These women were often groundbreakers, highly determined and very dedicated. They are shining examples of the fact that mathematics is not a “boys only” club, even if at many points in time it’s appeared that way on the surface. Today their work is recognized and appreciated, and they stand as fantastic sources of inspiration for a new generation of students and math enthusiasts – both female and male.

## 2008/09/29

### 10 remarkable female mathematicians

## 2008/09/26

### The work of teaching and the challenge for teacher education

Here is a very interesting presentation held by Deborah L. Ball on a visit to Vanderbilt University. Ball has been in charge of several major projects concerning the teachers' knowledge of mathematics, e.g. the LMT (Learning Mathematics for Teaching) project, and she was also a member of the National Mathematics Advisory Panel, which delivered an important report earlier this year. (See one of my previous posts about this!)

## 2008/09/25

### Reversibility of thought

In line with current efforts to understand the piece-by-piece structure and articulation of children’s mathematical concepts, this case study compares the reversibility schemes of two eighth-grade students. The aim of the study was to identify the mechanism through which students reverse their thought processes in a multiplicative situation. Data collected through clinical interviews depict the precise strategies that the participants used to work back to find the missing values in an inverse proportional task. This study also illustrates how a conceptual template generated by one of the participants afforded him considerable flexibility in the multiplicative task. Another outcome of the study is that it shows how the numerical characteristics of the parameters in the problem affected the students’ ability to reverse their thought processes. We infer that there is a need for further research on how students might represent their reversibility schemes in the form of algebraic equations.

### Learning about infinity

Based on an empirical study, we explore children’s primary and secondary perceptions on infinity. When discussing infinity, children seem to highlight three categories of primary perceptions: processional, topological, and spiritual. Based on their processional perception, children see the set of natural numbers as being infinite and endowwith a discrete structure by making transfers fromQtoN. In a continuous context, children are more likely to mobilize a topological perception. Evidence for a secondary perception of arises from students’ propensities to develop infinite sequences of natural numbers, and from their ability to prove thatQis infinite. Children’s perceptions on infinity change along the school years. In general, the perceptual dominance moves from sequential (processional) to topological across development. However, we found that around 11–13 years old, processional and topological perceptions interfere with each other, while before and after this age they seem to coexist and collaborate, one or the other being specifically activated by the nature of different tasks.N

## 2008/09/24

### Multiplication as original sin

This article describes the use of a case report, Multiplication as original sin (Corwin, R. B. (1989). Multiplication as original sin. Journal of Mathematical Behavior, 8, 223–225), as an assignment in a mathematics course for preservice elementary teachers. In this case study, Corwin described her experience as a 6th grader when she revealed an invented algorithm. Preservice teachers were asked to write reflections and describe why Corwin’s invented algorithm worked. The research purpose was: to learn about the preservice teachers’ understanding of Corwin’s invented multiplication algorithm (its validity); and, to identify thought-provoking issues raised by the preservice teachers. Rather than using mathematical properties to describe the validity of Corwin’s invented algorithm, a majority of them relied on procedural and memorized explanations. About 31% of the preservice teachers demonstrated some degree of conceptual understanding of mathematical properties. Preservice teachers also made personal connections to the case report, described Corwin using superlative adjectives, and were critical of her teacher.

### The fragility of group flow

This article considers two small groups of students in the same Grade 8 mathematics classroom whose approaches to the same mathematical problem result in very different experiences. Using videotapes and written transcripts, an analysis of the groups’ working processes was undertaken using Sawyer's pre-existing structures required for the presence of group flow, and Davis and Simmt's conditions for complex systems. It is suggested that although both groups had the prerequisite structures in place to experience group flow, the second group was not decentralized enough to enable all members to establish a working collaborative proximal zone of development in which they could develop their ideas as a collective, while the first group was sufficiently decentralized and appeared to demonstrate episodes of experiencing group flow. If teachers are aware of conditions that encourage the experience of group flow, this may help them in forming productive small groups within the classroom and developing successful group-oriented learning tasks.

