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This tweet from a friend of mine is one of my favorites.

My friend has taken a problem from the world that was personal to her, identified the variables that are essential to the problem, selected a model that describes those variables, performed operations on that model, and re-interpreted the result back into the world. And tweeted about it.

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That is modeling – the process of turning the world into math and then turning math back into the world. My friend probably wouldn’t wouldn’t label her experience like that but that’s what she’s doing. That’s what people who do math in the world do.

We know how this looks in many textbooks, though.

The amount of time (t) it takes a number of graduates (n) to cross the graduation stage can be modeled by the function t(n) = n/8. How long will it take all 288 graduates to cross the stage?

Here students would simply perform operations on real-world-flavored math while the important and interesting work is in turning the world into that math and turning that math back into the world.

Here is an alternate treatment, one that has students modeling as the practice is described in the Common Core.

Show this video.

Ask: “If I want to set an alarm that’ll let me take a long nap until just before my cousin Adarsh crosses the stage, how should I set the alarm?”

By design, it’s a short video. I’d like it to be boring enough to provoke my friend’s modeling but not terminally boring.

By design, it lacks mathematical structures because we’d like students to participate in the process of developing those structures. They won’t do that unassisted.

Before we get to the algebraic model, we can ask some important and interesting questions.

How long do you think it will take my cousin to graduate? Just estimate.

I asked that on Twitter and received the following estimates:

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These guesses interest us in a calculation and also prepare us to evaluate whether or not that calculation is correct.

Sketch the relationship between the number of graduates and time.

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Asking students to sketch the relationship, rather than plot it precisely, asks them to think relationally (“how do these two quantities change together?”) rather than instrumentally (“how do I plot these points?”).

Many students will assume the data is linear. But this prompt may invite some students to consider the possibility that the data is non-linear.

Collect data. Model the data. Get an answer.

Ask students to create a table of values. Ask students to plot the data in Desmos. Regress the data. Give them the graduation program. Calculate an answer.

I plotted the first ten names and modeled their times with a linear equation. (“Time v. names read” was my model, though commenter Josh thinks “time v. number of syllables read” would be more accurate.) The calculation for cousin Adarsh’s 157th name is 19 minutes. I would be foolish to rely on that calculation, however.

Ask your students to “Assume your answer is wrong, that something surprising actually happens. Anticipate that something and fix your mathematical answer.”

George Box: “All models are wrong, but some are useful.”

This is where we turn the math back into the world. This is where we make some math teachers uncomfortable, admitting that the world and the math don’t correspond exactly and that the math needs modification.

Watch all of these math teachers make exactly those modifications in the comments of the preview post. They perform mathematical operations and then proceed to describe why the results of those operations are wrong.

  • Scott: “Add the bit of time prior to starting and a few seconds for a switch in readers as tends to be customary in larger groups like this … “
  • Sadler: “14 minutes and 10 seconds but given that it is better to wake 10 seconds early than miss it, I would submit 14 minutes.”
  • Scott #2: “You would probably want to set your timer a little earlier so you are fully awake when your cousin’s name is called.”
  • Julie Wright: “As an embittered W, I am aware that there is lots of ponderous gravity for A’s and B’s, then everybody gets bored and speeds things up.”

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Validate (or invalidate) the answer.

Commenter Mark Chubb, at the end of his modeling cycle: “Can’t wait to see Act 3.” Act 3 is the reveal in this task framework I call three-act math. It isn’t enough for Mark to simply read the answer in the back of the book or hear it from me. He wants to see it. So:

Show the answer.

If you built a linear model from the first ten names, your answer winds up too large. Instead of 19 minutes, my cousin graduates at 17:12, sooner than the math predicted.

Why?

In the video, you can hear the validation of Julie Wright’s hypothesis above. The A’s and B’s get a lot of pomp, and then the commencement reader races through the rest.

Many congratulations to Megan Schmidt for her guess and to Scott and Kyle Pearce for their calculation. They all put down for 18 minutes. Special mention also to aga bey for 16.3 minutes. That commenter’s method? “I took the average of all submissions upthread.” Strong!

