Total 14 Posts

## Multiple Representations v. Best Representation

This is from a worksheet I assigned during my last year in the classroom:

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?

Featured Comment

Ok. First is bar chart, second is box plot.

All situations in statistics require some data, and the best data is that which students compile themselves. For this comparison a single set of data is best presented as a bar chart, but compare the data from five or more distinct groups of subjects, same measure, and the multiple strip bar chat is a bloody mess. Five box plots above the same numberline, and so much more is revealed, at a small cost of loss of detail.

I used to think that box plots were a waste of time until I saw the above usage.

The same is true of physical representations. I am thinking of many algebra growth problems that involve squares and growing patterns. It is valuable to ask students to go through the actions of adding squares to watch a pattern grow through the addition of tiles. This action can help them have the physical experience of a rate of change. But this representation also has its drawbacks. It is clearly cumbersome and not efficient.

I think of them all as connected to making predictions about data â€“ certain representations lend themselves to different ways in which data is presented, and certain representations help make predictions about that data.

Tables are great when you need to generate data from a scenario â€“ you have a situation that has been given to you and you need a place to start. Creating a table for some initial data helps you see the patterns in whats happening and helps you make littler predictions about where the data is going. If I want students to appreciate tables, I give them a visual pattern or a scenario problem with a starting condition and a rate of change, then ask them some questions about what will happen.

Graphs are great when youâ€™re given several random data points that, even when arranged as a table, donâ€™t indicate a clear pattern. Sometimes plotting these visually helps you predict what other points could be missing, or what other points exist as the pattern continues. This is especially true for situations whose solutions depend on two variables, such as only having 30 dollars to spend on item A that costs 2 dollars and item B that costs 1 dollar. When I want students to appreciate graphs, I give them one of these situations (which usually lends itself to standard form of an equation, but they donâ€™t know that) or I give them several data points and ask them whatâ€™s missing in the pattern. This is easier to see when organized visually and you you a particular shape to your points rather than a random collection.

Equations are the most efficient way to make predictions about patterns â€“ if youâ€™re given an equation, thereâ€™s no reason to have any other representation. Equations are useful for predicting far into the future for your data â€“ maybe you can figure out the first few terms of your pattern, but trying to generate the 100th term is inefficient. Using an equation is like being omnipotent with a set of data. When I want students to appreciate equations, I give them a scenario but ask for a data point in the absurd future where the table or graph necessary to find the point would be too large and unwieldy to use.

The order Iâ€™ve presented these in this comment is also my typical order for presenting these representations to students: tables are useful at the beginning to generate data; graphs are useful once you have lots of it that may or may not be organized and may be missing some points, and equations are good for predicting the future.

A curious consequence might be: itâ€™s not particular situations that necessitate one representation versus the other; rather, its what data you choose to give them at the beginning and what you ask them to do with it that makes one representation more valuable than another.

Jodi:

So very true. This skill seems to be neglected in our classrooms. Computers can take one representation and switch to others over and over again, much faster than humans can. If switching back and forth is your only skill, I can easily replace you with a \$100 calculator from Target. And the calculator will be faster and more accurate.

But if Iâ€™m training students to be problem solvers who are smarter than computers, the â€œwhich representation is needed hereâ€ is a much more important question. Iâ€™m not aware of a computer that can answer that question.

Draw a simple line on a graph.

Now what is the value at x = 1.37?

Now they see that the equation is quicker and more accurate than the graph â€” even when inside the graphed region.

Or draw two lines that do not meet at integer values. Where do they meet, exactly? Hence that simultaneous equations are better in some situations than graphs.

But again, we can draw y = log x crossing y = x2 quicker on our graphics calculator than we can solve it.

(Of course y = x2 doesnâ€™t cross y = log x, but they only know that if they graph it!)

BTW: Essential reading from Bridget Dunbar also: Effective v.Efficient.

## 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.

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.

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 â€“Â link it to the real world, link it to a job, link it to students’ lives â€“ 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.

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, â€œwhat did those steps accomplish?â€ blank stares are often seen.

By a creating a â€œheadacheâ€ using a theory of need, weâ€™re really looking back to the situations that prompted the development of the mathematics we intend students to learn. Weâ€™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 itâ€™s a mathematical habit of mind. Mathematicians donâ€™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 Iâ€™m afraid I simply donâ€™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. Oneâ€™s certainty is achieved when one determinesâ€”by whatever means he or she deems appropriateâ€” 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 â€“ proving and disproving each of them â€“ 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. Scott Farrar chases this same idea along a different path.

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 (hereâ€™s a list of 5 statements about rhombi â€“ 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?

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 â‰¤ or â‰¥. 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.

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 â€” 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 â€œtestâ€ like y > x2. They plot their point green or red depending on whether or not it passes the test â€” 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?