- CREATING OPTIMAL MATHEMATICS LEARNING ENVIRONMENTS: COMBINING ARGUMENTATION AND WRITING TO ENHANCE ACHIEVEMENT, by Dionne I. Cross
- APPROACHES TO TEACHING MATHEMATICS IN LOWER-ACHIEVING CLASSES, by Ruhama Even and Tova Kvatinsky
- ANALYSIS OF THE LEARNING EXPECTATIONS RELATED TO GRADE 1–8 MEASUREMENT IN SOME COUNTRIES, by Jung Chih Chen, Barbara J. Reys and Robert E. Reys
- EPISTEMOLOGICAL OBSTACLES IN COMING TO UNDERSTAND THE LIMIT OF A FUNCTION AT UNDERGRADUATE LEVEL: A CASE FROM THE NATIONAL UNIVERSITY OF LESOTHO, by Eunice Kolitsoe Moru

## 2009/09/26

### IJSME, October 2009

### Addition and subtraction of three-digit numbers

Empirical findings show that many students do not achieve the level of a flexible and adaptive use of arithmetic computation strategies during the primary school years. Accordingly, educators suggest a reform-based instruction to improve students’ learning opportunities. In a study with 245 German third graders learning by textbooks with different instructional approaches, we investigate accuracy and adaptivity of students’ strategy use when adding and subtracting three-digit numbers. The findings indicate that students often choose efficient strategies provided they know any appropriate strategies for a given problem. The proportion of appropriate and efficient strategies students use differs with respect to the instructional approach of their textbooks. Learning with an investigative approach, more students use appropriate strategies, whereas children following a problem-solving approach show a higher competence in adaptive strategy choice. Based on these results, we hypothesize that different instructional approaches have different advantages and disadvantages regarding the teaching and learning of adaptive strategy use.

### Flexible and adaptive use of strategies and representations

The flexible and adaptive use of strategies and representations is part of a cognitive variability, which enables individuals to solve problems quickly and accurately. The development of these abilities is not simply based on growing experience; instead, we can assume that their acquisition is based on complex cognitive processes. How these processes can be described and how these can be fostered through instructional environments are research questions, which are yet to be answered satisfactorily. This special issue on flexible and adaptive use of strategies and representations in mathematics education encompasses contributions of several authors working in this particular field. They present recent research on flexible and adaptive use of strategies or representations based on theoretical and empirical perspectives. Two commentary articles discuss the presented results against the background of existing theories.

## 2009/09/24

### How Do Parents Support Preschoolers’ Numeracy Learning Experiences at Home?

A new and interesting article has been published in Early Childhood Education Journal: How Do Parents Support Preschoolers’ Numeracy Learning Experiences at Home? The article is written by Sheri-Lynn Skwarchuk.

Abstract

This study described the kinds of early numeracy experiences that parents provide for their preschoolers, and determined the extent to which parental experiences and involvement in home activities enhanced preschoolers’ numeracy knowledge. Twenty-five parents completed a home activity questionnaire, a 2-week diary study, and a videotaped play session where they were asked to draw out numerical content. Preschoolers’ numeracy scores were predicted by: (1) parental reports of positive personal experiences with mathematics and (2) involvement in activities with complex (versus basic) numeracy goals. Parents felt that most activities had important or essential mathematical value, but focused on number sense goals. Finally, parents who reportedly spent more time on numeracy tasks received high quality interaction ratings in the videotaped sessions; but these variables did not predict numeracy scores. The findings are discussed in terms of educating parents about incorporating numeracy concepts.

### Finnish pre-service teachers’ and upper secondary students’ understanding of division and reasoning strategies used

New article in Educational Studies in Mathematics, written by Raimo Kaasila, Erkki Pehkonen and Anu Hellinen: Finnish pre-service teachers’ and upper secondary students’ understanding of division and reasoning strategies used

In this paper, we focus on Finnish pre-service elementary teachers’ (N= 269) and upper secondary students’ (N= 1,434) understanding of division. In the questionnaire, we used the following non-standard division problem: “We know that 498:6 = 83. How could you conclude from this relationship (without using long-division algorithm) what 491:6 = ? is?” This problem especially measures conceptual understanding, adaptive reasoning, and procedural fluency. Based on the results, we can conclude that division seems not to be fully understood: 45% of the pre-service teachers and 37% of upper secondary students were able to produce complete or mainly correct solutions. The reasoning strategies used by these two groups did not differ very much. We identified four main reasons for problems in understanding this task: (1) staying on the integer level, (2) an inability to handle the remainder, (3) difficulties in understanding the relationships between different operations, and (4) insufficient reasoning strategies. It seems that learners’ reasoning strategies in particular play a central role when teachers try to improve learners’ proficiency.

