In this paper, some fundamental issues on mathematics assessment and how they are related to the underlying cultural values in East Asia are discussed. Features of the East Asian culture that impact on mathematics assessment include the pragmatic nature of the culture, the social orientation of East Asian people, and the lop-sided stress on the utilitarian function of education. East Asians stress the algorithmic side of mathematics, and mathematics is viewed more as a set of techniques for calculation and problem solving. The notion of fairness in assessment is of paramount importance, and there is a great trust in examination as a fair method of differentiating between the able and the less able. The selection function of education and assessment has great impact on how mathematics is taught, and assessment constitutes an extrinsic motivation which directs student learning. Finally, the strengths and weaknesses of these East Asian values are discussed.
In summer 2006 the University of Education in Weingarten, Germany, and East China Normal University, Shanghai, performed a semi-virtual seminar with mathematics students on “Mathematics and Architecture”. The goal was the joint development of teaching materials for German or Chinese school, based on different buildings such as “Nanpu Bridge”, or the “Eiffel Tower”. The purpose of the seminar was to provide a learning environment for students supported by using information and communication technology (ICT) to understand how the hidden mathematics in buildings should be related to school mathematics; to experience the multicultural potential of the international language “Mathematics”; to develop “media competence” while communicating with others and using technologies in mathematics education; and to recognize the differences in teaching mathematics between the two cultures. In this paper we will present our ideas, experiences and results from the seminar.
This paper reports a part of a study on the construction of mathematical meanings in terms of development of semiotic systems (gestures, speech in oral and written form, drawings) in a Vygotskian framework, where artefacts are used as tools of semiotic mediation. It describes a teaching experiment on perspective drawing at primary school (fourth to fifth grade classes), starting from a concrete experience with a Dürer’s glass to the interpretation of a new artefact. We analyse the long term process of appropriation of the mathematical model of perspective drawing (visual pyramid) through the development of gestures, speech and drawings under the teacher’s guidance.
While reading the articles assembled in this volume, one cannot help asking Why gestures? What’s all the fuss about them? In the last few years, the fuss is, indeed, considerable, and not just here, in this special issue, but also in research on learning and teaching at large. What changed? After all, gestures have been around ever since the birth of humanity, if not much longer, but until recently, not many students of human cognition seemed to care. In this commentary, while reporting on what I saw while scrutinizing this volume for an answer, I will share some thoughts on the relationship between gesturing and speaking and about their relative roles in mathematical thinking.
The evaluation of teaching quality in mathematics has become increasingly important following research reports indicating that preschoolers are developmentally able to engage in mathematic thought and that child performance in mathematics at this level is a strong predictor of later school achievement. As attention turns to early mathematics education, so too does the focus on teaching quality. This paper reviews nine instruments designed to measure mathematics teaching quality—their theoretical bases, foci, and psychometrics—and examines their appropriateness for administration in early childhood settings. Three of the nine measures are identified as having highly desirable characteristics, with one of them specifically designed for early childhood administration. The measures, our review process, and our recommendations for practice are presented. As school divisions and teacher educators examine teaching quality, they will be better able to support their teachers’ practice, and better able to reap the benefits in improved child outcomes.
