There are lots of good reasons to ask students for multiple representations of relationships. But I worry that a consistent regiment of turning tables into equations into graphs and back and forth can conceal the fact that each one of these representations were invented for a purpose. Graphs serve a purpose that tables do not. And the equation serves a purpose that stymies the graph.

By asking for all three representations time after time, my students may have gained a certain conceptual fluency promised us by researchers like Brenner et al. But I’m not sure that knowledge was ever effectively *conditionalized*. I’m not sure those students knew when they could pick up one of those representations and leave the others on the table, except when the problem told them.

Otherwise, it’s possible they thought each problem required each of them.

The same goes for representations of one-dimensional data. We can take the same set of numbers and represent its mean, median, minimum, maximum, deviation, bar graph, column graph, histogram, pie chart, etc.

So here is the exercise. Take one representation. Now take another. Why did we invent that other representation? Now how do you put your students in a place to experience the *limitations* of the first representation such that the other one seems *necessary*, like aspirin to a headache?

**You can’t be in the business of creating headaches and offering the aspirin.**

That’s a conflict of interest and a moral hazard, claims Maya Quinn, one of the most interesting commenters to stop by my blog this summer. You can choose one or the other but choosing both seems a bit like a fireman starting fires just to give the fire department something to do.

But “creating headaches” was perhaps always a misnomer because the headaches exist whether or not we create them. New mathematical techniques were developed to resolve the limitations of old ones. Putting students in the way of those limitations, even briefly, results in those headaches. The teacher’s job isn’t to create the headaches, exactly, but to make sure students don’t *miss* them.

To briefly review, those headaches serve two purposes.

One, they satisfy cognitive psychologist Daniel Willingham’s observation that interesting lessons are often organized around *conflict*, specifically conflicts that are central to the discipline itself. (Harel identified those conflicts as needs for certainty, causality, computation, communication, and connection.)

Two, by tying our lessons to those five headaches we create several strong schemas for new learning. For example, many skills of secondary math were developed for the sake of efficiency in computation and communication. That is a theme that can be emphasized and strengthened by repeatedly putting students in a position to experience *inefficiency*, however briefly. If we instead begin every day by simply stating the new skill we intend to teach students, we will create lots and lots of *weak* schemas.

So creating these headaches is both useful for motivation and useful for learning.

Which brings me to my other critics.

**This One Weird Trick To Motivate All Of Your Students That THEY Don’t Want You To Know About**

There is a particular crowd on the internet who think the problem of motivation is overblown and my solutions are incorrect.

Some of them would like to dismiss concerns of motivation altogether. They are visibly and oddly celebratory when PISA revealed that students in many high-performing countries don’t look forward to their math lessons. They hypothesize that learning and motivation trade *against* each other, that we can choose one or the other but not both. Others even suggest that motivation accelerates inequity. They argue that we shouldn’t motivate students because their professors in college won’t be motivating.

I don’t doubt their sincerity. I believe they sincerely see motivation as a slippery slope to confusing group projects in which students spend too much time learning too much about birdhouses and not enough about the math behind the birdhouses. I share those concerns. Motivation, interest, and curiosity may assist learning but they don’t cause it. In the name of motivation, we have seen some of the worst innovations in education. (Though also some of the best.)

But there are also those who *do* care about motivation. They just think my solutions are overcomplicated and wrong. They have a competing theory that I don’t understand at all: just get students good at math. It’s that easy, they say, and anybody who tells you it’s any harder is selling something.

“Success in a skill is self-motivating.”

“Many forget that there’s intrinsic motivation to simply perform well in a subject.”

And, yeah, I’m sorry, friends, but I *do* have a hard time accepting such a simple premise. And I’m not alone. 62% of our nation’s Algebra teachers told the National Mathematics Advisory Panel that their *biggest* problem was “working with unmotivated students.”

I see two possibilities here. Either the majority of the nation’s Algebra teachers have never considered the option of simply speaking clearly about mathematics and assigning spiraled practice sets, or they’ve tried that pedagogy (perhaps even twice!) and they and their students have found it wanting.

Tell me that first possibility isn’t as crazy as it sounds to me. Tell me there’s another possibility I’m missing. If you can’t, I think we’re dealing with a failure of empathy.

I mean imagine it.

Imagine that an alien culture scrambles your brain and abducts you. You wake from your stupor and you’re sitting in a room where the aliens introduce you to their cryptic alphabet and symbology. They tell you the names they have for those symbols and show you lots of different ways to manipulate those symbols and how several symbols can be written more compactly as a single symbol. They ask you questions about all of this and you’re lousy at their manipulations at first but they give you feedback and you eventually understand those symbols and their basic manipulations. You’re competent!

