Merry Christmas

I wish all of my readers a merry Christmas! Things are going to be somewhat slow here on the blog for a few days, but I promise to be back in early January with more news about mathematics education research!

If you want to stay up to date, you might consider checking my shared articles on Google Reader, or you can go directly to the automatically updated articles within the field of mathematics education. Articles related to education research in general can be found here, and articles related to early childhood education can be found here. You might also consider following me on twitter, where I will also provide news and updates about mathematics education and other things of interest.

Teacher lust

Andrew M. Tyminski has written an article that was recently published online in Journal of Mathematics Teacher Education. The article is entitled Teacher lust: reconstructing the construct for mathematics instruction. Here is the abstract of Tyminski's article:

Two collegiate mathematics courses for prospective elementary and middle grades teachers provide the context for the examination of Mary Boole’s construct of teacher lust. Through the use of classroom observations and instructor interviews, the author presents a refined conception of teacher lust. Two working aspects of the construct were identified: (1) enacted teacher lust; an observable action that may remove an opportunity for students to think about or engage in mathematics for themselves; and (2) experienced teacher lust; an internal impulse to act in the manner described. Empirical examples of each facet, differences between conscious and unconscious interactions with teacher lust, along with potential antecedents are discussed.

Learning to teach mathematics through inquiry

Jo Towers has written an article entitled Learning to teach mathematics through inquiry: a focus on the relationship between describing and enacting inquiry-oriented teaching. The article was published online in Journal of Mathematics Teacher Education last week. Here is the abstract of the article:

This article is based on one of the several case studies of recent graduates of a teacher education programme that is founded upon inquiry-based, field-oriented and learner-focussed principles and practices and that is centrally concerned with shaping teachers who can enact strong inquiry-based practices in Kindergarten to Grade 12 classrooms. The analysis draws on interviews with one graduate, and on video data collected in his multi-aged Grade 1/2 classroom, to explore some of the ways in which this new teacher enacted inquiry-based teaching approaches in his first year of teaching and to consider his capacity to communicate his understanding of inquiry. This article presents implications for beginning teachers’ collaborative practices, for the assessment of new teachers and for practices in preservice teacher education.

TIMSS Advanced 2008

Last week, the results from the TIMSS Advanced 2008 were released. The TIMSS assessment is probably well known to most, and the TIMSS video studies might also be familiar to some, but what exactly is TIMSS Advanced? The following description from the official website might explain some of the confusion:

TIMSS Advanced 2008 assesses student achievement in advanced mathematics and physics in the final year of secondary school—the twelfth grade in many countries. TIMSS Advanced is part of IEA’s series of TIMSS international assessments designed to provide comparative information about educational achievement across countries. Because TIMSS Advanced assesses students in their last year of secondary school who have studied advanced mathematics or physics to prepare them for further study of mathematics and science at the tertiary level, the results are of particular importance for educational decision making. (Source: http://timss.bc.edu/timss_advanced/index.html)

If you want to take a closer look at the full report from this study, you can check out this link (this is a direct link to a 33MB pdf file!). In case you want to dig even deeper into all the details and documentation of this study, you might want to take a look at The TIMSS Advanced 2008 Technical Report (14MB).

References
Arora, A., Foy, P., Martin, M.O., & Mullis, I.V.S. (Eds.). (2009). TIMSS Advanced 2008 Technical Report. Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.

Mullis, I.V.S., Martin, M.O., Robitaille, D.F., & Foy, P. (2009). TIMSS Advanced 2008 International Report: Findings from IEA's Study of Achievement in Advanced Mathematics and Physics in the Final Year of Secondary School. Chestnut Hill, MA: TIMSS & PIRLS International Study Center, Boston College.

Math tutoring for low-achieving students

Ronnie Karsenty has written an article entitled Nonprofessional mathematics tutoring for low-achieving students in secondary schools: A case study. This article was published online in Educational Studies in Mathematics last week. The project that is reported in the article is part of a larger project (SHLAV - Hebrew acronym for Improving Mathematics Learning). The research questions in the study are:

1. Will nonprofessional tutoring be effective, in terms of improving students' achievements in mathematics, and if so, to what extent?
2. Which factors will be identified by tutors as having the greatest impact on the success or failure of tutoring?

Here is the abstract of the article:

This article discusses the possibility of using nonprofessional tutoring as means for advancing low achievers in secondary school mathematics. In comparison with professional, paraprofessional, and peer tutoring, nonprofessional tutoring may seem less beneficial and, at first glance, inadequate. The described case study shows that nonprofessional tutors may contribute to students' understanding and achievements, and thus, they can serve as an important assisting resource for mathematics teachers, especially in disadvantaged communities. In the study, young adults volunteered to tutor low-achieving students in an urban secondary school. Results showed a considerable mean gain in students' grades. It is suggested that affective factors, as well as the instruction given to tutors by a specialized counselor, have played a major role in maintaining successful tutoring.

