Abstracts of Review Talks

The Languages of Mathematics

Abraham Arcavi

The Weizmann Institute of Science, Rehovot, Israel

Students of mathematics have to juggle with at least three mathematical languages: rhetoric, symbolic and graphical. Each of these languages have distinct characteristics and can be used in different ways to support, or to alienate, sense-making. How can insights into the nature and characteristics of these languages enlighten mathematics education in all its branches . curriculum development, the practice of teaching, research on learning, teacher education? This talk will provide thoughts, proposals and open questions on these matters.

What Do "Good" Teachers Know? Investigating Teacher Professional Knowledge

Frank Banks

The Open University, Milton Keynes, U.K.

Everyone remembers a good teacher. was the theme of a recent teacher recruitment campaign in the United Kingdom, but what is it that is considered .good teaching. in science and technology? Drawing on empirical work carried out with teachers in Australia, Bangladesh, Canada, Finland, India, Iraq, New Zealand and the United Kingdom this paper sets out aspects of teacher professional knowledge by presenting a common frame of analysis.
What constitutes the school science and school technology curriculum has gone through considerable change in many countries over the last twenty years and the analytical framework can be shared with teachers to enable them to use it as a tool to focus on their own professional development needs and personal beliefs about successful teaching. Considering their subject knowledge, pedagogical knowledge, .school knowledge. and their own rationale for the teaching of science or technology, teachers are able to articulate their professional priorities. Satisfying those professional needs, however, in an environment where teachers are .time-poor. and under considerable pressure to be in school working day by day with their students to achieve examination results is an acute challenge for teacher educators and policy makers. The paper will consider open and distance learning as a model for effective school-based teacher professional development. By making the school itself the site of learning and the classroom, laboratory or workroom the arena of change, teacher professional growth can be not only effective but cost-effective.

Social Dimensions of Mathematics Education

Farida Abdulla Khan

Jamia Millia Islamia, New Delhi, India

Mathematics has been an integral part of the Indian school curriculum ever since the inception of modern schooling in the country. Policy documents have emphasised the importance of mathematics and its subject matter has been of much concern whenever changes in curricula or textbooks have taken place. Changes in school curricula have often been driven by the developments in Mathematics and the subsequent need to update mathematical knowledge. In recent years, the radical shift in understanding learning within models of child development and the attendant effort to reduce information load on children has therefore met with much resistance.
For parents and students mathematics as a school subject gains importance for its association with high-status professions . traditionally medicine and engineering and now increasingly the business professions . and the subsequent opportunities that it makes available. Mathematical knowledge therefore acquires importance not for its own sake but for what it can deliver. Success in school mathematics has little to do with the classroom or with the child.s interests or motivation. Its importance within the academic context is totally out of proportion to either its applicability in the real world, its ability to provide a knowledge base for scientific and technological understanding at more advanced levels of learning, or to the broad cognitive skills it is often claimed to foster.
This paper is an attempt to explore the factors that work subtly in classrooms, within schools and outside them to limit access to what the mathematics syllabi demand in terms of skill and competence. In reviewing research into the teaching and learning of school mathematics, curricular concerns and policy decisions, it will also examine the social and political contexts within which students achieve .success. and children and teachers get constructed as mathematically .competent. or .incompetent. and the ways in which this compels the focus of any analysis or intervention to be confined to the practice of learning and teaching within the classroom.

Universalization of Elementary Math and Science As a Scientific Problem

Vivek Monteiro

Navnirmiti, Mumbai, India

With universal access to education becoming a legal right of every child citizen of India, a number of questions arise. Firstly, what is the current status of the problem in the nation? How can this be reliably and accurately assessed? What standards must be set for compliance? Secondly, how can we systematically work towards achieving these standards in the shortest feasible time period?
It is proposed that the problem of universalization of education can, and must be perceived as a scientific problem, and engaged with at least the same seriousness with which some other mass scientific programmes were taken up in history. Science must begin by seeing the elephant, when it is visible. When universalization of primary mathematics is taken up as a scientific problem, it is clear that math pedagogy is only one of its important aspects. Other aspects like systems, administrative involvement, logistics, teacher involvement, assessment and training are no less important for delivering, or not delivering, outcomes. Some of the most basic issues of universalization are political and organizational, requiring political and organizational decisions. These cannot be avoided but must be addressed scientifically. We discuss some specific experiences and problems with implementing mass programmes.
The question of universalizing science has all the above mentioned dimensions of math education as well as some more. The concept of the real world as a school science laboratory is crucial to universalizing science. Can every school become a discovery school? We discuss how a terra-sun laboratory can be set up in each school at a cost that no school cannot afford. A comprehensive package of low cost and no cost experiments exists which can make world class science education accessible to every child. Mass science campaigns like Year of Planet Earth, and International Year of Astronomy 2009, can play a catalytic role in upgrading science education on the mass scale necessary to achieve universalization.

