particularly number, in grades pre-K–8, as well as the implications this knowledge has for teaching. How People Learn calls on recent findings in brain research to address issues of learning in general. In particular, it describes how skill and understanding in key subjects are acquired and discusses our growing knowledge about complex reasoning and problem solving.
Adding It Up is a research-based examination of pre-K–8 mathematics that focuses on what the mathematical content is with which students must develop proficiency, how instruction can help students develop this proficiency, and, most significant, the research that undergirds why these positions are taken. The report was developed by the Mathematics Learning Study Committee of the National Research Council. It arose out of concerns on the part of the National Science Foundation and the U.S. Department of Education about the shortage of reliable information on the learning of mathematics that could guide best practice.
The developers of Adding It Up based their findings and recommendations on research that is “relevant to important educational issues, sound in shedding light on the questions it sets out to answer, and generalizable in that it can be applied to circumstances beyond those of the study itself.” They also “looked for multiple lines of research that converge on a particular point” (p. 3). Using the analysis of this research, much of which is synthesized in the report, the committee states that “All young Americans must learn to think mathematically, and they must think mathematically to learn” (p. 16).
“Recognizing that no term captures completely all aspects of expertise, competence, knowledge, and facility in mathematics” the report chooses the phrase mathematical proficiency to capture “what it means for anyone to learn mathematics successfully.” Mathematical proficiency is then defined as having five interwoven and interdependent strands (p. 5):
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; and
productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy.