The Language of Mathematics

Bill Barton has written a book called The Language of Mathematics, which has been published by Springer recently. The connection between mathematics and language has been discussed a lot by others before, and this appears to be a nice contribution to this discussion. The book is written for researchers, graduate students and teachers of mathematics education. Unfortunately, I haven't got this book myself (yet), so I can only provide you with a copy of the publisher's description of it:

The Language of Mathematics: Telling Mathematical Tales emerges from several contemporary concerns in mathematics, language, and mathematics education, but takes a different stance with respect to language. Rather than investigating the way language or culture impacts mathematics and how it is learned, this book begins by examining different languages and how they express mathematical ideas. The picture of mathematics that emerges is of a subject that is much more contingent, relative, and subject to human experience than is usually accepted. Barton’s thesis takes the idea of mathematics as a human creation, and, using the evidence from language, comes to more radical conclusions than usual.

Everyday mathematical ideas are expressed quite differently in different languages. Variety occurs in the way languages express numbers, describe position, categorise patterns, as well as in the grammar of mathematical discourse. The first part of The Language of Mathematics: Telling Mathematical Tales explores these differences and thus illustrates the possibility of different mathematical worlds. This section both provides evidence of language difference with respect to mathematic talk and also demonstrates the congruence between mathematics as we know it and the English language. Other languages are not so congruent.

Part II discusses what this means for mathematics and argues for alternative answers to conventional questions about mathematics: where it comes from, how it develops, what it does and what it means. The notion that mathematics is the same for everyone, that it is an expression of universal human thought, is challenged. In addition, the relationship between language and mathematical thought is used to argue that the mathematical creativity embedded in minority languages should continue to be explored

The final section explores implications for mathematics education, discussing the consequences for the ways in which we learn and teach mathematics. The Language of Mathematics: Telling Mathematical Tales will appeal to those interested in exploring the nature of mathematics, mathematics educators, researchers and graduate students of mathematics education.

Challenging Mathematics in and Beyond the Classroom

Springer has published a new book related to mathematics education. The book has been entitled Challenging Mathematics In and Beyond the Classroom, and it is edited by Edward J. Barbeau and Peter J. Taylor. Here is a copy of the publisher's description of the book:

The last two decades have seen significant innovation both in classroom teaching and in the public presentation of mathematics. Much of this has centered on the use of games, puzzles and investigations designed to capture interest, challenge the intellect and encourage a more robust understanding of mathematical ideas and processes. ICMI Study 16 was commissioned to review these developments and describe experiences around the globe in different contexts, systematize the area, examine the effectiveness of the use of challenges and set the stage for future study and development. A prestigious group of international researchers, with collective experience with national and international contests, classroom and general contests and in finding a place for mathematics in the public arena, contributed to this effort. The result, Challenging Mathematics In and Beyond the Classroom, deals with challenges for both gifted as regular students, and with building public interest in appreciation of mathematics.

ESM, April 2009

The April issue of Educational Studies in Mathematics has been published, and it contains five articles (including a book review):

• The array representation and primary children’s understanding and reasoning in multiplication, by Patrick Barmby, Tony Harries, Steve Higgins and Jennifer Suggate'. Abstract:  We examine whether the array representation can support children’s understanding and reasoning in multiplication. To begin, we define what we mean by understanding and reasoning. We adopt a ‘representational-reasoning’ model of understanding, where understanding is seen as connections being made between mental representations of concepts, with reasoning linking together the different parts of the understanding. We examine in detail the implications of this model, drawing upon the wider literature on assessing understanding, multiple representations, self explanations and key developmental understandings. Having also established theoretically why the array representation might support children’s understanding and reasoning, we describe the results of a study which looked at children using the array for multiplication calculations. Children worked in pairs on laptop computers, using Flash Macromedia programs with the array representation to carry out multiplication calculations. In using this approach, we were able to record all the actions carried out by children on the computer, using a recording program called Camtasia. The analysis of the obtained audiovisual data identified ways in which the array representation helped children, and also problems that children had with using the array. Based on these results, implications for using the array in the classroom are considered.
  