Discourse and Learning in the Science Classroom

Eduardo F Mortimer

Federal University of Minas Gerais, Bela Horizonte, Brazil

Studies of discourse in science classrooms are relatively recent. If we consider the seminal work of Lemke (1990) as a starting point, from that time a variety of studies emerged in which discourse in science classrooms is considered from different perspectives. There are studies that consider the teacher-students interactions and how the patterns of discourse that emerged from these interactions framed the opportunities for learning science. Although many of these studies found that questions were used to control the classroom conversation, usually through the use of discursive patterns like IRA (Initiation-Response-Evaluation) (for example Carlsen, 1991, Lemke 1990), there are some studies showing that questioning can have a different purpose, allowing and encouraging student participation in the lessons (for example, van Zee & Minstrell, 1997). Others, yet, suggested that a balance between controlling, serving authoritative purposes, and encouragement of participation, serving dialogical purpose, is a way of conducting a lesson, as science is predominantly an authoritative discourse that has to be learned through dialogue (Scott, Mortimer & Aguiar, 2006).
In this article I am going to present a review of selective studies of discourse and learning in science classroom considering, beside those that analyze teacher-students interactions, the studies that focus on students. interactions and how the participant structure promotes student engagement in the classroom discourse practices (for example, Cornelius & Herrenkohl, 2004, Engle & Conant, 2002). And thirdly, we shall review some studies that analyze argumentation and the ways evidence is used in science classrooms (for example, Driver, Newton & Osborne, 2000). Although these studies tend to consider the logic and not the rhetoric of argumentation, for example through the use of Toulmin.s (1958) model, we shall analyze them because argumentation is central to reveal the nature of claims and warrants for scientific knowledge.
I shall finish this review giving an example of my own analysis of classroom discourse in which, following Kelly.s (2007) suggestion, I consider ways that discourse study can be used to inform teacher education. Accordingly, I will analyze a particular science teacher and the ways he alternates between authoritative and dialogic discourse to guide the students meaning making process in his classroom.
Carlsen, W.S. (1991). Questioning in the classrooms: A sociolinguistic perspective. Review of Educational Research, 61, 157-178.
Cornelius, L. L., & Herrenkohl, L. R. (2004). Power in the classroom: How the classroom environment shapes students. relationships with each other and with concepts. Cognition and Instruction, 22, 467. 498.
Driver, R. Newton, P., & Osborne, J. (2000). Establishing the norms of scientific argumentation in classrooms. Science Education, 84(3), 287. 312.
Engle, R. A., & Conant, F. R. (2002). Guiding principles for fostering productive disciplinary engagement: Explaining an emergent argument in a community of learners classroom. Cognition and Instruction, 20, 399. 484.
Kelly, G. (2007) Discourse in Science Classrooms. In S.K Abell and N.G. Lederman (ed.) Handbook of Research on Science Education.
Lemke, J. L. (1990). Talking science. Language, learning and values. Norwood, NJ: Ablex.
Toulmin, S. (1958). The uses of argument. Cambridge University Press: Cambridge.
Van Zee, E. H., & Minstrell, J. (1997). Reflective discourse: Developing shared understandings in a physics classroom. International Journal of Science Education, 19(2), 209- 228.

Design and Technology: An Emergent School Subject

Chitra Natarajan

Homi Bhabha Centre for Science Education, TIFR, Mumbai, India

How can one reconcile Gandhiji.s self-reliance principles envisioned in his Buniyadi Taleem (Basic Education) and Nayee Taleem (New Education) and iconised by the disciplined operation of a charkha, with Tagore.s dream of unleashing the nation.s inpidual and social creativity embodied in his conceptualisation of the Shantiniketan? The answer seems to suggest itself: a suitable education in design and technology. Technology and Design are organically linked. The latter stands for innovation and creativity, while the former is the very foundation of self-reliance.
The most compelling arguments for including Design and Technology (D&T) as part of school education arise from what it means to be human. I will argue for the inclusion of D&T as part of Indian school education in terms of its cultural and cognitive relevance. I will show that design and its practices are not subsumed either within the arts or the science school subjects. On the other hand, the cognitive benefits of designing are on par with and complimentary to the knowledge and skills gained from engaging in the sciences, and the humanities, including the arts and literature. Studies have been carried out at HBCSE on design and cognition as well as on the collaborative and communicative modes of working on D&T units. I will draw upon these studies carried out in different Indian school settings, and related studies from elsewhere, to illustrate how learning to design and make at the school level can empower students.
I will touch upon the need for a distinct model of D&T education for Indian schools to enable equitable participation of students from perse backgrounds, and propose the salient features of a possible D&T curriculum. I will briefly discuss the challenges of D&T education curriculum for Indian schools. Arguing that D&T is a vehicle for multiple modes of expression, creativity and design, I will discuss how Indian multicultural classrooms can benefit from communication and collaboration centred D&T activities.

People.s Knowledge of Proportions in Everyday Life and in the Classroom

Terezinha Nunes

University of Oxford, Oxford, U.K.

There are so many occasions for people to learn about proportions outside school that it has to be puzzling that many students find it difficult to solve mathematics and science problems involving proportional reasoning in school. People with little school instruction typically solve proportions problems in everyday life by methods that focus on quantities. These methods originate in the schema of one-to-many correspondence, which keeps the ratio between the quantities fixed, but as a theorem in action, not understood explicitly. This presentation will review studies that describe this informal knowledge and discuss how it could be transformed into formal mathematical knowledge, thus offering a better foundation for teaching students about proportionality in the mathematics and science classroom.

Gender Exclusion in Science: Questions about Epistemology, Policies and Institutional Frameworks

Veena Poonacha

Shreemati Nathibai Damodar Thackersey (SNDT) Women.s University, Mumbai, India

Critical epistemology has indicated that the process of generating scientific knowledge and the prevailing ethos within its institutions is not uninfluenced by the predominant norms and values in society. Influenced by the existing social, political and economic context, scientific institutions and ethos perpetuate certain exclusionary practices that prevent large sections of population including women from participating in the exciting process of knowledge creation. Such exclusionary practices raise questions about social equity and inclusion. There is also the possibility that the exclusionary practices restrict the growth of scientific enquiry and knowledge by not drawing upon other knowledge systems and pergent ways of knowing.
This paper draws upon Women.s Studies scholarship to indicate the gender blindness in India.s science policies and the embedded institutional practices that excludes women from science. The data is drawn from a study commissioned by the Indian National Science Academy, New Delhi, to understand the socio-economic barriers to Indian women.s entry into science. It begins with a brief review of Women.s Studies engagement with science to address the following questions: 1) The politics of knowledge generation that excludes women from scientific institutions and creative process; and 2) Would women.s entry into science alter the process of knowledge generation?
From this location, this paper examines the Indian science and technology policies. It also highlights some of the institutional mechanisms that are either gender blind or actively create structures of exclusion. Subsequently locating the organizations in the context of the current processes of change, it examines if the process of exclusion would be exacerbated in the wake of the current socio-political and economic changes. Finally it tentatively examines the possibilities of women bringing about a different perspective to science.

