It is not entirely new, but I just discovered it: a very nice little video from the National Science Foundation about "The struggle to 'fix' math education in the US". The video is interesting from many perspectives, but for me it is particularly interesting because two of the three people that are featured in this film played an important role in the symposium where I gave my own presentation at this year's AERA conference. Bill Schmidt was one of our two discussants, and Deborah Ball was chair of our session. Along with Joan Ferrini-Mundi from NSF, they raise some important issues for mathematics education research in this video:

The video was created in relation to the NSF special report, Math: What's the problem?

## 2009/04/30

### The struggle to "fix" math education

### Mathematics in early childhood education

- Elizabeth Dunphy has written an article called Early childhood mathematics teaching: challenges, difficulties and priorities of teachers of young children in primary schools in Ireland.
**Abstract:**Issues of pedagogy are critical in all aspects of early childhood education. Early childhood mathematics is no exception. There is now a great deal of guidance available to teachers in terms of high-quality early childhood mathematics teaching. Consequently, the characteristics of high-quality early childhood mathematics education are clearly identifiable. Issues such as building on young children's prior-to-school knowledge; engaging children in general mathematical processes; and assessing and documenting children's learning are some of the key aspects of high-quality early childhood mathematics education. The extent to which teachers of four- and five-year-old children in primary schools in Ireland incorporate current pedagogical guidance in early childhood mathematics education was explored in 2007 in a nationally representative questionnaire survey of teachers of four- and five-year-old children attending primary schools. This paper presents some of the findings of the study in relation to teachers' self-reported challenges, difficulties and priorities in teaching early childhood mathematics. Implications are drawn for professional development, curriculum guidance and educational policy. - Sally Howell and Coral Kemp have written an article called A participatory approach to the identification of measures of number sense in children prior to school entry.
**Abstract:**The research reported in this paper used a modified Delphi procedure in an attempt to establish a consensus on tasks proposed to assess components of number sense identified as essential for early mathematics success by a broad range of academics with expertise in the area of early mathematics. Tasks included as measures of these components were based on assessment tasks developed by early mathematics researchers. Eighteen questionnaires were returned by academics from Australia, the UK, New Zealand, The Netherlands and the USA, all with published work in the areas of early mathematics and/or number sense. Both the proposed components and tasks in the questionnaires were limited to the number domain. The study revealed considerable agreement with a number of the proposed tasks and thus provided a way forward for the development of an early number sense assessment to be trialled with young children prior to their first year of formal schooling. - A third article, entitled Numeracy-related exchanges in joint storybook reading and play, was written by Maureen Vandermaas-Peeler, Jackie Nelson, Charity Bumpass annd Bianca Sassine.
**Abstract:**Studies of the processes by which parents encourage early numerical development in the context of parent-child interactions during routine, culturally relevant activities at home are scarce. The present study was designed to investigate spontaneous exchanges related to numeracy during parent-child interactions in reading and play activities at home. Thirty-seven families with a four-year-old child (13 low-income) were observed. Two types of numeracy interactions were of interest: socio-cultural numeracy exchanges, explaining the use and value of money or numbers in routine activities such as shopping or cooking, and mathematical exchanges, including counting, quantity or size comparisons. Results indicated that high-income parents engaged in more mathematical exchanges during both reading and play than did low-income parents, though there were no differences in the initiation of socio-cultural numeracy exchanges. The focus of parental guidance related to numeracy was conceptual and embedded in the activity context, with few dyads focusing on counting or numbers per se. The findings suggest the importance of parent education efforts that incorporate numeracy-related discourse in the context of daily routines to augment young children's numeracy development.

## 2009/04/26

### Is it worth using CAS

The use of Computer Algebra Systems (CAS) in years 9 and 10 classrooms as a tool to support learning or in preparation for senior secondary mathematics is controversial. This paper presents an analysis of the positive and negative aspects of using CAS identified in the literature related to these year levels, along with the perceptions of 12 experienced secondary teachers who were working with years 9 and 10 students. The literature review shows that CAS is valued for calculation and manipulation capabilities, the option of alternative representations, the opportunity for systematic exploration and for prompting rich discussion. However, the technical overhead, initial workload for the teacher and unresolved questions about the perceived relative contribution of machine and by-hand work to learning currently pose obstacles to teaching with CAS in the middle secondary years. The teachers who contributed data to this study perceived that using CAS in their teaching is, on balance, worth the effort. However, they believed that CAS is of most benefit to their high ability students and may present an obstacle to their low ability students’ learning of mathematics.

