This article is concerned with how undergraduate students in their first abstract algebra course ... more This article is concerned with how undergraduate students in their first abstract algebra course learn the concept of group isomorphism. To probe the students' thinking, we interviewed them while they were working on tasks involving various aspects of isomorphism. Here are some of the observations that emerged from analysis of the interviews. First, students show a strong need for "canonical", unique, step-by-step procedures and tend to get stuck of having to deal with some degrees of freedom in their choices. Second, students exhibit various degrees of personification and localization in their language, as in "I can find a function that takes every element of G to every element of G J'' vs. "there exists a function from G to G'". Third, when having to deal with a list of properties, students choose first the properties they perceive as simpler; however, it turns out that their choice depends on the type of the task and the type of complexity involved. That is, in tasks involving groups in general, students mostly prefer to work with properties which are syntactically simple, whereas in tasks involving specific groups, students prefer properties which are computationally simpler.
International Journal of Mathematical Education in Science and Technology, 2003
This article examines the idea of 'following the flow of a proof with an example' in order to ass... more This article examines the idea of 'following the flow of a proof with an example' in order to assist the learner in the challenging task of understanding mathematical proofs. This strategy is termed 'mimicry of a proof'. However, such mimicry can be impractical or unreasonably demanding when the mathematical objects in the proof are difficult to manipulate without technological enhancement. This is the case with many proofs in Linear Algebra, in which the manipulated objects are vectors or matrices. Therefore, the article focuses on the idea of proof mimicry with a computer algebra system (CAS). As examples, this strategy is applied to the proofs of two theorems: the basis theorem and the orthogonalization theorem. In addition, pedagogical guidelines to be followed in constructing a set of computer activities for students are presented and examined.
Despite plethora of research that attends to the convincing power of different types of proofs, r... more Despite plethora of research that attends to the convincing power of different types of proofs, research related to the convincing power of counterexamples is rather slim. In this paper we examine how prospective and practicing secondary school mathematics teachers respond to different types of counterexamples. The counterexamples were presented as products of students' arguments, and the participants were asked to evaluate their correctness and comment on them. The counterexamples varied according to mathematical topic: algebra or geometry, and their explicitness. However, as we analyzed the data, we discovered that these distinctions were insufficient to explain why teachers accepted some counterexamples, but rejected others, with seemingly similar features. As we analyze the participants' perceived transparency of different counterexamples, we employ various theoretical approaches that can advance our understanding of teachers' conceptions of conviction with respect to counterexamples.
If our rea~>oning has logic, it~s fuuy at best. We have only one decision rule: I'll do it if it ... more If our rea~>oning has logic, it~s fuuy at best. We have only one decision rule: I'll do it if it feels tight The formal logic we first learn in tenth-grade geometry claS'S has little to do with it That's why we made it to tenth grade [BartKisko,p. 17]
ABSTRACT In this chapter, I share three stories of my personal learning that was triggered by int... more ABSTRACT In this chapter, I share three stories of my personal learning that was triggered by interaction with students. I analyze these stories using disaggregated perspective on learning, considering how the instructor’s mathematics and pedagogy is implemented to consider mathematical and pedagogical issues with prospective teachers. In the first story, I present a novel pedagogical approach in order to enhance teachers’ mathematics related to a translation of a parabola. In the second story, I share my learning of mathematics related to Affine transformations triggered by students’ errors that lead to correct results. In the third story, I explore several examples of numerical variation, presenting it as a powerful pedagogical tool.
Script-writing is a novel pedagogical approach and research tool in mathematics education. The go... more Script-writing is a novel pedagogical approach and research tool in mathematics education. The goal of this chapter is to introduce the approach and exemplify its implementation. A script-writing task presents a prompt, which usually includes an incomplete argument or erroneous claim of a student. Prospective teachers address the prompt by creating a script for a dialogue-presenting an imaginary interaction between a teacher and her students, or among different students. In this chapter I exemplify several results of implementing script-writing tasks and discuss advantages of this approach. In particular, I focus on the concepts related to elementary number theory, prime numbers and factors of a number, and demonstrate how the understanding of these concepts can be explored and refined, as scriptwriters create characters who discuss particular claims. I suggest that engaging prospective teachers in script-writing is one possible way to support and improve preparation of mathematics teachers.
