## 2008/08/29

### Proceedings from ICME-10

### Review of Math Investigations

After reading about the curriculum on the web, I find it quite interesting. The curriculum was developed in the 1990s, and it was developed with support from the National Science Foundation. From their website, I learn that the

*Investigations in Number, Data, and Space*(which is the official name of the curriculum, it appears) was designed to:

- Support students to make sense of mathematics and learn that they can be mathematical thinkers.
- Focus on computational fluency with whole numbers as a major goal of the elementary grades.
- Provide substantive work in important areas of mathematics—rational numbers, geometry, measurement, data, and early algebra—and connections among them.
- Emphasize reasoning about mathematical ideas.
- Communicate mathematics content and pedagogy to teachers.
- Engage the range of learners in understanding mathematics.

*students have mathematical ideas, (...) teachers are engaged in ongoing learning about mathematics content, pedagogy, and student learning (...) and that teachers collaborate with the students and curriculum materials to create the curriculum as enacted in the classroom*(quoted from their website). In many ways, the Investigations curriculum appears to have some common underlying ideas with the Everyday Math curriculum (which has also been strongly criticized by some). According to several impact studies, the Investigations curriculum appears to have a positive impact on the achievement of students, and Everyday Math is also a curriculum that is strongly based on research. As someone standing outside of this debackle, I am therefore somewhat amazed by the criticism these curricula has raised. Somewhat, but maybe not all that amazed after all. Our previous Norwegian curriculum (called L97) featured some of the same ideas about teaching and learning of mathematics, with a focus on letting the students discover and reinvent the mathematical ideas, having "mathematics in everyday life" as a main area of the curriculum, etc. After less than 10 years of implementations (evaluation reports showing that the curriculum had not really been implemented in the classrooms), it was replaced by a new curriculum called "Kunnskapsløftet" (Knowledge Promotion). This curriculum has a much stronger emphasis on basic skills, little or no mention of discovery and reinvention, little emphasis on connections with everyday life, etc. So, I guess this debate is not only typical for the US and in this case Loudoun county.

For me as a researcher, I think it is interesting to see how much resistance these "reform curriculum" efforts encounter, and it reminds me of something I read in

*The teaching gap*. Teaching of mathematics appears to be some kind of cultural entity, and I think Stigler and Hiebert used the notion: "cultural scripts". In order to implement a new curriculum, it is often necessary to change some of these cultural scripts, and that appears to be a rather cumbersome endeavor...

P.S. If any of you has some references to research, articles, etc. that relates to the above mentioned curriculum papers, please let me know!

## 2008/08/27

### New TMME monograph

- How can modelling activities be used to foster interdisciplinary projects in the school and university setting?
- How can the intricate connections between mathematics and physics be used to design and research interdisciplinary activities in schools and the university?
- How can research within the ethnomathematics domain of mathematics education be linked to critical mathematics education and interdisciplinary projects involving mathematics, art and culture?
- How can the push for mathematical and statistical literacy be connected to other subjects in the school curricula and emphasized via interdisciplinary activities?
- What are concrete examples of classroom experiments with empirical data that demonstrate new and unusual connections/relations between mathematics, arts and the sciences with implications for pedagogy?
- What is the role of technology and new ICT interfaces in linking communities of learners in interdisciplinary activities involving problem solving? The book is an important contribution to the literature on educational initiatives in interdisciplinary education increasing vital for emerging professions of the 21st century.

### Women of mathematics

Though women earn nearly half of the mathematics baccalaureate degrees in the United States, they make up a much smaller percentage of those pursuing advanced degrees in mathematics and those entering mathematics-related careers. Through semi-structured interviews, this study took a qualitative look at the beliefs held by five undergraduate women mathematics students about themselves and about mathematicians. The findings of this study suggest that these women held stereotypical beliefs about mathematicians, describing them to be exceptionally intelligent, obsessed with mathematics, and socially inept. Furthermore, each of these women held the firm belief that they do not exhibit at least one of these traits, the first one being unattainable and the latter two being undesirable. The results of this study suggest that although many women are earning undergraduate degrees in mathematics, their beliefs about mathematicians may be preventing them from identifying as one and choosing to pursue mathematical careers.

