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.     

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.     

Teachers' beliefs on teacher training contents and related characteristics of implementation—the example of introducing the topic study method in mathematics classrooms

So-called “bottom-up” strategies for implementation based on mathematics teachers' own developmental activities are considered to be a powerful approach when encouraging teachers to introduce alternative instructional practices. For evaluational research of in-service teacher training programs using “bottom-up” implementation strategies, the way how teachers implement contents of the teacher training is at the centre of interest. As the teachers' active role in the implementation process is necessary, their individual beliefs on the contents of the teacher training and their expectancies might influence the teachers' implementational activities. These beliefs can be considered as components of professional knowledge and pedagogical contents knowledge (Shulman, 1986) in particular. For this reason, the study focuses on the development of beliefs on contents of a teacher training program throughout the training on the one hand and relationships with characteristics of implementation on the other hand. We consider the example of introducing a student-centred learning environment, the so-called topic study method, in the teachers' classrooms. The results indicate that there are interdependencies between beliefs on the teacher training contents and characteristics of implementation.     

The beliefs of ‘Tomorrow's Teachers’ about mathematics: precipitating change in beliefs as a result of participation in an Initial Teacher Education programme

Mathematics education research has given increasing attention to the role of affective factors in the learning process. While 'affect' is used to refer to a variety of aspects including feelings, emotions, beliefs, attitudes and conceptions, this paper focuses on 'beliefs' of elementary pre-service teachers. In particular, the study evaluates the effect of participation in a reform-based elementary pre-service teacher education (referred to as Initial Teacher Education (ITE)) programme on participants' 'beliefs about the nature of mathematics'. This was completed using two (sub)scales of the Aiken's Revised Mathematics Scale measuring Enjoyment of Mathematics (E) and belief in the Value of Mathematics (V). Both scales were administered before and after participants completed the mathematics education programme, which consisted of 5 compulsory and consecutive modules. This study reveals that entry-level pre-service teachers report generally positive beliefs about the value of and enjoyment in doing mathematics. The findings challenge previous research, which report the tendency of teachers' beliefs to be resistant to change while in teacher education and suggest that it is possible for ITE mathematics education programmes to stimulate improvement in beliefs and attitudes among participants. Particular programme features are identified as instrumental in this positive change in beliefs about mathematics.     

Connecting experimental probability and theoretical probability

In this reflective paper, I explore the thinking of a group of pre-service teachers as they reason about experimental probability and theoretical probability. I am particularly interested in investigating whether pre-service teachers could construct a bidirectional link between the experimental probability and theoretical probability, similar to the tentative model I introduce elsewhere (2008) for coordinating the two perspectives on distribution. Overall, this research study contributes to understanding how pre-service students can build connections to help teachers conceptualize and support students to embrace elements that act as connections between the two approaches to probability.     

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.     

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.     

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.     

Mathematics learning and tools from theoretical, historical and practical points of view: the productive notion of mathematics laboratories

In our research work, we have looked at the way in which artefacts become, for teachers as well as for students, instruments of their mathematical activity. The issues related to the use of tools and technologies in mathematical education are now widely considered. A look to history highlights the different ways in which the same questions have been studied at different times and in different places. This suggests that the contribution of artefacts to mathematics learning should be considered in terms of various contexts. Our “visits” to these contexts will be guided by the coordination of two main theoretical frameworks, the instrumental approach and the semiotic mediation approach from the perspective of mathematics laboratory. This journey through history and schooling represents a good occasion to address some questions: Are there “good” contexts in which to develop mathematical instruments? Are there “good” teaching practices which assist students’ instrumental geneses and construct mathematical meanings? How is it possible to promote such teaching practices? Some study cases are discussed.     

Teachers attending to student reasoning: Do beliefs matter?

