Alignment, cohesion, and change

Dionne I. Cross has written an article called Alignment, cohesion, and change: Examining mathematics teachers’ belief structures and their influence on instructional practices. This article was recently published online in Journal of Mathematics Teacher Education. Here is the abstract of the article:

This collective case study reports on an investigation into the relationship between mathematics teachers’ beliefs and their classroom practices, namely, how they organized their classroom activities, interacted with their students, and assessed their students’ learning. Additionally, the study examined the pervasiveness of their beliefs in the face of efforts to incorporate reform-oriented classroom materials and instructional strategies. The participants were five high school teachers of ninth-grade algebra at different stages in their teaching career. The qualitative analysis of the data revealed that in general beliefs were very influential on the teachers’ daily pedagogical decisions and that their beliefs about the nature of mathematics served as a primary source of their beliefs about pedagogy and student learning. Findings from the analysis concur with previous studies in this area that reveal a clear relationship between these constructs. In addition, the results provide useful insights for the mathematics education community as it shows the diversity among the inservice teachers’ beliefs (presented as hypothesized belief models), the role and influence of beliefs about the nature of mathematics on the belief structure and how the teachers designed their instructional practices to reflect these beliefs. The article concludes with a discussion of implications of teacher education.

Blog reading tips - Poincaré's prize

Peter Ash has a nice blog about mathematics and education, and he has given a nice review of what appears to be an interesting book in a blog post about "Poincare's Prize". Here is the intro of his post, to tickle your interest:

I recently read Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles by George C. Szpiro. I recommend it highly. Some time back I recommended another book on the same topic, The Poincaré Conjecture: In Search of the Shape of the Universe by Donal O'Shea. If you can only read one book on the topic, I recommend the Szpiro book.

Tutored problem solving

Ron J.C.M. Salden, Vincent Aleven, Rolf Schwonke and Alexander Renki have written an article entitled The expertise reversal effect and worked examples in tutored problem solving. The article was printed online in Instructional Science on Thursday. Here is the abstract of their article:

Prior research has shown that tutored problem solving with intelligent software tutors is an effective instructional method, and that worked examples are an effective complement to this kind of tutored problem solving. The work on the expertise reversal effect suggests that it is desirable to tailor the fading of worked examples to individual students’ growing expertise levels. One lab and one classroom experiment were conducted to investigate whether adaptively fading worked examples in a tutored problem-solving environment can lead to higher learning gains. Both studies compared a standard Cognitive Tutor with two example-enhanced versions, in which the fading of worked examples occurred either in a fixed manner or in a manner adaptive to individual students’ understanding of the examples. Both experiments provide evidence of improved learning results from adaptive fading over fixed fading over problem solving. We discuss how to further optimize the fading procedure matching each individual student’s changing knowledge level.

An integrative learning experience

Barbra Melendez, Silas Bowman, Keith Erickson and Edward Swim have written an article called An integrative learning experience within a mathematics curriculum. The article was recently published online in Teaching Mathematics and its Applications. Here is the abstract of their article:

We developed four separate scenarios focusing on the connections between mathematics, biology, and social sciences. This structure facilitated the coordination of faculty from seven academic departments on campus. Each scenario had students from different majors build mathematical models, gather information from their respective disciplines, and develop a final presentation that included a committee consensus on how to approach the problem in a practical way. As a result, students learned how mathematics plays a role in other disciplines and how insight from different points of view affects the approach taken to a complex problem.

Interdisciplinary mathematics-physics approaches

Valérie Munier and Helene Merle have written an article that was published in the September issue of International Journal of Science Education. The article is entitled Interdisciplinary Mathematics-Physics Approaches to Teaching the Concept of Angle in Elementary School. Unfortunately, I don't have access to this article, but I find the topic interesting! Here is a copy of the abstract of their article:

The present study takes an interdisciplinary mathematics-physics approach to the acquisition of the concept of angle by children in Grades 3-5. This paper first presents the theoretical framework we developed, then we analyse the concept of angle and the difficulties pupils have with it. Finally, we report three experimental physics-based teaching sequences tested in three classrooms. We showed that at the end of each teaching sequence the pupils had a good grasp of the concept of angle, they had truly appropriated the physics knowledge at play, and many pupils are enable to successfully grasp new physics situations in which the angle plays a highly meaningful role. Using a physics framework to introduce angles in problem situations is then pertinent: by interrelating different spaces, pupils were able to acquire skills in the domains of mathematics, physics, and modelling. In conclusion, we discuss the respective merits of each problem situation proposed.

