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Journal of Mathematical Behavior
Journal Prestige (SJR): 0.97 Citation Impact (citeScore): 1 Number of Followers: 2 Hybrid journal (It can contain Open Access articles) ISSN (Print) 07323123 Published by Elsevier [3182 journals] 
 Investigating algorithmic and creative reasoning strategies by eye
tracking Abstract: Publication date: Available online 5 April 2019Source: The Journal of Mathematical BehaviorAuthor(s): Mathias Norqvist, Bert Jonsson, Johan Lithner, Tony Qwillbard, Linus Holm Imitative teaching and learning approaches have been dominating in mathematics education. Although more creative approaches (e.g. problembased learning) have been proposed and implemented, a main challenge of mathematics education research is to document robust links between teaching, tasks, student activities and learning. This study investigates one aspect of such links, by contrasting tasks providing algorithmic solution templates with tasks requiring students’ constructions of solutions and relating this to students’ learning processes and outcomes. Information about students’ task solving strategies are gathered by corneal eyetracking, which is related to subsequent posttest performances and individual variation in cognitive proficiency. Results show that students practicing by creative tasks outperform students practicing by imitative algorithmic tasks in the posttest, but also that students that perform less well on creative tasks tend to try ineffective imitative strategies.
 Abstract: Publication date: Available online 5 April 2019Source: The Journal of Mathematical BehaviorAuthor(s): Mathias Norqvist, Bert Jonsson, Johan Lithner, Tony Qwillbard, Linus Holm Imitative teaching and learning approaches have been dominating in mathematics education. Although more creative approaches (e.g. problembased learning) have been proposed and implemented, a main challenge of mathematics education research is to document robust links between teaching, tasks, student activities and learning. This study investigates one aspect of such links, by contrasting tasks providing algorithmic solution templates with tasks requiring students’ constructions of solutions and relating this to students’ learning processes and outcomes. Information about students’ task solving strategies are gathered by corneal eyetracking, which is related to subsequent posttest performances and individual variation in cognitive proficiency. Results show that students practicing by creative tasks outperform students practicing by imitative algorithmic tasks in the posttest, but also that students that perform less well on creative tasks tend to try ineffective imitative strategies.
 Pedagogical context and proof validation: The role of positioning as a
teacher or student Abstract: Publication date: Available online 3 April 2019Source: The Journal of Mathematical BehaviorAuthor(s): Erin E. Baldinger, Yvonne Lai Recent research shows the promise of using tasks that situate mathematics in a pedagogical context in courses for secondary teachers. Such tasks can engage teachers in connecting undergraduate mathematics and secondary mathematics teaching, even when pedagogical knowledge is not needed to solve the task. We examine the phenomenon that the presence of pedagogical context appears to change the work of a mathematical task. We presented 17 practicing secondary teachers with the same set of arguments to validate, first in the context of teaching secondary mathematics, and then in the context of a university mathematics course. We argue that the construct of social positioning – as a student or teacher – explains differences in secondary teachers’ validations as well as the problem of disconnection between undergraduate mathematics and secondary teaching.
 Abstract: Publication date: Available online 3 April 2019Source: The Journal of Mathematical BehaviorAuthor(s): Erin E. Baldinger, Yvonne Lai Recent research shows the promise of using tasks that situate mathematics in a pedagogical context in courses for secondary teachers. Such tasks can engage teachers in connecting undergraduate mathematics and secondary mathematics teaching, even when pedagogical knowledge is not needed to solve the task. We examine the phenomenon that the presence of pedagogical context appears to change the work of a mathematical task. We presented 17 practicing secondary teachers with the same set of arguments to validate, first in the context of teaching secondary mathematics, and then in the context of a university mathematics course. We argue that the construct of social positioning – as a student or teacher – explains differences in secondary teachers’ validations as well as the problem of disconnection between undergraduate mathematics and secondary teaching.
 Collaborative gesture as a case of extended mathematical cognition
 Abstract: Publication date: Available online 8 March 2019Source: The Journal of Mathematical BehaviorAuthor(s): Candace Walkington, Geoffrey Chelule, Dawn Woods, Mitchell J. Nathan Gestures have been shown to play a key role in mathematical reasoning and to be an indicator that mathematical understanding is embodied – inherently linked to action, perception, and the physical body. As learners collaborate and engage in mathematical discussions, they use discourse practices like explaining, refuting, and building on each other’s reasoning. Here we examine how gestural embodied actions become distributed over multiple learners confronting mathematical tasks. We define collaborative gestures as gestural exchanges that take place as learners discuss and explore mathematical ideas, using their bodies in concert to accomplish a shared goal. We identify several ways in which learners’ gestures can be used collaboratively and explore patterns in how collaborative gestures arise while proving geometric conjectures. Learners use collaborative gestures to extend mathematical ideas over multiple bodies as they explore, refine, and extend each other’s mathematical reasoning. With this work, we seek to add to notions of important talk moves in mathematical discussions to also include a consideration of important gesture moves.
 Abstract: Publication date: Available online 8 March 2019Source: The Journal of Mathematical BehaviorAuthor(s): Candace Walkington, Geoffrey Chelule, Dawn Woods, Mitchell J. Nathan Gestures have been shown to play a key role in mathematical reasoning and to be an indicator that mathematical understanding is embodied – inherently linked to action, perception, and the physical body. As learners collaborate and engage in mathematical discussions, they use discourse practices like explaining, refuting, and building on each other’s reasoning. Here we examine how gestural embodied actions become distributed over multiple learners confronting mathematical tasks. We define collaborative gestures as gestural exchanges that take place as learners discuss and explore mathematical ideas, using their bodies in concert to accomplish a shared goal. We identify several ways in which learners’ gestures can be used collaboratively and explore patterns in how collaborative gestures arise while proving geometric conjectures. Learners use collaborative gestures to extend mathematical ideas over multiple bodies as they explore, refine, and extend each other’s mathematical reasoning. With this work, we seek to add to notions of important talk moves in mathematical discussions to also include a consideration of important gesture moves.
 Epistemic issues in classroom mathematical activity: There is more to
students’ conversations than meets the teacher’s ear Abstract: Publication date: Available online 27 February 2019Source: The Journal of Mathematical BehaviorAuthor(s): Manuel Goizueta We report on a comparative investigation of student mathematical activity in two settings of a secondary mathematics classroom: peer interaction in small group and group interaction with the teacher. Our framework draws on the understanding of school mathematics as activity in between the historicity of knowledge and the situatedness of norms. We propose the term epistemic issue to refer to instances of knowledge about the construction and justification of school mathematics knowledge whose use is traceable in student activity. A major finding points to the presence of a number of epistemic issues during peer work and their omission in the communication of this work to the teacher; a phenomenon we call epistemic shift. To illustrate this finding and the methods of analysis, we take data from a group of students during a lesson of probability and the lessonbased video stimulated recall interview. We finish by discussing some implications for the development of the mathematical culture of the classroom.
 Abstract: Publication date: Available online 27 February 2019Source: The Journal of Mathematical BehaviorAuthor(s): Manuel Goizueta We report on a comparative investigation of student mathematical activity in two settings of a secondary mathematics classroom: peer interaction in small group and group interaction with the teacher. Our framework draws on the understanding of school mathematics as activity in between the historicity of knowledge and the situatedness of norms. We propose the term epistemic issue to refer to instances of knowledge about the construction and justification of school mathematics knowledge whose use is traceable in student activity. A major finding points to the presence of a number of epistemic issues during peer work and their omission in the communication of this work to the teacher; a phenomenon we call epistemic shift. To illustrate this finding and the methods of analysis, we take data from a group of students during a lesson of probability and the lessonbased video stimulated recall interview. We finish by discussing some implications for the development of the mathematical culture of the classroom.
 Analysing engineering students’ understanding of integration to propose
a genetic decomposition Abstract: Publication date: Available online 26 February 2019Source: The Journal of Mathematical BehaviorAuthor(s): Deonarain Brijlall, Nokwethemba Jubilee Ndlazi This paper reports on a study which explored engineering students’ understanding of the techniques of integration in calculus. There were 30 first year engineering students who participated in the project. The concepts were covered as part of a mathematics course at a university of technology in South Africa. Activity sheets, constructed with tasks based on integration were administered to the participants. Their written responses, which were used to identify the mental constructions of these concepts, were analysed using APOS (ActionProcessObjectSchema) theory and interviews were carried out to clarify the written responses. The discussions and written work indicated that students exhibit procedural tendencies in integration and that students could not define both definite and indefinite integrals. These findings raised some didactical implications for higher education and also provided applications of genetic decomposition design and modification.
 Abstract: Publication date: Available online 26 February 2019Source: The Journal of Mathematical BehaviorAuthor(s): Deonarain Brijlall, Nokwethemba Jubilee Ndlazi This paper reports on a study which explored engineering students’ understanding of the techniques of integration in calculus. There were 30 first year engineering students who participated in the project. The concepts were covered as part of a mathematics course at a university of technology in South Africa. Activity sheets, constructed with tasks based on integration were administered to the participants. Their written responses, which were used to identify the mental constructions of these concepts, were analysed using APOS (ActionProcessObjectSchema) theory and interviews were carried out to clarify the written responses. The discussions and written work indicated that students exhibit procedural tendencies in integration and that students could not define both definite and indefinite integrals. These findings raised some didactical implications for higher education and also provided applications of genetic decomposition design and modification.
 Using cycles of research in APOS: The case of functions of two variables
 Abstract: Publication date: Available online 22 February 2019Source: The Journal of Mathematical BehaviorAuthor(s): Rafael MartínezPlanell, María Trigueros This article reports on an ActionProcessObjectSchema Theory (APOS) based study consisting of three research cycles on student learning of the basic idea of a twovariable functions and its graphical representation. Each of the three research cycles used semistructured interviews with students to test a conjecture about mental constructions (genetic decomposition) students may use to understand functions of two variables, develop supporting classroom activities based on interview results, and successively improve the conjecture. The article brings together for the first time findings already reported in the literature from the first two research cycles, and the results of the third and final cycle. The final results show that students who were assigned special activities based on the research findings of the first two cycles were more likely to exhibit behavior consistent with a Process conception of function of two variables. An important contribution of the article is that it shows how different APOS research cycles may be used to successively improve students’ understanding of a mathematical notion. Also, the description of findings from the three research cycles, provides a potentially useful guide to improve student learning of function of two variables.
 Abstract: Publication date: Available online 22 February 2019Source: The Journal of Mathematical BehaviorAuthor(s): Rafael MartínezPlanell, María Trigueros This article reports on an ActionProcessObjectSchema Theory (APOS) based study consisting of three research cycles on student learning of the basic idea of a twovariable functions and its graphical representation. Each of the three research cycles used semistructured interviews with students to test a conjecture about mental constructions (genetic decomposition) students may use to understand functions of two variables, develop supporting classroom activities based on interview results, and successively improve the conjecture. The article brings together for the first time findings already reported in the literature from the first two research cycles, and the results of the third and final cycle. The final results show that students who were assigned special activities based on the research findings of the first two cycles were more likely to exhibit behavior consistent with a Process conception of function of two variables. An important contribution of the article is that it shows how different APOS research cycles may be used to successively improve students’ understanding of a mathematical notion. Also, the description of findings from the three research cycles, provides a potentially useful guide to improve student learning of function of two variables.
 Meaningfulness in representational fluency: An analytic lens for
students’ creations, interpretations, and connections Abstract: Publication date: Available online 21 February 2019Source: The Journal of Mathematical BehaviorAuthor(s): Nicole L. Fonger Representational fluency—the ability to create, interpret, translate between, and connect multiple representations—is key to meaningful understanding of mathematics. This research develops an analytic framework for meaningfulness in representational fluency in linear equation solving tasks. The analytic lens was developed by adapting a structure of observed learning outcome (SOLO) taxonomy. The framework advances a continuum of perspectives including disfluencies and fluencies both within and across representation types. Data from interviews with ninthgrade algebra students solving linear equations with computer algebra systems exemplify the finegrained analyses of problem solving made possible with this lens. Findings also reveal how lesser meaningfulness in representational fluency may be a productive starting point for more sophisticated reasoning. Implications for research and practice on the interplay between students’ representing and understanding of mathematical ideas are discussed.Graphical abstract
 Abstract: Publication date: Available online 21 February 2019Source: The Journal of Mathematical BehaviorAuthor(s): Nicole L. Fonger Representational fluency—the ability to create, interpret, translate between, and connect multiple representations—is key to meaningful understanding of mathematics. This research develops an analytic framework for meaningfulness in representational fluency in linear equation solving tasks. The analytic lens was developed by adapting a structure of observed learning outcome (SOLO) taxonomy. The framework advances a continuum of perspectives including disfluencies and fluencies both within and across representation types. Data from interviews with ninthgrade algebra students solving linear equations with computer algebra systems exemplify the finegrained analyses of problem solving made possible with this lens. Findings also reveal how lesser meaningfulness in representational fluency may be a productive starting point for more sophisticated reasoning. Implications for research and practice on the interplay between students’ representing and understanding of mathematical ideas are discussed.Graphical abstract
 Investigating secondary students’ generalization, graphing, and
construction of figural patterns for making sense of quadratic functions Abstract: Publication date: Available online 15 February 2019Source: The Journal of Mathematical BehaviorAuthor(s): Karina J. Wilkie An important aim in school mathematics is to help students experience algebra’s power to express generality. Researchers have been investigating figural growing pattern generalization as an early route to developing students’ understanding of functional relationships. More recently studies have been considering ‘open’ or ‘free’ tasks involving figural growing pattern construction. This study explored twelve Years 7–12 Australian students’ intuitions and connection of meanings for quadratic functions through growing pattern generalization, graphing, and construction activities during individual taskbased interviews. The students who evidenced complementary correspondence and covariation views when generalizing appeared to handle pattern construction activities in productive ways, regardless of their prior formal study of quadratic functions. Areas of difficulty with graphing growing patterns and potential benefits of pattern construction tasks for exploring quadratic equations and for formative assessment are discussed.
