This paper contains the comments of a discussant reviewing the paper, "The Role of Visual Representations in the Learning of Mathematics" (Abraham Arcavi). A research-based evaluation is made that focuses on the ideas of stochastics and the epistemological triangle. (Contains 28 references.) (DDR)

This essay is based on observation and interpretation of the mathematical language of children while they engage in situations involving fractional numbers. This interpretation includes the consideration of various levels of language use, informal metamorphic and metonymic uses of fractional number language, and the interplay between language use…

As biological and socio-cultural beings, humans are encouraged and aspire to "see" not only what comes "within sight," but also what is not possible to see. Visualization is described as offering a method of seeing the unseen. These ideas are organized into sections that focus on seeing the unseen in data, seeing the unseen in…

This paper gives a brief introduction to a discipline called the cognitive science of mathematics. The theoretical background of the arguments is based on embodied cognition and findings in cognitive linguistics. It discusses Mathematical Idea Analysis, a set of techniques for studying implicit structures in mathematics. Particular attention is…

This paper focuses on some cognitive and didactical phenomena that feature processes and products of pupils in grades 7-12 who learned mathematical proof within technological environments. The main issues in the analysis of students' performances consist of metaphors, deictics, mental times, narratives, functions of dragging, abductions, and…

SCALE-UP is an extension of the highly successful IMPEC (Integrated Math, Physics, Engineering, and Chemistry) project, one of North Carolina State's curricular reform efforts undertaken as part of the SUCCEED coalition. The authors utilize the interactive, collaboratively based instruction that worked well in smaller class settings and find ways…

This paper suggests that the systems of representations that we use in mathematics have a cultural origin and concludes that the knowledge produced with the help of these systems of representation likewise has a cultural origin. This assertion forces a reformulation of the issue of objectivity in terms that differ from those inherited from…

This paper focuses on some main distinctions that are necessary to analyze mathematical knowledge from a learning point of view and explain how many students come up against difficulties at each level of the curriculum. From a learning point of view, visualization cannot be used as an immediate and obvious support for understanding, even though it…

Hypermedia as a learning environment complements traditional instruction as it places the student at a higher level of control. Instead of "wait and see" what the teacher expects the student to learn next, hypermedia stimulates the student to become aware of his/her own cognitive needs and interests. This paper presents a combination of notions…

The Student-Centered Activities for Large Enrollment University Physics (SCALE-UP) project at North Carolina State University (NCSU) is developing a curriculum to promote learning through in-class group activities in introductory physics classes up to 100 students. The authors are currently in Phase II of the project using a specially designed…

Interest in mathematics learning that focuses on understanding, mathematical reasoning, and meaning-making underscores the need to develop ways of analyzing classrooms that foster certain types of learning. The goal of this paper is to show that the constructs of social and sociomathematical norms, which grew out of taking a symbolic…

This paper examines the novel problem solving actions of a pair of college students. The analysis highlights the role of the solvers' inferential processes including abductions, deductions, and inductions as structuring resources that contribute to both their understanding of the problems they face and the emerging novelty that constitutes their…

Students' academic achievement scores have been found to improve with the use of graphic organizers. Unfortunately, most previous research on graphic organizers has been constrained to English speakers in secondary and higher education. This study examines the effectiveness of graphic organizers when used by fourth grade students engaged in…

Cognitive flexibility is indispensable for applying knowledge to new situations. The development of this ability depends on certain conditions such as the attainment of a deep comprehension of the subject matter and the exposure to different knowledge representations. This paper focuses on these conditions and describes a study designed to foster…

Multidimensional knowledge structures, described from a constructivist perspective and aligned with the "Mind as Rhizome" metaphor, provide support for constructivist learning strategies. This qualitative study was conducted to seek empirical support for a description of multidimensional knowledge structures, focusing on the…