Again, if mathematical modeling requires the cycle of actions we find here, our textbooks typically only require one of them: performing operations. The purest mathematical action. The one that is often least interesting to students and the least useful in the world of work. So let’s offer students opportunities to experience the complete modeling cycle. Not just because those are the skills that most of the fun jobs require. But because modeling with math is fun for students now.

Featured Comment

Stacie:

I ran into this when working up an exponential growth problem for my son’s precalculus class. The CDC had data on the number of Ebola cases which could be modeled with an exponential growth curve at the time. However, the math needed correction because of a sudden increase in cases. The CDC readily admitted they believe the cases were unreported by a factor of 1.5 to 2.5. Thus, a human eye on the data to recognize that and make an adjustment was necessary.

Later, when the curve could be modeled nicely by a logistics curve, the equation was still incorrect in predicting the end of the epidemic. As teachers we would like to be able to button everything up and wrap it in a bow, but the real world seldom works that way.

This Week’s Skill

Exponent Rules.

Rules like these are too quickly abstracted, memorized, confused, and forgotten.

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We can attach to them meaning and purpose by asking ourselves, why did we come up with these shortcuts? If these shortcuts are aspirin, then how do we create the headache?

What a Theory of Need Recommends

Again, with Harel’s “need for computation,” students need to experience the “longcut” before they learn the shortcut. Otherwise it’s just another trick in the endless series of tricks students call “math class.”

Several people suggested the same in last week’s thread, of course. Ask students to calculate expressions like these the long way before discussing shortcuts the students may have noticed (or that you may have noticed as a member of the class also).

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Chris Hulitt was one of my workshop participants in Norristown, PA, and his group suggested an important addition to this idea. Ask students to calculate this expression instead

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Looks the same as the last, right? But whereas the last expression resolves to 16, this expression resolves to 1. That headache is a little bit sharper. How did this gangly mess of numbers result in such a simple answer? Could I have realized that in advance?

Again, this isn’t real world, or relevant, per our usual definitions of the term. And yet this approach may still endow exponent rules with a purpose they often lack.

Next Week’s Skill

Determining if a relationship is a function or not.

This is another skill that can become quickly instrumental (run a vertical line over the graph, etc.) and obscure why it is aspirin for a particular kind of headache.

Let us know your ideas for motivating the definition of a function in the comments.

What You Recommended

Tom Hall:

I think of my 6th grade students writing down the prime factorizations of whole numbers (Why learn that? Oh yeah, to improve number sense and seeing structure behind numbers, among other reasons). When you work with something simple like 24, writing out 2*2*2*3 is not so bad. When you up the stakes to something like 256, writing out 2*2*2*2*2*2*2*2 becomes annoying. It’s not so much a headache as a tedious process that any normal person would like to make quicker and easier. At this point, I discuss exponents as a means of communicating all of that multiplication without having to write it all.

Here are three minutes of a Harvard graduation ceremony and the relevant program. My cousin Adarsh is graduating and his name is quite a ways down.

I’d like to take a nap but I’ll set a timer first so I won’t sleep through my cousin’s walk across the stage.

What time should I set the timer for?

Tell us the time and your method in the comments. The winner is whoever comes closest to the time my cousin walks across the stage, without going over.

This Week’s Skill

Factoring quadratic trinomials.

eg. We can express quadratic trinomials like x2 – 7x – 18 as the product of the two binomials: (x – 9)(x + 2).

If you find that language disorienting, if it makes you wonder why anyone would even bother on a sunny day like today, you’re in good company with lots and lots of math students. At the secondary level, there are few skills that seem less necessary to students and few skills that seem harder to motivate for math teachers than factoring quadratic trinomials. (Sample their stress.)

What You Recommended

Mercifully, very few of the 70-ish comments on my last post suggested an instructional strategy for this skill without also describing the theory that gave rise to the strategy. We need more of that kind of conversation, not less.

Here are three theories I found particularly interesting:

The first two solutions seem to me very clearly defined and very easy to implement but also very far-fetched.

Why am I interested in the history of a topic that has terrorized me for years? I’d like to know less about its history than more. (Even Andy, who suggested the idea, admits its vulnerability.)

