- RESEARCH COMMENTARY: On "Gap Gazing" in Mathematics Education: The Need for Gaps Analyses, by Sarah Theule Lubienski
- RESEARCH COMMENTARY: A "Gap-Gazing" Fetish in Mathematics Education? Problematizing Research on the Achievement Gap, by Rochelle Gutiérrez
- RESEARCH COMMENTARY: Bridging the Gaps in Perspectives on Equity in Mathematics Education, by Sarah Theule Lubienski and Rochelle Gutiérrez
- Unpacking Pedagogical Content Knowledge: Conceptualizing and Measuring Teachers' Topic-Specific Knowledge of Students, by Heather C. Hill, Deborah Loewenberg Ball and Stephen G. Schilling
- Josh's Operational Conjectures: Abductions of a Splitting Operation and the Construction of New Fractional Schemes, by Anderson Norton
- How Mathematicians Determine if an Argument Is a Valid Proof, by Keith Weber

## 2008/06/30

### JRME, July 2008

## 2008/06/29

### ICMI newsletter

One of the many interesting news in this newsletter is concerning a new website about the history of ICMI. The website is edited by Fulvia Furinghetti and Livia Giacardi, and this site provides you with en excellent set of resources for information about the history of ICMI and, in many ways, the history of our field of research.

Another interesting information is concerning the so-called "ICMI Reading Room" at SpringerLink.

Up to December 31, 2008, members of the international community ofThese sholars represent some of the most important milestones in our field, and this is a very nice opportunity to learn more about the work of these four medallists.

mathematics educators will have open access, via SpringerLink.com, to

selected works published in Springer journals of the four most recent

ICMI medallists (Paul Cobb, Ubiratan D'Ambrosio, Jeremy Kilpatrick and

Anna Sfard).

The newsletter also announces the launcing of a new journal in mathematics education: Sutra - The International Journal of Mathematics Education. Sutra is the official journal of the Technomathematics Research Foundation, and the first issue will be published online in August this year.

You can read about this and much more in the lates issue of the ICMI newsletter. If you want to subscribe to the newsletter, there are two ways of doing that:

- Click on http://www.mathunion.org/ICMI/Mailinglist with a Web browser and go to the "Subscribe" button to subscribe to ICMI News online.
- Send an e-mail to icmi-news-request at mathunion.org with the Subject-line: Subject: subscribe

### Math history on the internet

## 2008/06/27

### Preparation of math teachers

No Common Denominator: The Preparation of Elementary Teachers in Mathematics by America's Education Schools, June 2008On the web site of NCTQ, you can download an executive summary, the test and answer key, or the full report.

American students' chronically poor performance in mathematics on international tests may begin in the earliest grades, handicapped by the weak knowledge of mathematics of their own elementary teachers. NCTQ looks at the quality of preparation provided by a representative sampling of institutions in nearly every state. We also provide a test developed by leading mathematicians which assesses for the knowledge that elementary teachers should acquire during their preparation. Imagine the implications of an elementary teaching force being able to pass this test.

### Triangles as intuitive non-examples

In this paper we examine the possibility of differentiating between two types of nonexamples. The first type, intuitive nonexamples, consists of nonexamples which are intuitively accepted as such. That is, children immediately identify them as nonexamples. The second type, non-intuitive nonexamples, consists of nonexamples that bear a significant similarity to valid examples of the concept, and consequently are more often mistakenly identified as examples. We describe and discuss these notions and present a study regarding kindergarten children’s grasp of nonexamples of triangles.

### Pythagorean approximations

In this article, we will show that the Pythagorean approximations of Formula coincide with those achieved in the 16th century by means of continued fractions. Assuming this fact and the known relation that connects the Fibonacci sequence with the golden section, we shall establish a procedure to obtain sequences of rational numbers converging to different algebraic irrationals. We will see how approximations to some irrational numbers, using known facts from the history of mathematics, may perhaps help to acquire a better comprehension of the real numbers and their properties at further mathematics level.