• Social constructivism and the Believing Game : a mathematics teacher’s practice and its implications, by Shelly Sheats Harkness. Abstract:  The study reported here is the third in a series of research articles (Harkness, S. S., D’Ambrosio, B., & Morrone, A. S.,in Educational Studies in Mathematics 65:235–254, 2007; Morrone, A. S., Harkness, S. S., D’Ambrosio, B., & Caulfield, R. in Educational Studies in Mathematics 56:19–38, 2004) about the teaching practices of the same university professor and the mathematics course, Problem Solving, she taught for preservice elementary teachers. The preservice teachers in Problem Solving reported that they were motivated and that Sheila made learning goals salient. For the present study, additional data were collected and analyzed within a qualitative methodology and emergent conceptual framework, not within a motivation goal theory framework as in the two previous studies. This paper explores how Sheila’s “trying to believe,” rather than a focus on “doubting” (Elbow, P., Embracing contraries, Oxford University Press, New York, 1986), played out in her practice and the implications it had for both classroom conversations about mathematics and her own mathematical thinking.
  
• Investigating imagination as a cognitive space for learning mathematics, by Donna Kotsopoulos and Michelle Cordy. Abstract:  Our work is inspired by the book Imagining Numbers (particularly the square root of minus fifteen), by Harvard University mathematics professor Barry Mazur (Imagining numbers (particularly the square root of minus fifteen), Farrar, Straus and Giroux, New York, 2003). The work of Mazur led us to question whether the features and steps of Mazur’s re-enactment of the imaginative work of mathematicians could be appropriated pedagogically in a middle-school setting. Our research objectives were to develop the framework of teaching mathematics as a way of imagining and to explore the pedagogical implications of the framework by engaging in an application of it in middle school setting. Findings from our application of the model suggest that the framework presents a novel and important approach to developing mathematical understanding. The model demonstrates in particular the importance of shared visualizations and problem-posing in learning mathematics, as well as imagination as a cognitive space for learning.
  
• Teachers’ perspectives on “authentic mathematics” and the two-column proof form, by Michael Weiss, Patricio Herbst and Chialing Chen. Abstract:  We investigate experienced high school geometry teachers’ perspectives on “authentic mathematics” and the much-criticized two-column proof form. A videotaped episode was shown to 26 teachers gathered in five focus groups. In the episode, a teacher allows a student doing a proof to assume a statement is true without immediately justifying it, provided he return to complete the argument later. Prompted by this episode, the teachers in our focus groups revealed two apparently contradictory dispositions regarding the use of the two-column proof form in the classroom. For some, the two-column form is understood to prohibit a move like that shown in the video. But for others, the form is seen as a resource enabling such a move. These contradictory responses are warranted in competing but complementary notions, grounded on the corpus of teacher responses, that teachers hold about the nature of authentic mathematical activity when proving.
  
• Book Review: The beautiful Monster by Mark Ronan (2006), Symmetry and the Monster, one of the greatest quests of mathematics. New York: Oxford University Press, 255 pp. ISBN 978-0-19-280723-6 £8.99 RRP

More about the Abel Prize winner

A few days ago, I wrote about the winner of this year's Abel Prize: Mikail Gromov. Since then, a couple of other posts or articles have been published about this. The Chronicle of Higher Education published a small article about it. Today, a very interesting article was posted by someone who would be able to understand Gromov's achievements better than most of us: Terence Tao. Professor Tao points to earlier articles he has written about Gromov's theories, and he presents one of Gromov's results along with a sketch of the original proof. So, if your mathematical skills are somewhat above average, you might be interested in taking a closer look at this :-)

Modes of reasoning

Kaye Stacey and Jill Vincent has written an article about Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks. This article was published online in Educational Studies in Mathematics a few days ago. Here is the abstract of their article:

Understanding that mathematics is founded on reasoning and is not just a collection of rules to apply is an important message to convey to students. Here we examined the reasoning presented in seven topics in nine Australian eighth-grade textbooks. Focusing on explanatory text that introduced new mathematical rules or relationships, we classified explanations according to the mode of reasoning used. Seven modes were identified, making a classification scheme which both refined and extended previous schemes. Most textbooks provided explanations for most topics rather than presenting “rules without reasons” but the main purpose appeared to be rule derivation or justification in preparation for practise exercises, rather than using explanations as thinking tools. Textbooks generally did not distinguish between the legitimacies of deductive and other modes of reasoning.

