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Math Teaching: words, symbols, and interests
Posted:
Jan 29, 1994 5:34 PM
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In a recent email conversation the topic turned to teaching and the
effect of posing math problems in "words" rather than using "abstract"
symbols. When do words confuse and illuminate? What is the role of
interest in learning?
For those who might be interested, pieces of the conversation are
abstracted below.
sseasto1@cc.swarthmore.edu (Sarah Seastone) writes:
>I went back to a book Ann Renninger edited (with others), about the
role of interest in learning, and began re-reading some results of
studies about how interest affects various measurable factors, and the
puzzling conclusion that interest seems to get in the way of certain
kinds of learning.
>One question that might be asked is about the role of "word
problems" in arousing interest in students and stimulating them to
burrow into the work. What is the effect/desirability of posing
problems in words rather than presenting them first using abstract
symbols?
>I don't think it can be stated simply. Let me try to describe my own
experience. I heard the Monty Hall 'choose among three doors'
probability problem -- as you posed it, there's a PowerPC behind one
door, and a goat behind each of the other two -- and it caught my
imagination, and I was immediately able to talk about it (and be
interested in it) because it came posed as a situation that I know (I've
seen the TV program) and that is familiar to most of us in the culture
(so there's less about it to have to explain when you first describe it to
others). Interest stimulating focus and attention. (Desirable, focuses
attention.)
>As I worked farther into it, however, and saw how hard it was for me
to understand, the words began to get in the way. Not universal
enough. Distracting because of constant mental images of computers
(yes!), goats (I'm not supposed to want a goat, but I like goats, in the
abstract; they are the source of cheese I love), Monty (UGH!),
memories of the whole TV show (designed to rot brain cells). Also, too
many syllables to have to generate every time we talk or write about
the problem. Rising annoyance at constant distraction. Loss of focus.
(Undesirable, gets in the way of staying with the problem.)
>I understand math as (among other things) about solving "real-life"
problems through creating more general abstractions which, once
deeply understood, can then be applied to other, new, analogous "real-
life" problems. True?
Steve - I think this way of putting it can participate in
misconceptions about math and abstraction/concreteness.
Doesn't mean you're wrong when you say this. And many
people might agree quite readily with your version.
I think math is about learning a language that makes visible
aspects of the world that are otherwise difficult to perceive and
use in making meaning. This language is abstract when you
have few relationships to it. The goal is not to move to higher
levels of abstraction but to make the abstract concrete in the
sense of forming many relationships to it (Uri Wilenski). This
language is generalizable/can often be applied to many
different "real-world" experiences and situations other than
those in which it was developed.
(Sarah continued) >In semantic terms, while "word problems"
stimulate interest by keying into people's prior experience, they
introduce the level of confusion we deal with when we say "chair" and
I see in my mind one picture of a chair, while you see another. Most
often the difference isn't important (unless we're trying to talk about
what to buy for a room we're both heavily invested in), but it's always
there.
>Such differences are reduced by using abstract symbols. Let C stand
for goat, and make it clear C is not the desired outcome. Link affect to
symbol. A level of confusion around words has been reduced. (A level
of interest may also have beeninfluenced [reduced] in the process,
although sometimes interest rises because problem-solving becomes
easier. Cognitive involvement stimulating interest. It's complex.)
>So my experience is that words first help catch my attention, but that
they may then produce unnecessary and perhaps distracting
dissonance, so that at some point it is more useful for me to move to
symbols to continue a process well begun -- though I may not be the
best judge of what that point is and an astute observer will want to
hold me to the words a while longer because built-up frustration
around word problems coming from early experiences of difficulty
with them is causing me to turn away too quickly, assume I can't
answer.
Steve - For you, maybe, probably not always because elsewhere
you've said certain kinds of stories would help. For others who
have no good relationships to those mathematical symbols or
stronger relationships to math through other experiences, it can
be an advantage to have the problem constructed in their own
field of "interest." Take the child involved with racing of some
sort (cars, horses, etc.) He or she may be quite able to do certain
sorts of equations concerning speed and distance without using
a formal mathematical language.
(Sarah continued) >Hard for me to imagine myself doing this, but I'll
take your word for it. :) But can he or she then take what's been
learned about speed and distance in racing and construct other
situations, use what's been done elsewhere?
Steve- Not necessarily. That's where the interest becomes
leverage, not just as motivator, but as scaffolding.
(Sarah continued) >Perhaps one task of the educator is to help the
student keep track of what steps have been learned, what steps are
being executed at a given moment, the whole pattern of the dance.
When it's time to move from words to symbols, or dance back and
forth between; when symbols have been internalized to the point of
deep understanding; when you can turn the student loose to dance
alone ("you have enough from the outside now, go inside and mull").
>Also, however, when you still need to keep a hand on the back of the
bike to avoid loss of trust, injury to the process of learning even
though learning to ride may take more time if you hold on a while
longer... If you let go too soon your student could learn to ride, but in
the process you can create a toxic environment in which unspoken
questions of trust and abandonment will muddy the waters and a lot
of energy will need to be spent to undo what's taken place. You can't
let go quickly unless there's a very solid, already well established
level of trust in place and you make it clear you're not going away for
good.
------------------------------------------------
-- steve
Stephen Weimar (sweimar1@cc.swarthmore.edu, weimar@aol.com)
Education Consultant:
- Geometry Forum Project, Swarthmore College, 800-756-7823
- Student Teacher Supervision, Swarthmore College, (215) 328-8344
- Conflict Education, (215) 328-1792, fax (215) 328-8446
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