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 – 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.

This Week’s Skill

Exponent Rules.

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

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.