The implementation of video modeling examples of mathematical hands-on experimentation may provide students with authentic and, at the same time, not too cognitively overwhelming experiences. However, the effectiveness of video modeling examples can be influenced by different characteristics of the observed models. On the one hand, based on the…

The purpose of this article is to demonstrate how exponential functions and patterns can be taught by well-structured lessons composed of classroom activities, mathematical questions, discussions, and solving problems. Sawyer (1943) used a rope-and-post system to introduce the concept of logarithms and to calculate a product by adding the…

The paper describes active learning strategies used in undergraduate college algebra and business calculus courses. There are a variety of active learning strategies described in the literature. We wanted to implement a few that together, satisfy the key characteristics of active learning strategies. The active learning strategies described in…

We describe some classic experiments on the Möbius strip, the projective plane band, and the Klein bottle band. We present our experience with freshmen college students, college teachers, high school students, and Mathematics Education graduate students. These experiments are designed to encourage readers to learn more about the properties of the…

We regularly compare magnitudes and describe these comparisons to other people. This article reports 9 experiments that examine how messages about the relative magnitude of two items affect inferences about the items' spatial arrangement. Native English speakers were given sentences such as "One tree is taller than the other," and their…

This article offers a new interpretation of Piaget's decanting experiments, employing the mathematical notion of equivalence instead of conservation. Some reference is made to Piaget's theories and to his educational legacy, but the focus in on certain of the experiments. The key to the new analysis is the abstraction principle, which…

This article documents the classroom mathematical practices observed in a collegiate level teacher education course related to the circle topic. The course, which was prepared as design research, utilized a dynamic geometry environment which influenced the type and nature of the evolved mathematical practices. The study uses emergent perspective…

In this article we present a didactic proposal for teaching linear algebra based on two compatible theoretical models: emergent models and mathematical modelling. This proposal begins with a problematic situation related to the creation and use of secure passwords, which leads students toward the construction of the concepts of spanning set and…

The sequence in which problems of different concepts are studied during instruction impacts concept learning. For example, several problems of a given concept can be studied together (blocking) or several problems of different concepts can be studied together (interleaving). In the current study, we demonstrate that the 2 sequences impact concept…

This current study examined human children's and adults' automatic processing of proportion using a Stroop-like paradigm. Preschool children and university students compared the areas of two sectors that varied not only in absolute areas but also in the proportions they occupied in their original rounds. A congruity effect was found in both age…

The purpose of this study is to investigate characteristics of limit concepts through the simultaneous use of historical and experimental epistemologies. Based on a historical epistemology which is an investigation of historical developments in a mathematical concept raised in the history of mathematics, four different developments of limit…

Many studies have shown the difficulties of learning and teaching the decimal number system for whole numbers. In the case of numbers bigger than one hundred, complexity is partly due to the multitude of possible relationships between units. This study was aimed to develop conditions of a resource which can help teachers to enhance their teaching…

To prepare students for participation in our society, where interpreting, representing, and manipulating of dynamic phenomena are becoming key activities, we believe that one should start developing a mathematical understanding of change at an early age. We therefore started a design research project to teach the concept of instantaneous speed in…

An experiment was conducted over three recent semesters of an introductory calculus course to test whether it was possible to quantify the effect that difficulty with basic algebraic and arithmetic computation had on individual performance. Points lost during the term were classified as being due to either algebraic and arithmetic mistakes…

Unlike a child's observable, physical interactions with mathematical tools (e.g., physically touching blocks in order to count them), the subtle manifestations of imagery construction can be considerably more challenging to identify and act upon. Although there have been substantive examinations of mental imagery in a variety of mathematical…