JMTE, April 2010

The April issue of Journal of Mathematics Teacher Education has recently been published. This issue contains 6 interesting articles:

• Crossing the divide: reflecting on the benefits of international collaboration, by Anne D. Cockburn
• Secondary mathematics cooperating teachers’ perceptions of the purpose of student teaching, by Keith R. Leatham and Blake E. Peterson
• Analyzing and attempting to overcome prospective teachers’ difficulties during problem-solving instruction, by Alexander Karp
• Developing teachers’ knowledge of students as learners of mathematics through structured interviews, by Oliver F. Jenkins
• The influence of video clubs on teachers’ thinking and practice, by Elizabeth A. van Es and Miriam Gamoran Sherin
• Prospective primary mathematics teachers’ learning from on-line discussions in a virtual video-based environment, by Salvador Llinares and Julia Valls

The article by van Es and Sherin is an Open Access article, so that one should be available even for non-subscribers!

New issue of IJMEST, vol 41, issue 3, 2010

A new issue of International Journal of Mathematical Education in Science and Technology has been released, and it contains a whole host of interesting articles and classroom notes. This issue appears to have a particular focus on the use of technology in mathematics teaching, and here is a list of the original articles that are contained in the issue:

• Sustainable student retention and gender issues in mathematics for ICT study, by Blazenka Divjak, Mirela Ostroski and Violeta Vidacek Hains
  
• A proposal for a variation on the axioms of classical geometry, by Bjorn Schellenberg
  
• Trends in performance of science and technology students (1997–2008) in Ireland, by Olivia Gill, John O’Donoghue, Fiona Faulkner and Ailish Hannigan
  
• Relational understanding and paths of reasoning through a Boolean lattice classification of quadrilaterals, by J. Karakonstantis and T. Patronis
  
• Using dynamic geometry to explore non-traditional theorems, by Arsalan Wares
  
• A new approach for proving or generating combinatorial identities, by Luis González
  

Review of my blog

After a slow month (on the blog - not at work!), it is great to see someone writing a nice review of this blog. This time it is Jerry Johnson from MathNEXUS who has written a very kind review. MathNEXUS is a web site particularly geared towards teachers of mathematics, and it presents itself as a mathematics portal with "news and ideas for teachers and learners of mathematics. So, if you're into teaching and/or learning of mathematics it might be worthwhile to check it out!

Thanks for the kind words, Jerry :-)

Appropriating geometric series as a cultural tool

A couple of new articles have been published online in Educational Studies in Mathematics lately, amongst those a very interesting one by my good colleague Martin Carlsen from the University of Agder, Norway. His article is entitled: Appropriating geometric series as a cultural tool: a study of student collaborative learning. Carlsen, along with other colleagues in Agder, have been influenced by the focus on small-group problem solving that was advocated by Neil Davidson and others some years ago. The Agder group is also strongly influenced by theories related to sociocultural perspectives of teaching and learning mathematics, and this article provides a nice overview of some of these theoretical foundations. The research reported in this article can be placed within a qualitative, naturalistic paradigm, and the data were analyzed using a dialogical approach (Carlsen here makes use of a framework developed by two other colleagues: Maria-Luiza Cestari and Raymond Bjuland). So, if you are interested in any of the perspectives referred to above, this article should be highly relevant for you! Here is the abstract of the article:

The aim of this article is to illustrate how students, through collaborative small-group problem solving, appropriate the concept of geometric series. Student appropriation of cultural tools is dependent on five sociocultural aspects: involvement in joint activity, shared focus of attention, shared meanings for utterances, transforming actions and utterances and use of pre-existing cultural knowledge from the classroom in small-group problem solving. As an analytical point of departure, four mathematical theoretical components are identified when appropriating the cultural tool of geometric series: (1) estimating of parameters, (2) establishing of the general term, (3) composing of the sum and (4) deciding on convergence. Analyses of five excerpts focused on the students’ social processes of knowledge objectification and the corresponding semiotic means, i.e., lecture notes, linguistic devices, gestures, head movements and gaze, to obtain shared foci and meanings. The investigation of these processes unveils the manner in which the students established links to pre-existing mathematical knowledge in the classroom and how they simultaneously combined the various mathematical theoretical components that go into appropriating the cultural tool of geometric series. From the excerpts, it is evident that the students’ participation changes throughout their involvement in the problem-solving process. The students are gaining mathematical knowing through a process of transforming and by establishing shared meanings for the concept and its theoretical components.

