Abstract
This study aimed to explore preservice elementary teachers’ use of a bar model as a heuristic for conceptualising relationships between quantities in situations involving ratio and percentages. As a part of a larger project, we focused on two preservice teachers, Maia and Jane, and investigated their solution paths in ratio and percentage problems and the role of the structural representations (i.e., bar model and percent line-bar model) in their solution paths. We analysed these focus participants’ written solutions and their think-aloud process during interviews using three coding phases with a constructivist grounded theory approach. We observed that the problem text fed the construction of the structural representations that served to make sense of the quantities and the relationships among them. Furthermore, the structural representations enhanced Maia and Jane’s mathematical insights about the problem situations. In addition, we observed that preservice teachers’ solution paths were neither linear nor similar, which indicates a mathematical richness in learning. In those solution paths, the model construction played a role in guiding the mathematical operations of constructing and operating on a unit rate in the ratio problem and identifying and operating on a referent whole unit in the percentage problem. We interpreted that variety and nonlinearity in solution paths suggested mathematical richness in learning, the educational implications of which were further discussed.
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Notes
Problem solver in this manuscript refers to an individual (at any age) involved in the process of solving a mathematical problem, irrespective of whether the problem is non-routine or procedural in nature. Therefore, we used the term in our review of the literature to refer to students engaged in problem-solving and throughout the manuscript to refer to the participants of the present study (i.e., the pre-service teachers) engaged in solving ratio and percentage problems.
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All procedures followed in conducting the research reported on in the manuscript are consistent with the principles of research ethics outlined by the American Psychological Association. For ethical considerations, IRB approval was received from Indiana University Bloomington (IRB Protocol No. 1401241233). Based on the regulations of the IRB Approval, informed consent forms were received from the participants. Since the participants are adults, there was no need for parental consent forms. This research is our original work and has not been published or under revision elsewhere.
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Appendix
Appendix
Initial list of codes and categories
Categories | Codes | Description |
|---|---|---|
The path of problem-solving | Text-Structure-Process (TSP) | A problem solution path where “children go through these three phases of problem necessarily in a fixed order” (Ng & Lee, 2009, p. 292). Students comprehend the problem situation, convert the problem into a model, and followed by calculations to reach a solution |
Text-Process (TP) | A solution path that students transform the problem into mathematical procedures, bypassing the structure phase | |
Text-Process-Structure (TPS) | A solution path that “where appropriate, children may first information into a set of arithmetic expressions (TP) and then use a model (TPS)” (Ng & Lee, 2009, p. 292) | |
Type of bar model | Comparison | A model showing the relationship between two quantities: “The arithmetic model represents the situation in which a = b + d and the sum of a and b is x.” (Ng & Lee, 2009, p. 287). An algebraic model represents the situation x + b = c where the unknown, x, is one of the quantities (Ng & Lee, 2009) |
Part-part-whole | A model involving at least two rectangles where either one of the rectangles or the whole (the sum of the rectangles) is unknown: “The part-whole model can be used to represent by a + b = x or the algebraic case of x + a = b” (Ng & Lee, 2009, p. 285) | |
Percent line only | A model showing the percentages and their associated values on a single line model | |
Incorporated percent line | A model involving two components: a percent line showing the percentages and aligned bar model showing the associated values of the quantities | |
Type of ratio model | Variable-part model (fixed number of variable parts) | A model involving “the ratio A to B if for some-sized part there are A parts of the first quantity and B parts of the second quantity… here A and B specify fixed numbers of parts that can vary in size.” (Beckmann & Izsák, 2015, p. 21) |
Multiple-batch model (multiple parts of fixed size) | “Given fixed units of measurement for two quantities (where the units can be the same or different) and given A units of the first quantity and B units of the second quantity (where A and B are positive numbers), such a pair of fixed quantities can be viewed as forming a composed unit or batch. From this perspective, another pair of fixed quantities is also in the ratio A to B if the new pair consists of a multiple of the original composed unit or batch.” (Beckmann & Izsák, 2015, p. 21) | |
Mathematical operations | Coordinating two quantities at a time | When model involves two parts such as two bars indicating a ratio between them or a percent line and a bar model, the problem-solver can maintain the ratio in the former one and the associated percentage in the latter one |
Keeping track of percentages with quantities | With an incorporated percent line model or percent line only model, the problem-solver can procedurally identify several percentages and their associated values in the quantity | |
Preserving ratio | Problem-solver’s maintaining the ratio in modifications of the model or in the calculations | |
Part-to-whole reasoning | Problem solvers’ identifying the ratio or percentage in relation to the whole referent quantity |
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Sevinc, S., Lizano, C. Bar model method as a problem-solving heuristic: an investigation of two preservice teachers’ solution paths in problems involving ratio and percentage. Math Ed Res J 36, 71–95 (2024). https://doi.org/10.1007/s13394-022-00427-9
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DOI: https://doi.org/10.1007/s13394-022-00427-9