- Mathematics education: new perspectives on gender, by Gilah Leder and Helen Forgasz
- Moving towards a feminist epistemology of mathematics, by Leone Burton†
- The emergence of women on the international stage of mathematics education, by Fulvia Furinghetti
- Israeli Jewish and Arab students’ gendering of mathematics, by Helen J. Forgasz and David Mittelberg
- Gender, technology and attitude towards mathematics: a comparative longitudinal study with Mexican students, by Sonia Ursini and Gabriel Sánchez
- On the role of computers and complementary situations for gendering in mathematics classrooms, by Helga Jungwirth
- Exploring gender factors related to PISA 2003 results in Iceland: a youth interview study, by Olof Bjorg Steinthorsdottir and Bharath Sriraman
- Gender differences in the mathematics achievements of German primary school students: results from a German large-scale study, by Henrik Winkelmann, Marja van den Heuvel-Panhuizen and Alexander Robitzsch
- Adolescent girls’ construction of moral discourses and appropriation of primary identity in a mathematics classroom, by Jae Hoon Lim
- Images of mathematicians: a new perspective on the shortage of women in mathematical careers, by Katrina Piatek-Jimenez
- Equity in mathematics education: unions and intersections of feminist and social justice literature, by Laura Jacobsen Spielman
- Progress and stagnation of gender equity: contradictory trends within mathematics research and education in Sweden, by Gerd Brandell
- Gender in mathematics relationality: counseling underprepared college students, by Jillian M. Knowles
- Stepping beyond high school mathematics: a case study of high school women, by Charlene Morrow and Inga Schowengerdt
- Goos, Stillman and Vale: teaching secondary mathematics: research and practice for the 21st century, by Gaye Williams
- Education for the knowledge to teach mathematics: it all has to come together, by Peter Sullivan
- Teachers’ perceptions of assessments of their mathematical knowledge in a professional development course, by Michelle T. Chamberlin, Jeff D. Farmer and Jodie D. Novak
- Learning mathematics for teaching in the student teaching experience: two contrasting cases, by Blake E. Peterson and Steven R. Williams
- Mathematical belief change in prospective primary teachers, by Peter Grootenboer
- Toward a framework for the development of mathematical knowledge for teaching, by Jason Silverman and Patrick W. Thompson
- Abstraction and consolidation of the limit procept by means of instrumented schemes: the complementary role of three different frameworks, by Ivy Kidron
- Students’ images and their understanding of definitions of the limit of a sequence, by Kyeong Hah Roh
- Deductive reasoning: in the eye of the beholder, by Michal Ayalon and Ruhama Even
- Signifying “students”, “teachers” and “mathematics”: a reading of a special issue, by Tony Brown
- On semiotics and subjectivity: a response to Tony Brown’s “signifying ‘students’, ‘teachers’, and ‘mathematics’: a reading of a special issue”, by Norma Presmeg and Luis Radford
- Review of the proceedings of the 2001, 2003 and 2005 French summer schools in Didactics of Mathematics, by Rudolf Sträßer
- Brian Griffiths (1927–2008) – his pioneering contributions to mathematics and education, by Keith Jones and Joanna Mamona-Downs
In this paper, I present an account of attempts to quantify deaths of Iraqis during the occupation by US and other forces since the invasion of March 2003, and of the reactions to these attempts. This story illuminates many aspects of current socio-political reality, particularly, but by no means exclusively, in the United States. Here, these aspects are selectively discussed in relation to the overarching themes of what the story illuminates about the uses of statistical information in society and about shortcomings in mathematics education.
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.
- Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell, by Mary Kay Stein, Randi A. Engle, Margaret S. Smith and Elizabeth K. Hughes
- Picture Books as an Impetus for Kindergartners' Mathematical Thinking, Marja van den Heuvel-Panhuizen and Sylvia van den Boogaard
- Mathematics Teaching and Learning as a Mediating Process: The Case of Tape Diagrams, by Aki Murata
- Do You Want Me to Do It with Probability or with My Normal Thinking? Horizontal and Vertical Views on the Formation of Stochastic Conceptions, by Susanne Prediger, Germany
- Teachers’ Perceptions of Mathematics Content Knowledge Assessments in Professional Development Courses, by Michelle T. Chamberlin, Robert A. Powers and Jodie D. Novak, USA
- Mathematics Anxiety Among 4th And 5th Grade Turkish Elementary School Students, by Fulya Yüksel-Şahin, Türkiye
- A Comparison of Placement in First-Year University Mathematics Courses Using Paper and Online Administration of a Placement Test, by Phyllis A. Schumacher and Richard M. Smith, USA
- Senior Student Teachers’ Understanding of Relations Between Function, Equation, and Polynomial Concepts as Conceptual Knowledge, Danyal Soybas, Yılmaz Aksoy and Hayri Akay, Türkiye
- Overview of research on teacher knowledge, with reference to pupil attainment
- Studying the mathematical work of teaching in order to evaluate construct equivalence of the teacher knowledge measures in new settings
- Using the mathematical quality of instruction to validate the multiple-choice measures of teacher knowledge
- Issues related to translation and cultural adaptation of measures
Delaney, S., Ball, D., Hill, H., Schilling, S., and Zopf, D. (2008). “Mathematical knowledge for teaching”: adapting U.S. measures for use in Ireland. Journal of Mathematics Teacher Education, 1(3):171-197.