## 2008/06/26

### Effectiveness of teacher education

Teacher-education research lacks a common theoretical basis, which prevents a convincing development of instruments and makes it difficult to connect studies to each other. Our paper models how to measure effective teacher education in the context of the current state of knowledge in the field. First, we conceptualize the central criterion of effective teacher education: “professional competence of future teachers”. Second, individual, institutional, and systemic factors are modeled that may influence the acquisition of this competence during teacher education. In doing this, we turn round the perspective taken by Cochran-Smith and Zeichner (Studying teacher education. The report of the AERA panel on research and teacher education. Lawrence Erlbaum, Mahwah 2005), who mainly take an educational-sociological perspective by focusing on characteristics of teacher education and looking for their effects. In contrast, we take an educational-psychological perspective by focusing on professional competence of teachers and examining influences on this. Challenges connected to an assessment of teacher-education outcomes are discussed as well.

## 2008/06/25

### Mathematics teaching during the early years in Hong Kong

This paper characterizes early mathematics instruction in Hong Kong. The teaching of addition in three pre-primary and three lower primary schools was observed and nine teachers were interviewed about their beliefs about early mathematics teaching. A child-centered, play-based approach was evident but teachers emphasized discipline, diligence and academic success. Observations also revealed practices reflective of both constructivist and instructivist pedagogies. Results from interviews suggest that teachers' traditional cultural beliefs about instruction were challenged by western ideologies introduced in continuing professional development courses and by notions promulgated by the educational reforms. Both consistencies and inconsistencies between teachers' beliefs and practices were identified. Implications of the findings are discussed.

### Students' problem solving behaviours

Abstract:

This paper reports one aspect of a larger study which looked at the strategies used by a selection of grade 6 students to solve six non-routine mathematical problems. The data revealed that the students exhibited many of the behaviours identified in the literature as being associated with novice and expert problem solvers. However, the categories of ‘novice’ and ‘expert’ were not fully adequate to describe the range of behaviours observed and instead three categories that were characteristic of behaviours associated with ‘naïve’, ‘routine’ and ‘sophisticated’ approaches to solving problems were identified. Furthermore, examination of individual cases revealed that each student's problem solving performance was consistent across a range of problems, indicating a particular orientation towards naïve, routine or sophisticated problem solving behaviours. This paper describes common problem solving behaviours and details three individual cases involving naïve, routine and sophisticated problem solvers.

## 2008/06/23

### TMME, July 2008

1. Jeff Babb & James Currie(Canada)

The Brachistochrone Problem: Mathematics for a Broad Audience via a Large Context

2. Michael Fried (Israel)

History of Mathematics in Mathematics Education: a Saussurean Perspective

3. Spyros Glenis (Greece)

Comparison of Geometric Figures

4. Giorgio T. Bagni (Italy)

“Obeying a rule”: Ludwig Wittgenstein and the foundations of Set Theory

5. Arnaud Mayrargue (France)

How can science history contribute to the development of new proposals in the teaching of the notion of derivatives?

6. Antti Viholainen (Finland)

Incoherence of a concept image and erroneous conclusions in the case of differentiability

7. Dores Ferreira & Pedro Palhares (Portugal)

Chess and problem solving involving patterns

8. Friðrik Diego & Kristín Halla Jónsdóttir (Iceland)

Associative Operations on a Three-Element Set

9. Jon Warwick (UK)

A Case Study Using Soft Systems Methodology in the Evolution of a Mathematics Module

10. Barbara Garii & Lillian Okumu (New York, USA)

Mathematics and the World: What do Teachers Recognize as Mathematics in Real World Practice?

11. Linda Martin & Kristin Umland (New Mexico, USA)

Mathematics for Middle School Teachers: Choices, Successes, and Challenges

12. Woong Lim (Texas, USA)

Inverses – why we teach and why we need talk more about it more often!

13. Steve Humble (UK)

Magic Math Cards

The issue also contains a couple of articles on logarithms in a historical perspective, a large section of articles with reactions on the report of the National Mathematics Advisory Panel, etc.

