Why Questions on Algebra? Because I Feel Like It.

When I read textbooks, whether in mathematics or other fields, these are usually as boring as watching a light bulb flicker. Searching the Internet for Algebra problems can get to be pretty depressing. (Whether someone who spends her spare time looking for Algebra problems might already have mental health issues is a separate question not to be discussed at this time.)

Seriously, though, I don’t believe math is inherently boring. Today, I am doing a repeated measures Analysis of Variance. The question I want to answer is how far you can go from the original plan for a training program before it ceases to be effective. No one would imagine that if, instead of teaching Algebra on-line for an entire semester, you walked  up to a group of students with a flat piece of slate and a rock, scratched out the Associative Property:

(a +bX) +cY = a + (bX + cY)

then went out for beer for the rest of the semester, that the students would learn an equivalent amount as in our full-semester, state-of-the-art course. Where is the dividing line, though? How many days could you skip? COULD you replace the computers with sharp rocks and flat pieces of slate and learn just as much? One way to test for this would be to check the significance of the interaction effect between type of class and the improvement on test scores.

I could go into great detail about what we are actually doing, and I probably will next time, but for now I am going to lament the sad state of Algebra. Here are a few examples of Algebra problems

The DeVry University page has questions about how much things cost if apples are fifteen cents and oranges are thirty-five cents or what the area of a circle is when r is increased by three.

The Broome Community College page asks you to factor 16x – 8.

This GRE practice site is a little better. It asks questions to problems that are mildly interesting, such as calculating total income from investments with different rates of return.

There are thousands of sites like those above, and these reflect nearly every Algebra textbook in America. One thing these all have in common is that I don’t much like them. We are asking students to apply a formula to a neat little problem. There are several reasons these are not the way I think we should teach Algebra.

1. Most real problems are messy. It is not immediately apparent which formula you should use.
2. Students are learning procedures rather than understanding mathematics. When a problem looks like this, apply the first formula. When it looks like that, apply the second formula. But why? I think there is a big difference between learning rules and thinking. A really big difference.
3. In life, you have to ask your own questions most of the time. Someone else doesn’t give them to you.
4. Questions that can be answered in 15 seconds aren’t the kind that really promote thinking.

THIS I like, from Drexel University, the Algebra problem of the week. For example,

” Find a function that expresses where a child sits on a seesaw in terms of her weight.”

This I like, from the Julia Forum,

If you woke up in the morning and everything was twice as big, how could you know?

Part of learning Algebra, I think, should be requiring students to come up with questions as well as answers. Questions could be either useful ones, such as about the effectiveness of changing course design, or simply interesting, like how you could know if the whole world doubled in size. You see, I absolutely believe that Algebra can be both interesting and useful. Unfortunately, the way it is generally taught, it is neither.