- Critical Mathematics Education: Recognizing the Ethical Dimension of Problem Solving, by Elizabeth de Freitas, USA
- Mathematics Teachers’ Interpretation of Higher-Order Thinking in Bloom’s Taxonomy, by Tony Thompson, USA
- Development of a Computerized Number Sense Scale for 3-rd Graders: Reliability and Validity Analysis, by Der-Ching Yang, Mao-neng Fred Li and Wei-Jin Li, Taiwan

## 2008/07/26

### IEJME, July 2008

### TMME monograph

Our innovative spirit and creativity lies beneath the comforts and security of today's technologically evolved society. Scientists, inventors, investors, artists and leaders play a vital role in the advancement and transmission of knowledge. Mathematics, in particular, plays a central role in numerous professions and has historically served as the gatekeeper to numerous other areas of study, particularly the hard sciences, engineering and business. Mathematics is also a major component in standardized tests in the U.S., and in university entrance exams in numerous parts of world.

Creativity and imagination is often evident when young children begin to develop numeric and spatial concepts, and explore mathematical tasks that capture their interest. Creativity is also an essential ingredient in the work of professional mathematicians. Yet, the bulk of mathematical thinking encouraged in the institutionalized setting of schools is focused on rote learning, memorization, and the mastery of numerous skills to solve specific problems prescribed by the curricula or aimed at standardized testing. Given the lack of research based perspectives on talent development in mathematics education, this monograph is specifically focused on contributions towards the constructs of creativity and giftedness in mathematics. This monograph presents new perspectives for talent development in the mathematics classroom and gives insights into the psychology of creativity and giftedness. The book is aimed at classroom teachers, coordinators of gifted programs, math contest coaches, graduate students and researchers interested in creativity, giftedness, and talent development in mathematics.

### How to stay up to date during the summer holidays

### Limits of a sequence

There are many studies on the role of images in understanding the concept of limit. However, relatively few studies have been conducted on how students’ understanding of the rigorous definition of limit is influenced by the images of limit that the students have constructed through their previous learning. This study explored how calculus students’ images of the limit of a sequence influence their understanding of definitions of the limit of a sequence. In a series of task-based interviews, students evaluated the propriety of statements describing the convergence of sequences through a specially designed hands-on activity, called the ɛ–strip activity. This paper illustrates how these students’ understanding of definitions of the limit of a sequence was influenced by their images of limits as asymptotes, cluster points, or true limit points. The implications of this study for teaching and learning the concept of limit, as well as on research in mathematics education, are also discussed.

## 2008/07/18

### Norwegian thesis: Tone Bulien

Tone Bulien has defended her thesis (dr. polit): Matematikkopplevelser i lærerutdanningen : en fenomenologisk orientert narrativ analyse av studenttekster (in Norwegian). The thesis is freely available as a pdf, and here is the abstract:

The thesis is a study of texts from and interviews with six Norwegian teacher students enrolled in a compulsory course in mathematics. It is a critical constructive descriptive investigation where the aim has been to listen to the students sharing their experiences studying mathematics. The thesis is not intended as an evaluation of the teacher education program, the students’ work or methodology, but rather as a contribution towards defining the didactic challenges teacher training is faced with. The thesis proceeds from a phenomenological perspective, using narratives as an important feature in both the analysis itself and the presentation of the results. Using phenomenologically oriented knowledge sociology and theories of narrative analysis, a description of the students’ perceptions of teaching and learning mathematics, both prior to and in the course of the compulsory course, is made visible through narratives. The methodology employed is narrative analysis. The students’ experiences are divided into four main areas of beliefs: beliefs about mathematics in general, beliefs about themselves as practitioners of mathematics, beliefs about teaching mathematics, and beliefs about how mathematics are learnt. One of the results indicated that the students’ experience of the compulsory course in mathematics did not depend on their previously held beliefs on mathematics education or their attitudes towards mathematics in general. Another result was that about 50% of all the students had higher expectations about their grade at the beginning of the semester than what they actually ended up with at the end. The reason for this remains to be conclusively demonstrated, but it seems likely that the way mathematics is taught in a teacher training program differs from the students’ previous experiences in how to learn mathematics. This should be taken into consideration in prospective mathematics programs, for instance by supervising the students about their own beliefs in a meta-perspective by analyzing their own narratives and how they are subject to alterations during the course.

