Buy Conditionally

May 7, 2009 — vlorbik

The 151 quizzes are good. I caught almost everyone with the trusty old “differentiate a constant” wheeze… $y = 11e^\pi \Rightarrow {{dy}\over{dx}}= 0$ of course, but you wouldn’t’ve known it by these guys on the day. In fact—math teacher at play— $\{11, \pi 11e^{\pi-1}, 11e^\pi, 11\root7\of{e^{22}}, e^\pi, 0\}$ is the “answer set” (sure as heck not the solution set!)—with $11e^\pi$ appearing several times (once as ${{11}\over{e^{-\pi}}}$ ) and a variant ( $11\pi e^{(\pi - 1)}$ … slightly better) on one other value.

Now. If I’m going to go in claiming that anybody “should have known” something… then how come they should’ve? Let’s see. 11e^x, 13e^\pi, cos(x), 1/x^2, polynomial, radical. How come two of ’em are alike? What’s different about ’em? This is pre-math stuff here: mere “test-taking skills”. Okay. What else?

Am I gonna say everybody oughta be trying to imagine a graph? Like, every single time? Well, maybe it’d be a good idea… at least right in here at first… to at least consider it. The trig function everybody darn well oughta know is some wavy thing; the polynomial’s a cubic so again one has a pretty good idea. The radical and the rational function are a little more esoteric and I wouldn’t blame a Calc I student a bit for not thinking too hard about it if nothing pops up for free in instant recall; if it comes to thinking about it, maybe it’d be a good time to break out the grapher. As for the problems with e‘s in ’em, I do feel that everybody oughta have a mental image of “exponential growth” ( $y = A_0 e^{Kx}$ , say [with A_0 and K positive constants]): it sure is a simple doggone thing to sketch…

So then. With any luck, one then asks oneself, “what’s the graph of e-to-the-pi then, eh?” and arrives somehow at the necessary insight. In fact, if one realizes only that one does not yet know the answer, one might, again, think of using the graphing feature of the calculator (or, of course, think of estimating e^pi directly on the command line [“homescreen”… whatever]). Once you see that flat line… on screen or paper or in imagination… you’ve got it.

OK, true enough. Is that all you got? Well, I suppose a drill-instructor style teacher might favor repeating over and over until blue in the face “pi and e are constants“. Might be worth a try…

Am I supposed to be, like, hanging on every bloody word then? Well, yes, ideally, but hanging on every bloody symbol would be a darn good place to start. One is always led to ask, “what does this symbol mean in this context?”—indeed, this sounds suspiciously like a description of reading itself (and not just of reading “math”—calculations and such).

One is curious at this point as to whether they’d’ve done any better with y= e^3… but enough.

Because the good news is that there were five otherwise-perfect papers and several other solid high scores; in particular, the class redeemed itself very nicely with the “epsilon-delta” proof. Nonetheless, here are some trouble spots.

$\bullet$ No “Let $\epsilon \rangle 0$ “. For this section, this line is mandatory. (One may of course “fix” epsilon or some other very slight variant.) I’m declaring by fiat that “all epsilon-delta proofs begin by making epsilon the name of a constant“.

$\bullet$ Equal sign for implication.

$\bullet$ No mention of $\delta$ .

$\bullet$ “ $\delta = {\epsilon\over{|slope|}}$ “—right there in the calculation as if english words were appropriate in the middle of algebraic expressions. In the actual exercise, the grader is hoping to see $\epsilon\over5$ . But hold on. I put the “mixed media” (algebra-and-plain-english) on the very blackboard myself! Ah, but it was in a marginal note about problems of this type generically… in particular problems (of this type—limits of linear functions) one will work with honest algebraic code. Should students be expected to just recognize slangy shorthand as such in its context while we’re preaching meticulous attention to detail in some other part of the work? Maybe. We can’t be explicit about everything.

$\bullet$ The very inequalities I most want to see… in reverse order. Or in variant orders harder to classify… “no apparent order at all” having made at least one appearance.

$\bullet$ Inequalities replaced by equations; reversed inequalities.

The thing is… at least some of the lack of clarity here ought to be considered my fault. In particular, of the calculations after fixing epsilon and writing delta as a function of epsilon… the calculations that we should be careful to present in the reverse order to the one we actually discover it in… I now feel one should explicitly say “(Then) The Following Are Equivalent” (and I’ll introduce the abbreviation TFAE for this situation in the next class). If we’re going to obsess over this definition… and we should… then it’s the logical structure that seems to present the biggest challenge; the equivalence of a certain set of inequalities is pretty close to the heart of the matter; we should be trying to spell this out as clearly as we know how.

