Math : Abstraction :: Karate : ?

There’s something particular to mathematics education I like to call the “wax on, wax off” problem.

Mr. Miyagi: Wax on, right hand. Wax off, left hand. Wax on, wax off. Breathe in through nose, out the mouth. Wax on, wax off. Don't forget to breathe, very important.  -- from The Karate Kid

Specifically, that there are some concepts learned at the lower levels of mathematics that seem inexplicable – both in a real life and an abstract sense – until they are fully explored at the higher levels.

Daniel: I learn plenty, yeah, I learned how to sand your decks maybe. I washed your car, paint your house, paint your fence. I learn plenty!

I will use functions as an example.

Officially, a function takes some set (the input) and associates it with some other set (the output) so that each input has only one output. Unofficially, in Algebra I students think of it as a regular equation

y = x² – 3

with a funny notation

f(x) = x² – 3

and an extra condition called the “vertical line test” which they learn dutifully (and can do well!) but never learn why that condition would be important or one should care.

In Algebra II, students learn to manipulate the functions by combining them using standard operations

f(x) + g(x)

f(x) – g(x)

but (most) students tend to consider this equivalent to adding equations to solve systems of equations.

Also in Algebra II a new symbol for composition of functions get introduced

f(x) o g(x)

which is new to students and seems unique to functions (although some might consider it equivalent to the substitution method of solving systems of equations). However, it tends to be taught for one day and then forgotten about as it doesn’t connect with any of the other Algebra II curriculum.

So at this point for many students functions seem like a funny notation that is designed to confuse and doesn’t allow them to solve anything new.

Then in Pre-Calculus comes the function transformations:

f(x) + k             Up

f(x) – k             Down

f(x-h)               Right

f(x+h)              Left

-f(x)                 Reflection over the x-axis

f(-x)                 Reflection over the y-axis

Finally the power of functions becomes apparent. Previously whenever students learn about how to graph an equation they have to relearn a new formula for each one. With the function transformations comes the power of generality; the same rules can be applied to any function.

In Calculus functions are used for the difference quotient

used in the definition of a derivative, arguably the crux of Calculus. Here not only are functions useful, they are essential.

The Issue

There’s a lot of rhetoric about how students don’t see how mathematics apply in real life, but this problem isn’t even that; at an abstract level the concept of a function seems like a meaningless exercise. Even to a novice, at some level it is immediately apparent why taking inverses to move elements of an equation might be useful.

2+x=5

2+x+(-2)=5+(-2)

x=3

Without the power that comes later, functions (and some other parts of mathematics) appear with no motivation.

Resolution?

Two easy solutions:

1. Take functions out of the early curriculum and only mention them later when students are ready.
2. Put the more powerful use of functions (at least the transformations) earlier in the curriculum.

Each has their problems:

1. The background students have when they reach functions – even if superficial – is quite helpful when students see get to abstract manipulation. Being able to parse f(x+4) in a natural way is hard; the leap is essentially the second layer of mathematics abstraction on top of the first layer of variables.
2. A full functions treatment at the Algebra I / II level might simply baffle students. At that level some still are percolating the idea of a variable, and that x + x = 2x and not x². The full introduction of functions thus may cause more harm than good.

I have a third compromise solution:

Teach functions at the superficial level for algebra students, but give students a preview of what is to come. Explain that functions allow us to manipulate equations at an even more abstract level than usual, and that higher abstraction equals more power. At the very least the students may realize there is purpose to what they’re doing even if they don’t realize the exact details yet.

There’s another use to functions that may also be helpful to a teacher. Most graphing calculators (without extra programs) can only graph functions. Graphing the (sideways) parabola y²=x requires two equations in the calculator.

Therefore as long as a teacher is using a graphing calculator there is an immediate real-life component.

Most discussion of “relevance” to students focus on real-life application, but even when topics are self-contained in mathematics there is a relevance issue. More attention needs to be paid to reassuring students that the future holds extraordinary things.

Mr. Miyagi: Look eye. Always look eye.

Jason Dyer holds degrees in Fine Arts Studies and Math and teaches at Pueblo High School in Arizona. His school mascot is the Warriors and his other blog of residence is The Number Warrior.

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