The Abel Prize 2009 - Mikhail Gromov

Russian mathematician Mikhail Gromov has been awarded the 2009 Abel Prize. The announcement of the first Abel Prize was made in 2002, in connection with the 200th anniversary of Norwegian mathematician Niels Henrik Abel's birth. Although the prize has a fairly recent history, it is already called "The Mathematicians' Nobel". The idea of having an annual mathematics prize like this was proposed as early as in 1899 by Norwegian mathematician Sophus Lie, when it was made clear that there would be no Nobel prize in mathematics. These early attempts ended for several reasons, amongst others because Sophus Lie himself died in this same year (1899), and the dissolution of the union between Sweden and Norway in 1905 also made it difficult to create such a prize.

Mikhail Gromov (born 1943) was announced as this year's winner today, by the President of the Norwegian Academy of Science and Letters, Øyvind Østerud. Gromov will receive the prize from His Majesty King Harald in a ceremony in Oslo, May 19. The prize carries a cash award of NOK 6,000,000 (about USD 950,000). Gromov was given the prize because of his revolutionary contributions to the field of geometry.

My guess is by the way, that the rather small Wikipedia article about Gromov will increase in the next couple of days :-)

Sources:
http://www.abelprisen.no/en/'
http://en.wikipedia.org/wiki/Mikhail_Gromov
http://en.wikipedia.org/wiki/Abel_prize

Dynamic graphs and student reasoning

Marshall Lassak has written an article about Using dynamic graphs to reveal student reasoning. This article was published earlier this month in International Journal of Mathematical Education in Science and Technology. Here is the (rather short) abstract of the article:

Using dynamic graphs, future secondary mathematics teachers were able to represent and communicate their understanding of a brief mathematical investigation in a way that a symbolic proof of the problem could not. Four different student work samples are discussed.

Histograms in teacher training

A. Bruno and M.C. Espinel have written an article called Construction and evaluation of histograms in teacher training. The article was published in International Journal of Mathematical Education in Science and Technology a couple of days ago. Their study shows, among other things, that students confuse histograms with bar diagrams. Here is their abstract:

This article details the results of a written test designed to reveal how education majors construct and evaluate histograms and frequency polygons. Included is a description of the mistakes made by the students which shows how they tend to confuse histograms with bar diagrams, incorrectly assign data along the Cartesian axes and experience difficulties in constructing the frequency polygon.

Epistemological beliefs

Dena L. Wheeler and Diane Montgomery have written an article about college students' epistemological beliefs. The article that is entitled Community college students’ views on learning mathematics in terms of their epistemological beliefs: a Q method study was published online in Educational Studies in Mathematics on Tuesday. Here is the abstract of their article:

The purpose of this study was to explore the views of students enrolled at a small United States Midwestern community college toward learning mathematics, and to examine the relationship between student beliefs about mathematic learning and educational experiences with mathematics using Q methodology and open-ended response prompts. Schommer’s (Journal of Educational Psychology, 82, 495–504, 1990) multidimensional theory of personal epistemology provided the structural framework for the development of 36 domain specific Q sort statements. Analysis of the data revealed three distinct but related views of learning mathematic which were labeled Active Learners, Skeptical Learners, and Confident Learners. Chi-square tests of independence revealed no significant differences based on gender. Additionally, there was no evidence for differences based on level of mathematics completed, age, or college hours accumulated. Student’s previous experiences in instructional environments, however, were closely associated with beliefs. Results are discussed in view of the implications for establishing learning environments and considerations in implementing Standards-based curricula in higher education.

Catwalk problems

Three articles have been published in The Journal of Mathematical Behavior recently that are all related to "the catwalk task".

1. Steven Case: The catwalk task: Reflections and synthesis: Part 1
Abstract: In this article I recount my experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, I reflect on my own mathematical work on the catwalk task, including my efforts to fit various algebraic models to the data. Second, I reflect on my experiences working with a group of high school students on the catwalk task and my interpretations of their mathematical thinking. Finally, I reflect on the entire experience with the catwalk problem, as a mathematics learner, as a teacher, and as a professional.

