Graphics calculators in examination

Roger G. Brown from the University of Leeds (UK) has written an article entitled Does the introduction of the graphics calculator into system-wide examinations lead to change in the types of mathematical skills tested? This article was published online in Educational Studies in Mathematics earlier this week. Here is the abstract of his article:
The paper reports on the introduction of the graphics calculator into three centralised examination systems, which were located in Denmark, Victoria (Australia) and the International Baccalaureate. The introduction of the graphics calculator required those responsible for writing examination questions to consider how to assess mathematical skills within this new environment. This paper illustrates the types of mathematics skills that have been assessed within the graphics-calculator-assumed environment. The analysis of the examination questions indicated that only two out of the six mathematics examinations considered demonstrated any significant change in the types of skills assessed in conjunction with the introduction of the graphics calculator. The results suggest that it is possible to reduce the use of questions assessing routine procedures (mechanical skills) with a graphics calculator, but it is also evident that there have not been major changes in the way that examination questions are written nor the mathematics skills which the questions are intended to assess.


Using live, online tutoring

Richard Lissaman, Sue de Pomerai and Sharon Tripconey have written an article that was recently published online in Teaching Mathematics and its Applications. The article is entitled Using live, online tutoring to inspire post 16 students to engage with higher level mathematics, and here is a copy of the article's abstract:
In recent years, there has been a decline in the number of students aged 16–18 studying and being able to access higher level mathematics in schools in the UK. The Further Mathematics Network (FMN) was set up to enable access to such mathematics to all students and to promote and encourage students to study at this level. The FMN has pioneered the use of Elluminate, a well established web-based package, for live mathematics tutoring. Small groups of students meet online with an experienced tutor to learn new aspects of mathematics and to look at ways to solve complex problems. There are also extensive online resources to support the students’ learning. The findings are discussed in the following article.


Pre-service teachers' teaching anxiety

Murat Peker has written an article about Pre-Service Teachers’ Teaching Anxiety about Mathematics and Their Learning Styles. This article was published in the last issue of Eurasia Journal of Mathematics, Science & Technology Education. A main issue in the article is the combination of focus on mathematics (teaching) anxiety and learning styles. When it comes to learning styles, Peker very much builds upon the theories of Kolb (see p. 337). The theoretical overview is quite interesting, and in many respects new to me.

The study included 506 pre-service teachers from Turkey, and two instruments were used in the study: the Learning Style Inventory and the Mathematics Teaching Anxiety Scale (both questionnaires). The first questionnaire is derived from Kolb's works, whereas the anxiety scale was developed by the researcher. I miss a discussion of the rationale behind the choice of methods/instruments in the study, and I think this is an important aspect of such a research article. I also think there are a couple of issues about the Learning Style Inventory that should be discussed somewhat. My main critique towards the statements from this questionnaire (as they are presented in the article) is that they appear very general. Being faced with a statement like "When I learn, I like to watch and listen", my response would vary according to the subject and teaching/learning context I had in mind. As with research on beliefs, I think it would make more sense to investigate views that teachers (pre-service or in-service) have on teaching and learning algebra, geometry, functions etc., rather than their views on teaching and learning in general. My response to a statement like "I learn best when I am practical" would also vary a lot according to what I had in mind when giving the response. I therefore think that the questionnaire has some severe weaknesses that need to be addressed. Other than that, I think the article is interesting, and Peker obviously points to some important issues!


The purpose of this study was to investigate the differences in the teaching anxiety of pre-service teachers in mathematics according to their learning style preferences. There were a total of 506 pre-service teachers involved in this study. Of the total, 205 were pre-service elementary school teachers, 173 were pre-service elementary mathematics teachers, and 128 were pre-service secondary mathematics teachers. In the collection of the data, the researcher employed two types of instruments: the Learning Style Inventory (LSI) and the Mathematics Teaching Anxiety Scale (MATAS). The LSI determined the participants’ learning style preference: divergent, assimilator, convergent, and accommodator. The MATAS found the participants’ mathematics teaching anxiety level. The researcher used the one-way ANOVA with α = 0.05 in the analysis of the data. The study revealed that there were statistically significant differences in mathematics teaching anxiety between
convergent and the other three types of learners: divergent, accommodator, and assimilator. The difference was in favour of convergent learners. In other words, convergent learners had less mathematics teaching anxiety than the other types of learners. The study also found that divergent learners showed the highest level of mathematics teaching anxiety.


