more than 4K characters for k. nowak

(i couldn’t put this in a comment
for this “log laws” post
at f(t); too long.)

it’s the old principles-versus-procedures problem.
students hate general principles.

i’ve had tutees pay me hundreds
to supervise their homework
who very clearly “tune out”
whenever i try to explain
what’s going in in a general way.

it’s worst when it’s about a topic
they’ve been exposed to before…
the famous “math phobia”. panic
sets in when you realize that
you’re going to expose your ignorance
yet again… about fractions, variables, or
vocabulary (“equations” and “expressions”,
for example)… to name some common
weaknesses for *beginning* algebra.

the tendency is to believe that whatever
this “general principle” thing is that
teachers keep wanting to come back to
is secret-math-head *code* for something
that, if we’d only present it in plain english,
then they’d know how to work the problems
on the test (which had darn well better
be *just like the ones we practiced*).

so we give in and show ’em the “procedures”:
FOIL instead of “repeated distributive law”,
for example… and they *pass* the tests
but get *further behind* in the grand scheme.

“i’ve done OK this far by disregarding
all the math-brains-only *theory*…
and now i’m failing. but i’ll *never*
be a math-brain… good heavens,
i’d have to go back and understand
an *awful lot* of stuff i’ve been
given a pass on all these years!…
it’s the pedagogy! it’s my learning
style! it’s my teacher! math sucks!”

the phenomenon persists at all levels
(that i’ve passed through in learning
math up to my published thesis…
when, sure enough, i still felt myself
an imposter compared to the “real
mathematicians” i’d been working
with in grad school).

“logs” are a particularly interesting case:
otherwise-well-prepared calculus students
are often very weak on logs. hell, every
tenth student has decided they’ll just
go ahead and pretend “ln(x)” is to be
replaced with “1/x” somewhere along
the line no matter how many times
it’s marked wrong.

when “should” students learn about
the abstract, proof-heavy style
of math-rightly-so-called?
the typical student will never
*feel* ready for *any* new idea.
(this should be considered in
any discussion of “developmental
barriers”).

do-the-same-thing-on-both-sides.
logs-are-inverse-exponentials.

here’s a common-log equation:

log(4) + log(25) = log(x).

and here’s its solution.

10^[log(4) + log(25)] = 10^[log(x)]
10^[log(4)] * 10^[log(25)] = x
4*25 = x
x=100.

now, kate’s worksheets are great
and i’m not suggesting that anyone
make my approach into their
exhibit A for day-two-of-logs.
so don’t get me wrong.

the point i set out to make when i started…
if i can remember that far back… is that
*no* approach can overcome the simple
fact that our courses are designed to
go *much too quickly* for median-level
students. the good news is that we
can spot talent pretty easily this way:
anybody who can keep up typically
hasn’t “hit the wall” where math
gets *hard*; give ’em an A and
pretend we’ve done something
to be proud of.

my “discrete mathematics” students
last quarter were all calc 2 vets:
future programmers and engineers
and whatnot. and *almost* all of
’em learned quite a bit about
writing proofs. the *easy* thing
here is (of course!) the logic.
(i mean the symbol manipulation;
what all the p’s and q’s have to do
with the “logical structure” of
a given proof is much more obscure
than it appears to non-teachers.)

if *anybody* i’ve ever worked with was
developmentally ready to take on
“show using the definition of logs
that log(a) + log(b) = log(ab)”
as an exercise, it’ll’ve been these
talented hardworking students.

but the resistance was palpable and i folded.
there was… as always… much too much
material (i also had to skip stuff like
“manipulations with sigmas” and
“complete induction”… the class
emerged ready to take on maybe
a third of the exercises in the sections
covered [mostly the easiest ones]).

there *is* no developmental appropriateness
in “real world problem solving”: one needs
a technique that *works* wherever it’s found.

our task as math-teachers-in-schools,
though, is pretty much “preparing students
for yet another class”… and one technique
that “works”, alas, is *hide your weaknesses”.

that idea of “apply some inverse-thingy
on both sides of the equation”? one of
a handful of big ideas in algebra
(along with “variables”, “graphs”,
and “roots of polynomials” and…
well, not much else…)

“look: to undo an *addition*, we do
a *subtraction*… to undo a multiplication,
we do a *division*… and these ideas
were *hard* when we tackled ’em
back in some earlier course.
a little later, we decided that
we wanted to ‘undo’ squares,
and introduced ‘square roots’;
more generally (and about half
the beginning-algebra folks
get lost around the bend in
here somewhere) we’ll need
to undo powers-of-the-variable
like x^K, and introduce Kth roots.
well, then, doggone it, why shouldn’t
the same undo-the-operation
strategy work for K^x?”

i’ve ranted out this mini-lecture
not just to get it out of my system
(for today), but to demonstrate
that one need *not* have any
*formal* understanding of
“inverse function” to use the
never-stop-harping-on-general-principles
method that i’m more or less advocating here.

