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True story: it’s possible to fly through your own secondary math education – honor roll bumper sticker on your mom’s minivan and all – but miss some of the Very Big Ideas of secondary math.

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.


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

Jason Dyer:

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.

Bowen Kerins:

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?

This Week’s Skill

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

Bill F:

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.

Tom Hall:

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.


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

I’ve thought about “modeling” more than I’ve thought about any other specialization in mathematics. I’m learning less and less about it each year so I’m hopping onto a different track for awhile, moving onto other questions. First, I wanted to collect these links in one location:



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 for students, leaving only the most mathematical, most abstract act behind.

Daniel Willingham, in his book Why Don’t Students Like School:

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.

This Week’s Skill


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

Scott Farrar:

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.

Mr Ruppel:

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.

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