We present the results of a quasi-experimental study of pre-service elementary teachers' learning to recognize students' mathematical reasoning from classroom videos. Researchers examined the nature of participants’ beliefs regarding mathematics education. We found that pre-service elementary teachers whose beliefs were consistent with NCTM Process Standards (NCTM, 2000), or that transitioned in the direction of consistency with the Standards, regarding the teaching and learning of mathematics, were more successful in recognizing students' reasoning than those whose beliefs were generally inconsistent. Predictive Analytics and Generalized Linear Regression modeling were used to quantify the magnitude of experimental pre-service teachers’ reasoning growth and combined pre/post study assessment reasoning success in contrast to that of the comparison groups. The resulting model explained nearly 90% of the variability in success on the reasoning assessment, showing that beliefs do indeed matter for recognition of reasoning.     

Turkish Preservice Primary School Teachers' Science Teaching Efficacy Beliefs and Attitudes Toward Science: The Effect of a Primary Teacher Education Program

The main purpose of this study was to investigate the effectiveness of a primary teacher education program in improving science teaching efficacy beliefs (personal science teaching efficacy beliefs and outcome expectancy beliefs) of preservice primary school teachers. The study also investigated whether the program has an effect on student teachers' attitudes toward science. Data were collected by administering the “Science Teaching Efficacy Beliefs Instrument” and “Attitudes toward Science Scale” to 282 preservice primary teachers (147 freshmen, 135 seniors). Statistical techniques such as means and t‐test were used to analyze the data. Results of the study showed that the primary teacher education program has a medium positive effect on science teaching efficacy beliefs of the primary preservice teachers (t = 4.791, p = .000) and that there were no gender differences in terms of efficacy beliefs. Results also indicated that preservice primary teachers' attitudes toward science were moderately positive and differ by class level. Fourth‐year preservice teachers' attitudes toward science were found to be significantly more positive than the first years (t = 5.494, p = .000). There were no gender differences in attitudes toward science.     

Communicative characteristics of teachers’ mathematical talk with children: from knowledge transfer to knowledge investigation

In-service teachers actively collaborated in a developmental research project. The main aim of the research project was the advancement of one central aspect of teacher professionalism: teachers’ diagnostic competencies. Conditions of understanding and possibilities of enriching teachers’ talk are of special interest because mathematics teaching is particularly affected by speech and communication (Söbbeke and Steinbring in Mathematik für Kinder—Mathematik von Kindern, pp. 26–38, 2004). One research focus was on the support of a productive enhancement of the teachers’ talk with one child. Is the teacher’s talk mainly a kind of knowledge transfer similar to traditional instruction or can it be seen as an investigation of the child’s own views and ideas of elementary mathematical knowledge? These teachers’ talks with one child should offer more reflective communication between teacher and child and result in a changed view of the child’s mathematical understanding. Using an elaborate interpretation based on a theoretical instrument of analysis, called “Forms of teachers’ mathematical Interaction (Formal-In)”, we describe the development from the first diagnostic talk with one child, at the beginning to the last talk at the end of the research project. Using an elaborate analysis of short episodes of teachers’ talk distinguishing the interactive and the epistemological dimensions, we can describe how both dimensions influence each other. The theoretically identified characteristics of teachers’ talk together with compatible video cases can be used in theory-based (in-service) teacher training aimed at enhancing professionalism.     

Science Attitudes of Selected Middle School Students in China: A Preliminary Investigation of Similarities and Differences as a Function of Gender

Evaluating the attitudes of science students is important for teachers, curriculum developers, and those working with preservice teachers. Although in the United States a great deal of attitudinal research has been conducted with regard to science education, in the People's Republic of China very little work concerning science attitudes has been completed. This study will report on an evaluation of Chinese boys' and girls' attitudes toward selected science topics. Students attended a middle school in the city of Shanghai. Analysis indicated that when the male and female Chinese students differ in their response patterns, females select more intense responses (“strongly agree” as opposed to “agree,”“strongly disagree” as opposed to “disagree”). Furthermore, the surveyed females often selected responses suggesting that they were more interested in the science topics and issues presented in the survey.     

