### midterm report

at the foundation of (an earlier version of)

this blog i ranted and rambled about

a tendency on the part of (lower-division

college math) textbooks to hamper the work

of the teacher by (deliberately!) suppressing

correct technical language.

of course things have continued to deteriorate.

but, by some miracle, i’m still earning

the random crust of bread by helping students

learn to *read* these ever-more-horribly flawed

documents. so far so good, then, i suppose.

anyhow, i’m grading linear algebra again

right in here (nothing *but* linear algebra

for something like a *year* now)…

and i’ve only recently become vividly aware

that this “tendency” has penetrated deeply into

the textbooks at this “higher level of the game”.

specifically, i hereby announce that some

satanic force has somehow (even here) replaced

*the sign of set-membership* ()

with its mindbendingly-wrong “plain english”

equivalent(s). the perfectly-correct

(and altogether-necessary) symbolism

(“x is an element of S”)

is now to be replaced, by the edict of

invisible (and mostly unimaginable)

entities, with “x is in S”.

[

this is a good place to skip ahead.

i’m going to geek out slightly here.

you *don’t* have to be an adept to follow.

i’m hoping to make a point that can be

at least *partly* understood by math laity.

is the set of natural numbers (more here…

much my most popular post here and probably

my best-read production of all time) “in” the set

of real numbers? loosely, yes. more precisely,

.

i can easily imagine myself talking to, say, another

teacher about, say, some “property” (like commutativity-

-of-addition; x+y=y+x [for all x & y]) that applies

in the natural numbers. “how do we *know* it applies?”,

i might say. and the answer might come: “because

the naturals are *in* the reals, and the *reals* enjoy

the property of commutativity”. “good answer!” i would

then reply, and move on to whatever i *really* wanted

to talk about.

again. are all possible probabilities *in* the reals?

well, yeah, (duh)! in “code”, one has ;

rephrasing, “all the numbers from zero to one (inclusive)

are *in* the reals” (but also, more precisely,

“the [closed] unit interval is a subset of the set of real numbers”)…

so. now i’m talking to some grad-school dropout (say): “is

the-interval-from-zero-to-one *in* the reals?” she asks;

“heck, yes” say i, and we get on with whatever we’re really doing.

is “pi” *in* the real numbers? sure! .

“pi is an element of the reals”.

but wait! being-a-subset is an *entirely different* relation

from being-an-element! pi is simply *not* “in” the reals

in the same way that is!

who cares? well, me and a few hundred thousand others or so.

if *you* don’t care? well, that’s why i invited you

to skip this part! read on!

]

the biggest problem from a practical standpoint

(if “how can we make this material better understood”

is a practical question) is simply that students

*hate writing* and at *every opportunity* will

replace “plain english” with (typically very

ill-understood) bits-and-pieces.

*nobody*… no student, no lecturer, no pro

mathematician… is going to write out the phrase

“is a real number” a whole lot more than

twenty or thirty times (in a given sitting-down)

without wanting *some* abbreviation for that

phrase.

and likewise for “is a subset of”… indeed,

*any* sufficiently common phrase *begs for

abbreviation* even in “plain english”

so there it is. mathophiles also… in some sense…

“hate writing”. anyhow, we *love abbreviating*.

“algebra is the science of equations”, i once

heard someone say (explicitly repeating something

he had learned “by rote” from a public-school

teacher during his own schooling… it went on

for another few lines but i didn’t learn that

part from listening to this guy say it three

or four times that one night). and i consider

this to’ve been very well said.

so i’ll hope to return to it.

but first. this history of elementary algebra

at w’edia summarizes the standard dogma of its subject (as i

understand it) well. the evolution from “rhetorical” algebra

(describe *everything* in plain-language words) through “syncopated”

algebra (where “shorthand” symbols [many still common today]

began to replace the most common techical terms… but the

actual *reasoning* was still natural-language based [and so,

by contemporary standards, “informal”]) into

“symbolic algebra” (the “science of equations” as we know it

today: a study of “formal” properties of [carefully-defined!]

symbolic “objects” [“variables” and “equations”, for example]).

what w’edia *doesn’t* make much of… but what matters to me

a great deal… is that the emergence of algebra pretty

closely *coincides* in (so-called) *western* history with

the (so-called) renaissance and the (no sneerword necessary)

scientific revolution.