Peker, M. (2009). Pre-Service Teachers' Teaching Anxiety about Mathematics and Their Learning Styles. Eurasia Journal of Mathematics, Science & Technology Education, 5(4), 335-345

NOMAD, October 2009

The Nordic Journal of Research in Mathematics Education (NOMAD) has recently released the October issue. This issue contains three research articles:

Graphic calculators and connectivity software

Ornella Robutti has written an article called Graphic calculators and connectivity software to be a community of mathematics practitioners. This article was recently published online in ZDM. Here is the abstract of the article:

In a teaching experiment carried out at the secondary school level, we observe the students’ processes in modelling activities, where the use of graphic calculators and connectivity software gives a common working space in the class. The study shows results in continuity with others emerged in the previous ICMEs and some new ones, and offers an analysis of the novelty of the software in introducing new ways to support learning communities in the construction of mathematical meanings. The study is conducted in a semiotic-cultural framework that considers the introduction and the evolution of signs, such as words, gestures and interaction with technologies, to understand how students construct mathematical meanings, working as a community of practice. The novelty of the results consists in the presence of two technologies for students: the “private” graphic calculators and the “public” screen of the connectivity software. Signs for the construction of knowledge are mediated by both of them, but the second does it in a social way, strongly supporting the work of the learning community.


Conceptions of effective mathematics ...

A new article about teachers' conception of effective mathematics teaching. The article investigates the perspectives of teachers from China and the U.S., and I find it particularly interesting because it focus on the issue of cultural beliefs. I think this is an interesting concept, and I've used it before in one of my own articles. The idea of cultural beliefs comes from results of cross-national studies where researchers have identified clear differences in the teaching practices of teachers from East-Asian and Western countries.

In the study referred to in the article below, 9 Chinese teachers and 11 U.S. teachers were interviewed. The semi-structured interviews that were used in the study were constructed according to Ernest's traditional framework of three aspects of mathematics teachers' beliefs. The study showed that the teachers from these two countries held quite different beliefs about good mathematics teaching. These views were also closely connected with their views on the nature of mathematics.

Conceptions of effective mathematics teaching within a cultural context: perspectives of teachers from China and the United States

Journal    Journal of Mathematics Teacher Education
Publisher    Springer Netherlands
ISSN    1386-4416 (Print) 1573-1820 (Online)
DOI    10.1007/s10857-009-9132-1
Subject Collection    Humanities, Social Sciences and Law
SpringerLink Date    Tuesday, November 17, 2009

By Jinfa Cai and Tao Wang

Abstract  This study investigates Chinese and U.S. teachers’ cultural beliefs concerning effective mathematics teaching from the teachers’ perspectives. Although sharing some common beliefs, the two groups of teachers think differently about both mathematics understanding and the features of effective teaching. The sample of U.S. teachers put more emphasis on student understanding with concrete examples, and the sample of Chinese teachers put more emphasis on abstract reasoning after using concrete examples. The U.S. teachers highlight a teacher’s abilities to facilitate student participation, manage the classroom and have a sense of humor, while the Chinese teachers emphasize a teacher’s solid mathematics knowledge and careful study of textbooks. Both groups of teachers agree that memorization and understanding cannot be separated. However, for the U.S. teachers, memorization comes after understanding, but for Chinese teachers, memorization can come before understanding. These differences of teachers’ beliefs are discussed in a cultural context.

Learning from video

The last couple of days, two articles with a focus on using video as a tool for teacher learning and development have been published in Journal of Mathematics Teaching Education. The first articleinvestigates how prospective primary mathematics teachers might learn from on-line discussions.

