One of the uses of a blog is as a drafting board available for public perusal. The following is my current attempt to sum up what I believe are the most important ideas which the student being introduced to mathematics proper – which is to say, in modern civilization, a high school student – must learn.
I. Prerequisites and Goals
For almost all students, for these things to make sense they must already be able to do arithmetic: certainly to have learned addition, subtraction, multiplication, and division; of whole numbers, fractions, and probably decimals; the methods by heart and the tables as well, preferably and probably by the tested method of rote memorization.
In my experience, the methods for whole numbers and decimals are generally well-learned by this age, but the tables are not, and, whether as an additional failure or because of this, lack of understanding of factoring, common denominators, and fraction work in general is quite common. The fraction methods are reviewed – I sometimes wonder if “taught” would be more applicable – almost as if new in most Algebra 1 texts (9th grade work), so either my standards are unrealistically high or arithmetic standards in general are drastically low.
II. The Three Ideas
A. Equal Change. One explanation I remember from my own days learning simple algebra (and have occasionally used myself) is the idea that the “equation must balance”. If you have two equal things, they must change the same way, or how do you know they are still equal? If two things are unequal, the same operations must be effected, or how do you know if the inequality is still present – or perhaps now reversed? This is the simplest example of this principle to illustrate; if x + 2 is 7, then by removing 2 from both parts we find that the variable x is here representing 5. (On which more later.) But it applies in many, many other places as well. One of the more difficult ideas for some students to grasp, for instance, is that a formula should do “the same thing” to any number it applies to. The most common path to enlightenment seems to be doing it again… and again… and again… “Evaluate x +2 if x = 3.” Ah, we get 5. “What if x = 7?” Er, what do we do with the 7? “Well, what does the formula tell you to do?” Um, add two? It sometimes takes a while for the idea to take.
B. Complete Variability. The counterpart of understanding the things are always the same, or need to be changed the same way, is realizing how much the things that change – that is, in basic algebra, the variables used to represent unknown quantities – can be changed. After the initial hurdle of realizing “x” is “any number we want it to be”, this usually is not a significant problem again until students hit trigonometry or calculus, where suddenly they start running into the mathematician’s favorite trick, “Let’s take everything in the denominator of this ratio and call it phi…” But at that point, it seems to completely stump some students, and most, my guess is, never really do enough work to really understand it. Personally, I would say I finally “got” the idea around my sophomore year of college. “We don’t care what it does by itself any more, let’s take this function and shove it in this other function and see what happens.” In short, the ability to abstract a concept almost completely from whatever reality it might represent. But while crucial, I wouldn’t say even this is the heart of mathematics.
C. Identifying Relationships. Most math courses including a bit on ratios when teaching fractions; and a lot more when dealing with probabilities; and “relations” are mentioned as a precursor to explaining “functions”, but the idea that one thing is specifically related to another thing, by an exact kind of relationship, is, I think, often lacking. Graphing, for instance, can provide a kind of concrete evidence that “this kind of operation ‘looks like’ this”, but it is time-consuming – and often lacking in examples of “this is the kind of thing it represents”. I’m perfectly happy, myself, playing the number games, but there’s an actuality to a fancy graph.
Then again, to go back to what is taught, you never really get away from numerators and denominators as you go on in math, and yet they’re mostly taught as “well this is how you deal with fractions”, and when they show up again in Final Boss form in (say) a rational expression, the student can sometimes be baffled.
This part I’m explaining really badly. In short, easy to understand words, you have to realize that, “It’s the relationship, stupid!” but you also need to understand what that relationship means… before you go on to abstracting it all for fun.
III. A Caveat
I teach at a school which is not “academically focused”. I have bright, serious students for the most part, but few of them have or have had academic ambitions. My guess is that, mathematically, I get not “the best of the best” but “the best of the rest”. I’m also an inexperienced teacher, so it’s difficult for me to say how many of the problems that drive me nuts are really my own fault. And finally, this is all fairly vague – as I said above, a draft – so if it’s a little bit incoherent in places please forgive me. That said:
IV. Suggested Improvements
The biggest, “Wait, what?” moment occurs regularly each year when we hit the part of the book that briefly reviews unit conversions, and it turns out several students have no idea what they’re doing. If I wrote an algebra textbook, it would start with the things. No, you don’t get variables yet. We’re going to convert kilometers to millimeters, and calculate volumes, and change it to feet, and convert that to inches, and then figure out how much it costs at $3 per cubic decimeter. And we’re going to do that for a month and a half. Then, and only then, when you’re thoroughly acclimated to the idea that we change take this number and change it in to whatever other number we want by throwing unchangeable fractions at it; when you’ve carried and canceled and squared and divided not just a few odd numbers to the tens’ place but bunches and bunches of units that end up in the final answer not at all; when you are able, out of sheer desperation if nothing else, to convert miles per gallon to knots per milliliter (okay, maybe that’s a bit far) just because I say so; then, then I give you your first variable. And it won’t be a number. It will be a really simple problem: 36 in = 3 x.
The difficulty, of course, is that I’m not sure how to go on from there keeping the same emphases I’ve just talked about. But observe that this at least would start students on what I’d consider the right track: they deal with equality (36 inches is 1 yard;100 centimeters is 1 meter; 5280 feet is 1760 yards) until it’s not a stranger but an assumption. The numerator and the denominator must be equivalent. Rates? They both have to change. They deal with ratio and fractions and the exact relationships until they start assuming everything is related. And yet, all those fine units and equivalences don’t matter, because they can be whatever we want them to be, and we can change them until they are.
I don’t claim it’s a perfect scheme. As I said, I don’t know how to implement it – I don’t have a plan, just a place to start, and a place that could probably use improvement. I’m also probably not going to do it, until I figure out (in my infinite wisdom) how to use a textbook to teach a subject instead of teaching a textbook on using a subject, or find time to actually write my own (instead of writing a blog).
But if it’s helpful, here’s my idea. And if it’s not helpful, at least it’s on the internet in all it’s fleeting perpetual glory, for you to laugh at and me to shake my head over in twenty years.