## 2008/06/19

### Does Mathematics Remediation Work?

Postsecondary remediation is a controversial topic. On one hand, it fills an important and sizeable niche in higher education. On the other hand, critics argue that it wastes tax dollars, diminishes academic standards, and demoralizes faculty. Yet, despite the ongoing debate, few comprehensive, large-scale, multi-institutional evaluations of remedial programs have been published in recent memory. The study presented here constitutes a step forward in rectifying this deficit in the literature, with particular attention to testing the efficacy of remedial math programs. In this study, I use hierarchical multinomial logistic regression to analyze data that address a population of 85,894 freshmen, enrolled in 107 community colleges, for the purpose of comparing the long-term academic outcomes of students who remediate successfully (achieve college-level math skill) with those of students who achieve college-level math skill without remedial assistance. I find that these two groups of students experience comparable outcomes, which indicates that remedial math programs are highly effective at resolving skill deficiencies.

## 2008/06/18

### Review of mathematics teaching in early years

## 2008/06/16

### IJMEST, June 2008

**Integrating supplementary application-based tutorials in the multivariable calculus course by I. M. Verner;**

S. Aroshas; A. Berman**If not, what yes? by Boris Koichu****Mathematical e-learning: state of the art and experiences at the Open University of Catalonia by A. Juan; A. Huertas; C. Steegmann; C. Corcoles; C. Serrat****Unique factorization in cyclotomic integers of degree seven by W. Ethan Duckworth****A college lesson study in calculus, preliminary report by Joy Becker; Petre Ghenciu; Matt Horak; Helen Schroeder**

## 2008/06/10

### Teaching and learning proof

- How do you teach proof?
- What place do you think proof has in the mathematics curriculum?
- At what age should proof be introduced to learners and how?

- Article 1: Students' Views of Proof, Celia Hoyles and Lulu Healy, Mathematics in School Issue 3 May 1999, published by The Mathematical Association;
- Article 2: Interpreting the Mathematics Curriculum: Developing reasoning through algebra and geometry, published by the Qualifications and Curriculum Authority, 2004;
- Article 3: Teaching Pythagoras' Theorem, Paul Chambers, Mathematics in Schools Issue 4 1999, published by The Mathematical Association.

## 2008/06/09

### What makes a problem mathematically interesting?

Mathematical problems are an integral part of mathematical learning, and although most pupils encounter mathematical problems as they are posed in textbooks, the teachers have an important role in assigning appropriate problems for the students to solve. Prospective teachers have had few opportunities to focus on problem posing in their studies, and their experience with mathematical problems are mostly in connection with the solving of problems that are posed by the teacher or a textbook. The authors of this article "consider the practice of problem posing to be especially important for prospective teachers because a great deal of the work of teaching entails the posing and generation of what the mathematics education community often refers to as “good” questions—questions that aim to support students’ mathematical work".

The main research questions in the study described in this article are:

- What is the role of exploration in the problem-posing process? (What happens when prospective teachers pose problems with and without first exploring the situation that could motivate their questions? What kinds of questions do they pose in each of these two kinds of structured problem-posing setting?)
- How do prospective elementary teachers decide on the quality of the questions they pose? (What rationale do they provide when asked to justify what makes their questions mathematically interesting? What is the effect of making explicit some of the qualities that make mathematics problems interesting and worth solving?)

Here is the abstract:

School students of all ages, including those who subsequently become teachers, have limited experience posing their own mathematical problems. Yet problem posing, both as an act of mathematical inquiry and of mathematics teaching, is part of the mathematics education reform vision that seeks to promote mathematics as an worthy intellectual activity. In this study, the authors explored the problem-posing behavior of elementary prospective teachers, which entailed analyzing the kinds of problems they posed as a result of two interventions. The interventions were designed to probe the effects of (a) exploration of a mathematical situation as a precursor to mathematical problem posing, and (b) development of aesthetic criteria to judge the mathematical quality of the problems posed. Results show that both interventions led to improved problem posing and mathematically richer understandings of what makes a problem ‘good.’

