Developing Preservice Teachers’ Mathematical and Pedagogical Knowledge Using an Integrated Approach

This paper describes how an integrated mathematics content and early field-experience course provides opportunities for preservice elementary teachers to develop understanding of mathematics and mathematics teaching. Engaging preservice teachers in solving and discussing mathematical tasks and providing opportunities to implement these tasks with elementary students creates an authentic context for the future teachers to reflect on their own understanding of mathematics, mathematics teaching, and students’ mathematical thinking. Essential elements of the cycle of events in the integrated model of instruction are discussed: preservice students’ acquisition of mathematical concepts in the context of selected tasks in the content course; subsequent posing of mathematical tasks in early field experiences; reflection on work with students; and response to instructors’ feedback

Exploring the Development of Core Teaching Practices in the Context of Inquiry-based Science Instruction: An Interpretive Case Study

This paper describes our reflection on a clinical-based teacher preparation program. We examined a context in which novice pre-service teachers and a mentor teacher implemented inquiry-based science instruction to help students make sense of genetic engineering. We utilized developmental models of professional practice that outline the complexity inherent in professional knowledge as a conceptual framework to analyze teacher practice. Drawing on our analysis, we developed a typography of understandings of inquiry-based science instruction that teachers in our cohort held and generated a two dimensional model characterizing pathways through which teachers develop core teaching practices supporting inquiry-based science instruction

Examining the Mathematical Knowledge for Teaching Involved in Pre-service Teachers’ Reflections

This study examines the mathematical knowledge for teaching involved in 24 pre-service teachers\u27 reflections on teaching the meaning of fractions to a small group of students in an elementary mathematics field experience. Excerpts from journals are used to describe the aspects of mathematical knowledge for teaching pre-service teachers include and emphasize in the content of their reflections. The article illuminates how mathematical knowledge for teaching assists pre-service teachers in analytically reflecting on multiple aspects of teaching and learning, thus making reflection more productive. Implications for teacher education are discussed

Pre-Service Teachers’ Knowledge of Algebraic Thinking and the Characteristics of the Questions Posed for Students

In this study, we explored the relationship between the strength of pre-service teachers’ algebraic thinking and the characteristics of the questions they posed during cognitive interviews that focused on probing the algebraic thinking of middle school students. We developed a performance rubric to evaluate the strength of pre-service teachers’ algebraic thinking across 130 algebra-based tasks. We used an existing coding scheme found in the literature to analyze the characteristics of the questions pre-service teachers posed during clinical interviews. We found that pre-service teachers with higher algebraic thinking abilities were able to pose probing questions that uncovered student thinking through the use of follow up questions. In comparison, pre-service teachers with lower algebraic thinking abilities asked factual questions; moving from one question to the next without posing follow up questions to probe student thinking

Pre-service Middle School Teachers’ Knowledge of Algebraic Thinking

In this study we examined the relationship between 18 pre-service middle school teachers’ own ability to use algebraic thinking to solve problems and their ability to recognize and interpret the algebraic thinking of middle school students. We assessed the pre-service teachers’ own algebraic thinking by examining their solutions and explanations to multiple algebra-based tasks posed during a semester-long mathematics content course. We assessed their ability to recognize and interpret the algebraic thinking of students in two ways. The first was by analyzing the preservice teachers’ ability to interpret students’ written solutions to open-ended algebra-based tasks. The second was by analyzing their ability to plan, conduct, and analyze algebraic thinking (AT) interviews of middle school students during a concurrent semester-long, field-based education class. We used algebraic habits of mind as a framework to identify the algebraic thinking that pre-service teachers exhibited in their own problem solving, and we asked students to use them to analyze the algebraic thinking of middle school students. The data revealed that pre-service teachers’ AT abilities varied across different features of algebraic thinking. In particular, their ability to justify a rule was the weakest of seven AT features. The ability to recognize and interpret the algebraic thinking of students was strongly correlated with the strength of the pre-service teachers’ own algebraic thinking. Implications for mathematics teacher education are discussed

Using Sociocultural Theory to Guide Teacher Use and Integration of Instructional Technology in Two Professional Development Schools

This article demonstrates how sociocultural theories can be used to support strategic structuring of professional development activities for preservice and practicing teachers on technology use and integration. Examples are drawn from the authors\u27 experiences with teachers in two professional development schools that participated in a four-year Preparing Tomorrow\u27s Teachers in Technology (PT3) project. After a review of sociocultural theory and their context, the authors describe three activity systems in these schools: one for practicing teachers, one for preservice teachers, and a joint preservice/practicing teacher system. Important supports for use and integration of technology built into each of these activity systems included varied activities aimed at both beginning and advanced technology users, multiple levels of assisted performance, and a collaborative culture that offered numerous opportunities for shared work. Lessons learned and implications for teacher educators involved in similar partnerships are outlined

Prospective K-8 Teachers’ Knowledge of Relational Thinking

The goal of this study was to examine two issues: First, pre-service teachers’ ability and inclination to think relationally prior to instruction about the role relational thinking plays in the K-8 mathematics curriculum. Second, to examine task specific variables possibly associated with pre-service teachers’ inclination to engage in relational thinking. The results revealed that preservice teachers engage in relational thinking about equality, however, their inclination to do so is rather limited. Furthermore, they tend to engage in relational thinking more frequently in the context of arithmetic than algebra-related tasks. Pre-service teachers’ inclination to engage in relational thinking appeared to also relate to the overall task complexity and the use of variables. Implications of these findings for pre-service teacher education are provided

K-8 Pre-service Teachers’ Algebraic Thinking: Exploring the Habit of Mind Building Rules to Represent Functions

In this study, through the lens of the algebraic habit of mind Building Rules to Represent Functions, we examined 18 pre-service middle school teachers\u27 ability to use algebraic thinking to solve problems. The data revealed that pre-service teachers\u27 ability to use different features of the habit of mind Building Rules to Represent Functions varied across the features. Significant correlations existed between 8 pairs of the features. The ability to justify a rule was the weakest of the seven features and it was correlated with the ability to chunk information. Implications for mathematics teacher education are discussed

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

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

Exploring Prospective 1-8 Teachers\u27 Number and Operation Sense in the Context of Fractions

This exploratory study examined prospective elementary teachers’ (PSTs’) number and operation sense (NOS) in the context of solving problems with fractions. Drawing on the existing literature, we identified seven skills that characterize fraction-related NOS. We analyzed 230 responses to 23 tasks completed by 10 PSTs for evidence of PSTs’ use of different fraction-related NOS skills. The analysis revealed that PSTs did not use all seven fractionrelated NOS skills to the same extent. PSTs’ responses documented their frequent reasoning about the meaning of symbols and formal mathematical language in the context of fractions. To a lesser extent, PSTs’ responses documented their reasoning about different representations of fractions and operations, about the composition of numbers, and about the effects of operations on pairs of fractions. We also examined possible relationships among the seven fraction-related NOS skills identified across the analyzed responses. The results reveal that some of the fraction-related NOS skills appear to support one another. Given that NOS skills provide a foundation for effective mental computation strategies, our study shows the need for explicit attention in teacher preparation programs to supporting PSTs in developing a strong awareness of and facility with a range of fraction-related NOS skills. Our study also raises questions about the relationship between PSTs’ conceptual understanding of fractions and their fraction-related NOS skills and provides suggestions for future research that explores further connections among the fraction-related NOS skills