In the article “On the Dual Nature of Mathematical
Conceptions: Reflections on Processes and Objects as Different Sides of the
Same Coin” Anna Sfard discusses the dichotomies present in mathemactical
concepts, contrasting them with such words as structural and operational,
abstract and algorithmic, and declarative and procedural. Sfard focuses on the
structural/operational dichotomy and attempts to find the bounds of these two
modes, to find where they begin to overlap.
She goes on to rescind her use,
and academia’s use, of the word dichotomy
claiming that, in fact, since these two terms are “inseparable, facets of the
same thing” it should be called a duality
instead as they complement each other.
A very interesting example of
this notion in action revolves around the concept of number. There exists a
one-to-one structural relationship between the number of objects in a set and
the words ‘one,’ ‘two,’ ‘three,’ but when a child is asked to determine how
many in a set, they will repeat the process of counting, showing the
operational side of the concept. So number represents both operation, (of
measuring how many) and it represents a structure (of representing an amount).
In what order should these two
different faces of a concept be learnt?
Or should it be strived to be simultaneous? Near the end of the article Sfard makes me
stop as she mentions that one of the ‘vexing issues’ facing math education is
the apparent trade-off between profiency and comprehension, between “promoting
the smooth performance of mathematical procedure and developing understanding
of how and why that procedure works and what it means.” What do you think? Is this an intrinsic issue in learning
mathematics, or is it extrinsic of learning math, introduced by ways of how we
approach math education?
That is a good question. It is certainly emphasized in math education, but could we have one without the other? Computers are increasingly capable of mathematical procedure, they can even prove things, but it would be a whole other debate about whether they actually `understand.' I tend to minimize procedure, flippantly joking that I'm not a human calculator, but rather capable of thinking and understanding the math behind it. That said, executing a particular procedure over and over is one of the most tried and true ways of reaching that point. I'm skeptical of it being entirely removable.
ReplyDeleteIt sounds like part of the article is trying to get at the heart of mathematical thinking. The is very much in the spirit of the article I read, entitled "psychology and mathematics education".
ReplyDeleteI like the distinction between declarative and procedure (notice I said distinction … not dichotomy ;). We know from studies of human memory that declarative and procedural memories do function differently and utilize different parts of the brain, and so it is not an artificial distinction.
The article I previously mentioned (my reading for this week) touches on this same distinction. More precisely, the author writes: "My argument is that, in psychology and in mathematics education especially, the object of our research cannot be reduced to the identification and description of the elementary components of a certain capacity or behavior, as Guilford's (1967) conception would require. If such a strategy would work, it would be possible to borrow from psychology the information already acquired with regard to the re- spective elements identified in other conditions and to apply it to mathematical conditions. Obviously, this is not the case." The author goes on to talk about, for example, the role of metaphor in understanding mathematical thinking, as well as different examples of suggested progressions in mathematical thinking. For example, "(a) the concept of "reification" (e.g., Sfard, 1994), inspired by the history of functions (the transition from the dynamic representation of function to a static, strictly formal one)."
In terms of ramifications for educational practices, I think we must draw on results from neuroscience. How often must a "procedure" be repeated before it will be "remembered" by the student? I know what you're thinking: it depends on the student, subject, etc… but still there must be meaningful averages and distributions that can be used. Another key question for me is: What gets stored into long term memory more easily, procedural or declarative memory? I can think of many other related questions, such as: What kind of knowledge (procedural vs declarative) is more likely to lead to "transfer" of knowledge to new situations?
I have my own guesses in regards to these questions, but I think that the answer might be "it depends". If that's the case, what we really need are "adaptive" methods of teaching. For example, we need to recognize that some students learn better by "doing" first, and others by "understanding" first. Technology could be very useful in providing this type of adaptability.
Looking into the future, I can imagine less emphasis placed on procedural knowledge in math education. I suspect the strong emphasis on procedural knowledge is mainly due to two factors: 1) It is easier to measure procedural knowledge, and so government agencies and other testing bodies trying to design standardized curricula feel comfortable with it. 2) The curriculum has not yet caught up with technology. As Sophie pointed out, many of the cookbook procedures taught to students can now easily be performed by computers.