## 2008/06/06

### IJMEST, new articles

Some new (iFirst) articles have been published in International Journal of Mathematical Education in Science and Technology:

The mean as the balance point: thought experiments with measuring sticks

Author: A. Flores

DOI: 10.1080/00207390701871655

Link: http://www.informaworld.com

An evaluation of the Supplemental Instruction programme in a first year calculus course

Authors: V. Fayowski; P. D. MacMillan

DOI: 10.1080/00207390802054433

Link: http://www.informaworld.com

The classical version of Stokes' theorem revisited

Author: Steen Markvorsen

DOI: 10.1080/00207390802091146

Link: http://www.informaworld.com

Unification and infinite series

Authors: J. V. Leyendekkers; A. G. Shannon

DOI: 10.1080/00207390802054474

Link: http://www.informaworld.com

### JMTE, Number 3, 2008

The June issue of Journal of Mathematics Teacher Education has arrived, with lots of interesting articles. Here is an overview of the 5 main articles in the issue:

- The tension between the general and the specific in an international mathematics teacher education by Dina Tirosh
- “Mathematical knowledge for teaching”: adapting U.S. measures for use in Ireland by Seán Delaney et al.
- Real-world connections in secondary mathematics teaching by Julie Gainsburg
- Sixth grade mathematics teachers’ intentions and use of probing, guiding, and factual questions by Alpaslan Sahin and Gerald Kulm
- Recruiting and retaining secondary mathematics teachers: lessons learned from an innovative four-year undergraduate program by Alice F. Artzt and Frances R. Curcio

## 2008/06/05

### The instructional triangle

AbstractAs mathematics educators think about teaching that

promotes students’ opportunities to learn, attention must be given to

the conceptualization of the professional development of teachers and

those who teach teachers. In this article, we generalize and expand the

instructional triangle to consider different interactions in a variety

of teacher development contexts. We have done so by addressing issues

of language for models of teachers’ professional development at

different levels and by providing examples of situations in which these

models can be applied. Through the expansion of our understanding and

use of the instructional triangle we can further develop the concept of

mathematics teacher development.

Teachers are professionals with a rich knowledge that is both content specific and general. They shape instruction by the way

*they interpret and respond to students and materials*(p. 2). The notion of "the instructional triangle" is based on the definition of instruction as (they refer to Cohen and Ball, 1999, p. 5 here):

*the interaction between*. These ideas are also shared by other researchers. One of them, Barbara Jaworski, created the

**teachers**and**students**around**educational material***teaching triad*, consisting of:

- management of student learning
- sensitivity to students
- engagement in challenging mathematics

## 2008/06/03

### How hints help speed up math performance

Campbell, J.I., Fuchs-Lacelle, S., Phenix, T.L. (2006). Identical elements model of arithmetic memory: Extension to addition and subtraction.

*Memory & Cognition, 34*(3), 633-647.

### Learning beginning algebra with spreadsheets

Michal Tabach, Rina Hershkowitza and Abraham Arcavi have written an article that was published online by The Journal of Mathematical Behavior yesterday. The article is entitled Learning beginning algebra with spreadsheets in a computer intensive environment. Here is the abstract:

This study is part of a large research and development project aimed at observing, describing and analyzing the learning processes of two seventh grade classes during a yearlong beginning algebra course in a computer intensive environment (CIE). The environment includes carefully designed algebra learning materials with a functional approach, and provides students with unconstrained freedom to use (or not use) computerized tools during the learning process at all times. This paper focuses on the qualitative and quantitative analyses of students’ work on one problem, which serves as a window through which we learn about the ways students worked on problems throughout the year. The analyses reveal the nature of students’ mathematical activity, and how such activity is related to both the instrumental views of the computerized tools that students develop and their freedom to use them. We describe and analyze the variety of approaches to symbolic generalizations, syntactic rules and equation solving and the many solution strategies pursued successfully by the students. On that basis, we discuss the strengths of the learning environment and the open questions and dilemmas it poses.

### The Pirie-Kieren theory

The study reported here extends the work of Pirie and Kieren on the nature and growth of mathematical understanding. The research examines in detail a key aspect of their theory, the process of ‘folding back’, and develops a theoretical framework of categories and sub-categories that more fully describe the phenomenon. This paper presents an overview of this ‘framework for folding back’, illustrates it with extracts of video data and elaborates on its key features. The paper also considers the implications of the study for the teaching and learning of mathematics, and for future research in the field.For another article discussing the Pirie-Kieren theory and related theories, you might want to take a look at this article by Droujkova et al. from PME29.