The introduction of the concept of the variable represents a critical point in the arithmetic–algebraic transition. This concept is complex because it is used with different meanings in different situations. Its management depends on the particular way of using it in problem-solving. The aim of this paper was to analyse whether the notion of “unknown” interferes with the interpretation of the variable “in functional relation” and the kinds of languages used by the students in problem-solving. We also wanted to study the concept of the variable in the process of translation from algebraic language into natural language. We present two experimental studies. In the first one, we administered a questionnaire to 111 students aged 16–19 years. Drawing on the conclusions of this research we carried out the second study with two pairs of students aged 16–17 years.
We examine teachers’ classroom activities with the spreadsheet, focusing especially on episodes marked by improvisation and uncertainty. The framework is based on Saxe’s cultural approach to cognitive development. The study considers two teachers, one positively disposed towards classroom use of technology, and the other not, both of them experienced and in a context in which spreadsheet use was compulsory: a new curriculum in France for upper secondary non-scientific classes. The paper presents and contrasts the two teachers in view of Saxe’s parameters, and analyzes their activity in two similar lessons. Goals emerging in these lessons show how teachers deal with instrumented techniques and the milieu under the influence of cultural representations. The conclusion examines the contribution that the approach and the findings can bring to understanding technology integration in other contexts, especially teacher education.
Here is the abstract of their article:
Shulman (1986, 1987) coined the term pedagogical content knowledge (PCK) to address what at that time had become increasingly evident—that content knowledge itself was not sufficient for teachers to be successful. Throughout the past two decades, researchers within the field of mathematics teacher education have been expanding the notion of PCK and developing more fine-grained conceptualizations of this knowledge for teaching mathematics. One such conceptualization that shows promise is mathematical knowledge for teaching—mathematical knowledge that is specifically useful in teaching mathematics. While mathematical knowledge for teaching has started to gain attention as an important concept in the mathematics teacher education research community, there is limited understanding of what it is, how one might recognize it, and how it might develop in the minds of teachers. In this article, we propose a framework for studying the development of mathematical knowledge for teaching that is grounded in research in both mathematics education and the learning sciences.
The purpose of the project reported in this article was to evaluate how assessing teachers’ mathematical knowledge within a professional development course impacted from the teachers’ perspective their learning and their experience with the course. The professional development course consisted of a 2-week summer institute and the content focus was geometry. We had decided to assess the mathematical learning of the teachers during this professional development course for various accountability reasons, but were concerned about possible negative by-products of this decision on the teachers and their participation. Thus, we worked to design assessment in ways that we hoped would minimize negative impacts and maintain a supportive learning environment. In addition, we undertook this evaluation to examine the impacts of the assessment, which included homework, quizzes, various projects, and an examination for program evaluation. Seventeen grade 5–9 teachers enrolled in the course participated in the study by completing written reflections and by describing their experiences in interviews. We learned that while our original intent was “to do no harm,” the teachers reported that their learning was enhanced by the assessment. The article concludes by describing the various properties of the assessments that the teachers identified as contributing to their learning of the geometry content, many of which align with current recommendations for assessing and evaluating grade K-16 mathematics students.