The increasing role of metacognitive skills in math

Manita Van der Stel, Marcel Veenman, Kim Deelen and Janine Haenen have written an article entitled The increasing role of metacognitive skills in math: a cross-sectional study from a developmental perspective. This article was published online in ZDM last week. The article is an Open Access article, so it is freely available for all to read, but here is a copy of the abstract to tickle your interest:

Both intelligence and metacognitive skillfulness have been regarded as important predictors of math performance. The role that metacognitive skills play in math, however, seems to be subjected to change over the early years of secondary education. Metacognitive skills seem to become more general (i.e., less domain-specific) by nature (Veenman and Spaans in Learn Individ Differ 15:159–176, 2005). Moreover, according to the monotonic development hypothesis (Alexander et al. in Dev Rev 15:1–37, 1995), metacognitive skills increase with age, independent of intellectual development. This hypothesis was tested in a study with 29 second-year students (13–14 years) and 30 third-year students (14–15 years) in secondary education. A standardized intelligence test was administered to all students. Participants solved math word problems with a difficulty level adapted to their age group. Thinking-aloud protocols were collected and analyzed on the frequency and quality of metacognitive activities. Another series of math word problems served as post-test. Results show that the frequency of metacognitive activity, especially those of planning and evaluation, increased with age. Intelligence was a strong predictor of math performance in 13- to 14-year-olds, but it was less prominent in 14- to 15-year-olds. Although the quality of metacognitive skills appeared to predict math performance in both age groups, its predictive power was stronger in 14- to 15-year-olds, even on top of intelligence. It bears relevance to math education, as it shows the increasing relevance of metacognitive skills to math learning with age.

Visual templates in pattern generalization activity

F.D. Rivera has written an article called Visual templates in pattern generalization activity. The article was published online in Educational Studies in Mathematics last Thursday. The study, which is described in the article, was carried out in an eighth-grade Algebra 1 class in California. Four and a half months after a teaching experiment on pattern generalization, 11 students were interviewed (clinical interviews). Clinical interviews were also made with these students directly before and after the teaching experiment. The article reports on results from the analyses of these clinical interviews.

Here is the abstract of the article:

In this research article, I present evidence of the existence of visual templates in pattern generalization activity. Such templates initially emerged from a 3-week design-driven classroom teaching experiment on pattern generalization involving linear figural patterns and were assessed for existence in a clinical interview that was conducted four and a half months after the teaching experiment using three tasks (one ambiguous, two well defined). Drawing on the clinical interviews conducted with 11 seventh- and eighth-grade students, I discuss how their visual templates have spawned at least six types of algebraic generalizations. A visual template model is also presented that illustrates the distributed and a dynamically embedded nature of pattern generalization involving the following factors: pattern goodness effect; knowledge/action effects; and the triad of stage-driven grouping, structural unit, and analogy.

Developing a 'leading identity'

Laura Black, Julian Williams, Paul Hernandez-Martinez, Pauline Davis, Maria Pampaka and geoff Wake have written an article called Developing a ‘leading identity’: the relationship between students’ mathematical identities and their career and higher education aspirations. This article was published online in Educational Studies in Mathematics last Wednesday. Here is the abstract of their article:

The construct of identity has been used widely in mathematics education in order to understand how students (and teachers) relate to and engage with the subject (Kaasila, 2007; Sfard & Prusak, 2005; Boaler, 2002). Drawing on cultural historical activity theory (CHAT), this paper adopts Leont’ev’s notion of leading activity in order to explore the key ‘significant’ activities that are implicated in the development of students’ reflexive understanding of self and how this may offer differing relations with mathematics. According to Leont’ev (1981), leading activities are those which are significant to the development of the individual’s psyche through the emergence of new motives for engagement. We suggest that alongside new motives for engagement comes a new understanding of self—a leading identity—which reflects a hierarchy of our motives. Narrative analysis of interviews with two students (aged 16–17 years old) in post-compulsory education, Mary and Lee, are presented. Mary holds a stable ‘vocational’ leading identity throughout her narrative and, thus, her motive for studying mathematics is defined by its ‘use value’ in terms of pursuing this vocation. In contrast, Lee develops a leading identity which is focused on the activity of studying and becoming a university student. As such, his motive for study is framed in terms of the exchange value of the qualifications he hopes to obtain. We argue that this empirical grounding of leading activity and leading identity offers new insights into students’ identity development.