### Realistic Mathematics Education in Indonesia

This paper reports an experimental study on the development of exemplary curriculum materials for the teaching of fractions in Indonesian primary schools. The study’s context is the current reform movement adopting realistic mathematics education (RME) theory, known as Pendidikan Matematika Realistik Indonesia (PMRI), and it looked at the role of design research in supporting the dissemination of PMRI. The study was carried out in two cycles of teaching experiments in two primary schools. The findings of the design research signified the importance of collaboration between mathematics educators and teachers in developing RME curriculum materials. The availability of RME curriculum materials is an important component in the success of the PMRI movement, particularly in supporting students and teachers in activity-based mathematics learning. Most of the students and teachers in the two schools positively appraised teaching and learning with the developed materials. Since the teachers were actively involved in developing the materials, they felt a sense of ownership and recognised that their students’ classroom experiences of the materials helped them avoid standard difficulties. That appears to be a particular benefit of the bottom-up approach characteristic of the PMRI movement.

## 2008/08/25

### Future teachers' competence to plan a lesson

The study "Mathematics Teaching in the 21st Century (MT21)" focuses beyond others on the measurement of teachers’ general pedagogical knowledge (GPK). GPK is regarded as a latent construct embedded in a larger theory of teachers’ professional competence. It is laid out how GPK was defined and operationalized. As part of an international comparison GPK was measured with several complex vignettes. In the present paper, the results of future mathematics teachers’ knowledge from four countries (Germany, South Korea, Taiwan, and the US) with very different teacher-education systems are presented. Significant and relevant differences between the four countries as well as between future teachers at the beginning and at the end of teacher education were found. The results are discussed with reference to cultural discourses about teacher education.

### Cognitive styles

This paper reports the outcomes of an empirical study undertaken to investigate the effect of students’ cognitive styles on achievement in measurement tasks in a dynamic geometry learning environment, and to explore the ability of dynamic geometry learning in accommodating different cognitive styles and enhancing students’ learning. A total of 49 6th grade students were tested using the VICS and the extended CSA-WA tests (Peterson, Verbal imagery cognitive styles and extended cognitive style analysis-wholistic analytic test—Administration guide. New Zealand: Peterson, 2005) for cognitive styles. The same students were also administered a pre-test and a post-test involving 20 measurement tasks. All students were taught a unit in measurement (area of triangles and parallelograms) with the use of dynamic geometry, after a pre-test. As expected, the dynamic geometry software seems to accommodate different cognitive styles and enhances students’ learning. However, contrary to expectations, verbalisers and wholist/verbalisers gained more in their measurement achievement in the environment of dynamic geometry than students who had a tendency towards other cognitive styles. The results are discussed in terms of the nature of the measurement tasks administered to the students.

### Embodied design

Design-based research studies are conducted as iterative implementation-analysis-modification cycles, in which emerging theoretical models and pedagogically plausible activities are reciprocally tuned toward each other as a means of investigating conjectures pertaining to mechanisms underlying content teaching and learning. Yet this approach, even when resulting in empirically effective educational products, remains under-conceptualized as long as researchers cannot be explicit about their craft and specifically how data analyses inform design decisions. Consequentially, design decisions may appear arbitrary, design methodology is insufficiently documented for broad dissemination, and design practice is inadequately conversant with learning-sciences perspectives. One reason for this apparent under-theorizing, I propose, is that designers do not have appropriate constructs to formulate and reflect on their own intuitive responses to students’ observed interactions with the media under development. Recent socio-cultural explication of epistemic artifacts as semiotic means for mathematical learners to objectify presymbolic notions (e.g., Radford, Mathematical Thinking and Learning 5(1): 37–70, 2003) may offer design-based researchers intellectual perspectives and analytic tools for theorizing design improvements as responses to participants’ compromised attempts to build and communicate meaning with available media. By explaining these media as potential semiotic means for students to objectify their emerging understandings of mathematical ideas, designers, reciprocally, create semiotic means to objectify their own intuitive design decisions, as they build and improve these media. Examining three case studies of undergraduate students reasoning about a simple probability situation (binomial), I demonstrate how the semiotic approach illuminates the process and content of student reasoning and, so doing, explicates and possibly enhances design-based research methodology.

