## Archive for March, 2011

without wanting to commit myself… here goes.

i’m *undercommitted* this quarter goodness knows.

my calculus blogging from spring ’09

might come in handy (but the Calc III stuff

is mixed in with the Calc I).

for that matter, the common errors page (not by me)

that i cited yesterday is *bound* to come in handy.

heck. learning math on the web?

just like learning anything else,

there’ll be plenty of good info… more than enough

to build a course around…

in the ever-amazing wikipedia. let’s see.

consider this list of calculus topics.

hmmm. it doesn’t refer, specifically and directly,

to the topics we looked at on tuesday…

exponential functions, inverse functions, log functions…

but *does* link to the precalculus page that

*does* treat of these topics specifically (though not directly;

instead it provides links to w’edia pages on each).

continuing in this vein, i’ve just looked (for the first time)

at the exponential function page. hmm.

i imagine myself a beginning student. what do i see?

forest-and-trees issues abounding… there’s an *awful lot*

of material here! but maybe it’s clearer than the textbook

even so. in particular, the article is (very rightly) about

*the* exponential; of course i mean the one with base “e”.

and, right out front, in the first sentence in fact, they’ve got

“the function e^x is its own derivative”.

every calculus *teacher* understands, at least to some extent,

that the importance of the number “e” is very closely tied to

this property… it’s the base that makes exponentiation

“work out conveniently” in “doing calculus”.

most of ’em, if pressed, would probably be able to tell you

that “y = e^x” is the only non-trivial solution

to… the world’s simplest interesting differential equation…

. (the trivial one is y’ = 0).

every calculus *book* obscures this point to some extent.

and there are reasons. one has not *defined* , after all.

*our* text—”stewart”—follows the usual pattern

of “discuss exponentials generally first”

(y = A^x for A a positive

number different from 1) and then singles out the case A=e

as the one having a tangent of slope 1 at its y-intercept.

this can hardly be very motivating for a beginner.

and, anyhow, neither has “tangent” (to a curve at a point)

been defined… so (as far as i can see) *nothing* is gained

in terms of “formal correctness” by focusing on this

particular *detail* of the fact that the exponential

is its own derivative.

okay. there are *better* reasons. textbooks *should*

review exponential-functions-generally (and provide

lots of practice problems). nobody’s going to

understand very much about y=e^x that doesn’t

know anything about its first cousin y=3^x.

still it seems to me that

somewhere pretty close to the moment that

the hugely-important constant “e” is introduced,

it would be helpful to at least some students…

students like i imagine myself to have been,

for example… to have some *succinct*

and *correct* justification (even if its details

can’t be spelled out fully with the concepts

already covered in the prerequisites-so-far).

if i recall correctly, my own experience of

learning-about-e was rather a horrible mishmosh

of formal-correctness and we’ll-learn-about-this-later.

i *did* learn about it later but it was an accident of history;

if i’d merely been a math *major*, it’d’ve been obscure

to me all my life, but since i went on to be a

*graduate student* in maths, i eventually

considered myself duty-bound to make sense

of it all (and had the “mathematical maturity”

to do ahead and do it).

it makes good sense *formally* to consider

the natural-logarithm function (“ln”)… defined

(of course!) as the integral-of-the-reciprocal…

*first*, and define “e” as the solution to

ln(x) =1.

hey, madeline just woke up. more later.

lindley’s paradox at *angrymath*.

eric schechter’s common errors file.

i’ve got a post for spring quarter. yay!

less money again and i suppose less

prestige… i’m running two sections

of a calc class. the students see lectures

(not by me) on mondays and wednesdays

and meet me for guided-problem-solving

(and to hand in homeworks and take quizzes;

exams have yet a different schedule;

i grade everything of course) tuesdays

and thursdays. also there are tutor-room duties.

but one-on-one tutoring is in many ways

the *best part* of the whole “math teacher”

thing. not by coincidence, the furthest away

from the show-me-the-money side:

grant applications and suchlike no-math

adminstrivia. pretending to participate

in curriculum design and suchlike policies:

worse-than-meaningless committee work.

there’s a great deal of online activity

associated with teaching at Big State U.

but almost everything has gone pretty well

in my two quarters here so far (this century).

the payroll-related interface was frustrating

to be sure… but there was an old-fashioned

paperwork over-ride that i eventually found

out about by showing up in person ready

to beg for help.

otherwise… filing grades and scheduling tutoring

duties for example… stuff works the first time

and the right buttons aren’t hidden among

dozens of useless ones with similar names.

new drawing today.

you can check my work…

as of course i already *have*:

start with any “cross”, C, and

verify on five cross-diagrams

(by shading one “circle” on each…

namely the one whose position

corresponds to that of C

in the “big picture”) that

the “Point-at-C” belongs

to each of the “Lines-through-C”.

i’ll edit in some algebra; probably soon.

but the point now is that i *don’t* need

to refer to any but purely *visual*

calculations.

somewhat to my surprise, i’ve decided

in middle age to become much more

of a “visual learner”. in my case, this

amounts to “how can i represent results

from abstract algebra as pictures?”.

it still counts.

according to EDUSolidarity (moving images

on the screen; yick) or this f’book page,

today is “teacher blog action day”. so i’ve signed on…

and should now post a piece on why i’m pro-union.