 Abstract: Publication date: Available online 15 February 2019Source: The Journal of Mathematical BehaviorAuthor(s): Karina J. Wilkie An important aim in school mathematics is to help students experience algebra’s power to express generality. Researchers have been investigating figural growing pattern generalization as an early route to developing students’ understanding of functional relationships. More recently studies have been considering ‘open’ or ‘free’ tasks involving figural growing pattern construction. This study explored twelve Years 7–12 Australian students’ intuitions and connection of meanings for quadratic functions through growing pattern generalization, graphing, and construction activities during individual taskbased interviews. The students who evidenced complementary correspondence and covariation views when generalizing appeared to handle pattern construction activities in productive ways, regardless of their prior formal study of quadratic functions. Areas of difficulty with graphing growing patterns and potential benefits of pattern construction tasks for exploring quadratic equations and for formative assessment are discussed.
 Computing as a mathematical disciplinary practice
 Abstract: Publication date: Available online 31 January 2019Source: The Journal of Mathematical BehaviorAuthor(s): Elise Lockwood, Anna F. DeJarnette, Matthew Thomas In this paper, we make a case for computing as a mathematical disciplinary practice. We present results from interviews with research mathematicians in which they reflected on the use of computing in their professional work. We draw on their responses to present evidence that computing is an inherent part of doing mathematics and is a practice they want their students to develop. We also discuss the mathematicians’ perspectives on how they learned and teach computing, and we suggest that much needs to be explored about how to teach computing effectively. Our overarching goal is to draw attention to the importance of the teaching and learning of computing, and we argue that it is an imperative topic of study in mathematics education research.
 Abstract: Publication date: Available online 31 January 2019Source: The Journal of Mathematical BehaviorAuthor(s): Elise Lockwood, Anna F. DeJarnette, Matthew Thomas In this paper, we make a case for computing as a mathematical disciplinary practice. We present results from interviews with research mathematicians in which they reflected on the use of computing in their professional work. We draw on their responses to present evidence that computing is an inherent part of doing mathematics and is a practice they want their students to develop. We also discuss the mathematicians’ perspectives on how they learned and teach computing, and we suggest that much needs to be explored about how to teach computing effectively. Our overarching goal is to draw attention to the importance of the teaching and learning of computing, and we argue that it is an imperative topic of study in mathematics education research.
 Using APOS theory as a framework for considering slope understanding
 Abstract: Publication date: Available online 12 January 2019Source: The Journal of Mathematical BehaviorAuthor(s): Courtney Nagle, Rafael MartínezPlanell, Deborah MooreRusso In this paper a framework for slope is proposed using APOS (ActionProcessObjectSchema) Theory and conceptualizations of slope previously identified in research. The proposed APOSslope framework allows for discussion of students’ cognitive development in relation to different conceptualizations of slope. As such, it may be adopted as a means to advance future research or as a way to plan instruction. In particular, the framework uses specific examples to consider interrelations between the ways of thinking about slope that have been reported to provide additional insight on how individuals understand this concept. The proposed framework contributes to the field by bringing together a number of past studies related to slope and providing a common ground under which these works might be interpreted.
 Abstract: Publication date: Available online 12 January 2019Source: The Journal of Mathematical BehaviorAuthor(s): Courtney Nagle, Rafael MartínezPlanell, Deborah MooreRusso In this paper a framework for slope is proposed using APOS (ActionProcessObjectSchema) Theory and conceptualizations of slope previously identified in research. The proposed APOSslope framework allows for discussion of students’ cognitive development in relation to different conceptualizations of slope. As such, it may be adopted as a means to advance future research or as a way to plan instruction. In particular, the framework uses specific examples to consider interrelations between the ways of thinking about slope that have been reported to provide additional insight on how individuals understand this concept. The proposed framework contributes to the field by bringing together a number of past studies related to slope and providing a common ground under which these works might be interpreted.
 Nonexamples of problem answers in mathematics with particular reference
to linear algebra Abstract: Publication date: Available online 11 January 2019Source: The Journal of Mathematical BehaviorAuthor(s): Igor’ Kontorovich In this paper, I introduce a theoretical conceptualization proposing that some mathematical problems can be decomposed into welldefined concepts, which allows one to conceive the correct final answer as belonging to the intersection of concept examples. Solvers can use the conceptualization to monitor their solutions in the following way: once an answer to a problem is devised, it can be checked for critical attributes of the concepts in the decomposition, and for contradictions with familiar properties and theorems. A failure to satisfy one of these checks allows concluding that the devised outcome is a nonexample of a problem answer. To illustrate the potential of the conceptualization, I analyzed the solutions to a problem in linear algebra that 421 undergraduates produced in their final course exam. The analysis showed that the proportion of nonexamples varied among students’ answers, when in two parts of the problem all incorrect answers could be rejected as they constituted nonexamples. The potential of the conceptualization for practice and research is drawn.
 Abstract: Publication date: Available online 11 January 2019Source: The Journal of Mathematical BehaviorAuthor(s): Igor’ Kontorovich In this paper, I introduce a theoretical conceptualization proposing that some mathematical problems can be decomposed into welldefined concepts, which allows one to conceive the correct final answer as belonging to the intersection of concept examples. Solvers can use the conceptualization to monitor their solutions in the following way: once an answer to a problem is devised, it can be checked for critical attributes of the concepts in the decomposition, and for contradictions with familiar properties and theorems. A failure to satisfy one of these checks allows concluding that the devised outcome is a nonexample of a problem answer. To illustrate the potential of the conceptualization, I analyzed the solutions to a problem in linear algebra that 421 undergraduates produced in their final course exam. The analysis showed that the proportion of nonexamples varied among students’ answers, when in two parts of the problem all incorrect answers could be rejected as they constituted nonexamples. The potential of the conceptualization for practice and research is drawn.
 Facilitating videobased discussions to support prospective teacher
noticing Abstract: Publication date: Available online 17 December 2018Source: The Journal of Mathematical BehaviorAuthor(s): Alison Castro Superfine, Julie Amador, John Bragelman Research points to the importance of attending to and understanding children’s mathematical thinking as an important aspect of what teachers need to know. Teachers determine where children need to go next instructionally, in part, by interpreting children’s thinking, often drawing on different sources of information as evidence of children’s current level of understanding. Yet, much of the extant research on preservice teacher noticing, in particular, has studied attending to and interpreting as components of noticing, without attention to prospective teachers’ use of evidence to support their noticing of children’s thinking. In this study, we examine the facilitation moves that seemed to support prospective teachers in providing evidence of their noticing of children’s mathematical thinking. We conclude with the implications of our work for facilitating videobased discussions of children’s mathematical thinking with prospective teachers.
 Abstract: Publication date: Available online 17 December 2018Source: The Journal of Mathematical BehaviorAuthor(s): Alison Castro Superfine, Julie Amador, John Bragelman Research points to the importance of attending to and understanding children’s mathematical thinking as an important aspect of what teachers need to know. Teachers determine where children need to go next instructionally, in part, by interpreting children’s thinking, often drawing on different sources of information as evidence of children’s current level of understanding. Yet, much of the extant research on preservice teacher noticing, in particular, has studied attending to and interpreting as components of noticing, without attention to prospective teachers’ use of evidence to support their noticing of children’s thinking. In this study, we examine the facilitation moves that seemed to support prospective teachers in providing evidence of their noticing of children’s mathematical thinking. We conclude with the implications of our work for facilitating videobased discussions of children’s mathematical thinking with prospective teachers.
 Students’ conceptions through the lens of a dynamic online geometry
assessment platform Abstract: Publication date: Available online 15 December 2018Source: The Journal of Mathematical BehaviorAuthor(s): Yael Luz, Michal Yerushalmy Despite the widespread use of dynamic geometry environments (DGEs) in mathematics classrooms, and despite the emergence of online assessment systems, DGEs still play only a minor role in assessment practices. In this paper, we examined the potential benefits and feasibility of using information gleaned from students' interactions with geometry interactive diagrams (GIDs) through the lens of an online assessment platform. We introduced two assessment design patterns: the first was designed to assess the comprehension of terms and concepts; the second, to assess comprehension of the logical status of statements. We present data collected from 63 middle school students who performed several tasks based on these design patterns. We demonstrate how GIDs in both design patterns were used to assess and categorize conceptions, problem solving approaches, and tendencies that can act as a barrier to geometry proving. Its automatic analysis was beneficial and revealed some of these categories. We envision that embedding GIDs in an online assessment platform promotes the use of DGEs in the classroom and in assessment procedures.
 Abstract: Publication date: Available online 15 December 2018Source: The Journal of Mathematical BehaviorAuthor(s): Yael Luz, Michal Yerushalmy Despite the widespread use of dynamic geometry environments (DGEs) in mathematics classrooms, and despite the emergence of online assessment systems, DGEs still play only a minor role in assessment practices. In this paper, we examined the potential benefits and feasibility of using information gleaned from students' interactions with geometry interactive diagrams (GIDs) through the lens of an online assessment platform. We introduced two assessment design patterns: the first was designed to assess the comprehension of terms and concepts; the second, to assess comprehension of the logical status of statements. We present data collected from 63 middle school students who performed several tasks based on these design patterns. We demonstrate how GIDs in both design patterns were used to assess and categorize conceptions, problem solving approaches, and tendencies that can act as a barrier to geometry proving. Its automatic analysis was beneficial and revealed some of these categories. We envision that embedding GIDs in an online assessment platform promotes the use of DGEs in the classroom and in assessment procedures.
 Matrix multiplication and transformations: an APOS approach
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Ana Paulina Figueroa, Edgar Possani, María Trigueros This study presents a contribution to research in undergraduate teaching and learning of linear algebra, in particular, the learning of matrix multiplication. A didactical experience consisting on a modeling situation and a didactical sequence to guide students’ work on the situation were designed and tested using APOS theory. We show results of research on students’ activity and learning while using the sequence and through analysis of student’s work and assessment questions. The didactic sequence proved to have potential to foster students’ learning of function, matrix transformations and matrix multiplication. A detailed analysis of those constructions that seem to be essential for students understanding of this topic including linear transformations is presented. These results are contributions of this study to the literature.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Ana Paulina Figueroa, Edgar Possani, María Trigueros This study presents a contribution to research in undergraduate teaching and learning of linear algebra, in particular, the learning of matrix multiplication. A didactical experience consisting on a modeling situation and a didactical sequence to guide students’ work on the situation were designed and tested using APOS theory. We show results of research on students’ activity and learning while using the sequence and through analysis of student’s work and assessment questions. The didactic sequence proved to have potential to foster students’ learning of function, matrix transformations and matrix multiplication. A detailed analysis of those constructions that seem to be essential for students understanding of this topic including linear transformations is presented. These results are contributions of this study to the literature.
 Difficult dialogs about degenerate cases: A proof script study
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Stacy Brown Contradictions play an important role in mathematics. Yet, much remains unknown about the ways undergraduate mathematics students’ attend to contradictions, especially outside of proofs by contradiction. To address this gap in the literature, 20 proof scripts were analyzed to explore students’ noticed proof problematics for a proof involving a logically degenerate case. Findings indicate that while students generally attended to the encountered contradiction as a problematic point, many struggled to address its role in the proof and held conceptions of proofs by cases that enabled them to dismiss the logically degenerate case rather than recognize an inconsistency in the emerging mathematical theory. Findings from the study not only support calls for further research on students’ reasoning about proofs by cases and the role of contradictions but also provide evidence of the viability of the proof script methodology as a mechanism for identifying difficulties observed by students but unseen by experts.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Stacy Brown Contradictions play an important role in mathematics. Yet, much remains unknown about the ways undergraduate mathematics students’ attend to contradictions, especially outside of proofs by contradiction. To address this gap in the literature, 20 proof scripts were analyzed to explore students’ noticed proof problematics for a proof involving a logically degenerate case. Findings indicate that while students generally attended to the encountered contradiction as a problematic point, many struggled to address its role in the proof and held conceptions of proofs by cases that enabled them to dismiss the logically degenerate case rather than recognize an inconsistency in the emerging mathematical theory. Findings from the study not only support calls for further research on students’ reasoning about proofs by cases and the role of contradictions but also provide evidence of the viability of the proof script methodology as a mechanism for identifying difficulties observed by students but unseen by experts.