## 2008/07/17

### Real-life connections in Japan and the Netherlands

The TIMSS 1999 Video Study revealed that Japan had the lowest (of the seven participating countries) amount of real-life connections in the eighth grade mathematics classrooms, whereas the Netherlands had the highest amount of connections with real life. This article examines more closely how these ideas were actually implemented by teachers in these two countries.

### PME 32

The PME conference this year is the 32. version of this annual research conference, and it is a joint meeting between the International Group and the North American Chapter of PME. The conference is held in Mexico. It starts today, and will finish on July 21. The program is downloadable as a pdf, and is voluminous. Take a look at the website, which contains lots of information, and feel free to tell me if you know about people who write about the conference in their blogs, twitter accounts, etc.

## 2008/07/14

### HPM 2008

International Study Group on the Relations between History and Pedagogy of Mathematics (HPM) is arranging their annual satellite meeting of 2008 in Mexico, and it starts the day after ICME 11 has finished. The meeting is held from July 14-18, in Mexico City.

These are the main themes of HPM 2008:

- Integrating the History of Mathematics in Mathematics Education.
- Topics in the History of Mathematics Education.
- Mathematics and its relation to science, technology and the arts: historical issues and educational implications.
- Cultures and Mathematics.
- Historical, philosophical and epistemological issues in Mathematics Education.
- Mathematics from the Americas

## 2008/07/13

### ICME 11 - Day 7

The last day of ICME 11 includes one plenary lecture and the final regular lectures. The plenary lecture, a report of Survey Team 4: "Representations of mathematical concepts, objects and processes in mathematics teaching and learning" is held by Gerald Goldin (USA).

If you know of anyone who has written about ICME 11 in their blogs, twitter accounts, etc., please let me know by posting a comment to this post!

## 2008/07/12

### ICME 11 - Day 6

The penultimate day of ICME 11 starts with a plenary presentation. This presentation includes a report from Survey Team 3: "The impact of research findings in mathematics education on students´ learning of mathematics". The presentation is held by Angel Gutiérrez (Spain).

## 2008/07/11

### ICME 11 - Day 5

The 5th day of ICME 11 starts with a panel debate. The topic being discussed is "Equal access to quality mathematics education". Here is the further description of the topic:

All students, regardless of age, race, ethnic group, religion, gender, socioeconomic status, geographic location, language, disability, or prior mathematics achievement, deserve equitable access to challenging and meaningful mathematics learning and achievement. This concept has profound implications for teaching and learning mathematics throughout the educational community. It suggests that ensuring equity and excellence must be at the core of systemic reform efforts in mathematics education.The panel debate is lead by Bill Atweh (Australia), and the other members are: Olimpia Figueras (Mexico), Murad Jurdak (Lebanon) and Catherine Vistro-Yu (The Philippines).

A necessary component for quality mathematics education is that all students receive an education that takes into account each student’s background, including prior learning, characteristics, and abilities in a way that maximizes his/her learning and does not diminish in any way the goals s/he is expected to achieve. This pertains to both high-achieving and low-achieving students.

In the afternoon, there is a plenary lecture which is held by two speakers: Toshiakira Fujii (Japan) and Ruhama Even (Israel). Their topic is: "Knowledge for teaching mathematics". Here is a short abstract:

Recent presentations at PME and elsewhere suggest that knowledge of mathematics teaching has been the focus of much activity in a variety of countries. The title was considered broad enough to allow the presenters to refer to current research into pedagogical content knowledge as well as to content knowledge. This also led us to consider two presenters who could ensure an extensive viewpoint.

