Studies that address preservice teachers' knowledge of area measurement emphasize their lack of knowledge and their tendency towards the use of formulas, without offering a body of knowledge that helps to address such difficulties. This study offers an approximation of the mathematical knowledge necessary for preservice teachers to solve area…

The benefits of technological and theoretical scaffolding were observed when pre-service teachers aiming to teach upper elementary grades were given three learning-based geometrical inquiry tasks involving inscribed circles. They were asked to collaboratively examine the accompanying geometrical illustration and data for some new or interesting…

A key topic throughout school geometry is measurement--namely, distance, area, and volume. This article focuses on one key idea for finding and justifying the area of two-dimensional (2D) shapes: area-preserving transformations. Although especially pertinent to geometry teachers, this article highlights a vertical connection of ideas progressing…

There are fundamental questions that teachers should ask themselves and their students as they prepare lessons on calculating area. Area measurement is commonly used in everyday life (perhaps to carpet a room or organize a space) and plays a foundational role in mathematics from multiplication all way up to calculus. Despite the usefulness of area…

The purpose of this study is to analyze the teaching strategies of the prospective teachers of mathematics for elementary school in relevance with the geometry formulas. The participants of the study are constituted by six senior students of the program of mathematics teaching for elementary school. In this study, where the qualitative research…

Solution of problems in mathematics, and in particular in the field of Euclidean geometry is in many senses a form of artisanship that can be developed so that in certain cases brief and unexpected solutions may be obtained, which would bring out aesthetically pleasing mathematical traits. We present four geometric tasks for which different proofs…

Equality is such an important concept for children to develop. In this article it is argued that a precise definition is needed to ensure that students are provided with a consistent "picture" of what it is that equality really means.

This study investigated middle school teachers' knowledge of students' misconceptions and the sources of these misconceptions related to the volume of prisms. This case study investigated four middle school teachers who taught 8th grade students in Ankara (Turkey). Based on the analysis of the data, two misconceptions related to the volume of…

The high school curriculum sometimes seems like a disconnected collection of topics and techniques. Theorems like the factor theorem and the remainder theorem can play an important role as a conceptual "glue" that holds the curriculum together. These two theorems establish the connection between the factors of a polynomial, the solutions…

The topics of decimals and polygons were taught to two classes by using challenging tasks, rather than the more conventional textbook approach. Students were given a pre-test and a post-test. A comparison between the two classes on the pre- and post-test was made. Prior to teaching through challenging tasks, students were surveyed about their…

Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in…

This article explores the notion of collateral learning in the context of classic ideas about the summation of powers of the first "n" counting numbers. Proceeding from the well-known legend about young Gauss, this article demonstrates the value of reflection under the guidance of "the more knowledgeable other" as a pedagogical method of making…

Formal requests were recently introduced for integration of ICT in secondary school mathematics. As the main issue, students must develop competence to decide when and how it is appropriate to use available ICT tools and to use them. These new requests put demands on those teachers who have not developed corresponding competencies themselves.…

It is widely attested that university students face considerable difficulties with reasoning in analysis, especially when dealing with statements involving two different quantifiers. We focus in this paper on a specific mistake which appears in proofs where one applies twice or more a statement of the kind "for all X, there exists Y such that R(X,…