ℙ n ℝ +in+a+technologyassisted+learning+environment&rft.title=Journal+of+Mathematical+Behavior&rft.issn=07323123&rft.date=&rft.volume=">Coordinating analytic and visual approaches: Math majors’ understanding
of orthogonal Hermite polynomials in the inner product space ℙ n ℝ
in a technologyassisted learning environment Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Günhan Caglayan The purpose of this research study was to understand how linear algebra students in a university in the United States make sense of the orthogonal Hermite polynomials as vectors of the inner product space ℙnℝ in a Dynamic Geometry Software (DGS) – MATLAB facilitated learning environment. Math majors came up with a diversity of innovative and creative ways in which they coordinated visual and analytic approaches (Zazkis, Dubinsky, & Dautermann, 1996) for visualizing inner products of Hermite polynomials along with other notions inherent in the inner product space, such as Triangle Inequality, Pythagorean Theorem, Parallelogram Law, Orthogonality and Orthonormality, Coordinates Relative to an Orthonormal Basis. Research participants not only produced such creative inner product space visualizations of the Hermite polynomials with the induced improper integral inner product 〈f,g〉=∫−∞∞e−x2f(x)g(x)dx on the DGS, but they also verified their findings both analytically and visually in coordination. The paper concludes by offering pedagogical implications along with implications for mathematics teaching profession and recommendations for future research.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Günhan Caglayan The purpose of this research study was to understand how linear algebra students in a university in the United States make sense of the orthogonal Hermite polynomials as vectors of the inner product space ℙnℝ in a Dynamic Geometry Software (DGS) – MATLAB facilitated learning environment. Math majors came up with a diversity of innovative and creative ways in which they coordinated visual and analytic approaches (Zazkis, Dubinsky, & Dautermann, 1996) for visualizing inner products of Hermite polynomials along with other notions inherent in the inner product space, such as Triangle Inequality, Pythagorean Theorem, Parallelogram Law, Orthogonality and Orthonormality, Coordinates Relative to an Orthonormal Basis. Research participants not only produced such creative inner product space visualizations of the Hermite polynomials with the induced improper integral inner product 〈f,g〉=∫−∞∞e−x2f(x)g(x)dx on the DGS, but they also verified their findings both analytically and visually in coordination. The paper concludes by offering pedagogical implications along with implications for mathematics teaching profession and recommendations for future research.
 Developing and refining a framework for mathematical and linguistic
complexity in tasks related to rates of change Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): William Zahner, Hayley Milbourne, Lynda Wynn We present and develop a preliminary framework for describing the relationship between the mathematical and linguistic complexity of instructional tasks used in secondary mathematics. The initial framework was developed through a review of relevant literature. It was refined by examining how 4 ninth grade mathematics teachers of linguistically diverse groups of students described the linguistic and mathematical complexity of a set of tasks from their curriculum unit on linear functions. We close by presenting our refined framework for describing the interaction of linguistic complexity and mathematical complexity in curriculum materials, and discuss potential uses of this framework in the design of more accessible classroom learning environments for linguistically diverse students.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): William Zahner, Hayley Milbourne, Lynda Wynn We present and develop a preliminary framework for describing the relationship between the mathematical and linguistic complexity of instructional tasks used in secondary mathematics. The initial framework was developed through a review of relevant literature. It was refined by examining how 4 ninth grade mathematics teachers of linguistically diverse groups of students described the linguistic and mathematical complexity of a set of tasks from their curriculum unit on linear functions. We close by presenting our refined framework for describing the interaction of linguistic complexity and mathematical complexity in curriculum materials, and discuss potential uses of this framework in the design of more accessible classroom learning environments for linguistically diverse students.
 An investigation of 6th graders’ solutions of Cartesian product problems
and representation of these problems using arrays Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Erik S. Tillema Two hourlong interviews were conducted with each of 14 sixthgrade students. The purpose of the interviews was to investigate how students solved combinatorics problems, and represented their solutions as arrays. This paper reports on 11 of these students who represented a balanced mix of students operating with two of three multiplicative concepts that have been identified in prior research (Hackenberg, 2007, 2010; Hackenberg & Tillema, 2009). One finding of the study was that students operating with different multiplicative concepts established and structured pairs differently. A second finding is that these different ways of operating had implications for how students produced and used arrays. Overall, the findings contribute to models of students’ reasoning that outline the psychological operations that students use to constitute product of measures problems (Vergnaud, 1983). Product of measures problems are a kind of multiplicative problem that has unique mathematical properties, but researchers have not yet identified specific psychological operations that students use when solving these problems that differ from their solution of other kinds of multiplicative problems (cf. Battista, 2007).
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Erik S. Tillema Two hourlong interviews were conducted with each of 14 sixthgrade students. The purpose of the interviews was to investigate how students solved combinatorics problems, and represented their solutions as arrays. This paper reports on 11 of these students who represented a balanced mix of students operating with two of three multiplicative concepts that have been identified in prior research (Hackenberg, 2007, 2010; Hackenberg & Tillema, 2009). One finding of the study was that students operating with different multiplicative concepts established and structured pairs differently. A second finding is that these different ways of operating had implications for how students produced and used arrays. Overall, the findings contribute to models of students’ reasoning that outline the psychological operations that students use to constitute product of measures problems (Vergnaud, 1983). Product of measures problems are a kind of multiplicative problem that has unique mathematical properties, but researchers have not yet identified specific psychological operations that students use when solving these problems that differ from their solution of other kinds of multiplicative problems (cf. Battista, 2007).
 Further elaboration of the Learning Through Activity research program
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin A. Simon This article was prompted by the thoughtful commentaries of Norton, Tzur, and Dreyfus. Their commentaries pointed out important ideas that were left implicit or only partially explained. The clarifications made here include how the Learning Through Activity (LTA) research program differs from related research programs, the relation of the LTA theoretical framework to scheme theory, our choice to not employ the construct of perturbation in explaining learning, and the structure of hypothetical learning trajectories. In addition, I discuss a type of mathematical concept that we have not discussed previously.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin A. Simon This article was prompted by the thoughtful commentaries of Norton, Tzur, and Dreyfus. Their commentaries pointed out important ideas that were left implicit or only partially explained. The clarifications made here include how the Learning Through Activity (LTA) research program differs from related research programs, the relation of the LTA theoretical framework to scheme theory, our choice to not employ the construct of perturbation in explaining learning, and the structure of hypothetical learning trajectories. In addition, I discuss a type of mathematical concept that we have not discussed previously.
 Simon’s team’s contributions to scientific progress in mathematics
education: A commentary on the Learning Through Activity (LTA) research
program Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Ron Tzur I discuss two ways in which the Learning Through Activity (LTA) research program contributes to scientific progress in mathematics education: (a) providing general and contentspecific constructs to explain conceptual learning and instructional design that corroborate and/or elaborate on previous work and (b) raising new questions/issues. The general constructs include using instructional design as testable models of learning and using theoretical constructs to guide realtime, instructional adaptations. In this sense, the general constructs promote understanding of linkages between conceptual learning and instruction in mathematics. The conceptspecific constructs consist of empiricallygrounded, hypothetical learning trajectories (HLTs) for fractional and multiplicative reasoning. Each HLT consists of specific, intended conceptual changes and tasks that can bring them forth. Questions raised for me by the LTA work involve inconsistencies between the stance on learning and reported teachinglearning interactions that effectively led to students’ abstraction of the intended mathematical concepts.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Ron Tzur I discuss two ways in which the Learning Through Activity (LTA) research program contributes to scientific progress in mathematics education: (a) providing general and contentspecific constructs to explain conceptual learning and instructional design that corroborate and/or elaborate on previous work and (b) raising new questions/issues. The general constructs include using instructional design as testable models of learning and using theoretical constructs to guide realtime, instructional adaptations. In this sense, the general constructs promote understanding of linkages between conceptual learning and instruction in mathematics. The conceptspecific constructs consist of empiricallygrounded, hypothetical learning trajectories (HLTs) for fractional and multiplicative reasoning. Each HLT consists of specific, intended conceptual changes and tasks that can bring them forth. Questions raised for me by the LTA work involve inconsistencies between the stance on learning and reported teachinglearning interactions that effectively led to students’ abstraction of the intended mathematical concepts.
 Frameworks for modeling students’ mathematics
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Anderson Norton This commentary addresses the role of theoretical frameworks in building models of students’ mathematics. Specifically, it compares ways that the Learning Through Activity framework (LTA) and scheme theory explain and predict students’ mathematical activity. Both frameworks rely on Piagetian constructs—especially reflective abstraction—to build explanatory models for teaching and learning. LTA attempts to provide the teacherresearcher with a greater degree of determination in student learning trajectories, but then the teacherresearcher must address constraints in the students’ available ways of operating. These issues are exemplified in the case of teaching students about multiplying fractions. Additional theoretical issues arise in explaining logical necessity in students’ ways of operating and the role of reflective abstraction in organizing new ways of operating.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Anderson Norton This commentary addresses the role of theoretical frameworks in building models of students’ mathematics. Specifically, it compares ways that the Learning Through Activity framework (LTA) and scheme theory explain and predict students’ mathematical activity. Both frameworks rely on Piagetian constructs—especially reflective abstraction—to build explanatory models for teaching and learning. LTA attempts to provide the teacherresearcher with a greater degree of determination in student learning trajectories, but then the teacherresearcher must address constraints in the students’ available ways of operating. These issues are exemplified in the case of teaching students about multiplying fractions. Additional theoretical issues arise in explaining logical necessity in students’ ways of operating and the role of reflective abstraction in organizing new ways of operating.
 Empiricallybased hypothetical learning trajectories for fraction
concepts: Products of the Learning Through Activity research program Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin A. Simon, Nicora Placa, Melike Kara, Arnon Avitzur The Measurement Approach to Rational Number (MARN) Project, a project of the ongoing Learning Through Activity (LTA) research program, produced eleven hypothetical learning trajectories (HLTs) for promoting fraction concepts. Four of these HLTs are the subject of research reports. In this article, we present the other seven HLTs We judged that the data and analyses of these seven would not separately make sufficient contributions to merit individual research reports. However, presenting these seven HLTs together was intended to meet the following goals:1. To give a broad set of examples of HLTs developed based on the LTA theoretical framework.2. To complete a set of HLTs that provide a comprehensive example of HLTs built on prior HLTs.3. To make available for future research and development the full set of HLTs generated by the MARN Project.LTA researchers have focused on how learners abstract a concept through their mathematical activity and how the abstractions can be promoted. The MARN Project continued this inquiry with rigorous singlesubject teaching experiments.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin A. Simon, Nicora Placa, Melike Kara, Arnon Avitzur The Measurement Approach to Rational Number (MARN) Project, a project of the ongoing Learning Through Activity (LTA) research program, produced eleven hypothetical learning trajectories (HLTs) for promoting fraction concepts. Four of these HLTs are the subject of research reports. In this article, we present the other seven HLTs We judged that the data and analyses of these seven would not separately make sufficient contributions to merit individual research reports. However, presenting these seven HLTs together was intended to meet the following goals:1. To give a broad set of examples of HLTs developed based on the LTA theoretical framework.2. To complete a set of HLTs that provide a comprehensive example of HLTs built on prior HLTs.3. To make available for future research and development the full set of HLTs generated by the MARN Project.LTA researchers have focused on how learners abstract a concept through their mathematical activity and how the abstractions can be promoted. The MARN Project continued this inquiry with rigorous singlesubject teaching experiments.
 Promoting reinvention of a multiplicationoffractions algorithm: A study
of the Learning Through Activity research program Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin A. Simon, Melike Kara, Nicora Placa Whereas proficiency in performing the canonic multiplicationoffractions algorithm is common, understanding of the algorithm is much less so. We conducted a teaching experiment with a fifthgrade student, based on an initial hypothetical learning trajectory (HLT), to promote reinvention of the multiplicationoffractions algorithm. The instructional intervention built on two concepts, recursive partitioning and distributive partitioning. As a study of the Learning Through Activity research program, our goal was to promote particular activity on the part of the student through which she could abstract the necessary concepts. The results of the teaching experiment were analyzed and, based on conclusions from the research, a revised HLT was generated. Recursive partitioning and distributive partitioning proved to be a strong foundation for construction of the algorithm.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin A. Simon, Melike Kara, Nicora Placa Whereas proficiency in performing the canonic multiplicationoffractions algorithm is common, understanding of the algorithm is much less so. We conducted a teaching experiment with a fifthgrade student, based on an initial hypothetical learning trajectory (HLT), to promote reinvention of the multiplicationoffractions algorithm. The instructional intervention built on two concepts, recursive partitioning and distributive partitioning. As a study of the Learning Through Activity research program, our goal was to promote particular activity on the part of the student through which she could abstract the necessary concepts. The results of the teaching experiment were analyzed and, based on conclusions from the research, a revised HLT was generated. Recursive partitioning and distributive partitioning proved to be a strong foundation for construction of the algorithm.