I agree that in this situation competence is preferable to *incompetence* but how is competence preferable to *not being abducted in the first place*?

If that exercise in empathy strikes you as nonsensical or irrelevant then I don’t think you’ve spent enough time with students who have failed math repeatedly and are still required to take it. If you *have* put in that time and *still* disagree, then at least we’ve identified the bedrock of our disagreement.

But just imagine how well these competing theories of motivation would hold up if math were an elective. Imagine what would happen if every student everywhere could suddenly opt out of their math education. If your theory of motivation suddenly starts to shrink and pale in your imagination, then you were never really thinking about motivation at all. You were thinking about *coercion*.

**Previously**

For one example: in our last post on simplifying rational expressions, the process of turning a lengthy rational expression into a simpler one, Bill F writes:

Another benefit of evaluating both expressions for a set of values is to emphasize the equivalence of both expressions. Students lose the thread that simplifying creates equivalent expressions. All too often the process is seen as a bunch-of-math-steps-that-the-teacher-tells-us-to-do. When asked, “what did those steps accomplish?” blank stares are often seen.

Past a certain point, those operations are trivial. But it’s only past a point much farther in the distance that the *understanding* – these two rational expressions are *equivalent* – becomes trivial.

For another example: I left high school adept at graphing functions. I could complete the square and change forms easily. I knew how to identify the asymptotes, holes, and limiting behavior of those thorny rational expressions. But it wasn’t until I had graduated university math and was several years into *teaching* that I really, really understood that the graph is a picture of all the points that make the function *true*. This was difficult for me because graphs don’t often *look* like a bunch of points. They look like a *line*

That’s one reason I’m excited about the Desmos Activity Builder and this activity I made in it last week, Loco for Loci!

It asks students to place a point anywhere on a graph so that it makes a particular relationship *true*. Then it asks the students to imagine what *all of our points* would look like if we pictured them on the same graph. Then the teacher can show the results, underscoring this Very Big Idea that I didn’t fully appreciate my first time through high school – what we eventually think of as a continuous line is a picture of lots and lots of points.

Here is what happened when 300 people on Twitter played along:

“Drag the green point so that it’s the same distance from both blue points.”

Trickier: “Drag the green point so that it’s five units from both blue points.”

Whimsical: “Drag the green point so it is the same distance from a) the line of dinosaurs and b) the big dinosaur.” I really couldn’t have hoped for better here.

And then a couple of very interesting misfires.

“Drag the green point so that it’s four units from the blue point.”

“Drag the green point so that a line segment is formed with a slope of .5.”

You could run a semester-long master’s seminar on the misconceptions in that last graph.

Well.

Maybe more like ten quick seconds at the start of your Algebra class.

If you’d like to run this activity with your own students, here is the teacher link.

**Questions for the Comments**

- Obviously, I didn’t invite hyperbolas and ellipses to the party. Which other loci should have received the same treatment?
- Which Very Big Ideas did you only fully understand once you started math teaching?

**Featured Comment**

I find this sort of gap fascinating [my inability to conceive of graphs as a picture of solutions –

dm] especially because it is likely somewhere along the line you were at least told this fact (you might even be able to track exactly where). But it still didn’t stick! It’s as if being told just isn’t enough.

The description you give about graphs is something we have to hit early and often in CME Project, it’s one of the top 3 things to learn in the entire curriculum. It’s amazing how that can get lost in the shuffle, but it does, and where it gets lost is the algorithmic way of graphing a function or equation: the A does this, the B does that, etc — all of this ignores the deeper fact that under the hood, this is all a relationship between two variables x and y.

The other two of Bowen’s top three things to learn in Algebra, according to Bowen on Twitter, are:

- Variables represent numbers, so test numbers to test ideas and build equations.
- Rules for new stuff should respect existing rules.

**Featured Tweets**

Amazing, all the people unburdening themselves on Twitter of math they only understood once they began teaching. *What does it all mean?*

Simplifying rational expressions.

In particular, adding rational expressions with unlike denominators, resulting in symbolic mish-mash of this sort here.

I’m not here to argue whether or not this skill should be taught or how *much* it should be taught. I’m here to say that if we *want* to teach it, we’re a bit stuck for our usual reasons *why*:

- It lacks real-world applications.
- It lacks job-world applications. (Unless you count “Algebra II teacher.”)
- It lacks relevance.

So our usual approaches to motivation fail us here.

**What a Theory of Need Recommends**

We have to ask ourselves, instead, why anyone would prefer the simplified form to the unsimplified form. If the simplified form is aspirin, then what is the headache?

