BuchananWWSept29

Response to Carpenter & Moser (1984) and Stein, Grober, & Henningsen (1996)

The domain of "doing mathematics" offers psychologists many potential insights into thinking and learning. Doing mathematics requires specific algorithmic skills, and more broad cognitive processing for selecting the best algorithms and evaluating their efficiency. In two similar studies, two groups of researchers sought to understand how students develop both algorithmic skills and more advanced understanding. Carpenter and Moser studied early elementary students' adoption of progressively sophisticated strategies for simple addition and subtraction, in "The Acquisition of Addition and Subtraction Concepts in Grades One Through Three." (1984) Stein, Grober, and Henningsen studied middle school students' development of more broad cognitive processing for doing mathematics, in "Building Student Capacity for Mathematical Thinking and Reasoning: An Analysis of Mathematical Tasks Used in Reform Classrooms." (1996) In each study, the researchers start with some assumptions about learning math, and try to document how students think about math. Both studies also offer valuable insights into possible ways to improve math instruction.

Carpenter and Moser assume that doing mathematics, particularly word problems,requires mastering simple algorithmic skills. In particular, they assume that "young children's solutions of word problems reflect the semantic structure of the problem." (p. 180) They further assume that children develop progressively sophisticated simple strategies for solving word problems. The series is progressive because some strategies are more efficient than others, taking less time or less energy. Students seem to abandon older, inefficient strategies for new ones. For example, students may initially use a counting-all strategy for addition, but progress to a counting-on from larger strategy. (p. 181) This progression leads to a more holistic cognitive processing, in which students "construct a representation of the relationships among all the pieces of information in the problem before solving it" (p. 183) Carpenter and Moser argue that students develop and replace these strategies somewhat independently of formal instruction, so math teachers should try "to capitalize on the rich informal mathematics that children bring to instruction" (p. 200)

Carpenter and Moser try to document how students think about math. They search for evidence of using and favoring progressively sophisticated strategies, through clinical interviews over a longitudinal study. They reasonably assert that the simple strategies are easy to distinguish by observing students solve carefully contrived word problems, and by questioning them about their approaches. Carpenter and Moser present aggregated data that shows a clear progression from simple strategies to more complex strategies, although they emphasize that students are not entirely consistent in their choice of strategies. In other words, a student may demonstrate skill in a more sophisticated strategy, but still occasionally use an older, less efficient strategy.

Carpenter and Moser's study offers valuable insight into understanding and improving math instruction. The elemental nature of the strategies and the plausibility of observing their use (e.g. counting on fingers) frame a clear account of elementary thinking and development in mathematics. Carpenter and Moser are reluctant to suggest specific reforms for math instruction, but they question some previous pedagogy. They explicitly hope their study leads to further inquiry, so that reformed math instruction can foster the progressive development of more sophisticated strategies.

Stein, Grober, and Henningsen studied much more sophisticated thinking and development. Like Carpenter and Moser, Stein et al. assume that students use algorithmic skills. But where Carpenter and Moser focus on the development of such algorithms, Stein et al. explore how teachers and students can strive for more broad cognitive processing. They assume that doing math requires a "deep and interconnected understanding of mathematical concepts, procedures, and principles." (p. 456) Stein et al. assume that learning mathematics depends on developing this understanding, and that several factors influence the "set up" and "implementation" of mathematical tasks that demand and foster such understanding. Most importantly, they assume that students construct this understanding, and teaching only "influences students' cognitive processes or thinking, which, in turn, influences their learning." (p. 457)

Stein et al. document how students think about math, using classroom observation over a longitudinal study. They select and code a representative sample of observed tasks. They deconstruct each task into set up and implementation, and identify the signs of holistic cognitive processing. Such processing is allegedly apparent when students use multiple strategies and representations, and provide mathematical explanations or justifications.

Stein et al. offer clear suggestions for understanding and improving math instruction. They are particularly interested in cases where the teacher sets up a cognitively demanding task, but the students' implementation shows "declined" holistic cognitive processing. (p. 462) They assert that the most common cause of such decline is a teacher providing the necessary holistic cognitive processing at the expense of the students' learning. For example, if a teacher sets up a cognitively demanding task, students' implementation will show declined holistic cognitive processing if the teacher suggests a specific strategy. Stein et al. suggest careful scaffolding, to press students to depend on their algorithmic skills as well as more "executive level" thinking.

In both their studies, Carpenter and Moser and Stein et al. demonstrate some of their assumptions about learning math, and try to document how students think about math. Both studies also offer valuable insights into how to understand and improve math instruction.

References

Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15, 179-202.

Stein, M. K., Grober, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33, 455-499.