### Teachers' perspectives on authentic mathematics

We investigate experienced high school geometry teachers’ perspectives on “authentic mathematics” and the much-criticized two-column proof form. A videotaped episode was shown to 26 teachers gathered in five focus groups. In the episode, a teacher allows a student doing a proof to assume a statement is true without immediately justifying it, provided he return to complete the argument later. Prompted by this episode, the teachers in our focus groups revealed two apparently contradictory dispositions regarding the use of the two-column proof form in the classroom. For some, the two-column form is understood to prohibit a move like that shown in the video. But for others, the form is seen as a resource enabling such a move. These contradictory responses are warranted in competing but complementary notions, grounded on the corpus of teacher responses, that teachers hold about the nature of authentic mathematical activity when proving.

### New roles for mathematics

A new concept, compulsory multi-disciplinary courses, was introduced in upper secondary school curriculum as a central part of a recent reform. This paper reports from a case study of such a triple/four-disciplinary project in mathematics, physics, chemistry and ‘general study preparation’ performed under the reform by a team of experienced teachers. The aim of the case study was to inquire how the teachers met the demands of the introduction of this new concept and, to look for signs of new relations established by the students between mathematics and other subjects, as a result of the multi-disciplinary teaching. The study revealed examples of good practice in planning and teaching. In addition, it served to illuminate interesting aspects of how students perceived the school subject mathematics and its relations to other subjects and to common sense.

## 2008/08/24

### YESS-4, Day 7

At the last day of YESS-4, Ferdinando Arzarello is going to deliver the main lecture. The topic for his talk is "Tools for analyzing learning processes in mathematics". He starts off with a discussion of problems concerning What, Why, How and Goals:

- What is necessary to observe in the classroom? (What)
- Which theoretical frames are suitable to answer the What-problem? (Why)
- How to observe all that is necessary? and How to interpret the observed data according to the assumed frame? (How)
- How to improve consequent didactical practices in the classroom? (Goal)

## 2008/08/22

### YESS-4, Day 5

At the 5th day of the YERME Summer School, Günter Törner is going to deliver the main lecture. His topic is "theory versus practice", and you can learn more from the paper that is published on the YESS-4 website.

Törner has published a multitude of papers and books in mathematics (algebra, geometry and discrete mathematics) as well as mathematics education. Several of them are available on his website, so take a look at the links I just gave you!

## 2008/08/21

### YESS-4, Day 4

The main lecture today at YESS-4 is held by Laurinda Brown. The theme of her main paper is: "Observing systems - how do we see what we see?". She aims at discussing issues concerning observations, and she points at the necessity of including discussions of theoretical, methodological and philosophical issues.

For more information, you should read her CERME-4 article and an article from Educational Studies in Mathematics. Both are published on the YESS-4 website.

## 2008/08/20

### YESS-4, Day 3

The main lecture at Day 3 of the YERME Summer School is held by Jean-Baptiste Lagrange. His talk will be concerning research about technology in mathematics education. Lagrange is going to look at different technologies with certain theoretical concerns:

- programming with the reification theories,
- microworlds with situated cognition,
- spreadsheets and computer symbolic systems with the instrumental and anthropological approaches,
- today fast developing web based technologies with the need for new approaches.

## 2008/08/19

### New book from Springer

In the new times of globalisation, international academic contacts and collaborations are ever increasing. They are taking many forms, from international conferences and publications, student and academic exchange, cross cultural research projects, curriculum development to professional development activities and affect every aspect of academic life from teaching, research to service. This book aims to:

- Develop theoretical frameworks of the phenomena of internationalisation and globalisation and identify related ethical, moral, political and economic issues facing mathematics and science educators.
- Provide a venue for the publication of results of international comparisons on cultural differences and similarities rather than merely on achievement and outcomes.
- Provide a forum for critical discussion of the various models and forms of international projects and collaborations.
- Provide a representation of the different voices and interests from around the world rather than consensus on issues.

### YESS-4, Day 2

The main lecture at day 2 of the YERME Summer School will be held by Guershon Harel from University of California, San Diego. Harel's talk will address two main issues in mathematics education:

- What is the mathematics that we should teach in school?
- How should we teach it?

### Use of examples in elementary mathematics

This empirical paper considers the different purposes for which teachers use examples in elementary mathematics teaching, and how well the actual examples used fit these intended purposes. For this study, 24 mathematics lessons taught by prospective elementary school teachers were videotaped. In the spirit of grounded theory, the purpose of the analysis of these lessons was to discover, and to construct theories around, the ways that these novice teachers could be seen to draw upon their mathematics teaching knowledge-base in their lesson preparation and in their observed classroom instruction. A highly-pervasive dimension of the findings was these teachers’ choice and use of examples. Four categories of uses of examples are identified and exemplified: these are related to different kinds of teacher awareness.

