Reasoning about relationships between quantities to reorganize inverse function meanings: The case of Arya

Researchers have argued high school students, college students, pre-service teachers, and in-service teachers do not construct productive inverse function meanings. In this report, I first summarize the literature examining students’ and teachers’ inverse function meanings. I then provide my theoretical perspective, including my use of the terms understanding and meaning and my operationalization of productive inverse function meanings. I describe a conceptual analysis of ways students may reorganize their limited inverse function meanings into productive meanings via reasoning about relationships between covarying quantities. I then present one pre-service teacher’s activity in a semester long teaching experiment to characterize how her quantitative, covariational, and bidirectional reasoning supported her in reorganizing her limited inverse function meanings into more productive meanings. I describe how this reorganization required her to reconstruct her meanings for various related mathematical ideas. I conclude with research and pedagogical implications stemming from this work and directions for future research.     

Using narrative inquiry for investigating the becoming of a mathematics teacher

This article presents narrative inquiry as a method for research in mathematics education, in particular the study of how pre-service teachers’ views of mathematics develop during elementary teacher education. I describe two different, complementary approaches to applying narrative analysis, one focusing on the content of a narrative, the other focusing on the form. The examples discussed are taken from interviews with and teaching portfolios compiled by four pre-service teachers. In analysing the content of the students’ narratives, I use emplotment to construct a retrospective explanation of how one pre-service teacher’s own experiences at school were reflected in the development of her mathematical identity. In analysing the form of the narratives, I also look at how the students told their stories, using linguistic features, for example, to identify core events in the accounts. This particular focus seems to be promising in locating turning points in the trainees’ views of mathematics.     

Revival of a classical topic in differential geometry: the exploration of envelopes in a computerized environment

Learning mathematics in a technology-rich environment enables us to revive classical topics which have been removed from the curriculum a long time ago. Both theoretical issues and applications can be studied with an experimental process. We present how envelopes of 1-parameter families of plane curves and some of their applications can be presented early in the curriculum either for pre-service teachers or for in-service teachers. This approach may be useful for students in an engineering curriculum. Working with technology yields important effects, such as reviving classical topics, broadening perspectives on already known topics, and enhancing the learner's experimental skills, where conversion between various registers of representation is an important issue.     

Game Design as an Interactive Learning Environment for Fostering Students'' and Teachers'' Mathematical Inquiry

Many learning environments, computer-based or not, have been developed for either students or teachers alone to engage them in mathematical inquiry. While some headway has been made in both directions, few efforts have concentrated on creating learning environments that bring both teachers and students together in their teaching and learning. In the following paper, we propose game design as such a learning environment for students and teachers to build on and challenge their existing understandings of mathematics, engage in relevant and meaningful learning contexts, and develop connections among their mathematical ideas and their real world contexts. To examine the potential of this approach, we conducted and analyzed two studies: Study I focused on a team of four elementary school students designing games to teach fractions to younger students, Study II focused on teams of pre-service teachers engaged in the same task. We analyzed the various games designed by the different teams to understand how teachers and students conceptualize the task of creating virtual game learning environment for others, in which ways they integrate their understanding of fractions and develop notions about students' thinking in fractions, and how conceptual design tools can provide a common platform to develop meaningful fraction contexts. In our analysis, we found that most teachers and students, when left to their own devices, create instructional games to teach fractions that incorporate little of their knowledge. We found that when we provided teachers and students with conceptual design tools such as game screens and design directives that facilitated an integration of content and game context, the games as well as teachers' and students' thinking increased in their sophistication. In the discussion, we elaborate on how the design activities helped to integrate rarely used informal knowledge of students and teachers, how the conceptual design tools improved the instructional design process, and how students and teachers benefit in their mathematical inquiry from each others' perspectives. In the outlook, we discuss features for computational design learning environments. This revised version was published online in August 2006 with corrections to the Cover Date.     

