Externalizing what you worked so hard to internalize

I've been writing quite a bit on the work of George Pólya, particularly focusing on what strike me as some common misconceptions and overlooked aspects. I started with his emphasis on building self-confidence and making the experience pleasant since these things are so much at odds with the tough-talk rhetoric that has become so popular in education circles over the last few years.

Pólya also spends a great deal of time talking about the importance of "getting inside the student's head," but compared to the parts about self-confidence, I think the point here is less emotional and more cognitive. One of the main ideas of the book How to Solve It is that people who work with mathematics professionally have almost invariably mastered and internalized a number of useful problem-solving tools. Unfortunately, by internalizing these tools we have also in effect hidden them from our students.

When faced with a problem, we quickly and in many cases unconsciously run through a number of techniques that we have found over the years to be helpful. We examine the problem, determine the unknown, compare the problem to those we've encountered in the past, perhaps draw a mental picture and run through any number of similar steps before deciding on the proper strategy. To the student this gives an unrealistically linear appearance to the process, as if "let u equal X cubed minus 8 and then factor by substitution" was the first thing that popped into our mind .

Polya's point, and I think it's a profound one, is that to explain a process to someone who is unfamiliar with it, you have to be self-aware enough to explain the whole process, not just the parts you are still conscious of.

Here's a great bad example from one of our recent whipping boys.





For the math people out there, this is not a bad explanation, but we aren't the target audience. For the target audience, this just terrible, particularly for a video. If you're working live, you can read the room. With a recorded medium, you have to anticipate the room.

Think of the world as being divided into two groups: people who can do this problem without help; and people who can't. The person doing this video is obviously in the first group, For the first group, setting up equations is second nature. It's obvious me, having been through this from every vantage, that the instructor took a few seconds to understand the problem and mentally outline the steps for solving problem, then he hit record and started filling that outline in.

The trouble is, for the overwhelming majority of the people in the second group, getting that outline is the part they were having trouble with. Very seldom do you see students who can effortlessly set up the equations for a word problem but who then get stuck on the basic algebra.

The students who need help look at this problem and see a lot of possible variables and equations. Maybe X should be the amount spent on the expensive paper. Maybe we should set up an equation to show the difference between the two amounts, cheap and expensive. You and I know these are dead ends because we've seen this show before. The key to explaining this type of problem is to imagine what it would look like if you were seeing it for the first time.

George Pólya was, of course, big on word problems. Here's a relevant passage from How to Solve It:

Pedagogical MacGuffins

What's a MacGuffin? A MacGuffin is the key or stolen diamonds or secret code or NOC list that the characters  desperately pursue. Audiences, pretty much by definition don't care about MacGuffins, but they do enjoy watching characters pursue them. Sometimes the audience isn't even clear on what the MacGuffin is.



Do you know what a NOC list is?

A pedagogical MacGuffin is a type of problem we pretend to care about even though we really don't. Like its fictional counterpart, what's important with a mathematical MacGuffin is not the thing but the pursuit.

The classic example is factoring polynomials. A standard part of most algebra classes is to learn how to take a trinomial like  

2x^2 - x - 15

and find two binomials you can multiply together to get it

(2x+5)(x-3)

Every once in a great while, you'll get a trinomial that won't factor but the rest of the time you'll get a nice clean answer where each binomial consists of an integer times x plus another integer. At least, that's how it works with the assignments. You may even be told that polynomial factoring is useful because it can help you solve equations. That part is a lie.

With a couple of notable exceptions (differences between two squares and perfect square trinomials), you will probably never even try to solve a problem by factoring a quadratic for the simple reason that most don't factor.

Not only does solving by factoring usually not work; we already have a simpler method that always works, the quadratic formula.

The truth is, we don't care whether or not you know how to factor a trinomial; we care about what you learned in the pursuit, things like problem solving skills and insights into how numbers work.

(2x+5)(x-3) is just something to keep the plot moving.

If you're interested, try a few randomly generated trinomials and see how many you can get to factor.