## 2008/09/23

### Aesthetics as a liberating force

This article investigates different meanings associated with contemporary scholarship on the aesthetic dimension of inquiry and experience, and uses them to suggest possibilities for challenging widely held beliefs about the elitist and/or frivolous nature of aesthetic concerns in mathematics education. By relating aesthetics to emerging areas of interest in mathematics education such as affect, embodiment and enculturation, as well as to issues of power and discourse, this article argues for aesthetic awareness as a liberating, and also connective force in mathematics education.

## 2008/09/22

### Mathematical imagination and embodied cognition

The goal of this paper is to explore qualities of mathematical imagination in light of a classroom episode. It is based on the analysis of a classroom interaction in a high school Algebra class. We examine a sequence of nine utterances enacted by one of the students whom we call Carlene. Through these utterances Carlene illustrates, in our view, two phenomena: (1) juxtaposing displacements, and (2) articulating necessary cases. The discussion elaborates on the significance of these phenomena and draws relationships with the perspectives of embodied cognition and intersubjectivity.

## 2008/09/19

### Performance and language proficiency

Ireland has two official languages—Gaeilge (Irish) and English. Similarly, primary- and second-level education can be mediated through the medium of Gaeilge or through the medium of English. This research is primarily focused on students (Gaeilgeoirí) in the transition from Gaeilge-medium mathematics education to English-medium mathematics education. Language is an essential element of learning, of thinking, of understanding and of communicating and is essential for mathematics learning. The content of mathematics is not taught without language and educational objectives advocate the development of fluency in the mathematics register. The theoretical framework underpinning the research design is Cummins’ (1976). Thresholds Hypothesis. This hypothesis infers that there might be a threshold level of language proficiency that bilingual students must achieve both in order to avoid cognitive deficits and to allow the potential benefits of being bilingual to come to the fore. The findings emerging from this study provide strong support for Cummins’ Thresholds Hypothesis at the key transitions—primary- to second-level and second-level to third-level mathematics education—in Ireland. Some implications and applications for mathematics teaching and learning are presented.

### Gendering of mathematics among Israel Jewish and Arab students

In English-speaking, Western countries, mathematics has traditionally been viewed as a “male domain”, a discipline more suited to males than to females. Recent data from Australian and American students who had been administered two instruments [Leder & Forgasz, in Two new instruments to probe attitudes about gender and mathematics. ERIC, Resources in Education (RIE), ERIC document number: ED463312, 2002] tapping their beliefs about the gendering of mathematics appeared to challenge this traditional, gender-stereotyped view of the discipline. The two instruments were translated into Hebrew and Arabic and administered to large samples of grade 9 students attending Jewish and Arab schools in northern Israel. The aims of this study were to determine if the views of these two culturally different groups of students differed and whether within group gender differences were apparent. The quantitative data alone could not provide explanations for any differences found. However, in conjunction with other sociological data on the differences between the two groups in Israeli society more generally, possible explanations for any differences found were explored. The findings for the Jewish Israeli students were generally consistent with prevailing Western gendered views on mathematics; the Arab Israeli students held different views that appeared to parallel cultural beliefs and the realities of life for this cultural group.

### Lesson study in Asia Pacific classrooms

If you are interested in the topic, this article gives a nice overview of the history and theoretical background of the Japanese Lesson Study approach, and there is also a nice list of references to dig into. In the conclusions of the article, they claim:

However, the significant features of Japanese Lesson Study, such as the use of collaborative work, working on common goals, sharing of ideas, team teaching, lesson observation and cooperation among peers seemed to exert similar impacts on all groups of participants. Participants from all glocal programs reported an improvement in their lesson planning, better pedagogical content knowledge and closer collegial relationship as a result of experiencing the Lesson Study process.Here is the abstract of the article:

Japanese Lesson Study is a model for teacher professional learning that has recently attracted world attention particularly within the mathematics education community. It is a highly structured process of teacher collaboration, observation, reflection and practice. The world focus has been mainly due to the work of American researchers such as Stigler and Hiebert (Am Educ Winter:1–10, 1998; The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. Free Press, New York 1999), Lewis and Tsuchida (Am Educ Winter:14–17; 50–52, 1998) and Fernandez [J Teach Educ 53(5):395–405, 2002]. These researchers have documented Lesson Study from the perspective of their social, cultural and educational contexts. In order to develop a deeper understanding of Lesson Study in a post-modern global world, there is a need to seek views beyond those presented from an American perspective. This paper will provide further additional perspectives from an Australian state view and a Malaysian state district view and a university view. The aim is to develop an understanding of how the different contexts have influenced the structure and implementation of the Japanese Lesson Study model.

