Vlorbik's Diner

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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”.

8 Responses to “the equality meaning of the equal sign”

  1. vlorbik said

    vlorbik sez check it out.

  2. vlorbik said

    “=” at KTM

  3. vlorbik said


    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.)

  4. vlorbik said

    Click to access AMSNotices_2014.pdf

    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.

  5. vlorbik said

    Click to access Interview-MM.pdf

    interview in _mathematical_medley_, 2012.

  6. vlorbik said

    Click to access RME_2012.pdf

    spotted by josh f.

  7. […] 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 […]

  8. […] 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 […]

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