And if learning this tough skill now helps me learn some tougher skill later (what Josh G. described as “passing the buck upwards“) I can save myself the trouble twice over and learn neither.

The last design theory has a lot of promise. People love puzzles after all, even the kind that are far from thereal world.” But it seems very difficult to implement.

I consulted two textbooks – one Algebra 1 text from McGraw-Hill and Pearson each. One attempted to tie the skill to the real world. The other said, “Look we’re just going to teach you this stuff okay.” The latter approach is honest, if uninspired. The former approach seems overly self-satisfied. After checking off the “real world” box, they proceed to teach the skill abstractly through a series of worked examples.

What a Theory of Need Recommends

First, ask yourself, if the skill of factoring quadratics is aspirin, what is the headache? How does the skill relieve a mathematician’s pain? The strongest answer, I think, is that the skill helps us locate zeros. The number that evaluates the expression x2 – 7x – 18 to zero isn’t obvious. In its factored form however – (x – 9)(x + 2) – the zero product property tells us the answer quickly: x = 9 and x = -2.

This is Guershon Harel’s “need for computation,” particularly the need for efficiency in computation.

Once that need is clear, the activity becomes much easier to design. Students should experience inefficient computation before we help them develop efficient computation.

So with nothing on the board, ask your students to: “Pick a number between 1 and 10. Write it down.”

Not a problem. Now put the expression x2 – 7x – 18 on the board and ask students to evaluate it for their number.

This is an unreal and irrelevant task, admittedly, but no one asks “When will I ever use this?” because students tend to ask that question when they feel disoriented and stupid. This prompt is relatively clear and accessible.

Now you ask: “Who got zero? Anybody? Anybody? Raise a hand. Nobody? Okay. Not a problem. Try a different number. Try a different number. Try a different number. Don’t stop until you get a zero. Call me over when you do.”

If someone did get a zero, ask them to get another one. (Later question: how do they know there are only two solutions to this equation.) Record the solution next to the quadratic on the board. Put up three more. Ask them to find more zeros.

Tease the possibility that a more efficient method than guess-and-check exists.

After 5-10 minutes of guess-and-check, help them learn that method.

What This Is and Isn’t

I’m not saying this activity will be your students’ favorite day in your class all year. Factoring quadratics was never going to be that.

But I’ll make a mild claim that this activity will be motivating for students. We’ve created a task with a clear goal state and a low entry and a high exit, a task that is iterative with timely feedback. These features are all common to the most intriguing puzzles. Of course a student could ask, “Why do I care about finding zero?” But they could ask similar questions about Sudoko, Tetris, and other puzzle games. They don’t because puzzles are, by definition, puzzling.

I’ll make a strong claim that this activity will endow factoring quadratics with a sense of purpose that it often lacks. Not purpose in the world of work or surfboards or trains leaving Philadelphia traveling west, but a purpose in the world of math. By tying the skill of factoring quadratics into a network of older skills (especially “guess and check”) we strengthen all of them.

I’ll make a strong claim that this is an example of taking a theory of instruction and enacting it. Finding a workable theory of instructional design is hard enough. Enacting it is even tougher. I love that work.

Doug Mackenzie asks an important question which I’m about to ignore:

Is it bad/good theory to expect that they will “construct” their own aspirin? (Do we leave them in disequilibrium until they get themselves out?) Is it good/bad theory for teachers to deliver the aspirin, or should students only get aspirin from other students

I’m not making any recommendations here about how students should learn that more efficient method for finding zeros. Tell them that method directly. Let them discover it. I know what I would do. We can draw from research. But that isn’t what this series about. This series is about creating the need for new learning, not satisfying it.

Next Week’s Skill

Exponent rules.

It’s like foldables were invented for exponent rules. Students can memorize a bunch of rules and write them down in something organized and pretty.

But why do we need them? If exponent rules are aspirin, then how do we create the headache?

Other Great Comments About Factoring Quadratics

It turns out I was on a similar frequency as Eric Fleming, Joshua Greene, Chuck Collins, and others.