Objectifying Symbols: The Uneasy Journey of the Mathematical Object from the World, to Mind, to Discourse

Anna Sfard

University of Haifa, Haifa, Israel & the Institute of Education, London, U.K.

Those who try to crack the puzzle of mathematical thinking cannot avoid asking the preliminary question of the nature of mathematical objects and of their relation to symbols. While seeking an answer, the investigators of mathematical thinking may wish to look at the history of modern semiotics, from its beginning in epistemologically-oriented work of Peirce, Saussure, Lacan and Jacobson to the research that is being done these days by those who call themselves social semioticians. The main questions asked by successive generations of semioticians is that of the nature and origins of the .referents. of symbols. In the case of mathematics, these referents are called mathematical objects. The transformations of semiotic thought first relocated mathematical objects from the .real world. to human mind, and then from the human mind to human communication. In this talk, after a brief summary of these developments, I will propose the view of mathematical objects as discursive constructs and will present some of the consequences of this stance for the research in mathematics education and for the practice of teaching and learning.

First Steps Toward Proof

Shailesh A Shirali

Rishi Valley School, Madanapalle, A.P., India

Proof is and has been for long a problematic area in the teaching of mathematics at the school level. While proof remains central to the discipline of mathematics (articles like Horgan, 1993, notwithstanding), its pedagogic role at the school level remains unclear. On perusing through the questions asked in the Math Forum site (http://mathforum.org/) one sees the demoralizing nature of the difficulties felt by students
The noted mathematics educator P K Srinivasan had listed the first exposure to .Proof. as one of three critical points at which children tend to switch off from the subject altogether.
A major contributory factor to this problem is surely that we introduce proofs at too late a stage. Moreover, it is done in too abrupt, too formal, and too stylized a manner. This results in a feeling of alienation for the child, who finds proofs unmotivated and unnatural. This feeling is added to if what is being proved looks .obvious.; and a majority of the early results encountered in geometry do indeed look .obvious.. (Recall some of the results we meet early in the study of geometry; e.g., the .bridge of asses. theorem.)
Whatever be the cause, the problem challenges us to respond with some effective pedagogy. The cost of not doing so is considerable. A child reaching the senior grades without a significant exposure to the culture of proof has lost a valuable opportunity to experience a central component of the discipline of mathematics. In this paper we report on a study done with children in grades 8 and 11, seeking their approaches and responses to the notion of proof in mathematics. We also explore some ways in which proof might be introduced to young children in a way that intrigues them and holds their attention, through non-geometric contexts like number theory and graph theory.
Hanna, Gila (2008). Beyond verification: Proof can teach new methods. http://www.unige.ch/math/EnsMath/Rome2008/WG1/Papers/HANNA.pdf
Hanna, Gila. The Ongoing Value of Proof. Web document last retrieved, June 17, 2008, http://fcis.oise.utoronto.ca/~ghanna/pme96prf.html
Horgan, J. (1993). The Death of Proof. Scientific American, 269(4), 93-103

Learning and Teaching Design and Technology: Meeting Needs, Developing Capability

Kay Stables

University of London, London, U.K.

Recent years have seen Technology Education grow and spread across regions, countries and provinces as a subject taught and learnt in mainstream schooling. Within this global growth, a distinct development has been of .Design and Technology. as a unified learning area. This paper will provide a rationale for the linking of Design with Technology within the curriculum, identifying the potential this affords, providing a conceptual framework for learning and teaching and exploring the issues and opportunities this raises for developing effective pedagogical approaches.
Through the paper I will explore reasons why learning Design and Technology is important for inpiduals and for societies, consider the debates about what needs to be learnt and outline how this learning might take place. Within this I will focus on the value of a capability approach and the importance of learning through doing by engaging in an iterative, responsive process. I will discuss the implications of this for pedagogical approaches to developing knowledge, skills and understanding, including issues of cognition, learning style and designing style. I will draw particularly on research that has developed our understanding both of the nature of Design and Technological capability and of ways of nurturing this capability through the learning experiences we can provide for all children in general education, at both primary and secondary levels.
Finally, I will explore the pedagogical issues this raises for teacher education, drawing particularly on experience gained through preparing new Design and Technology teachers in my own institution . Goldsmiths, University of London.

Strand I

Historical, Philosophical and Socio-cultural Issues Relevant to STM Education

Endorsing and Rejecting Scientific Claims: The Role of Evidence in Scientific Discourse

Leslie J. Atkins

California State University , Chico, U.S.A.

The view of science in the education community is shifting from a .rhetoric of conclusions. to a social process of knowledge construction via scientific argumentation. This emphasis on argument recasts the role of evidence and data in scientific classrooms: rather than being used to demonstrate scientific principles, it is the grounds on which claims are warranted. This understanding of science has launched curricula and research programs that are aimed at improving students. abilities to coordinate theory and evidence in scientific argumentation. In this paper, I examine a transcript of scientific discourse, exploring the rules by which participants in the discourse endorse or reject scientific claims. I appeal for a more nuanced understanding of direct evidence as one of many criteria by which scientific claims are evaluated, and that evidence, at times, is incommensurable with other criteria.

What is Mathematics Education for?

Brian Greer

Portland State University, Portland, U.S.A.

It is argued that familiar arguments for the status of mathematics as a major school subject, while retaining validity, need to be subjected to deep critique in relation to the roles played by mathematics in social, economic, and political aspects of our lives. Accordingly, mathematics educators (and mathematicians) should examine their ethical responsibilities in relation to the challenges facing humankind.