2. Emiliano Vega and Shawn Hicks: The catwalk task: Reflections and synthesis: Part 2
Abstract: In this article we recount our experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, we individually reflect on our own mathematical work on the catwalk task. Second, we reflect on our experiences working with a group of community college students on the catwalk task and our interpretations of their mathematical thinking. In so doing we also detail a number of innovative and novel student-generated representations of the catwalk photos. Finally, we each individually reflect on the entire experience with the catwalk problem, as mathematics learners, as teachers, and as professionals.

3. Chris Rasmussen: Multipurpose Professional Growth Sequence: The catwalk problem as a paradigmatic example
Abstract: An important concern in mathematics teacher education is how to create learning opportunities for prospective and practicing teachers that make a difference in their professional growth as educators. The first purpose of this article is to describe one way of working with prospective and practicing teachers in a graduate mathematics education course that holds promise for positively influencing the way teachers think about mathematics, about student learning, and about mathematics teaching. Specifically, I use the “catwalk” task as an example of how a single problem can serve as the basis for a coherent sequence of professional learning experiences. A second purpose of this article is to provide background information that contextualizes the subsequent two articles, each of which details the positive influence of the catwalk task sequence on the authors’ professional growth.

So, you may ask, what is this catwalk problem really about then? The problem is originated in a set of 24 time-lapse photographs of a running cat. The question is simply: How fast is the cat moving at frame 10? Frame 20? (See this pdf for a presentation of the problem!)

NOMAD, December 2008

The December issue of Nordic Studies in Mathematics Education (NOMAD) has already reached the subscribers (in the paper format). Now, it has also appeared online - or at least the abstracts. Here is the list of contents:

• Morten Blomhøj and Paola Valero: Bringing focus to mathematics education in multicultural and multilingual settings (Editorial)
• Kay Owens: Culturality in mathematics education: a comparative study
• Eva Norén: Bilingual students’ mother tongue: a resource for teaching and learning mathematics
• Troels Lange: Homework and minority students in difficulty with learning mathematics: the influence of public discourse
• Paola Valero, Tamsin Meaney, Helle Alrø, Uenuku Fairhall, Ole Skovsmose and Tony Trinick: School mathematical discourse in a learning landscape: understanding mathematics education in multicultural settings
• Barbro Grevholm: Activities for 2009 in the Nordic Graduate School in Mathematics Education
  

GeoGebra - freedom to explore and learn

Linda Fahlberg-Stojanovska and Vitomir Stojanovski have written an article called GeoGebra - freedom to explore and learn. The article was recently published in Teaching Mathematics and its Applications. GeoGebra is a very interesting piece of software, and if you are interested in more, the authors point you to this wiki. Here is the abstract of their article:

We start by visiting the maths section of the web site answers.yahoo.com. Here, anybody can ask a question from anywhere in the world at every possible level. Answers are given by anyone who wants to contribute and then askers/readers rate the responses. A brief look here and it is starkly clear that our young people are struggling and their ability to think logically—that is understand a problem, organize data into knowns and unknowns, explore possibilities and assess solutions is definitely on the decline. In our opinion, this is more insidious than the actual decline in their overall mathematics skills. Further, one is struck by the fact that technology seems to be contributing to this decline when in fact it should be the opposite. We then examine two question/answer cycles in detail and show how the freeware GeoGebra (www.geogebra.org GeoGebraWiki: www.geogebra.org/wiki GeoGebraForum: www.geogebra.org/forum)—which gives the freedom to explore and learn to everyone, everywhere and at any time—can be of tremendous value to pupils and students in their understanding of mathematics from the smallest ages on up.

An innovative system of lecture notes

E.J. Tonkes, P.S. Isaac and V. Scharaschkin have written an article entitled "Assessment of an innovative system of lecture notes in first-year mathematics". The article was recently published online in International Journal of Mathematical Education in Science and Technology. Here is the abstract of their article:

Lectures are a familiar component in the delivery of mathematical content. Lecturers are often challenged with presenting material in a manner that aligns with the various learning styles and abilities within a large class. Students complain that the old-fashioned lecture style of copying notes from a board hinders the learning process, as they simply concentrate on writing. In recent times, distributing elaborate lecture notes has become a widespread alternative, but has its own problems, alienating the audience with lack of participation. The authors have developed a system of lecture notes, we call partially populated lecture notes, that have enjoyed success with students and addressed these difficulties.