International Handbook of Research on Teachers and Teaching

Springer has published a new and interesting book: International Handbook of Research on Teachers and Teaching. This handbook has been edited by Lawrence J. Saha and A. Gary Dworkin, and it is a huge book of 1200 pages. Although the book is concerned with research on teachers and teaching in general, it should be interesting to researchers within the field of mathematics education as well. It also contains a chapter that is concerned with mathematics teaching in particular. Here is a copy of the publisher's info about the book:

• This book takes into account new research on both teachers and the nature of teaching
• Includes over 70 completely new and original articles covering many aspects of what we know about the teaching profession and about classroom teaching
• Treats teachers and teaching from a comparative perspective, highlighting similarities and differences across countries
• Addresses the role of culture in understanding variations in teaching practices
• Discusses both the changing levels of accountability for teachers and its effects

The International Handbook of Research on Teachers and Teaching provides a fresh look at the ever changing nature of the teaching profession throughout the world. This collection of over 70 original articles addresses a wide range of issues that are relevant for understanding the present educational climate in which the accountability of teachers and the standardized testing of students have become dominant.

The international collection of authors brings to the handbook a breadth of knowledge and experience about the teaching profession and a wealth of material across a number of comparative dimensions, such as between developed and developing countries and between Eastern and Western cultures. In addition, many articles address the emerging challenges to education and to the lives of teachers which are brought about by the globalization trends of the 21st Century.

New issue of Educational Studies in Mathematics

The September issue of Educational Studies in Mathematics was published last week, and - as always - it contains a number of interesting articles.

• Re-mythologizing mathematics through attention to classroom positioning, by David Wagner and Beth Herbel-Eisenmann
• Encrypted objects and decryption processes: problem-solving with functions in a learning environment based on cryptography, by Tobin White
• Using diagrams as tools for the solution of non-routine mathematical problems, by Marilena Pantziara, Athanasios Gagatsis and Iliada Elia
• Social representations as mediators of practice in mathematics classrooms with immigrant students, by Núria Gorgorió and Guida de Abreu
• Success in mathematics within a challenged minority: the case of students of Ethiopian origin in Israel (SEO), by Tiruwork Mulat and Abraham Arcavi
• Examining the supervision of mathematics student teachers through analysis of conference communications, by Maria Lorelei Fernandez and Evrim Erbilgin
• Approach to mathematical problem solving and students’ belief systems: two case studies, by María Luz Callejo and Antoni Vila
• Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction, by Doug M. Clarke and Anne Roche

I would like to point your interest to Tobin White's article in particular, since this is an Open Access article. So, regardless of whether you are a subscriber or not, this article is freely available to all!

Mathematically and practically based explanations

Esther Levenson has written an article called Fifth-grade students’ use and preferences for mathematically and practically based explanations. The article was published online in Educational Studies in Mathematics a few days ago. What Levenson refers to as "practice based explanations" are related to what others refer to as real-life connections, students' informal knowledge, etc. Practice based explanations do not rely on mathematical notions only, and include explanations that use manipulatives and explanations that are based on real-life contexts. Obviously, this implies that there is a variety of explanations to consider, and Levenson provides a nice overview of some relevant literature within this field. She also discusses students' evaluations of explanations, and she thereby enters a discussion of the different types of knowledge you need to have.

The study she reports from is a combination of quantitative and qualitative analysis of data from a total of 105 students in 5th grade (in Israel). Data were collected from two questionnaires, in addition to follow-up interviews with some of the students.

Here is the abstract of Levenson's article:

This paper focuses on fifth-grade students’ use and preference for mathematically (MB) and practically based (PB) explanations within two mathematical contexts: parity and equivalent fractions. Preference was evaluated based on three parameters: the explanation (1) was convincing, (2) would be used by the student in class, and (3) was one that the student wanted the teacher to use. Results showed that students generated more MB explanations than PB explanations for both contexts. However, when given a choice between various explanations, PB explanations were preferred in the context of parity, and no preference was shown for either type of explanation in the context of equivalent fractions. Possible bases for students’ preferences are discussed.

Children's strategies for division by fractions

Jaehoon Yim (South Korea) has written an article entitled Children’s strategies for division by fractions in the context of the area of a rectangle. The article was published online in Educational Studies in Mathematics on Tuesday. Here is the abstract of the article:

This study investigated how children tackled a task on division by fractions, and how they formulated numerical algorithms from their strategies. The task assigned to the students was to find the length of a rectangle given its area and width. The investigation was carried out as follows: First, the strategies invented by eight 10- or 11-year-old students, all identified as capable and having positive attitudes towards mathematics, were categorised. Second, the formulation of numerical algorithms from the strategies constructed by nine students with similar abilities and attitudes towards mathematics was investigated. The participants developed three types of strategies (making the width equal to 1, making the area equal to 1, and changing both area and width to natural numbers) and showed the possibility of formulating numerical algorithms for division by fractions referring to their strategies.

