## Archive for the ‘Math 151’ Category

by quotiant rule

EXHIBIT A

but this is *senior level* stuff…

“analysis” *not* freshman-calc.

and even *there*, one seldom encounters

such an *explicit* version of the

natural-log-equals-reciprocal fallacy

.

(the “quotiEnt rule”…

or rather, *the* quotient rule…

our subject has fallen into the

“leaving out articles makes it

harder to understand and so is

in my best interest” trap…

has *no bearing* on this [mis]-

calculation [we aren’t diff-

erentiating a fraction (or any-

thing else for that matter)].)

here’s the whole sad story.

math is hard and everybody knows it.

what they *don’t* know is that it’s

nonetheless easier than anything else.

*particularly* when one is trying

to do things like “pass math tests”.

there’s always a widespread (and *very*

persistent) belief abroad in math

classes that trying to understand

what the technical terms mean (for

example) is “confusing” and should

be dodged at every opportunity.

we teachers go on (as we must)

pretending that when we say things like

“an *equation*” or “the *product* law”

we believe that our auditors

are thinking of things like

“a string of symbols

representing the assertion

that a thing-on-the-left

*has the same meaning as*

a thing-on-the-right”

or “the rule (in its context)

about *multiplication*”.

but if we ever look at the documents

produced by these auditors in attempting

to carry out the calculations we only

wish we could still believe we have been

*explaining* for all these weeks?

we soon learn that they have been thinking

nothing of the kind.

anyhow. the example at hand.

calculus class is encountered by *most* of its students

as “practicing a bunch of calculating tricks”.

the “big ideas”… algebra-and-geometry, sets,

functions, sequences, limits, and so on…

are imagined as *never to be understood*.

math teachers will perversely insist on

*talking* in this language when demonstrating

the tricks.

well, the “big ideas” that *characterize*

freshman calc are “differentiation” and

“integration”. for example “differentiation”

transforms the expression “x^n” into the

expression “n*x^(n-1)” (we are suppressing

certain details more or less of course).

this “power law” is the one thing you can

count on a former calculus student to have

remembered (they won’t be able to supply

the context, though… those pesky “details”),

in my experience.

anyhow… long story longer… somewhere

along the line, usually pretty early on…

one encounters the weirdly mystifying

*natural logarithm* function. everything

up to this point could be understood as

glorified sets, algebra, and geometry…

and maybe i’ve been able to fake it pretty well…

but *this* thing depends on the “limit” concept

in a crucial way. so to heck with it.

and a *lot* of doing-okay-til-now students

just decide to learn *one thing* about

ell-en-of-ex

(the function [x |—-> ln(x)],

to give it its right name

[anyhow, *one* of its right

“coded” names; “the log”

and “the natural-log function”

serve me best, i suppose,

most of the time]):

“the derivative of ln(x) is 1/x”.

anything else will have to wait.

but here, just as i said i would,

i have given the student too much

credit for careful-use-of-vocabulary:

again and again and again and again,

one will see clear evidence that

whoever filled in some quiz-or-exam

instead “learned” that

“ln(x) is 1/x”.

because, hey look.

how am *i* supposed to know

that “differentiation” means

“take the derivative”?

those words have *no meaning*!

all i know… and all i *want* to know!…

is that somewhere along the line in

every problem, i’ll do one of the tricks

we’ve been practicing. and the only trick

i know…. or ever *want* to know!… about

“ln” is ell-en-is-one-over-ex. so there.

it gets pretty frustrating in calc I

as you can imagine. to see it in analysis II

would drive a less battle-hardened veteran

to despair; in me, to my shame, there’s a

tendency… after the screaming-in-agony

moments… to malicious glee. (o, cursed spite.)

because, hey, look. if we all we *meant* by

“ln(x)” was “1/x” what the devil would we have

made all this other *fuss* about? for, in

your case, grasshopper, several god-damn *years*?

did you think this was never going to *matter*?

in glorified-advanced-*calculus*? flunking fools

like this could get to be a pleasure.

but how would i know. i’m just the grader;

it’s just a two-point homework problem.

and anyway, that’s not really the *exact* thing

i sat down to rant about.

next ish: *more* bad freshman calc from analysis II.

blag.

at long last. this has been sitting

on the paperpile very-nearly-finished

for quite a while.

i started the “lectures without words”

series early on with 0.1: .

whose cover more or less announced

implicitly that it was one of a series

called . and that

was, like, five quarters ago.

and they’re only 8 micro-size pages.

a couple days ago i inked the graphs

and the corresponding code (the stuff

under the dotted line had *been* inked

and the whole rest of the issue was

entirely assembled). and zapped it off.

and yesterday i passed ’em around

at the end of class (to surprisingly few

students given that i’ve got freshly-

-graded exams). it went okay.

i did right away

(if i recall correctly), and in

high-art style, too (i used a

brush instead of a sharpie).

wasn’t much later.

