## Exploring the Relationship between K-8 Prospective Teachers’ Algebraic Thinking Proficiency and the Questions They Pose during Diagnostic Algebraic Thinking Interviews

Source: *Journal of Mathematics Teacher Education, Volume 17,* p. 429-461 (2014)

(Reviewed by the Portal Team)

This study aimed to explore the relationship between teachers’ mathematics content knowledge and the types of questions they pose to investigate students’ mathematical thinking.

**Methods **

This semester-long study was conducted at a large private university situated in the Midwestern United States in a mathematics content course and in a pedagogy course whose curricula were coordinated.

The participants were 18 prospective teachers and 18 middle school students from a nearby school selected as the site for the field experience. The prospective teachers were in their third or fourth year of their 4-year teacher education program, all candidates for a grades 1–8 teaching license.

The authors evaluated prospective teachers’ algebraic thinking proficiency across 125 algebra-based tasks and we analyzed the characteristics of questions they posed during the interviews.

**Prospective teachers’ mathematics content knowledge **The results provide insights about prospective teachers’ algebraic thinking ability and their readiness for fostering algebraic thinking in the K-8 students. The findings revealed 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.

This examination of prospective teachers’ algebraic thinking, one aspect of their mathematics content knowledge (CK), reinforces other research findings about prospective teachers’ algebraic thinking proficiency. Comparing the algebraic thinking proficiency of prospective teachers in the high and low groups revealed that the prospective teachers in the high group are most proficient at identifying, organizing, uncovering, and making sense of the regularities found in a problem, an important characteristic of algebraic thinking. In contrast, the prospective teachers in the low group demonstrated procedural aspects of symbolic algebra rather than the sense making aspects of algebraic thinking. The authors hypothesize, then, that prospective teachers in the high group are more prepared to use algebraic thinking to support student learning of algebra-related concepts.

This work provides several new directions to consider while using one-on-one diagnostic interviews as a mean of supporting prospective teachers’ in learning how to pose effective questions.

First, the authors think their debriefing interviews were less effective than they could have been because we asked the prospective teachers to simply find and highlight the questions they posed during their first interview.

The relationship between mathematics content knowledge and questioning Mathematics content knowledge (CK) is an important prerequisite in establishing pedagogical content knowledge (PCK). The prospective teachers in the high algebraic thinking group formulated some probing questions during their interviews, but the prospective teachers in the low group never asked probing questions in their interviews. Prospective teachers in the high group, on the other hand, were more often able (and willing) to pose probing questions to investigate their students’ understanding of the different features.

The authors believe that this study provides an important detail about the relationship between one aspect of CK (algebraic thinking proficiency) and one aspect of PCK (questioning about mathematical thinking), which can inform the work of mathematics educators in preparing prospective teachers to effectively teach mathematics. Knowing how to investigate students’ mathematical thinking through questioning is an important ability that prospective teachers need to develop.

The authors argue that supporting this ability by focusing on strengthening prospective teachers’ own mathematics content knowledge (CK) with their pedagogical content knowledge (PCK) should be one of the goals of teacher education programs.