Unmasking the Gods: Clarifying Social Construction Theses in Mathematics Education

Jagdish Madnani

Sahyadri School, Pune, India

The phrase .social construction. has raised hackles in academic debates and has rapidly gained currency among theorists of all stripes. Unfortunately, as almost always happens, once a phrase is used too often, it begins to lose its impact and its original meaning-in-use. One of the goals of this paper is to reinvigorate this metaphor by trying to look at some of the roles it has played in the various debates. Another goal is to try to clear the air regarding its use in mathematics. Although mathematical development is often seen as dialogical and historically contingent, it is a discipline that hides its nature under a monological mask that claims to be fixed, monolithic and eternally true. By explaining the use of this phrase in recent writing relating to the philosophy of mathematics and mathematics education, I hope to re-illuminate the contingent nature of the discipline and the mathematics classroom .In mathematical construction we are, as it were, gods.. . Salomon Maimon

Chinthamani Ragoonathachary and Secularisation of Time During the Late Nineteenth Century Madras Presidency

Venkateswaran T. V.

Vigyan Prasar, New Delhi, India

Chintamani Ragoonathachary, a .native. astronomer took the initiative to modify and publish a new Panchang (almanac) and thereby produced a change in the calanderical system followed in the Tamil region. Inspired by the modern astronomy, this effort towards modernization of Panchang is an effort towards secularization of time. To engender reform he utilized popularization of astronomy. It is argued that this project of modernization by Chinthamani Ragoonathachari is not a colonial project but a project of .native. elites to secularizing time in Tamil with the aim of meeting the needs of modern industrial society.

Historico-Critical Analysis of the Concept of Mass: From Antiquity to Newton

Mashood K.K.

Homi Bhabha Centre for Science Education (TIFR), Mumbai, India

What is carried out in this paper is a critical analysis of the conceptual evolution of .mass. from antiquity to Newton, showing how the concept has evolved out of the givens of experiences at various phases of human thought. The current significance and relevance of topics like this with regard to physics education is also discussed. The story of mass starts its course with the earliest notion of .measure of matter. which coincides with the measure of food, around the dawn of agricultural age. Matter, although intuitively obvious was an intractable morass for the thinkers that followed. The adumbrations in neoplatonic philosophy, followed by its mystic and still inarticulate presentation in theology gradually paved way for its manifestation in the physics of Kepler and Newton.

From One to Infinity: Historical Development and Student View of Large Numbers

Mala Saraswathy Nataraj & Michael O. J. Thomas

The University of Auckland, Auckland, New Zealand

The importance of a consideration of large numbers in primary and early secondary school should not be underestimated. Historically, in Indian mathematics (and Mayan), a study of large numbers seems to have provided the impetus for the development of a place value number system. Present day students do not have to create a number system, but they do need to understand its structure in order to develop number sense and operations. We believe that this can be done more quickly through reflection on large numbers. We consider different types of large numbers in use in India prior to the construction of the present number system and examine Year 9 student responses to a questionnaire on large numbers. These are categorised and the results suggest that many students show competence in naming and using large numbers, and that some are in the process of learning beyond their curriculum level.

Sculpted by Culture: Students. Embodied Images of Scientists

Ratna Narayan1, Soonhye Park2 & Deniz Peker3

1 Texas Tech University, Lubbock, U.S.A., 2 University of Iowa, Lowa, U.S.A., 3 Middle East Technical University, Ankara, Turkey

We conducted a cross cultural comparative analysis involving children from India, South Korea, Turkey and the United States. The study investigated children.s perceptions regarding scientists, the similarities and differences between their stereotypic perceptions of scientists and the cultural factors that contribute to them. The participant pool included students from grades 3 and 7 (120 per grade, per country) who were administered the Draw-A-Scientist-Test (Chambers, 1983). Randomly chosen students were also interviewed using a semi-structured interview protocol. A one-way ANOVA was performed to test for differences among the four countries. Results revealed some commonalities in the stereotypic perceptions regarding scientists and discussed the .value. placed on science in these countries.

Gender and Mathematics Education: Lessons from Pakistan

Anjum Halai

Aga Khan University Institute for Educational Development, Karachi, Pakistan

This paper reports from a large scale action research study1 on gender issues emerging in the context of mathematics education when teachers implement a new curriculum for improving gender equity in science and mathematics classrooms, and from field experiences when teacher educators promote gender equity teaching in mathematics.

Gender Differences and Mathematics Achievement of Rural Senior Secondary Students in Cross River State, Nigeria

Sam William Bassey1, M. T. Joshua2, Alice E. Asim2

1 Cross River University of Technology, Calabar, Nigeria, 2 University of Calabar, Calabar, Nigeria

To contribute to the realization of the Millennium Development Goal (MDG) by the United Nations on the promotion of gender equity, the researchers sought to empirically verify the existence or otherwise of gender inequality in the mathematics achievement of rural male and female students in Cross River State, Nigeria; and whether parental socio-economic status and school proprietorship, taken independently, are significant factors in the achievement of the students. By stratified and simple random sampling, 2000 students (50% males, 50% female) were selected and a 30-item four-option multiple choice mathematics achievement test (MAT) was constructed (KR20 of 0.87 and item difficulty, 0.40 < p < 0.82) and administered. The independent t-test analysis of significance revealed gender inequality in the entire sample as well as among the low socio economic students and within public schools. Educational implications have been highlighted.