## 2009/04/23

### Improving mathematics instruction through lesson study

This article presents a theoretical model of lesson study, an approach to instructional improvement that originated in Japan. The theoretical model includes four lesson study features (investigation, planning, research lesson, and reflection) and three pathways through which lesson study improves instruction: changes in teachers’ knowledge and beliefs; changes in professional community; and changes in teaching–learning resources. The model thus suggests that development of teachers’ knowledge and professional community (not just improved lesson plans) are instructional improvement mechanisms within lesson study. The theoretical model is used to examine the “auditable trail” of data from a North American lesson study case, yielding evidence that the lesson study work affected each of the three pathways. We argue that the case provides an “existence proof” of the potential effectiveness of lesson study outside Japan. Limitations of the case are discussed, including (1) the nature of data available from the “auditable trail” and (2) generalizability to other lesson study efforts.

### Sample space partitions

In this study subjects are presented with sequences of heads and tails, derived from flipping a fair coin, and asked to consider their chances of occurrence. In this new iteration of the comparative likelihood task, the ratio of heads to tails in all of the sequences is maintained. In order to help situate participants’ responses within conventional probability, this article employs unconventional set descriptions of the sample space organized according to: switches, longest run, and switches and longest run, which are all based upon subjects’ verbal descriptions of the sample space. Results show that normatively incorrect responses to the task are not devoid of correct probabilistic reasoning. The notion of alternative set descriptions is further developed, and the article contends that sample space partitions can act as an investigative lens for research on the comparative likelihood task, and probability education in general.

### Interpreting motion graphs

This article deals with the interpretation of motion Cartesian graphs by Grade 8 students. Drawing on a sociocultural theoretical framework, it pays attention to the discursive and semiotic process through which the students attempt to make sense of graphs. The students’ interpretative processes are investigated through the theoretical construct of knowledge objectification and the configuration of mathematical signs, gestures, and words they resort to in order to achieve higher levels of conceptualization. Fine-grained video and discourse analyses offer an overview of the manner in which the students’ interpretations evolve into more condensed versions through the effect of what is called in the article “semiotic contractions” and “iconic orchestrations.”

## 2009/04/22

### How learning and teaching of mathematics can be made interesting

In this article, we evaluate the true proportion of mathematics educators and teachers at under/post graduate levels in Karachi, Pakistan in making math courses lively to students. We use a random sample of 75 students of engineering and commerce studying in three different universities namely University of Karachi, Usman Institute of Technology (UIT) and Karachi Institute of Economics & Technology (PAF-KIET). A 95% confidence interval based on sample results reveals that the said proportion of math educators is in between 63 and 83%. Furthermore, we investigate with the help of students' responses how mathematics teachers at under/post graduate levels make their courses interesting-by showing their dedication in their subject, by giving logical reasoning and concrete examples or by making complex mathematical methods accessible to students giving them know-how of mathematical softwares. We find that the second technique is the most dominant and has a very strong impact (positive linear relationship) in achieving the said goal of a math-teacher. The linear correlation coefficient between students' opinion that math-teachers make their courses interesting and achieving this goal by giving logical reasoning and concrete examples is 0.989. Whereas the technique of using math softwares in attempt to make a math course lively has also a very strong but a cubic relationship and its multiple correlation coefficient is 0.984. Therefore, using technology in math classroom is also helpful in making math learning and teaching interesting but under some conditions that become apparent from our study made on the real data hence obtained.

### New TMME monographs

Two new monographs have been published from The Montana Mathematics Enthusiast:

- TMME Monograph 7 (May 2009) Interdisciplinarity, Creativity & Learning: Mathematics with literature, paradoxes, history, technology and modeling
- TMME Monograph 6 (June 2009) Critical Issues in Mathematics Education

Mono6 Preview

Mono7 Preview

## 2009/04/20

### Searching for good mathematics

In this article, we aim to provide a glimpse of what is counted as good mathematics instruction from Taiwanese perspectives and of various approaches developed and used for achieving high-quality mathematics instruction. The characteristics of good mathematics instruction from Taiwanese perspectives were first collected and discussed from three types of information sources. Although the number of characteristics of good mathematics instruction may vary from one source to another, they can be generally organized in three phases including lesson design before instruction, classroom instruction during the lesson and activities after lesson. In addition to the general overview of mathematics classroom instruction valued in Taiwan, we also analyzed 92 lessons from six experienced teachers whose instructional practices were generally valued in local schools and counties. We identified and discussed the characteristics of their instructional practices in three themes: features of problems and their uses in classroom instruction, aspects of problem–solution discussion and reporting, and the discussion of solution methods. To identify and promote high-quality mathematics instruction, various approaches have been developed and used in Taiwan including the development and use of new textbooks and teachers’ guides, teaching contests, master teacher training program, and teacher professional development programs.

