Mathematical expressions and equations are normally written on paper with a pencil, or on a blackboard with chalk, or on a whiteboard with markers. There’s a good reason for this: it’s really hard to type math, whether you’re using an old-fashioned typewriter or a modern device like a computer.
There are two distinct problems with typing math: the character set and left-to-right linearity. Consider a fairly typical mathematical expression:
(This was rendered by Microsoft Word’s Equation Editor, but more on that later.) If I wanted to write this by hand, it would look a lot like the expression above, except that my handwriting wouldn’t look so good. But if I wanted to type it into Excel or Facebook or a calculator or WolframAlpha or a programming language or a cell phone, I couldn’t handle the two-level fraction nor could I type the square root symbol. (Actually, WolframAlpha has a way out, but more on that later as well.) In all those cases I would type something like this:
(2 – sqrt(3))/5
It’s no wonder that most students, especially those who struggle with math, find this confusing at best and daunting at worst. They are confused by by the use of the slash instead of the fraction bar, by the use of the function sqrt instead of the familiar square root symbol, and especially by the two extra sets of parentheses. These difficulties are unsurprising: after all, the two expressions really don’t look much alike, do they?
So, is this issue an important one? I keep saying that we’re teaching ideas in math, not symbol manipulation. This issue, at first glance, looks like one that requires a focus on the trees rather than the forest. But it isn’t. The point is that we want students to make a thoughtful but eventually automatic connection between symbols and meaning. We want them to perceive this expression as a quotient, i.e. as something divided by something. The fraction bar serves as a grouping symbol, as do the outer set of parentheses. Those parentheses are not a matter of meaningless punctuation; they embody the very essence of what’s going on. They’re deep. They’re a reflection of big ideas.
Let’s continue…what about the innermost parentheses? Why do we type sqrt(3) when there are no parentheses around the 3 in √3? When we unpack the difficulty here, we find two different issues. One is the notation of square-root as a function rather than an operation. Unfortunately it’s both. Functions are perhaps the central idea of high-school mathematics, so we should emphasize them wherever we can. The standard convention is that the input to a function is enclosed in parentheses — except, for historical reasons, in the cases of sin, cos, tan, log, and a few others — so we type sqrt(3), where the function is sqrt and the input is 3.
Why is this on my mind now? It’s because I just finished grading CSA final exams, and I noticed many occurrences of this issue. Students had to record what they would type into WolframAlpha to achieve a certain result, and what they would type into Codea (a programming language on the iPad) to achieve some other result. The majority would write something like the handwritten expression above, not what they would actually type. If I were an optimistic mind-reader I might be pleased with this observation, imagining that the students were concentrating on meaning rather than form. But I don’t believe it. I’m quite sure that they were concentrating on form without meaning — the wrong form.
Before I return to wrapping up the two cliff-hangers at the top concerning the Equation Editor and WolframAlpha, I want to make a side comment about a related issue in trigonometry: the inverse trig functions. The inverse of the sine function, for example, is sometimes written sin–1 and sometimes written arcsin. I want to make a strong argument for the second choice, despite the popularity of the first one. I have several reasons:
- The practical reason is the one discussed above: the superscript –1 is impossible to type in many contexts, including Excel, Facebook, text messages, and programming languages.
- The pedagogical reason is that it is the world’s worst mathematical notation to have a superscript that looks just like an exponent. Most students — even some very good ones — confuse sin–1 with cosecant, since they know that an exponent of –1 is equivalent to a reciprocal, and they know that cosecant is the reciprocal of sine.
- The name arcsine reminds us that its output can be viewed as an arc. Many students are confused about the fact that a y-coordinate is the output of sine but the input to sin–1.
Finally, why can’t we just get technology to recognize the “normal” way to write mathematical expressions? Well, the fact is a lot of progress has been made in this area, but the change isn’t going to happen tomorrow. Or next year. Let’s look at a few situations where we don’t need to type linearized expressions:
- First is the aforementioned Equation Editor. Type an inline expression into a recent version of Word, and it will appear in the standard format. (But you still have to know how to type it!!!) Or use the Equation Editor add-on, which is really just a simplified version of MathType. (Then you can pick tools from a palette and lay out your work in two dimensions, just like writing.)
- Texas Instruments calculators also will display expressions properly — at least some of the time. Certain symbols, like √, are available on the keyboard. But the extra parentheses are still needed for my example expression.
- WolframAlpha, depending on your platform, offers extended on-screen keyboards with a variety of tools. But the most reasonable way to use it is just to type linear expressions, and if you want to use the new keyboards you still have to understand what you’re doing. (A good thing, of course.)
- Tablets offer significant promise for the future. For instance, the wonderful MyScript Calculator for iPads and similar devices lets you just draw on the screen with your finger the same way you would write on a whiteboard, and it converts what you write into standard mathematical notation — sometimes even accurately. As a bonus, it also gives you the approximate value if you have typed an expression, and it solves for x if you have typed an equation with a single unknown.