BOX 5-1 One boy, quite a good student, was working on the problem, “If you have 6 jugs, and you want to put 2/3 of a pint of lemonade into each jug, how much lemonade will you need?” His answer was 18 pints. I said, “How much in each jug?” “Two-thirds of a pint.” I said, “Is that more or less that a pint?” “Less.” I said, “How many jugs are there?” “Six.” I said, “But that [the answer of 18 pints] doesn’t make any sense.” He shrugged his shoulders and said, “Well, that’s the way the system worked out.” Holt argues: “He has long since quit expecting school to make sense. They tell you these facts and rules, and your job is to put them down on paper the way they tell you. Never mind whether they mean anything or not.”^{1} |
A recent report of the National Research Council,^{2}Adding It Up, reviews a broad research base on the teaching and learning of elementary school mathematics. The report argues for an instructional goal of “mathematical proficiency,” a much broader outcome than mastery of procedures. The report argues that five intertwining strands constitute mathematical proficiency:
Conceptual understanding—comprehension of mathematical concepts, operations, and relations
Procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
Strategic competence—ability to formulate, represent, and solve mathematical problems
Adaptive reasoning—capacity for logical thought, reflection, explanation, and justification
Productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy
These strands map directly to the principles of How People Learn. Principle 2 argues for a foundation of factual knowledge (procedural fluency), tied to a conceptual framework (conceptual understanding), and organized in a way to facilitate retrieval and problem solving (strategic competence). Metacognition and adaptive reasoning both describe the phenomenon of ongoing sense making, reflection, and explanation to oneself and others. And, as we argue below, the preconceptions students bring to the study of mathematics affect more than their understanding and problem solving; those preconceptions also play a major role in whether students have a productive