How do mathematicians learn math?: resources and acts for constructing and understanding mathematics

In this paper, we present an analytic framework for investigating expert mathematical learning as the process of building a network of mathematical resources by establishing relationships between different components and properties of mathematical ideas. We then use this framework to analyze the reasoning of ten mathematicians and mathematics graduate students that were asked to read and make sense of an unfamiliar, but accessible, mathematical proof in the domain of geometric topology. We find that experts are more likely to refer to definitions when questioning or explaining some aspect of the focal mathematical idea and more likely to refer to specific examples or instantiations when making sense of an unknown aspect of that idea. However, in general, they employ a variety of types of mathematical resources simultaneously. Often, these combinations are used to deconstruct the mathematical idea in order to isolate, identify, and explore its subcomponents. Some common patterns in the ways experts combined these resources are presented, and we consider implications for education.     

Intuitive vs analytical thinking: four perspectives

This article is an attempt to place mathematical thinking in the context of more general theories of human cognition. We describe and compare four perspectives—mathematics, mathematics education, cognitive psychology, and evolutionary psychology—each offering a different view on mathematical thinking and learning and, in particular, on the source of mathematical errors and on ways of dealing with them in the classroom. The four perspectives represent four levels of explanation, and we see them not as competing but as complementing each other. In the classroom or in research data, all four perspectives may be observed. They may differentially account for the behavior of different students on the same task, the same student in different stages of development, or even the same student in different stages of working on a complex task. We first introduce each of the perspectives by reviewing its basic ideas and research base. We then show each perspective at work, by applying it to the analysis of typical mathematical misconceptions. Our illustrations are based on two tasks: one from statistics (taken from the psychological research literature) and one from abstract algebra (based on our own research).

Students’ Learning with the Connected Chemistry (CC1) Curriculum: Navigating the Complexities of the Particulate World

The focus of this study is students’ learning with a Connected Chemistry unit, CC1 (denotes Connected Chemistry, chapter 1), a computer-based environment for learning the topics of gas laws and kinetic molecular theory in chemistry (Levy and Wilensky 2009). An investigation was conducted into high-school students’ learning with Connected Chemistry, based on a conceptual framework that highlights several forms of access to understanding the system (submicro, macro, mathematical, experiential) and bidirectional transitions among these forms, anchored at the common and experienced level, the macro-level. Results show a strong effect size for embedded assessment and a medium effect size regarding pre-post-test questionnaires. Stronger effects are seen for understanding the submicroscopic level and bridging between it and the macroscopic level. More than half the students succeeded in constructing the equations describing the gas laws. Significant shifts were found in students’ epistemologies of models: understanding models as representations rather than replicas of reality and as providing multiple perspectives. Students’ learning is discussed with respect to the conceptual framework and the benefits of assessment of learning using a fine-tuned profile and further directions for research are proposed. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.

Models of academic governance during a period of nation-building: the Hebrew University in the 1920s–1960s

This paper explores the development of the structures of university governance at the Hebrew University of Jerusalem between the 1920s and 1960s. The model that ultimately prevailed, a state-sponsored model of governance, dominated the higher education system in Israel until the early 2000s and was characterised by the dominance of academic faculty, a status that the government accorded to the faculty in exchange for their acceptance of the state’s normative vision for universities’ role in society. Two main governance models that were instituted at the HU are identified: (1) the Diaspora university, 1925–1950, a distinctive governance model that emerged in the pre-state period and was controlled by Jewish communities in the Diaspora, and (2) the state university, 1950s–2000s, which shifted the centre of gravity to the state. These models are further divided into sub-models and the processes described through which academic autonomy was institutionalised during a period of nation-building.     

Decision‐Making in Future Science and Technology Curricula

Summaries

English

In this article, the author argues in favour of a decision‐making orientated science and technology curriculum for secondary school students. To achieve this, the curriculum should (i) expose students to open‐ended problems within their natural setting, (ii) provide students with real decision‐making situations and (iii) involve them in scientific‐technological social actions, e.g. in community institutions or industrial plants.     

Examination-type preferences of secondary school students and their teachers in the science disciplines

A specially developed questionnaire was used: Types of Preferred Examinations (TOPE) to assess examination-type preferences of secondary school students in the Science disciplines according to school type affiliate and gender.Structured interviews were employed to assess both the rationale of students towards these preferences as well as teacher awareness about the preferences – in contrast to their actual examination practice.Our findings suggest that (a) secondary school students prefer written, unlimited time examinations which, according to their perception, stress learning with understanding rather than mechanical rote learning, and in which the use of supporting material (open book exams) is permitted; and (b) secondary school Science teachers are aware of student examination-type preferences, yet they continue to use the traditional written, time-limited – class examination which is definitely not preferred (disliked) by their students.In view of the special emphasis in current science education research on students' development of higher-order cognitive skills (HOCS) and the need for consonance between the new curriculum goals and examination types used, it is proposed that provisions be made to facilitate teachers' compliance with students' examination-type preferences provided the latter are congruent with learning objectives and our educational aspirations.     

Defining Computational Thinking for Mathematics and Science Classrooms

Science and mathematics are becoming computational endeavors. This fact is reflected in the recently released Next Generation Science Standards and the decision to include “computational thinking” as a core scientific practice. With this addition, and the increased presence of computation in mathematics and scientific contexts, a new urgency has come to the challenge of defining computational thinking and providing a theoretical grounding for what form it should take in school science and mathematics classrooms. This paper presents a response to this challenge by proposing a definition of computational thinking for mathematics and science in the form of a taxonomy consisting of four main categories: data practices, modeling and simulation practices, computational problem solving practices, and systems thinking practices. In formulating this taxonomy, we draw on the existing computational thinking literature, interviews with mathematicians and scientists, and exemplary computational thinking instructional materials. This work was undertaken as part of a larger effort to infuse computational thinking into high school science and mathematics curricular materials. In this paper, we argue for the approach of embedding computational thinking in mathematics and science contexts, present the taxonomy, and discuss how we envision the taxonomy being used to bring current educational efforts in line with the increasingly computational nature of modern science and mathematics.     

Higher and lower-order cognitive skills: The case of chemistry

A major driving force in the current effort to reform science education is the conviction that it is vital for our students to develop their higher-order cognitive skills capacity in order to function effectively in our modem, complex science and technology-based society. In line with this rationale, this study focuses on the use of examinations for studying student performance in chemistry examination on items that require higher-order cognitive skills (HOCS) or lower-order cognitive skills (LOCS). This usage of examinations is explored and demonstrated via “post-factum” data analysis of two case studies: the General Examination (in chemistry) and the Panhellenic Chemistry Competition administered natinally in Greece for secondary-school graduates in 1991. The main findings were: (a) students performed considerably lower on questions requiring HOCS than on those requiring LOCS; (b) performance on questions requiring HOCS may not correlate with that on questions requiring LOCS for which affective factors, LOCS-orientation in teaching and the extent of prior examination preparation may be responsible; and (c) examinations that contain intems of both types can be effectively used to identify HOCS- and LOCS- students within various contexts of chemistry teaching. Based on the above and previous related studies, the fostering of students' HOCS by appropriate teaching and assessment trategies is advocated.