Prospective primary mathematics teachers’ learning from on-line discussions in a virtual video-based environment

Journal    Journal of Mathematics Teacher Education
Publisher    Springer Netherlands
ISSN    1386-4416 (Print) 1573-1820 (Online)
DOI    10.1007/s10857-009-9133-0
SpringerLink Date    Wednesday, November 18, 2009

By Salvador Llinares and Julia Valls

Abstract The aim of this study was to investigate how participation and reification of ideas about mathematics teaching are constituted in on-line discussions when prospective primary mathematics teachers analysed video-cases about mathematics teaching. Prospective teachers enrolled in a mathematics methodology course participated for 4 weeks in two virtual learning environments that integrated the analysis of video-clips, on-line discussions and writing essays about key aspects of mathematics teaching. Three aspects were considered relevant to explain the prospective teachers’ learning: the way in which the theoretical information was used to frame and to interpret the events from mathematics teaching; the characteristics of engagement with others participating in the on-line discussions and the role played by prospective teachers’ beliefs. Possible reasons for the importance of these features include the specific questions posed in on-line discussions and the use of video-clips of mathematics teaching. These findings are considered useful in designing virtual learning environments and the kinds of tasks through which the understanding of mathematics teaching and learning-to-notice skills can be developed.

The other article also has a focus on using videos, by the use of so called "video clubs".

The influence of video clubs on teachers’ thinking and practice

Journal    Journal of Mathematics Teacher Education
Publisher    Springer Netherlands
ISSN    1386-4416 (Print) 1573-1820 (Online)
DOI    10.1007/s10857-009-9130-3
SpringerLink Date    Saturday, November 14, 2009

By Elizabeth A. van Es and Miriam Gamoran Sherin

Abstract This article examines a model of professional development called “video clubs” in which teachers watch and discuss excerpts of videos from their classrooms. We investigate how participation in a video club influences teachers’ thinking and practice by exploring three related contexts: (a) teachers’ comments during video-club meetings, (b) teachers’ self-reports of the effects of the video club, and (c) teachers’ instruction across the year. Data analysis revealed changes in all three contexts. In the video-club meetings, teachers paid increased attention to student mathematical thinking over the course of the year. In interviews, teachers reported having learned about students’ mathematical thinking, about the importance of attending to student ideas during instruction, and about their school’s mathematics curriculum. Finally, shifts were also uncovered in the teachers’ instruction. By the end of the year, teachers increasingly made space for student thinking to emerge in the classroom, probed students’ underlying understandings, and learned from their students while teaching.


Mathematical thinking of kindergarten boys and girls

Pnina S. Klein, Esther Adi-Japha and Simcha Hakak-Benizri have written an article called Mathematical thinking of kindergarten boys and girls: similar achievement, different contributing processes. This article was recently published online in Educational Studies in Mathematics. Here is the abstract of their article:
The objective of this study was to examine gender differences in the relations between verbal, spatial, mathematics, and teacher–child mathematics interaction variables. Kindergarten children (N = 80) were videotaped playing games that require mathematical reasoning in the presence of their teachers. The children’s mathematics, spatial, and verbal skills and the teachers’ mathematical communication were assessed. No gender differences were found between the mathematical achievements of the boys and girls, or between their verbal and spatial skills. However, mathematics performance was related to boys’ spatial reasoning and to girls’ verbal skills, suggesting that they use different processes for solving mathematical problems. Furthermore, the boys’ levels of spatial and verbal skills were not found to be related, whereas they were significantly related for girls. The mathematical communication level provided in teacher–child interactions was found to be related to girls’ but not to boys’ mathematics performance, suggesting that boys may need other forms of mathematics communication and teaching.
Several studies have focused on gender differences in mathematics education, but few have focused on gender differences with small children. The study of Klein and colleagues focus on gender differences in relation to "verbal skills, variables of spatial skills, and variables related to environmental factors, including teaching methods, quality of teaching, and mathematical communication". Four research questions are posed in the study:
  1. "Do kindergarten boys and girls differ mathematically?
  2. Are language and spatial skills related differently to mathematics achievements of boys and girls?
  3. Do boys and girls receive different mathematical communication by their teachers?
  4. Are the patterns of correlation between instructional behavior (mediation) and mathematics achievements different for boys and girls?"
A test called KeyMath was used to measure the mathematical thinking of a selection of children (n=80), half of the children were boys/girls. The test is supposed to cover an age range of 4.6-21 years. There are several subtests within this set of measures. Three tests were used to evaluate the verbal ability of the children, and two were used to evaluate their spatial skills. Observations of mathematical communications in teacher-child interactions were also made in the kindergartens. The actual testing was carried out by the authors of the paper.

