## the equality meaning of the equal sign

Posted by vlorbik on August 23, 2009

so these kids that refuse to learn

what we mean by “=”?

hugely important in my opinion.

and not just because i’ve been

going on and on about

“literal truth”: the word is not the world

and so literal truth interpreted strictly

never quite actually *happens* etcetera.

my tried-and-true “first thing that

fell in my hand via google” research strategy

gets us

Many students do not understand the mathematical meaning of the equals sign: that the expressions on either side have the same value. Instead they believe that an equals sign indicates where to write an answer. This has implications for their work in the Number dimension and also for their success with algebra. From the first years of school, teachers are encouraged to stress the equality meaning of the equals sign.

.

so this problem will have received considerable attention

in the literature. but there’s somehow this slippage.

the kids still ain’t *gettin* it. how is this *possible*?

they’ve gotta be learning *something* from all

this drumming in their dear little ears…

on my model it’s something like this.

“they’ll *never* tell me what they *really* want;

i clearly *see* that they’re *lying* about this

“equal sign” business for example when they tell

me that it *doesn’t* mean “put the answer here”

when my older sister (let’s say) told me it *does*

and it’s marked right *sometimes* and this teacher

is *picking on me* and i can never fucking win.”

the tragedy of course is that math “should” be

the great *leveler*: where the intellectual

underworld can meet the elite on something like,

yes, “equal” terms.

i myself lost all faith in school in 7th grade.

except math. in particular i quit doing homework

altogether with the first assignment: i could

just see it was going to take up a *lot* of time

uselessly. this reaction was something like despair;

i’d always been a “good student” in the sense

of doing all the work (but a class-clown style

troublemaker; i’m sort of a glutton for attention)

and there was this moment where i saw that i’d be

giving that up. teachers approving less of me?

well, i suppose it’s natural that this is where

i’d begin approving so much less of them.

but this isn’t what i mean really, not at all.

this was bloomington: i had plenty of talented teachers

and some of ’em i even loved. i sure still wanted to

impress ’em and would show off my erudition and whatnot

and even mostly try to pull my weight or better

in in-class work. i just could see that what we

were doing didn’t have a whole lot to do with

“learning the material”: if one were really

*interested* in any of the, anyhow *academic*

material, for example, one would have to read

one heck of a lot *more* than the assignments.

and write less. because i guess the real point

or anyway part of it is that i didn’t trust

*any* of ’em to (use the system we were in to)

teach me anything about writing. and they didn’t.

most of ’em left off trying pretty quick

and i’m grateful. and maybe they did after all.

some of ’em recommended some good books, for example.

but anyhow, the *real* point (…) is that

*math is different*. these guys knew

how to do some highly-prestigious(-for-school) stuff

that i’d need for geek cred; stuff that,

anyway as it felt to me at the time

(and i have no very good reason to

second guess this estimate now),

i’d *never* be able to just read and understand

without an experienced guide.

now, that’s not my *real* real point.

what i *mean* is that something *like*

my experience… becoming convinced

that “math is different” (because

her teachers *never lie* about important stuff

and that they’ll *stick to clearly stated rules*

and when you’re *right* everybody will *agree*)

is… maybe… pretty close to the elusive

phenomenon we’ve been halfheartedly tracking:

“having a ‘math brain’ “.

that such brains are earned not doled out at birth

we will hereby take for granted. my point right now

(elusive bugger) is that by the time one can be said

to *have* such a brain one will have long since

forgotten that the equal sign can be confusing.

no two things could be more equal, somebody said.

this is about as simple as we can expect anything

ever to get or so it seems and anyhow if you can’t

get *this*, there’s not much point in talking about

anything *else* because, well, as somebody said

somewhere “algebra is the science of equations”

and if you can’t be made to care enough about it

even to find out what the hell an equation even *is*,

to hell with it. really. shop class is down the hall.

and that something like *this*

accounts for the “people who understand math

can never explain math” paradox that drives

certain battles of the mathwars.

now, i’d like to believe that a *real* math brain

will be *eager* to explain *exactly this point*…

and be the *best possible* explainer in the bargain…

to any student *actually trying to find out*

what the heck these “users of equations” are on about.

and so on for any other topic in pedagogy of math.

now, it happens to be false.

“real” math brains fall into the “that’s so obvious,

there can be nothing to say; come back when you get it”

trap *more* often than most people because they’ve

*learned* more (and hence forgotten more). what i’ve

elsewhere called “the observer tautology”

(“nothing is obvious [until it is]”)

wreaks a great deal of havok here

and gives everybody a bad name.

(students, teachers, mathematicians…

all have come short of the glory…)

and it’s going to take amazing amounts

of caring and goodwill to get *past* these

difficulties. and that putting a bunch of fucking

cameras in the rooms so we can tell if the

fucking guard is letting too many people out

to take a fucking piss isn’t necessarily

the way to fucking get it.

and this may be obvious.

but maybe not to everybody.

and fewer will say it than think it.

and fewer still we be heard.

and that most of these will be

brushed off easily.

and that the few that are left will be bribed off.

or frightened off. or anyway controlled.

“committee work”.

## vlorbik said

http://www.angrymath.com/2010/08/more-equals-signs.html

vlorbik sez check it out.

## vlorbik said

http://kitchentablemath.blogspot.com/2011/05/equation.html

http://kitchentablemath.blogspot.com/2011/05/parker-on-equal-sign.html

“=” at KTM

## vlorbik said

http://www.kitchentablemath.net/twiki/bin/view/Kitchen/FallingOffTheMathCliffCartoon?skin=plain

with a comment from me.

(i am the *only* source

google gave me back

for “algebra is the science

of equations”… but i swear

i didn’t make it up. i don’t

even really *believe* it.)

## vlorbik said

Click to access AMSNotices_2014.pdf

wu

By writing “242 ÷ 16 = 15R2”, school textbooks send an unmistakable signal to elementary students that they may abuse the equal sign any which way they wish.

## vlorbik said

Click to access Interview-MM.pdf

interview in _mathematical_medley_, 2012.

## vlorbik said

Click to access RME_2012.pdf

spotted by josh f.

## cut & paste | the livingston review said

[…] of ex-math-teacher08/20/09 this will all make sense later pedagogical content knowledge08/23/09 the equality meaning of the equal sign try it you’ll like it08/27/09 why we fight definitions are […]

## Vlorbik On Math Ed | the livingston review said

[…] of ex-math-teacher 08/20/09 this will all make sense later pedagogical content knowledge 08/23/09 the equality meaning of the equal sign try it you’ll like it 08/27/09 why we fight definitions are […]