## 2008/07/09

### ICME 11 - Day 3

The day before excursion day at ICME 11 contains two plenary activities: a plenary lecture and a panel debate. The plenary lecture is held by José Antonio de la Peña (Mexico), who will talk about current trends in mathematics. The panel debate is entitled "History of the development of mathematics education in Latin American countries", and is lead by Fidel Oteiza (Chile). Members of the panel are: Eugenio Filloy (Mexico), Ubiratan D´Ambrosio (Brazil), Luis Campistrous (Cuba) and Carlos Vasco (Colombia).

## 2008/07/08

### ICME 11 - Day 2

The second day of ICME 11 includes several activities, and one plenary lecture. Celia Hoyles (UK) is going to make a presentation about technology and mathematics education. Her talk is entitled "Transforming the mathematical practices of learners and teachers through digital technology", and here is the online description of it:

My presentation takes inspiration from the work of Seymour Papert, Jim Kaput, Richard Noss and all the colleagues with whom I have been fortunate enough to collaborate in the area of mathematics education and technology over many years, in the U.K and beyond.

Drawing on the mass of evidence from research and practice, I will first set out what I see as the vision of the potential of Information and Communication Technologies (ICT) to transform the teaching and learning of mathematics. I suggest it can offer:Under each of the six headings, I will present research evidence and examples that illustrate their transformative potential. I will also identify: first, the costs and challenges at least partly to explain why in so many cases, impact has not reached expectations; and, second, actions that can be undertaken as contingencies against these risks. In this part of the talk, I will draw on some the outcomes of the recent ICMI Study 17, Technology Revisited that considered these questions from the important and under-represented vantage point of the situation of developing countries: how technology could be used for the benefit of these countries rather than serve as yet another source of disadvantage.

- dynamic & visual tools that allow mathematics to be explored in a shared space - changing how mathematics is learned and taught;
- tools that outsource processing power that previously could only be undertaken by humans - changing the collective focus of attention during mathematics learning;
- new representational infrastructures for mathematics - changing what can be learned and for whom;
- connectivity - opening new opportunities for shared knowledge construction and for student autonomy over their mathematical work;
- connections between school mathematics and learners’ agendas and culture - bridging the gap between school mathematics and problem solving ‘in the real world’;
- some intelligent support to the teacher while learners are engaged in an exploratory environment;

Taken together, the overriding evidence suggests that in order for ICT to move from the periphery to centre stage in mathematics teaching and learning and for its potential for transforming mathematical practice for the benefit of all learners to be realised, teachers must be part of the transformative process:

i) to do mathematics for themselves with the digital tools (before and alongside thinking about pedagogy and embedding in practice) thus allowing teachers, regardless of experience, the time and space to take on the role of learner,

ii) to co-design activity sequences that embed the ICT tools and make explicit appropriate didactic strategies,

iii) to try out iteratively in classrooms as a collective effort and debug together.

This design process is challenging, not least because at every phase the dialectical influence of tools on mathematical expression and communication must be taken into account.

A further challenge facing innovations using ICT is scaling up, since, all too often, design experiments while reporting positive results wither away soon after any funding ends. One way we are working in England to break this cycle is through the National Centre for Excellence in the Teaching of Mathematics. The National Centre was set up in England in 2006 (see www.ncetm.org.uk, and I have been its director since June 2007. Its major aim is to develop a sustainable national infrastructure for subject-specific professional development of teachers of mathematics that will enable the mathematical potential of learners to be fully realised. The NCETM offers a blend of approaches to effective Continuing Professional development (CPD): national and regional face-to-face meetings, and tools and resources on its portal designed to promote and sustain collaborative CPD among teachers of mathematics (for example through on-line communities). These networks and communities include the use of ICT in classrooms.

A major challenge faced by the NCETM is to reach out to all teachers of mathematics across all the phases of education in ways that develop ownership of NCETM’s CPD offer and, in particular, ownership of and fluency with the tools available on the portal. If this ownership is achieved, the tools will grow with use, as teachers contribute to the content and to the on-line communities and in so doing support each other in transforming their practice. It is my contention that it is only through this process of mutual support that the potential of ICT will be realised - not only the potential already on offer, but also through new technological innovations such as personal and mobile technology, and all that will become available in the future.