Estimation of Gender Bias through Specially Developed Learning Material (Interactive C.D.) . A Study in Nashik City

Nikhila Bhagwat1 & Hemant Rajguru2

1S.M.R.K. Women.s College, S.N.D.T. University, Nashik, India, 2Yashwantrao Chavan Maharashtra Open University, Nashik, India

Gender discrimination pervades Indian society. Even after 60 years of independence, women in India suffer from inequity and bias. The horrific instances of infanticide and other forms of violence against women has cultural considerations at its root, coupled with failure to enforce legislation. There is a dire need to introduce an attitude of equality and free people from their prejudiced perspective. To address this problem, it was planned to give .Gender Education. by direct intervention method. An innovative self-learning multimedia package comprised of a .documentary film. and .self-learning interactive multimedia package. was devised with a systematic process that disseminated information on gender education. It was implemented with 833 adult men and women in the Nashik city. The experiment proved effective in reducing the bias and communicating concepts important to redefine gender values.

Strand II

Cognitive Studies of STM Learning

An Indigenous Approach to Elementary Astronomy - How Cognitive Research Can Help

Shamin Padalkar & Jayashree Ramadas

Homi Bhabha Centre for Science Education, TIFR, Mumbai, India

We report a study on Grade 8 students. understanding of elementary astronomy. We aim to use recent research in visuospatial reasoning to develop a locally appropriate low-cost teaching methodology for elementary astronomy. Our Experimental sample is drawn from students of three different schools (tribal, rural, urban-slum) in educationally disadvantaged areas in the State of Maharashtra, India. The study consists of three parts: (1) investigation of students. initial knowledge, (2) exposing students at the end of Grade 7 to various problem situations over one year, requiring them to explain or predict some daily astronomical phenomena with the help of tools for visuospatial reasoning such as concrete models, gestures and diagrams and (3) assessment of their progress at the end of Grade 8. We present the rationale and general features of the pedagogical sequence which we developed with the aim to help students use visuospatial reasoning to explain daily astronomical phenomena. We also report some preliminary findings about students. development.

Refined Concept Maps for Science Education: A Feasibility Study

Meena Kharatmal & Nagarjuna G.

Homi Bhabha Centre for Science Education, TIFR, Mumbai, India

Refined concept map (RCM) is comprised of node names and a well-defined, invariant, minimal set of relation names. Using RCM as a methodology, it can be applied to study the changes in the knowledge structure, as a tool for analysis of forms of representations. In this paper, we discuss the study conducted to test the ease and feasibility of RCM by comparing it with other modes of representation. A homogeneous sample of school students were assigned the same task from a specific domain. The analysis shows that it was easy and feasible to use RCM by the school students. The fixed set of relation names, does not affect the expression of knowledge and at the same time helps in representing accurate knowledge. The constraints in the RCM served as an anchoring and a facilitator for representing scientific knowledge.

Studying Indian Middle School Students' Attitudes towards Technology

Ritesh Khunyakari, Swati Mehrotra, Chitra Natarajan & Sugra Chunawala

Homi Bhabha Centre for Science Education, TIFR, Mumbai, India

Understanding students. ideas is a key step towards meaningful learning. Perception and attitudinal studies in education have been used to unravel crucial aspects about a particular issue, concept or an idea. Technology is a construct little explored in the Indian context. The paper reports development of an instrument and study of students. perceptions of technology and attitudes toward it. Some salient findings from the study are reported. The study brings out the influence of urban and rural settings, medium of learning and gender differences on the several items probed. The differences highlight the necessity to incorporate some of the ideas in developing units for technology education. Insights from the study can be channelled to making technology education units more inclusive and interesting.

Evidences of Learning Through Collaboration in Design and Technology Tasks in Indian Classrooms

Swati Mehrotra, Ritesh Khunyakari, Sugra Chunawala & Chitra Natarajan

Homi Bhabha Centre for Science Education, TIFR, Mumbai, India

Working together in a group or teamwork is a soft skill that is highly valued in the job market. Technology tasks provide an opportunity where teamwork could be encouraged, not merely for completing a project with pre-determined goals but also for learning to work and operate as a team. The paper focuses on the results of a study (that was part of a larger project) conducted with middle school students in three different socio-cultural settings in India. This paper will report the evidences of collaborative learning that occurred while the students engaged in the design and technology units.

Moving from Analysing to Designing Artefacts: Studying Middle School Students. Ideas about Design and Designers

Farhat Ara, Sugra Chunawala & Chitra Natarajan

Homi Bhabha Centre for Science Education, TIFR, Mumbai, India

The paper presents an investigation of middle school students. nave ideas about design and designers and details of the trial of design-related activities aimed at sensitizing students to issues of design. Twenty five students studying in class 7 participated in the study conducted over a period of 5 days. A questionnaire was administered to all students at two occasions, pre and post intervention and a few students (eight) were interviewed in the pre-intervention stage. The activities ranged from analyzing familiar and unfamiliar artefacts to designing artefacts. Preliminary analysis of the survey revealed students. intuitive ideas about design and designer. Most students stated that animals are also designers and referred to nest and home building activities. However, when referring to humans they focused mainly on aesthetic aspects of design. In the designing activity students generated various creative solutions to a given real-life problem. The present study would provide opportunities to develop and try out more and different activities with students and even with teachers.

Middle School Students' Knowledge about Static and Dynamic Artefacts Studied through their Drawings and Descriptions

Gandhimathy Selvaraj, Ritesh Khunyakari & Chitra Natarajan

Homi Bhabha Centre for Science Education, TIFR, Mumbai, India

This paper reports a study of students' understanding of artefacts through their drawings and descriptions. The experimental design of the study carried out with 12 middle school students (ages 11- 13 years) involved four stages: pre-test, intervention and post-test, followed by a semi-structured interview of each student. The intervention activities engaged students in filling a questionnaire by estimating and measuring dimensions of a variety of artefacts of common shapes and sizes, writing their descriptions, and in repairing a bicycle. The study analysed students' paper-pencil productions in the tests and the questionnaire, and audio and video data collected during the intervention and interviews. The effect of the intervention on the nature of depictions of proportions and dimensional attributes in the drawings depended on the context of problem solving. Interviews helped to make explicit the meanings ascribed by students to the descriptions and the strategies used by them in their object depictions. The study highlights the im-portance of engaging students in authentic contexts of problem solving, and making drawings in such contexts.