## 2008/09/18

### Relation between students' behaviors and their mathematical ideas

The purpose of this study is to analyze the relationship between student behaviors and the growth of mathematical ideas (using the Pirie–Kieren model). This analysis was accomplished through a series of case studies, involving middle school students of varying ability levels, who were investigating a combinatorics problem in after-school problem-solving sessions. The results suggest that certain types of student behaviors appear to be associated with the growth of ideas and emerge in specific patterns. More specifically, as understanding grows, there is a general shift from behaviors such as students questioning each other, explaining and using their own and others’ ideas toward behaviors involving the setting up of hypothetical situations, linking of representations and connecting of contexts. Recognizing that certain types of student behaviors tend to emerge in specific layers of the Pirie–Kieren model can be important in helping us to understand the development of mathematical ideas in children.Warner focus a lot on the Pirie-Kieren model in her theoretical framework (see the article of Susan Pirie and Thomas Kieren from 1994). The main focus of Warner's article is to address the following questions:

Are different types of student behaviors associated with the growth of mathematical ideas in specific ways? If so, how?In her conclusions, Lisa Warner suggests that for the students in her study, "certain types of behaviors appeared to be associated with the growth of mathematical ideas in certain ways". She also suggests that further research is needed in order to investigate whether these findings correspond with findings in similar studies of other students, different types of tasks, etc.

### Navigating Numeracies

The book aims to further understanding of why some pupils have low achievement in numeracy in the school context. The authors aim to achieve this by a relatively original view that focuses on numeracy as a social practice. They report on their investigations into the meanings and uses of numeracy in school and home and community contexts, using ethnographic-style approaches, including formal and informal interviews and observations. The book will be useful for policy, practice and further research into the teaching and learning of mathematics in schools. It will therefore be of interest to policy makers, teachers and practitioners, academics and practitioners in teacher education, education researchers, and parents and community leaders.

## 2008/09/16

### Beliefs seminar with Jeppe Skott

Skott started off with a session on the historical background of research on beliefs in mathematics education research. He talked about the development of teacher training in the Scandinavian countries, and he pointed to some of the major international studies in recent years. Then he lead us back to the OEEC study from the early sixties, and in this connection, he introduced Bauersfeld's three levels:

- Matter meant
- Matter taught
- Matter learnt

Significant changes in school mathematics will only be achieved if there are marked changes in the perceptions and attitudes of these teachers and if they are assisted to develop necessary new skills.A strong focus was thereby put on the teachers' perceptions and attitudes. The focus on the teacher as the main problem in the implementation process was thereby presented, and much of the research did (and still do) refer to Ernest's model of the relationship between the espoused and enacted beliefs of the mathematics teacher. A main issue here, according to Skott, is that the premise for this research is taken for granted, and it is not based on analysis of data!

As a further theoretical background for the discussion, he introduced theories concerning constructivism (radical and social) and other.

Skott then introduced us with some of his own research in this field, and he introduced the case of Christopher as an example. (See his 2001 article for more on this!) In relation to this example, Skott introduced some of his own concepts: school mathematics images (SMI) and critical incidents of practice (CIP).

On Friday, Skott brought up the difficult and interesting discussion about the nature and existence of beliefs, and how we investigate them. His initial claim was that "traditional beliefs research" had made it impossible to give a reasonable answer to the question about the

relationship between a teacher's conceptions about a subject on the one hand, and the teaching practice on the other hand. The main reason for that is that the answer has already been given as a premise for the research: there is a strong relationship between the two. This has not

been based on empirical evidence, Skott claims.

He then introduced a discussion about methods in beliefs research, and he pointed to the study he and Tine Wedege made of the Nordic KappAbel contest as an example (PDF version of the report). In a discussion of data analysis, Skott introduced the constructivist version of grounded theory presented by Charmaz (2006) as an example.