“modern times”, then, *began* when certain humans (*finally*!)

figured out how collections of symbols-on-paper (representing

certain abstractly-defined-objects), produced according to

various “rules”, could be interpreted to reveal previously-obscure

*laws of nature* (so-called). this *was* the “scientific revolution”.

and how does it work? *equations* is how.

so one of the *first* things in understanding

what’s going on the contemporary philosophical

environment is to find out *what an equation is*

(for all literate people): “the equality meaning

of the equals sign”.

when we’re being sloppy, we can confuse “=” with “is”…

but when we actually get to work *using* equations,

we have to *much more precise* to get any value from

the procedure at all. plain-english “is” is *always*

in some sense metaphorical (except in empty utterances

like “it is what it is”)… whereas the equal-sign

rightly-used is as far away from metaphor as we know

how to get.

how does algebra work? (equations and *what* else?)

by *the method of substitution* is how.

“in a context such that A=B is taken as ‘true’,

a properly-written piece of code

including (the symbol) B

*does not change its truth-value*

when (the symbol) A is *substituted* for B.”

this “method” *characterises* algebra.

i first became aware of some its awesome power

in about seventh grade.

and *what* else? “doing the same thing to

both sides of an equation”. and what else?

that’s about it. that’s algebra. the rest

is commentary.

now, for *set theory*, two of the main ideas

are caught up in set-inclusion and set-containment:

and , for example.

and one must be every bit as careful in the use

of these symbols when studying sets as one must be

in the use of the sign-of-equality in studying,

say, polynomial equations (i.e., pretty much,

in algebra).

but about forty percent of the class already don’t

take *equations* seriously. and they’re morally

certain that “sets” are meaningless traps designed

to distract them from “how do i get the *answer*?”.

and the textbook does a great deal to encourage them

to *maintain* this view.

and i don’t like it. please stop.

November 2, 2012 at 8:48 am

I am still wondering whether you would like the David Lay textbook.

I’ll be teaching discrete math next semester. Logic comes up. I’ll start with truth-tellers and liars perhaps, and eventually wend my way, and my class’ [hmm, I’d say class-es, does this notation convey it right?], to the notation that gets us precise. It would be great to have a list of things that are hard to state well without the notation.

Limits might be one. In my calculus course, I wanted students to see the power of calculus before being asked to struggle with the technicalities of the definition of limit. So I am just getting ready to teach limits next week, as part of the 4th unit of the course. The books all start with ‘informal definitions’, and eventually get to a formal definition. We’ve been using our own very informal notions (I won’t say definition here) for 2 1/2 months. I still don’t have examples to show students why we need the precision, but maybe I can find some this weekend.

Good to hear from you.

November 8, 2012 at 1:31 pm

hey, sue v. !

thanks for commenting.

you probably know already that i banged out

some drunken rant-&-ramble reply

a few days back… though, with any luck,

i’ll have deleted it before very many *others*

found it.

i’m kind of curious about the linear-algebra

text in question myself. it’ll have been a

couple hundred hours or more that i’ve spent

*grading* linear algebra by now and i’m developing

an expert’s knowledge about *what goes wrong*.

my theory… and i suppose after all this time

i have some sort of *duty* to present it,

as clearly as i know how, to as many readers

as i can find (without actually going out and

*paying* for ’em… what kind of a blockhead

*am* i at long last?)…

my theory… is something like this.

so-called “higher education” has become

more and more corrupted by the party

of “money, money, money” over the

course of my entire lifetime… and

*particularly* over the course of my

academic lifetime (roughly the second

half… from my mid-twenties till now).

the entities controlling these enterprises

are *hostile* to plain facts stated clearly.