The results of the study are quite interesting. They did not find any differences in mathematical achievements between the boys and girls in the study. There was, however, significant gender differences in some of the factors that were related to these results. As they state: "The boys’ mathematical achievement was significantly related to their spatial reasoning, whereas the girls’ mathematical achievement was related to their verbal skills."

I find this study interesting in many ways, but there are a few issues that I would have liked to learn more about (and that the article does not address):
  • Were the measures translated from English into Hebrew? (If so, I would like to learn more about this process)
  • What are the reasons for deciding on this particular method, and using these particular measures, in the study?


Developing flexibility for teaching algebra

Christopher Yakes and Jon R. Star have written an article that was recently published online in Journal of Mathematics Teacher Education. The article is entitled Using comparison to develop flexibility for teaching algebra. Here is the abstract of their article:
In this paper, we describe a one-day professional development activity for mathematics teachers that promoted the use of comparison as an instructional tool to develop students’ flexibility in algebra. Effective use of comparison in mathematics instruction involves using side-by-side presentation of problems and solution methods and subsequent student discussion of these multiple solution methods to highlight the similarities and differences among problem-solving techniques. The goals of the professional development activity were to make teachers aware of how to use comparison effectively in their instruction, as well as to impact teachers’ own flexibility in algebra by using comparison instructionally during the professional development. Our analysis of teachers’ experiences in the professional development activity suggests that when teachers were presented with techniques for effective use of comparison, their own understanding of multiple solution methods was reinforced. In addition, teachers began to question why they relied exclusively on one familiar method over others that are equally effective and perhaps more efficient and started to draw new connections between problem-solving methods. Finally, as a result of experiencing instructional use of comparison, teachers began to see value in teaching for flexibility and reported changing their own teaching practices.


Teachers' metacognitive and heuristic approaches to word problem solving

Fien Depaepe, Erik De Corte and Lieven Verschaffel have written an interesting article about Teachers’ metacognitive and heuristic approaches to word problem solving: analysis and impact on students’ beliefs and performance. The article was published online in ZDM last Friday. Here is the abstract of their article:
We conducted a 7-month video-based study in two sixth-grade classrooms focusing on teachers’ metacognitive and heuristic approaches to problem solving. All problem-solving lessons were analysed regarding the extent to which teachers implemented a metacognitive model and addressed a set of eight heuristics. We observed clear differences between both teachers’ instructional approaches. Besides, we examined teachers’ and students’ beliefs about the degree to which metacognitive and heuristic skills were addressed in their classrooms and observed that participants’ beliefs were overall in line with our observations of teachers’ instructional approaches. In addition, we investigated how students’ problem-solving skills developed as a result of teachers’ instructional approaches. A positive relationship between students’ spontaneous application of heuristics to solve non-routine word problems and teachers’ references to these skills in their problem-solving lessons was found. However, this increase in the application of heuristics did not result in students’ better performance on these non-routine word problems.


JMTE, December 2009

The December issue of Journal of Mathematics Teacher Education has been published (volume 12, number 6). This issue is a special issue with focus on social justice perspectives in matheamtics teacher education, and it contains the following articles:

Mathematics and positive sciences

On Friday, a new article by Giorgio T. Bagni was released from Educational Studies in Mathematics. The article is entitled Mathematics and positive sciences: a reflection following Heidegger. Bagni takes Heidegger's Being and Time as a starting point in an examination of Heidegger's ideas about sciences in general and mathematics in particular. Here is the abstract of Bagni's article:
In this article, I make a case for the inputs that Martin Heidegger's theoretical perspective offers to current concerns about the nature of mathematics, its teaching and learning, and the problem of subjectivity. In particular, I consider Heidegger's notion of positive science and discuss both its applicability to mathematics and its importance to mathematics education. I argue that Heidegger's ontological position is consonant with some sociocultural approaches in mathematics education and that Heidegger's work can shed some light on the problem of knowing and being. Finally, I raise some questions concerning subjectivity and the link between language and mathematical objects.


December issue of Educational Studies in Mathematics

The December issue of Educational Studies in Mathematics has just been published (volume 72, number 3), and it contains the following articles:

Instructional Science, November 2009

The November issue of Instructional Science has been released (Volume 37, Number 6, 2009). Here is a list of articles that is published in this issue:


New journal in mathematics education!