Tim Hatman does some really impressive work exploiting Harel’s “need for certainty”:

So here’s my headache. Graph y=(x2 + 7x + 10)/(x + 5)

Without factoring, the only way to graph this is to just start plugging in x’s and making a table – that’s a headache! But when you start plotting the points…Whaaaaaaaat?!? It’s a straight line! How did that happen? What’s the equation of that line (why is one point missing) and how can I get there through a shortcut?

Malcolm Roberts names a central dilemma to all theories of instruction, not just this one:

Given that all learners are different, and that the context of learning varies every time we teach, it seems to me to be a near impossible task to create a situation that will be headache inducing for all (maybe even the majority of) students all (maybe most of) the time.

Simon names one key misunderstanding of factoring quadatics:

I think one of things that’s important is that our students understand that 1) factorising doesn’t change the value of the expression and 2) why it is more useful. Too often I find students thinking (x-3)(x+2) only ‘works’ for x = {-2,3}.

Chuck Collins names a second:

you’d be surprised how many college students don’t realize that the quadratic formula gives the same solutions that you get from factoring

Scott Hills recommends “diamond puzzles” in the weeks running up to this instruction:

I start out, about 2 weeks before factoring becomes a part of the math lexicon, with “diamond puzzles” in which students must first identify what 2 numbers add to a particular sum while multiplying to a particular product. The puzzle being the point, no mention is made of factoring.

Several months ago, I asked you, “You’re about to plan a lesson on concept [x] and you’d like students to find it interesting. What questions do you ask yourself as you plan?”

There were nearly 100 responses and they said a great deal about the theories of learning and motivation that hum beneath everything we do, whether or not we’d call them “theories,” or call them anything at all.

  • “How can [x] help them to see math in the world around them?”
  • “How can I connect [x] to something they already know?”
  • “How can I explain [x] clearly?”
  • “What has led up to [x] and where does [x] lead?”

You can throw a rock in the math edublogosphere and hit ten lessons teaching [x]. They might all be great but I’d bet against even one of them describing some larger theory about learning or mathematics or describing how the lesson enacts that theory.

Without that theory, you’re left with one (maybe) great lesson you found online. Add theory, though, and you start to notice other lessons that fit and don’t fit that theory. When great lessons don’t fit your theory about what makes lessons great, you modify your theory or construct another one. The wide world of lesson plans starts to shrink. It becomes easier to find great lessons and avoid not great ones. It becomes easier to create great ones. Your flywheel starts spinning and you miss your highway exit because you’re mentally constructing a great lesson.

Here is the most satisfying question I’ve asked about great lessons in the last year. It has led to some bonkers experiences with students and I want more.

  • “If [x] is aspirin, then how do I create the headache?”

I’d like you to think of yourself for a moment not as a teacher or as an explainer or a caregiver though you are doubtlessly all of those things. Think of yourself as someone who sells aspirin. And realize that the best customer for your aspirin is someone who is in pain. Not a lot of pain. Not a migraine. Just a little.

Piaget called that pain “disequilibrium.” Neo-Piagetians call it “cognitive conflict.” Guershon Harel calls it “intellectual need.” I’m calling it a headache. I’m obviously not originating this idea but I’d like to advance it some more.

One of the worst things you can do is force people who don’t feel pain to take your aspirin. They may oblige you if you have some particular kind of authority in their lives but that aspirin will feel pointless. It’ll undermine their respect for medicine in general.

Math shouldn’t feel pointless. Math isn’t pointless. It may not have a point in job [y] or [z] but math has a point in math. We invented new math to resolve the limitations of old math. My challenge to all of us here is, before you offer students the new, more powerful math, put them in a place to experience the limitations of the older, less powerful math.

I’m going to take the summer and work out this theory, once per week, with ten skills in math that are a poor fit for other theories of interest and motivation. As with everything I have ever done in math education, your comments, questions, and criticism will push this project farther than I could push it on my own.

The first skill I’ll look at it is factoring trinomials with integer roots, ie. turning x2 + 7x + 10 into (x + 5)(x + 2). All real world applications of this skill are a lie. So if your theory is “math is interesting iff it’s real world,” your theory will struggle for relevance here.

Instead, ask yourself, “Why did mathematicians think this skill was worth even a little bit of our time? If the ability to factor that trinomial is aspirin for a mathematician, then how do we create the headache?”

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