 Fostering construction of a meaning for multiplication that subsumes
wholenumber and fraction multiplication: A study of the Learning Through
Activity research program Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin A. Simon, Melike Kara, Anderson Norton, Nicora Placa We report on a teaching experiment intended to foster a concept of multiplication that would both subsume students’ multiplegroups concept of whole number multiplication and provide a conceptual basis for understanding multiplication of fractions. The teaching experiment, which used a rigorous singlesubject methodology, began with an attempt to build on students’ multiplegroups concept by promoting generalizing assimilation. This was not totally successful and led to a redesign aimed at promoting reflective abstraction. Analysis of this latter phase led to several significant conclusions, which in turn led to a revised hypothetical learning trajectory. The revised trajectory aims to foster a concept of multiplication as a change in units.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin A. Simon, Melike Kara, Anderson Norton, Nicora Placa We report on a teaching experiment intended to foster a concept of multiplication that would both subsume students’ multiplegroups concept of whole number multiplication and provide a conceptual basis for understanding multiplication of fractions. The teaching experiment, which used a rigorous singlesubject methodology, began with an attempt to build on students’ multiplegroups concept by promoting generalizing assimilation. This was not totally successful and led to a redesign aimed at promoting reflective abstraction. Analysis of this latter phase led to several significant conclusions, which in turn led to a revised hypothetical learning trajectory. The revised trajectory aims to foster a concept of multiplication as a change in units.
 An empiricallybased trajectory for fostering abstraction of
equivalentfraction concepts: A study of the Learning Through Activity
research program Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Melike Kara, Martin A. Simon, Nicora Placa Promoting deep understanding of equivalentfractions has proved problematic. Using a oneonone teaching experiment, we investigated the development of an increasingly sophisticated, sequentially organized set of abstractions for equivalent fractions. The article describes the initial hypothetical learning trajectory (HLT) which built on the concept of recursive partitioning (anticipation of the results of taking a unit fraction of a unit fraction), analysis of the empirical study, conclusions, and the resulting revised HLT (based on the conclusions). Whereas recursive partitioning proved to provide a strong conceptual foundation, the analysis revealed a need for more effective ways of promoting reversibility of concepts. The revised HLT reflects an approach to promoting reversibility derived from the empirical and theoretical work of the researchers.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Melike Kara, Martin A. Simon, Nicora Placa Promoting deep understanding of equivalentfractions has proved problematic. Using a oneonone teaching experiment, we investigated the development of an increasingly sophisticated, sequentially organized set of abstractions for equivalent fractions. The article describes the initial hypothetical learning trajectory (HLT) which built on the concept of recursive partitioning (anticipation of the results of taking a unit fraction of a unit fraction), analysis of the empirical study, conclusions, and the resulting revised HLT (based on the conclusions). Whereas recursive partitioning proved to provide a strong conceptual foundation, the analysis revealed a need for more effective ways of promoting reversibility of concepts. The revised HLT reflects an approach to promoting reversibility derived from the empirical and theoretical work of the researchers.
 Learning Through Activity – Basic research on mathematical cognition
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Tommy Dreyfus Simon and colleagues propose a line of pure basic research in what is fundamentally an applied domain, mathematics education. Learning Through Activity (LTA) has all the advantages and many of the disadvantages that a pure theory in an applied domain is likely to have. The seven papers in this special issue aptly demonstrate many of the advantages. An additional advantage is that explicating the theory, as they do, raises many important questions and issues. In this commentary, I discuss some of these questions and issues concerning both, the pure research and its potential applicability to real classrooms.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Tommy Dreyfus Simon and colleagues propose a line of pure basic research in what is fundamentally an applied domain, mathematics education. Learning Through Activity (LTA) has all the advantages and many of the disadvantages that a pure theory in an applied domain is likely to have. The seven papers in this special issue aptly demonstrate many of the advantages. An additional advantage is that explicating the theory, as they do, raises many important questions and issues. In this commentary, I discuss some of these questions and issues concerning both, the pure research and its potential applicability to real classrooms.
 Promoting a concept of fractionasmeasure: A study of the Learning
Through Activity research program Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin A. Simon, Nicora Placa, Arnon Avitzur, Melike Kara Promoting deep understanding of fraction concepts continues to be a challenge for mathematics education. Research has demonstrated that students whose concept of fractions is limited to partwhole have difficulty with advanced fraction concepts. We conducted teaching experiments to study how students can develop a measurement concept of fractions and how task sequences can be developed to promote the necessary abstractions. Building particularly on the work of Steffe and colleagues and aspects of the ElkoninDavydov curriculum, we focused on fostering student reinvention of a measurement concept of fractions. As a study of the Learning Through Activity research program, our goal was to promote particular activity on the part of the students through which they could abstract the necessary concepts.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin A. Simon, Nicora Placa, Arnon Avitzur, Melike Kara Promoting deep understanding of fraction concepts continues to be a challenge for mathematics education. Research has demonstrated that students whose concept of fractions is limited to partwhole have difficulty with advanced fraction concepts. We conducted teaching experiments to study how students can develop a measurement concept of fractions and how task sequences can be developed to promote the necessary abstractions. Building particularly on the work of Steffe and colleagues and aspects of the ElkoninDavydov curriculum, we focused on fostering student reinvention of a measurement concept of fractions. As a study of the Learning Through Activity research program, our goal was to promote particular activity on the part of the students through which they could abstract the necessary concepts.
 An emerging methodology for studying mathematics concept learning and
instructional design Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin A. Simon Our microgenetic research methodology was developed for two purposes: 1) to develop greater understanding of conceptual learning and instructional design that promotes conceptual learning and 2) to develop empiricallybased hypothetical learning trajectories for particular mathematical concepts. The challenge in developing the methodology was to afford analysis of the learning process (e.g., the transition), not just identification of conceptual steps through which learners progress. The methodology, developed by the Learning Through Activity research program, was based on constructivist teaching experiment methodology. Modifications were made to data collection for the purposes outlined and a multilevel retrospective analysis was developed.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin A. Simon Our microgenetic research methodology was developed for two purposes: 1) to develop greater understanding of conceptual learning and instructional design that promotes conceptual learning and 2) to develop empiricallybased hypothetical learning trajectories for particular mathematical concepts. The challenge in developing the methodology was to afford analysis of the learning process (e.g., the transition), not just identification of conceptual steps through which learners progress. The methodology, developed by the Learning Through Activity research program, was based on constructivist teaching experiment methodology. Modifications were made to data collection for the purposes outlined and a multilevel retrospective analysis was developed.
 Towards an integrated theory of mathematics conceptual learning and
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin A. Simon, Melike Kara, Nicora Placa, Arnon Avitzur We discuss the theoretical framework of the Learning Through Activity research program. The framework includes an elaboration of the construct of mathematical concept, an elaboration of Piaget’s reflective abstraction for the purpose of mathematics pedagogy, further development of a distinction between two stages of conceptual learning, and a typology of different reverse concepts. The framework also involves instructional design principles built on those constructs, including steps for the design of task sequences, development of guided reinvention, and ways of promoting reversibility of concepts. This article represents both a synthesis of prior work and additions to it.
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin A. Simon, Melike Kara, Nicora Placa, Arnon Avitzur We discuss the theoretical framework of the Learning Through Activity research program. The framework includes an elaboration of the construct of mathematical concept, an elaboration of Piaget’s reflective abstraction for the purpose of mathematics pedagogy, further development of a distinction between two stages of conceptual learning, and a typology of different reverse concepts. The framework also involves instructional design principles built on those constructs, including steps for the design of task sequences, development of guided reinvention, and ways of promoting reversibility of concepts. This article represents both a synthesis of prior work and additions to it.
 Introduction
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin Simon
 Abstract: Publication date: December 2018Source: The Journal of Mathematical Behavior, Volume 52Author(s): Martin Simon
 Variability in the formal and informal content instructors convey in
lectures Abstract: Publication date: Available online 28 November 2018Source: The Journal of Mathematical BehaviorAuthor(s): Alon Pinto This paper investigates the variability in the content of lectures by different university mathematics instructors teaching concurrently in the same course. The instructors enacted a shared lessonplan and coordinated their teaching to ensure students in different sections of the course will receive equivalent academic preparation. However, classroom observations and interviews with the instructors indicate that while explicitly discussing the same definitions and theorems and working out similar examples, the instructors were implicitly trying to convey different informal content. An analysis of the instructors’ mathematical discourse during the lectures and the interviews reveals that the instructors coordinated with each other content corresponding with objectlevel learning, while implicitly orienting their lectures towards fostering metalevel learning. This found gap highlights miscommunication between the instructors and points to the need of a shared and explicit discourse on metalevel learning at the collegiate level.
 Abstract: Publication date: Available online 28 November 2018Source: The Journal of Mathematical BehaviorAuthor(s): Alon Pinto This paper investigates the variability in the content of lectures by different university mathematics instructors teaching concurrently in the same course. The instructors enacted a shared lessonplan and coordinated their teaching to ensure students in different sections of the course will receive equivalent academic preparation. However, classroom observations and interviews with the instructors indicate that while explicitly discussing the same definitions and theorems and working out similar examples, the instructors were implicitly trying to convey different informal content. An analysis of the instructors’ mathematical discourse during the lectures and the interviews reveals that the instructors coordinated with each other content corresponding with objectlevel learning, while implicitly orienting their lectures towards fostering metalevel learning. This found gap highlights miscommunication between the instructors and points to the need of a shared and explicit discourse on metalevel learning at the collegiate level.
 A disability studies in mathematics education review of intellectual
disabilities: Directions for future inquiry and practice Abstract: Publication date: Available online 22 November 2018Source: The Journal of Mathematical BehaviorAuthor(s): Paulo Tan, Rachel Lambert, Alexis Padilla, Rob Wieman We employ a Disability Studies in Mathematics Education perspective to explore current research involving students with intellectual disabilities. Such perspective affords a critical and socialcultural angle into the construction of disability in mathematics education. Results of our exploration suggest that current research on students with intellectual disabilities largely focused on finding deficits and neglected students’ capacity for abstract thought. Moreover, while sociocultural or sociopolitical (e.g., equitybased) foci are prevalent in contemporary mathematics education, such foci were largely absent in research involving students with intellectual disabilities. Yet, we identify features of several articles that can inform future sociopolitical mathematics education research related to individuals with intellectual disabilities. These features are critical for the continual advancement of new knowledge, equitable practices, and global, inclusive education agenda.
 Abstract: Publication date: Available online 22 November 2018Source: The Journal of Mathematical BehaviorAuthor(s): Paulo Tan, Rachel Lambert, Alexis Padilla, Rob Wieman We employ a Disability Studies in Mathematics Education perspective to explore current research involving students with intellectual disabilities. Such perspective affords a critical and socialcultural angle into the construction of disability in mathematics education. Results of our exploration suggest that current research on students with intellectual disabilities largely focused on finding deficits and neglected students’ capacity for abstract thought. Moreover, while sociocultural or sociopolitical (e.g., equitybased) foci are prevalent in contemporary mathematics education, such foci were largely absent in research involving students with intellectual disabilities. Yet, we identify features of several articles that can inform future sociopolitical mathematics education research related to individuals with intellectual disabilities. These features are critical for the continual advancement of new knowledge, equitable practices, and global, inclusive education agenda.
 Students’ reasons for introducing auxiliary lines in proving
situations Abstract: Publication date: Available online 9 November 2018Source: The Journal of Mathematical BehaviorAuthor(s): Alik Palatnik, Tommy Dreyfus This paper focuses on reasons students’ give while introducing auxiliary lines in geometry proofs. Three cases of pairs of high school students participating in proving activities are presented. The overarching theme of the tasks proposed to the students is the comparison of areas of triangles and/or parallelograms. The students gave a wide spectrum of reasons when introducing an auxiliary line. Two main groups of reasons are discerned: The students introduced auxiliary lines recalling some known results or definitions and modified the given diagrams accordingly, as a part of a learned procedure. The students introduced auxiliary lines anticipating to receive more information from a modified situation. Students may combine reasons of recalling and anticipating nature when introducing an auxiliary line.
 Abstract: Publication date: Available online 9 November 2018Source: The Journal of Mathematical BehaviorAuthor(s): Alik Palatnik, Tommy Dreyfus This paper focuses on reasons students’ give while introducing auxiliary lines in geometry proofs. Three cases of pairs of high school students participating in proving activities are presented. The overarching theme of the tasks proposed to the students is the comparison of areas of triangles and/or parallelograms. The students gave a wide spectrum of reasons when introducing an auxiliary line. Two main groups of reasons are discerned: The students introduced auxiliary lines recalling some known results or definitions and modified the given diagrams accordingly, as a part of a learned procedure. The students introduced auxiliary lines anticipating to receive more information from a modified situation. Students may combine reasons of recalling and anticipating nature when introducing an auxiliary line.
 Adolescent perceptions of being known in the mathematics classroom
 Abstract: Publication date: Available online 6 November 2018Source: The Journal of Mathematical BehaviorAuthor(s): Tanner LeBaron Wallace, Charles Munter This study expands prior research on mathematics classrooms by investigating students’ (N = 373, 59% selfidentified as a student of color) perceptions of their interactions with their current mathematics teacher, and appraisals of value, expectancy, and cost of participation and experiences with race and mathematics. Our analyses revealed students’ perceptions of how well their teacher knew them were related to all four of those variables—and that relations to selfreported value of mathematics and experiences with race and mathematics were stronger for students of color than for white students. Results suggest assessing adolescents’ perceptions of being known in studies of mathematical learning may explain variability in making meaning of experience, particularly for students of color.
 Abstract: Publication date: Available online 6 November 2018Source: The Journal of Mathematical BehaviorAuthor(s): Tanner LeBaron Wallace, Charles Munter This study expands prior research on mathematics classrooms by investigating students’ (N = 373, 59% selfidentified as a student of color) perceptions of their interactions with their current mathematics teacher, and appraisals of value, expectancy, and cost of participation and experiences with race and mathematics. Our analyses revealed students’ perceptions of how well their teacher knew them were related to all four of those variables—and that relations to selfreported value of mathematics and experiences with race and mathematics were stronger for students of color than for white students. Results suggest assessing adolescents’ perceptions of being known in studies of mathematical learning may explain variability in making meaning of experience, particularly for students of color.