This study examines future secondary physics teachers' knowledge related to the teaching of sound... more This study examines future secondary physics teachers' knowledge related to the teaching of sound waves, and specifically the topics of sound level and sound intensity. The data is comprised of future teachers' responses to a task in which they had to compose a script for an imaginary dialogue between a teacher and a group of students and to provide a commentary elaborating on their instructional choices. The topics selected for the task were chosen intentionally as they provide authentic and rich opportunities to bridge mathematics and science concepts, while challenging future teachers to consider the logarithmic measurement scale and its role in science. The task provided the participants with the beginning of a dialogue that featured student confusion about the measurement of sound level using a decibel scale. Future physics teachers were asked to extend this dialogue through describing envisioned instructional interactions that could have ensued. The instructional interchange related to the relationship between sound intensity and sound level, and particular teachers' responses to the student ideas related to the meaning of a decibel sound level scale were analysed. These responses were categorized as featuring superficial or deep, and conceptual or procedural knowledge for teaching. We describe each category using illustrative excerpts from the participants' scripts. We conclude with highlighting the affordances of scriptwriting for teachers, teacher educators, and researchers.
This article is concerned with how undergraduate students in their first abstract algebra course ... more This article is concerned with how undergraduate students in their first abstract algebra course learn the concept of group isomorphism. To probe the students' thinking, we interviewed them while they were working on tasks involving various aspects of isomorphism. Here are some of the observations that emerged from analysis of the interviews. First, students show a strong need for "canonical", unique, step-by-step procedures and tend to get stuck of having to deal with some degrees of freedom in their choices. Second, students exhibit various degrees of personification and localization in their language, as in "I can find a function that takes every element of G to every element of G J'' vs. "there exists a function from G to G'". Third, when having to deal with a list of properties, students choose first the properties they perceive as simpler; however, it turns out that their choice depends on the type of the task and the type of complexity involved. That is, in tasks involving groups in general, students mostly prefer to work with properties which are syntactically simple, whereas in tasks involving specific groups, students prefer properties which are computationally simpler.
International Journal of Mathematical Education in Science and Technology, 2003
This article examines the idea of 'following the flow of a proof with an example' in order to ass... more This article examines the idea of 'following the flow of a proof with an example' in order to assist the learner in the challenging task of understanding mathematical proofs. This strategy is termed 'mimicry of a proof'. However, such mimicry can be impractical or unreasonably demanding when the mathematical objects in the proof are difficult to manipulate without technological enhancement. This is the case with many proofs in Linear Algebra, in which the manipulated objects are vectors or matrices. Therefore, the article focuses on the idea of proof mimicry with a computer algebra system (CAS). As examples, this strategy is applied to the proofs of two theorems: the basis theorem and the orthogonalization theorem. In addition, pedagogical guidelines to be followed in constructing a set of computer activities for students are presented and examined.
Despite plethora of research that attends to the convincing power of different types of proofs, r... more Despite plethora of research that attends to the convincing power of different types of proofs, research related to the convincing power of counterexamples is rather slim. In this paper we examine how prospective and practicing secondary school mathematics teachers respond to different types of counterexamples. The counterexamples were presented as products of students' arguments, and the participants were asked to evaluate their correctness and comment on them. The counterexamples varied according to mathematical topic: algebra or geometry, and their explicitness. However, as we analyzed the data, we discovered that these distinctions were insufficient to explain why teachers accepted some counterexamples, but rejected others, with seemingly similar features. As we analyze the participants' perceived transparency of different counterexamples, we employ various theoretical approaches that can advance our understanding of teachers' conceptions of conviction with respect to counterexamples.
If our rea~>oning has logic, it~s fuuy at best. We have only one decision rule: I'll do it if it ... more If our rea~>oning has logic, it~s fuuy at best. We have only one decision rule: I'll do it if it feels tight The formal logic we first learn in tenth-grade geometry claS'S has little to do with it That's why we made it to tenth grade [BartKisko,p. 17]
ABSTRACT In this chapter, I share three stories of my personal learning that was triggered by int... more ABSTRACT In this chapter, I share three stories of my personal learning that was triggered by interaction with students. I analyze these stories using disaggregated perspective on learning, considering how the instructor’s mathematics and pedagogy is implemented to consider mathematical and pedagogical issues with prospective teachers. In the first story, I present a novel pedagogical approach in order to enhance teachers’ mathematics related to a translation of a parabola. In the second story, I share my learning of mathematics related to Affine transformations triggered by students’ errors that lead to correct results. In the third story, I explore several examples of numerical variation, presenting it as a powerful pedagogical tool.
Script-writing is a novel pedagogical approach and research tool in mathematics education. The go... more Script-writing is a novel pedagogical approach and research tool in mathematics education. The goal of this chapter is to introduce the approach and exemplify its implementation. A script-writing task presents a prompt, which usually includes an incomplete argument or erroneous claim of a student. Prospective teachers address the prompt by creating a script for a dialogue-presenting an imaginary interaction between a teacher and her students, or among different students. In this chapter I exemplify several results of implementing script-writing tasks and discuss advantages of this approach. In particular, I focus on the concepts related to elementary number theory, prime numbers and factors of a number, and demonstrate how the understanding of these concepts can be explored and refined, as scriptwriters create characters who discuss particular claims. I suggest that engaging prospective teachers in script-writing is one possible way to support and improve preparation of mathematics teachers.