(indeed, inverses-treated-formally
is a classical crux. “never mind
*why* i solve x = f(y) when i want
a formula for f^{-1}(x)… just show
me *how*!”… one of the most
glaring examples of the
i’ll-never-understand-the-reasons
phenomenon known to me.)

okay. this was fun.
haven’t i got a *job* or something?

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  1. Could you have made 4k by removing a
    few
    carriage returns
    ? : {)> Jonathan

  2. vlorbik

    i
    don’t think so… it counts out at 5563.
    sure and i coulda cut it by a few hundred
    characters… and probably improved it
    considerably. but i’m damned if i’ll
    actually *work* at this silly blogging
    thing for no pay (and very few readers).
    it’s fun to see people discussing actual issues
    we face in classroom work in specific terms
    and i’m thrilled to participate i suppose.
    but really it’s just a way of making the time
    go by without having to think about money.
    and, lo, it isn’t working. bye.

    • Anonymous

      Thanks for ranting my rant, Owen. I get scared that I’m not doing anything of real value, because they want to parse math in such a limited way. But in the one class (of 3) that went well this past semester, I think they may have gotten a glimpse of what it could be about. Just a tiny glimpse, no real progress toward the real math I know, not yet.

      I think about Dan’s wolverine post in this context. I think I’ll have them read it next semester. (A bunch of them are taking the next class with me.)

  3. Anonymous

    (Oops! I forgot to put my name. But it doesn’t take many words to become recognizable, does it?)

  4. i’m very touchy on the subject of wolverines
    so i’ll let that go by i think.

    the whole framework that even calls for
    “log *laws*” has this wrong-itude to it.
    those born-again-in-the-faith,
    when presented with a bunch of rules
    to be used in contexts yet-to-be-supplied,
    will look around immediately for some
    “higher-level” rule from which the big ugly
    list will all follow as corollaries
    (failing this, shrink the list;
    “the log of a product is the sum of the logs”
    *implies*
    “the log of quotient is the difference of the logs”
    [and so only one “formula”
    should be “memorized”…], e.g.).

    the “one true rule” of logs is then
    log_a(x) = y :\equiv a^y = x;
    in plain english, “logs are inverse-exponentials”
    (and, in teacherese, alas, likely to be
    “logs [to a given base… which should
    be a positive number different from one]
    are inverse-exponentials [to the same base]”).

    and these *other* facts should be, sure,
    learned as needed (and then, and this
    is a good deal of the point i suppose,
    forgotten when forgotten… in the serene
    confidence that they can be derived as needed).

    anyhow, when you get stuck, *bang on it*
    with the *one thing you know* (if you only
    know one thing) rather than dig yourself
    into some shame-pit where there’s this
    “law” you’re supposed to follow…
    and a culture-wide cliche to the effect
    that “ignorance of the law is no excuse”.

    i failed to make it explicit in my post…
    and it probably wasn’t clear to me when
    i posted it… but part of the point’ll’ve been
    that i make no appeal to any log law
    other than the (defining) inverse-exponential
    property.

    and students usually come to teachers-of-logs
    with the needed “law of exponents”
    — a^(x+y) = a^x * a^y —
    pretty firmly in their grasp
    (as these things go); i conjecture
    that this usually comes about
    from working “multiply these
    polynomials right here” much more
    than it does from “practice these *laws*”.

    i conjecture further that there’s something
    to be learned from this state of affairs.
    (problems that “make sense” only as
    tools-used-in-some-“higher-level”-problem
    appear to be very common around here.)

    typically bosses… layers-down-of-rules…
    don’t even *know* how the work really
    gets done. and typically they lie about
    it even when they *do* know.
    (
    “you can’t *handle* the truth!”
    “not developmentally appropriate.”
    “speak no evil.”
    et-everlasting-cetera.
    )

    i’m here advocating, not math *anarchism* quite,
    but math “minimism”… “that government is best
    that governs least” and all that. not “do what thou
    wilt shall be the whole of the law” but rather
    “the axiom system before us shall be…”.

    something i saw in a zine somewhere in the 90’s:
    “figure it out for yourself or obey without thinking”.

  5. it turns out after a tiny bit more research
    that i’m actually touchy on the subject
    of wolverine wranglers, not wolverines
    per se.

  6. So, I hadn’t seen that post, and needless to say (but here I am saying it) but giving kids tools earlier makes some sense, especially when they will suffer for not having them later.

    Hell, you know I deny my algebra I students the exponent laws, right? I do example after example where x(x^3)^2 becomes x(x^3)(x^3) which becomes, elegantly, x(xxx)(xxx). Count ’em up, rite ’em out. Numbers instead of x’s work too.

    And every damned log lesson opens not with your highly abstract formula, but with the far gentler: log_2(8) = 3 :\equiv 2^3 = 8 And be assured that I expect them to MEMORIZE that one, and jot it down at the moment they hear the word “logarithm”.

    Did you know that if I cut/paste your formulas from the comments, it comes out not like the formula, but like latex. Cool.

    Jonathan




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