Knowledge and beliefs

Much of my own research the last years has been related to knowledge and beliefs concerning mathematics, teaching and learning of mathematics. In the most recent issue of Instructional Science, Angela Boldrin and Lucia Mason have written an article that caught my attention: Distinguishing between knowledge and beliefs: students’ epistemic criteria for differentiating. Here is the abstract of this highly interesting article:

“I believe that he/she is telling the truth”, “I know about the solar system”: what epistemic criteria do students use to distinguish between knowledge and beliefs? If knowing and believing are conceptually distinguishable, do students of different grade levels use the same criteria to differentiate the two constructs? How do students understand the relationship between the two constructs? This study involved 219 students (116 girls and 103 boys); 114 were in 8th grade and 105 in 13th grade. Students had to (a) choose which of 5 graphic representations outlined better the relationship between the two constructs and to justify their choice; (b) rate a list of factual/validated, non-factual/non-validated and ambiguous statements as either knowledge or belief, and indicate for each statement their degree of truthfulness, acceptance and on which sources their views were based. Qualitative and quantitative analysis were performed. The data showed how students distinguish knowledge from belief conceptually and justify their understanding of the relationship between the two constructs. Although most students assigned a higher epistemic status to knowledge, school grade significantly differentiated the epistemic criteria used to distinguish the two constructs. The study indicates the educational importance of considering the notions of knowledge and belief that students bring into the learning situation.

Obama on Math

Yesterday, president Obama held a speech relating to education. The speech also mentioned math education a couple of times. Michael Alison Chandler gives a nice overview of this in an interesting blog post over at The Washington Post.

The emergence of "speaking with meaning"

Phillip G. Clark, Kevin C. Moore and Marilyn P. Carlson have written an article that was recently published online in The Journal of Mathematical Behavior. The article is entitled Documenting the emergence of “speaking with meaning” as a sociomathematical norm in professional learning community discourse. Here is the abstract of their article:

We introduce the sociomathematical norm of speaking with meaning and describe its emergence in a professional learning community (PLC) of secondary mathematics and science teachers. We use speaking with meaning to reference specific attributes of individual communication that have been revealed to improve the quality of discourse among individuals engaged in discourse in a PLC. An individual who is speaking with meaning provides conceptually based descriptions when communicating with others about solution approaches. The quantities and relationships between quantities in the problem context are described rather than only stating procedures or numerical calculations used to obtain an answer to a problem. Solution approaches are justified with logical and coherent arguments that have a conceptual rather than procedural basis. The data for this research was collected during a year-long study that investigated a PLC whose members were secondary mathematics and science teachers. Analysis of the data revealed that after one semester of participating in a PLC where speaking with meaning was emphasized, the PLC members began to establish their own criteria for an acceptable mathematical argument and what constituted speaking with meaning. The group also emerged with common expectations that answers be accompanied by explanations and mathematical operations be explained conceptually (not just procedurally). The course and PLC design that supported the emergence of speaking with meaning by individuals participating in a PLC are described.

Working with schools

Alan H. Schoenfeld has written an article that was published in the March issue of American Mathematical Monthly. The article has been entitled: Working with Schools: The Story of a Mathematics Education Collaboration. Here is a copy of the article abstract:

Working for meaningful mathematical change in the schools isn't easy. There are issues of politics, turf, and sometimes unreasonable expectations on the part of the school district and the volunteers who work with it. But with good intentions, goodwill, and tenacity, there are ways to make a difference. This paper describes some of the ups, the downs, and the ultimate progress in a collaboration between U.C. Berkeley and the Berkeley Unified School district. It offers lessons to mathematicians who want to understand and/or work with their local schools.

