Miriam Amit

The ‘What is a system’ reflection interview as a knowledge integration activity for high school students’ understanding of complex systems in human biology

ABSTRACT This study examined the reflection interview as a tool for assessing and facilitating the use of ‘systems language’ amongst 11th grade students who have recently completed their first year of high school biology. Eighty-three students composed two concept maps in the 10th grade—one at the beginning of the school year and one at its end. The first part of the interview is dedicated to guiding the students through comparing their two concept maps and by means of both explicit and non-explicit teaching. Our study showed that the explicit guidance in comparing the two concept maps was more effective than the non-explicit, eliciting a variety of different, more specific, types of interactions and patterns (e.g. ‘hierarchy’, ‘dynamism’, ‘homeostasis’) in the students’ descriptions of the human body system. The reflection interview as a knowledge integration activity was found to be an effective tool for assessing the subjects’ conceptual models of ‘system complexity’, and for identifying those aspects of a system that are most commonly misunderstood.

High-stakes assessment as a tool for promoting mathematical literacy and the democratization of mathematics education

Abstract The extensive use of National Completion Examinations in Mathematics (NCEM) as a critical filter for educational and social advancement highlights the tension between society’s ambition for high achievement and its ambition for general mathematical literacy. This paper presents an innovative model for a NCEM that demonstrates how such examinations can serve to promote mathematical literacy for all, without losing its ability to promote high achievement. In the course of the paper, the meaning of mathematical literacy for all is also discussed; it is stressed that at the heart of this notion lies students’ openness to mathematics, so that the success of a program directed towards mathematical literacy for all is reasonably measured by the number of students who persist in their study of mathematics and who develop a confident attitude towards it. Data were given suggesting that the new model for the NCEM is successful in this respect.

The co-development and interrelation of proof and authority: The case of Yana and Ronit

Students’ mathematical lives are characterized not only by a set of mathematical ideas and the engagement in mathematical thinking, but also by social relations, specifically, relations of authority. Watching student actions and speaking to students, one becomes cognizant of a ‘web of authority’ ever present in mathematics classrooms. In past work, it has been shown how those relations of authority may sometimes interfere with students’ reflecting on mathematical ideas. However, “…by shifting the emphasis from domination and obedience to negotiation and consent…” (Amit & Fried, 2005, p.164) it has also been stressed that these relations are fluid and are, in fact, asine qua non in the process of students’ defining their place in a mathematical community. But can these fluid relations be operative also in the formation of specific mathematical ideas? It is my contention that they may at least coincide with students’ thinking about one significant mathematical idea, namely, the idea ofproof. In this talk, I shall discuss both the general question of authority in the mathematics classroom and its specific connection with students’ thinking about proof in the context of work done in two 8th grade classrooms.

Assessing a Modeling Process of a Linear Pattern Task

This research investigated the process of generalizing a pictorial linear pattern problem, as done by fifty-three mathematically promising students participating in an after school math club. The students’ work revealed a range of solution paths and representations, and a cycle of expressing – testing – revising. While the majority of them found the constant difference property of the pattern, they experienced difficulties in expressing the general rule. The majority of students applied recursive strategies, even when more global strategies were called for. Although the aforementioned task lacks a real-life context that is essential for modeling problems, the advantages of such problems in multi-cultural classes are discussed.

Model Eliciting Environments as “Nurseries” for Modeling Probabilistic Situations

This study presents an extension of model-eliciting problems into model-eliciting environments which are designed to optimize the chances that significant modeling activities will occur. Our experiment, conducted in such an environment, resulted in the modeling of a probabilistic situation. Students in grades 6–9 participated in competitive games involving rolling dice. These tasks dealt with the concept of fairness, and the desire to win connected students naturally to familiar “real life” situations. During a “meta-argumentation” process, results were generalized, and a model was formed. In this case, it was a model describing a “fair game” created by the differential compensation of different events to “even the odds.” The strength of this model can be seen in its ability to first reject preexisting knowledge which is partial or incorrect, and second to verify the knowledge that survives the updating and refining process. Thus, a two-directional process is created – the knowledge development cycles lead to a model, and the model helps to retroactively examine the knowledge in previous stages of development.

Weaving Culture and Mathematics in the Classroom: The Case of Bedouin Ethnomathematics

Our study attempted to address young Bedouin students’ persistent difficulties with mathematics by integrating ethnomathematics into a standard curriculum. First, we conducted extensive interviews with 35 Bedouin elders to identify the mathematical elements of their daily lives—particularly traditional units of length and weight. We then combined these with the standard curriculum to make an integrated 30-hour 7th grade teaching unit that was implemented in two Bedouin schools. Comparisons between the experimental group (75) and the control group (70) showed that studying the integrated curriculum improved the students’ self-perception and motivation, but had almost no effect on achievements in school tests that were conducted immediately after the experiment. The experiment had an extra social impact, changing students’ attitudes to their own culture and the tribe’s older generation.

Commentary 1 on Re-conceptualizing Mathematics Education as a Design Science

The term design research was borrowed from “design sciences” such as architecture or engineering—where: (a) many of the most important kinds of systems that need to be understood were designed or developed by humans, (b) the conceptual systems that humans develop in order to design or understand the preceding systems also are used to make new adaptations, and (c) multi-disciplinary perspectives usually are needed to solve most realistically complex problems.

Classroom note: Area of convex regions as a unifying theme for the review of some topics in first-year college courses

Teachers of first-year college mathematics and engineering courses must often spend considerable time reviewing material originally taught in high school. Instead of this being a mere exercise in repetition, this article suggests that such a review can enrich and revitalize by unifying some of the subjects that need to be re-taught. In the example presented, the subjects in question are absolute values, graphs and solutions of equations, and domains of definition. These are unified by the problem of finding an analytic expression for a square and triangle and their interiors. In the course of the development, basic notions such as the additive property of areas and convexity are introduced. The approach presented in the article was tried with secondary school teachers participating in professional development workshops and with students at a technical college; the teachers and students responded enthusiastically to the material.