Balancing mathematics education research and the NCTM standards

The release of the Principles and Standards for School Mathematics in the United States by the National Council of Teachers of Mathematics (NCTM) brought to the forefront the debate of whether research should determine the validity of the espoused Standards? Or conversely whether the Standards should influence the research agenda of the mathematics education community? How should university teacher educators address this issue? Should pre-service and practicing teachers blindly, accept the Standards as well as the research, or do we cultivate the critical thinking skills that will allow preparing teachers to resolve this dilemma? In this article a university mathematics educator and an idealistic pre-service elementary teacher try to resolve the dilemma of balancing the Standards with research and personal beliefs about the teaching and learning of mathematics.     

Incorporating the dynamic mathematics software GeoGebra into a history of mathematics course

The purpose of this study was to investigate pre-service teachers’ views about the history of mathematics course in which GeoGebra was used. The qualitative research design was used in this study. The participants of the study consisted of 23 pre-service mathematics teachers studying at a state university in Turkey. An open-ended questionnaire was used as a data collection tool. Qualitative data obtained from the pre-service teachers were analyzed by means of content analysis. As a result, it was determined that GeoGebra software was an effective tool in the learning and teaching of the history of mathematics.     

The ostensive dimension through the lenses of two didactic approaches

The paper presents how two different theories—the APC-space and the ATD—can frame in a complementary way the semiotic (or ostensive) dimension of mathematical activity in the way they approach teaching and learning phenomena. The two perspectives coincide in the same subject: the importance given to ostensive objects (gestures, discourses, written symbols, etc.) not only as signs but also as essential tools of mathematical practices. On the one hand, APC-space starts from a general semiotic analysis in terms of “semiotic bundles” that is to be integrated into a more specific epistemological analysis of mathematical activity. On the other hand, ATD proposes a general model of mathematical knowledge and practice in terms of “praxeologies” that has to include a more specific analysis of the role of ostensive objects in the development of mathematical activities in the classroom. The articulation of both theoretical perspectives is proposed as a contribution to the development of suitable frames for Networking Theories in mathematics education.     

The cultural dimension of beliefs: an investigation of future primary teachers’ epistemological beliefs concerning the nature of mathematics in 15 countries

Beliefs constitute a central part of a person’s professional competencies and are crucial to the perception of situations as they influence our choice of actions. This paper focuses on epistemological beliefs about the nature of mathematics of future primary teachers from an international perspective. The data reported are part of a larger sample originating from the TEDS-M study which compares primary mathematics teacher education in 15 countries. In this paper we examine the pattern of beliefs of future teachers aiming to teach mathematics at primary level. We explore whether and to what extent beliefs concerning the nature of mathematics are influenced by cultural factors, in our case the extent to which a country’s culture can be characterized by an individualistic versus collectivistic orientation according to Hofstede’s terminology. In the first part of the paper, the literature on epistemological beliefs is reviewed and the role of culture and individualism/collectivism on the formation of beliefs concerning the nature of mathematics will be discussed. In the empirical part, means and distributions of belief ratings will be reported. Finally, multilevel analyses explore how much of the variation of belief preferences between countries can be explained by the individualistic orientation of a country.     

Epistemological beliefs concerning the nature of mathematics among teacher educators and teacher education students in mathematics

Beliefs constitute a central part of a person’s professional competences as beliefs are crucial to the perception of situations and as they influence our choice of actions. The present article focuses on epistemological beliefs about the nature of mathematics among future teachers and their educators at university and post-university teacher-training institutions in Germany. The data reported are part of a larger sample originating from the MT21 study [supported by the National Science Foundation through a grant to W. S. Schmidt and M. T. Tatto (REC-0231886). MT21 started in 2003] which explores and compares mathematics teacher education in Bulgaria, Germany, Mexico, South Korea, Taiwan, and the United States. In this article, we examine the structure and level of beliefs concerning the nature of mathematics for teacher education students in Germany both at the beginning (n = 368) and the end of their education (n = 286) as well as their educators (n = 77) in three academic disciplines (mathematics, mathematics pedagogy and general pedagogy). In the first part of the article, the literature on epistemological beliefs and their structure will be reviewed. In the empirical part, analyses on the level and the structure of beliefs for our samples and subsamples will be presented. Relations between educators’ and students’ beliefs will be explored.