Some teaching projects in which the learning of mathematics was combined with mainly theatrical productions are reported on. They are related and opposed to an approach of drama in education by Pesci and the proposals of Sinclair for mathematics teaching and beauty. The analysis is based on the distinction between aesthetics as related to beauty or as related to sensual perception. The usefulness of concepts of model and metaphor for the understanding of aesthetic representations of mathematical subject matter is examined. It is claimed that the Peircean concept of the interpretant contributes to a concise analytical approach. The pedagogical attitude is committed to a balanced relationship of scientific and aesthetic values.
Two questions are on the mind of many mathematics educators; namely: What is the mathematics that we should teach in school? and how should we teach it? This is the second in a series of two papers addressing these fundamental questions. The first paper (Harel, 2008a) focuses on the first question and this paper on the second. Collectively, the two papers articulate a pedagogical stance oriented within a theoretical framework called DNR-based instruction in mathematics. The relation of this paper to the topic of this Special Issue is that it defines the concept of teacher’s knowledge base and illustrates with authentic teaching episodes an approach to its development with mathematics teachers. This approach is entailed from DNR’s premises, concepts, and instructional principles, which are also discussed in this paper.
In the late seventies, Guy Brousseau set himself the goal of verifying experimentally a theory he had been building up for a number of years. The theory, consistent with what was later named (non-radical) constructivism, was that children, in suitable carefully arranged circumstances, can build their own knowledge of mathematics. The experiment, carried out by a team of researchers and teachers that included his wife, Nadine, in classrooms at the École Jules Michelet, was to teach all of the material on rational and decimal numbers required by the national programme with a carefully structured, tightly woven and interdependent sequence of “situations.” This article describes and discusses the third portion of that experiment.
A quick read revealed that it wasn’t. The correct failure rate was 10.2 percent, but the error made by the journalist, and presumably approved by the editor, was perhaps more revealing about the true position of mathematics nationally. They reasoned that since 4.5 percent of students had failed the higher level paper, 5.7 percent had failed the foundation level and 12.3 percent had failed the ordinary level paper, it must follow that 4.5 + 5.7 + 12.3 = 22.5 percent of students had failed mathematics. The enormity of such an error and its ability to reach the front page illustrates clearly that many of us are functionally innumerate.The article further gives a nice insight into the situation for mathematics education in Ireland, and although it is more of a news article than a scientific paper, it might be worth reading.
Are only certain people destined to be multicreative—capable of unique and meaningful contributions across unrelated domains? In this article, we argue that all students have multicreative potential. We discuss this argument in light of different conceptions of creativity and assert that the likelihood of expressing multicreative potential varies across levels of creativity (most likely at smaller-c levels of creativity; least likely at professional and eminent levels of creativity). We close by offering considerations for how math educators might nurture the multicreative potential of their students.
Research interest in the professional knowledge of mathematics teachers has grown considerably in recent years. In the COACTIV project, tests of secondary mathematics teachers’ pedagogical content knowledge (PCK) and content knowledge (CK) were developed and implemented in a sample of teachers whose classes participated in the PISA 2003/04 longitudinal assessment in Germany. The present article investigates the validity of the COACTIV constructs of PCK and CK. To this end, the COACTIV tests of PCK and CK were administered to various “contrast populations,” namely, candidate mathematics teachers, mathematics students, teachers of biology and chemistry, and advanced school students. The hypotheses for each population’s performance in the PCK and CK tests were formulated and empirically tested. In addition, the article compares the COACTIV approach with related conceptualizations and findings of two other research groups.
In this article, I consider the history of the International Commission on Mathematical Instruction (ICMI) from its inception until the International Congress on Mathematical Education (ICME) held in 1969. In this period, mathematics education developed as a scientific discipline. My aim is to study the presence and the contribution of women (if any) in this development. ICMI was founded in 1908, but my history starts before then, at the end of the nineteenth century, when the process of internationalization of mathematics began, thanks to the first International Congress of Mathematicians. Already in those years, the need for internationalizing the debate on mathematics teaching was spreading throughout the mathematical community. I use as my main sources of information the didactics sections in the proceedings of the International Congresses of Mathematicians and the proceedings of the first ICME. The data collected are complemented with information from the editorial board of two journals that for different reasons are linked to ICMI: L’Enseignement Mathématique and Educational Studies in Mathematics. In particular, as a result of my analyses, I have identified four women who may be considered as pioneer women in mathematics education. Some biographical notes on their professional life are included in the paper.