Interplay between School Mathematics and Work Place Mathematics

Nirmala Naresh

Miami University, Oxford, OHIO, USA

Sociocultural dimensions of mathematical knowledge have greatly influenced research in the field of mathematics education in the past few decades, resulting in the rise of different areas of research that include ethnomathematics, everyday mathematics, situated cognition, and workplace mathematics. The line of research on everyday mathematics has pointed out the importance of situations that evoke superior performance in quantitative reasoning in everyday settings and researchers have called for further investigation of everyday practices that involve mental mathematics. The general aim of this study is to develop a better understanding of the mathematics used in the workplace of bus conductors in Chennai, India. In particular, this study focuses on investigating the mental mathematics involved in bus conductors. work.

The Use of Convenient Value Strategies among Young Train Vendors in Mumbai, India

Yasmin Sitabkhan

University of California, Berkeley, USA

This study analyzes the mathematical understandings young unschooled vendors develop through participation in the culturally based practice of selling small items on the local trains in Mumbai, India. A core assumption of the study is that inpiduals. goals and the strategies they construct to accomplish those goals take form in relation to social interactions, activity structures, and the understandings people bring to their practices. Analyses of the interviews and observations highlight the role of commonly used values, which I have termed convenient value strategies, in the mathematical understandings as well as in the lived practice of the child sellers. Convenient value strategies provide a link between sellers. mathematical understandings and the practices with which they are engaged.

Strategic Content Learning Approach to Promote Self- Regulated Learning in Mathematics

Haneet Gandhi1 & M. Varma2

1 Delhi University, Delhi, India, 2 University of Lucknow, Lucknow, India

Advocacy that inpiduals learn when they are proactive, self-organized, self-reflecting, and, in turn, self-regulating has been examined and summarized in this paper. An instructional approach: .Strategic Content Learning. was adapted to promote self-regulated learning skills for problem solving in mathematics. Understanding average mathematics performers. beliefs and knowledge about mathematics as a subject, their understanding of .mathematical problems., and helping these average performers to select, adapt or invent strategies that help them in becoming better self- regulated solvers in non-routine mathematical problems was the main objective of the study. Concomitantly the students were also helped to develop their personalized strategies that they could transfer across problems and time. As a result of strategic content learning, they improved in their monitoring skills, circumventing on their weaknesses and capitalizing their strengths to achieve active control over their chosen tasks.

Rote and Algorithmic Techniques in Primary Level Mathematics Teaching in the Light of Gagne.s Hierarchy

Amarendra Narayan

Patna University, Patna, India

The rote and algorithmic methods based on .Sutras. in traditional Indian schools are looked down upon by the majority of modern educationists. This paper examines these techniques in the light of Robert Gagne.s theory of hierarchy of concepts and meaning. A semantic net model has been presented and the techniques of rote and algorithmic learning as well as the principle of constructivism are examined in its light. It is shown that the traditional techniques prepare the foundation necessary for concept formation. On the basis of this, it is suggested that teaching method be developed that properly utilizes the insights on cognition provided by Gagne.

Strand III

Curriculum and Pedagogical Studies

The Introduction of Angles

Usha Menon

National Institute of Science, Technology and Development Studies, CSIR, Delhi, India

This paper reviews some of the problems with the current frameworks for studying geometry curriculum at the primary level. This is done based on the experience of an on-going project involving the development of an alternative geometry trajectory. The results presented indicate the possibility of introducing the angle concept much earlier than what is currently the case. A leading role for instruction to create a zone of proximal development is suggested.

Introducing Fractions Using Share and Measure Interpretations: A Report from Classroom Trials

Jayasree Subramanian1 & Brijesh Verma2

1Eklavya, Hoshangabad, India, 2Muskan, Bhopal, India

It is well acknowledged that .fractions. is one of the most complex topics in the primary and middle school curriculum. There has been a considerable amount of research on teaching and learning of fractions in the last few decades and the curriculum design in the West. A Non-Governmental Organisation (NGO) in collaboration with an education research institute is engaged in similar attempts in the Indian context. This paper is a report of our trials in two schools on introducing fractions to primary school children using a combination of share and measure interpretation.

Exploring the Confusions: Bar Graphs

Susan Hillman

Saginaw Valley State University, Bay City, USA

Professional development with practicing teachers of mathematics in the middle grades from the U.S. and India revealed confusions about the construction and interpretation of data with bar graphs when working with certain kinds of data. Textbooks often influence teachers. decisions about what and how mathematics is taught. Textbooks used in both countries were examined for definitions, descriptions, and examples that highlighted (or obscured) features of bar graphs. Potential sources of confusion for teachers (and ultimately their students) regarding this type of graph were identified. Implications and recommendations for teaching data analysis with bar graphs are provided.

The Curriculum is More than Textbooks and Technology: Project M3: Mentoring Mathematical Minds

Linda Jensen Sheffield

Northern Kentucky University, Highland Heights, USA

Project M3: Mentoring Mathematical Minds is a United States Department of Education Javits research grant project designed to provide challenging, motivational curriculum units for students with mathematical promise in grades two through six. In addition to the print materials for students and teachers, the project is designed to develop problem solving heuristics and strategies through the use of rich learning tasks, questioning strategies, hints for students having difficulties, .Think Beyond. questions to assist with the differentiation of the tasks, and verbal and written discourse.