## 2008/08/18

### YERME Summer School

This week, the 4th version of YERME Summer School (YESS-4) is organized in Turkey. The venue for the summer school is Karadeniz Technical University in Trabzon, near the Black Sea. KTU is a public research university with 30.000 students. There are about 40 master and PhD-students in mathematics education.

The summer school has a very interesting program, and although I am not able to attend it myself, I will try and cover it in my blog.

YESS-4 features a panel of distinguished experts, who will deliver the main lectures:

- Prof.Dr. Guershon Harel, University of California (USA)
- Prof.Dr. Linda Brown, University of Bristol (England)
- Prof.Dr. Jean-Baptiste Lagrange, IUFM De Reims Paris VII University (France)
- Prof.Dr. Günter Törner, Universität Duisburg-Essen Standort (Germany)
- Prof.Dr. Ferdinando Arzarello, Università di Torino (Italy)

The opening talk will be held this afternoon by Barbara Jaworski.

Links:

### JMTE, August 2008

- Researchers and their roles in teacher education, by Konrad Krainer
- Investigating changes in prospective teachers’ views of a ‘good teacher’ while engaging in computerized project-based learning, by Ilana Lavy and Atara Shriki
- Teaching experiments and professional development, by Anderson Hassel Norton and Andrea McCloskey
- Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving, by Andreas J. Stylianides and Deborah L. Ball
- Expanding the instructional triangle: conceptualizing mathematics teacher development, by Kelli Nipper and Paola Sztajn

## 2008/08/15

### Showing you're working

To help students view mathematics in a more favourable light, a number of former pupils were contacted and asked to give details of how they use mathematics in their daily lives. This information was gathered through an online questionnaire or visits to the school to talk to pupils—a booklet of responses was also given to students. Attitudinally pre- and post-testing students suggested that this initiative helped address pupils’ concerns regarding the purpose of classroom mathematics. The diversity of professions also helped dispel many myths about the usefulness of mathematics. Subsequently, the project has proven to be a catalyst for a range of cross-curricular projects and events inspired by the former pupils’ case studies, all of which serve to continue to address the initial aims of the project regarding pupil perception of the subject, in the light of both workplace and everyday life.

## 2008/08/14

### Mathematical belief change

Grootenboer provides a nice overview of previous research in this area, and that alone is reason enough to read this article. In addition, the study he reports is very interesting. Unlike many other studies of teachers' beliefs, Grootenboer has conducted a naturalistic study (in his own classroom), and he collected data from different sources: observation, interviews and assignments. If, like me, you are interested in teachers' beliefs in mathematics education, you should definitely read this article! Here is the abstract:

The development and influence of beliefs in teacher education has been a topic of increasing interest for researchers in recent years. This study explores the responses of a group of prospective primary teachers to attempts to facilitate belief change as part of their initial teacher education programme in mathematics. The students’ responses seemed to fall into three categories: non-engagement; building a new set of beliefs and; reforming existing beliefs. In this article the participants’ responses are outlined and illustrated with stories from three individuals. This study suggests that belief reform is complex and fraught with ethical dilemmas. Certainly there is a need for further research in this area, particularly given the pervasive influence of beliefs on teaching practice.

### Learning mathematics for teaching

Here is the abstract:

Student teaching (guided teaching by a prospective teacher under the supervision of an experienced “cooperating” teacher) provides an important opportunity for prospective teachers to increase their understanding of mathematics in and for teaching. The interactions between a student teacher and cooperating teacher provide an obvious mechanism for such learning to occur. We report here on data that is part of a larger study of eight student teacher/cooperating teacher pairs, and the core themes that emerged from their conversations. We focus on two pairs for whom the core conversational themes represent disparate approaches to mathematics in and for teaching. One pair, Blake and Mr. B., focused on controlling student behavior and rarely talked about mathematics for teaching. The other pair, Tara and Mr. T., focused on having students actively participating in the lesson and on mathematics from the students’ point of view. These contrasting experiences suggest that student teaching can have a profound effect on prospective teachers’ understanding of mathematics in and for teaching.