Interesting AERJ articles

The latest issue of American Educational Research Journal contains several articles that are interesting for the mathematics education research community. Here are three that I find particularly interesting:

• National Income, Income Inequality, and the Importance of Schools: A Hierarchical Cross-National Comparison, by Amita Chudgar and Thomas F. Luschei. Abstract: The international and comparative education literature is not in agreement over the role of schools in student learning. The authors reexamine this debate across 25 diverse countries participating in the fourth-grade application of the 2003 Trends in International Mathematics and Science Study. The authors find the following: (a) In most cases, family background is more important than schools in understanding variations in student performance; (b) schools are nonetheless a significant source of variation in student performance, especially in poor and unequal countries; (c) in some cases, schools may bridge the achievement gap between high and low socioeconomic status children. However, schools’ ability to do so is not systematically related to a country’s economic or inequality status.
• Assessing the Contribution of Distributed Leadership to School Improvement and Growth in Math Achievement, by Ronald H. Heck and Philip Hallinger. Abstract: Although there has been sizable growth in the number of empirical studies of shared forms of leadership over the past decade, the bulk of this research has been descriptive. Relatively few published studies have investigated the impact of shared leadership on school improvement. This longitudinal study examines the effects of distributed leadership on school improvement and growth in student math achievement in 195 elementary schools in one state over a 4-year period. Using multilevel latent change analysis, the research found significant direct effects of distributed leadership on change in the schools’ academic capacity and indirect effects on student growth rates in math. The study supports a perspective on distributed leadership that aims at building the academic capacity of schools as a means of improving student learning outcomes.
• The Hispanic-White Achievement Gap in Math and Reading in the Elementary Grades, by Sean F. Reardon and Claudia Galindo. Abstract: This article describes the developmental patterns of Hispanic-White math and reading achievement gaps in elementary school, paying attention to variation in these patterns among Hispanic subgroups. Compared to non-Hispanic White students, Hispanic students enter kindergarten with much lower average math and reading skills. The gaps narrow by roughly a third in the first 2 years of schooling but remain relatively stable for the next 4 years. The development of achievement gaps varies considerably among Hispanic subgroups. Students with Mexican and Central American origins—particularly first- and second-generation immigrants—and those from homes where English is not spoken have the lowest math and reading skill levels at kindergarten entry but show the greatest achievement gains in the early years of schooling.
  

Transitional stages and students' motivation

Eirini Geraniou has written an article called The transitional stages in the PhD degree in mathematics in terms of students’ motivation. This article was published online in Educational Studies in Mathematics on Friday. Here is the abstract of Geraniou's article:

This paper presents results of a longitudinal study in the transition to independent graduate studies in mathematics. The analysis of the data collected from 24 students doing a PhD in mathematics revealed the existence of three transitional stages within the PhD degree, namely Adjustment, Expertise and Articulation. The focus is on the first two transitional stages, since the data collection focused mainly on these. Based on the first two transitional stages and the students’ ways of dealing with them, which were called ‘survival strategies’, three types of students were identified. The importance of motivation for each transitional stage and the successful transition overall are considered as well.

ZDM, August 2009

Summer is over, and I am back at work (and blogging)! I am not going to try and catch up with everything that has been published and done during my vacation, but rather start with what is new now.

One of the major journals - ZDM - has recently released a new issue: Volume 41, Number 4. This issue contains 11 articles, in addition to the introduction by Stephen J. Hegedus and Luis Moreno-Armella.

• Intersecting representation and communication infrastructures, by Stephen J. Hegedus and Luis Moreno-Armella
• Sounds and pictures: dynamism and dualism in Dynamic Geometry, by Nicholas Jackiw and Nathalie Sinclair
• Artifacts and signs after a Vygotskian perspective: the role of the teacher, by Maria Alessandra Mariotti
• Time for telling stories: narrative thinking with dynamic geometry, by Nathalie Sinclair, Lulu Healy and Cassia Osorio Reis Sales
• Potential scenarios for Internet use in the mathematics classroom, by Marcelo C. Borba
• “No! He starts walking backwards!”: interpreting motion graphs and the question of space, place and distance, by Luis Radford
• Dynamic mathematics and the blending of knowledge structures in the calculus, by David O. Tall
• Broadening the sense of ‘dynamic’: a microworld to support students’ mathematical generalisation, by Richard Noss, Celia Hoyles, Manolis Mavrikis, Eirini Geraniou, Sergio Gutierrez-Santos and Darren Pearce
• Co-action with digital technologies, by Luis Moreno-Armella and Stephen J. Hegedus
• Book review: Menghini, M., Furinghetti, F., Giacardi, L. and Azarello, F. (eds.): The first century of the International Commission on Mathematical Instruction (1908–2008): reflecting and shaping the world of mathematics education, by Michael N. Fried
• ICMI Study 20: educational interfaces between mathematics and industry, by Alain Damlamian and Rudolf Sträßer