journal-articles

Students’ understanding of a logical structure in the definition of limit

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Kyeong Hah Roh has written an article entitled An empirical study of students’ understanding of a logical structure in the definition of limit via the ε-strip activity. This article was published online in Educational Studies in Mathematics last Thursday. Here is the abstract of the article:

This study explored students’ understanding of a logical structure in defining the limit of a sequence, focusing on the relationship between ε and N. The subjects of this study were college students who had already encountered the concept of limit but did not have any experience with rigorous proofs using the ε–N definition. This study suggested two statements, each of which is written by using a relationship between ε and N, similar to the ε–N definition. By analyzing the students’ responses to the validity of the statements as definitions of the limit of a sequence, students’ understanding of such a relationship was classified into five major categories. This paper discusses some essential components that students must conceptualize in order properly to understand the relationship between ε and N in defining the limit of a sequence.

Working like real mathematicians

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Atara Shriki has written an interesting article called Working like real mathematicians: developing prospective teachers’ awareness of mathematical creativity through generating new concepts. This article was recently published online in Educational Studies in Mathematics. The author reports from a study related to a methods course, where a strong focus is on creativity in mathematics. The article has a particular focus on prospective teachers’ awarenes of creativity in mathematics.

Here is the abstract of Shriki’s article.

This paper describes the experience of a group of 17 prospective mathematics teachers who were engaged in a series of activities aimed at developing their awareness of creativity in mathematics. This experience was initiated on the basis of ideas proposed by the participants regarding ways creativity of school students might be developed. Over a period of 6 weeks, they were engaged in inventing geometrical concepts and in the examination of their properties. The prospective teachers’ reflections upon the process they underwent indicate that they developed awareness of various aspects of creativity while deepening their mathematical and didactical knowledge.

Exploration of technologies

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Paulus Gerdes has written an article called Exploration of technologies, emerging from African cultural practices, in mathematics (teacher) education. This article was recently published online in ZDM. In this article, Gerdes provides an interesting overview of how the cultural practices of African mathematics (teacher) education has developed, and he makes a seemingly (to me) impossible connection between traditional basket weaving and exploration of technologies.

Here is the abstract of the article:

The study at teacher education institutions in Africa of mathematical ideas, from African history and cultures, may broaden the horizon of (future) mathematics teachers and increase their socio-cultural self-confidence and awareness. Exploring educationally mathematical ideas embedded in, and derived from, technologies of various African cultural practices may contribute to bridge the gap between ‘home’ and ‘school’ culture. Examples of the study and exploration of these technologies and cultural practices will be presented. The examples come from cultural practices as varied as story telling, basket making, salt production, and mat, trap and hat weaving.

IJSME, August 2009

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The August issue (Volume 7, Number 4) of International Journal of Science and Mathematics Education has been published. This issue contains 9 articles:

School mathematics curriculum materials for teachers’

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Gwendolyn M. Lloyd has written an article that was recently published online in ZDM. The article is entitled School mathematics curriculum materials for teachers’ learning: future elementary teachers’ interactions with curriculum materials in a mathematics course in the United States. Here is the abstract of her article:

This report describes ways that five preservice teachers in the United States viewed and interacted with the rhetorical components (Valverde et al. in According to the book: using TIMSS to investigate the translation of policy into practice through the world of textbooks, Kluwer, 2002) of the innovative school mathematics curriculum materials used in a mathematics course for future elementary teachers. The preservice teachers’ comments reflected general agreement that the innovative curriculum materials contained fewer narrative elements and worked examples, as well as more (and different) exercises and question sets and activity elements, than the mathematics textbooks to which the teachers were accustomed. However, variation emerged when considering the ways in which the teachers interacted with the materials for their learning of mathematics. Whereas some teachers accepted and even embraced changes to the teaching–learning process that accompanied use of the curriculum materials, other teachers experienced discomfort and frustration at times. Nonetheless, each teacher considered that use of the curriculum materials improved her mathematical understandings in significant ways. Implications of these results for mathematics teacher education are discussed.

