Wednesday, December 17, 2008

Students learn what they need, not what is assigned

A friend recently wrote about her experience with the currently fashionable style of teaching where students are expected to reinvent mathematics.

She wrote:
So much attention is given as to why - and to having students figure out algorithms themselves. Some teachers feel this is a waste of time - time that could be used for the students to be practicing the algorithms and becoming more fluent.

I think that this is largely a waste as well. We cannot expect students of all abilities to recapitulate in 6 years the history of more than 2000 thousand years of thought by the best minds the earth has ever seen. To imagine that they can is simply ludicrous. To imagine that what they can invent in this short time and that they don't need the benefit of those great thoughts is similarly ludicrous.

At the risk of sounding self-contradictory, I also think that when the student owns the task of learning, they will learn enormously better than when the task is imposed. This does not imply that the student discover everything, however. It merely implies that they need to discover the need for the things that they learn. Students who need to learn something can learn from almost any source, even from something as currently unfashionable as, say, sitting quietly thorugh traditional lecture.

The last time I taught in the classroom was as a member of a two-person teaching team teaching a software engineering class on machine level programming. In the past, this had been done by lecture and assignment and was truly a stunningly boring class. On the first day, I turned the structure of the class upside-down and assigned the entire final exam. This consisted of a single question in the form of a task (to build a robot that would drive around as fast as possible following a line on the floor). I then passed out soldering irons, computer components and kits of lego parts and told them to get to work. For the record, I had never tried to build such a 'bot myself.

This tactic resulted, as you would expect, in panic. The students complained that they didn't know how to solder, that they didn't know anything about the computer I had given them and that they didn't know how to build robots. I told them that they would have to learn all this and much more and that I would try to help them find out all this information, but they would have to tell me what they wanted to learn. Early on, the questions were about soldering. Over time, the questions became more and more sophisticated. At the beginning of the class, we had a list of lectures that we wanted to give, but we held them back until somebody asked a related question. At that point we would have a vote among the class whether they would like to have a lecture on the subject or would rather continue work on the robots.

By the end of the semester, I was getting complaints from the department because my students were (voluntarily) spending so much time on my class that they were neglecting their other classes. Some were spending 40 hours or more in the computer lab and many had built remarkable contraptions little related to the impending exam. This enthusiasm translated into perfect line-following performance on sample lines.

But ...

What I hadn't told the students was that the final would (for the first time) involve a line that crossed itself. Essentially all of the robots would fail on this line because they hadn't been designed or programmed to deal with that case. The real exam was whether they could deal with the unexpected and redesign and reprogram their robots during the 3 hour exam period to succeed on the harder problem.

All students succeeded.

Moreover two thirds of the class came back a week after the official final to repeat it in front of a friend who flew down to see how the class had worked out. I think, but do not have much data to support the assertion that it is rare for students to ask to be allowed to put off summer vacation so that they can repeat a final exam.

The point of it all was that these students could learn vastly more than was expected of them if they just wanted to. They could learn this material almost effortlessly in a traditional setting where somebody (me) wrote illegibly on the blackboard on difficult and abstract topics. But previous classes with exactly the same lectures given by better teachers than me had failed abysmally. I think the difference was that my students felt that they absolutely needed to learn what was in the lecture ... indeed, they had to ask for the lecture before I would give it. They also felt very much the owners of their task.

In the end, the students felt that they had invented almost everything they needed. In fact, I had spoon-fed fed them almost all of the key material. I told them how to make motors turn, how to make lights turn on, how to assemble and load software and how to design a software project. That didn't matter because they didn't remember that. What they remembered was that they initiated their learning of all of the material. They owned the class. I was their assistant, not their master.

And they learned. A lot.

Tuesday, December 16, 2008

So what about the question of why two negatives multiplied give a positive?

The short answer is that if this were not so, the world would not exist.

That is, the fundamental reason that multiplication works that way is because otherwise the definition would be broken. By broken, I mean that it would lead you to nonsensical conclusions if you followed it as far as you could. Any definition multiplication in a system with negative numbers has to work pretty much that way.

On Tue, Dec 16, 2008 at 6:42 AM, wrote:
Why does a negative number times a negative number equal a positive number?
How can you show that using a manipulative?

Let's take the concrete (but abstract) example of arithmetic on a 7 hour clock. Here is the addition table:

+ 0 1 2 3 4 5 6
0 0 1 2 3 4 5 6
1 1 2 3 4 5 6 0
2 2 3 4 5 6 0 1
3 3 4 5 6 0 1 2
4 4 5 6 0 1 2 3
5 5 6 0 1 2 3 4
6 6 0 1 2 3 4 5

OK. Now we can build multiplication on top of that. I will use * instead of x to indicate multiplication because I don't have really good fonts.

* 0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6
2 0 2 4 6 1 3 5
3 0 3 6 2 5 1 4
4 0 4 1 5 2 6 3
5 0 5 3 1 6 4 2
6 0 6 5 4 3 2 1

OK. Not much to see here except that in the multiplication every row has all of the integers. Moreover, every row has these integers in a different order. This is a consequence of the fact that 7 is a prime number. It doesn't happen that way on the 12 hour clock.