"Me and maths"

Pietro Di Martino and Rosetta Zan have written an article entitled ‘Me and maths’: towards a definition of attitude grounded on students’ narratives. The article was published online in Journal of Mathematics Teacher Education on Friday. Here is a copy of the abstract of their article:

The attitude construct is widely used by teachers and researchers in mathematics education. Often, however, teachers’ diagnosis of ‘negative attitude’ is a causal attribution of students’ failure, perceived as global and uncontrollable, rather than an accurate interpretation of students’ behaviour, capable of steering future action. In order to make this diagnosis useful for dealing with students’ difficulties in mathematics, it is necessary to clarify the construct attitude from a theoretical viewpoint, while keeping in touch with the practice that motivates its use. With this aim, we investigated how students tell their own relationship with mathematics, proposing the essay “Me and maths” to more than 1,600 students (1st to 13th grade). A multidimensional characterisation of a student’s attitude towards mathematics emerges from this study. This characterisation and the study of the evolution of attitude have many important consequences for teachers’ practice and education. For example, the study shows how the relationship with mathematics is rarely told as stable, even by older students: this result suggests that it is never too late to change students’ attitude towards mathematics.

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!

Abstract

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.

Reference:

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:

• Leif Bjørn Skorpen: Nokre spesielle trekk ved arbeidet med matematikkfaget i begynnaropplæringa (in Norwegian)
  
• Frode Olav Haara and Kari Smith: Practical activities in mathematics teaching – mathematics teachers’ knowledge based reasons
• Diana Stentoft and Paola Valero:  Identities-in-action. Exploring the fragility of discourse and identity in learning mathematics
  

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:

• Working with mathematics teachers and immigrant students: an empowerment perspective, by Núria Planas and Marta Civil
• ‘Gender games’: a post-structural exploration of the prospective teacher, mathematics and identity, by Anna Llewellyn
• Engaging with issues of emotionality in mathematics teacher education for social justice, by Mark Boylan
• ‘The conference was awesome’: social justice and a mathematics teacher conference, by Tamsin Meaney, Tony Trinick and Uenuku Fairhall
  

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:

• Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks, by Kaye Stacey and Jill Vincent
• Community college students’ views on learning mathematics in terms of their epistemological beliefs: a Q method study, by Denna L. Wheeler and Diane Montgomery
• Constructing mathematics in an interactive classroom context, by Paul Ngee-Kiong Lau, Parmjit Singh and Tee-Yong Hwa
• The effects of cooperative learning on preschoolers’ mathematics problem-solving ability, by Kamuran Tarim
• Students’ perceptions of institutional practices: the case of limits of functions in college level Calculus courses, by Nadia Hardy
• Mathématiques de la vie quotidienne au Burkina Faso: une analyse de la pratique sociale de comptage et de vente de mangues, by Kalifa Traoré and Nadine Bednarz
• The challenge of self-regulated learning in mathematics teachers' professional training, by Bracha Kramarski and Tali Revach
  

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:

• The effects of representational format on learning combinatorics from an interactive computer simulation, by Bas Kolloffel, Tessa H. S. Eysink, Ton de Jong and Pascal Wilhelm
• Uncertainty and engagement with learning games, by Paul A. Howard-Jones and Skevi Demetriou
• Are instructional explanations more effective in the context of an impasse? by Emilio Sánchez, Héctor García-Rodicio and Santiago R. Acuña
• Teacher and student intrinsic motivation in project-based learning, by Shui-fong Lam, Rebecca Wing-yi Cheng and William Y. K. Ma
• Graduate students’ conceptions of university teaching and learning: formation for change, by Alenoush Saroyan, Joyce Dagenais and Yanfei Zhou
  

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:

• Gesture, conceptual integration and mathematical talk, by Laurie Edwards
• Learning in virtual environments: a methodology for the analysis of teacher discourse, by Marcello Bairral
• Teacher's semiotic games in mathematics laboratory, by Ornella Robutti
  

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.

JMTE - October 2009

Journal of Mathematics Teacher Education has just released the October issue (Volume 12, Number 5). This issue contains 5 interesting articles:

• Conditions of progress in mathematics teacher education, by João Pedro da Ponte
• Teachers’ innovative change within countrywide reform: a case study in Rwanda, by Alphonse Uworwabayeho
• Alignment, cohesion, and change: Examining mathematics teachers’ belief structures and their influence on instructional practices, by Dionne I. Cross
• Multiple representations as sites for teacher reflection about mathematics learning, by Amy E. Ryken
• Understanding the influence of two mathematics textbooks on prospective secondary teachers’ knowledge, by Jon D. Davis

Alphonse Uworwabayeho's article abat the Rwandan case study is an Open Access article.