### Why do gestures matter?

The goal of this article is to present a sketch of what, following the German social theorist Arnold Gehlen, may be termed “sensuous cognition.” The starting point of this alternative approach to classical mental-oriented views of cognition is a multimodal “material” conception of thinking. The very texture of thinking, it is suggested, cannot be reduced to that of impalpable ideas; it is instead made up of speech, gestures, and our actual actions with cultural artifacts (signs, objects, etc.). As illustrated through an example from a Grade 10 mathematics lesson, thinking does not occur solely in the head but also in and through a sophisticated semiotic coordination of speech, body, gestures, symbols and tools.Luis Radford is a distinguished scholar, and he has published a large number of important articles over the years. If you want to read more about his work, you should visit his list of publications. Most of his articles are freely available in pdf-format!

### Exploring gender factors

Students’ mathematical achievement in Iceland, as reported in PISA 2003, showed significant and (by comparison) unusual gender differences in mathematics: Iceland was the only country in which the mathematics gender gap favored girls. When data were broken down and analyzed, the Icelandic gender gap appeared statistically significant only in the rural areas of Iceland, suggesting a question about differences in rural and urban educational communities. In the 2007 qualitative research study reported in this paper, the authors interviewed 19 students from rural and urban Iceland who participated in PISA 2003 in order to investigate these differences and to identify factors that contributed to gender differences in mathematics learning. Students were asked to talk about their mathematical experiences, their thoughts about the PISA results, and their ideas about the reasons behind the PISA 2003 results. The data were transcribed, coded, and analyzed using techniques from analytic induction in order to build themes and to present both male and female student perspectives on the Icelandic anomaly. Strikingly, youth in the interviews focused on social and societal factors concerning education in general rather then on their mathematics education.

## 2008/08/12

### Gestures and conceptual integration

Spontaneous gesture produced in conjunction with speech is considered as both a source of data about mathematical thinking, and as an integral modality in communication and cognition. The analysis draws on a corpus of more than 200 gestures collected during 3 h of interviews with prospective elementary school teachers on the topic of fractions. The analysis examines how gestures express meaning, utilizing the framework of cognitive linguistics to argue that gestures are both composed of, and provide inputs to, conceptual blends for mathematical ideas, and a standard typology drawn from gesture studies is extended to address the function of gestures within mathematics more appropriately.A key idea in the article is that mathematics is seen as "an embodied, socially constructed human product", and gestures therefore might provide a relevant contribution to the mathematical thinking and communication. Edwards provides a nice explanation for the role of research on gestures:

(...) gesture constitutes a particular modality of embodied cognition, and, along with oral speech, written inscriptions, drawings and graphing, it can serve as a window on how learners think and talk about mathematics.The article provides a good overview of the theoretical framework for this area of research, and the study itself is also interesting. The participants (all women) were twelve volunteers from a course for prospective elementary school teachers, and the course was taught by Edwards herself. The participants were interviewed in pair, and the interview sessions were videotaped. The gestures that were caught on videotape were classified by McNeill's scheme.

## 2008/08/11

### Construction of moral discourses

This qualitative study examines the way three American young adolescent girls who come from different class and racial backgrounds construct their social and academic identities in the context of their traditional mathematics classroom. The overall analysis shows an interesting dynamic among each participant’s class and racial background, their social/academic identity and its collective foundation, the types of ideologies they repudiate and subscribe to, the implicit and explicit strategies they adopt in order to support the legitimacy of their own position, and the ways they manifest their position and identity in their use of language referring to their mathematics classroom. Detailed analysis of their use of particular terms, such as “I,” “we,” “they,” and “should/shouldn’t” elucidates that each participant has a unique view of her mathematics classroom, developing a different type of collective identity associated with a particular group of students. Most importantly, this study reveals that the girls actively construct a social and ideological web that helps them articulate their ethical and moral standpoint to support their positions. Throughout the complicated appropriation process of their own identity and ideological standpoint, the three girls made different choices of actions in mathematics learning, which in turn led them to a different math track the following year largely constraining their possibility of access to higher level mathematical knowledge in the subsequent schooling process.