i have plenty of notes for ,

too, and could knock out a version

on any day here at the studio

(given a couple hours and some

peace of mind) that’d fit right in.

anyhow, what we have here are,

first of all, obviously, a couple graphs

and a bunch of code. here, at risk

of verbosity, is some line-by-line

commentary.

in the upper left is

part of the graph of

the linear equation

y=(x+1)/2…

namely the part whose x’s

(x co-ordinates) are between

-1 and 1.

and my students (like all

deserving pre-calculus graduates)

are familiar with *most* of the

notations… and *all* the ideas…

in this first line.

that funky *arrow*, though.

well, i can’t easily put it in here

(my wordpress skills are but weak)

but i’m talking about the one

looking otherwise like

.

and in the actual *zine*, it’s

a Bijection Arrow.

something like ” >—->>”.

as explained (or, OK, “explained”)

*below* the dotted line.

where *three* set-mapping “arrows”

are defined (one in each line;

the in each line

denotes “is equivalent-by-definition to”).

the Injective Arrow >—> denotes that

f:D—->R is “one-to-one” (as such

functions are generally known

in college maths [and also in

the pros for that matter; “injective”

and its relatives aren’t *rare*,

but their plain-language versions

still get used oftener]).

“one-to-one”, defined informally,

means “different x’s always get

different y’s”. coding this up

(“formally”), with D for the “domain”

and R for the “range” (though i

prefer “target” in this context

when i’m actually present to

*explain* myself) means that

when d_1 and d_2 are in D,

and d_1 \not= d_2

(“different x’s”), one has

f(d_1) \not= f(d_2)

(“different y’s”).

likewise the Surjective Arrow —>>

denotes what is ordinarily called

an “onto” function:

every range element

(object in R)

“gets hit by” some domain element.

and of course the Bijective Arrow >—>>

denotes “one-to-one *and* onto”.

there’s quite a bit of symbolism

here that’s *not* familiar to

typical college freshmen.

the Arrows themselves.

“for all”

“there exists”

logical “and”

and the seldom-seen-even-by-me

“such that” symbol that, again,

i’m unable to reproduce here

in type.

still, i hope i’m making a point

worth making by writing out

these “definitions without words”.

anyhow… worth doing or not…

it’s out of the way and i can return

to the main line of exposition:

the mapping from (-1, 1) to (0,1)

in the top line of my photo here

is a “bijection”, meaning that it’s

a “one-to-one and onto” function.

for *finite* sets A and B,

a bijection f: A —-> B

exists if and only if

# A = # B…

another unfamiliar notation

i suppose but readily understood…

A and B have

*the same number of elements*.

we extend this concept to *infinite*

sets… when A and B are *any*

sets admitting a bijection

f:A—>B, we again write

#A = #B

but now say that

A and B have the same **cardinality**

(rather than “number”;

in the general case, careful

users will pronounce #A

as “the cardinality of A”).

i’m *almost* done with the top line.

i think. but there’s one more notation

left to explain (or “explain”).

the “f:D—>R” convention i’ve been

using throughout this discussion

is in woefully scant use in textbooks.

but it *is* standard and (as i guess)

often pretty easily made out even

by beginners when introduced;

one has been *working* with

“functions” having “domains”

and “ranges”, so fixing the notation

in this way should seem pretty natural.

but replacing the “variable function”

symbol “f” by *the actual name

of the function* being defined?

this is *very rare* even in the pros.

alas.

the rest is left as an exercise.

two hours fly by like nothing… working exercises

with one student at a time is generally the best part

of the whole lifetime-math-teacher trip.

there’s quite a bit of homework for 151 students

to work out this week… and then to digitize.

putting the answers into the computer interface

can be expected to present *lots* of problems.

but this level of coding is easy enough for *me*,

so part of how i was able to help with

in a couple of instances was translating.

handwritten work leading to

might be entered as

(-I, -3*2^.5/2)U(-3*2^.5/2,0)U(0,3*2^.5/2)U(3*2^.5,I)

—which is anyway much more trouble to type

than the handwritten version is to write out.

the student bringing this up already had (essentially)

correct code; maybe this is easier than i think.

we’ll see how it goes i guess.

PS

first quiz.

my latest rant mentioned the reason

this was postponed from tuesday to thursday

(or from T to R as i’m likely to use in handwriting):

computer security snafu. so i got a hardcopy on W

and created two slight variants in .

got those run off in the printing office.

and i spread ’em around around the room

with the allotted 15 minutes left to go

(versions alpha and beta to alternate columns).

took the roll for the first time ever and copied

names into “seating chart” order in my notes;

an old habit. everybody handed in with time

to spare; good.

i haven’t graded ’em yet (more or less of course).

but i glanced at a few of the papers and saw

*lots* of right answers including those on

at least one “perfect” paper. looks like it’ll be

“so far, so good” when i’m passing ’em around.

here’s hoping.

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.

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.