Category: headaches

Total 13 Posts

A Response To Critics

Let me wrap up this summer’s series by offering some time at the microphone to two groups of critics.

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

If Simplifying Rational Expressions Is Aspirin Then How Do You Create The Headache?

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.

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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.

Tracy:

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.

If Proof Is Aspirin, Then How Do You Create the Headache?

This Week’s Skill

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.

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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:

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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.

If Graphing Linear Inequalities Is Aspirin, Then How Do You Create The Headache?

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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:

  1. Graph the boundary. Use a solid line when the inequality contains ≤ or ≥. Use a dashed line when the inequality contains < or >.
  2. Use a test point to determine which half-plane should be shaded.
  3. 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.

Chris Hunter:

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.

Bowen Kerins:

One problem I like is having each kid pick a point, then running it through a “test” like y > x2. 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?

If Functions Are Aspirin, Then How Do You Create The Headache?

This Week’s Skill

Determining if a relationship is a function or not.

A relationship that maps one set to another can be confusing. Questions like, “What single element does 2 map to in the output set below?” are impossible to answer because 2 maps to more than one element.

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By contrast, a function is a relationship with certainty. Take any element of the input and ask yourself, “Where does this function say that element maps?” You aren’t confused about any of them. Every input element maps to exactly one element in the output.

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Pearson and McGraw-Hill’s Algebra 1 textbooks simply provide a definition of a function. Pearson’s definition refers to a previous worked example. McGraw-Hill has students apply the definition to a worked example immediately afterwards. Khan Academy dives straight into an abstract explanation of the concept. In none of these cases is the need for functions apparent. Students are given functions without ever feeling the pain of not having them.

What a Theory of Need Recommends

If we’d like students to experience the need for the certainty functions offer us, it’s helpful to put students in a place to experience the uncertainty of non-functional relationships first. Here is what I’m talking about.

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Put the letters A, B, C, and D on your back wall, spaced evenly apart.

Ask every student to stand up. Then give them a series of instructions.

Transportation

If you walked to school today, stand under A.
If you rode your bike to school today, stand under B.
If you drove or rode in a vehicle today, stand under C.
If you got to school any other way, stand under D.

Duration

If it took you fewer than 10 minutes to get to school today, stand under B.
If it took you 10 or more minutes to get to school today, stand under D.

Class

If you’re in seventh grade, stand under A.
If you’re in eighth grade, stand under B.
If you’re in ninth grade, stand under C.
If you’re in any other grade, stand under D.

These instructions are all clear and easy to follow. Students are certain where they should go. Then give two other sets of instructions.

Clothes

If you’re wearing blue, stand under A.
If you’re wearing red, stand under B.
If you’re wearing black, stand under C.
If you’re wearing white, stand under D.

Birthday

If you were born in January, stand under A.
If you were born in February, stand under B.
If you were born in March, stand under C.
If you were born in April, stand under D.

Perhaps you see how these last two examples generate a lack of certainty. Students were lulled by the first examples and may now feel a headache.

“I’m wearing white and red. Where do I go?”

“I was born in August. There’s no place for me to stand.”

Now we gather back together and apply formal language to the concepts we’ve just felt. “Mathematicians call these three relationships ‘functions.’ Here’s why. Why do you think these relationships aren’t functions?” Invite students to interrogate the concept of a function in different contexts. Try to keep the focus on certainty – can you predict the output for any input with certainty? – rather than on the vertical line test or other rules that expire.

Next Week’s Skill

Graphing linear inequalities. It’s extraordinarily easy to turn questions like “Graph y < -2x + 5" into the following series of steps:

  1. Graph the line.
  2. If the inequality includes the boundary, make the line solid. Otherwise, make the line dashed.
  3. Test a point on either side of the line. Use (0,0) if possible.
  4. If that point is a solution to the inequality, shade that side of the line.
  5. If that point isn’t a solution to the inequality, then shade the other side of the line.

Students can become quite capable at executing that algorithm without understanding its necessity or how it figures into algebra’s larger themes.

What can you do with this?

BTW

Kate Nowak encouraged me to look at other textbooks beyond McGraw-Hill and Pearson’s. She recommended CME, which, it turns out, does some great work highlighting this need for functions. It asks students to play a “guess my rule” game, one which has a great deal of certainty. Each input corresponds to exactly one output. Then the CME authors offer a vignette where a partner reports multiple outputs for the same input, making the game impossible to play. Strong work, CME buds.