International Journal of Studies in Mathematics Education is a new international peer-reviewed journal within the field of mathematics education. The editors of the journal are from Brazil, and the journal web-site is in both Portuguese and English. The journal also accepts submission of articles in Portuguese, English, French and Spanish. According to their own description of the journal:
The journal aims to stimulate reflection on mathematics education at all levels: to generate productive discussion; to encourage enquiry and research; to promote criticism and evaluation of ideas and procedures current in the field
It is intended for the mathematics educator who is aware that the learning and teaching of mathematics are complex enterprises about which much remains to be revealed and understood
It reflects both the variety of research concerns within the field and the range of methods used to study them. We accept for submission articles in Portuguese, English, French and Spanish. The journal emphasizes high-level articles that go beyond local or national interest.
The journal has an online submission system, and the Open Journal Systems is being used. The journal is an online journal, and it appears to have an Open Access philosophy, so that the articles will be freely available for everyone to read/download. The aims of the journal are:
  • to stimulate reflection on mathematics education at all levels;
  • to generate productive discussion;
  • to encourage enquiry and research;
  • to promote criticism and evaluation of ideas and procedures current in the field
Unfortunately, I cannot find an RSS feed yet, but I am definitely going to keep an eye on this journal even though!

The first issue of the journal is already available, and it contains several interesting articles. The following articles are in English:


New IJMEST articles

A couple of new articles have been published online in International Journal of Mathematical Education in Science and Technology. Several are so-called classroom notes (see this link for full list), and then there are these two original articles:
  • Pilot study on algebra learning among junior secondary students, by Kin-Keung Poon and Chi-Keung Leung. Abstract: The purpose of the study reported herein was to identify the common mistakes made by junior secondary students in Hong Kong when learning algebra and to compare teachers' perceptions of students' ability with the results of an algebra test. An algebra test was developed and administered to a sample of students (aged between 13 and 14 years). From the responses of the participating students (N = 815), it was found that students in schools with a higher level of academic achievement had better algebra test results than did those in schools with a lower level of such achievement. Moreover, it was found that a teacher's perception of a student's ability has a correlation with that student's level of achievement. Based on this finding, an instrument that measures teaching effectiveness is discussed. Last but not least, typical errors in algebra are identified, and some ideas for an instructional design based on these findings are discussed.
  • Student connections of linear algebra concepts: an analysis of concept maps, by Douglas A. Lapp,  Melvin A. Nyman and John S. Berry. Abstract: This article examines the connections of linear algebra concepts in a first course at the undergraduate level. The theoretical underpinnings of this study are grounded in the constructivist perspective (including social constructivism), Vernaud's theory of conceptual fields and Pirie and Kieren's model for the growth of mathematical understanding. In addition to the existing techniques for analysing concept maps, two new techniques are developed for analysing qualitative data based on student-constructed concept maps: (1) temporal clumping of concepts and (2) the use of adjacency matrices of an undirected graph representation of the concept map. Findings suggest that students may find it more difficult to make connections between concepts like eigenvalues and eigenvectors and concepts from other parts of the conceptual field such as basis and dimension. In fact, eigenvalues and eigenvectors seemed to be the most disconnected concepts within all of the students' concept maps. In addition, the relationships between link types and certain clumps are suggested as well as directions for future study and curriculum design.


Insight into the fractional calculus via a spreadsheet

David A. Miller and Stephen J. Sugden have written an article called Insight into the Fractional Calculus via a Spreadsheet. The article was published in the most recent issue of electronic journal Spreadsheets in Education. The article is freely available as a pdf download, but here is a copy of the abstract:
Many students of calculus are not aware that the calculus they have learned is a special case (integer order) of  fractional calculus. Fractional calculus is the study of arbitrary order derivatives and integrals and their applications. The article begins by stating a naive question from a student in a paper by Larson (1974) and establishes, for polynomials and exponential functions, that they can be deformed into their derivative using the μ-th order fractional derivatives for 0<μ<1. Through the power of Excel we illustrate the continuous deformations dynamically through conditional formatting. Some applications are discussed and a connection made to mathematics education.