## 2008/07/07

### ICME 11 - Day 1

The first plenary lecture of ICME 11 is held by two distinguished scholars in our field: Michèle Artigue and Jeremy Kilpatrick. Their lecture is entitled: "What do we know? And how do we know it?" Here is the description of their presentation:

The International Program Committee of ICME-11 proposed that we launch the academic activities of this congress through a dialogue on issues of crucial interest for mathematics education today, such as the following: What do we know that we did not know ten years ago in mathematics education, and how have we come to know it? What kind of evidence is needed and available in mathematics education? What are society's expectations regarding our field, and how do we respond to them? How far can visions of teaching and learning mathematics and evidence in the field transcend the diversity of educational contexts and cultures? In the plenary, we will engage in such a dialogue, presenting our respective views of the dynamics of the field and its outcomes in the last ten or fifteen years, the main challenges we have to face today, and how we can address them.This plenary presentation is followed up by a panel debate after lunch. The debate is chaired by David Clarke (Australia), and the panel consists of: Paul Cobb (USA), Mariolina Bartolini Bussi (Italy), Teresa Rojano (Mexico) and Shiqi Li (China).

## 2008/07/06

### ICME 11

The International Congress on Mathematical Education - ICME - is arguably the largest and most important conference/congress in mathematics education research. The congress is arranged every four years, and ICME-11 is arranged in Monterrey, Mexico (July 6-13). Around 4000 participants are expected from 100 countries!

These are the official goals of ICME-11:

- To provide a forum for mathematics education professionals from all over the world, where they can exchange ideas, information and viewpoints and develop productive dialog with their peers. By M.E. professionals we mean to include teachers, teacher assistants, researchers, curriculum designers, textbook and materials authors, academic administrators, and others whose work and interests are strongly related to mathematics education.
- To provide M.E. professionals with opportunities for professional development by presenting their work and receiving immediate feedback and to establish or strengthen working relationships with their peers.
- To promote collaboration between educators from different countries, in a wide and inclusive manner, regardless of gender, ethnic origin, religion, political ideology, citizenship, or any other difference between groups or individuals.
- To improve the practice and research of mathematics education in all the countries represented at the congress, inasmuch as we believe that this is an expected outcome of the type of study, learning, dialog, and collaboration that the work developed prior to and at the congress promotes.

If you plan on following the conference online, all plenary activities are broadcasted online.

## 2008/07/05

## 2008/07/04

### Numerical problems on energy

This study was designed to assess whether the level of performance of selected Jamaican 11th-grade physics students on some numerical problems on the energy concept was satisfactory and if there were significant differences in their performance linked to their gender, socioeconomic background (SEB), school location, English language and mathematical abilities. The 331 sampled students consisted of 213 boys and 118 girls; 197 students were from a high SEB and 134 students from a low SEB; 296 students were from seven urban schools and 35 students from three rural schools; 112, 153 and 66 of the students had high, average and low English language abilities, respectively, while 144, 81 and 106 of the students had high, average and low mathematical abilities, respectively. An Energy Concept Test (ECT) consisting of six structured numerical questions was employed for data collection. The results indicated that although the students’ level of performance was regarded as fairly satisfactory, there was a lot of room for improvement. There were statistically significant differences in the students’ performance on the ECT linked to SEB, and mathematical abilities in favour of students from a high SEB, and high mathematical abilities, respectively. There was a positive, statistically significant but weak correlation between the students’ (a) mathematical abilities, and (b) English language abilities and their performance on the ECT, while there were no correlations among their gender, school location, and SEB and their performance on the ECT.