I don’t believe the answer is “elegance” or “beauty” or any of the abstract ideals we often attribute to mathematicians. Talking about “efficiency” gets us closer, but still and again, we’re just *talking* about motivation here. Let’s ask students to *do* something.

We simplify because it makes life *easier*. It makes all kinds of *operations* easier. So students need to experience the relative difficulty of performing even simple operations on the unsimplified rational expression before we help them learn to simplify.

*Like evaluation.*

So with nothing on the board, ask students to call out three numbers. Put them on the board. And then put up this rational expression.

Ask students to evaluate the numbers they chose. It’s like an opener. It’s review. As they’re working, you start writing down the answers on Post-It notes, which you do quickly because you know the simplified form. You place one Post-It note beneath each number the students chose. You’re finished with all three before anybody has finished just one.

As students reveal *their* answers and find out that you got *your* answers more efficiently and with more accuracy than they did, it is likely they’ll experience a headache for which the process of simplification is the aspirin.

Again we find that this approach does more than just motivate the simplification process. It makes that process *easier*. That’s because students are performing the same process of finding common denominators and adding fractions with *numbers*, they’ll shortly perform with *variables*. We’ve made the abstract more concrete.

Again, I don’t mean to suggest this would be *the most interesting lesson ever!* I’m suggesting that our usual theories of motivating a skill – link it to the real world, link it to a job, link it to students’ lives – crash hard on this huge patch of Algebra that includes rational expressions. That isn’t to say we shouldn’t teach it. It’s to say we need a stronger theory of motivation, one that draws strength from the development of math itself rather than from a student’s moment-to-moment interests.

**Next Week**

Wrapping up.

**Featured Comments**

Another benefit of evaluating both expressions for a set of values is to emphasize the equivalence of both expressions. Students lose the thread that simplifying creates equivalent expressions. All too often the process is seen as a bunch-of-math-steps-that-the-teacher-tells-us-to-do. When asked, “what did those steps accomplish?” blank stares are often seen.

By a creating a “headache” using a theory of need, we’re really looking back to the situations that prompted the development of the mathematics we intend students to learn. We’re attempting to place students in the position of the mathematician/scientist/logician/philosopher who was originally staring down a particular set of mathematics without a clue about where to go and developing a massive headache from his hours of attempt. I love this idea because it transcends any subject and students learn the value of the learning process.

I feel like it’s a mathematical habit of mind. Mathematicians don’t like drudgery either. But what makes them different from a typical American math student is, rather than passively accepting the work as tedious and plowing ahead anyway, they do something about it. They look for a workaround, or another approach.

Mike:

]]>It is elegance, it is beauty, and I’m afraid I simply don’t buy the efficiency argument at all.

*Print*

**Missing the Promise of Mathematical Modeling**. My Spring 2015 article in*Mathematics Teacher*analyzes the meager opportunities for modeling in school mathematics and recommends some solutions.**The Checkout Line A Scam. (Or Is It? [Yes, It Is.])**My blog post for Heinemann illustrating modeling through the lens of grocery express lines, which are a scam.

*Video*

**Math Class Needs a Makeover**. A 11-minute talk I gave for TEDxNYED in 2010. Couple million views.**Fake-World Math: When Mathematical Modeling Goes Wrong and How to Get it Right**My TEDxNYED talk + five more years of study + 50 extra minutes.

Summarizing all of the above in a single paragraph:

]]>Modeling asks students a) to take the world and turn it into mathematical structures, then b) to operate on those mathematical structures, and then c) to take the results of those operations and turn them back into the world. That entire cycle is some of the most challenging, exhilarating, democratic work your students will ever do in mathematics, requiring the best from all of your students, even the ones who dislike mathematics. If traditional textbooks have failed modeling in any one way, it’s that they perform the first and last acts

forstudents, leaving only the most mathematical, most abstract act behind.

Sometimes I think that we, as teachers, are so eager to get to the answers that we do not devote sufficient time to developing the question.

Scott Farrar, on my last post on motivating proof:

]]>I think this latches onto the structure of the geometry course: we develop tool (A) to study concept (B). But curriculum can get too wrapped up in tool (A), losing sight of the very reason for its development. So we lay a hook by presenting concept (B) first.

Proof.

This is too big for a blog post, obviously.

**What a Theory of Need Recommends**

If proof is the aspirin, then *doubt* is the headache.

In school mathematics, proof can feel like a game full of contrived rules and fragile pieces. Each line of the proof must interlock with the others *just so* and the players must write each of them using tortured, unnatural syntax. The saddest aspect of this game of proof is that the outcome of the game is already known *every time*.