### Conceptualizing and organizing content for teaching and learning

In this study, selected Chinese, Japanese and US mathematics textbooks were examined in terms of their ways of conceptualizing and organizing content for the teaching and learning of fraction division. Three Chinese mathematics textbook series, three Japanese textbook series, and four US textbook series were selected and examined to locate the content instruction of fraction division. Textbook organization of fraction division and other content topics were described. Further analyses were then conducted to specify how the content topic of fraction division was conceptualized and introduced. Specific attention was also given to the textbooks’ uses of content constructs including examples, representations, and exercise problems in order to show their approaches for the teaching and learning of fraction division. The results provide a glimpse of the metaphors of mathematics teaching and learning that have been employed in Chinese, Japanese, and US textbooks. In particular, the results from the textbook analyses demonstrate how conceptual underpinnings were developed while targeting procedures and operations. Implications of the study are then discussed.

### Productive failure in mathematical problem solving

This paper reports on a quasi-experimental study comparing a “productive failure” instructional design (Kapur in Cognition and Instruction 26(3):379–424, 2008) with a traditional “lecture and practice” instructional design for a 2-week curricular unit on rate and speed. Seventy-five, 7th-grade mathematics students from a mainstream secondary school in Singapore participated in the study. Students experienced either a traditional lecture and practice teaching cycle or a productive failure cycle, where they solved complex problems in small groups without the provision of any support or scaffolds up until a consolidation lecture by their teacher during the last lesson for the unit. Findings suggest that students from the productive failure condition produced a diversity of linked problem representations and methods for solving the problems but were ultimately unsuccessful in their efforts, be it in groups or individually. Expectedly, they reported low confidence in their solutions. Despite seemingly failing in their collective and individual problem-solving efforts, students from the productive failure condition significantly outperformed their counterparts from the lecture and practice condition on both well-structured and higher-order application problems on the post-tests. After the post-test, they also demonstrated significantly better performance in using structured-response scaffolds to solve problems on relative speed—a higher-level concept not even covered during instruction. Findings and implications of productive failure for instructional design and future research are discussed.

### Instructional Science, May 2009

- The use of language in understanding subject matter, by Lennart Svensson, Elsie Anderberg, Christer Alvegård and Thorsten Johansson
- Online but off-topic: negotiating common ground in small learning groups, by Trena M. Paulus
- The building of pre-service primary teachers’ knowledge of mathematics teaching: interaction and online video case studies, by Salvador Llinares and Julia Valls
- The learners’ experience of variation: following students’ threads of learning physics in computer simulation sessions, by Åke Ingerman, Cedric Linder and Delia Marshall
- The development of science activities via on-line peer assessment: the role of scientific epistemological views, by Chin-Chung Tsai and Jyh-Chong Liang

### ESM, May 2009

- Acquisition and use of shortcut strategies by traditionally schooled children, by Joke Torbeyns, Bert De Smedt, Pol Ghesquière and Lieven Verschaffel
- From arithmetical thought to algebraic thought: The role of the “variable”, by Elsa Malisani and Filippo Spagnolo
- The relationship between performance on mathematical word problems and language proficiency for students learning through the medium of Irish, by Máire Ní Ríordáin and John O’Donoghue
- Teachers’ emergent goals in spreadsheet-based lessons: analyzing the complexity of technology integration, by Jean-Baptiste Lagrange and Emel Ozdemir Erdogan
- Book review: mathematics classrooms in twelve countries, Clarke, D., Keitel, C., & Shimizu, Y. (Eds.). (2006). Mathematics classrooms in twelve countries: The insider’s perspective. Rotterdam, The Netherlands: Sense Publishers.

## 2009/04/18

### Concept mapping in mathematics

Concept Mapping in Mathematics: Research into Practice is the first comprehensive book on concept mapping in mathematics. It provides the reader with an understanding of how the meta-cognitive tool, namely, hierarchical concept maps, and the process of concept mapping can be used innovatively and strategically to improve planning, teaching, learning, and assessment at different educational levels. This collection of research articles examines the usefulness of concept maps in the educational setting, with applications and examples ranging from primary grade classrooms through secondary mathematics to pre-service teacher education, undergraduate mathematics and post-graduate mathematics education. A second meta-cognitive tool, called vee diagrams, is also critically examined by two authors, particularly its value in improving mathematical problem solving.