Exploring kindergarten teachers' pedagogical content knowledge of mathematics

Last week, an interesting article was published online in the International Journal of Early Childhood. The article is entitled Exploring Kindergarten Teachers’ Pedagogical Content Knowledge of Mathematics, and it has been written by Joohl Lee. The combination of teachers knowledge of mathematics and kindergarten is very interesting, and while a lot of research has been done to learn more about the type of knowledge mathematics  teachers need in school, little has been done to learn more about this in kindergarten. This is also mentioned by Lee in the article. As the title of the article reveals, Lee builds upon Shulman's traditional framework of teachers' professional knowledge. What I don't understand, however, is how it is possible to write an article about teachers' pedagogical content knowledge of mathematics without making any reference to the MKT (Mathematical Knowledge for Teaching) framework, or any of the work done by Deborah Ball and her colleagues at the University of Michigan. I understand that this article has a focus on kindergarten, but still... I also think there should be some mention of how the teachers in the study were selected. 81 kindergarten teachers were assessed in the study, and 55% of these had a master's degree. I would like to know more about how representative this sample was. Still, I think it is an interesting article, and I think it is a good thing that the issue of kindergarten teachers' knowledge of mathematics is addressed.

Here is the abstract of the article:

The purpose of this study was to assess 81 kindergarten teachers’ pedagogical content knowledge of mathematics on six subcategory areas such as number sense, pattern, ordering, shapes, spatial sense, and comparison. The data showed participants possessed a higher level of pedagogical content knowledge of “number sense” (M = 89.12) compared to other mathematics pedagogical content areas. The second highest scores among six subcategories of pedagogical content knowledge of mathematics was for the pedagogical content area of “pattern” (M = 82.33). The lowest scores among those six subcategories of kindergarten teachers’ pedagogical content knowledge were obtained from the subcategory of “spatial sense” (M = 44.23), which involved the means to introduce children to spatial relationships. The second lowest score was obtained for the subcategory of “comparison” (M = 50.40) which involved the means to introduce the concept of graphing and the use of a balance scale for measurement.

Teachers attending to students' reasoning

John M. Francisco and Carolyn A. Maher have written an article about Teachers attending to students’ mathematical reasoning: lessons from an after-school research program. This article was published online in Journal of Mathematics Teacher Education last Thursday. This article is interesting in several respects, amongst others because awareness of and knowledge about students' mathematical reasoning is something teachers need, and it should be part of every mathematics teacher's professional knowledge. Theoretically, it builds upon Shulman's classic framework, but they also make interesting links to a focus on practitioner-researcher collaboration. The article reports on a study that was made of "elementary and middle school teachers who participated as interns in the 1-year NSF-funded Informal Mathematical Learning Project (IML)". Here is a copy of the abstract of their article:

There is a documented need for more opportunities for teachers to learn about students’ mathematical reasoning. This article reports on the experiences of a group of elementary and middle school mathematics teachers who participated as interns in an after-school, classroom-based research project on the development of mathematical ideas involving middle-grade students from an urban, low-income, minority community in the United States. For 1 year, the teachers observed the students working on well-defined mathematical investigations that provided a context for the students’ formation of particular mathematical ideas and different forms of reasoning in several mathematical content strands. The article describes insights into students’ mathematical reasoning that the teachers were able to gain from their observations of the students’ mathematical activity. The purpose is to show that teachers’ observations of students’ mathematical activity in research sessions on students’ development of mathematical ideas can provide opportunities for teachers to learn about students’ mathematical reasoning.

Busy days...

The last couple of weeks have been extremely busy - for many reasons - and I haven't been able to follow up on all the latest articles and news in the field. I apologize for this, and I hope that all the readers of the blog have patience with me! I promise that I will catch up :-)

In the meantime, you always have a couple of good options in order to stay really up-to-date:

• Pay attention to my Google Reader blog (shared items)
• Follow me on twitter

These two sites are a bit easier for me to update, and when I don't manage to update my blog as often as I would, I will probably continue to push new updates to these two other services. Still, as soon as I get my head above water again, I will keep providing you with information here as well :-)