## 2009/09/21

### Teachers' conceptions of creativity

Here is the abstract of their article:

Teachers in the UK and elsewhere are now expected to foster creativity in young children (NACCCE, 1999; Ofsted, 2003; DfES, 2003; DfES/DCMS, 2006). Creativity, however, is more often associated with the arts than with mathematics. The aim of the study was to explore and document pre-service (in the UK, pre-service teachers are referred to as ‘trainee’ teachers) primary teachers’ conceptions of creativity in mathematics teaching in the UK. A questionnaire probed their conceptions early in their course, and these were supplemented with data from semi-structured interviews. Analysis of the responses indicated that pre-service teachers’ conceptions were narrow, predominantly associated with the use of resources and technology and bound up with the idea of ‘teaching creatively’ rather than ‘teaching for creativity’. Conceptions became less narrow as pre-service teachers were preparing to enter schools as newly qualified, but they still had difficulty in identifying ways of encouraging and assessing creativity in the classroom. This difficulty suggests that conceptions of creativity need to be addressed and developed directly during pre-service education if teachers are to meet the expectations of government as set out in the above documents.

### Self-efficacy beliefs regarding mathematics and science teaching

This study investigated Turkish preservice, elementary teachers’ personal mathematics teaching efficacy (PMTE), and science teaching efficacy (PSTE) beliefs at the end of their teacher education program. A majority of the participants believed they were well prepared to teach both elementary mathematics and science, but their PSTE scores were significantly lower than their PMTE scores. However, a significant correlation was found between the PMTE and PSTE scores. No significant gender effect on PMTE and PSTE scores was observed, but unlike the results from other countries, Turkish female preservice elementary teachers were found to have slightly higher PMTE and PSTE scores than their male peers. High school major area was found to be a significant predictor of participants’ PMTE and PSTE scores. Participants with mathematics/science high school majors were found to have significantly higher PMTE and PSTE scores than those with other high school majors.

## 2009/09/20

### Three new ZDM articles

The purpose of this study was to identify the pedagogical knowledge relevant to the successful completion of a pie chart item. This purpose was achieved through the identification of the essential fluencies that 12–13-year-olds required for the successful solution of a pie chart item. Fluency relates to ease of solution and is particularly important in mathematics because it impacts on performance. Although the majority of students were successful on this multiple choice item, there was considerable divergence in the strategies they employed. Approximately two-thirds of the students employed efficient multiplicative strategies, which recognised and capitalised on the pie chart as a proportional representation. In contrast, the remaining one-third of students used a less efficient additive strategy that failed to capitalise on the representation of the pie chart. The results of our investigation of students’ performance on the pie chart item during individual interviews revealed that five distinct fluencies were involved in the solution process: conceptual (understanding the question), linguistic (keywords), retrieval (strategy selection), perceptual (orientation of a segment of the pie chart) and graphical (recognising the pie chart as a proportional representation). In addition, some students exhibited mild disfluencies corresponding to the five fluencies identified above. Three major outcomes emerged from the study. First, a model of knowledge of content and students for pie charts was developed. This model can be used to inform instruction about the pie chart and guide strategic support for students. Second, perceptual and graphical fluency were identified as two aspects of the curriculum, which should receive a greater emphasis in the primary years, due to their importance in interpreting pie charts. Finally, a working definition of fluency in mathematics was derived from students’ responses to the pie chart item.The other is written by Alan T. Graham, Maxine Pfannkuch and Michael O.J. Thomas. Their article is called Versatile thinking and the learning of statistical concepts. In the abstract you learn more about the main ideas in this article:

