by Dave Maier
Math is pretty easy when you’re just starting out. You’re just adding and subtracting and multiplying and dividing. They might even let you use calculators, but even if they don’t, you’re just dealing with whole numbers, the kind you use when you’re counting on your fingers. (Sometimes they spring some newfangled versions of the multiplication algorithm on you, but it’s still just multiplication.)
Some students first run into trouble when they get to fractions, usually in sixth grade or so. Now we are writing the same number in rather different ways (1/2 = 2/4 = 0.5, and so on), and we can’t really count on our fingers either. All of a sudden there are a whole bunch of numbers between 2 and 3. In fact, as it turns out, there are an infinite number of such numbers. Infinity was okay when it was the biggest number of all, all the way on the end (or ends) of the number line and thus safely out of the way, but now we’re using it to count things, and those things are themselves not only the things we count with, but the numbers between what we seem now to be calling the “counting” numbers. (It even turns out – although they don’t make a big deal of this in sixth grade, thank goodness – that there are more numbers between 2 and 3 than there are “counting” numbers on the whole number line, even though both numbers are infinite. Yikes!)
Again, though, in arithmetic at least we’re just talking about numbers. Every problem has a single right answer, even if we now get to write that answer in different ways. But then, all of a sudden, straight up ahead: algebra.