In the final round, Skott made a strong emphasis on the importance of context in beliefs research, and the implications this has on choice of research methods, etc. Some of his main points were:

- Inconsistency between beliefs and practice is from the point of view of the observer
- Consistency is situated in practice
- It is NOT the teacher's practice

## 2008/09/12

### Opportunity to learn in the preparation of teachers

Cross-national research studies such as the Program for International Student Assessment and the Third International Mathematics and Science Study (TIMSS) have contributed much to our understandings regarding country differences in student achievement in mathematics, especially at the primary (elementary) and lower secondary (middle school) levels. TIMSS, especially, has demonstrated the central role that the concept of opportunity to learn plays in understanding cross-national differences in achievement Schmidt et al., (Why schools matter: A cross-national comparison of curriculum and learning 2001). The curricular expectations of a nation and the actual content exposure that is delivered to students by teachers were found to be among the most salient features of schooling related to academic performance. The other feature that emerges in these studies is the importance of the teacher. The professional competence of the teacher which includes substantive knowledge regarding formal mathematics, mathematics pedagogy and general pedagogy is suggested as being significant—not just in understanding cross-national differences but also in other studies as well (Hill et al. in Am Educ Res J 42(2):371–406, 2005). Mathematics Teaching in the 21st Century (MT21) is a small, six-country study that collected data on future lower secondary teachers in their last year of preparation. One of the findings noted in the first report of that study was that the opportunities future teachers experienced as part of their formal education varied across the six countries (Schmidt et al. in The preparation gap: Teacher education for middle school mathematics in six countries, 2007). This variation in opportunity to learn (OTL) existed in course work related to formal mathematics, mathematics pedagogy and general pedagogy. It appears from these initial results that OTL not only is important in understanding K-12 student learning but it is also likely important in understanding the knowledge base of the teachers who teach them which then has the potential to influence student learning as well. This study using the same MT21 data examines in greater detail the configuration of the educational opportunities future teachers had during their teacher education in some 34 institutions across the six countries.

### Investigating imagination

Our work is inspired by the book Imagining Numbers (particularly the square root of minus fifteen), by Harvard University mathematics professor Barry Mazur (Imagining numbers (particularly the square root of minus fifteen), Farrar, Straus and Giroux, New York, 2003). The work of Mazur led us to question whether the features and steps of Mazur’s re-enactment of the imaginative work of mathematicians could be appropriated pedagogically in a middle-school setting. Our research objectives were to develop the framework of teaching mathematics as a way of imagining and to explore the pedagogical implications of the framework by engaging in an application of it in middle school setting. Findings from our application of the model suggest that the framework presents a novel and important approach to developing mathematical understanding. The model demonstrates in particular the importance of shared visualizations and problem-posing in learning mathematics, as well as imagination as a cognitive space for learning.

### Towards a feminist epistemology

There is, now, an extensive critical literature on gender and the nature of science, three aspects of which, philosophy, pedagogy and epistemology, seem to be pertinent to a discussion of gender and mathematics. Although untangling the inter-relationships between these three is no simple matter, they make effective starting points in order to ask similar questions of mathematics to those asked by our colleagues in science. In the process of asking such questions, a major difference between the empirical approach of the sciences, and the analytic nature of mathematics, is exposed and leads towards the definition of a new epistemological position in mathematics.

This is a version of a paper first presented at the ICME7 theme group of the International Organisation on Women and Mathematics Education, Quebec, 1992. Its present content owes much to discussion with and comments from members of that network. In addition, I would particularly like to thank Mary Barnes, Leonie Daws, Stephen Lerman and the anonymous reviewers for challenging and provoking re-working of the ideas.