Using Research on English to Understand Mathematical Writing

Byung-In Seo

Saginaw Valley State University, Bay City, USA

As a secondary teacher, I was in a unique position, for I taught both English and mathematics to high school students. Through my observations, I learned that my students employed similar writing practices when they learned mathematical skills. This study explored how high school students addressed the audience when they wrote mathematically.

Virtual Manipulatives: Potential Instructional Hazards and Possible Design-based Solutions

William Speer

University of Nevada, Las Vegas, USA

Virtual manipulatives are employed by both preservice and inservice teachers to enhance the instructional effectiveness of physical manipulatives and related tools by addressing limitations of access, cost, and adaptability. While research into the use of emerging technologies continues, there are several variables to consider when using, or measuring the effects of virtual manipulative use. Research design, sampling characteristics, and the type of manipulative used may influence achievement. For example, some studies that have shown evidence of increased achievement were administered when classroom teachers believed they fit in with the natural flow of the curriculum. Other studies with no noticeable increase in student achievement were administered at times that interrupted the normal curriculum. Other variables that may influence the effectiveness of using virtual manipulatives include: previous experience with computers, grade level, mathematical topic, treatment length, student attitudes toward mathematics, and computer-to-student ratio.

Problem Solving In Mathematics: A Tool for Cognitive Development

Preety N. Tripathi

State University of New York, Oswego, USA

How can problem solving be used as a tool for cognitive development? Can problem solving be used to effect a change in learners. attitudes and beliefs about mathematics so that they come to view mathematics as a discipline founded on reasoning? What are some strategies that instructors may use in the process? To seek answers to these questions, I conducted a study with a group of prospective elementary school teachers. In this article, I describe briefly my attempts to answer the above questions.

Achilles and the Tortoise Paradox - Finding the Zone of Proximal Development in Understanding Limit of a Sequence

Bronislaw Czarnocha1 &Vrunda Prabhu2

1Hostos CC, CUNY, NYC, USA, 2Bronx CC, CUNY, NYC, USA

We discuss here the outcome of a Teaching Experiment conducted in the classes of Freshman calculus in CUNY colleges of the Bronx, NYC, which was supported by the NSF-ROLE grant #0126141, Introducing Inpisibles into Calculus Instruction. One of the main issues investigated was student understanding of the concept of the limit of sequences. The teaching used the guided inquiry (or guided discovery) method. Here we report students' work with the Achilles and the Tortoise paradox which involves an infinite converging sequence. Attempts to identify the zone of proximal development with respect to this problem, together with appropriate modifications to the presentation of the paradox in the form of assignment problems are described.

Students Using a Mathematica Learning Project Work to Conceptualise the Derivative

Kristie Naidoo & Ramu Naidoo

Durban University of Technology, Durban, South Africa

This study is based on a constructivist approach to the learning of the concept of the derivative by discovery and by self-pacing. To determine whether the new learning and teaching environment had an impact on the students. understanding of the derivative, two groups, each consisting of 34 students, comprised the control and experimental groups respectively. The experimental group participated in a Mathematica Learning Project whilst the control group was taught traditionally. Both groups were tested. It was envisaged that Mathematica Learning project may minimize the cognitive overload experienced by students during a traditional lecture. The project work in the laboratory formed part of the assessment for the experimental group. To evaluate students. responses, errors made by students during the project and the paper-pencil test were analysed. Findings revealed a greater number of structural errors in the control group as compared to the experimental group. Further the experimental group exhibited more deep structures than surface structures whilst the traditional group exhibited more superficial structures than deep structures.

What Do Cells Really Look Like? Children.s Resistance to Accepting a 3-D Model

Aisha Kawalkar, Jyotsna Vijapurkar

Homi Bhabha Centre for Science Education, TIFR, Mumbai, India

In our explorations of children.s concepts in science we have found that children of grades 6 and 7 visualize cells as 2-D objects. This aspect of children.s understanding of the cell has not been reported before, to our knowledge. In this paper we describe our motivation for exploring this concept and report our discovery of children.s resistance to accepting the idea of the cell as a 3-D object. We discuss the implications of our results for teaching this fundamental concept in biology. Further, we give recommendations of teaching strategies that we developed and found to be successful.

Stepping into Science in Small Schools: Together with Tools, Techniques and Toys

Lalit Kishore

Centre for Unfolding Learning Pontentials, Jaipur, India

This intervention action research paper describes the development and implementation of primary level science experiences in the project mode for small schools in the non-governmental organization (NGO) sector located in dispersed small habitations in the desert area of an Indian State called Rajasthan. The project was developed in a participatory manner through a need analysis with the author functioning as designer and mentor. The project includes development, training, implementation and recurrent reviews of science activities in the schools combining the use of hand tools and low-cost science experiments. The components of the science learning at primary level had the following aspects: (a) making improvised measuring instruments; (b) familiarization with elementary hand tools; (c) making models; (d) performing experiments; (e) doing investigations; (f) making toys; (g) science-related drawing skill. The project was meant for grades three, four and five with 20 science-related activities at each grade level. The evaluation of the project was done through quarterly participatory reviews by the teachers and evaluation of children.s performance in the annual test. The project after one year resulted in establishing the science activity corner, and annual science exhibition for parents and village community. The innovative features of the programme are the use of readily available material; comprehensive experiences; participatory processes and community linkages. The project resulted in improving the achievement levels in the project schools at grade three as compared to the previous year.s achievement scores at 0.01 levels of significance as revealed by the t-test analysis.