Statistics was for a long time a domain where calculation dominated to the detriment of statistical thinking. In recent years, the latter concept has come much more to the fore, and is now being both researched and promoted in school and tertiary courses. In this study, we consider the application of the concept of flexible or versatile thinking to statistical inference, as a key attribute of statistical thinking. Whilst this versatility comprises process/object, visuo/analytic and representational versatility, we concentrate here on the last aspect, which includes the ability to work within a representation system (or semiotic register) and to transform seamlessly between the systems for given concepts, as well as to engage in procedural and conceptual interactions with specific representations. To exemplify the theoretical ideas, we consider two examples based on the concepts of relative comparison and sampling variability as cases where representational versatility may be crucial to understanding. We outline the qualitative thinking involved in representations of relative density and sample and population distributions, including mathematical models and their precursor, diagrammatic forms.Finally, George Gadanidis and Vince Geiger have written an article about A social perspective on technology-enhanced mathematical learning: from collaboration to performance. Here is the abstract of their article:

This paper documents both developments in the technologies used to promote learning mathematics and the influence on research of social theories of learning, through reference to the activities of the International Commission on Mathematical Instruction (ICMI), and argues that these changes provide opportunity for the reconceptualization of our understanding of mathematical learning. Firstly, changes in technology are traced from discipline-specific computer-based software through to Web 2.0-based learning tools. Secondly, the increasing influence of social theories of learning on mathematics education research is reviewed by examining the prevalence of papers and presentations, which acknowledge the role of social interaction in learning, at ICMI conferences over the past 20 years. Finally, it is argued that the confluence of these developments means that it is necessary to re-examine what it means to learn and do mathematics and proposes that it is now possible to view learning mathematics as an activity that is performed rather than passively acquired.

## 2009/09/17

### A study on the teaching of the concept of negative numbers

This study mainly aims to develop an effective strategy to overcome the known difficulties in teaching negative numbers. Another aim is to measure the success of this teaching strategy among a group of elementary level pupils in Idotzmir, Turkey. Learning negative concepts are supported by computer animations. The academic achievement test developed by the researchers was administered to 150 sixth-grade pupils at the beginning of and following the learning period. The teaching strategy was applied to the experiment group (n = 75) as stated above, while the traditional teaching model most frequently used in Turkey was applied to the control group (n = 75). At the end of the study, a significant difference was found in favour of the experiment group (t = 17.51, df = 148, p = 0.000 < 0.05).

### Honoring Paul Ernest

Paul Ernest’s name is synonymous with social constructivism as a philosophy of mathematics. His contributions to mathematics education have occurred at a very fundamental level and to a extent shaped theory development in this field. His research addresses fundamental questions about the nature of mathematics and how it relates to teaching, learning and society. For the last three decades Paul has been a prolific scholar who has published in a wide array of topics such as the relationship between the philosophy of mathematics and mathematics education, and more generally the philosophy of mathematics education, ethics and values in mathematics education, and the philosophy of research methodology.

The title of this Festschrift is meant to be a pun to convey the sometimes relativistic dimension to mathematical certainty that Paul argued for in developing his philosophy, and also a play on words for the fact that absolute “earnestness” may perhaps be a Platonic construct, and not possible in the realm of language and human discourse! Paul Ernest’s scholarly evolution and life can best be summarized in the words of Walt Whitman “Do I contradict myself? Very well then I contradict myself” (I am large, I contain multitudes). Indeed his presence has been large and multitudinous and this Festschrift celebrates his 65th Birthday with numerous contributions coming from the mathematics, philosophy and mathematics education communities around the world.