- Ancient accounting in the modern mathematics classroom, by Kathleen Clark and Eleanor Robson
- The influence of Amatino Manucci and Luca Pacioli, by Fenny Smith
- A teaching module on the history of public-key cryptography and RSA, by Uffe Thomas Jankvist
- The history of symmetry and the asymmetry of history, by Peter M. Neumann
- A mathematical walk in Surrey, by Simon R. Blackburn
Educational Studies in Mathematics has released the October issue of this year, with a special focus on "The role and use of examples in mathematics education". The articles in the issue include:
- Intuitive nonexamples: the case of triangles, by Pessia Tsamir, Dina Tirosh and Esther Levenson
- Using learner generated examples to introduce new concepts, by Anne Watson and Steve Shipman
- Doctoral students’ use of examples in evaluating and proving conjectures, by Lara Alcock and Matthew Inglis
- Exemplifying definitions: a case of a square, by Rina Zazkis and Roza Leikin
- The purpose, design and use of examples in the teaching of elementary mathematics, by Tim Rowland
- Characteristics of teachers’ choice of examples in and for the mathematics classroom, by Iris Zodik and Orit Zaslavsky
- Shedding light on and with example spaces, by Paul Goldenberg and John Mason
Journal of Mathematics Teacher Education has released the September issue with the following highlights:
- How can research be used to inform and improve mathematics teaching practice? by Anne D. Cockburn
- Promoting student collaboration in a detracked, heterogeneous secondary mathematics classroom, by Megan E. Staples
- Using a video-based curriculum to develop a reflective stance in prospective mathematics teachers, by Shari L. Stockero
- What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems, by Sandra Crespo and Nathalie Sinclair
- Mathematical preparation of elementary teachers in China: changes and issues, by Yeping Li, Dongchen Zhao, Rongjin Huang and Yunpeng Ma
International Journal of Science and Mathematics Education has released the September issue of this year with the following articles:
- Effects of advance organiser strategy during instruction on secondary school students’ mathematics achievement in Kenya’s Nakuru district, by Bernard N. Githua and Rachel Angela Nyabwa
- Examining Reflective Thinking: A Study of Changes in Methods Students’ Conceptions and Understandings of Inquiry Teaching, by Jing-Ru Wang and Sheau-Wen Lin
- Following Young Students’ Understanding of Three Phenomena in which Transformations of Matter Occur, by Lena Löfgren and Gustav Helldén
- Secondary School Students’ Construction and Use of Mathematical Models in Solving Word Problems, by Salvador Llinares and Ana Isabel Roig
- Cognitive Incoherence of Students Regarding the Establishment of Universality of Propositions through Experimentation/Measurement, by Mikio Miyazaki
- Differentials in Mathematics Achievement among Eighth-Grade Students in Malaysia, by Noor Azina Ismail and Halimah Awang
- THAI GRADE 10 AND 11 STUDENTS’ UNDERSTANDING OF STOICHIOMETRY AND RELATED CONCEPTS, by Chanyah Dahsah and Richard Kevin Coll
- The Inquiry Laboratory as a Source for Development of Metacognitive Skills, by Mira Kipnis and Avi Hofstein
Otherwise, For the learning of mathematics has released issue 2 of this year.