Conceptual variation and coordination in probability reasoning

This study investigates students’ conceptual variation and coordination among theoretical and experimental interpretations of probability. In the analysis we follow how Swedish students (12-13 years old) interact with a dice game, specifically designed to offer the students opportunities to elaborate on the logic of sample space, physical/geometrical considerations and experimental evidence when trying to develop their understanding of compound random phenomena.The analytical construct of contextualization was used as a means to provide structure to the qualitative analysis performed. Within the frame of the students’ problem encounters during the game and how they contextualized the solutions of the problems in personal contexts for interpretations, the analysis finds four main forms of appearance, or of limitations in appearance, of conceptual variation and coordination among theoretical and experimental interpretations of probability.     

Pre-service mathematics teachers’ concept images of radian

This study investigates pre-service mathematics teachers’ concept images of radian and possible sources of such images. A multiple-case study was conducted for this study. Forty-two pre-service mathematics teachers completed a questionnaire, which aims to assess their understanding of radian. Six of them were selected for individual interviews on the basis of theoretical sampling. The data indicated that participants’ concept images of radian were dominated by their concept images of degree. As the data in this study suggested, pre-service mathematics teachers were reluctant to accept trigonometric functions with the inputs of real numbers but rather they use value in degrees. More interestingly, they have two distinct images of π : π as an angle in radian and π as an irrational number.     

Pre-service Teachers’ Developing Conceptions about the Nature and Pedagogy of Mathematical Modeling in the Context of a Mathematical Modeling Course

Adopting a multitiered design-based research perspective, this study examines pre-service secondary mathematics teachers’ developing conceptions about (a) the nature of mathematical modeling in simulations of “real life” problem solving, and (b) pedagogical principles and strategies needed to teach mathematics through modeling. Unlike other studies that have focused on single-topic and lesson-sized research sites, a course-sized research site was used in this study. Having been through several iterations over three teaching semesters, the 15-week long course was implemented with 25 pre-service secondary mathematics teachers. Findings revealed that pre-service teachers developed ideas about the nature of mathematical modeling involving what mathematical modeling is, the relationship between mathematical modeling and meaningful understanding, and the nature of mathematical modeling tasks. They also realized the changing roles of teachers during modeling implementations and diversity in students’ ways of thinking. The researchers’ conceptual development, on the other hand, involved realizing the critical aspect of the “teacher role” played by the instructor during modeling implementations, and the need for more experience of modeling implementations for pre-service teachers.     

Characteristics of Pre-Service Primary School Teachers’ Configural Reasoning

The goal of this study is to identify the characteristics of pre-service primary teachers’ configural reasoning, understood as the relationships between concepts and figures set to solve geometrical proof problems. Ninety-seven primary teachers were asked to solve two geometrical proof problems in which a geometrical figure was provided. The results suggest the existence of two levels of the pre-service primary teachers’ discursive organization. The first, when pre-service primary teachers link the geometrical facts to the figure by discursive apprehensions, and the second, when several geometrical facts are related by logical chains to infer new information. The identification of a relevant configuration and the way in which geometrical facts are logically organized from discursive apprehensions are key factors in the shift between these moments. We contend that these factors help to explain how the image component stimulates thought and how conceptual constraints control the formal rigor of the process.     

Problem posing via “what if not?” strategy in solid geometry — a case study

The study describes the kinds of problems posed by pre-service teachers on the basis of complex solid geometry tasks using the “what if not?” strategy and the educational value of such an activity. Twenty-eight pre-service teachers participated in two workshops in which they had to pose problems on the basis of given problems. Analysis of the posed problems revealed a wide range of problems including those containing a change of one of the numerical data to another specific one, to a proof problem. Different kinds of posed problems enlightened some phenomena such as a bigger frequency of posed problems with another numerical value and a lack of posed problems including formal generalization. We also discuss the educational strengths of problem posing in solid geometry using the “what if not?” strategy, which could make the learner rethink the geometrical concepts he uses while creating new problems, make connections between the given and the new concepts and as a result deepen his understanding of them.     

