Are you a math teacher? If you are then you probably have built up a tolerance to the funny looks; a tolerance to the subtle comments that are meant to infer that you are VERY different from everyone else. You must have this tolerance or how else could you continue in your profession. I understand. I have it too.
But here is the thing. It dulls us to the messages around us. Something is wrong in in the state of math education. Most of us know this, though some still fight it. Most of us know that it is the charge of math teachers to make it better. We must raise an entire generation of children that meet math teachers and say, "Yeah, math was always too easy for me so I decided to do something difficult like teach grammar instead." That is our charge, and to make it happen we need to assiduously evaluate everything that we do. We must find the mistakes and fix them. I don't know if what I am about to propose a solution to is the most pressing problem, but it definitely is one. We may be able to learn something from the problem, and I hope something from my proposed solution. Here goes...
Complex Numbers
Let's take what a child is supposed to know about complex numbers after Algebra 2. Algebra 2 is or was a terminating course in high school math. So this may be all that any one every learns about these crazy things. Here are the objectives from the Algebra 2 book I am teaching out of during summer school:
- Simplify radicals containing negative radicands
- multiply pure imaginary numbers
- solve quadratic equations that have pure imaginary solutions
- add, subtract, and multiply complex numbers
- simplify rational expressions containing complex numbers in the denominator
This is really the first time that students have been exposed to these kinds of numbers. Some have heard about them and wondered what they meant. And what do get to find out about them. That the purpose of complex numbers is to be added, multiplied, subtracted and simplified. Imagine that you were trying to sell and innumerate person on the integers.
Stone Age Math Teacher: Hey, have you heard about this great new number -1?
Stone Age English Major: Cool what can you do with it?
Stone Age Math Teacher: You can add, subtract, multiply and divide with it. Pretty cool huh? And you can solve any subtraction problem if you allow a whole class of new numbers called the integers!
Stone Age English Major: Wow. Major. Let's go invent beer, so I can invent poetry.
A purpose for complex numbers must be the ground work to any introduction to these numbers. Many of you may feel that these numbers have no purpose. Or that maybe the purpose is to allow us to say that yes, every quadratic does have two solutions (if you count multiplicities). But no, the purpose is real. It is part of our everyday existence. Just as the reason that 1, 2, 3 came about was to help sheep herders keep track of the flock. We are talking real.
What is it. Yawn.... I will write that tomorrow.
7 comments:
We're waiting... :-)
Seriously, I wanna know.
I heard someone recently say that if parents were promised their kids would score an 800 on their verbal SATs, but as part of the deal they would never read another book for pleasure, no parent would go for that. Yet we make a deal with that devil every day in mathematics. You and that person should get together and fix math ed. Wish I could remember who it was.
That is a great analogy for what we do. I often don't think about the standardized testing regime because I work at a private school. It is important for us to prepare kids for it, but I know that kids would be more likely to remember simple math if they believed it had a purpose.
It doesn't begin to make sense to teach complex numbers in high school. They're not going to see them again until differential equations in college (and how many end up there?). I spend a lot of time on the history of math to demonstrate the why/where/how of complex numbers. I feel guilt as sin for having to teach complex numbers but I want them to realize that this isn't an arbitrary collection of numbers. Rather, they provide means of solutions that are otherwise unattainable.
But all that is beside the point. I say, use the TI-83/84 and teach how to do complex arithmetic on it. Then spend a day on rotations, just to take the pain away. Then move on.
What we need to do is find out how to keep these damn test scores high enough but also address relevant content. complex numbers are not relevant to high school students.
Kevin,
I think your outline is for what to cover when it comes to complex numbers is right on track. The only part of your post that worries me is: "Rather, they provide means of solutions that are otherwise unattainable." Why would I want to attain these solutions. Math should have more connection to reality then simply creating solutions, so that we can then test students to see if they can find "meaningless" solutions.
This is not to say that I think that complex numbers should be taught. They should, but we, the teachers, should learn more about what those solutions might mean. We need to all be taught more about complex numbers and a way to make it meaningful. This is why I think that the rotational nature of complex numbers is so important. It is possible to imagine a situation where a 90 degree is the solution. Then the math should come up with i as the solution.
I agree, I'd rather not teach complex numbers at all at this level. But historically they came about without application. They were just work arounds. I do point out that they inseparable from modern technology and talk a bit about those applications. But as with so many applications, the science is too much. In mathematics, we generally outstrip the pace of science in schools so we're left with really powerful tools but little meaningful content to apply them to.
I think that any application is enough. The more it connects the better, but with imaginary numbers/complex numbers we have the issue of their NAME. It is easy to believe that imaginary numbers are made up. The truth, in my opinion, is that all numbers are made up and that imaginary numbers are no more made up than natural numbers. I think using something like free transform on a photoshop does a good job of showing what complex numbers do through multiplication. This is a good enough application. It is as real as solving when a projectile lands, but it is more important in this case because we have to prove that imaginary numbers are not just "made up".
Certainly the labels: Real, Complex and Imaginary don't help the problem. I'd argue that natural numbers are at least perceived as more real than imaginary numbers. As for rotations you don't need complex numbers, you can use matrices which ties in with systems of equations. Moreover matrices lead into transformations in general and vectors.
The photoshop example is clever, but just because we can apply an idea doesn't mean we should or would. I'd argue that we're just looking for reasons to use these things because they're in our state standards.
But given that we are going to teach them, yes we should do something with them. I agree. For my part I do a week or so on fractals, concluding with the Mandelbrot set. It ties in complex numbers and also iteration and its pretty.
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