In this article we report on a study focused on revealing and categorizing the arguments that preservice mathematics teachers put forward when they are asked about why mathematics is taught, which is a question closely related to the justification problem in mathematics education. Another focus of the study is the identification of myths within…

The purpose of this article is to show how the philosophical theory of inferentialism can be used to understand students' conceptual development in the field of mathematics. Based on the works of philosophers such as Robert Brandom, an epistemological theory in mathematics education is presented that offers the opportunity to trace students'…

Supporting young students in developing meanings for the set-theoretic function definition is emphasized in 6th-12th grade curricula around the world. In prior work, we have highlighted how covariational reasoning supports college students in constructing relationships that afford considering mathematical properties important for a set-theoretic…

In this paper, the application of fuzzy logic in mathematical education is viewed from the perspective of pre- school education. The aim of the paper is to give a brief overview of examples from the literature related to fuzzy logic and to point out the presence of fuzzy linguistic variables in the everyday life of a preschool child, as well as…

In the field of human cognition, language plays a special role that is connected directly to thinking and mental development (e.g., Vygotsky, "1938"). Thanks to "verbal thought", language allows humans to go beyond the limits of immediately perceived information, to form concepts and solve complex problems (Luria, "1975"). So, it appears language…

SProblem solving has a long and successful history in mathematics education and is valued by many teachers as a way to engage and facilitate learning within their classrooms. The potential benefit for using problem solving in the development of algebraic thinking is that "it may broaden and develop students' mathematical thinking beyond the…

Critical thinking is the sort of mental activity that uses facts to plan, order, and work toward an end; seeks meaning or an explanation; is self-reflective; and uses reason to question claims and make judgments. Any subject--be it physics, algebra, or auto repair--can promote critical thinking as long as teachers teach the subject matter in…

Analyzes one student's thinking using the Van Hiele levels of geometric thinking. (KHR)

Explores the development of understanding of the concept of average with students from grades 3 to 9 through interviews. Observed six levels of response based on an hierarchical model of cognitive functioning. Documents usage of ideas associated with the three standard measures of central tendency and representation as strategies for problem…

Fifty 1-to 3-year-olds were asked to imitate: (1) the rolling of a cylinder down an incline and (2) the making of an incline with a board and a block. Their incorrect imitations and progress were videotaped and analyzed, and interpreted in light of Piaget's theory about the elaboration of logico-mathematical relationships in an interrelated way.…

This paper reports on a portion of a 14-year study of the mathematical thinking of a cohort group of students that was based on an identifiable perspective on how mathematical ideas are built. Three videotape episodes are presented so that groups of students working together can be viewed at three points in time: as fourth graders (ages 9-10) and…

Sixty Japanese children ranging in age from 3 years 4 months to 7 years 5 months were individually interviewed with three Piagetian tasks. Children's levels of representation were assessed by asking for a graphic representation of 4 dishes, 6 pencils, 8 small blocks etc. A conservation-of-number task was then given to assess children's level of…

Elementary school children were interviewed about how gears move on a gearboard and how they work in commonplace machines. Children's reasoning became more general, formal, and mathematical as problem complexity increased, suggesting that mathematical forms of reason may develop when they provide a clear advantage over simple causal…

The major idea in this paper is the formulation of a theory of three distinct but interrelated worlds of mathematical thinking each with its own sequence of development of sophistication, and its own sequence of developing warrants for truth, that in total spans the range of growth from the mathematics of new-born babies to the mathematics of…

Discusses the nature of tasks used to elicit the development of such systems by middle school students. Analyzes mathematical reasoning development of students across tasks and the diversity of thinking patterns identified on problem tasks. Discusses student reasoning about the relationships between and among quantities and their application in…