This study examines future secondary physics teachers' knowledge related to the teaching of sound... more This study examines future secondary physics teachers' knowledge related to the teaching of sound waves, and specifically the topics of sound level and sound intensity. The data is comprised of future teachers' responses to a task in which they had to compose a script for an imaginary dialogue between a teacher and a group of students and to provide a commentary elaborating on their instructional choices. The topics selected for the task were chosen intentionally as they provide authentic and rich opportunities to bridge mathematics and science concepts, while challenging future teachers to consider the logarithmic measurement scale and its role in science. The task provided the participants with the beginning of a dialogue that featured student confusion about the measurement of sound level using a decibel scale. Future physics teachers were asked to extend this dialogue through describing envisioned instructional interactions that could have ensued. The instructional interchange related to the relationship between sound intensity and sound level, and particular teachers' responses to the student ideas related to the meaning of a decibel sound level scale were analysed. These responses were categorized as featuring superficial or deep, and conceptual or procedural knowledge for teaching. We describe each category using illustrative excerpts from the participants' scripts. We conclude with highlighting the affordances of scriptwriting for teachers, teacher educators, and researchers.
This book offers multiple interconnected perspectives on the largely untapped potential of elemen... more This book offers multiple interconnected perspectives on the largely untapped potential of elementary number theory for mathematics education: its formal and cognitive nature, its relation to arithmetic and algebra, its accessibility, its utility and intrinsic merits, to name just a few. Its purpose is to promote explication and critical dialogue about these issues within the international mathematics education community. The studies comprise a variety of pedagogical and research orientations by an international group of researchers that, collectively, make a compelling case for the relevance and importance of number theory in mathematics education in both pre K-16 settings and mathematics teacher education.
Topics variously engaged include:
*understanding particular concepts related to numerical structure and number theory;
*elaborating on the historical and psychological relevance of number theory in concept development;
*attaining a smooth transition and extension from pattern recognition to formative principles;
*appreciating the aesthetics of number structure;
*exploring its suitability in terms of making connections leading to aha! insights and reaching toward the learner's affective domain;
*reexamining previously constructed knowledge from a novel angle;
*investigating connections between technique and theory;
*utilizing computers and calculators as pedagogical tools; and
*generally illuminating the role number theory concepts could play in developing mathematical knowledge and reasoning in students and teachers.
Overall, the chapters of this book highlight number theory-related topics as a stepping-stone from arithmetic toward generalization and algebraic formalism, and as a means for providing intuitively grounded meanings of numbers, variables, functions, and proofs.
Number Theory in Mathematics Education: Perspectives and Prospects is of interest to researchers, teacher educators, and students in the field of mathematics education, and is well suited as a text for upper-level mathematics education courses.
This study investigates procedural and conceptual aspects in preservice elementary school teacher... more This study investigates procedural and conceptual aspects in preservice elementary school teachers' understanding of the Fundamental Theorem of Arithmetic. The data were collected by the means of a written questionnaire and individual interviews. The results suggest that the idea of the uniqueness of prime decomposition is very difficult to grasp.
Journal for Research in Mathematics Education, Jan 1, 1996
This study contributes to a growing body of research on teachers' content knowledge in mathematic... more This study contributes to a growing body of research on teachers' content knowledge in mathematics. The domain under investigation was elementary number theory. Our main focus concerned the con- cept of divisibility and its relation to division, multiplication, prime and composite numbers, fac- torization, divisibility rules, and prime decomposition. We used a constructivist-orientedtheoretical framework for analyzing and interpreting data acquired in clinical interviews with preservice teach- ers. Participants' responses to questions and tasks indicated pervasive dispositions toward procedural attachments, even when some degree of conceptual understanding was evident. The results of this study provide a preliminary overview of cognitive structures in elementary number theory.
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Papers by Rina Zazkis
Topics variously engaged include:
*understanding particular concepts related to numerical structure and number theory;
*elaborating on the historical and psychological relevance of number theory in concept development;
*attaining a smooth transition and extension from pattern recognition to formative principles;
*appreciating the aesthetics of number structure;
*exploring its suitability in terms of making connections leading to aha! insights and reaching toward the learner's affective domain;
*reexamining previously constructed knowledge from a novel angle;
*investigating connections between technique and theory;
*utilizing computers and calculators as pedagogical tools; and
*generally illuminating the role number theory concepts could play in developing mathematical knowledge and reasoning in students and teachers.
Overall, the chapters of this book highlight number theory-related topics as a stepping-stone from arithmetic toward generalization and algebraic formalism, and as a means for providing intuitively grounded meanings of numbers, variables, functions, and proofs.
Number Theory in Mathematics Education: Perspectives and Prospects is of interest to researchers, teacher educators, and students in the field of mathematics education, and is well suited as a text for upper-level mathematics education courses.