### Dynamic geometry meets variation theory

What makes Dynamic Geometry Environment (DGE) a powerful mathematical knowledge acquisition microworld is its ability to visually make explicit the implicit dynamism of thinking about mathematical geometrical concepts. One of DGE’s powers is to equip us with the ability to retain the background of a geometrical configuration while we can selectively bring to the fore dynamically those parts of the whole configuration that interest us. That is, we can visually study the variation of an aspect of a DGE figure while keeping other aspects constant, hence anticipating the emergence of invariant patterns. The aim of this paper is to expound the epistemic value of variation of the Dragging tool in DGE in mathematical discovery. Functions of variation (contrast, separation, generalization, fusion) proposed in Marton’s theory of learning and awareness will be used as a framework to develop a discernment structure which can act as a lens to organize and interpret dragging explorations in DGE. Such a lens focuses very strongly on mathematical aspects of dragging in DGE and is used to re-interpret known dragging modalities (e.g., Arzarello et al.) in a potentially more mathematically-relevant way. The exposition will centre about a specific geometrical problem in which two dragging trajectories are mapped out, consequently resulting in a DGE theorem and a visual theorem. In doing so, a new spectral dragging strategy will be introduced that literally allows one to see the drag mode in action. A model for the lens of variation in the form of a discernment nest structure is proposed as a meta-tool to interpret dragging experiences or as a meta-language to relate different dragging analyses which consequently might give rise to pedagogical and epistemological implications.

## 2008/07/03

### Project organised science studies

The 2-year introductory study programme in the natural sciences (Nat-Bas) at Roskilde University is an example of a project organised, participant directed, problem oriented, and interdisciplinary science study programme. The paper gives an account of the organisational framework around the project work, and discusses in particular, the thematic organisation of project work, the notion of exemplarity, the problem orientation, the interdisciplinary nature of the problems, the assessment of the project work, and the students’ individual learning. Based on descriptions and analyses of six selected project reports from the Nat-Bas in 2005-2007, we illustrate the multiple perspectives of science and mathematics and the learning potentials found in the project work. The paper is concluded with a general discussion of the quality of the project work and its educational function in the Nat-Bas programme.

### Integrating history and philosophy

Through the last three decades several hundred problem-oriented student-directed projects concerning meta-aspects of mathematics and science have been performed in the 2-year interdisciplinary introductory science programme at Roskilde University. Three selected reports from this cohort of project reports are used to investigate and present empirical evidence for learning potentials of integrating history and philosophy in mathematics education. The three projects are: (1) a history project about the use of mathematics in biology that exhibits different epistemic cultures in mathematics and biology. (2) An educational project about the difficulties of learning mathematics that connects to the philosophy of mathematics. (3) A history of mathematics project that connects to the sociology of multiple discoveries. It is analyzed and discussed in what sense students gain first hand experiences with and learn about meta-aspects of mathematics and their mathematical foundation through the problem-oriented student-directed project work.

## 2008/07/02

### Algebra beginners in computer intensive environment

The transition from arithmetic to algebra in general, and the use of symbolic generalizations in particular, are a major challenge for beginning algebra students. In this article, we describe and analyze students’ learning in a “computer intensive environment” designed ad hoc and implemented in two seventh grade classrooms throughout two consecutive school years. In particular, this article focuses on the description and analysis of how students initial generalizations (which relied on computerized tools that enabled different students’ to work with different strategies) shifted to recursive and explicit symbolic generalizations.

### Abstraction and consolidation of the limit procept

I investigate the contributions of three theoretical frameworks to a research process and the complementary role played by each. First, I describe the essence of each theory and then follow the analysis of their specific influence on the research process. The research process is on the conceptualization of the notion of limit by means of the discrete continuous interplay. I investigate the influence of the different perspectives on the research process and realize that the different theoretical approaches intertwine. Moreover, I realize that the research study demanded the contribution of more than one theoretical approach to the research process and that the differences between the frameworks could serve as a basis for complementarities.

## 2008/07/01

### From lessons to lectures

Given the recent radical overhaul of secondary school qualifications in New Zealand, similar in style to those in the UK, there has been a distinct change in the tertiary entrant profile. In order to gain insight into this new situation that university institutions are faced with, we investigate some of the ways in which these recent changes have impacted upon tertiary level mathematics in New Zealand. To this end, we analyse the relationship between the final secondary school qualifications in Mathematics with calculus of incoming students and their results in the core first-year mathematics papers at Canterbury since 2005, when students entered the University of Canterbury with these new reformed school qualifications for the first time. These findings are used to investigate the suitability of this new qualification as a preparation for tertiary mathematics and to revise and update entrance recommendations for students wishing to succeed in their first-year mathematics study.