“why teachers like me support unions.”

this is harder than it sounds.

it’s my old “imagine an appropriate audience” problem.

everybody *knows* (or they think they do)

the reasons union supporters support unions;

everybody knows how they feel about ’em.

why should *my* testimony change anybody’s mind…

or even excite their interest?

in face-to-face interviews, by exercising incredible

amounts of patience, one can… very rarely in my

experience… sometimes discuss *local* political

issues in what seems to me like a meaningful way.

but almost all of what passes for “talking about politics”

is worse than useless.

this is pretty well-known; anyhow, it’s a cliche

that *religion* and politics are taboo subjects

for polite conversation in some contexts.

both for the same reasons: everybody’s already

picked their side and listens to the *other* side

*only* to try to spot weaknesses that will allow

them to vary their “i’m so right” speeches with

some “you’re so wrong” seasoning.

you’ve got to already have made some commitments

to function as a social being… even if only to the

very common “a plague on both houses” stance

i’m illustrating here. for god’s sake, let’s talk

about something where you won’t be tempted

to try to bully me into saying things i don’t mean.

thus.

“i’m pro-union because i don’t like being pushed around”

is probably about as good as i can do. but if someone

should respond with “how is *another layer of bureaucracy*

going to get you pushed around *less*?” i won’t have

an answer that satisfies even *me*… i have to fall back

on “hope is better than despair” or some such

*attitudinal* thing. anyhow, i do unless i’m talking

about a *particular* workplace at some *particular* time.

together, *maybe*, we can do something about

some particular abuse-of-power by management;

one that we feel powerless to do anything about

by individual action. if your ideological commitments

won’t let you go *this* far, you’re much too far gone

for there to be any value in this part of our conversation

at all. even those who *oppose* unions have to admit

that they can sometimes be effective in changing policies

(even if it’s only about *which* squeaky wheels get greased).

when i worked as an organizer, one of the clearest messages

i got from the veteran organizers during “training” was this.

listen sympathetically while the prospect talks about

their problems with their jobs. then, when you’re

about to ask them to sign on, you can come back to

“we’ve got a plan for dealing with [your issue]:

stick together and *make* ’em listen.”

it’s the “all politics is local” phenomenon.

people talking about *their own lives*

don’t *engage* in the same ways as when

they’re talking about the perfect-world fantasies

that make up almost all “religion and politics”

as encountered in day-to-day life… and almost

the *only* way to get through the fog of

“that’s not how i was taught to believe”

is to speak of *very particular* incidents

from our very particular lives.

so… sorry, internet… i can’t *tell* you

“why i support unions”.

here’s a cliche dump for ya just the same.

because my enemy’s enemy is (sometimes) my friend.

because “love your neighbor” is a lovelier law

than “might makes right”.

because individual human beings

are openly treated as *resources*

by money-machines determined

to respect *only* much larger

entities in making their decisions.

(even the courts now admit that

mere human rights can’t be allowed

to trump *corporate* rights in

our money-uber-alles political system.)

because hope is better than despair;

because co-operation is better than violence.

enough. anyhow, there’s plenty of pro-labor

stuff by me out there i think. i did a blog once.

yadda, yadda, though, i guess. in real life

i don’t pay dues to a union and, as these

remarks suggest, i avoid talking about ’em.

mostly because i never get anything *out*

of talking about ’em but frustration.

much of my working life persuades me vividly

that it’s much harder to get people to change

their minds… about *anything*… than we

usually let ourselves think. even when i’m

armed with *certain knowledge*… in algebra,

say, where the opinion i’m trying to convert

the student to is *universal* among

those-informed-about-the-subject…

students will pick up new ideas only

*very slowly* and typically only then

by *getting them wrong* over and over

until finally finding a way to get rid of

whatever conceptual block it was that

led to our discussion in the first place.

if i can’t get people who have actually

*paid good money* to hear what i think

to believe that, say, “clearing fractions

is helpful in solving certain kinds of equations”…

why, then, how am i going to get ’em to

believe me if i said “your god is an idol”?

never by acting alone, is how. persuasion?

over-rated. come over to our side.

treating people as ends-in-themselves

is more *fun* than treating them as cannon fodder.

math ed issue of the *notices*.

also: ben’s back (at research in practice).

alison wolf in her own words, at micromath (borovik).

not a projective space though

(there are “parallel lines”, for example).

ten “lines” through ten “points”.

you can get this by using

2-element subsets of {0,1,2,3,4} as points.

the lines are then triples

(like {0,1}–{1,2}–{0,2})

such that each point of the triple

is the symmetric difference

()

of the other two.

the point associated to {0,1}–{1,2}–{0,2}

in the points-to-lines correspondence

i’ve illustrated here is then {3,4}…

the complement of the union of

the three points of the line.

so you can re-create this at will.

you just have to fiddle out a

nice symmetric version of the picture.

anyhow. pick a big circle.

look at the *three* big circles

matching the dark dots inside.

the three “lines” meet at the

original point.

