The study we present here concerns the consequences of integrating online resources into the teaching of mathematics. We focus on the interaction between teachers and specific online resources they draw on: e-exercise bases. We propose a theoretical approach to study the associated phenomena, combining instrumental and anthropological perspectives. For given didactical tasks, we observe teachers’ instrumental geneses, and the didactical techniques they develop. We exemplify our approach with the analysis of a case study of trigonometry in grade 9.

## 2009/02/26

### Online resources in mathematics

### Supervision of mathematics student teachers

Student teaching is often a capstone experience in the preparation of mathematics teachers. Thus, it is essential to better understand key aspects of the experience. We conducted a qualitative study of post-lesson conferences led by supervisors (classroom cooperating teachers and a university supervisor) working with mathematics student teachers. Analysis of conference communications revealed differences in the types and content of communications in conferences led by the cooperating teachers and by the university supervisor. Cooperating teachers tended toward evaluative supervision that lacked a focus on the mathematics of the lessons while the university supervisor tended toward educative supervision, guiding student teachers to reflect on and learn from their own classroom experiences including the mathematics of their lessons. Differences are discussed, and suggestions concerning the supervision of student teachers are made along with recommendations for further research.

### Mathematics classrooms with immigrant students

This article suggests that a critical perspective of the notion of social representations can offer useful insights into understanding practices of teaching and learning in mathematics classrooms with immigrant students. Drawing on literature using social representations, previous empirical studies are revisited to examine three specific questions: what are the dominant social representations that permeate the mathematics classroom with immigrant students? What impact do these social representations have on classroom practices? What are the spaces for changing these practices through becoming reflective and critically aware of these representations? These questions are addressed mostly in relation to teachers’ representations, though the article also draws on data from research with students and parents to illustrate the diversity of representations and to argue for a critical and reflective perspective.

## 2009/02/25

### IJSME, Vol 7, Number 2

- Language and Student Performance in Junior Secondary Science Examinations: The Case of Second Language Learners in Botswana, by Robert B. Prophet and Nandkishor B. Badede
- The system of coordinates as an obstacle in understanding the concept of dimension, by Constantine Skordoulis, Theodore Vitsas, Vassilis Dafermos and Eugenia Koleza
- Misconceptions of Turkish Pre-Service Teachers about Force and Motion, by Sule Bayraktar
- Variable Relationships among Different Science Learners in Elementary Science-Methods Courses, by Robert E. Bleicher
- Efficacy of Two Different Instructional Methods Involving Complex Ecological Content, by Christoph Randler and Franz X. Bogner
- Correlations Among Five Demographic Variables and the Performance of Selected Jamaican 11th-graders on Some Numerical Problems on Energy, by Nicholas Emepue and Kola Soyibo
- From "exploring the middle zone" to "constructing a bridge": Experimenting in the Spiral Bianshi mathematics curriculum, by Ngai-Ying Wong, Chi-Chung Lam, XuHua Sun and Anna Mei Yan Chan
- Number Sense Strategies Used by Pre-Service Teachers in Taiwan, by Der-Ching Yang, Robert E. Reys and Barbara J. Reys
- Listen to the silence: The left-behind phenomenon as seen through classroom videos and teachers' reflections, by Hagar Gal, Fou-Lai Lin and Jia-Ming Ying

### Transition between different coordinate systems

The main objective of this paper is to apply the onto-semiotic approach to analyze the mathematical concept of different coordinate systems, as well as some situations and university students’ actions related to these coordinate systems. The identification of objects that emerge from the mathematical activity and a first intent to describe an epistemic network that relates to this activity were carried out. Multivariate calculus students’ responses to questions involving single and multivariate functions in polar, cylindrical, and spherical coordinates were used to classify semiotic functions that relate the different mathematical objects.

### Changing practice, changing minds

This study examines the process of change among grade 4 teachers (students aged 9–10 years) who participated in a yearlong Teacher Quality Grant innovation program. The concerns-based adoption model (CBAM), which informed the design and implementation of the program, was used to examine the process of change. Two questions guided the investigation: (1) How did teachers’ concerns about and levels of use of the innovation evolve during the course of the project? (2) What changes in teachers’ perceptions and practices arose as a result of the innovation? Results showed that several of the teachers’ concerns evolved from self/task toward impact. With continued support, several participants achieved routine levels of use, which they sustained beyond the project.

