One of the most salient arguments in favor of a semiotic approach, put forward on various occasions among others by Luis Radford, claims that semiotics is most appropriate to treat the interaction between socio-cultural and objective aspects of knowledge problems. But if we want to take such claims seriously, we have to undertake revisions of our…

This article addresses an important problem of graphing quantitative data: should one include zero on the scale showing magnitude? Based on a real time series example, the problem is discussed and some recommendations are proposed.

We describe a set of manipulatives developed to help students learn three-dimensional Cartesian, cylindrical, and spherical coordinate systems. (Contains 6 figures.)

This paper explores how models can support productive thinking. For us a model is a "thing", a tool to help make sense of something. We restrict attention to specific models for whole-number multiplication, hence the wording of the title. They support evolving thinking in large measure through the ways their users redesign them. They assume new…

In this paper, a focus is made on defining processes at stake in an unfamiliar situation coming from discrete mathematics which brings surprising mathematical results. The epistemological framework of Lakatos is questioned and used for the design and the analysis of the situation. The cognitive background of Vergnaud's approach enriches the study…

The purpose of this article is to explore student-generated connections among counting problems. The literature indicates that such problems pose difficulties for students, who struggle to detect common structures and identify models of underlying problem types. A case study is presented here, in which students elaborate upon connections they make…

There is an over-arching consensus that the use of the history of mathematics should decidedly improve the quality of mathematics teaching. Mathematicians and mathematics educators show here a rare unanimity. One deplores, however, and in a likewise general manner, the scarcity of positive examples of such a use. This paper analyses whether there…

This study investigated the conceptual understanding of measures of spread among community college students in an introductory statistics course. The course is centered around deemphasizing computational skills and focused, rather, on development of conceptual understanding. Open-ended questions were developed to explore and assess students'…

Drawing from sources in gesture studies, cognitive science, the anthropology of religion and art/architecture history, this article explores cultural, bodily and cosmological resonances carried (unintentionally) by mathematical graphs on Cartesian coordinates. Concepts of asymmetric bodily spaces, grids, orthogonality, mapping and sacred spaces…

The focus of this study in which the theoretical framework of APOS was used is students' generalizing function notion from single variable to two-variable function concepts in Analysis II course in the elementary mathematics education program. In the teaching process, teaching activities that support generalizing the function notion with multiple…

This paper explores misconceptions of the number line which are revealed when pre-service primary teachers locate negative decimals on a number line. Written test responses from 94 pre-service primary teachers provide an initial data source which is supplemented by group responses to worksheets completed during a lesson and individual interviews.…

The authors wanted their students to consider mathematical notions of space and dimension, which are typically not part of the middle school curriculum. They developed a two-week lesson that was drawn from the curriculum unit "Exploring the Shape of Space" (Weeks 2001). They began by considering implications of living in a two-dimensional…

Our view of learning takes into account how learners, when learning, are engaged in both doing science and mathematics and becoming mathematicians and scientists. Drawing on our work on a number of European projects that developed inquiry approaches across mathematics and science, we propose a new research and development agenda seeking new ways…

This study examined ways in which students make use of a graphing calculator and how use relates to comfort and understanding with mathematical symbols. Analysis involved examining students' words and actions in problem solving to identify evidence of algebraic insight. Findings suggest that some symbols and symbolic structures had strong…

This article provides an initial comparison of the Principles and Standards for School Mathematics and the Common Core State Standards for Mathematics by examining the fundamental notion of slope. Each set of standards is analyzed using eleven previously identified conceptualizations of slope. Both sets of standards emphasize Functional Property,…