## 2008/09/10

### Acquisition and use of shortcut strategies

This study aimed at analysing traditionally taught children’s acquisition and use of shortcut strategies in the number domain 20–100. One-hundred-ninety-five second, third, and fourth graders of different mathematical achievement levels participated in the study. They were administered two tasks, both consisting of a series of two-digit additions and subtractions that maximally elicit the use of the compensation (45 + 29 = _; 45 + 30 - 1 = 75 - 1 = 74) and indirect addition strategy (71 - 68 = _; 68 + 2 = 70, 70 + 1 = 71, so the answer is 2 + 1 or 3). In the first task, children were instructed to solve all items as accurately and as fast as possible with their preferred strategy. The second task was to generate at least two different strategies for each item. Results demonstrated that children of all grades and all achievement levels hardly applied the compensation and indirect addition strategy in the first task. Children’s strategy reports in the second task revealed that younger and lower achieving children did not apply these strategies because they did not (yet) discover these strategies. By contrast, older and higher achieving children appeared to have acquired these strategies by themselves. Results are interpreted in relation to cognitive psychological and socio-cultural perspectives on children’s mathematics learning.

### Social constructivism and the Believing Game

The study reported here is the third in a series of research articles (Harkness, S. S., D’Ambrosio, B., & Morrone, A. S.,in Educational Studies in Mathematics 65:235–254, 2007; Morrone, A. S., Harkness, S. S., D’Ambrosio, B., & Caulfield, R. in Educational Studies in Mathematics 56:19–38, 2004) about the teaching practices of the same university professor and the mathematics course, Problem Solving, she taught for preservice elementary teachers. The preservice teachers in Problem Solving reported that they were motivated and that Sheila made learning goals salient. For the present study, additional data were collected and analyzed within a qualitative methodology and emergent conceptual framework, not within a motivation goal theory framework as in the two previous studies. This paper explores how Sheila’s “trying to believe,” rather than a focus on “doubting” (Elbow, P., Embracing contraries, Oxford University Press, New York, 1986), played out in her practice and the implications it had for both classroom conversations about mathematics and her own mathematical thinking.

### JMTE, September 2008

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

### New IJMEST articles

- Extraction of roots of quintics by division method. Author: Raghavendra G. Kulkarni
- Modelling and inverse-modelling: experiences with O.D.E. linear systems in engineering courses. Author: Victor Martinez-Luaces

## 2008/09/08

### Online geometry resources

Dan Meyer is a high-school mathematics teacher from Santa Cruz, California. He recently decided to put his entire Geometry curriculum online. This includes every lesson plan, every handout, more than 2000 slides (in Keynote, Powerpoint and PDF) ... everything from an entire year of geometry teaching! Everything is nicely ordered for the web, so that you can follow his plans from week 1 to week 38.

In my view, as a researcher and mathematics educator, this is an exemplary action! I know, there might be several teachers out there who are going to copy his ideas, and that is okay. On the other side, this provides a very nice insight into one teacher's ideas and thinking, and being able to follow a course for an entire year like this is an excellent opportunity for a researcher as well. I only wish more teachers would follow up what Dan has done, because I think this provides an excellent example of how our "new" technologies can be used to improve our teaching profession!

I am still thinking about how I could make use of this as a researcher, and if you have ideas concerning this, please post a comment below!

## 2008/09/05

### EECERA - Using powerful mathematical ideas

An important practical technique in the project was the use of so-called "learning stories", which I personally found very fascinating! These learning stories have three important elements:

- Descriptions of what the child/children had done
- Evaluation of what the child/children had done
- Reflections concerning: What next?

Here is the abstract of Perry's presentation:

Young children can be powerful mathematical learners. This paper reports work done with preschool educators in South Australia in which powerful ideas in mathematics were identified, linked to the Developmental Learning Outcomes in the mandated South Australian curriculum, and celebrated and extended through narrative assessment.. It emphasises the processes involved in building the educators’ confidence and competence in the observation, development, implementation and assessment of meaningful mathematical learning for young children and suggests ways in which this approach can improve the mathematics education of these children without weakening the strongly held traditional principles of sound early childhood practice.

The key construct in this project is a numeracy matrix—a 56-cell table linking the powerful mathematical ideas with the Developmental Learning Outcomes through pedagogical inquiry questions. These questions are designed to ask preschool educators about their practice and how it relates to their children’s mathematics learning. Using this matrix, preschool and first year of school educators have devised and begun to trial a detailed assessment process through which they can access children’s powerful mathematical ideas, show progress as these ideas grow, and answer planning questions for each child as to what would be the most appropriate next instructional step.

The value of the paper is that it illustrates how work undertaken at the preschool level can inform similar work undertaken in the first year of school and it provides educators at both levels with an innovative way for considering the mathematics education of young children in both preschool and school.