Today, I discovered in Carlos Torres' blog that the keynote presentations are actually available online, on Slideshare! (Take a look at Cartoni21's slideshows!) These were the main presentations:
1. Barbara Jaworski's opening talk:
2. Guershon Harel's presentation
3. The presentation of Jean-Baptiste Lagrange
4. Laurinda Brown's talk
5. Günther Törner's presentation
We study in this article mathematics teachers’ documentation work: looking for resources, selecting/designing mathematical tasks, planning their succession, managing available artifacts, etc. We consider that this documentation work is at the core of teachers’ professional activity and professional development. We introduce a distinction between available resources and documents developed by teachers through a documentational genesis process, in a perspective inspired by the instrumental approach. Throughout their documentation work, teachers develop documentation systems, and the digitizing of resources entails evolutions of these systems. The approach we propose aims at seizing these evolutions, and more generally at studying teachers’ professional change.
Jo Boaler wrote an interesting article in Education Week, which was published online on Friday. The article is entitled Where Has All the Knowledge Gone? The Movement to Keep Americans at the Bottom of the Class in Math. In the article she gives some interesting reflections concerning the report of the National Math Panel, about the "anti-knowledge movement" in the U.S., about the Math Wars, and about the development of mathematics education in the U.S. in general. Boaler claims that:
There is a movement at work across America that smothers research knowledge, gives misleading data to parents, and substantially undermines our ability to improve American children’s mathematical understanding.And she claims that this movement has had a strong impact - even into the White House...
Non-attendance to meaning by students is a prevalent phenomenon in school mathematics. Our goal is to investigate features of instruction that might account for this phenomenon. Drawing on a case study of two high school algebra teachers, we cite episodes from the classroom to illustrate particular teaching actions that de-emphasize meaning. We categorize these actions as pertaining to (a) purpose of new concepts, (b) distinctions in mathematics, (c) mathematical terminology, and (d) mathematical symbols. The specificity of the actions that we identify allows us to suggest several conjectures as to the impact of the teaching practices observed on student learning: that students will develop the belief that mathematics involves executing standard procedures much more than meaning and reasoning, that students will come to see mathematical definitions and results as coincidental or arbitrary, and that students’ treatment of symbols will be largely non-referential.
In this study, we investigated the extent of knowledge in mathematics and pedagogy that Chinese practicing elementary mathematics teachers have and what changes teaching experience may bring to their knowledge. With a sample of 18 mathematics teachers from two elementary schools, we focused on both practicing teachers’ beliefs and perceptions about their own knowledge in mathematics and pedagogy and the extent of their knowledge on the topic of fraction division. The results revealed a gap between these teachers’ limited knowledge about the curriculum they teach and their solid mathematics knowledge for teaching, as an example, fraction division. Moreover, senior teachers used more diverse strategies that are concrete in nature than junior teachers in providing procedural justifications. The results suggested that Chinese practicing teachers benefit from teaching and in-service professional development for the improvement of their mathematics knowledge for teaching but not their knowledge about mathematics
In this paper we focus on an instructional sequence that aims at supporting students in their learning of the basic principles of rate of change and velocity. The conjectured process of teaching and learning is supposed to ensure that the mathematical and physical concepts will be rooted in students’ understanding of everyday-life situations. Students’ inventions are supported by carefully planned activities and tools that fit their reasoning. The central design heuristic of the instructional sequence is emergent modeling. We created an educational setting in three tenth grade classrooms to investigate students’ learning with this sequence. The design research is carried out in order to contribute to a local instruction theory on calculus. Classroom events and computer activities are video-taped, group work is audio-taped and student materials are collected. Qualitative analyses show that with the emergent modeling approach, the basic principles of calculus can be developed from students’ reasoning on motion, when they are supported by discrete graphs.
