April 13, 2020:
Introducing the Harry Potter School of Mathematics.
Intrigued by the idea of a Hogwarts Academy for algebraic wizardry? Unfortunately, not only is this a bad idea, it’s an all too common feature of current mathematical education. Math requires an interesting combination of high level abstract reasoning skills linked with basic rote learning factual skills. In some ways, this is a lot like literature.
Literature? My colleagues in the arts might jump at a chance to contradict me on that, but it’s not so far fetched. Mathematics requires a good grounding in basic arithmetic skills just like the study of literature requires a good basic vocabulary. You also need to understand rules and processes: of math operations in mathematics, the rules and processes of grammar, syntax, and so on in literature. In both subjects you need skills of abstract reasoning to recognize larger patterns. It’s those patterns and how they are applied that are key to understanding algebra and English literature.
That said, some people are better at mathematical reasoning than they are at literary reasoning and vice versa. You tend to like what you are good at and work harder at it. On the other hand, neither mathematics nor literature is so hard that any educated person should be excused from not having a basic understanding of both. Yes, work will be involved, but any worthwhile endeavor—including education—requires a solid foundation of a variety of skills.
In mathematics, the key is to understand what you are doing with mathematical manipulations. One of my favorite ways to do this, mainly because it was aggravating as well as useful, was to ask students, “Why?”
For example, I would write an algebra equation to solve on the board. Say: x + 4 = 12.
I would ask, “How do we solve this equation for x?
Obnoxious student in the front row: “Ooo! Oh! I know! Subtract four.”
Me (blinking in confusion): “Where?”
Student (sighing in frustration): “From both sides of the equation.”
Me: “Sure.” (pause to do it)
Me again: “Okay, good. Why?”
Student (funny look.) “What?”
Me: “Why did you subtract four from both sides?”
My idea was not to give students a hard time (that was a side benefit). I was trying to reinforce a couple of things. First, when solving an equation, we are trying to find a mystery number involved in a given computation. The key point is that the equation to be solved is a true statement. Since it’s true, I like to think of the equation as balanced on a teeter-totter with the balance point at the equals sign. To solve the true equation, we must get the x by itself on one side to see what it balances with on the other. That will be the answer. Keep it balanced, and you keep it true.
There are a couple of poor ways to learn how to solve an equation like this. One is to see it as an addition riddle: What added to four makes twelve? A quick review of addition facts yields the answer. But what happens if the numbers get a little bigger? Say: x + 59 = 96.
Another flawed approach, and one I’ve seen all too often, is to memorize the fact that if something is added to the unknown, in this case the four, you always subtract it—from both sides. This, dear reader, is a prime example of math taught by the Harry Potter School of Magic method. It appears to be a rule that, if memorized, will solve algebra problems—at least ones involving addition.
What about subtraction: x – 15 = 22? H.P. student: “Add fifteen to both sides.” In short order we have new rules for solving subtracts (add), multiplies (divide), and divides (multiply). Four rules? That’s not too bad. However, this idea is a seductive path to being poor at mathematics.
This is the wrong way to learn math because this method looks at math as a series of “magic tricks” that have to be memorized—hence the Harry Potter reference. Very quickly there are new rules for when the operations are combined, when there is more than one group of variables, and so on, and so on, and so on . . . Students trying to learn math this way are quickly overwhelmed with rules, regulations, and requirements.
The right way to learn algebra involves the idea of inverse operations: subtraction is the opposite of addition. In other words, subtraction undoes an addition operation. In that equation: x + 4 = 12, to get the x by itself, to undo the + 4, subtract 4. So far, there’s not a lot of difference between that and the Harry Potter method—except a little more complicated language.
Here’s where it gets good. You’ve probably noticed operations come in pairs: Addition—Subtraction, Multiplication—Division, and so on. They pair up because the operations are inverses: one does, the other undoes.
Addition undoes subtraction.
Subtraction undoes addition.
Notice the symmetry? Well, that basic idea can be extended:
Multiplication is the inverse of division.
Powers are the inverse of roots.
Logarithms are the inverse of exponentials.
Arcsine is the inverse of sine.
Integration is the inverse of differentiation.
Yes, I started out talking about algebra, but this idea can take you all the way through Calculus and beyond. But that one idea remains the same: Solve an equation for x by undoing whatever has been done to the poor thing.
Now there is a lot more to learn about the operations I’ve listed, and there are many rules, regulations, codicils, subparagraphs, and footnotes that have to be considered. There are other ideas along the way that help the solution process. However, I think if you look at this bigger picture in mathematics, the basic idea is straight forward. Details are important, but you have to get your head up out of the weeds so you can see where you are going.
The big problem with mathematics education is that so much of it is presented as they would at the Harry Potter School of Mathematics. With just a properly memorized swish and flick, we solve?
No, no, no. Stop looking at the weeds. Ask WHY. It makes mathematics so much easier.