### Children's arithmetical thinking

This article focuses on spontaneous and progressive knowledge building in “the arithmetic of the child.” The aim is to investigate variations in the behavior patterns of eight pupils attending a school for the intellectually disabled. The study is based on the epistemology of radical constructivism and the methodology of multiple clinical interviews. Theoretical models elucidate behavior patterns and the corresponding mental structures underlying them. The individual interviews of the pupils were video recorded. The results show that the activated behavior patterns, which are responses to well-adapted contexts presented by the researcher, are compatible with findings in Swedish compulsory schools. Six of the pupils’ mental structures in the study are numerical. A substantial implication for special education is the harmonization of the content in teaching with the children's own ways of operating, which implies a triadic teaching process.

## 2008/06/02

### The effects of designing Webquests

The purpose of this study was to examine the effects of webquest-based applications on the pre-service elementary school teachers' motivation in mathematics. There were a total of 202 pre-service elementary school teachers, 125 in a treatment group and 77 in a control group. The researcher used a Likert-type questionnaire consisting of 34 negative and positive statements. This questionnaire was designed to evaluate a situational measure of the pre-service teachers' motivation. This questionnaire was used as pre- and post-tests in the study that took place in two semesters. It was administered to the participants by the researcher before and after the instruction during a single class period. The paired-samplest-test, the independent-samplest-test and analysis of covariance with= 0.05 were used to analyse the quantitative data. The study showed that there was a statistically significant difference found in participants' motivation between treatment and control groups favouring the treatment group. In other words, the participants who designed the webquest-based applications indicated positive attitudes towards mathematics course than the others who did the regular course work.

## 2008/06/01

### The particular and the general

Vicenç Font and Ángel Contreras wrote an article that was recently published in Educational Studies in Mathematics. The article is entitled "The problem of the particular and its relation to the general in mathematics education", and here is the abstract:

### Two IJSME articles

Two articles has recently been published online in International Journal of Science and Mathematics Education. Here are the titles and abstracts:

- Lene Møller Madsen and Carl Winsløw have written an article called "RELATIONS BETWEEN TEACHING AND RESEARCH IN PHYSICAL GEOGRAPHY AND MATHEMATICS AT RESEARCH-INTENSIVE UNIVERSITIES". Abstract: We examine the relationship between research and teaching practices as they are enacted by university professors in a research-intensive university. First we propose a theoretical model for the study of this relationship based on Chevallard’s anthropological theory. This model is used to design and analyze an interview study with physical geographers and mathematicians at the University of Copenhagen. We found significant differences in how the respondents from the two disciplines assessed the relationship between research and teaching. Above all, while geography research practices are often and smoothly integrated into geography teaching even at the undergraduate level, teaching in mathematics may at best be ‘similar’ to mathematical research practice, at least at the undergraduate level. Finally, we discuss the educational implications of these findings.
- Muammer Çalik, Alipaşa Ayas and Richard K. Coll wrote an article called "INVESTIGATING THE EFFECTIVENESS OF AN ANALOGY ACTIVITY IN IMPROVING STUDENTS’ CONCEPTUAL CHANGE FOR SOLUTION CHEMISTRY CONCEPTS". Abstract: This paper reports on an investigation on the use of an analogy activity and seeks to provide evidence of whether the activity enables students to change alternative conceptions towards views more in accord with scientific views for aspects of solution chemistry. We were also interested in how robust any change was and whether these changes in conceptual thinking became embedded in the students’ long-term memory. The study has its theoretical basis in an interpretive paradigm, and used multiple methods to probe the issues in depth. Data collection consisted of two concept test items, one-on-one interviews, and student self-assessment. The sample consisted of 44 Grade 9 students selected from two intact classes (22 each), from Trabzon, Turkey. The interviews were conducted with six students selected because of evidence as to their significant conceptual change in solution chemistry. The research findings revealed statistically significant differences in pre-test and post-test scores, and pre-test and delayed post-test scores (p<0.05), but no differences between post-test and delayed test scores (p>0.05). This suggests that the analogy activity is helpful in enhancing students’ conceptual understanding of solution chemistry, and that these changes may be stored in the students’ long-term memory.

### Structures of argumentation

Christine Knipping wrote an article that was recently published online in ZDM. The article is entitled: A method for revealing structures of argumentations in classroom proving processes. Here is the abstract: