Mathematics assessment in East Asia

Frederick K.S. Leung from The University of Hong Kong has written an article in ZDM about assessment in East Asia. The article is entitled In the books there are golden houses: mathematics assessment in East Asia, and it was published online on Tuesday. The paper is an adaption of a plenary lecture that Leung presented at the Third East Asian Regional Conference on Mathematics Education in Shanghai, August 2005. Here is the article abstract:
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.

Semi-virtual seminar in mathematics education

Matthias Ludwig, Wolfgang Müller and Binyan Xu have written an article about A Sino-German semi-virtual seminar in mathematics education. The article was recently published in ZDM. Here is the abstract of their article:
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.

Working with artefacts

Michela Maschietto and Maria G. Bartolini Bussi have written an article entitled Working with artefacts: gestures, drawings and speech in the construction of the mathematical meaning of the visual pyramid. The article was published online in Educational Studies in Mathematics two days ago. Here is a copy of the abstract:
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.


Empirical research on mathematics teachers

Sigrid Blömeke, Gabriele Kaiser, Rainer Lehmann and William H. Schmidt have written an article that has been entitled: Introduction to the issue on Empirical research on mathematics teachers and their education. The article was published in ZDM some days ago. The article is without an abstract, and it appears to be the editorial of the forthcoming issue of ZDM. This issue will have a main focus on results from the international comparative study: "Mathematics Teaching in the 21st Century (MT21)". So, it appears as if those of us who are interested in the preparation of teachers, teacher education, teacher knowledge, etc. are up for an interesting next issue of ZDM!

What's all the fuss about gestures?

Over the last years, the focus on gestures in mathematics education research has been growing. Anna Sfard has now written an article that was published in Educational Studies in Mathematics a couple of days ago. The article has a focus on this particular field of research, and it is entitled: What’s all the fuss about gestures? A commentary. Here is the abstract:
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.

Measuring quality of mathematics teaching in early childhood

Carolyn R. Kilday and Mable B. Kinzie have written an article called An Analysis of Instruments that Measure the Quality of Mathematics Teaching in Early Childhood. This article was published online in Early Childhood Education Journal on Friday. A starting point for this article (both authors work at the University of Virginia, in the U.S.) is that "the National Mathematics Advisory Panel (2008) has recently called for more research to determine the skills and practices underlying teacher effectiveness, and on methods for developing this capacity". The article gives an interesting overview of some of the major instruments for evaluating and measuring teaching quality in the U.S. Here is the abstract of the article:
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.


ZDM, November 2008

Along with Educational Studies in Mathematics and Journal of Mathematics Teacher Education, ZDM has also recently published their November issue of this year. This issue contains a long list of interesting articles:
The theme of this issue is: Mathematics Education: New Perspectives on Gender.

JMTE, November 2008

The November issue of Journal of Mathematics Teacher Education has been published, and it contains the following set of articles:
Personally, I find this issue particularly interesting, as it has a strong focus on mathematical content knowledge as well as beliefs. These are the main focus areas of my own research as well. I especially find the article by Silverman and Thompson interesting, and their attempt to approach a framework for the development of mathematical knowledge for teaching provides a nice overview of the research that has been done after Lee Shulman presented his ideas about Pedagogical Content Knowledge.

ESM, November issue

The November issue of Educational Studies in Mathematics has arrived. It contains the following articles:


Estimating Iraqi deaths

Brian Greer's article, which was published in ZDM two days ago, surely has an interesting title: Estimating Iraqi deaths: a case study with implications for mathematics education. The focus of this article is also interesting:
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.

Knowledge and confidence of pre-service mathematics teachers

Yeping Li and Gerald Kulm have written an interesting article that was published in ZDM on Tuesday. The article is entitled Knowledge and confidence of pre-service mathematics teachers: the case of fraction division. Here is the abstract of the article:
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.

MTL, Volume 10 Issue 4 2008

Issue 4 of Mathematical Thinking and Learning has been published with the following main articles:


IEJME, October 2008

The October issue of International Electronic Journal of Mathematics Education has been published. It has the following articles (links to the article abstracts):


Seminar with Sean Delaney

Thursday and Friday last week, we had an interesting seminar at University of Stavanger with Seán Delaney from Marino Institute of Education, Ireland. The seminar had four themes, all within the topic of mathematical knowledge for teaching (MKT):
  1. Overview of research on teacher knowledge, with reference to pupil attainment
  2. Studying the mathematical work of teaching in order to evaluate construct equivalence of the teacher knowledge measures in new settings
  3. Using the mathematical quality of instruction to validate the multiple-choice measures of teacher knowledge
  4. Issues related to translation and cultural adaptation of measures
Seán Delaney has been part of the Learning Mathematics for Teaching (LMT) Project at University of Michigan, and he finished his PhD earlier this year. His thesis was entitled Adapting and using U.S. measures to study Irish teachers' mathematical knowledge for teaching, and he had Deborah Ball as his main supervisor. In the June issue of Journal of Mathematics Teacher Education, an article about the pilot phase of Delaney's study was published:

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.

From arithmetical thought to algebraic thought

Elsa Malisani and Filippo Spagnolo have written an article called From arithmetical thought to algebraic thought: The role of the “variable”. This article was published online in Educational Studies in Mathematics last week. Here is the article abstract:
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.


Teachers' goals in spreadsheet-based lessons

Jean-Baptiste Lagrange and Emel Ozdemir Erdogan have written an article called Teachers’ emergent goals in spreadsheet-based lessons: analyzing the complexity of technology integration, which was published in Educational Studies in Mathematics on Tuesday. Here is the abstract of the article:
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.

Mathematical knowledge for teaching

Jason Silverman and Patrick W. Thompson have written an interesting article entitled Toward a framework for the development of mathematical knowledge for teaching. This article was published online in Journal of Mathematics Teacher Education on October 14. In the article, they draw upon the research that has been done in the area of Mathematical Knowledge for Teaching (MKT), and they try to navigate towards a framework for this. Silverman and Thompson present a framework that is "not only informed by the work of mathematics teaching, but also a developmental trajectory for mathematics learning and the learning sciences" (from their concluding comments).

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.

Teachers' perceptions of assessments

Michelle T. Chamberlin, Jeff D. Farmer and Jodie D. Novak have written an article called Teachers’ perceptions of assessments of their mathematical knowledge in a professional development course. The article was published online in Journal of Mathematics Teacher Education a couple of days ago. Here is the abstract:
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.

Mathematics learning and aesthetic production

Herbert Gerstberger has written an interesting article about the connection between Mathematics learning and aesthetic production. In the article, he introduces several interesting aspects concerning aesthetics, arts, metaphor, semiotics, etc. The article was published online in ZDM, two days ago. Here is the article abstract:
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.

A DNR perspective on mathematics curriculum and instruction

Guershon Harel has written an article called A DNR perspective on mathematics curriculum and instruction. Part II: with reference to teacher’s knowledge base, which was published online in ZDM on Tuesday this week. In this article, Harel touches upon many interesting issues concerning the teaching and learning of mathematics. Here is the abstract of the article:
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.


Rationals and decimals

Guy Brousseau, Nadine Brousseau and Virginia Warfield have written an article called Rationals and decimals as required in the school curriculum Part 3. Rationals and decimals as linear functions. The article was published in The Journal of Mathematical Behavior a few days ago. Here is the abstract of the article:
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.


Is there a crisis in maths education

Brendan Goldsmith, Professor at Dublin Institute of Technology has written an interesting article about the crisis in maths education in Ireland. This article was published under the "Opinion" section of Trinity News.ie. The introduction deals with what a Dublin paper referred to as a crisis, where more than 20 percent of the students had failed mathematics when the "Leaving Certificate" results were published. A quick review of Professor Goldsmith revealed that the crisis was more severe on the newspaper's side:
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.


Do we all have multicreative potential?

Ronald A. Beghetto and James C. Kaufman have written an article in ZDM that was published on Friday. The article is entitled Do we all have multicreative potential? and it deals with the issue of creativity and multicreativity. Here is the abstract of the article:
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.

Secondary mathematics teachers' pedagogical content knowledge

Stefan Krauss, Jürgen Baumert and Werner Blum have written an article entitled Secondary mathematics teachers’ pedagogical content knowledge and content knowledge: validation of the COACTIV constructs. The article (which is an Open Access article!) was published online in ZDM last week. This is a very interesting article, which gives a nice contribution to the field of research related to teachers' knowledge. It builds upon the framework of Shulman, and it gives a nice overview of these theories, as well as an overview of some of the other research projects that have been contributing to this field (like the study of Ball, Hill, Schilling et al. in Michigan). Here is the abstract of the article:
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.