I’ll probably have more to say about this…

Posted in 151, Rambling. 7 Comments »

jd2718 Says:
May 9, 2009 at 4:57 pm

I was part of as good a high school class as you could imagine teaching, and we got nailed (all but one of us) by e^271 or something like that. Can’t fix that without deducting points (if it happens again, that’s another story)

And on the details, I’m not so sure what’s bad about the word in the middle of the algebra, except that we don’t like it. Do we have a good reason not to like it?

Jonathan

vlorbik Says:
May 9, 2009 at 6:00 pm

i think we want to imagine algebraic “code”
as something that in principle can be “understood”
even by a machine… so everything is *imagined*
(the reality is far different) as having a definition
lying around nearby that can be looked up when
some part of the code isn’t understood.
“slope” is of course a perfectly good variable name,
better than “M” or “f'(x)” in some contexts…
so we sure as heck don’t have a reason
to *abhor* it… but a variable name in algebraic code
with lots of x’s and y’s and whatnot oughtn’t to be
five letters long *without a darn good reason*…
and anyway the *symbol* “slope” (pronounced,
let’s say, “s,l,o,p,e”) *hasn’t* been defined by us
(as code) in *this* context.

i guess i’m saying in part that the distinction
between the *meta*-language and the “real code”
*is* important… and that it’s *convenient*
to use “plain english words” as sort of a *marker*
for “not actual code”. obviously, in computer code
(for example) one sees words-as-variables
quite commonly (also in other fields even further
from maths). in maths… calculations are hard.
if we’re actually going to *use* the variables
in by-hand calculations, we’ll want ’em short.

i’ll admit to a sort of bigotry *against*
mixing in words with algebraic symbols:
even sin(x) and cos(x)… but particularly
stuff like Aut(G) (for the set of “automorphisms”
of a group)… used to sort of annoy me.
somewhere along the line i realized that,
lacking any better idea, maybe i’d learn
to love it the way it is. and maybe i have.

but yeah. learning to read code is hard.
“keep it simple”. i can’t teach ’em to write
good english… nobody can. by narrowing
our focus to almost nothing… a few lousy
little algebraic symbols and some logic…
we can be so precise about *one little thing*
as to know for sure when something’s
exactly right. it’s just bound to be a good idea,
in this famously make-it-or-break-it moment,
to lead ’em in the direction of the actual practices
of existing texts.

if $f(x) = Mx + B$ is a linear function,
we can compute the limit of f at a
by letting $\epsilon > 0$ and putting $\delta = {\epsilon\over{|M|}}$ .

now, *that*’s how you mix english with your code.
and then, *out loud* you say
“divide epsilon by the absolute value of the slope”.

i make no effort to write at the blackboard in this style
usually. but i darn well have it in mind that ideally
a student will learn from me how to *approach* writing
in this style (if they’re that one-in-a-thousand
that actually *wants* to).

vlorbik Says:
May 9, 2009 at 6:12 pm

oh, p.s.

like i said in the post…
it was “5” i wanted
in the actual event
(not “M” or “slope”…).

jd2718 Says:
May 10, 2009 at 2:22 am

I’ll take it as a transition, a good place to move further from…

But then again, I spend time teaching kids, explicitly, to translate from phrases into algebraic expressions. I expect them occasionally to leave phrases in their work.

Denise Says:
May 11, 2009 at 3:38 pm

It’s probably because I mostly teach middle school kids, but I like the word-as-a-variable approach. It helps my students to think about “what does this symbol [or this equation] mean in this context?” Too many letters and symbols crowded together tends to produce that deer-in-the-headlights reaction. But then, your students are not supposed to be beginners like mine.

I like your “answer set.” I probably did some of those in my college days, too—although it’s been so long, I don’t remember any specifics.

vlorbik Says:
May 11, 2009 at 4:02 pm

thanks for the feedback.
choosing an appropriate level of formality
for a given context is looking to me like
one of the most important things for instructors
to *do*. ideally this would be done
at higher levels (textbook committees
and suchlike course-designing entities)
but in my experience these can’t be trusted.

if students *only* realize that *whatever* we do,
it’s never the final, one-best-way, for-all-time
ONLY way to do it… we’ll have done at least
*some* service… if they realize that careful choices
of notations can yield spectactular payoffs
when calculations become necessary,
we’ve got a big win.

Math At Play Blog Carnival #7 - Onomatopoeia Says:
May 15, 2009 at 10:02 am

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