### Constructing competence

Constructing competence: an analysis of student participation in the activity systems of mathematics classrooms. The article was published online in Educational Studies in Mathematics a couple of days ago. Here is the abstract of the article:

This paper investigates the construction of systems of competence in two middle school mathematics classrooms. Drawing on analyses of discourse from videotaped classroom sessions, this paper documents the ways that agency and accountability were distributed in the classrooms through interactions between the teachers and students as they worked on mathematical content. In doing so, we problematize the assumption that competencies are simply attributes of individuals that can be externally defined. Instead, we propose a concept of individual competence as an attribute of a person's participation in an activity system such as a classroom. In this perspective, what counts as “competent” gets constructed in particular classrooms, and can therefore look very different from setting to setting. The implications of the ways that competence can be defined are discussed in terms of future research and equitable learning outcomes.

### Progress and stagnation of gender equity

During the last decade women in Sweden have reduced men’s lead in participation in mathematics education and in professional careers as mathematicians. However, the development is uneven and slow overall. In some areas and at the highest levels women have increased their participation only marginally. Why, one may ask, is progress so slow after almost 20 years of active work from the Women and Mathematics movement in Sweden and within a society in which gender equity is highly valued at the societal and political levels? The development is described in quantitative measures going back 20 years. Several concrete and successful initiatives from the last decade intended to “de-gender” mathematics and to involve women and men alike in mathematics are described. In contrast a gender-blind position or a view of women as problems in mathematics seems to reign within some influential bodies.

### Distinguishing between mathematics classrooms

The research reported in this paper examined spoken mathematics in particular well-taught classrooms in Australia, China (both Shanghai and Hong Kong), Japan, Korea and the USA from the perspective of the distribution of responsibility for knowledge generation in order to identify similarities and differences in classroom practice and the implicit pedagogical principles that underlie those practices. The methodology of the Learner’s Perspective Study documented the voicing of mathematical ideas in public discussion and in teacher–student conversations and the relative priority accorded by different teachers to student oral contributions to classroom activity. Significant differences were identified among the classrooms studied, challenging simplistic characterisations of ‘the Asian classroom’ as enacting a single pedagogy, and suggesting that, irrespective of cultural similarities, local pedagogies reflect very different assumptions about learning and instruction. We have employed spoken mathematical terms as a form of surrogate variable, possibly indicative of the location of the agency for knowledge generation in the various classrooms studied (but also of interest in itself). The analysis distinguished one classroom from another on the basis of “public oral interactivity” (the number of utterances in whole class and teacher–student interactions in each lesson) and “mathematical orality” (the frequency of occurrence of key mathematical terms in each lesson). Classrooms characterized by high public oral interactivity were not necessarily sites of high mathematical orality. In particular, the results suggest that one characteristic that might be identified with a national norm of practice could be the level of mathematical orality: relatively high mathematical orality characterising the mathematics classes in Shanghai with some consistency, while lessons studied in Seoul and Hong Kong consistently involved much less frequent spoken mathematical terms. The relative contributions of teacher and students to this spoken mathematics provided an indication of how the responsibility for knowledge generation was shared between teacher and student in those classrooms. Specific analysis of the patterns of interaction by which key mathematical terms were introduced or solicited revealed significant differences. It is suggested that the empirical investigation of mathematical orality and its likely connection to the distribution of the responsibility for knowledge generation and to student learning ourcomes are central to the development of any theory of mathematics instruction and learning.

### Learning community of problem solvers

Virtual multidisciplinary learning communities can become an important resource helping school teachers and students to foster a culture of communication, problem solving, and technology integration. Not only does the community concept virtually enlarge the mathematical learning space, it also opens several innovative ways to connect mathematics to other subjects, namely science and language arts. This article reflects on theoretical foundations of the new interactive virtual science and mathematics learning community, CASMI, as well as the first results of its implementation. The process of designing, enacting, and analyzing virtual problem solving communities, their technological, pedagogical and social aspects as a common ground for integrating mathematical, science and reading literacy into classroom practice and pre-service teacher training in an innovative and efficient way will be discussed.