Knowledge and confidence of pre-service mathematics teachers: the case of fraction division

To make teacher preparation and professional development effective, it is important to find out possible deficiencies in teachers’ knowledge as well as teachers’ own perceptions about their needs. By focusing on pre-service teachers’ knowledge of fraction division in this article, we conceptualize the notion of pre-service teachers’ knowledge in mathematics and pedagogy for teaching as containing both teachers’ perceptions of their preparation and their mathematics knowledge needed for teaching. With specific assessment instruments developed for pre-service middle school teachers, we focus on both pre-service teachers’ own perceptions about their knowledge preparation and the extent of their mathematics knowledge on the topic of fraction division. The results reveal a wide gap between sampled pre-service middle school teachers’ general perceptions/confidence and their limited mathematics knowledge needed for teaching fraction division conceptually. The results suggest that these pre-service teachers need to develop a sound and deep understanding of mathematics knowledge for teaching in order to build their confidence for classroom instruction. The study’s findings indicate the feasibility and importance of conceptualizing the notion of teachers’ knowledge in mathematics and pedagogy for teaching to include teachers’ perceptions. The applicability and implications of this expanded notion of teachers’ knowledge is then discussed.     

Middle school students' understanding of the role sample size plays in experimental probability

In this study, we examined the impact of an instructional program on sixth-grade students' understanding of experimental probability as it relates to sample size. As the number of trials in an experiment increases, the experimental probability is more likely to reflect the parent distribution; thus, smaller samples are more likely to yield unusual results. Results of this study indicate that, while typical middle school students are seemingly unaware of the relationship between experimental probability and sample size, appropriate cognitive activity focused on results of simulations of random phenomena can foster conceptual development. We witnessed growth that occurred as a result of key instructional tasks and concomitant mental activity.     

Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers

This mixed-methods study describes classroom characteristics and student outcomes from university mathematics courses that are based in mathematics departments, targeted to future pre-tertiary teachers, and taught with inquiry-based learning (IBL) approaches. The study focused on three two-term sequences taught at two research universities, separately targeting elementary and secondary pre-service teachers. Classroom observation established that the courses were taught with student-centred methods that were comparable to those used in IBL courses for students in mathematics-intensive fields at the same institutions. To measure pre-service teachers' gains in mathematical knowledge for teaching, we administered the Learning Mathematics for Teaching (LMT) instrument developed by Hill, Ball and Schilling for in-service teacher professional development. Results from the LMT show that pre-service teachers made significant score gains from beginning to end of their course, while data from interviews and from surveys of learning gains show that pre-service teachers viewed their gains as relevant to their future teaching work. Measured changes on pre-/post-surveys of attitudes and beliefs were generally supportive of learning mathematics but modest in magnitude. The study is distinctive in applying the LMT to document pre-service teachers' growth in mathematical knowledge for teaching. The study also suggests IBL is an approach well suited to mathematics departments seeking to strengthen their pre-service teacher preparation offerings in ways consistent with research-based recommendations.     

Inductive and Deductive Justification of Knowledge: Flexible Judgments Underneath Stable Beliefs in Teacher Education

Personal epistemological beliefs are considered to play an important role for processes of learning and teaching. However, research on personal epistemology is confronted with theoretical issues as there is conflicting evidence regarding the structure, stability, and context-dependence of epistemological beliefs. We give evidence how theoretical and methodological issues can partly be resolved by distinguishing between relatively stable “epistemological beliefs” and situation-specific “epistemological judgments.” A qualitative content analysis of a series of semistructured interviews (study 1) with pre-service teachers, teachers, and teacher educators as well as a statistical analysis of pre-service teachers’ extensive answers in questionnaires (study 2), both on the topic of “mathematical discovery,” reveal not only beliefs of the participants but also different qualities of judgments. Therefore, in further research both aspects of beliefs should be considered in a more differentiated manner when categorizing belief structures.     