## 2009/09/16

### What the eyes already know

Angela Heine and colleagues have written an article called: What the eyes already 'know': using eye movement measurement to tap into children's implicit numerical magnitude representations. The article has recently been published in Infant and Child Development. The authors make interesting links between eye movements and childrens understanding of numbers. Here is the abstract of their article:

*implicit*and

*explicit*knowledge at certain points in development that become manifest by their gestures and gaze orientation in different problem solving contexts. Stimulated by this research, we used eye movement measurement to investigate the development of basic knowledge about numerical magnitude in primary school children. Sixty-six children from grades one to three (i.e. 6-9 years) were presented with two parallel versions of a number line estimation task of which one was restricted to behavioural measures, whereas the other included the recording of eye movement data. The results of the eye movement experiment indicate a quantitative increase as well as a qualitative change in children's implicit knowledge about numerical magnitudes in this age group that precedes the overt, that is, behavioural, demonstration of explicit numerical knowledge. The finding that children's eye movements reveal substantially more about the presence of implicit precursors of later explicit knowledge in the numerical domain than classical approaches suggests further exploration of eye movement measurement as a potential early assessment tool of individual achievement levels in numerical processing.

## 2009/09/11

### Students' understanding of a logical structure in the definition of limit

Kyeong Hah Roh has written an article entitled An empirical study of students’ understanding of a logical structure in the definition of limit via the ε-strip activity. This article was published online in Educational Studies in Mathematics last Thursday. Here is the abstract of the article:

This study explored students’ understanding of a logical structure in defining the limit of a sequence, focusing on the relationship between ε and N. The subjects of this study were college students who had already encountered the concept of limit but did not have any experience with rigorous proofs using the ε–N definition. This study suggested two statements, each of which is written by using a relationship between ε and N, similar to the ε–N definition. By analyzing the students’ responses to the validity of the statements as definitions of the limit of a sequence, students’ understanding of such a relationship was classified into five major categories. This paper discusses some essential components that students must conceptualize in order properly to understand the relationship between ε and N in defining the limit of a sequence.

## 2009/09/10

### Working like real mathematicians

Atara Shriki has written an interesting article called Working like real mathematicians: developing prospective teachers’ awareness of mathematical creativity through generating new concepts. This article was recently published online in Educational Studies in Mathematics. The author reports from a study related to a methods course, where a strong focus is on creativity in mathematics. The article has a particular focus on prospective teachers' awarenes of creativity in mathematics.

Here is the abstract of Shriki's article.

This paper describes the experience of a group of 17 prospective mathematics teachers who were engaged in a series of activities aimed at developing their awareness of creativity in mathematics. This experience was initiated on the basis of ideas proposed by the participants regarding ways creativity of school students might be developed. Over a period of 6 weeks, they were engaged in inventing geometrical concepts and in the examination of their properties. The prospective teachers’ reflections upon the process they underwent indicate that they developed awareness of various aspects of creativity while deepening their mathematical and didactical knowledge.

## 2009/09/09

### Exploration of technologies

Here is the abstract of the article:

The study at teacher education institutions in Africa of mathematical ideas, from African history and cultures, may broaden the horizon of (future) mathematics teachers and increase their socio-cultural self-confidence and awareness. Exploring educationally mathematical ideas embedded in, and derived from, technologies of various African cultural practices may contribute to bridge the gap between ‘home’ and ‘school’ culture. Examples of the study and exploration of these technologies and cultural practices will be presented. The examples come from cultural practices as varied as story telling, basket making, salt production, and mat, trap and hat weaving.

### Theories of Mathematics Education

Looking at the table of contents is enough to make me believe that this is definitely going to be an important book, and it will make an impact on our field of research! If you won't take my word for it, please take the time to read through the table of contents yourself:

Theories of Mathematics Education - TOC

I especially like the way it is built up, with introductions and commentaries to all the parts of the book. This will give the reader a feeling of how the field has evolved, and how it is still in a process of evolving.

The publisher has given the following description of the book:

This inaugural book in the new series Advances in Mathematics Education is the most up to date, comprehensive and avant garde treatment of Theories of Mathematics Education which use two highly acclaimed ZDM special issues on theories of mathematics education (issue 6/2005 and issue 1/2006), as a point of departure. Historically grounded in the Theories of Mathematics Education (TME group) revived by the book editors at the 29th Annual PME meeting in Melbourne and using the unique style of preface-chapter-commentary, this volume consist of contributions from leading thinkers in mathematics education who have worked on theory building.