## 2008/09/04

### EECERA - symposium session

We have just finished the first symposium session at the EECERA conference in Stavanger, and I attended a session with focus on mathematics and natural science. All three presentations focused on mathematics, so I guess they could have taken away the last part of the title.

Elizabeth Dunphy from St. Patrick's College in Ireland did an excellent job to chair the session, and since I took part in one of the presentations myself, I can say that on behalf of the presenters as well as the audience.

The first presenter, Oliver Thiel from Germany, had an interesting presentation about a research project concerning teachers' attitudes towards mathematics in early childhood. He had used interviews with children, based on already developed questionaires and scales. One part, for instance, was related to mathematical beliefs, and he had taken some scales developed by Grigutsch, Raatz and Törner as a starting point. Here is the abstract of Oliver's paper:

Over the past few years the nursery school in Germany is increasingly perceived as an educational establishment instead of a child care centre. This can be seen in establishing curricula for young children, including mathematics as a domain of learning skills. In the past mathematics has not been part of the curriculum for training young children's teachers. Therefore it is not clear, what actually their understanding is concerning mathematics. Van Oers (2004) has proven that teachers would support the mathematical development of the children only on the basis of their mathematical epistemology. The study reported here investigated teachers’ attitudes towards mathematics. The questions risen are:The second presentation was held by Janne Fauskanger and myself. You can see our presentation below:A questionnaire has been developed, which included four scales, suggested by Grigutsch, Raatz and Toerner (1998). This form has been filled in by 100 teachers in Germany. For the evaluation of the questionnaires confirmatory factor analysis and structural equation modeling were used.

- Do nursery school teachers feel open or reluctant towards mathematics?
- Is mathematics seen as an abstract system of terms, rules and formulas?
- Or do the teachers see mathematics reflected in the collection and sequencing of experiences and in problem solving?
- And what activities are expected to further the development of the child’s mathematical ideas?

The most important result of this survey is that the teachers show an ambivalent behavior towards mathematics, but in general they underline the benefit for the daily living. Concerning the activities of children, only those are seen as mathematical experiences, which include numbers and shapes.

Here is our abstract:

In 2006, Norwegian schools and kindergartens were faced with new curriculum reforms. For the first time in Norway the curriculum for kindergartens has a chapter on mathematics. As these reforms are now being put into action, teachers, schools, kindergartens and local governments are asking for in-service education. Evaluation of the previous curriculum reform in compulsory school indicates that there has been little change in the way teachers teach. Our aim is to investigate and try to identify features of ‘the best’ in-service education. A natural point of departure for such a project is to analyse teachers’ knowledge (MKT;mathematical knowledge for teaching) and beliefs to be able to adjust the in-service education to the participants’ needs. Our project is therefore divided into two parts. In the first part, we are researching teachers' knowledge and beliefs, and in the second part we plan on using this knowledge to design a working model for in-service education. This presentation will focus on the research regarding teachers' knowledge, and we would like the discussion to focus on pre-school teachers' knowledge. The first step in our project will be to participate in the translation, adjustment and use of an American measuring system developed at the University of Michigan. It is important to know more about teachers’ knowledge when planning and evaluating in-service education and the measures would allow professional developers to measure teacher learning rather than just teachers’ level of satisfaction with professional workshops and in-service mathematics education can be improved. What about pre-school teachers' knowledge?The third and last presenter was Marc Wantz from Luxembourg, who talked about "Gender differences in mathematical competencies". Here is the abstract of his paper:

In the present paper we use theories from research on the structure of cognitive abilities to conceive a comprehensive measurement conception of mathematical competencies. Specifically, our measurement conception allows disentangling specific arithmetical knowledge as well as the analysis of gender differences in these competencies. Data were obtained from 151 children who participated in a longitudinal study spanning the age range from kindergarten (5 years olds) to second grade (8 years olds). Our results revealed that gender differences in the competencies under investigation were not distinct concerning their static aspects as well as their developmental dynamics.His entire presentation can be found on this link.

### EECERA - day 1

The programme book for the conference can be found here, and the abstract book here.