## 2009/02/24

### ESM, March 2009

*Gestures and Multimodality in the Construction of Mathematical Meaning*. It contains 10 interesting articles:

10 Articles |

91-95 | ||

97-109 | ||

111-126 | ||

127-141 | ||

143-157 | ||

159-174 | ||

175-189 | ||

191-200 | ||

201-210 | ||

211-215 |

## 2009/02/23

### Geometric and algebraic approaches

The present study explores students’ abilities in conversions between geometric and algebraic representations, in problem- solving situations involving the concept of “limit” and the interrelation of these abilities with students’ constructed understanding of this concept. An attempt is also made to examine the impact of the “didactic contract” on students’ performance through the processes they employ in tackling specific tasks on the concept of limit. Data were collected from 222 12th-grade high school students in Greece. The results indicated that students who had constructed a conceptual understanding of limit were the ones most probable to accomplish the conversions of limits from the algebraic to the geometric representations and the reverse. The findings revealed the compartmentalized way of students’ thinking in non-routine problems by means of their performance in simpler conversion tasks. Students who did not perform under the conditions of the didactic contract were found to be more consistent in their responses for various conversion tasks and complex problems on limits, compared to students who, as a consequence of the didactic contract, used only algorithmic processes.

### Ethiopian students in Israel

Many studies have reported on the economical, social, and educational difficulties encountered by Ethiopian Jews since their immigration to Israel. Furthermore, the overall academic underachievement and poor representation of students of Ethiopian origin (SEO) in the advanced mathematics and science classes were highlighted and described. Yet, studies focusing on differential achievements within SEO and on students who succeed against all odds are scarce. In this study, we explored success stories of five SEO studying in a pre-academic program at a prestigious technological university in Israel. Our goal was to understand how these students frame and interpret their success in mathematics and to identify elements perceived as fostering their mathematics and academic trajectories. Using qualitative methodology, we identified perceived personal motivational variables, effective learning and coping strategies, and students’ immediate environment as key elements contributing to achieving and maintaining success. We discuss possible theoretical contributions and practical implications of the findings.

### Mathematical interaction in different social settings

The study presented in this article investigates forms of mathematical interaction in different social settings. One major interest is to better understand mathematics teachers’ joint professional discourse while observing and analysing young students mathematical interaction followed by teacher’s intervention. The teachers’ joint professional discourse is about a combined learning and talking between two students before an intervention by their teacher (setting 1) and then it is about the students learning together with the teacher during their mathematical work (setting 2). The joint professional teachers’ discourse constitutes setting 3. This combination of social settings 1 and 2 is taken as an opportunity for mathematics teachers’ professionalisation process when interpreting the students’ mathematical interactions in a more and more professional and sensible way. The epistemological analysis of mathematical sign-systems in communication and interaction in these three settings gives evidence of different types of mathematical talk, which are explained depending on the according social setting. Whereas the interaction between students or between teachers is affected by phases of a process-oriented and investigated talk, the interaction between students and teachers is mainly closed and structured by the ideas of the teacher and by the expectations of the students.

### Teachers' reflective thinking skills

Here is the article abstract:

In this study, we examined prospective middle school mathematics teachers’ reflective thinking skills to understand how they learned from their own teaching practice when engaging in a modified lesson study experience. Our goal was to identify variations among prospective teachers’ descriptions of students’ thinking and frequency of their interpretations about how teaching affected their students’ learning. Thirty-three participants responded to open-ended questionnaires or interviews that elicited reflections on their own teaching practice. Prospective teachers used two forms of nuance when describing their students’ thinking: (1) identifying students’ specific mathematical understandings rather than general claims and (2) differentiating between individual students’ thinking rather than characterizing students as a collective group. Participants who described their students’ thinking with nuance were more likely to interpret their teaching by posing multiple hypotheses with regard to how their instruction affected their students’ learning. Implications for supporting continued growth in reflective thinking skills are discussed in relation to these results.

## 2009/02/20

### Anniversary!!!

I can hardly believe that it is only a little more than a year since I started this blog! It has been a great learning experience for me, and hopefully for someone else as well. I just found out that I have actually reached my 400^{th} post, which is quite an anniversary! So, happy 400 :-)

### IJMEST, volume 40, issue 2, 2009

International Journal of Mathematical Education in Science and Technology has just released issue 2 of 2009. Here is a list of the original articles included in the issue:

Authors: Ann Kajander; Miroslav Lovric DOI: 10.1080/00207390701691558 | ||

Authors: T. Vilkomir; J. O'Donoghue DOI: 10.1080/00207390802276200 | ||

Author: Nevin Mahir DOI: 10.1080/00207390802213591 | ||

Authors: Valsa Koshy; Paul Ernest; Ron Casey DOI: 10.1080/00207390802566907 | ||

Authors: Sinead Breen; Joan Cleary; Ann O'Shea DOI: 10.1080/00207390802566915 | ||

Authors: Sonya Ellouise Sherrod; Jerry Dwyer; Ratna Narayan DOI: 10.1080/00207390802566923 | ||

