Conceptual Understanding of Problem Solving
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.
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Individual differences in children's addition and subtraction knowledge
Cognitive Development, Vol. 19, Issue: 1, January - March, 2004. pp. 81-93
Canobi, Katherine H.
Relations among patterns of conceptual and procedural knowledge and grade were examined in 90 six- to eight-year-olds in order to explore addition and subtraction development. Conceptual knowledge was assessed by examining children's responses to pairs of problems reflecting various part-whole relations. Children solved related problems as part of a Problem-solving Task, judged, and explained part-whole relations in a Judgement Task. Children also solved a random set of addition and.
Acquiring an understanding of design: evidence from children's insight problem solving
Cognition, Vol. 89, Issue: 2, September, 2003. pp. 133-155
Defeyter, Margaret Anne; German, Tim P.
The human ability to make tools and use them to solve problems may not be zoologically unique, but it is certainly extraordinary. Yet little is known about the conceptual machinery that makes humans so competent at making and using tools. Do adults and children have concepts specialized for understanding human-made artifacts? If so, are these concepts deployed in attempts to solve novel problems? Here we present new data, derived from problem-solving experiments, which support the following..
The Cyclic Nature of Problem Solving: An Emergent Multidimensional Problem-Solving Framework
Educational Studies in Mathematics, Vol. 58, Issue: 1, January 2005. pp. 45 - 75
Carlson, Marilyn P.; Bloom, Irene
This paper describes the problem-solving behaviors of 12 mathematicians as they completed four mathematical tasks. The emergent problem-solving framework draws on the large body of research, as grounded by and modified in response to our close observations of these mathematicians. The resulting Multidimensional Problem-Solving Framework has four phases: orientation, planning, executing, and checking. Embedded in the framework are two cycles, each of which includes at least three of the four.
This paper analyzes the results of a national examination from the perspective of conceptual learning versus algorithmic problem solving.
A framework for thinking about knowledge and its organization is presented that can account for known expert-novice differences in knowledge storage and problem solving behavior. Interpreting any relationship between ability to answer qualitative and quantitative questions requires a model of cognition and that research should seek to develop assessments that monitor component aspects of developing expertise.