- Prove angle B is congruent to angle D.
- Prove triangle BCD is congruent to triangle ACB.
- Decide if angle B and angle C are congruent. If they are, prove why they are. If they aren’t, prove why they aren’t.
- Prove line l and line m are parallel.
- Prove that corresponding angles are congruent.

One of those proof prompts is not like the others. Its most important difference is that it leaves open the very question of its truth, where the other prompts leave no doubt.

The act of proving has many purposes. It doesn’t do us any favors to pretend there is only one. But one purpose for proof that is frequently overlooked in school mathematics is the need to dispel doubt, or as Harel put it, the “need for certainty“:

The need for certainty is the need to prove, to remove doubts. One’s certainty is achieved when one determines—by whatever means he or she deems appropriate— that an assertion is true. Truth alone, however, may not be the only need of an individual, who may also strive to explain why the assertion is true.

So instead of giving students a series of theorems to prove about a rhombus (implicitly verifying in advance that those theorems are *true*) consider sowing doubt first. Consider giving each student a random rhombus, or asking your students to construct their own rhombus (if you have the time, patience, and capacity for heartache that activity would require).

Invite them to measure all the segments and angles in their shapes. Do they notice anything? Have them compare their measurements with their neighbors’. Do they notice anything now?

Now create a class list of conjectures. Interject your own, if necessary, so that the conjectures vary on two dimensions: true & false; easy to prove & hard to prove.

For example:

“Diagonals intersect at perpendicular angles” is true, but not as easy to prove as “opposite sides are congruent,” which is also true. “A rhombus can never have four right angles” meanwhile is false and easy to disprove with a counterexample. “A rhombus can never have side lengths longer than 100 feet” is false but requires a different kind of disproof than a counterexample.

With this cumulative list of conjectures, ask your students now to decide which of them are true and which of them are false. Ask your students to try to disprove each of them. Try to draw a rhombus, for example, even a sketch, where the diagonals *don’t* intersect at perpendicular angles.

If they *can’t* draw a counterexample, then we need to prove why a counterexample is impossible, why the conjecture is in fact true.

This approach accomplishes several important goals.

**It motivates proof.**When I ask teachers about their rationale for teaching proof, I hear most often that it builds students’ skills in logic or that it trains students’ mind. (“I tell them, when you see lawyers on TV arguing in front of a judge, that’s a proof,” one teacher told me last week.) Forgive me. I’m not hopeful that our typical approach to proof accomplishes any of those transfer goals. I’m also unconvinced that lawyers (or even*mathematicians*) would persist in their professions if the core job requirement were working with two-column proofs.**It lowers the threshold for participation in the proof act**. Measuring, noticing, and speculating are easier actions (and more interesting too) than trying to recall the abbreviation “CPCTC.”**It allows students to familiarize themselves with formal vocabulary and with the proof act.**Students I taught would struggle to prove that “opposite sides of a rhombus are congruent.” This is because they’re essentially reading a foreign language, but also because mathematical argumentation, even the*informal*kind, is a foreign*act*. Offering students the chance to prove trivial conjectures puts them in arm’s reach of the feeling of*insight*which all non-trivial proofs require.**It makes proving easier.**When students try to disprove conjectures by drawing lots of different rhombi, they stand a better chance of noticing the aspects of the rhombus that vary and don’t vary. They stand a better chance of noticing that they’re drawing an*awful*lot of isosceles triangles, for example, which may become an essential line in their formal proof.

Resolving this list of conjectures about the rhombus – proving and disproving each of them – will take more than a single period. Not every proof needs this kind of treatment, certainly. But occasionally, and especially early on, we should help students understand *why* we bother with the proof act, why proof is the aspirin for a particular kind of headache.

**Next Week’s Skill**

Simplifying sums of rational expressions with unlike denominators. Like this worked example from PurpleMath:

If that simplified form is aspirin, then how do we create the headache?

**BTW**. For anybody not on board this “headache -> aspirin” thing, I want to clarify: totally fine. Thanks for contributing anyway. But please name your priors. Why that task instead of another? Some of these tasks you all suggest in the comments seem great and full of potential, but tasks aren’t generative of other tasks. I need fewer interesting tasks and more interesting *theories* about what make tasks tick. These kinds of theories, when properly beaten into shape, have the capacity to generate lots of other tasks.

**BTW**. Scott Farrar chases this same idea along a different path.

**Featured Comments**

I think this latches onto the structure of the geometry course: we develop tool (A) to study concept (B). But curriculum can get too wrapped up in tool A losing sight of the very reason for its development. So, we lay a hook by presenting concept B first.

