### Feb

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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.

- Most real problems are messy. It is not immediately apparent which formula you should use.
- 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.
- In life, you have to ask your own questions most of the time. Someone else doesn’t give them to you.
- 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.

### Feb

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# Not All Mathematical Ideas are Created Equal

Filed Under Algebra | 2 Comments

I was reading a book this week, Mathematics for the Intelligent Non-mathematician. If it was a person, this book would be your grandmother, not terribly exciting but pleasant to spend time with and if you paid attention you were likely to learn something.

Since I use mathematics for my living, you might reasonably wonder why I would be reading this book. The answer is that I believe in considering different perspectives. I’ve never really quite “got” the whole humanities thing. When I took history in school, I was secretly thinking, “They’re all dead. Get over it.” In English class, I was the kind that made teachers throw up their hands in despair. They wanted me to discuss, “The deep meaning of Moby Dick, what do you think it is really about?”

What did I think it was really about. I thought it was about a big white whale, for crying out loud, because it said that on the first page and about seven hundred more times throughout the book. The title? That’s the name of the whale, hello? Apparently, that was not the correct answer and you are supposed to say that it is a metaphor for the universal struggle of man against the sea, or man against himself or for man’s domination of marmots.

As you might guess, the second I had the opportunity for classes in college like Accounting, Calculus and Statistics where the questions had actual answers, like 42, I jumped at the chance. This isn’t to say that I made A’s in all of those classes initially, as that would have interfered with my plan of going to parties at night and sleeping through the morning. This plan was ended through a talk with the dean and some threatening words about losing my scholarship and having to find $20,000 under a mattress. Heck, I didn’t even own a mattress, much less $20,000 to find under it.

So, here I am thirty years after graduation looking at mathematics from a more naive point of view, which brought out a couple of points I had never really given much thought.

The first is that mathematics is the most general thing in the world. You cannot apply psychology to rocks or biology to building a space shuttle or oceanography to orthopedic surgery. However, as the author said, you can count devils or angels, whales or stars. In fact, when I went from being an industrial engineer to studying for my Ph.D. in Educational Psychology I used the exact same equations I had applied to predict which cruise missile would fail testing before launch to predict which child with a disability would die within the next five years. (Yeah, I wasn’t a lot of fun at parties back then.)

The second interesting point was one that is obvious after someone else states it, i.e., some ideas in mathematics are more important than others. For example, it is a fact that the digits in multiples of nine always add up to nine, e.g., 2x 9 = 18 and 1+8 = 9. This is not a key fact on which a lot of mathematics is based. So, this led me to thinking about the ideas in mathematics that I think are crucial and wondering about what other people think.

I always thought that the basic properties of real numbers, such as the distributive property –

A x B = B x A or A+ B = B +A was one of the most fundamental ideas in mathematics.

A second really important idea was the associative property, –

A(B+ C) = AB + AC

and the commutative property is a third

(4A + 2B) + 11C = 4A + (2B + 11C)

Once a student understands these properties, it opens up an enormous number of problems that he or she can now solve.

And that is why I like teaching Algebra.