A Couple Videos

A long time ago, minute physics posted these excellent videos (yes, the "proof" in the first doesn't really work, but yes, that series can be said to converge to -1):

Recently, someone posted this response:

This third video isn't mathematically correct--though there is some really really interesting mathematics in it. Our initial reaction is, I believe, to tell the author of the third video that he's just wrong. But I don't think that's the right approach--indeed, I think he's doing something we all should do more of, thinking about math. Yes, his argument doesn't quite work. But, he's reasoning about math. He's found something that doesn't make sense to him--that sounds wrong to him--and he's grappling with it, trying to show it's wrong. And he's found something substantive, not something trivial, like in so many math facebook memes. (What's 1+1+1+1+1+1x0+1= ? Does anyone really care?) Isn't this what we would want people to do, in any other subject--to make arguments, and thereby, to learn?

Number

What's a number?

That's a surprisingly difficult question that I plan to revisit a few times. But for now, we can go with this quick definition. Numbers are how we keep track of things. We make a one-to-one relationship between particular marks or actions, and something we want to count. For instance, when keeping score, we often use tally marks to count. I scored ||||| points, you scored |||| points, so I won.

But this is very very impractical. In game 3 of this years NBA finals, the Thunder scored ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| points,
the Heat scored ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| points.

Who won?

Each month, my rent is
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How do I know if I've paid the right amount? What if I'm ||||||||||||||||| short? Care to check if I've paid in full?

As I think you can see, things get ridiculously complicated.

Addition and subtraction are super easy (just delete tallies, or add more), but what about multiplication and division?

I have ||||||||||||| pieces of candy, and I want to share them between ||||| students. How many pieces does each student get? (No fair converting to our notation!)

All these problems would be infinitely easier with our modern decimal notation. But our notation didn't fall from heaven. People invented our notation, and invented the algorithms we use. (And, perhaps surprisingly to some of us, there are different algorithms, often used in different parts of the world.)

The first step was relatively natural. "Instead of writing all those lines, let's group them." The Thunder scored XIXXIIIXXIXXX (order doesn't matter in the notation I'm using here). But how do we go from something like XXIXXIIIXXIXXX or LXXXV to 85?--Historically, before  there was our modern decimal notation, that is. That was an incredibly difficult mathematical step. And now that we have this new way of writing--instead of LXXXV, we write 85--how do we do arithmetic in this new system? And how do we know that system will always work? Can you guarantee me that my books will always come out right if I use this new notation of yours, and these new algorithms? That I won't owe more (or less) that I actually pay? These are the questions Leonardo of Pisa, better known as Fibonacci, had to answer when he published his Liber Abaci which popularized this new numeration system. And he's rightly considered one of the greatest mathematicians ever, mostly due to his work with number systems.

See what we've done? Our children are in a position to relive the questions Leonardo of Pisa asked. To work through the same mathematics he did. To conjecture, and play with numbers. To work out our notation, and to come up with algorithms. And what do we do? We think they're being mathematical if they can find:

 537
-142
 395

Or that somehow magically they'll learn to think like Fibonacci if they can just write 395 under the line, and not 415. (A common error.) But what is 395? And what are 537 and 142? And what is subtraction? And why do we get that particular answer, and not a different one?

It's probably impossible for us, as adults, to relive these questions. But we can perhaps get something toward the struggle if we try to work them backwards. What's DXXXVII-CXLII? Are you sure? What's XXXVII times XLIII? (No fair converting to our notation!) Can you come up with an algorithm that always works?

Right Answer?

At the start of the Math for Elementary Teachers class I'm mentoring in, we asked the students how many of them thought they knew enough math to teach elementary mathematics. Almost all of them said they did. Everyone knows subtraction, right?

Actually, not really. Everyone can do the following problem, but it's a really difficult algorithm to understand (E. Thanheiser "Preservice Elementary Teachers' Conceptions of Multidigit Whole Numbers, Journal for Research in Mathematics Education 40, 251-281.):

 537
-142

In order to follow best, you should probably get out a piece of paper and work the problem. (Comment below!) Or, at least, figure out how you wold work the problem.

When you did it, you probably did something like the following: 7 minus 2 is 5. You can't take 4 from 3, so you have to borrow from the 5. So you cross out the 5, write a 4, and a little one next to the 3. 13 minus 4 is nine. Four minus 1 is 3. So the answer is 395.

Ok, now the hard part--the essay question part: What does that little 1 you wrote next to the 3 mean? Why can you cross out the 5 and write a 4, and why can you treat the second column as 13-4? How does that make sense? And do the 3 the 9 and the 5 say "Let our powers combine" and summon three hundred ninety-five? Could you explain it to a first grader? (Again, feel free to leave a comment below explaining your answers.)

Surprisingly, these latter questions are more closely related to the actual practice of being a mathematician, than the rote application of the algorithm is. Or perhaps even more closely related: Come up with a way of doing the subtraction problem, and show it will always give you the right answer. (Presumably, asked of someone who doesn't know the standard algorithm.) Again, if you'd like to try to come up with an algorithm, and to explain why it will always work, feel free to leave a comment. (Though, if it's an easy question for you, don't spoil it for everyone.) Oh, and there's definitely more than one right answer to this last question.

Mathematics Education

"Oh, you're studying mathematics? Well, at least there's only one right answer, right?"

I don't know how many times I've been told that. The short answer:"No, there isn't only one right answer. You've just been taught wrong." The long answer? Well, that's what I hope to address a bit on this blog.

First, a little background on me:

Though I have an undergraduate degree in mathematics, my real interests were in philosophy and, especially literature. I even took the GRE topic exam in literature way back in 2004. (More recently, my interests have shifted more toward music rather than literature.) But in 2005, a local private school needed me to teach a couple of more advanced mathematics courses--presumably only for a year before I left for graduate school. But, to my surprise, I found teaching really interesting, and decided to stay on. Four years later, I decided that I wanted to be a teacher, but at a more advanced level--and so I returned to school to pursue a Masters in Mathematics, hoping to get a job at a Community College after I graduated.

During my master's program, I was introduced to the discipline of mathematics education, and several professors encouraged me not to stop with a masters, but to pursue a Ph.D. I applied for Ph.D. programs in mathematics and mathematics education, eventually setteling on mathematics education. At the time, I thought collegiate mathematics education research sounded really interesting, but research into lower levels of mathematics education sounded boring.

That was before I got to graduate school. After I arrived, I fell in love with some of the aspects of elementary mathematics education, particularlly preservice teacher education. Most likely, I'll do my dissertation work on collegiate education--I'm currently working on a NSF Calculus grant--but I definitely have interests in lower level mathematics.

Returning to the mathematics education. Presumably, when we teach "mathematics", we should teach people to be good at mathematics--to begin to do the thing mathematicians do. (Or perhaps to use mathematics as engineers and scientists do.)

But what is mathematics? It's the study of numbers, memorization of long bizarre, nearly nonsensical algorithms, and confusing symbols? It's not exactly supposed to make sense, what we're really after is the right answer. Get that, you succeed, don't get that, you fail. Right?

Wrong. (Though I should confess, that describes my classes far more than I would like it to.) What is mathematics? That's actually a surprisingly difficult question to answer. It's the studey of numbers? But what's a number? And what about branches of mathematics, like Topology, and Category Theory, that hardly use numbers at all?

Hopefully in the following posts we'll begin to investigate mathematics, and make some guesses about its nature. But for now, the important point is: That's not what it is. Mathematicians almost never have the right answer, or an already existing formula, or anything like that. The subject you learned in school probably doesn't have terribly much to do with what mathematicians do.