 The complex interplay between examples and proving: Where are we and where
should we head' Abstract: Publication date: Available online 27 October 2018Source: The Journal of Mathematical BehaviorAuthor(s): Orit Zaslavsky, Eric Knuth
 Abstract: Publication date: Available online 27 October 2018Source: The Journal of Mathematical BehaviorAuthor(s): Orit Zaslavsky, Eric Knuth
 Valuethinking and locationthinking: Two ways students visualize points
and think about graphs Abstract: Publication date: Available online 26 October 2018Source: The Journal of Mathematical BehaviorAuthor(s): Erika J. David, Kyeong Hah Roh, Morgan E. Sellers The purpose of this study is to examine the characteristics of students’ thinking about aspects of graphs in the context of evaluating statements about realvalued functions from Calculus. We conducted clinical interviews in which undergraduate students evaluated mathematical statements using graphs to explain their reasoning. From our data analysis, we found two ways students think about graphs, valuethinking and locationthinking. These two ways of thinking were rooted in students’ attention to different attributes of points on graphs we provided: either the values represented by the points or the locations of the points in space. In this paper, we report our classification of students’ thinking about aspects of graphs in terms of valuethinking and locationthinking. Our findings indicate that students’ thinking about aspects of graphs accounts for key differences in their understandings of mathematical statements. We discuss some implications of our findings for instruction and curriculum development in Calculus and beyond.
 Abstract: Publication date: Available online 26 October 2018Source: The Journal of Mathematical BehaviorAuthor(s): Erika J. David, Kyeong Hah Roh, Morgan E. Sellers The purpose of this study is to examine the characteristics of students’ thinking about aspects of graphs in the context of evaluating statements about realvalued functions from Calculus. We conducted clinical interviews in which undergraduate students evaluated mathematical statements using graphs to explain their reasoning. From our data analysis, we found two ways students think about graphs, valuethinking and locationthinking. These two ways of thinking were rooted in students’ attention to different attributes of points on graphs we provided: either the values represented by the points or the locations of the points in space. In this paper, we report our classification of students’ thinking about aspects of graphs in terms of valuethinking and locationthinking. Our findings indicate that students’ thinking about aspects of graphs accounts for key differences in their understandings of mathematical statements. We discuss some implications of our findings for instruction and curriculum development in Calculus and beyond.
 Undergraduate mathematics students’ athome exploration of a
proveordisprove task Abstract: Publication date: Available online 9 October 2018Source: The Journal of Mathematical BehaviorAuthor(s): Kristen Lew, Dov Zazkis A common genre of task in proofcentered mathematics courses involves prompting students to evaluate the veracity of a mathematical claim by either proving the claim, or providing a proof that the claim is false (disproving the claim). The way in which students interact with these proveordisprove tasks is not well understood. We examine students’ athome work as a way of learning about their processes of generating disproofs of claims. In particular, we study the interactions between example/counterexample generation activities, attempts to prove the (false) claim, and attempts to prove related results.
 Abstract: Publication date: Available online 9 October 2018Source: The Journal of Mathematical BehaviorAuthor(s): Kristen Lew, Dov Zazkis A common genre of task in proofcentered mathematics courses involves prompting students to evaluate the veracity of a mathematical claim by either proving the claim, or providing a proof that the claim is false (disproving the claim). The way in which students interact with these proveordisprove tasks is not well understood. We examine students’ athome work as a way of learning about their processes of generating disproofs of claims. In particular, we study the interactions between example/counterexample generation activities, attempts to prove the (false) claim, and attempts to prove related results.
 The plot thickens: The aesthetic dimensions of a captivating mathematics
lesson Abstract: Publication date: Available online 27 September 2018Source: The Journal of Mathematical BehaviorAuthor(s): Andrew S. Richman, Leslie Dietiker, Meghan Riling We present an analysis of a sixthgrade mathematics lesson in which an aestheticallyrich moment of mathematical surprise, inspired by a decontextualized integer addition problem, spurred students to ask mathematical questions and actively sustain inquiry into the lesson’s central ideas. In order to understand how the unfolding mathematical content enabled this moment, we interpret the lesson as a mathematical story. Using this narrative framework, we describe the aesthetic dimensions of the story including its plot, density, coherence, and rhythm, and connect them to the unfolding mathematical content. This analysis demonstrates how these aesthetic elements of a lesson can be recognized and how they help explain the students’ productive engagement. This framework offers a potential tool for researchers and practitioners who seek to understand, design, and enact captivating mathematical experiences.
 Abstract: Publication date: Available online 27 September 2018Source: The Journal of Mathematical BehaviorAuthor(s): Andrew S. Richman, Leslie Dietiker, Meghan Riling We present an analysis of a sixthgrade mathematics lesson in which an aestheticallyrich moment of mathematical surprise, inspired by a decontextualized integer addition problem, spurred students to ask mathematical questions and actively sustain inquiry into the lesson’s central ideas. In order to understand how the unfolding mathematical content enabled this moment, we interpret the lesson as a mathematical story. Using this narrative framework, we describe the aesthetic dimensions of the story including its plot, density, coherence, and rhythm, and connect them to the unfolding mathematical content. This analysis demonstrates how these aesthetic elements of a lesson can be recognized and how they help explain the students’ productive engagement. This framework offers a potential tool for researchers and practitioners who seek to understand, design, and enact captivating mathematical experiences.
 Key memorable events: A lens on affect, learning, and teaching in the
mathematics classroom Abstract: Publication date: Available online 27 September 2018Source: The Journal of Mathematical BehaviorAuthor(s): Ofer Marmur This paper proposes a theoretical construct termed Key Memorable Events (KMEs): classroom events that are perceived by many students as memorable and meaningful in support of their learning, and are typically accompanied by strong emotions, either positive or negative. Though the approach taken in this paper is theoretical, the construct is illustrated and supported by empirical data collected in the undergraduate mathematics classroom. I address the potential value of the construct, and demonstrate its fit with existing constructs. Theoretical and pedagogical implications are discussed in terms of lesson design, data analysis, and our conceptualization of student learning, whilst enriching the existing metaphor of learning as a trajectory.
 Abstract: Publication date: Available online 27 September 2018Source: The Journal of Mathematical BehaviorAuthor(s): Ofer Marmur This paper proposes a theoretical construct termed Key Memorable Events (KMEs): classroom events that are perceived by many students as memorable and meaningful in support of their learning, and are typically accompanied by strong emotions, either positive or negative. Though the approach taken in this paper is theoretical, the construct is illustrated and supported by empirical data collected in the undergraduate mathematics classroom. I address the potential value of the construct, and demonstrate its fit with existing constructs. Theoretical and pedagogical implications are discussed in terms of lesson design, data analysis, and our conceptualization of student learning, whilst enriching the existing metaphor of learning as a trajectory.
 How elementary and collegiate instructors envision tasks as supportive of
mathematical argumentation: A comparison of instructors’ task
constructions Abstract: Publication date: Available online 10 September 2018Source: The Journal of Mathematical BehaviorAuthor(s): Kimberly Cervello Rogers, Karl W. Kosko We examine how instructors at different grade levels proposed mathematical tasks to support students’ engagement in constructing viable arguments and critiquing others’ reasoning. During small group interviews with pairs of practicing elementary and collegiate mathematics instructors, they created tasks intended to support students’ ability to create and critique mathematical arguments. From the anslysis of their written and verbal work, we found that although all instructors requested explanations in their created tasks, there were key differences in the nature of reasoningandproving expected. Grade 1 instructors tended to focus on empirical justifications and procedural explanation; Grade 3 and 4 instructors included similar requirements, but had more emphasis on soliciting rationales; and college instructors required students to ultimately develop more formalized arguments, consistently including requests for conjectures and generalizations. The findings provide evidence for key differences in argumentation norms at specific levels in schooling, and implications for research, teaching, and curricular design.
 Abstract: Publication date: Available online 10 September 2018Source: The Journal of Mathematical BehaviorAuthor(s): Kimberly Cervello Rogers, Karl W. Kosko We examine how instructors at different grade levels proposed mathematical tasks to support students’ engagement in constructing viable arguments and critiquing others’ reasoning. During small group interviews with pairs of practicing elementary and collegiate mathematics instructors, they created tasks intended to support students’ ability to create and critique mathematical arguments. From the anslysis of their written and verbal work, we found that although all instructors requested explanations in their created tasks, there were key differences in the nature of reasoningandproving expected. Grade 1 instructors tended to focus on empirical justifications and procedural explanation; Grade 3 and 4 instructors included similar requirements, but had more emphasis on soliciting rationales; and college instructors required students to ultimately develop more formalized arguments, consistently including requests for conjectures and generalizations. The findings provide evidence for key differences in argumentation norms at specific levels in schooling, and implications for research, teaching, and curricular design.
 Investigating the relationships among elementary teachers’ perceptions
of the use of students’ thinking, their professional noticing skills,
and their teaching practices Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Mi Yeon Lee, Dionne Cross Francis This study is an investigation of relationships among elementary teachers’ perceptions of the use of students’ thinking in instructional decisionmaking, their professional noticing skills, and their use of students’ thinking during instruction. Interviews were conducted with 33 participants using a twopart, semistructured protocol and 25 teachers’ instructional videos were collected. The data were analyzed using the Mathematical Quality of Instruction instrument and grounded theory techniques including open coding, identification of themes, and the development and description of categories. Preliminary findings suggest that there is a relationship between elementary teachers’ perceptions of the use of students’ thinking and their professional noticing skills, but misalignment was found between teachers’ perceptions of the use of students’ thinking and their practices as observed in videos of their own teaching. Implications are discussed for teacher knowledge and the design of effective professional development programs to encourage productive use of students’ thinking in lesson planning and teaching.
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Mi Yeon Lee, Dionne Cross Francis This study is an investigation of relationships among elementary teachers’ perceptions of the use of students’ thinking in instructional decisionmaking, their professional noticing skills, and their use of students’ thinking during instruction. Interviews were conducted with 33 participants using a twopart, semistructured protocol and 25 teachers’ instructional videos were collected. The data were analyzed using the Mathematical Quality of Instruction instrument and grounded theory techniques including open coding, identification of themes, and the development and description of categories. Preliminary findings suggest that there is a relationship between elementary teachers’ perceptions of the use of students’ thinking and their professional noticing skills, but misalignment was found between teachers’ perceptions of the use of students’ thinking and their practices as observed in videos of their own teaching. Implications are discussed for teacher knowledge and the design of effective professional development programs to encourage productive use of students’ thinking in lesson planning and teaching.
 The alignment of student fraction learning with textbooks in Korea and the
United States Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Jaehong Shin, Soo Jin Lee A variety of international comparison studies of textbooks have been conducted on the assumption that a mathematics textbook is a critical factor in preparing students for mathematical achievement. The idea suggests that mathematics textbooks be written in a way that promotes students’ cognitive developmental continuum of particular mathematical concepts. In the present study, we investigate how mathematics textbooks in Korea and the United States (in particular, Everyday Mathematics) entail students’ learning progressions in terms of three operations (recursive partitioning, common partitioning, and distributive partitioning) that we assert undergird students’ constructions of higher levels of fraction knowledge. Using the three operations and hierarchical relationships among them as an analytical framework, we select relevant topics with the operations and show how those topics are well aligned with our framework. We also discuss how the tasks included in the topics afford and constrain students’ development of coherent fraction learning in terms of the framework.
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Jaehong Shin, Soo Jin Lee A variety of international comparison studies of textbooks have been conducted on the assumption that a mathematics textbook is a critical factor in preparing students for mathematical achievement. The idea suggests that mathematics textbooks be written in a way that promotes students’ cognitive developmental continuum of particular mathematical concepts. In the present study, we investigate how mathematics textbooks in Korea and the United States (in particular, Everyday Mathematics) entail students’ learning progressions in terms of three operations (recursive partitioning, common partitioning, and distributive partitioning) that we assert undergird students’ constructions of higher levels of fraction knowledge. Using the three operations and hierarchical relationships among them as an analytical framework, we select relevant topics with the operations and show how those topics are well aligned with our framework. We also discuss how the tasks included in the topics afford and constrain students’ development of coherent fraction learning in terms of the framework.
 Exploring experiences for assisting primary preservice teachers to extend
their knowledge of student strategies and reasoning Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Sharyn Livy, Ann Downton Exploring how students learn mathematics and what types of tasks promote mathematical reasoning may assist preservice teachers to develop their pedagogical content knowledge (PCK) for teaching. This study reports on an experience of a cohort of Australian primary (elementary) preservice teachers, destined to become specialist mathematics teachers associated with the topics of measurement and geometry. The first author presented a lesson with Year 5/6 students related to geometric reasoning and calculating the size of angles. The lesson was launched with no instructions and students were expected to attempt a problem without using a protractor or help from the teacher. The preservice teachers, classroom teacher and second author were nonparticipant observers. Findings suggest course experiences provided an opportunity to extend preservice teachers’ knowledge of how students learn and their knowledge of lesson structure and PCK whilst also considering how students might learn to reason and solve a challenging mathematical task.