(orwell wasn’t spinning some wild fantasy

back there when he explained so clearly

why parties-in-power *always* develop

some form of “newspeak”.)

so some god-damn fool will *always* be found…

while the empire of lie-about-everything prevails…

to claim loudly and repeatedly that doing things

*clearly* is “confusing” and… for the children!…

“we” should *replace* plain language with the

*baby talk* they’ve been taught to expect

(and never mind whose career gets destroyed;

don’t you know there’s a *war* on?).

here’s the news of the day, oh my fellow

linear-algebra teacher. there’s this

section i’m about to begin marking the papers for.

whose equivalent has been *skipped* hitherto

in all the sections i’ve graded for.

it’s the “linear spaces considered abstractly”

stuff. i *promise* you these kids (grownups

of course, but with gradeschool “math maturity”

in many cases) are *not* ready.

they won’t write out their ideas in plain language

no matter how i beg ’em or penalize ’em;

they’re playing “hide your weakness” when the

actual game should be “show your strength”.

but if they go on pretending to believe i’m wrong

about careful use of “code” and technical terms,

they can go on blaming *me* for not having told ’em

what they need to do to succeed.

and it’s the textbook’s *job* to *help* them blame

me by muddying the waters into thick scummy cloudiness.

at long *last* i’ll have permission to say

“f:D—>R”. as i should have been saying

since remedial-baby-talk one-oh-friggin-one

when they introduced the *names* “domain”

and “range” and then went out of their way

to be sure nobody would ever learn the point.

look! sets of numbers! integers! rationals!

reals! too bad your very *teachers* don’t

know why we’re talking about these damn

things, ain’t it? hey, we don’t need no

stinking well-defined *symbols*… flat-out

wrong “plain english” is *ever* so much

more confusing! you can’t tell an

“answer” from a “solution”? good!

a “solution” from a “solution *set*”?…

hey, that doesn’t *matter*! some expert

is getting all *picky*!

and then our text has “is a real number”

*over* and *over* and *over*

as if *every actual experienced user*

of linear algebra… or *any* advanced

maths!… didn’t spell this .

(for damn good reasons! our medium

is handwriting! [but this must never be

acknowledged publicly lest fewer worthless

god-damn databoxes be sold pointlessly

to the ruination of their so-called users.])

and i beg ’em to at least goddamn it *try*

to get it right but i’m obviously a crank

who doesn’t respect their right to make

the very “equals” sign mean *whatever

they vaguely imagine*.

ah well. the simple fact is, i’m absurdly happy

with this crazy lifestyle. why *shouldn’t* i

spend a few dozen hours every week studying

student work? it’s reading & writing, ain’t it?

and at quite a “high level” at that! there are

*very* few people who could do what i do

(even if they were crazy enough to *want* to).

limits?

i’ve taught considerably less calc than algebra

(& set-theory & suchlike “discrete” topics…).

but i think i’ve had pretty good success

in this particular arena in

helping classes see the forest of

“each-stroke-of-the-pencil-

-has-a-carefully-designed-*meaning*”

through the trees of “*this* part of

*this* picture is ‘getting close’ to

this *other* part” (or what have you…

we begin by using the handwaving

to clarify the code, but have *not*

succeeded until our students can

use the *code* to clarify the *handwaving*).

epsilon-delta is of course another whole can of worms.

(mmmm… yummy *worms*….)

still. the next thing you know, what do they go

and “use” ’em for? so-called “improper integrals”,

is what. where, what else, they make easy

things look hard. sic transit glorious monday.

finding out about derivatives (from greg (“mister”)

peters (in high-school)… in the full-blown

“limit of a difference-quotient” form…

was a thrilling, nay life-altering, experience

for *me* (more or less of course, i suppose).

by now i’ll have failed *many thousand times*

to’ve passed on my excitement about this

gorgeous cluster of ideas. and, with any luck,

succeeded in a small handful of cases.

once more into the breech, dear friends!

getting back to work.

November 8, 2012 at 1:50 pm

I like this: we begin by using the handwaving to clarify the code, but have *not* succeeded until our students can use the *code* to clarify the *handwaving*

I presented limits on Tuesday, and was very happy with how well my students engaged with the proper definition of limits. I should blog about that day in class.

Today I was trying to explain l’hopital’s rule and bungled it. The books always prove the case for 0/0 but not inf/inf. I’m thinking that shouldn’t be much harder. I might be thinking about this over the weekend, so I can get it right for the students next semester.

November 8, 2012 at 1:53 pm

On Tuesday, I used a Cookie Crispness Index to illustrate the epsilon-delta strips used in the definition. It was good fun, and got them to let themselves think about it.

June 27, 2013 at 5:28 pm

Click to access VariablesInMathEd.pdf

sussana epp on variables. spotted at charles wells’

http://www.abstractmath.org/

.