The theoretical underpinnings of concept mapping and of the studies in the book include Ausubel’s cognitive theory of meaningful learning, constructivist and Vygotskian psychology to name a few. There is evidence which suggests that students’ mathematical literacy and problem solving skills can be enhanced through students collaborating and interacting as they work, discuss and communicate mathematically. This book proposes the meta-cognitive strategy of concept mapping as one viable means of promoting, communicating and explicating students’ mathematical thinking and reasoning publicly in a social setting as they engage in mathematical dialogues and discussions.

Concept Mapping in Mathematics: Research into Practice is of interest to researchers, graduate students, teacher educators and professionals in mathematics education.

### Instructional beliefs

The purposes of this study were to determine preservice physics teachers’ instructional beliefs and to investigate the relationship between their beliefs and practices. The theoretical framework was based on the combination Haney & McArthur’s (Science Education, 86(6):783–802, 2002) research and Ford’s (1992) motivation systems theory. A multicase study design was utilized for the research in order to focus on a belief–practice relationship within several examples. Semistructured interviews, observations, and preservice teachers’ written documents were used to collect data. Results showed that most preservice teachers held instructional beliefs aligned with constructivist philosophy. Some of the preservice teachers’ beliefs were consistent with their practices while some of them presented different practices from their beliefs in different placements.

## 2009/04/17

### Why do I blog?

Today, I am giving a presentation at AERA, in a Public Communication Workshop. I have been invited to participate in this session because I am an education researcher who blog about the field that I am in. I have been asked to focus on six questions, and I thought it might be nice to share my thoughts about this with all my readers.

### Mathematics teachers' practices and thinking

Yeping Li, Xi Chen and Gerald Kulm have written an article called Mathematics teachers’ practices and thinking in lesson plan development: a case of teaching fraction division. The article was recently published online in ZDM. Here is their article abstract:

In this study, we aimed to examine mathematics teachers’ daily lesson plans and associated practices and thinking in lesson plan development. By focusing on teachers’ preparation for teaching fraction division, we collected and analyzed a sequence of four lesson plans from each of six mathematics teachers in six different elementary schools in China. Interviews with these teachers were also analyzed to support the lesson plan analysis and reveal teachers’ thinking behind their practices. The results show that Chinese teachers placed a great consideration on several aspects of lesson planning, including content, process, and their students’ learning. Teachers’ lesson plans were similar in terms of some broad features, but differed in details and specific approaches used. While the textbook’s influence was clearly evident in these teachers’ lesson plans, lesson planning itself was an important process for Chinese teachers to transform textbook content into a script unique to different teachers and their students. Implications obtained from Chinese teachers’ lesson planning practices and their thinking are then discussed in a broad context.

### In-service teacher training in Botswana

Teaching is a field that is dynamic, with innovations necessitating upgrading of skills and education of teachers for the successful implementation of reforms. The behaviour and attitudes of teachers towards teaching and learning and their knowledge banks are the result of the impact of in-service training. This study investigated the perceptions of mathematics and science teachers in Botswana towards in-service provision by the Department of Mathematics and Science Education In-service Training unit (DMSE-INSET), whose mandate is to improve the quality of teaching by supporting teachers through training programmes that enable them to take ownership of their professional development. Data were collected from a sample of 42 senior Mathematics and Science secondary school teachers, using structured interviews with open-ended questions, which were analyzed qualitatively. The findings show that teachers’ concerns included the lack of impact of current in-service training programmes on the education system, no regular follow-up activities to support the one-off workshops and insufficient skills acquired to sustain the implementation of the strategies solicited by the workshops.

## 2009/04/16

### Drag with a worn-out mouse

We consider what a concern for social justice in terms of social inclusion might mean for teacher education, both practising and prospective, with particular reference to the use of information and communication technology (ICT) in mathematics education taking place at a borderland school. Our discussion proceeds through the following steps: (1) We explore what a borderland position might denote to address what social inclusion might mean. (2) We consider the significance of mathematics education and the use of ICT for processes of social inclusion. (3) We briefly refer to the Interlink Network, as many of our observations emerge as reflections on this project. (4) We present different issues that will be of particular importance with respect to teacher education if we want to establish a mathematics education for social inclusion. These issues concern moving away from the comfort zone, establishing networks, identifying new approaches, moving beyond prototypical research, and getting in contact. This brings us to (5) final considerations, where we return to the notion of social justice.