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Sharyn Livy, Ann Downton Exploring how students learn mathematics and what types of tasks promote mathematical reasoning may assist preservice teachers to develop their pedagogical content knowledge (PCK) for teaching. This study reports on an experience of a cohort of Australian primary (elementary) preservice teachers, destined to become specialist mathematics teachers associated with the topics of measurement and geometry. The first author presented a lesson with Year 5/6 students related to geometric reasoning and calculating the size of angles. The lesson was launched with no instructions and students were expected to attempt a problem without using a protractor or help from the teacher. The preservice teachers, classroom teacher and second author were nonparticipant observers. Findings suggest course experiences provided an opportunity to extend preservice teachers’ knowledge of how students learn and their knowledge of lesson structure and PCK whilst also considering how students might learn to reason and solve a challenging mathematical task.
 Supporting teachers in improving their knowledge of mathematics
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Peter Sullivan
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Peter Sullivan
 Teaching the language of mathematics: What the research tells us teachers
need to know and do Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Louise C. Wilkinson The focus of this paper is on what research suggests primary school teachers need to know about the specialized language of mathematics teaching and learning: the mathematics register. When teachers act effectively on this knowledge, students have multiple opportunities to construct mathematical understandings and to demonstrate what they know in the variety of tasks that are required by formal schooling. First, the national and policy context within which mathematics is taught and learned is considered. Secondly, the mathematics register is defined, followed by a consideration the significance for teaching, learning, and assessment in school; examples are provided. The educational implications of these analyses include guidelines for what teachers need to know and to do, so that their students appropriately utilize the mathematics register in the classroom. In so doing, students are provided with optimal circumstances to learn, where they may deploy all of their cognitivelinguistic resources to the particular mathematics tasks at hand.
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Louise C. Wilkinson The focus of this paper is on what research suggests primary school teachers need to know about the specialized language of mathematics teaching and learning: the mathematics register. When teachers act effectively on this knowledge, students have multiple opportunities to construct mathematical understandings and to demonstrate what they know in the variety of tasks that are required by formal schooling. First, the national and policy context within which mathematics is taught and learned is considered. Secondly, the mathematics register is defined, followed by a consideration the significance for teaching, learning, and assessment in school; examples are provided. The educational implications of these analyses include guidelines for what teachers need to know and to do, so that their students appropriately utilize the mathematics register in the classroom. In so doing, students are provided with optimal circumstances to learn, where they may deploy all of their cognitivelinguistic resources to the particular mathematics tasks at hand.
 A framework for identifying mathematical arguments as supported claims
created in daytoday classroom interactions Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): AnnaKarin Nordin, Lisa Björklund Boistrup This article addresses how to distinguish mathematical arguments created during whole class discussions in grades 3–5 in Sweden, while taking a broad range of communicational resources, such as speech, drawings and symbols, into account. We present a stepbystep framework of how to systematically reconstruct mathematical arguments. The framework is developed drawing on Toulmin’s model of argumentation and a multimodal approach. When giving account for the framework, we show how various communicational resources convey the mathematical meaning of the arguments created. The framework can be used for further research investigating interaction in classroom settings, for teacher students as a basis for reflection during practicum periods, as well as a lens for teachers in identifying informal and formal mathematical arguments in daytoday communication in the mathematics classroom.
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): AnnaKarin Nordin, Lisa Björklund Boistrup This article addresses how to distinguish mathematical arguments created during whole class discussions in grades 3–5 in Sweden, while taking a broad range of communicational resources, such as speech, drawings and symbols, into account. We present a stepbystep framework of how to systematically reconstruct mathematical arguments. The framework is developed drawing on Toulmin’s model of argumentation and a multimodal approach. When giving account for the framework, we show how various communicational resources convey the mathematical meaning of the arguments created. The framework can be used for further research investigating interaction in classroom settings, for teacher students as a basis for reflection during practicum periods, as well as a lens for teachers in identifying informal and formal mathematical arguments in daytoday communication in the mathematics classroom.
 Perceptions of social issues as contexts for secondary mathematics
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Ami Mamolo This paper investigates preservice secondary teachers’ perceptions of learning and teaching mathematics through extended explorations that are contextualized in issues of social importance. The study is situated within a research program concerned with mathematical knowledge used in, and useful for, teaching, and how such knowledge may be fostered in teacher education programs.
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Ami Mamolo This paper investigates preservice secondary teachers’ perceptions of learning and teaching mathematics through extended explorations that are contextualized in issues of social importance. The study is situated within a research program concerned with mathematical knowledge used in, and useful for, teaching, and how such knowledge may be fostered in teacher education programs.
 The role of linguistic features when reading and solving mathematics tasks
in different languages Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Ewa Bergqvist, Frithjof Theens, Magnus Österholm The purpose of this study is to deepen the understanding of the relation between the language used in mathematics tasks and the difficulty in reading and solving the tasks. We examine issues of language both through linguistic features of tasks (word length, sentence length, task length, and information density) and through different natural languages used to formulate the tasks (English, German, and Swedish). Analyses of 83 PISA mathematics tasks reveal that tasks in German, when compared with English and Swedish, show stronger connections between the examined linguistic features of tasks and difficulty in reading and solving the tasks. We discuss if and how this result can be explained by general differences between the three languages.
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Ewa Bergqvist, Frithjof Theens, Magnus Österholm The purpose of this study is to deepen the understanding of the relation between the language used in mathematics tasks and the difficulty in reading and solving the tasks. We examine issues of language both through linguistic features of tasks (word length, sentence length, task length, and information density) and through different natural languages used to formulate the tasks (English, German, and Swedish). Analyses of 83 PISA mathematics tasks reveal that tasks in German, when compared with English and Swedish, show stronger connections between the examined linguistic features of tasks and difficulty in reading and solving the tasks. We discuss if and how this result can be explained by general differences between the three languages.
 Preservice teacher proficiency with transformationsbased congruence
proofs after a college proofbased geometry class Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Meredith Hegg, Dimitri Papadopoulos, Brian Katz, Timothy FukawaConnelly This report explores preservice teachers’ proficiency with concepts of transformational geometry at the end of a semesterlong advanced geometry course. In the course, the instructor incorporated transformational geometry content, including congruence proofs, in an attempt to prepare the preservice teachers to teach high school geometry in alignment with the Common Core State Standards for Mathematics. At the conclusion of the course, students expressed a preference for using traditional triangle congruence criteria (SAS, ASA, SSS, and AAS) over using transformations to complete proofs, but were nevertheless generally successful in completing proofs using transformations. Similarly, while the students often described thinking of transformations in terms of analytic forms, they were successfully able to prove triangle congruences in synthetic contexts. Finally, some evidence indicates that students may have motion or process conceptions of transformations, but not map or object conceptions, but this evidence is not conclusive.
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Meredith Hegg, Dimitri Papadopoulos, Brian Katz, Timothy FukawaConnelly This report explores preservice teachers’ proficiency with concepts of transformational geometry at the end of a semesterlong advanced geometry course. In the course, the instructor incorporated transformational geometry content, including congruence proofs, in an attempt to prepare the preservice teachers to teach high school geometry in alignment with the Common Core State Standards for Mathematics. At the conclusion of the course, students expressed a preference for using traditional triangle congruence criteria (SAS, ASA, SSS, and AAS) over using transformations to complete proofs, but were nevertheless generally successful in completing proofs using transformations. Similarly, while the students often described thinking of transformations in terms of analytic forms, they were successfully able to prove triangle congruences in synthetic contexts. Finally, some evidence indicates that students may have motion or process conceptions of transformations, but not map or object conceptions, but this evidence is not conclusive.
 Primary mathematics teachers’ responses to students’ offers: An
‘elaboration’ framework Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Lawan Abdulhamid, Hamsa Venkat Responding constructively ‘inthe moment’ to student offers is described as a critical, and yet difficult, aspect of skilful and responsive teaching. South African evidence points to limited evaluation of student offers in schools serving poor communities. In this paper, we present and discuss an ‘elaboration’ framework emerging from a grounded analysis of data drawn from video recordings of 18 mathematics lessons prepared and conducted by four inservice primary school teachers in South Africa. This analysis led to a categorization of the situations in which teacher responses to student offers occurred, and the nature and range of these responses. Three response situations are identified within the framework: breakdown, sophistication, and individuation/collectivization, with a range of response (and nonresponse) categories in each situation. Literature on responsive feedback is drawn in to explore hierarchies and relationships between the emergent categories within situations of elaboration. The elaboration framework provides a tool for lesson observation, and a model for thinking about developments in responsive teaching.
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Lawan Abdulhamid, Hamsa Venkat Responding constructively ‘inthe moment’ to student offers is described as a critical, and yet difficult, aspect of skilful and responsive teaching. South African evidence points to limited evaluation of student offers in schools serving poor communities. In this paper, we present and discuss an ‘elaboration’ framework emerging from a grounded analysis of data drawn from video recordings of 18 mathematics lessons prepared and conducted by four inservice primary school teachers in South Africa. This analysis led to a categorization of the situations in which teacher responses to student offers occurred, and the nature and range of these responses. Three response situations are identified within the framework: breakdown, sophistication, and individuation/collectivization, with a range of response (and nonresponse) categories in each situation. Literature on responsive feedback is drawn in to explore hierarchies and relationships between the emergent categories within situations of elaboration. The elaboration framework provides a tool for lesson observation, and a model for thinking about developments in responsive teaching.
 Generating different lesson designs and analyzing their effects: The
impact of representations when discerning aspects of the derivative Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Ulf Ryberg This paper reports results that concern the way in which the design of instruction may influence students’ opportunities to discern the relationship between a graph and its derivative graph. Two studies were conducted that together included 144 Swedish uppersecondary students who were enrolled in an introductory calculus course. In both studies, all students participated in a 120min intervention. The first study used a qualitative approach and aimed at generating, and analyzing the outcomes of, three different lesson designs. The second study used a quantitative approach and aimed at testing the validity of the results from the first study. The results of the studies are compatible and suggest that teaching should initially focus exclusively on graphs and also include a variety of graphs. In contrast, using graphs in conjunction with formulas and/or using only graphs of polynomial functions decreased students’ opportunities to discern graphical aspects of the derivative.
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Ulf Ryberg This paper reports results that concern the way in which the design of instruction may influence students’ opportunities to discern the relationship between a graph and its derivative graph. Two studies were conducted that together included 144 Swedish uppersecondary students who were enrolled in an introductory calculus course. In both studies, all students participated in a 120min intervention. The first study used a qualitative approach and aimed at generating, and analyzing the outcomes of, three different lesson designs. The second study used a quantitative approach and aimed at testing the validity of the results from the first study. The results of the studies are compatible and suggest that teaching should initially focus exclusively on graphs and also include a variety of graphs. In contrast, using graphs in conjunction with formulas and/or using only graphs of polynomial functions decreased students’ opportunities to discern graphical aspects of the derivative.
 Using contextualized tasks to engage students in meaningful and worthwhile
mathematics learning Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Doug Clarke, Anne Roche A teacher’s choice and use of tasks are major determinants of the nature and quality of students’ learning. Teachers of 11–15 yearolds in a project called Task Types and Mathematics Learning used a range of contextualized tasks and reported affordances and disadvantages of the use of such tasks. We offer a rationale for the use of contextualized tasks, examples of tasks providing insightful student thinking, and teacher feedback on affordances and constraints of the use of these tasks. For three different types of tasks, students provided feedback on the relative extent to which they enjoyed, learned from, and found difficult each type of task, respectively. Finally, we report on a follow up project which studied teacher actions supporting persistence on cognitively demanding contextualized tasks. Findings inform our understanding of the teacher’s role in providing engaging and worthwhile mathematics for all students.
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Doug Clarke, Anne Roche A teacher’s choice and use of tasks are major determinants of the nature and quality of students’ learning. Teachers of 11–15 yearolds in a project called Task Types and Mathematics Learning used a range of contextualized tasks and reported affordances and disadvantages of the use of such tasks. We offer a rationale for the use of contextualized tasks, examples of tasks providing insightful student thinking, and teacher feedback on affordances and constraints of the use of these tasks. For three different types of tasks, students provided feedback on the relative extent to which they enjoyed, learned from, and found difficult each type of task, respectively. Finally, we report on a follow up project which studied teacher actions supporting persistence on cognitively demanding contextualized tasks. Findings inform our understanding of the teacher’s role in providing engaging and worthwhile mathematics for all students.
 Enhancing and analyzing kindergarten teachers’ professional knowledge
for early mathematics education Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Hedwig Gasteiger, Christiane Benz While early childhood mathematics education did not play an important role for many years, today the necessity to be attentive to children’s early mathematics learning is no longer in doubt. There is a broad consensus that mathematical learning in kindergarten is more informal than learning in school. To guarantee coherent mathematical learning in kindergarten, pre and inservice education of kindergarten teachers in mathematics education is becoming more important. Therefore, a sound theoretical foundation and examination of professional knowledge to realize early mathematics education are necessary. We present a theoretically and empirically founded domainspecific model of kindergarten teachers’ professional knowledge. It serves as a framework to plan and realize professional development programs – as an example shows. A qualitative study – worked out in the German context – indicates that this structure can help to analyze kindergarten teachers’ professional knowledge. We indicate how our findings and interpretations connect to issues shared across national contexts.