We almost always do an always-sometimes-never to motivate a particular proof. Mine are usually teacher-generated (here’s a list of 5 statements about rhombi – tell me if they are always, sometimes, or never true). Then we prove the always and the never.

Michael Paul Goldenberg and Michael Serra offer some very convincing criticism of the ideas in this post.

]]>Fake-World Math was the talk I gave for most of 2014, including at NCTM. It looks at mathematical modeling as it’s defined in the Common Core, practiced in the world of knowledge work, and maligned in print textbooks. I discuss methods for helping students become proficient at modeling and methods for helping them *enjoy* modeling, which are not the same set of methods.

Also, a note on process. I recorded my screen throughout the entire process of *creating the talk*. Then I sped it up and added some commentary.

**This Week’s Skill**

Here is the first paragraph of McGraw-Hill’s Algebra 1 explanation of graphing linear inequalities:

The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality. An equation defines a boundary, which divides the coordinate plane into two half-planes.

This is mathematically correct, sure, but how many novices have you taught who would sit down and attempt to parse that expert language?

The text goes on to offer three steps for graphing linear inequalities:

- Graph the boundary. Use a solid line when the inequality contains ≤ or ≥. Use a dashed line when the inequality contains < or >.
- Use a test point to determine which half-plane should be shaded.
- Shade the half-plane that contains the solution.

The text offers aspirin for a headache no one has felt.

The shading of the half-plane emerges from nowhere. Up until now, students have represented solutions graphically by plotting points and graphing lines. This shading representation is new, and its motivation is opaque. The fact that the shading is *more efficient* than a particular alternative, that the shading was invented to *save time*, isn’t clear.

We can fix that.

**What a Theory of Need Recommends**

My commenters save me the trouble.

Ask students to find two numbers whose sum is less than or equal to ten (or, alternatively, points that satisfy your 2x + y < 5 above). The headache is caused by asking them to list ’em all. The aspirin is another way to communicate all of these points — the graph determined by the five steps listed above. Rather than present the steps, have students plot their points as a class.

One problem I like is having each kid pick a point, then running it through a “test” like y > x

^{2}. They plot their point green or red depending on whether or not it passes the test — and a rough shape of the graph emerges.

John Scammell writes about a similar approach. Nicole Paris offers the same idea, and adds hooks into later lessons in a unit.

Great work, everybody. My only addition here is to connect all of these similar lessons with two larger themes of learning and motivation. One large theme in Algebra is our efforts to find solutions to questions about numbers. Another large theme is our efforts to *represent* those solutions as concisely and efficiently as possible. My commenters have each *knowingly* invited students to represent solutions using an existing inefficient representation, all to prepare them to use and appreciate the more efficient representation they can offer.

They’re linking the new skill (graphing linear inequalities) to the old skill (plotting points) and the new representation (shading) to the old representation (points). They’re tying new knowledge to old, strengthening both, motivating the new in the process.

**Next Week’s Skill**

Proofs. Triangle proofs. Proving trigonometric identities. If proof is aspirin, then how do you create the headache?

]]>Zak Champagne, Mike Flynn, and I are all NCTM conference presenters and we were all concerned about the possibility that a) none of our participants did much with our sessions once they ended, b) lots of people who might benefit from our sessions (and whose questions and ideas might benefit *us*) weren’t in the room.

The solution to (b) is easy. Put video of the sessions on the Internet. Our solution to (a) was complicated and only partial:

**Build a conference session so that it prefaces and provokes work that will be ongoing and online.**

To test out these solutions, we set up Shadow Con after hours at NCTM. We invited six presenters each to give a ten-minute talk. Their talk *had* to include a “call to action,” some kind of closing homework assignment that participants could accomplish when they went home. The speakers each committed to help participants with that homework on the session website we set up for that purpose.

Then we watched and collected data. There were two major surprises, which we shared along with other findings with the NCTM president, president-elect, and executive director.

Here is the five-page brief we shared with them. We’d all benefit from your feedback, I’m sure.

**Featured Comments**

Marilyn Burns on her reasons for attending conferences like NCTM:

I don’t expect an NCTM conference to provide in-depth professional development, but act more like a booster shot for my own learning.

Elham Kazemi, one of our Shadow Con speakers, tempers expectations for online professional development:

]]>I have a different set of expectations about conferences and whether going to them with a team allows you to go back to your own contexts and continue to build connections there. Can we expect conferences and the internet to do that — to feed our local collaborations? I get a lot of ideas from #mtbos and from my various conversations and conferences. But really making sense of those ideas takes another level of experience.