What is mathematics?
It’s worth saying something here about what I’m hoping these four children learn. Not the specifics — which I’ll write about later — but, more generally, what it means to engage in math. I like the National Research Council’s definition of mathematical proficiency in terms of five “strands”: procedural fluency, conceputal understanding, strategic competence, adaptive reasoning, and productive disposition. You can read descriptions of each strand here, if you haven’t already. The metaphor of strands is a nice one because it emphasizes how interrelated these competencies are. It is impossible to teach any one in isolation.
To illustrate this point rather dramatically, I’d like to tell you about a fifth-grade girl my research team and I worked with in a previous study. I’ll call her Lemesha. To assess her understanding of math, I gave Lemesha a problem-solving interview. In this assessment, I read problems to her (or she could read them herself), and she solved them using whatever strategy she wanted to. We were assessing how children solved word problems that involved multiplicative situations. Altogether there were about 10 of these type problems. Lemesha didn’t solve a single one correctly. (She attempted to use keyword strategies, which didn’t help because the problems weren’t designed to be solvable by keywords). But when we gave her a multiplication-fact recall test, we found she knew well over half of her multiplication facts by recall. What a contrast! I suspect that having learned her multiplication facts in isolation from problem solving prevented her from understanding how these facts were linked to multiplicative situations.
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