Abstract
In this study, we explored the relationship between prospective teachers’ algebraic thinking and the questions they posed during one-on-one diagnostic interviews that focused on investigating the algebraic thinking of middle school students. To do so, we evaluated prospective teachers’ algebraic thinking proficiency across 125 algebra-based tasks and we analyzed the characteristics of questions they posed during the interviews. We found that prospective teachers with lower algebraic thinking proficiency did not ask any probing questions. Instead, they either posed questions that simply accepted and affirmed student responses or posed questions that guided the students toward an answer without probing student thinking. In contrast, prospective teachers with higher algebraic thinking proficiency were able to pose probing questions to investigate student thinking or help students clarify their thinking. However, less than half of their questions were of this probing type. These results suggest that prospective teachers’ algebraic thinking proficiency is related to the types of questions they ask to explore the algebraic thinking of students. Implications for mathematics teacher education are discussed.
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Appendices
Appendix 1: Debriefing interview questions
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1.
What questions did you pose for your student during your first interview?
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2.
Why did you ask these questions?
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3.
How did the student react to the questions you posed?
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4.
Thinking about your interview experience, what questions do you wish you had asked the student during the first interview? Why?
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Would you change the way you questioned your student? Why or why not?
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Are there any questions you would like to include in your second interview? Why?
Appendix 2: Rubric for assessing prospective teachers’ use of Building Rules to Represent Functions
Not Evident (1) | Emerging (2) | Proficient (3) | |
|---|---|---|---|
Organizing Information | The solution does not indicate that the prospective teacher organized the information in the problem in a way that is useful for discovering the underlying patterns and relationships | The solution indicates that the prospective teacher organized the information in the problem in a way that is useful for discovering the underlying patterns and relationships; BUT, the organizational scheme used is not explicitly connected to the context of the problem (e.g., problem information is organized in a table but table entries are not contextualized, i.e., their meaning explained with links to the context of the problem) | The solution indicates that the prospective teacher organized the information in the problem in a way that is useful for discovering underlying patterns and relationships; AND, the organizational scheme used is explicitly connected to the context of the problem (e.g., uses a table to organize information in the problem and clearly relates table entries to the context of the problem) |
Predicting Patterns | The solution does not indicate the prospective teacher’s understanding of how the pattern works; OR, the pattern is identified incorrectly | The solution indicates the prospective teacher’s understanding of how the pattern works (e.g., terms beyond the perceptual field are identified correctly, explicit or recursive rule that describes the pattern is correct); BUT, the pattern or discovered regularities are not explicitly connected to the context of the problem | The solution indicates the prospective teacher’s understanding of how the pattern works (e.g., terms beyond the perceptual field are identified correctly, explicit or recursive rule that describes the pattern is correct); AND, the pattern or discovered regularities are explicitly connected to the context of the problem |
Chunking Information | The solution does not indicate that the prospective teacher identified repeated chunks of information that explain how the pattern works, OR repeated chunks of information in the pattern are identified incorrectly | The solution indicates that the prospective teacher identified repeated chunks of information to explain how the pattern works; BUT, the identified repeated chunks are not explicitly connected to the context of the problem | The solution indicates that the prospective teacher identified repeated chunks of information that explain how the pattern works; AND, the identified repeated chunks of information are explicitly connected to the context of the problem |
Describe a rule | The solution does not indicate that the prospective teacher identified and described the steps of a rule through which the relationship embedded in the problem can be represented | The solution indicates that the prospective teacher described the rule (verbal or symbolic) to represent the uncovered relationship; BUT the rule is not explicitly connected to the context of the problem | The solution indicates that the prospective teacher described the rule (verbal or symbolic) to represent the uncovered relationship; AND, the rule is explicitly connected to the context of the problem |
Different Representations | The solution does not indicate that the prospective teachers used different verbal, numerical, graphical, or algebraic representations to uncover different information about the problem | The solution indicates that the prospective teacher used Different Representations (e.g., verbal, numerical graphical, or algebraic) to uncover and explore information embedded in the problem; BUT, the representations used are not explicitly connected to the context of the problem (e.g., uses a list of numbers without contextualizing their meaning) | The solution indicates that the prospective teacher used Different Representations (e.g., verbal, numerical graphical, or algebraic) to uncover and explore information embedded in the problem; AND, the representations used are explicitly connected to the context of the problem |
Describing Change | The solution does not indicate that the prospective teacher considered change in a process or relationship as a function of the relationship between variables in the problem, i.e., change in the input variable with respect to the change in the output variable | The solution indicates that the prospective teacher described the change in a process or relationship as a function of the relationship between variables in the problem, (i.e., change in the input variable with respect to the change in the output variable); BUT the described change is not explicitly connected to the context of the problem | The solution indicates that the prospective teacher described the change in a process or relationship as a function of the relationship between variables in the problem (i.e., change in the input variable with respect to the change in the output variable); AND, the described change is explicitly connected to the context of the problem |
Justifying a Rule | The solution does not indicate that the prospective teacher explained why the rule found in the problem works for any number | The solution indicates that the prospective teacher explained why the rule found in the problem works for any number. The justification is not explicitly connected to the context of the problem | The solution indicates that the prospective teacher explained why the rule found in the problem works for any number. The justification is explicitly connected to the context of the problem |
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van den Kieboom, L.A., Magiera, M.T. & Moyer, J.C. Exploring the relationship between K-8 prospective teachers’ algebraic thinking proficiency and the questions they pose during diagnostic algebraic thinking interviews. J Math Teacher Educ 17, 429–461 (2014). https://doi.org/10.1007/s10857-013-9264-1
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DOI: https://doi.org/10.1007/s10857-013-9264-1