Author: Victor Martinez-Luaces DOI: 10.1080/00207390802276291 |

### Exemplary mathematics lessons

## 2009/02/19

### Khayyam with Cabri

The study reported in this article deals with the observed actions of Turkish pre-service mathematics teachers in dynamic geometry environment (DGE) as they were learning Khayyam's method for solving cubic equations formed as x^{3}+ ax = b. Having learned the method, modelled it in DGE and verified the correctness of the solution, students generated their own methods for solving different types of cubic equations such as x^{3}+ ax^{2}= b and x^{3}+ a = bx in the light of Khayyam's method. With the presented teaching experiment, students realized that Khayyam's mathematics is different from theirs. We consider that this gave them an opportunity to have an insight about the cultural and social aspects of mathematics. In addition, the teaching experiment showed that dynamic geometry software is an excellent tool for doing mathematics because of their dynamic nature and accurate constructions. And, it can be easily concluded that the history of mathematics is useful resource for enriching mathematics learning environment.

## 2009/02/18

### BSHM Bulletin

- The hunt for the lost cities of Ptolemy, by Daniel Mintz
- A puzzle rhyme from 1782, by Kristin Bjarnadottir
- International mathematical journals published in Poland between the Wars, by Malgorzata Przenioslo
- The contribution of M H A Newman and his mathematicians to the creation of the Manchester 'Baby', by David Anderson

### Free journal article

Educational programs for young children emerged reasonably early in the history of the United States of America. The movements of Child-Centered Education, the Nursery School, the Project Method, Curriculum Reform, and contemporary research have all influenced mathematics in early childhood education. The Froebelian kindergarten and the Montessori Casa die Bambini (Children’s House) included approaches to teaching mathematics. This article reviews the history of mathematics education in relation to the history of early childhood education from the turn of the twentieth century. It also discusses how research in mathematics education attempted to gain its own identity. Throughout history, researchers have identified mathematics issues and addressed them, defining the field, and generating a cadre of mathematics researchers.

## 2009/02/17

### Hidden lessons

Amy B. Ellis and Paul Grinstead have written an article that was published in The Journal of Mathematical Behavior last week. The article is entitled Hidden lessons: How a focus on slope-like properties of quadratic functions encouraged unexpected generalizations. Here is a copy of their article abstract:

This article presents secondary students’ generalizations about the connections between algebraic and graphical representations of quadratic functions, focusing specifically on the roles of the parameters a, b, and c in the general form of a quadratic function, y = ax^{2}+ bx + c. Students’ generalizations about these connections led to a surprising finding: two-thirds of the students interviewed identified the parameter a as the “slope” of the parabola. Analysis of qualitative data from interviews and classroom observations led to the development of three focusing phenomena in the classroom environment that inadvertently supported a focus on slope-like properties of quadratic functions: (a) the use of linear analogies, (b) the rise over run method, and (c) viewing a as dynamic rather than static.

### IJMEST, issue 1, 2009

Issue 1 of International Journal of Mathematical Education in Science and Technology has been published. The issue contains several articles that I find really interesting! Here is a list of all the articles in this issue:

Original Articles | ||

Authors: Derek Holton; Eric Muller; Juha Oikkonen; Oscar Adolfo Sanchez Valenzuela; Ren Zizhao DOI: 10.1080/00207390802597621 | ||

Authors: Jan Thomas; Michelle Muchatuta; Leigh Wood DOI: 10.1080/00207390802597654 | ||

Authors: Laura Fenwick-Sehl; Marcella Fioroni; Miroslav Lovric DOI: 10.1080/00207390802568192 | ||

Authors: Pierre Arnoux; Daniel Duverney; Derek Holton DOI: 10.1080/00207390802586145 | ||

Authors: Hong Kian Sam; Ting Lang Ngiik; Hasbee Hj Usop DOI: 10.1080/00207390802514519 | ||

Authors: Johann Engelbrecht; Ansie Harding DOI: 10.1080/00207390802597738 | ||

Authors: Cristina Varsavsky; Marta Anaya DOI: 10.1080/00207390802514543 | ||

Authors: B. Barton; L. Sheryn DOI: 10.1080/00207390802576807 | ||

Authors: A. C. Croft; M. C. Harrison; C. L. Robinson DOI: 10.1080/00207390802542395 | ||

Author: Juha Oikkonen DOI: 10.1080/00207390802582961 | ||

Authors: Eric Muller; Chantal Buteau; Mihly Klincsik; Ildik Perjsi-Hmori; Csaba Srvri DOI: 10.1080/00207390802551602 | ||

Authors: Gary Harris; Jason Froman; James Surles DOI: 10.1080/00207390802514493 |