much less than Sam Shah and Gordi Hamilton. I care about sociological importance much less than Mathalicious. I care about applications of math to art and creativity much less than Anna Weltman.

But I love how each one of them testifies on behalf of their project. When any of them takes the stand to testify, I’m locked in. They make their project my own.

Again:

Why are you here? What is your project? How do you testify on its behalf?

## 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.

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.

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.

“Methods”

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.

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.

## Blue Point Rule

My biggest professional breakthrough this last year was to understand that every idea in mathematics can be appreciated, understood, and practiced both formally and also informally.

In this activity, students first use their informal home language to describe how the red point turns into the blue point. Then, more formally, I ask them to predict where I’ll find the blue point given an arbitrary red point. Finally, and most formally, I ask them to describe the rule in algebraic notation. Answer: (a, b) -> (a/2, b/2).

It’s always harder for me to locate the informal expression of a idea than the formal. That’s for a number of reasons. It’s because I learned the formal most recently. It’s because the formal is often easier to assess, and easier for machines to assess especially. It’s because the formal is often more powerful than the informal. Write the algebraic rule and a computer can instantly locate the blue point for any red point. Your home language can’t do that.

But the informal expressions of an idea are often more interesting to students, if for no other reason than because they diversify the work students do in math and, consequently, diversify the ways students can be good at math.

The informal expressions aren’t just interesting work but they also make the formal expressions easier to learn. I suspect the evidence will be domain specific, but I look to Moschkovich’s work on the effect of home language on the development of mathematical language and Kasmer’s work on the effect of estimation on the development of mathematical models.

Therefore:

• Before I ask for a formal algebraic rule, I ask for an informal verbal rule.
• Before I ask for a graph, I ask for a sketch.
• Before I ask for a proof, I ask for a conjecture.
• David Wees: Before I ask for conjectures, I ask for noticings.
• Before I ask for a calculation, I ask for an estimate.
• Before I ask for a solution, I ask students to guess and check.
• Bridget Dunbar: Before I ask for algebra, I ask for arithmetic.
• Jamie Duncan: Before I ask for formal definitions, I ask for informal descriptions.
• Abe Hughes: Before I ask for explanations, I ask for observations.
• Maria Reverso: Before I ask for standard algorithms, I ask for student-generated algorithms.
• Maria Reverso: Before I ask for standard units, I ask for non-standard units.
• Kent Haines: Before I ask for definitions, I ask for characteristics.
• Avery Pickford: Before I ask for complete mathematical propositions, I ask for incomplete propositions.
• Dan Finkel: Before I ask for the general rule, I ask for a specific instance of the rule.
• Dan Finkel: Before I ask for the literal, I ask for an analogy.
• Jim Murray: Before I ask for algorithms, I ask for patterns.
• Nicola Vitale: Before I ask for proofs, I ask for conjectures, questions, wonderings, and noticings.
• Natalie Cogan: Before I ask for an estimation, I ask for a really big and really small estimation.
• Eileen Quinn Knight: Before I ask for algorithms, I ask for shorthand.
• Bill Thill: Before I ask for definitions, I ask for examples and non-examples.
• Larry Peterson: Before I ask for symbols, I ask for words.
• Andrew Gael: Before I ask for “regrouping” and “borrowing,” I ask for grouping by tens and place value.

At this point, I could use your help in three ways:

• Offer more shades between informal and formal for the blue dot task. (I offered three.)
• Offer more SAT-style analogies. sketch : graph :: estimate : calculation :: [your turn]. That work has begun on Twitter.
• Or just do your usual thing where you talk amongst yourselves and let me eavesdrop on the best conversation on the Internet.

BTW. I’m grateful to Jennifer Wilson and her post which lodged the idea of a secret algebraic rule in my head.

Featured Comment

Allison Krasnow points us to Steve Phelp’s Guess My Rule activities.

David Wees reminds us that the van Hiele’s covered some of this ground already.