Proof constructions and evaluations

Andreas J. Stylianides and Gabriel J. Stylianides have written an article called Proof construction and evaluations. The article was published online in Educational Studies in Mathematics on Friday. Here is a copy of their article abstract:

In this article, we focus on a group of 39 prospective elementary (grades K-6) teachers who had rich experiences with proof, and we examine their ability to construct proofs and evaluate their own constructions. We claim that the combined “construction–evaluation” activity helps illuminate certain aspects of prospective teachers’ and presumably other individuals’ understanding of proof that tend to defy scrutiny when individuals are asked to evaluate given arguments. For example, some prospective teachers in our study provided empirical arguments to mathematical statements, while being aware that their constructions were invalid. Thus, although these constructions considered alone could have been taken as evidence of an empirical conception of proof, the additional consideration of prospective teachers’ evaluations of their own constructions overruled this interpretation and suggested a good understanding of the distinction between proofs and empirical arguments. We offer a possible account of our findings, and we discuss implications for research and instruction.

Free access to special issue of ESM!

SpringerLink has announced that the recent special issue of Educational Studies in Mathematics will be freely available to all. The special issue has a focus on Gestures and Multimodality in the Construction of Mathematical Meaning, and it contains 10 interesting articles. All are freely available to anyone before April 30, 2009.

See also my earlier post about the contents of this issue!

Sociocultural complexity in mathematics teaching

Barbara Jaworski and Despina Potari have written an article called Bridging the macro- and micro-divide: using an activity theory model to capture sociocultural complexity in mathematics teaching and its development. The article was published in Educational Studies in Mathematics a few days ago. Here is a copy of their abstract:

This paper is methodologically based, addressing the study of mathematics teaching by linking micro- and macro-perspectives. Considering teaching as activity, it uses Activity Theory and, in particular, the Expanded Mediational Triangle (EMT) to consider the role of the broader social frame in which classroom teaching is situated. Theoretical and methodological approaches are illustrated through episodes from a study of the mathematics teaching and learning in a Year-10 class in a UK secondary school where students were considered as “lower achievers” in their year group. We show how a number of questions about mathematics teaching and learning emerging from microanalysis were investigated by the use of the EMT. This framework provided a way to address complexity in the activity of teaching and its development based on recognition of central social factors in mathematics teaching–learning.

Exemplary mathematics instruction in Japanese classrooms

Yoshinori Shimizu has written an article that I think will be of great interest to many: Characterizing exemplary mathematics instruction in Japanese classrooms from the learner’s perspective. For more than a decade, researchers have had a focus on teaching practice in East-Asia, and in particular in Japan. Shimizu aims at examining some key characteristics of exemplary mathematics instruction in Japanese eigth-grade classrooms. The article was published online in ZDM on Wednesday. Here is the abstract:

This paper aims to examine key characteristics of exemplary mathematics instruction in Japanese classrooms. The selected findings of large-scale international studies of classroom practices in mathematics are reviewed for discussing the uniqueness of how Japanese teachers structure and deliver their lessons and what Japanese teachers value in their instruction from a teacher’s perspective. Then an analysis of post-lesson video-stimulated interviews with 60 students in three “well-taught” eighth-grade mathematics classrooms in Tokyo is reported to explore the learners’ views on what constitutes a “good” mathematics lesson. The co-constructed nature of quality mathematics instruction that focus on the role of students’ thinking in the classroom is discussed by recasting the characteristics of how lessons are structured and delivered and what experienced teachers tend to value in their instruction from the learner’s perspective. Valuing students’ thinking as necessary elements to be incorporated into the development of a lesson is the key to the approach taken by Japanese teachers to develop and maintain quality mathematics instruction.

Conditional inference and advanced mathematical study

Matthew Inglis and Adrian Simpson have written an article that was recently published online in Educational Studies in Mathematics. The article is entitled Conditional inference and advanced mathematical study: further evidence. Here is the article abstract:

In this paper, we examine the support given for the ‘theory of formal discipline’ by Inglis and Simpson (Educational Studies Mathematics 67:187–204, 2008). This theory, which is widely accepted by mathematicians and curriculum bodies, suggests that the study of advanced mathematics develops general thinking skills and, in particular, conditional reasoning skills. We further examine the idea that the differences between the conditional reasoning behaviour of mathematics and arts undergraduates reported by Inglis and Simpson may be put down to different levels of general intelligence in the two groups. The studies reported in this paper call into question this suggestion, but they also cast doubt on a straightforward version of the theory of formal discipline itself (at least with respect to university study). The paper concludes by suggesting that either a pre-university formal discipline effect or a filtering effect on ‘thinking dispositions’ may give a better account for the findings.

