Theses Doctoral

# Effects of Diagrams on Strategy Choice in Probability Problem Solving

Xing, Chenmu

The role of diagrammatic representations and visual reasoning in mathematics problem solving has been extensively studied. Prior research on visual reasoning and problem solving has provided evidence that the format of a diagram can modulate solvers’ interpretations of the structure and concept of the represented problem information, and influence their problem solving outcomes. In this dissertation, two studies investigated how different types of diagrams influence solvers’ choice of solution strategy and their success rate in solving probability word problems. Participants’ solution strategies suggested that problem solvers tended to construct solutions that reflect the structure of a provided diagram, resulting in different representations of the mathematical structure of the problem. For the present set of problems, a binary tree or a binary table tends to steer solvers to use a sequential-sampling strategy, which defines simple or conditional probabilities for each selection stage and calculates the intersection of these probabilities as the final probability value, using the multiplication rule of probability. This strategy choice is structurally matched with the diagrammatic structure of a binary tree or a binary table, which represents unequally-likely outcomes at the event level. In contrast, an N-by-N (outcome) table steers solvers to use of an outcome-search strategy, which involves searching for the total number of target outcomes and all the possible outcomes at the equally-likely outcome level, and calculates the part-over-the-whole value as the final probability, using the classical definition of probability. This strategy is strongly cued by the N-by-N (outcome) table, because the table structure represents all equally-likely outcomes for a probability problem, and organizes the information so that the target outcomes can be seen as a subset embedded in the whole outcome space. When an N-ary (outcome) tree was provided, choices were split between the two solutions, because the N-ary tree structure not only cues searching for equally-likely outcomes but also organizes the problem information in a sequential-sampling, stage-by-stage way. Furthermore, different diagrams seem to be associated with different patterns of characteristic errors. For example, solving a combinations problem with an N-by-N table tended to elicit erroneous solutions involving miscounting those self-repeated combinations represented by the table’s diagonal cells as valid outcomes. Typical errors associated with the use of a binary tree involved incorrect value definitions of the conditional probability of the outcome of a selection. And the N-ary tree may lead to less successful coordination of all the target outcomes for the studied problems, because the target outcomes were dispersed in the outcome space depicted by the tree, thus not salient.
The findings support arguments (e.g., Tversky, Morrison, & Betrancourt, 2002) that in order to promote problem solving success, a diagrammatic representation must be carefully selected or designed so that its structure and content can be well-matched to the problem structure and content. And for computational efficiency, information should be spatially organized so that it can be processed readily and accurately. In addition to the implications for effective diagram design for problem solving activities, the findings also offer important insights for probability education. It is suggested that a variety of diagram types be utilized in the educational activities for novice learners of probability, because they tend to highlight different probability concepts and structures even for the same probability topic.

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