### CERME 6

### NOMAD, June 2008

As we are about to shift from June to July, it is time to point your attention to the June issue of NOMAD (Nordic Studies in Mathematics Education). The issue contains an interesting editorial concerning the development of the journal, some information from Barbro Grevholm about the Nordic graduate school in mathematics education, and three research articles:

- Matematikopfattelser hos 2g’ere: fokus på de ‘tre aspekter‘ by Uffe Thomas Jankvist. Abstract: Based on the so-called ‘three aspects‘ from the 1987-regulations for the Danish upper secondary mathematics programme this article discusses second-year upper secondary students’ beliefs about the nature of mathematics. That is to say, it investigates the students’ beliefs concerning the historical evolution of mathematics, the application of mathematics in society, and the inner structures of mathematics as a scientific discipline. Firstly, the article examines the origin of the ‘three aspects‘ as well as the role they play in both the KOM-project of 2002 and the new regulations for the Danish upper secondary mathematics programme of 2007. Secondly, it discusses how the students in a concrete second-year class of upper secondary level seem to fulfil the goals of the ‘three aspects’. Thirdly, the results of this study are compared to a similar study from 1980 and differences and similarities between the two are discussed. It is concluded that there still is room for improvement concerning the fulfilment of the three aspects, and that the students’ beliefs in the 1980-study and in the 2007-study are very similar. In the end, the article speculates upon why the ‘three aspects’ do not seem to have had a larger impact on the mathematics teaching on upper secondary level when they have been in the regulations for twenty years now.
- Interrater reliability in a national assessment of oral mathematical communication by Torulf Palm. Abstract: Mathematical communication, oral and written, is generally regarded as an important aspect of mathematics and mathematics education. This implies that oral mathematical communication also should play a part in various kinds of assessments. But oral assessments of subject matter knowledge or communication abilities, in education and elsewhere, often display reliability problems, which render difficulties with their use. In mathematics education, research about the reliability of oral assessments is comparably uncommon and this lack of research is particularly striking when it comes to the assessment of mathematical communication abilities. This study analyses the interrater reliability of the assessment of oral mathematical communication in a Swedish national test for upper secondary level. The results show that the assessment does suffer from interrater reliability problems. In addition, the difficulties to assess this construct reliably do not seem to mainly come from the communication aspect in itself, but from insufficiencies in the model employed to assess the construct.
- Finnish mathematics teacher students’ informal and formal arguing skills in the case of derivative by Antti Viholainen. Abstract: In this study, formal and informal reasoning skills of 146 Finnish subject-teacher students in mathematics are investigated. The students participated in a test in which they were asked to argue two claims concerning derivative both informally and formally. The results show that the success in the formal tasks and the success in the informal tasks were dependent. However, there were several students who did well in the formal tasks despite succeeding poorly in the informal tasks. The success both in the formal tasks and in the informal tasks was dependent also on the amount of passed studies in mathematics and on the success in these studies. Moreover, these factors could have a stronger effect on the formal than on the informal reasoning skills.

### How much math does a teacher need to know to teach math?

So, how much should a teacher know? The following quote from the blog post touches this:

It seems obvious that teachers must have knowledge of the subject matter they will actually teach. But how much more knowledge should a teacher have than what she or he is seeking to assist students in learning? The case of secondary school mathematics is instructive. Is it enough for a high school trigonometry teacher to know trigonometry cold – but not, say, real analysis, or ordinary differential equations?This issue was exactly the one that was raised in the LMT project (Learning Mathematics for Teaching) at University of Michigan. This was also the main issue in an article written by Heather Hill, Deborah Ball and Stephen Schilling in the last issue of Journal for Research in Mathematics Education. (The LMT team has also written several other scientific articles about the issue.)