 Abstract: Publication date: September 2018Source: The Journal of Mathematical Behavior, Volume 51Author(s): Hedwig Gasteiger, Christiane Benz While early childhood mathematics education did not play an important role for many years, today the necessity to be attentive to children’s early mathematics learning is no longer in doubt. There is a broad consensus that mathematical learning in kindergarten is more informal than learning in school. To guarantee coherent mathematical learning in kindergarten, pre and inservice education of kindergarten teachers in mathematics education is becoming more important. Therefore, a sound theoretical foundation and examination of professional knowledge to realize early mathematics education are necessary. We present a theoretically and empirically founded domainspecific model of kindergarten teachers’ professional knowledge. It serves as a framework to plan and realize professional development programs – as an example shows. A qualitative study – worked out in the German context – indicates that this structure can help to analyze kindergarten teachers’ professional knowledge. We indicate how our findings and interpretations connect to issues shared across national contexts.
 Using mobile puzzles to exhibit certain algebraic habits of mind and
demonstrate symbolsense in primary school students Abstract: Publication date: Available online 31 August 2018Source: The Journal of Mathematical BehaviorAuthor(s): Ioannis Papadopoulos Given the growing concern for developing students’ algebraic ideas and thinking in earlier grades (NCTM, 2000) it is important for students to have experiences that better prepare them for their formal introduction to algebra. Mobile puzzles seem to be an opportunity for exhibiting certain algebraic habits of mind as well as for demonstrating symbolsense which might support students in their transition from arithmetic to algebra. These puzzles include multiple balanced collections of objects whose weights must be determined by the solver. The arms/beams must be perfectly balanced for it to hang properly. Therefore, they represent, in a pictorial way, systems of equations. Each arm/beam that balances two sets of objects (representing variables as unknown “weights”) represents an equation. The data derived from Grade6 students who were asked to solve a collection of tasks reflect the presence of the “Puzzling and Persevering” and “Seeking and Using Structure” habits of mind. At the same time these data incorporate instances of some main components of symbolsense such as “friendliness with symbols”, “manipulating and ‘reading through’ symbolic expressions”, and “choice of symbols”. Also discussed is the way this experience contributes to an intuitive application of the conventional rules for solving equations that will be later introduced to the students as the standard algebraic “moves”.
 Abstract: Publication date: Available online 31 August 2018Source: The Journal of Mathematical BehaviorAuthor(s): Ioannis Papadopoulos Given the growing concern for developing students’ algebraic ideas and thinking in earlier grades (NCTM, 2000) it is important for students to have experiences that better prepare them for their formal introduction to algebra. Mobile puzzles seem to be an opportunity for exhibiting certain algebraic habits of mind as well as for demonstrating symbolsense which might support students in their transition from arithmetic to algebra. These puzzles include multiple balanced collections of objects whose weights must be determined by the solver. The arms/beams must be perfectly balanced for it to hang properly. Therefore, they represent, in a pictorial way, systems of equations. Each arm/beam that balances two sets of objects (representing variables as unknown “weights”) represents an equation. The data derived from Grade6 students who were asked to solve a collection of tasks reflect the presence of the “Puzzling and Persevering” and “Seeking and Using Structure” habits of mind. At the same time these data incorporate instances of some main components of symbolsense such as “friendliness with symbols”, “manipulating and ‘reading through’ symbolic expressions”, and “choice of symbols”. Also discussed is the way this experience contributes to an intuitive application of the conventional rules for solving equations that will be later introduced to the students as the standard algebraic “moves”.
 Belief structure as explanation for resistance to change: The case of
Robin Abstract: Publication date: Available online 30 August 2018Source: The Journal of Mathematical BehaviorAuthor(s): AnnaMarie Conner, Carlos Nicolas Gomez One goal of teacher mathematics education is to transform teachers’ beliefs to more closely align with beliefs that promote desired teaching practices. However, beliefs are difficult to change. A construct that helps to explain how and why mathematics teachers’ beliefs change is that of belief structures (Cooney et al., 1998). We expand upon belief structures to add a new construct: adaptive idealist. In this case study, we used information gleaned from interviews and reflective papers to introduce a prospective secondary mathematics teacher, describe her beliefs and shifts in beliefs, identify a resistance to change that resonated with our other experiences, and conclude that this new belief structure allows us to explain her reactions and interpretations within our secondary mathematics education program.
 Abstract: Publication date: Available online 30 August 2018Source: The Journal of Mathematical BehaviorAuthor(s): AnnaMarie Conner, Carlos Nicolas Gomez One goal of teacher mathematics education is to transform teachers’ beliefs to more closely align with beliefs that promote desired teaching practices. However, beliefs are difficult to change. A construct that helps to explain how and why mathematics teachers’ beliefs change is that of belief structures (Cooney et al., 1998). We expand upon belief structures to add a new construct: adaptive idealist. In this case study, we used information gleaned from interviews and reflective papers to introduce a prospective secondary mathematics teacher, describe her beliefs and shifts in beliefs, identify a resistance to change that resonated with our other experiences, and conclude that this new belief structure allows us to explain her reactions and interpretations within our secondary mathematics education program.
 Conventions, habits, and U.S. teachers’ meanings for graphs
 Abstract: Publication date: Available online 23 August 2018Source: The Journal of Mathematical BehaviorAuthor(s): Kevin C. Moore, Jason Silverman, Teo Paoletti, Dave Liss, Stacy Musgrave In this paper, we use relevant literature and data to motivate a more detailed look into relationships between what we perceive to be conventions common to United States (U.S.) school mathematics and individuals’ meanings for graphs and related topics. Specifically, we draw on data from preservice (PST) and inservice (IST) teachers to characterize such relationships. We use PSTs’ responses during clinical interviews to illustrate three themes: (a) some PSTs’ responses implied practices we perceive to be conventions of U.S. school mathematics were instead inherent aspects of PSTs’ meanings; (b) some PSTs’ responses implied they understood certain practices in U.S. school mathematics as customary choices not necessary to represent particular mathematical ideas; and (c) some PSTs’ responses exhibited what we or they perceived to be contradictory actions and claims. We then compare our PST findings to data collected with ISTs.
 Abstract: Publication date: Available online 23 August 2018Source: The Journal of Mathematical BehaviorAuthor(s): Kevin C. Moore, Jason Silverman, Teo Paoletti, Dave Liss, Stacy Musgrave In this paper, we use relevant literature and data to motivate a more detailed look into relationships between what we perceive to be conventions common to United States (U.S.) school mathematics and individuals’ meanings for graphs and related topics. Specifically, we draw on data from preservice (PST) and inservice (IST) teachers to characterize such relationships. We use PSTs’ responses during clinical interviews to illustrate three themes: (a) some PSTs’ responses implied practices we perceive to be conventions of U.S. school mathematics were instead inherent aspects of PSTs’ meanings; (b) some PSTs’ responses implied they understood certain practices in U.S. school mathematics as customary choices not necessary to represent particular mathematical ideas; and (c) some PSTs’ responses exhibited what we or they perceived to be contradictory actions and claims. We then compare our PST findings to data collected with ISTs.
 Accounting for mathematicians’ priorities in mathematics courses for
secondary teachers Abstract: Publication date: Available online 16 August 2018Source: The Journal of Mathematical BehaviorAuthor(s): Yvonne Lai Recent studies suggest that change is needed in undergraduate mathematics capstone courses for prospective secondary teachers. One promising but infrequently used strategy for improvement is to incorporate tasks that explicitly focus on pedagogical content knowledge (PCK). Expectancyvalue theory provides an account for why instruction of these courses does not more regularly employ this strategy. To make this argument, this paper uses an interviewbased study of 9 mathematicians that investigated the process of prioritizing tasks and goals for these courses. As the study found, these mathematicians valued developing teachers’ PCK. However, they were unconfident of their ability to teach with tasks and goals focused on developing teachers’ PCK relative to more purely mathematical tasks and goals. The central implication is that interventions in mathematicians’ teaching must take into account the possibility that it may be just as important to improve confidence and resources as it is to change values.
 Abstract: Publication date: Available online 16 August 2018Source: The Journal of Mathematical BehaviorAuthor(s): Yvonne Lai Recent studies suggest that change is needed in undergraduate mathematics capstone courses for prospective secondary teachers. One promising but infrequently used strategy for improvement is to incorporate tasks that explicitly focus on pedagogical content knowledge (PCK). Expectancyvalue theory provides an account for why instruction of these courses does not more regularly employ this strategy. To make this argument, this paper uses an interviewbased study of 9 mathematicians that investigated the process of prioritizing tasks and goals for these courses. As the study found, these mathematicians valued developing teachers’ PCK. However, they were unconfident of their ability to teach with tasks and goals focused on developing teachers’ PCK relative to more purely mathematical tasks and goals. The central implication is that interventions in mathematicians’ teaching must take into account the possibility that it may be just as important to improve confidence and resources as it is to change values.
 Strengths and inconsistencies in students’ understanding of the
roles of examples in proving Abstract: Publication date: Available online 14 August 2018Source: The Journal of Mathematical BehaviorAuthor(s): Orly Buchbinder, Orit Zaslavsky We report on a study investigating highschool students’ understanding of the interplay between examples and proving or disproving of statements drawn from highschool algebra and geometry curriculum. The data, collected through a series of taskbased interviews with 6 pairs of students, were analyzed in terms of alignment or misalignment of students’ responses with a conventional mathematical perspective summarized in the Roles of Examples in Proving (REP) framework. The results provide insights into strengths and weaknesses in students’ understanding of the status of examples in proving. In addition, we identified three types of inconsistencies in students’ responses: inconsistency with respect to the type of example, inconsistency with respect to the type of statement, and inconsistency with respect to the type of inference.
 Abstract: Publication date: Available online 14 August 2018Source: The Journal of Mathematical BehaviorAuthor(s): Orly Buchbinder, Orit Zaslavsky We report on a study investigating highschool students’ understanding of the interplay between examples and proving or disproving of statements drawn from highschool algebra and geometry curriculum. The data, collected through a series of taskbased interviews with 6 pairs of students, were analyzed in terms of alignment or misalignment of students’ responses with a conventional mathematical perspective summarized in the Roles of Examples in Proving (REP) framework. The results provide insights into strengths and weaknesses in students’ understanding of the status of examples in proving. In addition, we identified three types of inconsistencies in students’ responses: inconsistency with respect to the type of example, inconsistency with respect to the type of statement, and inconsistency with respect to the type of inference.
 Exploring unfamiliar paths through familiar mathematical territory:
Constraints and affordances in a preservice teacher’s reasoning about
fraction comparisons Abstract: Publication date: Available online 14 August 2018Source: The Journal of Mathematical BehaviorAuthor(s): Ian Whitacre, Şebnem Atabaş, Kelly Findley Preservice elementary teachers (PSTs) have been described as having difficulties with fractions, relying on standard procedures, and experiencing math anxiety. We are interested in productive ways in which PSTs can and do use their prior knowledge when exploring unfamiliar paths through familiar mathematical territory. We conducted interviews with PSTs in which we challenged them with various fraction comparison tasks and encouraged them to develop new strategies. In this paper, we present a case study focused on one PST who made considerable progress in her reasoning about fraction comparisons during such an interview. We use Greeno’s (1991) environment metaphor to conceptualize number sense as situated knowing in a conceptual domain. This perspective helps us account for both cognitive and affective factors. We highlight 4 themes concerning features of the mathematical environment that Jennifer (pseudonym) appeared to inhabit during the interview: (a) the interview context created a safe space that emphasized the interviewer’s interest in Jennifer’s ideas, as opposed to correct answers; (b) Jennifer used her prior knowledge of parts and wholes to ground her arguments meaningfully and as building blocks to invent new strategies; (c) she used her prior knowledge of cross multiplication as an established path that provided reassurance and facilitated her exploration of unfamiliar paths; (d) it was beliefs and affective factors, not deficiencies in knowledge, that constrained Jennifer’s exploration of unfamiliar paths through familiar mathematical territory. We discuss the implications of these findings for research concerning PSTs’ mathematical thinking and learning and for mathematics teacher education.
 Abstract: Publication date: Available online 14 August 2018Source: The Journal of Mathematical BehaviorAuthor(s): Ian Whitacre, Şebnem Atabaş, Kelly Findley Preservice elementary teachers (PSTs) have been described as having difficulties with fractions, relying on standard procedures, and experiencing math anxiety. We are interested in productive ways in which PSTs can and do use their prior knowledge when exploring unfamiliar paths through familiar mathematical territory. We conducted interviews with PSTs in which we challenged them with various fraction comparison tasks and encouraged them to develop new strategies. In this paper, we present a case study focused on one PST who made considerable progress in her reasoning about fraction comparisons during such an interview. We use Greeno’s (1991) environment metaphor to conceptualize number sense as situated knowing in a conceptual domain. This perspective helps us account for both cognitive and affective factors. We highlight 4 themes concerning features of the mathematical environment that Jennifer (pseudonym) appeared to inhabit during the interview: (a) the interview context created a safe space that emphasized the interviewer’s interest in Jennifer’s ideas, as opposed to correct answers; (b) Jennifer used her prior knowledge of parts and wholes to ground her arguments meaningfully and as building blocks to invent new strategies; (c) she used her prior knowledge of cross multiplication as an established path that provided reassurance and facilitated her exploration of unfamiliar paths; (d) it was beliefs and affective factors, not deficiencies in knowledge, that constrained Jennifer’s exploration of unfamiliar paths through familiar mathematical territory. We discuss the implications of these findings for research concerning PSTs’ mathematical thinking and learning and for mathematics teacher education.
