Total 8 Posts

## 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.

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.

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.

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 â€“ can you predict the output for any input with certainty? â€“Â 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.

## Graduation & What It Means To “Model With Mathematics”

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.

That is modeling â€“ 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.

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.

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.

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 cousinâ€™s name is called.”
• Julie Wright: “As an embittered W, I am aware that there is lots of ponderous gravity for Aâ€™s and Bâ€™s, then everybody gets bored and speeds things up.”

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:

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

I ran into this when working up an exponential growth problem for my sonâ€™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.

## If Exponent Rules Are Aspirin, Then How Do You Create The Headache?

This Week’s Skill

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

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).

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

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

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. Itâ€™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.