### 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?

December 30, 2010 at 1:16 am

Could you have made 4k by removing a

few

carriage returns

? : {)> Jonathan

December 30, 2010 at 11:04 am

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.

December 31, 2010 at 1:46 am

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.)

December 31, 2010 at 1:47 am

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

December 31, 2010 at 6:13 pm

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

;

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”.

December 31, 2010 at 6:31 pm

it turns out after a tiny bit more research

that i’m actually touchy on the subject

of wolverine wranglers, not wolverines

per se.January 1, 2011 at 4:34 am

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 becomes which becomes, elegantly, . 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: 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

August 25, 2012 at 2:03 pm

http://function-of-time.blogspot.com/2012/06/are-you-sure-you-want-to-do-this.html

(just another amazing piece by ms. nowak.)