ZDM, November 2009

ZDM - The International Journal on Mathematics Education - has published the November issue of 2009 (Volume 41, Number 6). The issue contains the following nine articles:

• Curriculum research to improve teaching and learning: national and cross-national studies, by Gerald Kulm and Yeping Li
• Mathematics teachers’ practices and thinking in lesson plan development: a case of teaching fraction division, by Yeping Li, Xi Chen and Gerald Kulm
• Approaches and practices in developing school mathematics textbooks in China, by Yeping Li, Jianyue Zhang and Tingting Ma
• Mathematics curriculum: a vehicle for school improvement, by Christian R. Hirsch and Barbara J. Reys
• School mathematics curriculum materials for teachers’ learning: future elementary teachers’ interactions with curriculum materials in a mathematics course in the United States, by Gwendolyn Monica Lloyd
• How a standards-based mathematics curriculum differs from a traditional curriculum: with a focus on intended treatments of the ideas of variable, by Bikai Nie, Jinfa Cai and John C. Moyer
• Cross-cultural issues in linguistic, visual-quantitative, and written-numeric supports for mathematical thinking, by Karen C. Fuson and Yeping Li
• Conceptualizing and organizing content for teaching and learning in selected Chinese, Japanese and US mathematics textbooks: the case of fraction division, by Yeping Li, Xi Chen and Song An
• Cross-national comparisons of mathematics curriculum materials: what might we learn? by Edward A. Silver

The article by Karen C. Fuson and Yeping Li is an Open Access article, so that one should be available to all, even non-subscribers :-)

Seminar with Bharath Sriraman

On Thursday, we had a very nice seminar with Bharath Sriraman, or rather we had him as a visitor at University of Stavanger for the whole day! Bharath is professor of mathematical sciences at the University of Montana. He is also founder and editor of The Montana Mathematics Enthusiast, a great online journal within the field of mathematics education research, and editor of numerous other journals, book series, etc.

When he visited us on Thursday, he held a lecture with a focus on gifted students, one of his specialties. Here are my notes from the lecture:

Gifted students - presentation by Bharath Sriraman

How do we figure out if a student is gifted? Nature versus nurture - is it genetic, or is it due to upbringing. Why is it okay for a child to be talented in sports and not so much so in a subject like mathematics?

When it comes to funding, little money is spent on gifted education. (Less than 1% of the funding for special needs education - giftedness is viewed as a special need!)

In the U.S. there is an east versus west debate. Why are they doing so much better in the eastern systems? The western system is viewed as fostering creativity and freedom, but why is it that so many of the prodigies are from the eastern part of the world?

In the U.S., public schools are poorly funded, teachers are not held in high regard or paid well, etc.

Interesting fact:  U.S. has the highest prison population proportion in the western, developed world - 30% of the prisoners are high school dropouts.

In the Asian countries, there is a lot of focus on moral, hard work, perseverance, etc. Exams are very competitive! In the East, the point of an exam is to stratify the society. Late bloomers do not have a chance within the Eastern system! The U.S. (and Western) system, however, allows for a second chance.

As a teacher, there is always the potential conflict between equity and excellence! This could be seen as a false dichotomy! Alternative perspectives:

• The Hamilton tradition stressed elitism,
  
• whereas the Jacksonian tradition suggests that everyone is equal no matter what
• The Jeffersonian tradition stresses that you should give people equal opportunities, and then it is up to them to use these opportunities

Mathematical intelligence is considered as:

• a strong indicator of general intelligence
• numerical and spatial reasoning is part of the IQ score
• ...
  

The construct of intelligence is controversial in many ways since the psychometric testing. History of IQ-test: Binet-Simon tests --> Stanford-Binet tests, etc. The modern Stanford-Binet test is very much like a mathematical test, and you don't take into account socio-cultural and environmental variables. Still, it is very much used as a test for giftedness.

There are, however, some alternative views when it comes to discussing giftedness. Usiskin (Uni. Chicago) tried to classify the mathematical talent in the world in a hierarchy of Level 0 to Level 7.

• Level 0 - no talent. Adults who know very little mathematics
• Level 1 - culture level. Adults who have some number sense (comparable to grades 6-9), and they have learned it through usage
• Level 2 - represent the honors high school student
• Level 3 - the "terrific" student, those who score 750-800 on the SAT.
• Level 4 - the "exceptional" students, those who excel in math competitions
• Level 5 - represents the productive mathematician
• Level 6 - the exceptional mathematician
• Level 7 - the all-time greats, Fields medal winners in mathematics

Krutetskii's book is very nice - about gifted students!