## 2009/04/15

### Tuesday sessions at AERA

Today, I have attended three sessions at AERA, including the symposium session where I made my own presentation.

## 2009/04/14

### My AERA presentation

I am giving my presentation on Tuesday, April 14, in a symposium session from 10:35am to 12:05pm. Here is the slideshow for my presentation:

AERA_Mosvold-Fauskanger

(Direct link to paper)

Here is the article I am presenting:

Mosvold-Fauskanger, AERA 2009 paper

(Direct link to the article)

### Preparation for our symposium session

I have just been to a preparation meeting for our symposium session at AERA tomorrow. The session is called Adapting and Using U.S. Measures of Mathematical Knowledge for Teaching in Other Countries: Lessons and Challenges. The session is going to be chaired by Deborah L. Ball, and there are going to be five presentations of papers:

- I am going to make the first presentation after the chair's introductoin, and I am going to present a paper that I have written in collaboration with my colleague, Janne Fauskanger: Challenges of Translating and Adapting the MKT Measures for Norway
- The next presentation is going to be held by Minsung Kwon from South Korea. She is going to present her paper: Validating the Adapted MKT Measures in Korea
- Dicky Ng is following up with a presentation of his study in Indonesia. The title of his paper is: Translating and Adapting the Geometry Measures for Indonesia
- Yaa Cole unfortunately couldn't make it, but there has been prepared a video presentation of her paper: Studying the Work of Teaching Mathematics in Ghana
- The final presentation is made by Sean Delaney from Ireland. He was the one who invited us all to participate in this symposium, and he has been in charge of the entire process. He is presenting his paper: Using Qualitative and Quantitative Methods to Study Construct Equivalence of a Teacher Knowledge Construct

## 2009/04/13

### AERA 2009 Annual Meeting

When the American Educational Research Association (AERA) hosts the AERA Annual Meeting next month, more than 14,000 education research scholars will convene in San Diego, California where 2,000 peer-reviewed sessions are scheduled from April 13 to17.AERA was founded in 1916, and it is:

(...) the most prominent international professional organization, with the primary goal of advancing educational research and its practical application (Source).As of today, it has more than 26,000 members worldwide, and the membership represents a broad range of disciplines like:

- education
- psychology
- statistics
- sociology
- history
- economics
- philosophy
- anthropology
- political science

## 2009/04/11

### Preparations for AERA

I am spending the last few days at home before I leave for the AERA conference in San Diego. This is the first time I go to this conference, and I am really looking forward to it!

## 2009/04/10

### Sexy maths

(...) some of the most revolutionary contributtions to geometry since those of Euclid.The article gives an interesting insight into some of the most important aspects of the historical development of geometry, with Euclid's parallel postulate as a pivotal point. An excellent article by du Sautoy, who is a mathematician himself.

### Solutions of linear equations

The solution of linear ordinary differential equations (ODEs) is commonly taught in first-year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognizing what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to tables of solutions, is an important skill for students to carry with them to advanced courses in mathematics. In this study, we describe a teaching and learning strategy that replaces the traditional algorithmic, transmission presentation style for solving ODEs with a constructive, discovery-based approach where students employ their existing skills as a framework for constructing the solutions of first and second-order linear ODEs. We elaborate on how the strategy was implemented and discuss the resulting impact on a first-year undergraduate class. Finally, we propose further improvements to the strategy as well as suggesting other topics which could be taught in a similar manner.

## 2009/04/09

### Supervision of teachers

Göta Eriksson has written an article that was recently published online in The Journal of Mathematical Behavior. The article is entitled Supervision of teachers based on adjusted arithmetic learning in special education. Here is the abstract:

This article reports on 20 children's learning in arithmetic after teaching was adjusted to their conceptual development. The report covers periods from three months up to three terms in an ongoing intervention study of teachers and children in schools for the intellectually disabled and of remedial teaching in regular schools. The researcher classified each child's current counting scheme before and after each term. Recurrent supervision, aiming to facilitate the teachers’ modelling of their children's various conceptual levels and needs of learning, was conducted by the researcher. The teaching content in harmony with each child's ability was discussed with the teachers. This approach gives the teachers the opportunity to experience the children's own operational ways of solving problems. At the supervision meetings, the teachers theorized their practice together with the researcher, ending up with consistent models of the arithmetic of the child. So far, the children's and the teachers’ learning patterns are promising.