Understanding the complexities of student motivations

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Janet G. Walter and Janelle Hart have written an article about the interesting issue of Understanding the complexities of student motivations in mathematics learning. The article was recently published in The Journal of Mathematical Behavior. Here is the abstract of their article:

Student motivation has long been a concern of mathematics educators. However, commonly held distinctions between intrinsic and extrinsic motivations may be insufficient to inform our understandings of student motivations in learning mathematics or to appropriately shape pedagogical decisions. Here, motivation is defined, in general, as an individual’s desire, power, and tendency to act in particular ways. We characterize details of motivation in mathematical learning through qualitative analysis of honors calculus students’ extended, collaborative problem solving efforts within a longitudinal research project in learning and teaching. Contextual Motivation Theory emerges as an interpretive means for understanding the complexities of student motivations. Students chose to act upon intellectual-mathematical motivations and social-personal motivations that manifested simultaneously. Students exhibited intellectual passion in persisting beyond obtaining correct answers to build understandings of mathematical ideas. Conceptually driven conditions that encourage mathematical necessity are shown to support the growth of intellectual passion in mathematics learning.

How to develop mathematics for teaching and understanding

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Susanne Prediger has written an article about How to develop mathematics-for-teaching and for understanding: the case of meanings of the equal sign. The article was published online in Journal of Mathematics Teacher Education on Thursday last week. Point of departure in her article is the very important question about what mathematical (content) knowledge prospective teachers need. A main claim which is raised already in the introduction is: “Listen to your students!” In the theoretical background, Prediger takes Shulman’s classic conceptualization of three main categories of content knowledge in teaching as point of departure:

  1. Subject-matter knowledge
  2. Pedagogical-content knowledge
  3. Curricular knowledge

She continues to build heavily on the work done by Hyman Bass and Deborah Ball (e.g. Ball & Bass, 2004), and she goes on to place her own study in relation to the work of Bass and Ball:

Whereas Bass and Ball (2004) concentrate on the first part of their program, namely, identifying important competences, this article deals with both parts, the analytical study of identifying, and the developmental study of constructing a sequence for teacher education, exemplified by a sequence in the course entitled school algebra and its teaching and learning for second-year, prospective middle-school teachers.

Here is the abstract of Prediger’s article:

What kind of mathematical knowledge do prospective teachers need for teaching and for understanding student thinking? And how can its construction be enhanced? This article contributes to the ongoing discussion on mathematics-for-teaching by investigating the case of understanding students’ perspectives on equations and equalities and on meanings of the equal sign. It is shown that diagnostic competence comprises didactically sensitive mathematical knowledge, especially about different meanings of mathematical objects. The theoretical claims are substantiated by a report on a teacher education course, which draws on the analysis of student thinking as an opportunity to construct didactically sensitive mathematical knowledge for teaching for pre-service middle-school mathematics teachers.

References:
Bass, H., & Ball, D. L. (2004). A practice-based theory of mathematical knowledge for teaching: The case of mathematical reasoning. In W. Jianpan & X. Binyan (Eds.), Trends and challenges in mathematics education (pp. 107–123). Shanghai: East China Normal University Press.

"The conference was awesome"

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Tamsin Meaney, Tony Trinick and Uenuku Fairhall have written an article with an interesting focus on professional development and mathematics teacher conferences. The title of their article is ‘The conference was awesome’: social justice and a mathematics teacher conference. The article was recently published online in Journal of Mathematics Teacher Education. Here is the abstract of their article:

Professional development comes in many forms, some of which are deemed more useful than others. However, when groups of teachers are excluded, or exclude themselves, from professional development opportunities, then there is an issue of social justice. This article examines the experiences of a group of teachers from a Māori-medium school who attended a mathematics teacher conference. By analysing the teachers’ sense of belonging through their ideas about engagement, alignment and imagination, we are able to describe how different kinds of relationships influence the inclusion/exclusion process. This leads to a discussion about what can be done by the teachers as well as conference organisers to increase these teachers’ likelihood of attending further conferences in the future.

Alignment, cohesion, and change

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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.

Tutored problem solving

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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.