# Alert 51 – ESM juni 2013

Inhoudsopgave Educational Studies in Mathematics (Volume 83 Number 3):

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Generating and using examples in the proving process

J. Sandefur, J. Mason, G. J. Stylianides & A. Watson.

We report on our analysis of data from a dataset of 26 videotapes of university students working in groups of 2 and 3 on different proving problems. Our aim is to understand the role of example generation in the proving process, focusing on deliberate changes in representation and symbol manipulation. We suggest and illustrate four aspects of situations in which example generation seems to play a positive role in proving. These aspects integrate qualities of students and of problems: experience of utility of examples in proving, personal example spaces and technical tools, formulation of the problem, and relational necessity. Our analysis led to integrating two theoretical ideas: the alignment of conceptual insight and technical handle when trying to prove; and manipulating, getting-a-sense-of, and articulating as phases of work associated with example construction.

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The co-construction of learning difficulties in mathematics—teacher–student interactions and their role in the development of a disabled mathematical identity

Einat Heyd-Metzuyanim

Leaning on a communicational framework for studying social, affective, and cognitive aspects of learning, the present study offers a new look at the construction of an identity of failure in mathematics as it occurs through teaching–learning interactions. Using the case of Dana, an extremely low-achieving student in 7th grade mathematics, I attempt to unearth the mechanisms of interaction between Dana and myself, her teacher, that instead of advancing Dana, perpetuated her failure. Through examining the interactional routines followed by Dana and me, I show how Dana’s deviations from normative routines resulted in my identification of Dana as “clueless” in mathematics. This identification, shared both by Dana and by me, was accompanied by adherence to ritual rule following that did not enable Dana’s advancement in mathematical discourse. This case points to the need to re-examine permanent difficulties in mathematics in light of the reciprocal nature of such difficulties’ (re)construction in teaching-and-learning interactions.

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Examining calculator use among students with and without disabilities educated with different mathematical curricula

Emily C. Bouck, Gauri S. Joshi & Linley Johnson

This study assessed if students with and without disabilities used calculators (fourfunction, scientific, or graphing) to solve mathematics assessment problems and whether using calculators improved their performance. Participants were sixth and seventh-grade students educated with either National Science Foundation (NSF)-funded or traditional mathematics curriculum materials. Students solved multiple choice and open-ended problems based on items from the State’s released previous assessments. A linear mixed model was conducted for each grade to analyze the factors impacting students’ self-reported calculator use. Chi Square tests were also performed on both grade’s data to determine the relationship between using a calculator and correctly solving problems. Results suggested only time as a main factor impacting calculator use and students who self-reported using a calculator were more likely to answer questions correctly. The results have implications for practice given the controversy over calculator use by students both with and without disabilities.

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How secondary level teachers and students impose personal structure on fractional expressions and equations—an expert-novice study

Christian Rüede

While an algebraic expression is typically assigned a regular structure, this article introduces the concept of personal structure (Strukturierung); here, the structuring of an algebraic expression is understood as the act of forming relationships between its parts. This concept is used for the analysis of interviews in which experts and novices talk about the personal structures they produced when looking at fractional expressions and equations. The results show, firstly, that experts will structure a given fractional expression in various different ways. Secondly, four categories have been identified: structuring can mean that an expression is made more visually straightforward, changed, reinterpreted or classified. This is meaningful for negotiating the appropriateness of the personal structures in teaching algebra.

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University students’ grasp of inflection points

Pessia Tsamir & Regina Ovodenko

This paper describes university students’ grasp of inflection points. The participants were asked what inflection points are, to mark inflection points on graphs, to judge the validity of related statements, and to find inflection points by investigating (1) a function, (2) the derivative, and (3) the graph of the derivative. We found four erroneous images of inflection points: (1) f ′ (x) = 0 as a necessary condition, (2) f ′ (x) ≠ 0 as a necessary condition, (3) f ″ (x) = 0 as a sufficient condition, and (4) the location of “a peak point, where the graph bends” as an inflection point. We use the lenses of Fischbein, Tall, and Vinner and Duval’s frameworks to analyze students’ errors that were rooted in mathematical and in real-life contexts.

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The juxtaposition of instructor and student perspectives on mathematics courses for elementary teachers

Lynn C. Hart, Susan Oesterle & Susan L. Swars

This qualitative study examined perspectives of two key stakeholder groups, instructors and students, on mathematics content courses for prospective elementary teachers (Mathematics for Teachers [MFT] courses). A collective case study approach, which drew from the data of two cases in different but comparable settings, contributed to the robustness of the findings. Cross-case analysis of the interview data revealed several convergent themes: the role of affect in student learning, pedagogy and instructor disposition, connections to the elementary classroom, and mathematics content. The findings included both conflicting and complementary perspectives between the two key stakeholder groups. When juxtaposed, the multiple viewpoints offer insights into some of the central issues related to teaching and learning in MFT courses and suggest potential avenues for improving experiences in these courses.

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New materialist ontologies in mathematics education: the body in/of mathematics

Elizabeth de Freitas & Nathalie Sinclair

In this paper we study the mathematical body as an assemblage of human and non-human mathematical concepts. We argue that learners’ bodies are always in the process of becoming assemblages of diverse and dynamic materialities. Following the work of the historian of science Karen Barad, we argue that mathematical concepts must be considered dynamic material, and we suggest a “pedagogy of the concept” that animates concepts as both logical and ontological. We draw on the philosopher of mathematics Gilles Châtelet in order to pursue this argument, elaborating on the way that mathematical concepts partake of the mobility of the virtual, while learners, in engaging with this mobility, enter a material process of becoming. We show how the concept of virtuality allows us to look at mathematical concepts in school curriculum in new ways.

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“Bigger number means you plus!”—Teachers learning to use clinical interviews to understand students’ mathematical thinking

Mary Anne Heng & Akhila Sudarshan

This paper examines the perceptions and understandings of ten grades 1 and 2 Singapore mathematics teachers as they learned to use clinical interviews (Ginsburg, Human Development 52:109–128, 2009) to understand students’ mathematical thinking. This study challenged teachers’ pedagogical assumptions about what it means to teach for student understanding. Clinical task-based interviews opened a window into students’ knowledge, problem-solving and reasoning, and helped teachers reflect on their teaching and assessment of student learning. Teachers also learnt about what it means to establish a culture of thoughtful questioning in the classroom and developed an emerging awareness that this requires a readiness to hear students’ ideas and connect informal or invented strategies with classroom mathematics.

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Book Review: Expertise in mathematics instruction: an international perspective. Yeping Li and Gabriele Kaiser (Eds.) (2011)

Kristin King Hess

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