Integrating Concepts and Skills Through Design of Learning Activities

Deepak S. Paranjape and Sudhakar C. Agarkar

Homi Bhabha Centre for Science Education, TIFR, Mumbai, India

Understanding physical geography involves graphicacy as well as cognitive skills and concepts from the domain of science and mathematics. Empirical data necessary for formulating teaching strategies that develop students. skills, or for developing appropriate instructional materials, is however inadequate. Moreover, pedagogic content knowledge of practicing teachers needs to be developed for facilitating effective learning of physical geography. A research approach involving an informed design of pedagogic innovations can provide the means for addressing these concurrent issues of practice. Hence, learning activities are designed for integrated development of graphicacy skills and transdisciplinary concepts; particularly involved in understanding the physical processes associated with rain. The design and planning of the learning activities has involved a synthesis of educational theory and pedagogic content knowledge of physical geography. This paper discusses the details of this prospective pedagogic design; evolving through reflective practice. Learning activities are found to provide the context for explicating the significant elements of student learning, with the corresponding teacher knowledge.

Strand IV

Systemic Change and Teacher Professional Development in STM Education

The Role of Formative Assessment in Enhancing Independent Learning and Reflective Teaching: Some Results of the Austrian IMST-Project

Thomas Stern

IUS / Institute of Instructional and School Development, University of Klagenfurt, Klagenfurt, Austria

IMST (Innovations in Mathematics and Science Teaching) is a long term research and development project aimed at establishing an effective support system for Austrian schools. In this framework three case studies of secondary school innovations are analysed, in which the students learned independently and new forms of assessment were introduced. These small scale innovations triggered a series of changes in the classroom and on the school level and led the teachers to question some of their fundamental beliefs. By investigating their practice, reflecting about teaching priorities and sharing their thoughts with colleagues, they were embarking upon a process of self-directed professional development which needed little outside support. Their creativity and problem-solving competencies raise hopes that their knowledge and experience might be valuable resources for the imminent reform of the Austrian school system.

The STEM Agenda in the Context of Initial Teacher Education: Challenges and Potential Ways Forward

Graham Hardy, Andrew Howes, David Spendlove & Geoff Wake

University of Manchester, Manchester, England

The STEM agenda within the UK .is a series of initiatives geared towards creating a strong supply of scientists, engineers and technologists. (QCA, 2007). One curriculum strategy being tried is the fostering of links between STEM subjects. This paper explores the potential of such a strategy by researching a small scale curriculum initiative within a one year full time post graduate teacher education programme in England. CHAT analysis served to provide a theoretical lens showing how trainee teachers had to make difficult boundary crossings between the different demands of the university based and school based elements of training. The paper concludes that the STEM initiative needs to be proactive in reconciling the tensions that exist when boundary crossing occurs whilst also providing opportunities for establishing boundary zones for key players to operate in.

Science Teacher Educators: A Shift towards Student-centredness

Sathiaseelan Pillay

University of KwaZulu-Natal, South Africa

The introduction of modularization at universities and an outcomes-based education at schools in South Africa were based on the expectation that science teacher educators would make attempts at changing their teaching styles from a teacher-centred approach to a learner-centred approach. I conducted research as a science teacher educator to establish the extent to which eleven science teacher educators (lecturers) at three universities in a province in South Africa responded to the policy changes, and to which policy change in particular. An analysis of the observations showed that some science teacher educators had made appropriate changes for learner-centredness through a role-modeling process while others continued in a traditional teacher-centred approach. It was obvious that the shift to learner-centredness was more a response to school-related expectations of change and not necessarily to those of modularization at a higher education (university) level.

'It makes Sense to Ask That Question': A Closer Look at Student Questions

Anita Balasubramanian

University of Illinois, Chicago, USA

Several studies have investigated classroom mathematical discourse and how teachers and students participate in it to create meaningful interactions. In this paper I consider the questions that students ask in a high school math classroom in relation to the teacher.s expectations of the kinds of student questions he wants to encourage in the discourse. Using the framework of reification, I show that there is a relationship between students developing a structural outlook to asking questions that the teacher expects from them.

Focus: Women of Rural Communities of Tamil Nadu

Vrunda Prabhu1 & Bronislaw Czarnocha2

1Bronx Community College,City University of New York ,New York City,USA, 2Hostos Community College,City University of New York ,New York City,USA

An ongoing teaching-research project in mathematics classrooms of community colleges of the Bronx at City University of New York meets with community development projects in rural Tamil Nadu, India, initiated by grassroots organizer and mathematics historian at epiSTEME-1. Over the course of three years, the partnership forms. Mathematicans, physicists, psycho-social workers, action researchers of CUNY work hand-in-hand with the Arunthatiar communities of the Salem and Erode districts of Tamil Nadu. In a unique collaboration, that starts with education and mathematics education in particular, both groups learn from each other, and a project sparks to life. The women of the communities led by Alamelu, a young woman from Salem have created hundreds of self-help-groups of women. This article reports on the teaching-action-research project, which at its present stage integrates the needs of the community for quality mathematics learning with the daily teaching-research activities of the teacher-researchers at CUNY. The self-help-groups develop the idea of Montessori-for-Mothers, and a dream of Communities of the Future is born, whose focus is on preserving the spark of the child.

Initiatives Under the Sarva Shiksha Abhiyan for Improvement in Basic Numeracy Skills Among Children in the Early Grades

Binay Pattanayak

Sarva Shiksha Abhiyan, MHRD, New Delhi, India

The Sarva Shiksha Abhiyan (SSA) aiming for universalisation of quality elementary education for each child in the 6 to 14 age group has undertaken several initiatives to improve the quality of mathematics education both at Primary and Upper Primary level. This paper attempts to highlight some of such initiatives. Recently the SSA norms have been revised to strengthen the quality related interventions in a more rigorous manner. However learning achievement surveys undertaken by National Council of Education Research and Training (NCERT) and other agencies show that mathematics pedagogy calls for more attention to help children acquire the basic skills in mathematics. At present the attempt is to strengthen the early reading and mathematics skill development programmes at the Primary level and Mathematics teaching at Upper Primary level to prepare the students in a better manner. The paper attempts to highlight such initiatives, their strengths and limitations and indicates further possibilities to give it the desired direction.