Abstraction and Things Students Care About

Kids don’t ask “When will I ever use this?” when the math is easy. My students have no complaints if I ask them to solve 2x+9=21, or to find one-third of 900, or to find the square root of 225. That’s because, at some point, they got really good at all of those, and stopped caring whether they’d use it — it’s better than being asked to read Shakespeare, or generate their own science project. And these questions are all the way at the top of the ladder of abstraction. I’m throwing them on homework, no context, no meaning besides I want you to spend a few minutes at home practicing math every day. No problem.

The bottom rung of the ladder of abstraction normally goes pretty well for me too. Stacking cups. Graphing stories. You have a penny on day one, and every day you double your money. Fun stuff. Not too many complaints on those days either — and my students know they aren’t getting $10,000,000, and that they will never have a use for calculating the number of cups needed to stack to the top of their head.

But there’s something about that transition from the concrete to the abstract that begs the question. I don’t think it has anything to do with students’ desire to know what the math will be used for — I’m skeptical that an answer that relates to their life will make them a content, hardworking student for the next 45 minutes. They’re asking because they’re frustrated, and asking “when will I ever use this?” is a time-honored tradition of frustrated math students.

I had one of those days today. I’m doing a quick survey of quadratics right now. Nothing fancy — no factoring, no completing the square, nothing really beyond graphing them and looking at some of their properties, in particular symmetry, to reinforce function concepts and provide some background and formative assessment for later this year when we dive deep. Don’t ask me why I’m doing it. It was a bad idea, I don’t want to talk about it.

But we’re sticking with it for now. And today my goal was to move beyond graphing and naming parts of a quadratic (which they were pretty good at, notwithstanding integer operations mistakes) to the abstraction of the axis of symmetry always being -b/2a. I wrote up what I thought was a clever Desmos activity (focused on this graph), watched it bomb, switched to pencil and paper, saw mostly blank stares, and just felt like I had swung and missed on a great opportunity. Anyway, I had a prep at the end of the day, and I was pretty frustrated. I grabbed my copy of “What Is Mathematics For?” by Underwood Dudley and read through it for the umpteenth time. It was unbelievably comforting, and with nothing else useful to say tonight, I’m going to leave a few of Dudley’s thoughts

On why the “you’ll need this for x job” spiel is a lie:

I am glad that we do not have to depend on
workers’ ability to solve algebra problems to get
through the day because, as every teacher of
mathematics knows, students don’t always get
problems right. The chair of the department of a
Big Ten university once observed, probably after
a bad day, that it was possible for a student to
graduate with a mathematics major without ever
having solved a single problem correctly. Partial
credit can go a long way. This was in the 1950s,
looked on by many as a golden age of mathematics

In one of those international tests of mathematical
achievement appeared the problem of finding
which of two magazine subscriptions was cheaper:
24 issues with (a) the first four issues free and $3
each for the remainder or (b) the first six issues
free and $3.50 each for the remainder. This is not
a tough problem, so I leave its solution to you. As
easy as it is, only 26% of United State eighth-graders
could do it correctly. That percentage was above
the international average of 24%. Even the Japanese
eighth-graders could manage only 39%. No doubt
when the eighth-graders become adults they will
be better at solving such problems, but even so I
do not want them having to solve problems that
when solved incorrectly can do me harm.

People seem to think that because something
involves mathematics it is necessary to know
mathematics to use it. Radio does indeed involve
sines and cosines, but the person adjusting the dials
needs no trigonometry. Geologists searching for oil
do not have to solve differential equations, though
differential equations may have been involved in
the creation of the tools that geologists use.

But despite the truths that we might not want to hear, we cannot, and will not, abandon mathematics education —

What mathematics education is for is not for
jobs. It is to teach the race to reason. It does not,
heaven knows, always succeed, but it is the best
method that we have. It is not the only road to
the goal, but there is none better. Furthermore,
it is worth teaching. Were I given to hyperbole I
would say that mathematics is the most glorious
creation of the human intellect, but I am not given
to hyperbole so I will not say that. However, when I
am before a bar of judgment, heavenly or otherwise,
and asked to justify my life, I will draw myself up
proudly and say, “I was one of the stewards of
mathematics, and it came to no harm in my care.”
I will not say, “I helped people get jobs.”

I hope to say the same. And against perhaps my better judgment I am going to do my very best to help my students connect the graphs of quadratic functions that they are comfortable with the abstract equations they represent. And I don’t care what they ever use it for.


One thought on “Abstraction and Things Students Care About

  1. howardat58

    I love the quote from Dudley !
    Here is a suggestion, unless you did it this way already:
    1: View the parabola as fixed in space, with a coordinate system in place, and an equation y=x^2+5x+2
    2. “Wouldn’t it be a nicer equation if we had chosen a different set of axes ?”
    3. “How about moving the y axis so that it passes through the apex (vertex, whatever) ?”
    4. “But we don’t know where that is !”
    5. “So we give it a name. That will be the x value fro the new axis. Call it p.”
    6. Now we have a different x coordinate system, we measure distance from x=p and call this the X coordinate, with x = X + p
    7. Now we can write the equation in X and y … y=(X+p)^2 + 5(X+p) + 3
    8. The new y axis will pass through the vertex if there is no X term, so find the value of p which does this………
    9. The result is effectively “completing the square”, but with a graspable purpose, and the min value falls out, as expected.

    Change of variable (linear) is effectively the same as doing a translation of the object. it appears to be completely overlooked in the Common core. Why, I have no idea.


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