 Grade 5 children’s drawings for integer addition and subtraction
open number sentences Abstract: Publication date: Available online 3 July 2018Source: The Journal of Mathematical BehaviorAuthor(s): Nicole M. WessmanEnzinger Three Grade 5 children participated in a microgenetic study embedded in 12week teaching experiment on integer addition and subtraction. They solved open number sentences in four individual sessions across the 12weeks and produced drawings. Through the lens of learnergenerated drawings and qualitative analysis, these drawings provide perspective into the children’s thinking about integer addition and subtraction. The following categories are described: Single and Double Set of Objects, Number Sequences, Empty Number Lines, Number Lines, Number Sentences, Sign Emphasis, and Answer in Box Only. One student drew sets of objects frequently and the other students drew number lines more. Descriptions of how use of their drawings changed over time are provided. Implications point to a reexamination of integer instructional models and insight into potential learning progressions.
 Abstract: Publication date: Available online 3 July 2018Source: The Journal of Mathematical BehaviorAuthor(s): Nicole M. WessmanEnzinger Three Grade 5 children participated in a microgenetic study embedded in 12week teaching experiment on integer addition and subtraction. They solved open number sentences in four individual sessions across the 12weeks and produced drawings. Through the lens of learnergenerated drawings and qualitative analysis, these drawings provide perspective into the children’s thinking about integer addition and subtraction. The following categories are described: Single and Double Set of Objects, Number Sequences, Empty Number Lines, Number Lines, Number Sentences, Sign Emphasis, and Answer in Box Only. One student drew sets of objects frequently and the other students drew number lines more. Descriptions of how use of their drawings changed over time are provided. Implications point to a reexamination of integer instructional models and insight into potential learning progressions.
 Improving realistic word problem solving by using humor
 Abstract: Publication date: Available online 29 June 2018Source: The Journal of Mathematical BehaviorAuthor(s): Wim Van Dooren, Stephanie Lem, Hannelore De Wortelaer, Lieven Verschaffel Research has shown that children have a strong tendency to exclude real world considerations when solving word problems. In this study, we investigated a novel way to try to change this tendency. We tested whether children would adapt their behavior when solving word problems in which realistic considerations are required (Pitems) when these problems are embedded in a humoristic context as compared to when they are offered in a typical word problem solving context. 148 sixth graders solved four Pitems in a humor condition versus a word problem condition. It was found that overall significantly more realistic reactions were given in the humor condition, and this was the case for three of the four problems. Implications of these findings for further research and for classroom instruction are discussed.
 Abstract: Publication date: Available online 29 June 2018Source: The Journal of Mathematical BehaviorAuthor(s): Wim Van Dooren, Stephanie Lem, Hannelore De Wortelaer, Lieven Verschaffel Research has shown that children have a strong tendency to exclude real world considerations when solving word problems. In this study, we investigated a novel way to try to change this tendency. We tested whether children would adapt their behavior when solving word problems in which realistic considerations are required (Pitems) when these problems are embedded in a humoristic context as compared to when they are offered in a typical word problem solving context. 148 sixth graders solved four Pitems in a humor condition versus a word problem condition. It was found that overall significantly more realistic reactions were given in the humor condition, and this was the case for three of the four problems. Implications of these findings for further research and for classroom instruction are discussed.
 Reasoning within quantitative frames of reference: The case of Lydia
 Abstract: Publication date: Available online 27 June 2018Source: The Journal of Mathematical BehaviorAuthor(s): Hwa Young Lee, Kevin C. Moore, Halil Ibrahim Tasova Quantitative reasoning is important in the development of K–16 mathematical ideas such as function and rate of change. Coordinate systems are used to coordinate sets of quantities by establishing frames of reference and constructing representational spaces in which sets of quantities are joined. Despite the critical role of coordinate systems in mathematics, much is left to understand about how students construct and reason within frames of reference and associated coordinate systems. In this report, we draw from a teaching experiment to discuss how an undergraduate student, Lydia, constructed and reasoned within frames of reference when graphing in noncanonical coordinate systems. We pay specific attention to distinctions between figurative and operative aspects of thought in her committing to reference points and directionality of measure comparison within frames of reference. In this regard, we present shifts in Lydia’s reasoning during the teaching experiment and consider implications and future research directions.
 Abstract: Publication date: Available online 27 June 2018Source: The Journal of Mathematical BehaviorAuthor(s): Hwa Young Lee, Kevin C. Moore, Halil Ibrahim Tasova Quantitative reasoning is important in the development of K–16 mathematical ideas such as function and rate of change. Coordinate systems are used to coordinate sets of quantities by establishing frames of reference and constructing representational spaces in which sets of quantities are joined. Despite the critical role of coordinate systems in mathematics, much is left to understand about how students construct and reason within frames of reference and associated coordinate systems. In this report, we draw from a teaching experiment to discuss how an undergraduate student, Lydia, constructed and reasoned within frames of reference when graphing in noncanonical coordinate systems. We pay specific attention to distinctions between figurative and operative aspects of thought in her committing to reference points and directionality of measure comparison within frames of reference. In this regard, we present shifts in Lydia’s reasoning during the teaching experiment and consider implications and future research directions.
 From the classification of quadrilaterals to the classification of prisms:
An experiment with prospective teachers Abstract: Publication date: Available online 27 June 2018Source: The Journal of Mathematical BehaviorAuthor(s): Lina Brunheira, João Pedro da Ponte This article reports a research in the context of a K6 prospective teacher education experiment developed in a geometry course in the 2nd year of their preparation program. This course included the study of the classification of quadrilaterals and prisms. The research is guided by the following question: how does the learning of hierarchical classification of geometric figures evolve from a teacher education experiment that includes the classification of quadrilaterals and prisms and follows an exploratory approach to teaching' Data was collected from audio and video records from the lessons and from the participants’ written reports about the classification of quadrilaterals and prisms. The results show that, in the first stage that focused on quadrilaterals, the participants’ difficulties in classifying derived mainly from their inexperience with the process of classifying geometrical objects and from their strong conceptualization of some quadrilaterals, very attached to prototypical images. In the second stage, the classification of prisms showed a positive and significant evolution, with a lower influence of prototypical images and a higher understanding about the classification process and the identification of hierarchical relationships among “close” and “distant” figures. However, the final evaluation test showed that the prospective teachers still had misunderstandings, most often related to the interpretation of the discourse and logical reasoning than to limited figural concepts.
 Abstract: Publication date: Available online 27 June 2018Source: The Journal of Mathematical BehaviorAuthor(s): Lina Brunheira, João Pedro da Ponte This article reports a research in the context of a K6 prospective teacher education experiment developed in a geometry course in the 2nd year of their preparation program. This course included the study of the classification of quadrilaterals and prisms. The research is guided by the following question: how does the learning of hierarchical classification of geometric figures evolve from a teacher education experiment that includes the classification of quadrilaterals and prisms and follows an exploratory approach to teaching' Data was collected from audio and video records from the lessons and from the participants’ written reports about the classification of quadrilaterals and prisms. The results show that, in the first stage that focused on quadrilaterals, the participants’ difficulties in classifying derived mainly from their inexperience with the process of classifying geometrical objects and from their strong conceptualization of some quadrilaterals, very attached to prototypical images. In the second stage, the classification of prisms showed a positive and significant evolution, with a lower influence of prototypical images and a higher understanding about the classification process and the identification of hierarchical relationships among “close” and “distant” figures. However, the final evaluation test showed that the prospective teachers still had misunderstandings, most often related to the interpretation of the discourse and logical reasoning than to limited figural concepts.
 Students’ understanding of the concepts involved in onesample
hypothesis testing Abstract: Publication date: Available online 15 June 2018Source: The Journal of Mathematical BehaviorAuthor(s): Harrison E. Stalvey, Annie BurnsChilders, Darryl Chamberlain, Aubrey Kemp, Leslie J. Meadows, Draga Vidakovic Hypothesis testing is a prevalent method of inference used to test a claim about a population parameter based on sample data, and it is a central concept in many introductory statistics courses. At the same time, the use of hypothesis testing to interpret experimental data has raised concerns due to common misunderstandings by both scientists and students. With statistics education reform on the rise, as well as an increasing number of students enrolling in introductory statistics courses each year, there is a need for research to investigate students’ understanding of hypothesis testing. In this study we used APOS Theory to investigate twelve introductory statistics students’ reasoning about onesample population hypothesis testing while working two realworld problems. Data were analyzed and compared against a preliminary genetic decomposition, which is a conjecture for how an individual might construct an understanding of a concept. This report presents examples of Actions, Processes, and Objects in the context of onesample hypothesis testing as exhibited through students’ reasoning. Our results suggest that the concepts involved in hypothesis testing are related through the construction of higherorder, coordinated Processes operating on Objects. As a result of our data analysis, we propose refinements to our genetic decomposition and offer suggestions for instruction of onesample population hypothesis testing. We conclude with appendices containing a comprehensive revised genetic decomposition along with a set of guided questions that are designed to help students make the constructions called for by the genetic decomposition.
 Abstract: Publication date: Available online 15 June 2018Source: The Journal of Mathematical BehaviorAuthor(s): Harrison E. Stalvey, Annie BurnsChilders, Darryl Chamberlain, Aubrey Kemp, Leslie J. Meadows, Draga Vidakovic Hypothesis testing is a prevalent method of inference used to test a claim about a population parameter based on sample data, and it is a central concept in many introductory statistics courses. At the same time, the use of hypothesis testing to interpret experimental data has raised concerns due to common misunderstandings by both scientists and students. With statistics education reform on the rise, as well as an increasing number of students enrolling in introductory statistics courses each year, there is a need for research to investigate students’ understanding of hypothesis testing. In this study we used APOS Theory to investigate twelve introductory statistics students’ reasoning about onesample population hypothesis testing while working two realworld problems. Data were analyzed and compared against a preliminary genetic decomposition, which is a conjecture for how an individual might construct an understanding of a concept. This report presents examples of Actions, Processes, and Objects in the context of onesample hypothesis testing as exhibited through students’ reasoning. Our results suggest that the concepts involved in hypothesis testing are related through the construction of higherorder, coordinated Processes operating on Objects. As a result of our data analysis, we propose refinements to our genetic decomposition and offer suggestions for instruction of onesample population hypothesis testing. We conclude with appendices containing a comprehensive revised genetic decomposition along with a set of guided questions that are designed to help students make the constructions called for by the genetic decomposition.
 Analysis of the argumentation of nineyearolds engaged in discourse about
comparing fraction models Abstract: Publication date: Available online 7 June 2018Source: The Journal of Mathematical BehaviorAuthor(s): Cheryl K. Van Ness, Carolyn A. Maher In this paper we analyze the argumentation of nineyearolds as they justify their reasoning for a fraction comparison problem. We illustrate how their prooflike arguments evolve in a class discussion and are backed by the rod models they produce, showing both valid and invalid arguments.
 Abstract: Publication date: Available online 7 June 2018Source: The Journal of Mathematical BehaviorAuthor(s): Cheryl K. Van Ness, Carolyn A. Maher In this paper we analyze the argumentation of nineyearolds as they justify their reasoning for a fraction comparison problem. We illustrate how their prooflike arguments evolve in a class discussion and are backed by the rod models they produce, showing both valid and invalid arguments.
 Gesturing standard deviation: Gestures undergraduate students use in
describing their concepts of standard deviation Abstract: Publication date: Available online 28 May 2018Source: The Journal of Mathematical BehaviorAuthor(s): Fey Parrill, Alison McKim, Kimberly Grogan This descriptive study explores the gestures undergraduate students produce when talking about the concept of standard deviation (SD). Gestures can be an important source of information about underlying internal representations of mathematical concepts, but thus far have not been examined in research exploring why concepts of variability (such as standard deviation) are difficult for students to learn. Thirteen undergraduates who had taken at least one course in statistics were asked to explain how they thought about SD. Speech and gesture were analyzed for their possible interpretation as encoding distance from the mean and frequency of values at different distances. We describe the different types of gestures that occurred in the data and also present a coding scheme that can be utilized for future research linking these gestures to students’ understanding of SD. We discuss how such a multimodal approach can enhance mathematics pedagogy.
 Abstract: Publication date: Available online 28 May 2018Source: The Journal of Mathematical BehaviorAuthor(s): Fey Parrill, Alison McKim, Kimberly Grogan This descriptive study explores the gestures undergraduate students produce when talking about the concept of standard deviation (SD). Gestures can be an important source of information about underlying internal representations of mathematical concepts, but thus far have not been examined in research exploring why concepts of variability (such as standard deviation) are difficult for students to learn. Thirteen undergraduates who had taken at least one course in statistics were asked to explain how they thought about SD. Speech and gesture were analyzed for their possible interpretation as encoding distance from the mean and frequency of values at different distances. We describe the different types of gestures that occurred in the data and also present a coding scheme that can be utilized for future research linking these gestures to students’ understanding of SD. We discuss how such a multimodal approach can enhance mathematics pedagogy.