Problem: a pole is 15 meters tall, another one is 10 meters tall. You have a rope from the top of one to the bottom of the other, and vice versa. How tall is the crossing point of the ropes from the ground?

There is a difference between Creativity and creativity (everyone has the latter, the former is related to being creative within a certain field).

There are lots of way to adapt the curriculum so that the gifted students get what they need.

Research shows that there are no harmful effect on early college admission - the students manage well, and they adapt well.

In the U.S. there is a lot of emphasis on the modeling-based curricula nowadays, and this gets a lot of funding. Several programs are made which are based on real-world situations. (one from Montana!)


After this interesting lecture, he gave a presentation of a new book that he has been editing together with Lyn English: Theories of Mathematics Education: Seeking new frontiers. The book is published by Springer, and has just been released. Bharath told that the book took him five years to finish, and it is definitely going to become an important contribution to our field!

Thanks a lot for the visit, Bharath, and for sharing this day with us! Hopefully, this is only going to be the first in a series of visits to Stavanger!

Mathematics curriculum: a vehicle for school improvement

Christian R. Hirsch and Barbara J. Reys have written an article entitled Mathematics curriculum: a vehicle for school improvement. This article was recently published online in ZDM. Here is a copy of their article abstract:

Different forms of curriculum determine what is taught and learned in US classrooms and have been used to stimulate school improvement and to hold school systems accountable for progress. For example, the intended curriculum reflected in standards or learning expectations increasingly influences how instructional time is spent in classrooms. Curriculum materials such as textbooks, instructional units, and computer software constitute the textbook curriculum, which continues to play a dominant role in teachers’ instructional decisions. These decisions influence the actual implemented curriculum in classrooms. Various curriculum policies, including mandated end-of-course assessments (the assessed curriculum) and requirements for all students to complete particular courses (e.g., year-long courses in algebra, geometry, and advanced algebra or equivalent integrated mathematics courses) are also being implemented in increasing numbers of states. The wide variation across states in their intended curriculum documents and requirements has led to a historic and precedent-setting effort by the Council of Chief State School Officers and the National Governors Association Council for Best Practices to assist states in the development and adoption of common College and Career Readiness Standards for Mathematics. Also under development by this coalition is a set of common core state mathematics standards for grades K-12. These sets of standards, together with advances in information technologies, may have a significant influence on the textbook curriculum, the implemented curriculum, and the assessed curriculum in US classrooms in the near future.

CAS calculators in algebra instruction

S. Aslι Özgün-Koca has written an article called Prospective teachers’ views on the use of calculators with Computer Algebra System in algebra instruction. This article has recently been published online in Journal of Mathematics Teacher Education. Here is the abstract of the article:

Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This study investigated the views of Turkish prospective secondary mathematics teachers on the use of advanced calculators with CAS in algebra instruction. An open-ended questionnaire and group interviews revealed prospective teachers’ views and beliefs about when and why they prefer three possible uses of CAS—black box, white box, or Symbolic Math Guide (SMG). The results showed that participants mainly preferred the white box methods and especially SMG to the black box method. They suggested that while the black box method could be used after students mastered the skills, the general white box method and SMG could be used to teach symbolic manipulation.

MTL, Volume 11, Issue 4

A new issue of Mathematical Thinking and Learning has appeared: Volume 11, Issue 4, 2009. The issue contains three research articles (along with an editorial by Lyn D. English and a book review):

• Learning Mathematics via a Problem-Centered Approach: A Two-Year Study, by Candice L. Ridlon 
• Efficacy of Different Concrete Models for Teaching the Part-Whole Construct for Fractions, by Kathleen Cramer; Terry Wyberg 
• Reasoning-and-Proving in School Mathematics Textbooks, by Gabriel J. Stylianides
  

Teachers' perceptions about the purpose of student teaching

Keith Leatham from Brigham Young University in Utah, U.S., is one of the scholars who have made important contribution to research of teachers' beliefs in mathematics education research in the last couple of years. I very much like his proposed framework for viewing teachers' beliefs as sensible systems (from his 2006 article in Journal of Mathematics Teacher Education). Now he has written a new article with focus on beliefs (or this time it is referred to as perceptions), and he has co-written this article with a colleague from Brigham Young University: Blake E. Peterson. Their article is entitled Secondary mathematics cooperating teachers’ perceptions of the purpose of student teaching, and it was published online in Journal of Mathematics Teacher Education last week. Here is their article abstract:

This article reports on the results of a survey of 45 secondary mathematics cooperating teachers’ perceptions of the primary purposes of student teaching and their roles in accomplishing those purposes. The most common purposes were interacting with an experienced, practising teacher, having a real classroom experience, and experiencing and learning about classroom management. The most common roles were providing the space for experience, modeling, facilitating reflection, and sharing knowledge. The findings provided insights into the cooperating teachers’ perceptions about both what should be learned through student teaching and how it should be learned. These findings paint a picture of cooperating teachers who do not see themselves as teacher educators—teachers of student teachers. Implications for mathematics teacher educators are discussed.

