“Is that the kind of thing you just have to memorize?”
That’s a question that should make a math teacher nervous. There are certainly procedures and formulas that are occasionally useful to memorize. But not “just” memorize! If we don’t aim for a deeper understanding, then we have lost sight of why we are doing any of this. In fact, if we never even try to have our students understand some particular piece of mathematics, then that piece is a good candidate to drop from the curriculum. (I’m looking at you, Descartes Rule of Signs).
On the other hand, we do want our students to be able to do a lot of things! And the reasons behind some of them are sometimes more subtle or more advanced than what students need to hear at the moment they are first learning them. So it can be a bit of a balancing act — finding the right degree of explanation and rigor at every level.
One somewhat disheartening thing I see at the high school level is that my students don’t seem to expect to understand the rules. They have been graded on their ability to replicate a startling number of procedures. That these things can be understood seems almost beside the point to them. I wish we taught a whole lot more about a whole lot less…
So what has me started on this rant on a lovely snow day in the new year? Well, while not grading a pile of papers, I came across a twitter discussion of this question:
Why is a negative times a negative a positive?
As you ponder this issue, you see that it is related to:
Why is a positive times a negative a negative?
I would like to offer an answer that requires no algebra — so it can be used with the students who are first learning these rules.
Often in math, we start with an idea that has one interpretation and then we extend it into new areas that cause us to expand or even give up the original interpretation. But we try to extend the idea in ways that continue the rules and patterns that we had before.
So for example, when you first learned to multiply, say 5 × 4 = ?
You can think of that as 5 sets of 4. (Or 4 sets of 5!)
You can make rows of dots:
• • • •
• • • •
• • • •
• • • •
• • • •
But now someone asks you to multiply a positive times a negative: 5 × -4 = ?
I don’t know how to make 5 rows with -4 dots in a row. Or -4 rows with 5 dots in each. So to answer this question, I am going to have to extend what it means to multiply. But I want to keep patterns intact. What kind of patterns? Well, here’s one:
5 × 4 = 20
5 × 3 = 15
5 × 2 = 10
5 × 1 = 5
5 × 0 = 0
Hmm. When you are multiplying 5 by successively smaller whole numbers, your answer decreases by 5 each time. What if we want that pattern to continue?
5 × -1 = -5
5 × -2 = -10
5 × -3 = -15
5 × -4 = -20 Aha! That was the question we started with. We just found a way to multiply a positive by a negative.
[If all you care about is getting the answer, then you memorize the procedure: to multiply a positive by a negative, just multiply the “regular” way and then make your answer negative. But without some understanding, it feels so hollow.]
But wait! There’s more…
Now someone asks you to multiply, say: -4 × -5 = ?
I still don’t know how to draw -5 rows each with -4 dots. But I can look for patterns!
We have learned how to multiply positives and negatives. So for example…
4 × -5 = -20
3 × -5 = -15
2 × -5 = -10
1 × -5 = -5
0 × -5 = 0
Any pattern? Hmmm. When you multiply -5 by successively decreasing whole numbers, the products INCREASE by 5 each time…and if we continue the pattern:
-1 × -5 = +5
-2 × -5 = +10
-3 × -5 = +15
-4 × -5 = +20 Aha (again)! We have our answer — not because we “just” memorized that a negative times a negative is a positive, but because that is the way it has to be for our pattern to be extended.
The idea of developing new rules in a way that maintains old patterns will certainly be useful again. In fact, I just remembered another example of this. I once gave a guest lesson to a 5th grade class that wanted to know why a number raised to the power of zero is 1. I came up with a kind of lengthy explanation involving the laws of exponents…I don’t know how many hearts and minds I won that day. Then I went home and thought about it a little more. I came back the following day with a much shorter lesson:
2^5 = 32
2^4 = 16
2^3 = 8
2^2 = 4
2^1 = 2
What happens to the answer each time you lower the exponent by 1? Keep that pattern going…
And keep it going further, to learn more than you expected!