This book is as much summative and synthetic as well as forward-looking by highlighting theories from psychology, philosophy and social sciences that continue to influence theory building. In addition a significant portion of the book includes newer developments in areas within mathematics education such as complexity theory, neurosciences, modeling, critical theory, feminist theory, social justice theory and networking theories. The 19 parts, 17 prefaces and 23 commentaries synergize the efforts of over 50 contributing authors scattered across the globe that are active in the ongoing work on theory development in mathematics education.

You might also be interested in taking a look at the cover of the book

Theories of Mathematics Education - Cover

To me, at least, this is definitely a book I am looking forward to read. And after all, October is not that far away :-)

## 2009/09/08

### IJSME, August 2009

- INVESTIGATING THE EFFECTIVENESS OF AN ANALOGY ACTIVITY IN IMPROVING STUDENTS’ CONCEPTUAL CHANGE FOR SOLUTION CHEMISTRY CONCEPTS, by Muammer Çalik, Alipaşa Ayas and Richard K. Coll
- INSTRUCTIONAL ACTIVITIES AND GROUP WORK IN THE US INCLUSIVE HIGH SCHOOL CO-TAUGHT SCIENCE CLASS, by Laura J. Moin, Kathleen Magiera and Naomi Zigmond
- DESIGNING AND EVALUATING RESEARCH-BASED INSTRUCTIONAL SEQUENCES FOR INTRODUCING MAGNETIC FIELDS, by Jenaro Guisasola, Jose Manuel Almudi, Mikel Ceberio and Jose Luis Zubimendi
- ENGINEERING IN CHILDREN’S FICTION - NOT A GOOD STORY? by Allyson Holbrook, Lisa Panozza and Elena Prieto
- RELATIONS BETWEEN TEACHING AND RESEARCH IN PHYSICAL GEOGRAPHY AND MATHEMATICS AT RESEARCH-INTENSIVE UNIVERSITIES, by Lene Møller Madsen and Carl Winsløw
- GEOMETRIC AND ALGEBRAIC APPROACHES IN THE CONCEPT OF “LIMIT” AND THE IMPACT OF THE “DIDACTIC CONTRACT”, by Iliada Elia, Athanasios Gagatsis, Areti Panaoura, Theodosis Zachariades and Fotini Zoulinaki
- IDENTIFYING SENIOR HIGH SCHOOL STUDENTS’ MISCONCEPTIONS ABOUT STATISTICAL CORRELATION, AND THEIR POSSIBLE CAUSES: AN EXPLORATORY STUDY USING CONCEPT MAPPING WITH INTERVIEWS, by Tzu-Chien Liu, Yi-Chun Lin and Chin-Chung Tsai
- PROMOTING EFFECTIVE SCIENCE TEACHER EDUCATION AND SCIENCE TEACHING: A FRAMEWORK FOR TEACHER DECISION-MAKING, by Michael P. Clough, Craig A. Berg and Joanne K. Olson
- VARIABLE RELATIONSHIPS AMONG DIFFERENT SCIENCE LEARNERS IN ELEMENTARY SCIENCE-METHODS COURSES, by Robert E. Bleicher and Joan S. Lindgren

## 2009/09/07

### School mathematics curriculum materials for teachers’

This report describes ways that five preservice teachers in the United States viewed and interacted with the rhetorical components (Valverde et al. in According to the book: using TIMSS to investigate the translation of policy into practice through the world of textbooks, Kluwer, 2002) of the innovative school mathematics curriculum materials used in a mathematics course for future elementary teachers. The preservice teachers’ comments reflected general agreement that the innovative curriculum materials contained fewer narrative elements and worked examples, as well as more (and different) exercises and question sets and activity elements, than the mathematics textbooks to which the teachers were accustomed. However, variation emerged when considering the ways in which the teachers interacted with the materials for their learning of mathematics. Whereas some teachers accepted and even embraced changes to the teaching–learning process that accompanied use of the curriculum materials, other teachers experienced discomfort and frustration at times. Nonetheless, each teacher considered that use of the curriculum materials improved her mathematical understandings in significant ways. Implications of these results for mathematics teacher education are discussed.