## 2008/09/03

### Some interesting news flashlights

The other article was published in Washington Post on Monday, and it aims at giving an overview of issues related to mathematics education. Some of the main issues in the article are:

- How is math taught?
- How much math is taught?
- What's the fuss over math?
- When should kids learn algebra?

### Doctoral students' use of examples

This paper discusses variation in reasoning strategies among expert mathematicians, with a particular focus on the degree to which they use examples to reason about general conjectures. We first discuss literature on the use of examples in understanding and reasoning about abstract mathematics, relating this to a conceptualisation of syntactic and semantic reasoning strategies relative to a representation system of proof. We then use this conceptualisation as a basis for contrasting the behaviour of two successful mathematics research students whilst they evaluated and proved number theory conjectures. We observe that the students exhibited strikingly different degrees of example use, and argue that previously observed individual differences in reasoning strategies may exist at the expert level. We conclude by discussing implications for pedagogy and for future research.

### Research reports

## 2008/09/01

### The array representation

We examine whether the array representation can support children’s understanding and reasoning in multiplication. To begin, we define what we mean by understanding and reasoning. We adopt a ‘representational-reasoning’ model of understanding, where understanding is seen as connections being made between mental representations of concepts, with reasoning linking together the different parts of the understanding. We examine in detail the implications of this model, drawing upon the wider literature on assessing understanding, multiple representations, self explanations and key developmental understandings. Having also established theoretically why the array representation might support children’s understanding and reasoning, we describe the results of a study which looked at children using the array for multiplication calculations. Children worked in pairs on laptop computers, using Flash Macromedia programs with the array representation to carry out multiplication calculations. In using this approach, we were able to record all the actions carried out by children on the computer, using a recording program called Camtasia. The analysis of the obtained audiovisual data identified ways in which the array representation helped children, and also problems that children had with using the array. Based on these results, implications for using the array in the classroom are considered.

### Bodily experience and mathematical conceptions

Mathematical concepts and conceptions have been theorized as abstractions from—and therefore transcending—bodily and embodied experience. In this contribution, we re-theorize mathematical conceptions by building on recent philosophical work in dialectical phenomenology. Accordingly, a conception exists only in, through, and as of the experiences that the individual realizes it. To exemplify our reconceptualization of mathematical conceptions, we draw on an episode from a study in a second-grade classroom where the students learned about three-dimensional geometrical objects.

### What really matters?

Here is the abstract:

The learner’s perspective study, motivated by a strong belief that the characterization of the practices of mathematics classrooms must attend to learner practice with at least the same priority as that accorded to teacher practice, is a comprehensive study that adopts a complementary accounts methodology to negotiate meanings in classrooms. In Singapore, three mathematics teachers recognized for their locally defined ‘teaching competence’ participated in the study. The comprehensive sets of data from the three classrooms have been used to explore several premises related to the teaching and learning of mathematics. In this paper the student interview data and the teacher interview data were examined to ascertain what do students attach importance to and what do teachers attach importance to in a mathematics lesson? The findings of the student interview data showed that they attached importance to several sub-aspects of the three main aspects, i.e., exposition, seatwork and review and feedback of their teachers’ pedagogical practices. The findings of the teacher interview data showed that they attached importance to student’s self assessment, teacher’s demonstration of procedures, review of prior knowledge and close monitoring of their student’s progress in learning and detailed feedback of their work. It was also found that teachers and students did attach importance to some common lesson events.

### Gender differences in Germany

In Germany, national standards for mathematics for the end of primary school were established in 2004. In the present study, data were collected to evaluate these standards, and were used to compare the mathematical skills of girls and boys. Many studies have shown that gender differences are strongest at the highest levels of education. The findings from primary school are less consistent. Thus, in our study we analyzed achievement differences in a sample of approximately 10,000 third and fourth graders, representative of the German elementary school population. Gender-specific competencies were compared in the different content domains, both for the general mathematical competence, and for the cognitive levels of the tasks. Overall, boys outperformed girls, but substantial variation was found between the content domains and general mathematical achievement. Differences were higher in grade three than in grade four. The proportion of boys in the classroom did not appear to affect the individual level of performance. Analysis of the items on which boys or girls clearly outperformed each other reproduced a pattern of specific item characteristics predicting gender bias consistent with those reported in previous studies in other countries.