### Learning math by thinking

Michael Paul Goldenberg over at the Rational Mathematics Education blog has written an interesting post about LEARNING MATH BY THINKING - Hassler Whitney, Louis P. Benezet, and how many more wasted lives and decades will it take?

## 2009/04/08

### 6 out of 10 university students have math anxiety

I learned about this through Deb Russel's blog over at About.com. A Spanish study reveals that:

Six out of every 10 university students, regardless their field of study, present symptoms of anxiety when it comes to dealing with mathematics

The researchers assessed the students using the Fennema-Sherman Mathematics Attitudes Scales, a questionnaire validated by experts from all over the world which has been used since the 70s. The students took the questionnaire at the beginning of the second four-month period of school.These are interesting results. Math anxiety should definitely be taken seriously, and a person's attitudes towards mathematics are important, regardless if they are related to anxiety or not. I have done a much more informal study of my own students in early childhood education over the last couple of years, and almost half of them find mathematics boring and/or difficult. If some of them even have math anxiety, I think this will strongly impact their work as future teachers, kindergarten teachers or whatever they will end up doing!

## 2009/04/02

### Effect of personalization

This study investigated the effect of personalized print-based instruction on the achievement and self-efficacy regarding mathematics word problems of 320 senior secondary students in Nigeria. The moderator effect of gender was also examined on independent variable (personalization) and dependent variables (mathematics word problem achievement and self-efficacy). The t-test statistic was used to analyse the data collected for the study. The results showed that significant differences existed in the mathematics word problem achievement and self-efficacy beliefs of personalized and non-personalized groups, male and female personalized groups and male and female non-personalized groups.

### The problem of the pyramid

We consider Egyptian mathematics from a postmodern perspective, by which we mean suspending judgement as to strict correctness in order to appreciate the genuine mathematical insights which they did have in the context in which they were working. In particular we show that the skill which the Egyptians possessed of obtaining the general case from a specific numerical example suggests a complete solution to the well-known, but hitherto not completely resolved, question of how the volume of the truncated pyramid given in Problem 14 of the Moscow papyrus was derived. We also point out some details in Problem 48 of the Rhind papyrus, on the area of the circle, which have previously gone unnoticed. Finally, since many of their mathematical insights have long been forgotten, and fall within the modern school syllabus, we draw some important lessons for contemporary mathematics education.

### Students discovering spherical geometry

Dynamic geometry software (DGS) such as Cabri and Geometers' Sketchpad has been regularly used worldwide for teaching and learning Euclidean geometry for a long time. The DGS with its inductive nature allows students to learn Euclidean geometry via explorations. However, with respect to non-Euclidean geometries, do we need to introduce them to students in a deductive manner? Do students have quite different experiences in non-Euclidean environment? This study addresses these questions by illustrating the student mathematics teachers' actions in dynamic spherical geometry environment. We describe how student mathematics teachers explore new conjectures in spherical geometry and how their conjectures lead them to find proofs in DGS.

### Performance of undergraduate students in the limit concept

In this work, we investigated first-year university students' skills in using the limit concept. They were expected to understand the relationship between the limit-value of a function at a point and the values of the function at nearby points. To this end, first-year students of a Turkish university were given two tests. The results showed that the students were able to compute the limit values by applying standard procedures but were unable to use the limit concept in solving related problems.

### Students' experiences with mathematics teaching and learning

This study documents students' views about the nature of mathematics, the mathematics learning process and factors within the classroom that are perceived to impact upon the learning of mathematics. The participants were senior secondary school students. Qualitative and quantitative methods were used to understand the students' views about their experiences with mathematics learning and mathematics classroom environment. Interviews of students and mathematics lesson observations were analysed to understand how students view their mathematics classes. A questionnaire was used to solicit students' views with regards to teaching approaches in mathematics classes. The results suggest that students consider learning and understanding mathematics to mean being successful in getting the correct answers. Students reported that in the majority of cases, the teaching of mathematics was lecture-oriented. Mathematics language was considered a barrier in learning some topics in mathematics. The use of informal language was also evident during mathematics class lessons.

## 2009/04/01

### When two circles determine a triangle

Visualization of mathematical relationships enables students to formulate conjectures as well as to search for mathematical arguments to support these conjectures. In this project students are asked to discover the sufficient and necessary condition so that two circles form the circumscribed and inscribed circle of a triangle and investigate how this condition effects the type of triangle in general and its perimeter in particular. Its open-ended form of the task is a departure from the usual phrasing of textbook’s exercises “show that…”.