As a young field in its own right (unlike the ancient discipline of mathematics), mathematics education research has been eclectic in drawing upon the established knowledge bases and methodologies of other fields. Psychology served as an early model for a paradigm that valorized psychometric research, largely based in the theoretical frameworks of cognitive science. More recently, with the recognition of the need for sociocultural theories, because mathematics is generally learned in social groups, sociology and anthropology have contributed to methodologies that gradually moved away from psychometrics towards qualitative methods that sought a deeper understanding of issues involved. The emergent perspective struck a balance between research on individual learning (including learners’ beliefs and affect) and the dynamics of classroom mathematical practices. Now, as the field matures, the value of both quantitative
and qualitative methods is acknowledged, and these are frequently combined in research that uses mixed methods, sometimes taking the form of design experiments or multi-tiered teaching experiments. Creativity and rigor are required in all mathematics education research, thus it
is argued in this paper, using examples, that characteristics of both the arts and the sciences are implicated in this work.
In this paper we analyse the language used by kindergarten children and their teacher while they discuss the fairness of two games that involved the concept of chance. Their discussions show that the children are able to overcome their primary intuitions concerning the fairness of a game and to comprehend the important role of materials. The children mostly used counting strategies in order to justify their opinion; this reveals the establishment of a primary discursive community based on the premise that each opinion should be justified in order to be accepted by the other children and the teacher.
In this paper, we describe how the combination of two theories, each embedded in a different realm, may contribute to evaluating teachers’ knowledge. One is Shulman’s theory, embedded in general, teacher education, and the other is Fischbein’s theory, addressing learners’ mathematical conceptions and misconceptions. We first briefly describe each of the two theories and our suggestions for combining them, formulating the Shulman–Fischbein framework. Then, we present two research segments that illustrate the potential of the implementation of the Shulman–Fischbein framework to the study of mathematics teachers’ ways of thinking. We conclude with general comments on possible contributions of combining theories that were developed in mathematics education and in other domains to mathematics teacher education.
In the past decades, the CHC (Confucian heritage culture) learner’s phenomenon has spawned one of the most fruitful fields in educational research. Despite the impression that CHC learners are brought up in an environment not conducive to learning, their academic performances have been excelling their Western counterparts (Fan et al. in How Chinese learn mathematics: perspectives from insiders, 2004). Numerous explanations were offered to reveal the paradox (Morrison in Educ J, 2006), and there were challenges of whether there is “over-Confucianisation” in all these discussions (Chang in J Psychol Chin Soc, 2000; Wong and Wong in Asian Psychol, 2002). It has been suggested that the East and the West should come and discuss at the “middle zone” so that one can get the best from the two worlds. On the other hand, at the turn of the new millennium, discussions on mathematics curriculum reform proliferate in many places. One of the foci of the debate is the basic skills—higher-order thinking “dichotomy”. Viewing from the perspective of the process of mathematisation, teaching mathematics is more than striking a balance between the two, but to bridge basic skills to higher-order thinking competences. Such an attempt was explored in recent years and the ideas behind will be shared in this paper.
This interview study takes place in the context of a single small district in the United States. In the algebra curriculum of this district, there was a shift in the conception of equation, from a statement about unknown numbers to a question about the comparison of two functions over the domain of the real numbers. Using two of Shulman’s [Shulman, L. S. (1986). Paradigms and research programs in the study of teaching: A contemporary perspective. In Wittrock, M. C. (Ed.), Handbook of research in teaching (3rd ed., pp. 3–36). New York: Macmillan] categories of teachers’ knowledge – pedagogical content knowledge and curricular content knowledge – we explore whether in this context teachers’ content knowledge give signs of being reorganized. Our findings suggest that the teachers see this conception of equation as useful for equations in one variable. They struggle with its ramifications for equations in two variables. Nonetheless, this conception of equation leads them to reflect on the algebra curriculum in substantial ways; two of the three teachers explicitly spoke about their curricular ideas as being associated with this conception of an equation or with their earlier views. The third teacher seems so taken with these curricular ideas that he explored their ramifications throughout the interview. We argue that the consideration of this new conception of equation was an important resource that the teachers used to construct their understandings of alternative curricular approaches to school algebra. As they work with this new conception of an equation, we find an analogy to their situation in Kuhn’s description of the individual scientist in the process of adopting a new paradigm.