### RME, September 2008

- How persuaded are you? A typology of responses
**Authors:**Matthew Inglis; Juan Pablo Mejia-Ramos - To be or to become: how dynamic geometry changes discourse
**Authors:**Nathalie Sinclair; Violeta Yurita - A diagrammatic view of the equals sign: arithmetical equivalence as a means, not an end
**Author:**Ian Jones - Paradoxes as a window to infinity
**Authors:**Ami Mamolo; Rina Zazkis - The effect of graphic calculators on Negev Arab pupils' learning of the concept of families of functions
**Author:**Muhammad Abu-Naja - The mathematical competence of adults returning to learning on a university

foundation programme: a selective comparison of performance with the

CSMS study**Author:**Mary Dodd - Mathematics and dyslexics: classroom management skills and children's response to noise
**Author:**Mari Palmer - A synthesis of existing frameworks used to analyse mathematics curricula
**Author:**Nusrat Fatima Rizvi - Beginning elementary teachers' use of representations in mathematics teaching
**Author:**Fay Turner - Observing students' use of images through their gestures and gazes
**Author:**Tracy Wylie

## 2008/08/09

### MTL, new issue

A new issue of Mathematical Thinking and Learning has been published:

> Mathematical Thinking and Learning: Volume 10 Issue 3 (http://www.informaworld.com/openurl?genre=issue&issn=1098-6065&volume=10&issue=3&uno_jumptype=alert&uno_alerttype=new_issue_alert,email

> ) is now available online at informaworld (http://

> www.informaworld.com).

>

> This new issue contains the following articles:

>

> Turnaround Students in High School Mathematics: Constructing

> Identities of Competence Through Mathematical Worlds, Pages 201 - 239

> Author: Ilana Seidel Horn

> DOI: 10.1080/10986060802216177

> Link: http://www.informaworld.com/openurl?genre=article&issn=1098-6065&volume=10&issue=3&spage=201&uno_jumptype=alert&uno_alerttype=new_issue_alert,email

>

> Toddlers' Spontaneous Attention to Number, Pages 240 - 270

> Authors: Arthur J. Baroody; Xia Li; Meng-lung Lai

> DOI: 10.1080/10986060802216151

> Link: http://www.informaworld.com/openurl?genre=article&issn=1098-6065&volume=10&issue=3&spage=240&uno_jumptype=alert&uno_alerttype=new_issue_alert,email

>

> The Interplay Between Gesture and Discourse as Mediating Devices in

> Collaborative Mathematical Reasoning:A Multimodal Approach, Pages

> 271 - 292

> Authors: Raymond Bjuland; Maria Luiza Cestari; Hans Erik Borgersen

> DOI: 10.1080/10986060802216169

> Link: http://www.informaworld.com/openurl?genre=article&issn=1098-6065&volume=10&issue=3&spage=271&uno_jumptype=alert&uno_alerttype=new_issue_alert,email

>

> A Modeling Perspective on the Teaching and Learning of Mathematical

> Problem Solving, Pages 293 - 304

> Authors: Nicholas G. Mousoulides; Constantinos Christou; Bharath

> Sriraman

> DOI: 10.1080/10986060802218132

> Link: http://www.informaworld.com/openurl?genre=article&issn=1098-6065&volume=10&issue=3&spage=293&uno_jumptype=alert&uno_alerttype=new_issue_alert,email

>

> A Critique on the Role of Social Justice Perspectives in Mathematics

> Education, Pages 305 - 312

> Author: Bettina Dahl

> DOI: 10.1080/10986060802216185

> Link: http://www.informaworld.com/openurl?genre=article&issn=1098-6065&volume=10&issue=3&spage=305&uno_jumptype=alert&uno_alerttype=new_issue_alert,email

#ens

## 2008/08/08

### Semiotics and subjectivity

In this response we address some of the significant issues that Tony Brown raised in his analysis and critique of the Special Issue of Educational Studies in Mathematics on “Semiotic perspectives in mathematics education” (Sáenz-Ludlow & Presmeg, Educational Studies in Mathematics 61(1–2), 2006). Among these issues are conceptualizations of subjectivity and the notion that particular readings of Peircean and Vygotskian semiotics may limit the ways that authors define key actors or elements in mathematics education, namely students, teachers and the nature of mathematics. To deepen the conversation, we comment on Brown’s approach and explore the theoretical apparatus of Jacques Lacan that informs Brown’s discourse. We show some of the intrinsic limitations of the Lacanian idea of subjectivity that permeates Brown’s insightful analysis and conclude with a suggestion about some possible lines of research in mathematics education.