A coloured window on pre-service teachers’ conceptions of rational numbers

In undergraduate mathematics courses, pre-service elementary school teachers are often faced with the task of re-learning some of the concepts they themselves struggled with in their own schooling. This often involves different cognitive processes and psychological issues than initial learning: pre-service teachers have had many more opportunities to construct understandings and representations than initial learners, some of which may be more complex and engrained; pre-service teachers are likely to have created deeply-held–and often negative–beliefs and attitudes toward certain mathematical ideas and processes. In our recent research, we found that pre-service teachers who used a particular computer-based microworld, one emphasising visual representations of and experimental interactions with elementary number theory concepts, overcame many cognitive and psychological difficulties reported in the literature. In this study, we investigate the possibilities of using a similarly-designed microworld that involves a set of rational number concepts. We describe the affordances of this microworld, both in terms of pre-service teacher learning and research on pre-service teacher learning, namely, the helpful “window” it gave us on the mathematical meaning-making of pre-service teachers. We also show how their interactions with this microworld provided many with a new and aesthetically-rich set of visualisations and experiences.     

Mathematical paradoxes as pathways into beliefs and polymathy: an experimental inquiry

This paper addresses the role of mathematical paradoxes in fostering polymathy among pre-service elementary teachers. The results of a 3-year study with 120 students are reported with implications for mathematics pre-service education as well as interdisciplinary education. A hermeneutic-phenomenological approach is used to recreate the emotions, voices and struggles of students as they tried to unravel Russell’s paradox presented in its linguistic form. Based on the gathered evidence some arguments are made for the benefits and dangers in the use of paradoxes in mathematics pre-service education to foster polymathy, change beliefs, discover structures and open new avenues for interdisciplinary pedagogy.     

Using dynamic software in mathematics: the case of reflection symmetry

This study was carried out to examine the effects of computer-assisted instruction (CAI) using dynamic software on the achievement of students in mathematics in the topic of reflection symmetry. The study also aimed to ascertain the pre-service mathematics teachers’ opinions on the use of CAI in mathematics lessons. In the study, a mixed research method was used. The study group of this research consists of 30 pre-service mathematics teachers. The data collection tools used include a reflection knowledge test, a survey and observations. Based on the analysis of the data obtained from the study, the use of CAI had a positive effect on achievement in the topic of reflection symmetry of the pre-service mathematics teachers. The pre-service mathematics teachers were found to largely consider that a mathematics education which is carried out utilizing CAI will be more beneficial in terms of ‘visualization’, ‘saving of time’ and ‘increasing interest/attention in the lesson’. In addition, it was found that the vast majority of them considered using computers in their teaching on the condition that the learning environment in which they would be operating has the appropriate technological equipment.     

概念图——概率统计教学的新途径

简述了概念图的起源、国内外发展现状,介绍了概念图的三个组成部分:结点、连线和连接语词,指出了相对于传统教学的概念图教学,在概率统计教学中的突出之处:概念图可作为教师教学的先行组织者;可作为学生复习时整理知识的工具;可作为教师检测学生学习的工具,提出了概率统计知识概念图制作的策略,并用实例说明了概念图应用于概率统计教学中的良好效果.     

Implications of Informal Education Experiences for Mathematics Teachers' Ability to Make Connections Beyond Formal Classroom

The Common Core Standard for Mathematical Practice 4: Model with Mathematics specifies that mathematically proficient students are able to make connections between school mathematics and its applications to solving real‐world problems. Hence, mathematics teachers are expected to incorporate connections between mathematical concepts they teach and their applications to solving problems arising in real‐world situation. Clearly, it is assumed that the teachers themselves are able to make such connections. On the other hand, research shows that mathematics teachers find it difficult to make those connections. In this paper, we present the results of the study that investigated the ways in which exploring mathematics in informal sites, and in particular science museum, assist teachers with making connections between school mathematics and its applications in real world.