The emergence of women

Fulvia Furinghetti has written an article about The emergence of women on the international stage of mathematics education. This article was published online in ZDM last week. The article has a particular focus on women in the history of ICMI. Here is the article abstract:
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.

BSHM Bulletin


Updates on the major journals

I have written a lot about new articles that have been published in the major journals lately, but not so much about updates on new issues of these journals. Here is an overview of some of the latest news from the major journals:

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:

Journal of Mathematics Teacher Education has released the September issue with the following highlights:

International Journal of Science and Mathematics Education has released the September issue of this year with the following articles:

Otherwise, For the learning of mathematics has released issue 2 of this year.


YESS-4 revisited

In August, the 4th version of the YERME Summer School (YESS-4) was organized in Turkey. I wrote about this event in several blogposts. (Click on all the marked words for links to the various articles!)

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:

Yess4 Barbara Jaworski
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2. Guershon Harel's presentation

Guershon HAREL
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3. The presentation of Jean-Baptiste Lagrange

Yess4 Jean-baptiste Lagrange
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4. Laurinda Brown's talk

Yess 4 Laurinda Brown
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5. Günther Törner's presentation

Yess 4 Günter Törner
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Documentation systems

Ghislaine Gueudet and Luc Trouche have written an article about mathematics teachers' documentation work. The article is called Towards new documentation systems for mathematics teachers? In my Master thesis, I wrote about genesis principles - in particular historical genesis (the use of history of mathematics in an indirect approach) - and Gueudet and Trouche introduce the concept of "documentational genesis" which I find interesting! The article was published online in Educational Studies in Mathematics a couple of days ago. Here is the abstract of their article:
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.


Where has all the knowledge gone?

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...


Attention to meaning by algebra teachers

Guershon Harel, Evan Fuller and Jeffrey M. Rabin have written an article that was published online in The Journal of Mathematical Behavior on Wednesday. The article is entitled Attention to meaning by algebra teachers. Here is the article abstract:
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.

Chinese teachers' knowledge

Yeping Li and Rongjin Huang have written an article called Chinese elementary mathematics teachers’ knowledge in mathematics and pedagogy for teaching: the case of fraction division. The article was published online in ZDM on Wednesday. Here is the abstract of the article:

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

Emergent modeling

L.M. Doorman and K.P.E. Gravemeijer have written an article entitled Emergent modeling: discrete graphs to support the understanding of change and velocity. The article was recently published online in ZDM. This article was published as an Open Access article, so it should be freely available to all! Here is the article abstract:
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.

Embracing arts and sciences

Norma Presmeg has written an article with the interesting perspective: Mathematics education research embracing arts and sciences. The article was published in ZDM on Wednesday.Here is the article abstract:

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.

The fairness of probabilistic games

Konstantinos Tatsis, Sonia Kafoussi and Chrysanthi Skoumpourdi have written an article called Kindergarten Children Discussing the Fairness of Probabilistic Games: The Creation of a Primary Discursive Community. The article was recently published in Early Childhood Education Journal. Here is the abstract of the article:
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.


Combining theories

Pessia Tsamir and Dina Tirosh have written an article about Combining theories in research in mathematics teacher education. This article was published in ZDM two days ago. In this interesting article, they examine how the combination of the theories of Shulman and Fischbein "may contribute to the evaluation of mathematics teachers' (prospective and inservice) knowledge". Here is the article abstract:
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.

Confucian heritage culture learner's phenomenon

Ngai-Ying Wong from The Chinese University of Hong Kong, has written an article with the interesting title: Confucian heritage culture learner’s phenomenon: from “exploring the middle zone” to “constructing a bridge”. The article was published online in ZDM on Tuesday. The article gives some interesting insight into aspects of the Chinese culture, and it did represent several new issues and aspects to me. Besides, it is the first scientific article that I have ever seen (within our field, at least) that includes martial-art pictures. In the article, Wong also draws upon variation theory (which derives from the work of Swedish scholar Ference Marton and colleagues). Here is the abstract of the article:
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.


An analytic conception of equation

Daniel Chazan, Michael Yerushalmy and Roza Leikin have written an article that was published online in The Journal of Mathematical Behavior yesterday. The article is entitled An analytic conception of equation and teachers’ views of school algebra, and here is the abstract:
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.