### Stepping beyond high school mathematics

The Summer Explorations and Research Collaborations for High School Girls (SEARCH) Program, held annually since 2004 at Mount Holyoke College in the US, was created for talented high school girls to explore mathematics beyond that taught in high school. Our study, which focuses on factors that facilitate or inhibit the pursuit of higher level mathematics by girls, is centered on the 2006 SEARCH Program. We present a combination of qualitative and quantitative data drawn from student journals written during SEARCH, program evaluations written at the end of SEARCH, post-program interviews, and comparisons with two peer group samples. From this data we point to important factors, such as developing a mathematical voice, gaining a broader view of advanced mathematics, being challenged in a supportive atmosphere, and having a positive stance toward risk-taking, that may help to maintain the interest of talented girls in advanced mathematical studies.

### Supporting mathematical literacy

Mathematical literacy implies the capacity to apply mathematical knowledge to various and context-related problems in a functional, flexible and practical way. Improving mathematical literacy requires a learning environment that stimulates students cognitively as well as allowing them to collect practical experiences through connections with the real world. In order to achieve this, students should be confronted with many different facets of reality. They should be given the opportunity to participate in carrying out experiments, to be exposed to verbal argumentative discussions and to be involved in model-building activities.

This leads to the idea of integrating science into maths education. Two sequences of lessons were developed and tried out at the University of Education Schwäbisch Gmünd integrating scientific topics and methods into maths lessons at German secondary schools. The results show that the scientific activities and their connection with reality led to well-based discussions. The connection between the phenomenon and the model remained remarkably close during the entire series of lessons. At present the sequences of lessons are integrated in the European ScienceMath project, a joint project between universities and schools in Denmark, Finland, Slovenia and Germany (see www.sciencemath.ph-gmuend.de).

## 2008/08/07

### A mathematician's lament

This month's column is devoted to an article called A Mathematician's Lament, written by Paul Lockhart in 2002. Paul is a mathematics teacher at Saint Ann's School in Brooklyn, New York. His article has been circulating through parts of the mathematics and math ed communities ever since, but he never published it. I came across it by accident a few months ago, and decided at once I wanted to give it wider exposure. I contacted Paul, and he agreed to have me publish his "lament" on MAA Online. It is, quite frankly, one of the best critiques of current K-12 mathematics education I have ever seen. Written by a first-class research mathematician who elected to devote his teaching career to K-!2 education.

### Construction of mathematical meaning of motion graphs

This paper examines the relation between bodily actions, artifact-mediated activities, and semiotic processes that students experience while producing and interpreting graphs of two-dimensional motion in the plane. We designed a technology-based setting that enabled students to engage in embodied semiotic activities and experience two modes of interaction: 2D freehand motion and 2D synthesized motion, designed by the composition of single variable function graphs. Our theoretical framework combines two perspectives: the embodied approach to the nature of mathematical thinking and the Vygotskian notion of semiotic mediation. The article describes in detail the actions, gestures, graph drawings, and verbal discourse of one pair of high school students and analyzes the social semiotic processes they experienced. Our analysis shows how the computerized artifacts and the students’ gestures served as means of semiotic mediation. Specifically, they supported the interpretation and the production of motion graphs; they mediated the transition between an individual’s meaning of mathematical signs and culturally accepted mathematical meaning; and they enable linking bodily actions with formal signs.The article gives a nice introduction to the theoretical foundations concerning the connections between bodily movement and semiotics. The study being described in the article was a learning experiment, and the use of illustrative photos and figures in the article makes it easy to understand the discussion of the different motions and pointing gestures that were used.

## 2008/08/06

### IJSME, September 2008

- 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

The article by Llinares and Roig has a focus on students' problem solving, with a particular focus on word problems. Connections are made with research on mathematical modelling (e.g. the research of Danish colleague and editor of NOMAD, Morten Blomhøj), and the article gives a nice overview of research concerning problem solving and mathematical modelling. The study that is reported in the article is a survey/test where students were faced with five questions/problems. Llinares and Roig discuss the problem-solving strategies that were used to solve the three word problems in this test.

