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* (\in)
with its mindbendingly-wrong “plain english”
equivalent(s). the perfectly-correct
(and altogether-necessary) symbolism
x \in S (“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,
{\Bbb N} \subset {\Bbb R}.

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 [0, 1] \subset {\Bbb R};
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 \in {\Bbb R}.
“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 \Bbb N 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

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:
\pi \in \Bbb R and {\Bbb N} \subset \Bbb R, 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.


  1. suevanhattum

    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.

  2. 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 \in \Bbb R.
    (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).


    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
    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.

  3. 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.

  4. 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.

  5. somebody else

    Click to access VariablesInMathEd.pdf

    sussana epp on variables. spotted at charles wells’

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