A case study in Rwanda

I haven't read many scientific articles in mathematics education from or about Rwanda, but here is one! Alphonse Uworwabayeho from Kigali Institute of Education in Rwanda, and University of Bristol, UK, has written an article entitled Teachers’ innovative change within countrywide reform: a case study in Rwanda. The article was published online in Journal of Mathematics Teacher Education on Wednesday. This is even an Open Access article, so everyone should have full access to it! Here is the abstract of the article:

This article presents practical perspectives on mathematics teacher change through results of collaborative research with two mathematics secondary school teachers in order to improve the teaching and learning of mathematics in Rwanda. The 2006 national mathematics curriculum reform stresses pedagogies that enhance problem-solving, critical thinking and argumentation. Teachers need to use new teaching strategies. This article is a case study looking at issues around developing teachers’ use of interactions in mathematics classrooms independently of the national programme. Outputs of the study include teachers’ awareness of the need for change and their increased flexibility to accept learners’ autonomy in shifting from teacher-centred to learner-centred pedagogy. Geometer’s Sketchpad challenged teachers’ practice and then provoked reflection to improve student learning.

Teachers' use of representation

An article called Prospective elementary teachers use of representation to reason algebraically has recently been published online in The Journal of Mathematical Behavior. The article was written by Kerri Richardson, Sarah Berenson and Katrina Staley. Here is the abstract of their article:

We used a teaching experiment to evaluate the preparation of preservice teachers to teach early algebra concepts in the elementary school with the goal of improving their ability to generalize and justify algebraic rules when using pattern-finding tasks. Nearly all of the elementary preservice teachers generalized explicit rules using symbolic notation but had trouble with justifications early in the experiment. The use of isomorphic tasks promoted their ability to justify their generalizations and to understand the relationship of the coefficient and y-intercept to the models constructed with pattern blocks. Based on critical events in the teaching experiment, we developed a scale to map changes in preservice teachers’ understanding. Features of the tasks emerged that contributed to this understanding.

Students’ perceived sociomathematical norms

Esther Levenson, Dina Tirosh and Pessia Tsamir (all from Tel Aviv University in Israel) have written an article that was recently published in The Journal of Mathematical Behavior. The article is entitled Students’ perceived sociomathematical norms: The missing paradigm. Here is the article abstract:

This study proposes a framework for research which takes into account three aspects of sociomathematical norms: teachers’ endorsed norms, teachers’ and students’ enacted norms, and students’ perceived norms. We investigate these aspects of sociomathematical norms in two elementary school classrooms in relation to mathematically based and practically based explanations. Results indicate that even when the observed enacted norms are in agreement with the teachers’ endorsed norms, the students may not perceive these same norms. These results highlight the need to consider the students’ perspective when investigating sociomathematical norms.

ESM - November issue

A new issue of Educational Studies in Mathematics has been released, Volume 72, Number 2 / November, 2009. This issue contains seven interesting articles:

• Using the onto-semiotic approach to identify and analyze mathematical meaning when transiting between different coordinate systems in a multivariate context, by Mariana Montiel, Miguel R. Wilhelmi, Draga Vidakovic and Iwan Elstak
• Changing practice, changing minds, from arithmetical to algebraic thinking: an application of the concerns-based adoption model (CBAM), by Jeanne Tunks and Kirk Weller
• Conditional inference and advanced mathematical study: further evidence, by Matthew Inglis and Adrian Simpson
• Didactical designs for students’ proportional reasoning: an “open approach” lesson and a “fundamental situation”, by Takeshi Miyakawa and Carl Winsløw
• Bridging the macro- and micro-divide: using an activity theory model to capture sociocultural complexity in mathematics teaching and its development, by Barbara Jaworski and Despina Potari
• Proof constructions and evaluations, by Andreas J. Stylianides and Gabriel J. Stylianides
• Researchers’ descriptions and the construction of mathematical thinking, by Richard Barwell
  

Maths week in Ireland

The 4th annual Maths Week Ireland takes place this week, and it is presented as an "all island celebration of mathematics" at the official website. The Irish Times published an interesting article about this yesterday. There are several activities going on during the week, and it is all "timed to coincide with the annual Hamilton Day" on October 16th. William Rowan Hamilton (1805-1865) was one of Ireland's most famous mathematicians, and he was also a physicist and astronomer.

