In this article, the author asserts that asking and responding to diagnostic questions is the single most important part of teaching secondary school mathematics. He notes the importance of formative assessment and recommends a formative assessment strategy that requires students to be public about their answers to questions, displaying their…

In this Research Commentary, the author explores what is meant by "teaching for understanding" and delves into these questions: How does teaching for understanding interact with the backgrounds of the students who experience it or the attributes of the contexts in which they learn? Which empirical findings are context dependent, and…

In 2001, Kilpatrick, Swafford, and Findell proposed a new way to look at what it means for students to be mathematically proficient. They described mathematical proficiency as comprising five intertwined strands: procedural fluency, conceptual understanding, adaptive reasoning, strategic competence, and productive disposition. The vision is that…

There are infinitely many ways of talking about infinity. The assortment of discourses on learning infinity is infinite as well. When the author says "way of talking" or "discourse," she is concerned with much more than the question of how words are chosen and combined. Ways of talking are not just innocent "auxiliaries" to thinking--they shape…

I begin by appreciating the contributions in the volume that indirectly and directly address the questions: Why do gestures and embodiment matter to mathematics education, what has understanding of these achieved and what might they achieve? I argue, however, that understanding gestures can in general only play an important role in "grasping" the…

This paper describes a model of student attainment of mathematical concepts and its development. In this model three types of activities (developmental, connecting, and abstract) are considered as an overlay on the three ways of representing mathematical concepts (physical/visual, oral, and symbolic). Each activity type involves some means of…

Limitations in children's understanding of the symbols of arithmetic may inhibit choice of appropriate solution procedures. The teacher's role involves negotiation of new meanings for words and symbols to match extensions to solution procedures. (MKR)

Summarizes research related to models of understanding mathematics, describes the fundamental conception of understanding mathematics, discusses basic components substantially common to the process models of understanding mathematics, and presents a theoretical framework of a process model consisting of two axes. (26 references) (MKR)

Too many approaches to mathematics instruction at the elementary level focus only on the production of correct answers rather than on the development of mathematical thinking skills. Author urges encouragement of children's own thought processes. (Author/PGD)

Briefly discusses the importance of the idea of sets in the development of children's number concepts. (RH)

Applies methods of formal concept analysis to an analysis of objectives in the domain of educational mathematics, including sequences and hierarchies of topics within the syllabi, establishing plans for single lessons, analysis of students' mistakes in calculating, and the development of students' mathematical concepts. (33 references) (Author/MKR)

Several experimental studies are reviewed, the results of which suggest that students often fail to acquire an understanding of some of the concepts, relationships, principles, and processes that are fundamental to traditional high school course material. Questions of what it means to understand something and of how to assess understanding are…

Argues the importance of teaching fractions and discusses examples of fundamental problems in students' conceptualization of common fractions and decimal fractions. (SS)

The author shows some causes of failure in the creation of mathematical concepts. One is the stiffening of concept cores, which prevents identification of atypical exemplars and solution of atypical problems and causes a bifurcation between the natural system of everyday concepts and the formal system of school concepts. (Author/MDH)

Examines one preschooler's conversation with author to determine what concepts educators consider "difficult" for this age group. Finds that concepts such as "infinity" and "negative numbers," typically avoided by primary educators, were brought into view in the task-based conversations. Argues that individual…