But let's look again at the addition table. Note that 1 + 6 = 0 and that 2 + 5 = 0. Notice also that 0 only appears once in each row. That means that we can consider 6 to be a way of writing -1. Or perhaps -1 is a way to write 6. This lets us solve addition problems such as 3 + x = 5 by adding 4 (which is -3) to both sides.

But what happens when we multiply 3 * (-2)? Do we get -6?

Well, -2 = 5 so this should be the same as 3 * 5 which is 1. But 1 is also -6 !

And all of our properties like the distributive law should still work.

Thus 3 * (-2) + 3 * 4 = 1 + 5 = 6 should be 3 * (-2 + 4) which is 3 * 2 = 6. So that works.

What about (-3) * (-2) = 6?

Well, this should be (and is) the same as 4 * 5 = 6.

That's cool.

In a very limited and geeky dull kind of way.

What is more cools is that this is all just a very concrete example of how the laws or arithmetic imply a certain kind of order. We could test this out for all the different kinds of arithmetic that have the properties that we like about integers. I don't mind starting infinite tasks, but I do mind having to finish them.


Let's do much better by reasoning from those properties directly.

For instance, assume that we have the following:
  1. an additive unit, 0 such that x + 0 = 0 + x = x and thus 0 * x = 0
  2. an additive inverse, -x = 0 - x
  3. left distributive law, (a + b) * x = a * x + b * x
  4. right distributive law, x * (a + b) = x * a + x * b

Now take x * (0 - y). This has to be equal to (x * 0) - (x * y) = 0 - (x * y) = - (x * y)

Or (0 - x) * (0 - y) = 0 - x * (0 - y) = 0 - (0 - x * y) = x * y

This means that ANY kind of arithmetic where multiplication and addition have an additive inverse and left and right distributive laws will follow the pattern that (-x) * (-y) = x * y


if that system doesn't have that, then it won't have one or all of the properties mentioned.

This works for a spinning globe, for quantum mechanics, for clocks and rubik's cubes.

And THAT is why abstraction is cool. But not why it is useful. That comes next.

Mathematics as an article of faith

A friend asked me a wonderful question recently.

She asked:
> I feel like math is so wonderful and useful to a certain point. (Like sixth grade.)
> Then it seems abstract... I can't see the purpose.

She was very much right that mathematics becomes abstract at that point.

And that is very much the point of it. It allows us to think abstractly. It allows us to find patterns in many different things that work the same way. That allows us to think a problem through in mathematically form and then recognize that form again and again.

One good example is modular arithmetic. We could call it clock arithmetic and talk about integral hours that wrap around the 12 hour clock. We could build an addition table that tells us everything there is to know about that kind of arithmetic. We could extend that addition table to be multiplication by repeated multiplication.

That is all well and good. But it only tells us about clock arithmetic on a 12 hour clock. It doesn't tell us as it stands about clock arithmetic on an 11 or 13 or 24 hour clock. It doesn't tell us about arithmetic on a clock where the time isn't just an integer, but can be between the hours. Maybe we don't care about those things because we don't have to tell time on a 17 hour clock very often.

But what about the fact that the original arithmetic that we started with on our 12 hour clock also works just fine for .... music on a western scale.

Should we spend the same amount of time reinventing all of what we learned about the 12 hour clock when it is exactly the same as for musical scales? Or should we re-use that knowledge?

Well, to re-use that knowledge, we have to stop a moment and erase all the places where we originally said "clock" or "hour" or "one day later" and replace them with the integers from 0 to 11 inclusive and replace "move clockwise one hour" with "add one, reduce by removing 12's". This abstracts the original system which can make it harder to learn, but it also makes it much more useful because we don't have to learn it again and again.

But what if somebody asks about pentatonic scales? Or about microtones? Or the 35 note scales that were talked about in the sixties?

Well, if we had started by abstracting away the number of hours on the clock and abstracting away whether the numbers were integers or real numbers, then we would be able to answer those questions instantly.

And if we had abstracted the idea of rotation a bit more, we would see that rotation around a circle can be generalized into rotation around a sphere or even more complex things. Now our clock arithmetic can solve navigational problems and get us home on a dark night.

But with just a bit more extension, that same clock arithmetic (now with 4 hours and 9 dimensions) could solve a Rubik's cube with about 5 minutes of thought.

None of the details here matter. For instance, the fact that 12 is not prime plays a big role in the nature of the arithmetic that we get on clocks, but that isn't the point of all of this.

Now, this abstraction is really frustrating because you can't always point to something and say that it is what you are talking about. In fact, the point of abstraction is that you aren't talking about something specific. This is exactly what makes it hard to teach abstraction using manipulatives. Manipulatives are all about grounding intuition in naive physics. Abstraction is all about UNgrounding your intuitions.

The point is that mathematics is all about abstraction. It is just a mechanism so that we can repeat what once appeared the work of genius without having ourselves to be genii.

So is it useful for most people to understand these wheels within wheels that turn behind the world we see?

To me, yes, it is useful and beautiful and wondrous. But I can't speak for others.

I do think that anybody with a spiritual tendency is turning away from the hand of god if they choose to not see all the kinds of order in the world.