Make sure to check out the official website for a list of events!

Sudoku: Strategy versus structure

"You don't have to know any mathematics to solve a Sudoku puzzle," one of the major Norwegian newspaper claimed when the Sudoku wave started a couple of years ago. I have always disagreed strongly with this, and it seems that Provan J. Scott, who has written an article about this in American Mathematical Monthly, has a similar opinion. The article is entitled Sudoku: strategy versus structure. Here is the article abstract:

Sudoku puzzles, and their variants, have become extremely popular in the last decade. They can now be found in major U.S. newspapers, puzzle books, and web sites; almost as pervasive are the many guides to Sudoku strategy and logic. We give a class of solution strategies-encompassing a dozen or so differently named solution rules found in these guides-that is at once simple, popular, and powerful. We then show the relationship of this class to the modeling of Sudoku puzzles as assignment problems and as unique nonnegative solutions to linear equations. The results provide excellent applications of principles commonly presented in introductory classes in finite mathematics and combinatorial optimization, and point as well to some interesting open research problems in the area.

Teachers' difficulties during problem-solving instruction

Alexander Karp has written an article that was published online in Journal of Mathematics Teacher Education on Saturday. The article is called Analyzing and attempting to overcome prospective teachers’ difficulties during problem-solving instruction. Here is the abstract of Karp's article:

This article analyzes the experiences of prospective secondary mathematics teachers during a teaching methods course, offered prior to their student teaching, but involving actual teaching and reflexive analysis of this teaching. The study focuses on the pedagogical difficulties that arose during their teaching, in which prospective teachers lacked pedagogical content knowledge and skills. It also analyzes the experience of the course itself, which was aimed at scaffolding the work of prospective teachers on developing their pedagogical content knowledge and skills.

Curriculum research to improve teaching and learning

Gerald Kulm and Yeping Li have written an article entitled Curriculum research to improve teaching and learning: national and cross-national studies. The article was recently published online in ZDM. Here is the abstract of their article:

Curriculum, as a cultural and system-specific artifact, outlines mathematics teaching and learning activities in school education. Studies of curriculum and its changes are thus important to reveal the expectations, processes and outcomes of students’ school learning experiences that are situated in different cultural and system contexts. In this article, we aim to propose a framework that can help readers to develop a better understanding of curriculum practices and changes in China and/or the USA that have been reported and discussed in articles published in this themed issue. Going beyond the selected education systems, further studies of curriculum practices and changes are much needed to help ensure the success of educational reforms in the different cultural and system contexts.

Mathematics instruction for students with learning disabilities

Russel Gersten and colleagues have written an article called Mathematics Instruction for Students With Learning Disabilities: A Meta-Analysis of Instructional Components. This article was published in the recent issue of Review of Educational Research. Here is the abstract of their article:

The purpose of this meta-analysis was to synthesize findings from 42 interventions (randomized control trials and quasi-experimental studies) on instructional approaches that enhance the mathematics proficiency of students with learning disabilities. We examined the impact of four categories of instructional components: (a) approaches to instruction and/or curriculum design, (b) formative assessment data and feedback to teachers on students' mathematics performance, (c) formative data and feedback to students with LD on their performance, and (d) peer-assisted mathematics instruction. All instructional components except for student feedback with goal-setting and peer-assisted learning within a class resulted in significant mean effects ranging from 0.21 to 1.56. We also examined the effectiveness of these components conditionally, using hierarchical multiple regressions. Two instructional components provided practically and statistically important increases in effect size–teaching students to use heuristics and explicit instruction. Limitations of the study, suggestions for future research, and applications for improvement of current practice are discussed.

Multiple solution methods and multiple outcomes

Pessia Tsamir, Dina Tirosh, Michal Tabach and Esther Levenson have written an article about Multiple solution methods and multiple outcomes—is it a task for kindergarten children? This article was recently published online in Educational Studies in Mathematics. Here is a copy of their article abstract:

Engaging students with multiple solution problems is considered good practice. Solutions to problems consist of the outcomes of the problem as well as the methods employed to reach these outcomes. In this study we analyze the results obtained from two groups of kindergarten children who engaged in one task, the Create an Equal Number Task. This task had five possible outcomes and five different methods which may be employed in reaching these outcomes. Children, whose teachers had attended the program Starting Right: Mathematics in Kindergartens, found more outcomes and employed more methods than children whose teachers did not attend this program. Results suggest that the habit of mind of searching for more than one outcome and employing more than one method may be promoted in kindergarten.