Teaching contests

Yeping Li and Jun Li have written an interesting article called Mathematics classroom instruction excellence through the platform of teaching contests. The article was published online in ZDM on Tuesday. Here is a copy of their abstract:

In this study, we aimed to examine features of mathematics classroom instruction excellence identified and valued through teaching contests in the Chinese mainland. By taking a case study approach, we focused on a prize-winning lesson as an exemplary lesson that was awarded the top prize in teaching contests at both the district and the city level. The analyses of the exemplary lesson itself revealed important features on the lesson’s content treatment, students’ engagement, and the use of multiple methods to facilitate students’ learning. These features are consistent with what the contest evaluation committees valued and what seven other mathematics expert teachers focused in their comments. The Chinese teaching culture in identifying and promoting classroom instruction excellence is then discussed in a broader context.

HPM newsletter, March 2009

HPM is a study group affiliated to ICMI, and it has a focus on the relations between the History and Pedagogy of Mathematics. HPM has now published their newsletter No. 70. The newsletter is freely available as PDF download, and it contains lost of useful information for those interested in the relationship between history and teaching/learning of mathematics.

Didactical designs

Takeshi Miyakawa and Carl Winsløw have written an article called Didactical designs for students’ proportional reasoning: an “open approach” lesson and a “fundamental situation”. The article was published online in Educational Studies in Mathematics on Saturday. Here is their abstrac:

In this paper, we analyze and compare two didactical designs for introducing primary school pupils to proportional reasoning in the context of plane polygons. One of them is well-documented in the literature; the other one is based on our own data and is accordingly presented and discussed in more detail in this paper. The two designs come from different cultural and intellectual environments: lesson study in Japan (implicitly based on the “open approach method”) and “didactical engineering” in France (based on the theory of didactical situations). The general aim of our paper is to compare these two environments and their approaches to didactical design, basing our discussion on the concrete designs mentioned above. Clear differences among them are presented, while we also identify links which hold potential for integrating research and practice.

Teaching research groups in China

Yudong Yang has written an interesting article that was recently published online in ZDM. The article has been entitled How a Chinese teacher improved classroom teaching in Teaching Research Group: a case study on Pythagoras theorem teaching in Shanghai. The Teaching Research Group system seems to be somewhat similar to the Japanese Lesson Study approach, and I find this very interesting. Here is the article abstract:

In China, a school-based teaching research system was built since 1952 and Teaching Research Group (TRG) exists in every school. In the paper, a teacher’s three lessons and the changes in each lesson were described, which might show a track of how lessons were continuously developed in TRG. The Mathematical Tasks Framework, The Task Analysis Guide, and Factors Associated with the Maintenance and the Decline of High-level Cognitive Demands developed in the Quantitative Understanding: Amplifying Student Achievement and Reasoning project (Stein and Smith in Math Teach Middle School 3(4):268–275, 1998; Stein et al. in Implementing stardards-based mathematics instruction. Teachers College Press, NY, pp. 1–33, 2000), were employed in this study. Based on the perspective of Mathematical Task Analysis, changes of three lessons were described and the author provided a snapshot for understanding how a Chinese teacher gradually improved his/her lessons in TRG activities.

Black-white gap in mathematics course taking

Sean Kelly has written an article about The Black-White Gap in Mathematics Course Taking. This article has been published in a recent issue of the journal Sociology of Education. Here is the abstract of Kelly's article:

Using data from the National Education Longitudinal Study, this study investigated differences in the mathematics course taking of white and black students. Because of lower levels of achievement, prior course taking, and lower socioeconomic status, black students are much more likely than are white students to be enrolled in low-track mathematics courses by the 10th grade. Using multilevel models for categorical outcomes, the study found that the black-white gap in mathematics course taking is the greatest in integrated schools where black students are in the minority and cannot be entirely accounted for by individual-level differences in the course-taking qualifications or family backgrounds of white and black students. This finding was obscured in prior research by the failure to model course taking adequately between and within schools. Course placement policies and enrollment patterns should be monitored to ensure effective schooling for all students.