## 2009/09/04

### ICMI News

- Editorial: Continuing Professional Development and Effective integration of Digital Technologies in Teaching and Learning Mathematics: Two Challenges for ICMI
- A XXIst century Felix Klein's follow up workshop
- Deadline Extended: ICMI / ICIAM STUDY
- EARCOME5: First Announcement
- Chilean Journal of Statistics (ChJS)
- Calendar of Events of Interest to the ICMI Community
- ICMI encounters: Hassler Whitney, Laurence C. Young and Dirk J. Struik: Personal recollections
- Subscribing to ICMI News

### Algebra - the birthplace and graveyard for many

Although algebra to many represents a hurdle, or even the graveyard in their mathematical careers, the article claims that:

Algebraic thinking is done even by people who don't realize they're using algebra.After a series of examples, Chute goes on to quote Michele Burgess, who claims that students should not be confronted with algebra for the first time in the Algebra 1 course. This leads me to think about the debate (and research) concerning early algebra, although this is not referred to in this article in particular. If you are interested, I recommend the chapter on early algebra by David Carraher and Analucia Schliemann in NCTM's Second Handbook of Research on Mathematics Teaching and Learning (Lester, 2007), or even Carolyn Kieran's chapter on algebra in the same handbook.

**Reference:**

Lester, F. K. (Ed.) (2007). Second handbook of research on mathematics teaching and learning. Charlotte, NC: Information Age Pub.

## 2009/09/01

### Understanding the complexities of student motivations

Student motivation has long been a concern of mathematics educators. However, commonly held distinctions between intrinsic and extrinsic motivations may be insufficient to inform our understandings of student motivations in learning mathematics or to appropriately shape pedagogical decisions. Here, motivation is defined, in general, as an individual's desire, power, and tendency to act in particular ways. We characterize details of motivation in mathematical learning through qualitative analysis of honors calculus students’ extended, collaborative problem solving efforts within a longitudinal research project in learning and teaching. Contextual Motivation Theory emerges as an interpretive means for understanding the complexities of student motivations. Students chose to act upon intellectual-mathematical motivations and social-personal motivations that manifested simultaneously. Students exhibited intellectual passion in persisting beyond obtaining correct answers to build understandings of mathematical ideas. Conceptually driven conditions that encourage mathematical necessity are shown to support the growth of intellectual passion in mathematics learning.

### How to develop mathematics for teaching and understanding

- Subject-matter knowledge
- Pedagogical-content knowledge
- Curricular knowledge

Whereas Bass and Ball (2004) concentrate on the first part of their program, namely, identifying important competences, this article deals with both parts, the analytical study of identifying, and the developmental study of constructing a sequence for teacher education, exemplified by a sequence in the course entitled school algebra and its teaching and learning for second-year, prospective middle-school teachers.Here is the abstract of Prediger's article:

What kind of mathematical knowledge do prospective teachers need for teaching and for understanding student thinking? And how can its construction be enhanced? This article contributes to the ongoing discussion on mathematics-for-teaching by investigating the case of understanding students’ perspectives on equations and equalities and on meanings of the equal sign. It is shown that diagnostic competence comprises didactically sensitive mathematical knowledge, especially about different meanings of mathematical objects. The theoretical claims are substantiated by a report on a teacher education course, which draws on the analysis of student thinking as an opportunity to construct didactically sensitive mathematical knowledge for teaching for pre-service middle-school mathematics teachers.

**References:**

Bass, H., & Ball, D. L. (2004). A practice-based theory of mathematical knowledge for teaching: The case of mathematical reasoning. In W. Jianpan & X. Binyan (Eds.),

*Trends and challenges in mathematics education*(pp. 107–123). Shanghai: East China Normal University Press.

### "The conference was awesome"

Professional development comes in many forms, some of which are deemed more useful than others. However, when groups of teachers are excluded, or exclude themselves, from professional development opportunities, then there is an issue of social justice. This article examines the experiences of a group of teachers from a Māori-medium school who attended a mathematics teacher conference. By analysing the teachers’ sense of belonging through their ideas about engagement, alignment and imagination, we are able to describe how different kinds of relationships influence the inclusion/exclusion process. This leads to a discussion about what can be done by the teachers as well as conference organisers to increase these teachers’ likelihood of attending further conferences in the future.