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.
>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. ------------------------------------------------