### It’s Not The Cheat, It’s The Futility

The transformations section of the text begins, to my predictable chagrin, with a graphing-calculator “Exploration”: adding (or subtracting) a constant value at the end of a function (in the Function Editor [i.e., the “Y=” screen] of the grapher) produces the by-now familiar (to any student in regular attendance) vertical shift. “We are led to the following conclusion:

If a real number k is added to the right side of a function y = f(x), the graph of the new function y = f(x) + k is the graph of f shifted vertically up k units (if $k > 0$) or down |k| units (if $k < 0$)."
And this is about as clear as can be expected: this stuff is hard. But, doggone it, we’re adding to the right side of an equation, aren’t we. And shouldn’t we take a hint from the doggone grapher and refer to $y_1$ and $y_2$ (the authors go on, in effect, to call $y_1$ by the name f—but what’s the name for the new graph?)?

As for the cases on the sign of k (and the use of absolute value), well, it’s not the way I’d handle it but it’s not obviously worse. I usually say something to the effect that the graph of [the equation] y = f(x) + h is obtained by shifting the graph of y = f(x) “up” by k units with the understanding that “shifting up by a negative number” is interpreted—in a very familiar way—as shifting down. It can be helpful to mention, say “y = Mx + B” here—the point being that we don’t need to know the sign of B for the formula to be valid. (Or the words: for the case of y = f(x-h) we add h to each x co-ordinate of the graph—regardless of the sign of h.)

Again, it’s not obvious that treating the “up” and “down” cases explicitly right there in the display that (effectively) defines “vertical shifting” was a bad idea … but I believe it was a bad idea (sort of): anyway, by the time a student reaches the Conic Sections stuff (next quarter) some of the “formula” displays have become much more complicated than they have any good reason to be (and goodness knows, Conics are hard enough already). The sign issues have to be discussed somewhere; using inequalites and absolute values to do it is probably very much the right idea … what I’m after here is that the definitions be definitions. Which would mean, first of all, identifying ’em as such; and then, as concise as you can be while getting the job done.

It would appear that somebody made an editorial decision not to discuss the kind of “understanding”s I mentioned a moment ago; the results are disastrous. This is particularly true for students inclined to the so-called “rote memory” strategy (learn the formulas by heart before even beginning to work at “seeing the big picture”). Of course, most of us probably consider it something of a duty to try to talk such students out of relying heavily on this strategy … but this hardly seems like the right way to do it. And yes, there sure does seem to be quite a bit of resistance (anyway at the level just below 148) to the idea that, say, “dividing by a number is just multiplying by its reciprocal”—this comes across as “mumblemumble” to many a 102 student (for example).

I call it “teacher talk”: the student somehow knows for sure that this technical stuff you’re always so careful about saying cannot possibly have anything to do with their existing ideas about how to solve equations; they’ll move heaven and earth to try to find some list of “rules” they can memorize if only they can avoid using the word “reciprocal” (or, of course, what’s worse, “multiplicative inverse”). And if you’d only stop trying to pretend any of this means anything and just tell them what to do, the scales would probably fall from their eyes in a minute flat. Of course it’s damn-right a duty to try to disabuse these poor souls of their misapprehensions as to the nature of our art. And this duty doesn’t fall on me alone.

Overlooking the power of the symbols until we end up with different formulas for, say, ellipses with vertical and horizontal major axes feels to me like pandering to the worst instincts of our weakest students and, I keep having to apologize (“Well, if they’d done this right, it’d be much easier … well, look: forget it. Here’s what they want you to do …”).

So. Let’s try and make this very difficult task as easy as it can (reasonably) be. Thinking of, say “stretching” and “compressing” graphs as two aspects of the same process (requiring only a single formula), is just a flat out good idea: the notation is beginning to do some of our thinking for us. It’s not quite the Heart of the Matter—an awful lot of people have learned about Transformations and Conics (for example) with the kind of overly-detailed treatment of cases I’m complaining of here, after all. But for me it’s pretty close. And, believe it or not, I don’t necessarily want to do constant battle with textbooks.

1. If they’re going to do like they’re doing in the definition, they shouldn’t conflate “up” and “down” in a way that forces them to say abs(k) units. If someone is trying to split things down to make it easier, they shouldn’t go half way.

Just calling it “vertical shift” without worry about the sign is likely better, but I’ve seen enough classroom confusion over f(x-h) I don’t mind listing out the four cases while they’re getting used to it. But certainly they should eventually have

f(x-h) + k

all as one thing, with k defined vertical and h horizontal and no mention of sign at all, right?

re: ellipses. I teach only one formula. Focus formula is now c^2=abs(a^2-b^2) with the extra abs().

If I did eccentricity I would have to tweak to make sure they used the major axis, but that’s still easy.

Don’t even get me started on hyperbolas. I think the need to avoid starting formulas with a negative is a holdover from the … 15th century? … before negative numbers were considered acceptable so expressions had to be written with “minus” instead?

2. This deep desire to simplify things makes a mess yet again.

Better to say it well and clearly, and then to provide clarifying examples and illustrations, as necessary.

Jonathan

3. i consider it a *good idea* to avoid
starting a formula with a negative …
to the extent that sometimes i’ll prefer
“5 – 6x – x^2” to “-x^2 – 6x +5” …
but for an extramathematical reason
(namely, it’s easier to lose track of

on a related issue, texts routinely claim that
Ax + By = C
is Standard Form for linear equations in two variables
(which is okay by me if that’s the way they want it);
quite often they’ll actually *want* A >= 0
and integer coefficients with no common factor
(which is *only* okay if it’s made explicit …
as it seldom is …).

of course they don’t dare *insist*
on integer coefficients since in principle
this isn’t always possible … one is led
to wonder why they made such a big fuss
in the beginning about “sets of numbers”
($\Bbb Z, \Bbb Q, \Bbb R, \Bbb C$) when they plan
on *ignoring* the distinctions whenever
the opportunity to *use* them arises.

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Vlorbik On Math Ed ('07—'09)
(a good place to start!)