Conceptual Understanding of Problem Solving

Research Findings:
Research at the secondary and even post-secondary level on understanding of basic concepts that are involved in solving biology, chemistry, and physics problems (many of which require the application of algebraic or other mathematical concepts) indicates that students do not understand the concepts. This is confirmed by many research studies on problem solving in which students solve problems aloud. Research shows that even though students frequently solve mathematical problems correctly, they are unable to answer conceptual questions on which the problems are based.

Although there is a limited amount of research to indicate that understanding basic concepts qualitatively improves mathematical problem solving, it appears that this would be the case, especially for solving higher-level problems. Problem- solving research has led to the identification of commonly held scientific misconceptions, and to the conclusion that addressing these misconceptions in instruction may help to improve students' problem-solving ability.

In The Classroom:
Many secondary students use algorithms to solve biology, chemistry, and physics problems that require the use of mathematics. They substitute data given in a problem into a formula (use the factor-label method, or a Punnett Square), perform appropriate mathematical operations, and arrive at a correct solution. However, when asked about the meaning of what they have done or requested to describe the variables and the relationship among the variables involved, they are unable to do so.

There is some evidence that having students perform numerous problems in this manner does not necessarily lead to conceptual understanding. If conceptual understanding is an expected outcome of science instruction, a more reasonable approach would be to first emphasize a qualitative understanding of the underlying concepts, including clarification of related student misconceptions. Then the use of mathematical problem solving should help provide students with deeper insight into the concepts.

For example, many students can calculate the density of a solid, yet when shown samples of identical mass but different volumes, are unable to serial order the samples by density. It is unlikely that having students solve numerous density problems by substituting values into the density formula will help them distinguish between density and volume.