Somebody’s preparing a lecture or something and mentions the angle-sum laws. There are two of ’em sharing the office with me doing the same class (different from mine but one I’ve taught many times). Says he probably won’t get to it tonight.

Me:

“I could never remember the doggone things until I found out about the “cis” function… e-to-the-i-theta equals cos-theta plus i-sine-theta. I always make it a point to show it to the students this way too… if you can just take this bit on faith, you can remember what’s what forever. After a while it even begins to make sense… cosines are x-coördinates (so we should think of “real parts”) and sines are y’s (and so, “imaginary parts”)… then things multiply out accordingly.”

And the *other* guy seemed to know pretty well what I was talking about. But the guy who brought it up, not. But he’s in too much of a hurry to sit still for any actual explanation, alas (not that I blame him). So I gave up and prepared my *own* stuff or something and did my lecture and all that. But had it on my mind.

So, first, here’s the part I’ve known since, I forget, early grad school days.

Prerequisite: A little trig and a little faith. (Actually, you could take it *all* on faith and still have the best way to memorize the angle-sum laws… but of course there’s no *point* in memorizing the angle-sum laws if you’re not already studying trig.) On faith, you should believe that there’s a Complex Number Field “containing” the Reals and a number, *i*, satisfying *i*^2 = -1. This isn’t much of a leap… anyhow, students will have heard rumors about this situation and may even have done a few exercises.

The *leap* of faith comes with the proclamation

This is admittedly something of a whopper. To *prove* this admittedly-weird equation one requires some Analysis (that’s “Calculus” to you, I suppose). The only proof I’ve worked out in detail involves so-called “infinite series”… oh, what the hell.

[

Shove in i-times-theta for “t” and turn the crank; equate real and imaginary parts.

]

The function

is sometimes called “cis()”, by the way. So the theorem I’ve outlined—the one I ask my students to “take on faith” if they want to learn my favorite trig-mnemonic—can be stated as = cis(). OK. Everybody’s willing to suspend disbelief thus far, right? Because, look. The “angle sum” law for the cis function goes like this.

Equate real and imaginary parts for the results. To wit:

and

the “angle sum” laws.

Amazing isn’t it.

(Summary: when you know “cis” and how to work with exponents [“*add* exponents to *multiply* exponentials-with-matching-bases]… and, oh yeah, how to use *i*^2 = -1 [to multiply “complex numbers”]… you get the [matchings and {s, i, g, n}-signs for] the trig functions “for free”. And you really *oughta* know those things: this isn’t all they’re good for [by a long shot]. )

Now, that’s essentially the lecture I’ve given many times: the part I’ve known for years. But I’ve dusted off some ODE books lately and think I have a better idea than I ever did about what’s “really” going on.

Bump up the prerequisites. To get much of anything out of what follows, the reader should know a little Calculus… derivatives for exponentials and trig-functions… and be willing to believe an existence-and-uniqueness theorem from ODE (“ordinary differential equations”). I plan to get *around* the “infinite series” argument… or maybe just to *hide* it…

Let’s imagine that we *know* about e^x for Real x and want to investigate e^z for Complex z. Write z = x + iy.

Now e^(x+iy) = e^x * e^(iy), of course. And we already understand e^x by assumption. So we can focus in on the function

f(t) = e^(it).

Differentiate f twice: f”(t) = -e^(it).

So f is a solution to the ODE f” + f = 0.

But one already knows (or readily checks) that sin(t) and cos(t) are *also* solutions to the same ODE. Moreover (here’s the “high theory”), since ours is a *second-order* ODE, these *two* (“linearly independent”) functions “span the solution space”: *every* solution can be written as a sum-of-scalar-multiples of these two.

Assuming this is true. There exist constants K_1 and K_2 satisfying

Putting t = 0 shows that K_1 = 1 (recall that cos(0)=1 and sin(0)=0); it then follows that (by a similar calculation).

Now consider . Our sum-of-scalar-multiples equation gives us …

but we can also calculate (by our previous paragraph). So K_2 is a square-root-of-minus-one: .

OK. It’s late. Getting rid of the \pm (plus-or-minus) is an exercise. The point, if I still have a reader, is clear, I hope: assuming “good behavior” for Complex solutions to (second-order linear) Differential Equations, we’ve *bypassed* the series (which in some sense are just a complicated expression of the facts that (e^x)’ – (e^x) = 0, (sin(x))” + sin(x) = 0, and (cos(x))” + cos(x) = 0 [along with, now that I think of it, the values of e^0, cos(0), and sin(0)])… to obtain the e-to-the-i-theta-equals-cis theorem.

If I’d’ve known you could actually *do* stuff with ODE, maybe I’d’ve learned about it when I was a sophomore. Or the one time I taught it…