The article by Githua and Nyabwa provides insight into mathematics teaching in Kenya, and the article builds heavily on Ausubel's theory of advance organisers. The objectives of the reported study were to investigate whether or not there were statistical significant differences in mathematics achievement between students who had been taught using advance organisers or not, and they also wanted to investigate whether gender affected achievement when advance organisers were used.

Another interesting article was the one by Ismail and Awang, which provides more insight into factors that influenced the achievement of Malaysian students in the TIMSS 1999 student assessment.

## 2008/08/05

### New IJMEST articles

- A note on variance components model, by Anant M. Kshirsagar and R. Radhakrishnan
- An elementary proof of a converse mean-value theorem, by Ricardo Almeida
- Bionomic exploitation of a ratio-dependent predator-prey system, by Alakes Maiti, Bibek Patra and G.P. Samanta

## 2008/08/04

### ZDM, August 2008

- Introduction to the special issue on didactical and epistemological perspectives on mathematical proof, by Maria Alessandra Mariotti and Nicolas Balacheff
- Proofs as bearers of mathematical knowledge, by Gila Hanna and Ed Barbeau
- Proof as a practice of mathematical pursuit in a cultural, socio-political and intellectual context, Man-Keung Siu
- Theorems that admit exceptions, including a remark on Toulmin, by Hans Niels Jahnke
- Truth versus validity in mathematical proof, by Viviane Durand-Guerrier
- Argumentation and algebraic proof, by Bettina Pedemonte
- Indirect proof: what is specific to this way of proving?, by Samuele Antonini and Maria Alessandra Mariotti
- Students’ encounter with proof: the condition of transparency, by Kirsti Hemmi
- A method for revealing structures of argumentations in classroom proving processes, by Christine Knipping
- Strategies to foster students’ competencies in constructing multi-steps geometric proofs: teaching experiments in Taiwan and Germany, by Aiso Heinze, Ying-Hao Cheng, Stefan Ufer, Fou-Lai Lin and Kristina Reiss
- Reasoning and proof in geometry: effects of a learning environment based on heuristic worked-out examples, by Kristina Maria Reiss, Aiso Heinze, Alexander Renkl and Christian Groß
- When, how, and why prove theorems? A methodology for studying the perspective of geometry teachers, by Patricio Herbst and Takeshi Miyakawa
- DNR perspective on mathematics curriculum and instruction, Part I: focus on proving, by Guershon Harel
- The role of the researcher’s epistemology in mathematics education: an essay on the case of proof, by Nicolas Balacheff

### ESM, September 2008

- School mathematics and its everyday other? Revisiting Lave’s ‘Cognition in Practice’, by Christian Greiffenhagen and Wes Sharrock
- Beyond ‘blaming the victim’ and ‘standing in awe of noble savages’: a response to “Revisiting Lave’s ‘cognition in practice’”, by David W. Carraher
- The problem of the particular and its relation to the general in mathematics education, by Vicenç Font and Ángel Contreras
- Transitions among different symbolic generalizations by algebra beginners in a computer intensive environment, by Michal Tabach, Abraham Arcavi and Rina Hershkowitz
- Centenary birth anniversary of E. W. Beth (1908–1964), by Giorgio T. Bagni

### Equity in mathematics education

Traditional models of gender equity incorporating deficit frameworks and creating norms based on male experiences have been challenged by models emphasizing the social construction of gender and positing that women may come to know things in different ways from men. This paper draws on the latter form of feminist theory while treating gender equity in mathematics as intimately interconnected with equity issues by social class and ethnicity. I integrate feminist and social justice literature in mathematics education and argue that to secure a transformative, sustainable impact on equity, we must treat mathematics as an integral component of a larger system producing educated citizens. I argue the need for a mathematics education with tri-fold support for mathematical literacy, critical literacy, and community literacy. Respectively, emphases are on mathematics, social critique, and community relations and actions. Currently, the integration of these three literacies is extremely limited in mathematics.

### Exemplifying definitions

In this study we utilize the notion of learner-generated examples, suggesting that examples generated by students mirror their understanding of particular mathematical concepts. In particular, we explore examples generated by a group of prospective secondary school teachers for a definition of a square. Our framework for analysis includes the categories of accessibility and correctness, richness, and generality. Results shed light on participants’ understanding of what a mathematical definition should entail and, moreover, contrast their pedagogical preferences with mathematical considerations.

### Mathematical paradoxes

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