How Mathematics Propels the Development of Physical Knowledge

Abstract

Three studies examined whether mathematics can propel the development of physical understanding. In Experiment 1, 10-year-olds solved balance scale problems that used easy-to-count discrete quantities or hard-to-count continuous quantities. Discrete quantities led to age typical performances. Continuous quantities caused performances like those of 5-year-olds. In Experiment 2, 11-year-olds solved problems with feedback. They were encouraged to use math or words to justify their answers. Children who used math developed an understanding superior to most adults, whereas children who used words did not. In Experiment 3, 9-year-olds solved problems with or without prompts to use math. Children encouraged to use math exhibited greater qualitative understanding, even though they were unable to discover metric proportions. The results indicate it is possible to design symbolic experiences to propel the development of physical understanding, thereby relating developmental psychology to instructional theory.Mathematics, diagrams, and other explicit representations help scientists discover and organize complex empirical relations and this also may be true for the developing child. The proposal that external representations and organizing activities contribute to development comes from Vygotsky's (1978) foundational insight that JOURNAL

Practicing versus inventing with contrasting cases: The effects of telling first on learning and transfer.

“…Without countable quantities, the students would probably make comparisons with terms like "more" and "less," which lack the precision required to find the ratio structure. Schwartz, Martin, and Pfaffman (2005), for example, found that children who were asked to give verbal explanations to balance scale problems did not learn to relate the dimensions of weight and distance. In contrast, children who were asked to invent mathematics to explain their answers were more likely to discover that proportionate ratios determine whether the scale balances.…”

Learning to Think: Cognitive Mechanisms of Knowledge Transfer

“…We see that students who have learned the language of algebra are much more likely to solve a particular class of complex story problems (which do not absolutely require equations) than students who have not learned the language of algebra. 11 Some research has shown that prompting students to think with a symbolic language can enhance learning (Roll, Aleven, & Koedinger, 2009;Schwartz, Martin, & Pfaff man, 2005). In both of these studies, students who were asked to reason using mathematical symbols acquired a more general, transferable representation knowledge than students who were not instructed to use mathematical notations.…”

“…However, other examples suggest the reverse route. For example, asking students to reason with data prior to giving them instruction may facilitate mental representations that support the subsequent acquisition of procedural competencies (Roll, Aleven, & Koedinger, 2009;Schwartz & Martin, 2004;Schwartz, Martin, & Pfaff man, 2005).…”

Instructional Control of Cognitive Load in the Design of Complex Learning Environments

“…These methods constrain the problem spaces within which learners work, and then make it more difficult for them to generate creative solutions or "think outside the box." An example is provided by a study of Schwartz, Martin, and Pfaffman (2005), in which children learned to manipulate pieces to help solve fraction problems. One group learned with pie pieces with different sizes, with a focus on routine building because the pieces are easily seen as fractions of a whole; the other group learned with tile pieces of equal sizes, with a focus on interpretation because the pieces should be interpreted as parts of a whole rather than just units.…”