Good mathematics instruction in South Korea

JeongSuk Pang has written an article called Good mathematics instruction in South Korea. The article has recently been published online in ZDM. Here is the article abstract:

There have been only a few studies of Korean mathematics instruction in international contexts. Given this, this paper describes in detail a sixth grade teacher’s mathematics instruction in order to investigate closely what may be counted as high-quality teaching and learning in Korea. This paper then discusses several key characteristics of good mathematics instruction along with some background information on Korean educational practice. This paper concludes with remarks that good mathematics instruction may be perceived differently with regard to underlying social and cultural norms.

Teaching Mathematics and its Applications, issue 1, 2009

The first issue (of 2009) of Teaching Mathematics and its Applications has been published. Here is an overview of the contents:

Section A

Get checked abstract Adnan Baki and Bulent Guven

Khayyam with Cabri: experiences of pre-service mathematics teachers with Khayyam's solution of cubic equations in dynamic geometry environment
Teaching Mathematics and its Applications Advance Access published on February 17, 2009
Teaching Mathematics Applications 2009 28: 1-9; doi:10.1093/teamat/hrp001 [Abstract] [PDF] [Request Permissions]

Get checked abstract Paul Glaister and Elizabeth M. Glaister

HMS—harmonic motion by shadows
Teaching Mathematics and its Applications Advance Access published on November 3, 2008
Teaching Mathematics Applications 2009 28: 10-15; doi:10.1093/teamat/hrn022 [Abstract] [PDF] [Request Permissions]

Get checked abstract Yiu-Kwong Man

On Feynman's Triangle problem and the Routh Theorem
Teaching Mathematics and its Applications Advance Access published on January 30, 2009
Teaching Mathematics Applications 2009 28: 16-20; doi:10.1093/teamat/hrn024 [Abstract] [PDF] [Request Permissions]

Get checked abstract John Monaghan, Peter Pool, Tom Roper, and John Threlfall

Open-start mathematics problems: an approach to assessing problem solving
Teaching Mathematics and its Applications Advance Access published on January 30, 2009
Teaching Mathematics Applications 2009 28: 21-31; doi:10.1093/teamat/hrn023 [Abstract] [PDF] [Request Permissions]

Get checked abstract Keith Parramore

Enlisting excel—again
Teaching Mathematics Applications 2009 28: 32-37; doi:10.1093/teamat/hrp004 [Abstract] [PDF] [Request Permissions]

Get checked abstract Tanja Van Hecke

Minimizing the delay at traffic lights
Teaching Mathematics and its Applications Advance Access published on February 17, 2009
Teaching Mathematics Applications 2009 28: 38-42; doi:10.1093/teamat/hrp002 [Abstract] [PDF] [Request Permissions]

Section B

Get checked abstract Yiu-Kwong Man

A study of two-term unit fraction expansions via geometric approach
Teaching Mathematics and its Applications Advance Access published on October 19, 2008
Teaching Mathematics Applications 2009 28: 43-47; doi:10.1093/teamat/hrn020 [Abstract] [PDF] [Request Permissions]

Get checked abstract Chris Sangwin

The wonky trammel of Archimedes
Teaching Mathematics and its Applications Advance Access published on November 28, 2008
Teaching Mathematics Applications 2009 28: 48-52; doi:10.1093/teamat/hrn019 [Abstract] [PDF] [Request Permissions]

IJCML, volume 13, issue 3

The December issue (2008) of International Journal of Computers for Mathematical Learning has recently been made available - obviously a bit late. These are the five articles included:

• Playing with Representations: How Do Kids Make Use of Quantitative Representations in Video Games? by Tom Satwicz and Reed Stevens
• Graphic Calculators and Micro-Straightness: Analysis of a Didactic Engineering, by Michela Maschietto
• An ‘Emergent Model’ for Rate of Change, by Sandra Herbert and Robyn Pierce
• Using Dynamic Geometry Software to Gain Insight into a Proof, by Bulent Guven
• Computational Diversions: Julia Variations, by Michael Eisenberg