How syntactic reasoners can develop understanding

Keith Weber has written an article that was recently published in The Journal of Mathematical Behavior. The article is entitled How syntactic reasoners can develop understanding, evaluate conjectures, and generate counterexamples in advanced mathematics. Here is the abstract of Weber's article:

This paper presents a case study of a highly successful student whose exploration of an advanced mathematical concept relies predominantly on syntactic reasoning, such as developing formal representations of mathematical ideas and making logical deductions. This student is observed as he learns a new mathematical concept and then completes exercises about it. The paper focuses on how Isaac developed an understanding of this concept, how he evaluated whether a mathematical assertion is true or false, how he generated counterexamples to disprove a statement, and the general role examples play for him in concept development and understanding.

What's sophisticated about elementary mathematics?

The current issue of American Educator includes a very interesting article about mathematics teaching in the elementary grades. The article is entitled What's Sophisticated about Elementary Mathematics? Plenty—That's Why Elementary Schools Need Math Teachers, and it is written by Hung-Hsi Wu, who is professor emeritus of mathematics at UC Berkeley. I should probably say a lot more about him, because his list of merits is impressive, but I will leave it to you to find out more.

It appears to be a rather common impression that teaching elementary mathematics is ... well, rather elementary. I mean, the mathematics is quite simple, so how hard can it be? In this article, Wu provides a very nice introduction to how challenging it can actually be. In the introductory part of the article, he claims: "The fact is, there's a lot more to teaching math than teaching how to do calculations." In the article, he provides examples of how hard it can actually be to teach something as "elementary" as place value and fractions.

I am tempted to quote more or less the entire article, because so many interesting issues are presented here, but I will not. I am, however, going to recommend that you take the time and read this excellent article. If you are somewhat interested in teaching mathematics, I am sure you will find this interesting!

Thanks a lot to Assistant Editor Jennifer Dubin for telling me about this article, by the way! I appreciate it :-)

Developing school mathematics textbooks in China

Yeping li, Jianyue Zhang and Tingting Ma have written an article entitled Approaches and practices in developing school mathematics textbooks in China. This article was published online in ZDM a couple of days ago. This article gives a very interesting overview of how Chinese textbooks are developed, but it also provides nice insights into how mathematics teaching has developed (or is developing) in China. Here is the abstract of this article:

In this study, we aim to examine and discuss approaches and practices in developing mathematics textbooks in China, with a special focus on the development of secondary school mathematics textbook in the context of recent school mathematics reform. Textbook development in China has its own history. This study reveals some common practices and approaches developed and used in selecting, presenting and organizing content in mathematics textbooks over the years. With the recent curriculum reform taking place in China, we also discuss some new developments in compiling and publishing high school mathematics textbooks. Implications obtained from Chinese practices in textbook development are then discussed in a broad context.

The productive notion of mathematics laboratories

Michela Maschietto and Luc Trouche have written an article called Mathematics learning and tools from theoretical, historical and practical points of view: the productive notion of mathematics laboratories. The article was published online in ZDM on Monday. Although they start with a glimpse from a babylonian clay tablet, their main focus is on the development of tools and use of tools in the last century. In the main part of their article, they have a strong focus on the so-called mathematics laboratories. Here is the abstract of their article:

In our research work, we have looked at the way in which artefacts become, for teachers as well as for students, instruments of their mathematical activity. The issues related to the use of tools and technologies in mathematical education are now widely considered. A look to history highlights the different ways in which the same questions have been studied at different times and in different places. This suggests that the contribution of artefacts to mathematics learning should be considered in terms of various contexts. Our “visits” to these contexts will be guided by the coordination of two main theoretical frameworks, the instrumental approach and the semiotic mediation approach from the perspective of mathematics laboratory. This journey through history and schooling represents a good occasion to address some questions: Are there “good” contexts in which to develop mathematical instruments? Are there “good” teaching practices which assist students’ instrumental geneses and construct mathematical meanings? How is it possible to promote such teaching practices? Some study cases are discussed.

IJSME, October 2009

A new issue of International Journal of Science and Mathematics Education has been published, Volume 7, Number 5, October 2009. The issue contains nine articles, several of which are related to mathematics education:

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

Addition and subtraction of three-digit numbers

Aiso Heinze, Franziska Marschick and Frank Lipowsky have written an article that was published in the recent issue of ZDM. The article is entitled Addition and subtraction of three-digit numbers: adaptive strategy use and the influence of instruction in German third grade. Here is the abstract of their article:

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