Mathematically based and practically based explanations

Esther Levenson, Pessia Tsamir and Dina Tirosh have written an interesting article about Mathematically based and practically based explanations in the elementary school: teachers’ preferences. Their article was published online in Journal of Mathematics Teacher Education on Friday. In this article, the authors make interesting connnections between research on teachers' knowledge and beliefs. Although their focus is on knowledge and beliefs in relation to the use of explanations (and they distinguish between mathematically and practically based explanations) in the classroom, the article makes a nice contribution to extending our understanding of the way these concepts are related. The part of teachers' knowledge (and beliefs) that the authors discuss is related to students' thinking, or even a sub-category of that. In this respect, they make valuable contributions to what Deborah Ball and her colleagues refer to as Knowledge of Content and Students, but their focus is also in the borderline of what is referred to as Knowledge of Content and Teaching. The links to research concerning teachers' beliefs is also interesting, althought the authors don't go into great detail here. They are, of course, aware of this, and explain that they have only given "a glimpse into the complexity of the relationship between teachers' knowledge and beliefs", in particular with focus on teachers' use of explanations.

Here is the abstract of their article:

This article focuses on elementary school teachers’ preferences for mathematically based (MB) and practically based (PB) explanations. Using the context of even and odd numbers, it explores the types of explanations teachers generate on their own as well as the types of explanations they prefer after reviewing various explanations. It also investigates the basis for these preferences. Results show that teacher-generated explanations include more MB explanations than PB explanations. However, many still choose to use mostly PB explanations in their classrooms, believing that these explanations will be most convincing to their students. The implications for teacher education are discussed.

ZDM, February, 2010

A new issue of ZDM - The International Journal on Mathematics Education has been published. This issue has a special focus on Historical aspects of the use of technology and devices in ICMEs and ICMI, and guest editors have been Maria G. Bartolini Bussi and Marcelo C. Borba. The issue contains 15 articles which covers many aspects of the theme, from historical perspectives on the use of technology to a social perspective on technology enhanced learning. Here is an overview of the contents of this issue:

• The role of resources and technology in mathematics education, by Maria G. Bartolini Bussi and Marcelo C. Borba
• Historical comments on the use of technology and devices in ICMEs and ICMI, by Gert Schubring
• Exploration of technologies, emerging from African cultural practices, in mathematics (teacher) education, by Paulus Gerdes
• Concrete models and dynamic instruments as early technology tools in classrooms at the dawn of ICMI: from Felix Klein to present applications in mathematics classrooms in different parts of the world, by Maria G. Bartolini Bussi, Daina Taimina and Masami Isoda
• Mathematics learning and tools from theoretical, historical and practical points of view: the productive notion of mathematics laboratories, by Michela Maschietto and Luc Trouche
• Collectives of humans-with-media in mathematics education: notebooks, blackboards, calculators, computers and … notebooks throughout 100 years of ICMI, by Mónica E. Villarreal and Marcelo C. Borba
• Charting the microworld territory over time: design and construction in mathematics education, by Lulu Healy and Chronis Kynigos
• Graphic calculators and connectivity software to be a community of mathematics practitioners, by Ornella Robutti
• A social perspective on technology-enhanced mathematical learning: from collaboration to performance, by George Gadanidis and Vince Geiger
• Integrating technology into mathematics teaching at the university level, by Zsolt Lavicza
• Place and use of new technology in the teaching of mathematics: ICMI activities in the past 25 years, by Colette Laborde and Rudolf Sträßer
• Clarkson, P., Presmeg, N. (eds) (2008): Critical Issues in Mathematics Education: Major Contributions of Alan Bishop, Springer, New York, 257 pp., ISBN 978-0-387-09672-8, by Irit Peled
• Acknowledgements to reviewers 2009, by Gabriele Kaiser
• Acknowledgments to the members of the Editorial Board, by Gabriele Kaiser
• Advances in mathematics education: new book series connected to ZDM—The International Journal on Mathematics Education, by Gabriele Kaiser and Bharath Sriraman
  

Combining constructions of knowledge

Ivy Kidron and Tommy Dreyfus have written an article entitled Justification enlightenment and combining constructions of knowledge. The article was published online in Educational Studies in Mathematics last week. Here is a copy of the abstract of their article:

This case study deals with a solitary learner’s process of mathematical justification during her investigation of bifurcation points in dynamic systems. Her motivation to justify the bifurcation points drove the learning process. Methodologically, our analysis used the nested epistemic actions model for abstraction in context. In previous work, we have shown that the learner’s attempts at justification gave rise to several processes of knowledge construction, which develop in parallel and interact. In this paper, we analyze the interaction pattern of combining constructions and show that combining constructions indicate an enlightenment of the learner. This adds an analytic dimension to the nested epistemic actions model of abstraction in context.