Based on the premise that meaningful learning in mathematics occurs when children link their understandings with the symbols and procedures they are taught, three places are identified where children can link symbols and their referents, procedural rules and conceptual knowledge, and answers to problems with other knowledge about what is…

Discusses students' inability to make the connection between manipulative materials and pencil-and-paper calculations in mathematics instruction. Outlines the development of mathematical ideas through the concrete, representational, and abstract phases of instruction. An annotated bibliography listing teacher resources for representational-level…

Investigated long-term retention of mathematical concepts in a Keller plan course conducted without a final examination as compared to a conventional lecture course with a final examination. Results indicate greater long-term retention of concepts for students taking a traditional lecture course with final examination rather than the Keller plan…

Presents objectives, instructional strategies, recommended grade levels, and student worksheets (which can be duplicated) for activities focusing on developing concepts of longer/shorter; estimating and measuring distances between points; estimating and measuring perimeters of polygons; and practice in scale drawing and developing a basis of…

Stresses that powers are all based on geometric progressions which start at unity. Various patterns are discussed. (MNS)

Relationships between mathematical and humanistic cultures and the development of mathematical concepts are presented. The author suggests that courses highlighting the history and relevance of mathematics be developed; possible instructional resources are listed. (SD)

Presented is the second of a two part series on the use of exponential functions. Discussed, with examples, is the exponential growth of compound interest problems. (SL)

Mathematics departments rarely require students to study very much logic before working with proofs. Normally, the most they will offer is contained in a small portion of a "bridge" course designed to help students move from more procedurally-based lower-division courses (e.g., abstract algebra and real analysis). What accounts for this seeming…

To ensure all students have every opportunity to succeed, Indiana adopted world-class academic standards in English-language arts, mathematics, science, and social studies and an assessment system to measure student progress. These standards clearly outline what students should know and be able to do in each subject, at each grade level. This…

This paper considers a group of preservice teachers' construction of concept maps derived from y = x + 5 and y = [pi]x[squared] with emphasis on their conceptual understanding of function. The two statements are perceived to represent a number of different concepts with indications of compartmentalized knowledge structures that might prevent the…

One of the aims of mathematics instruction is to achieve the understanding of mathematical concepts through the development of rich and well organized cognitive representations (Goldin, 1998; NCTM, 2000; DeWindt-King, & Goldin, 2003). In this study the term representation is interpreted as the tool used for representing mathematical ideas such…

In this report, we give a sociocultural account of the mediating functions handheld graphing calculators and social interaction play in students' mathematical understanding. We discuss the evolution of students' abilities to symbolize, model, and develop collective mathematical practices about polynomial inequalities in instrumented activity. In…

Starting from a situated cognition perspective, this paper reports on the activity of 9th grade students who are interpreting the shape of a graph arising from the motion of a bouncing ball. In an unfamiliar context, informed by previous knowledge of similar experiments, the obstacle of understanding why the graph does not start from the origin is…

The teaching and learning of early algebra draw heavily on arithmetic and the relationship between these two forms of activity is much debated. Drawing on interviews with 12 year old pupils we consider the ways in which some pupils used substitution of numbers for letters to extend their ability to manipulate algebraic expressions. We argue that…

Increasing numbers of mathematics educators, policy makers, and researchers believe that algebra should become part of the elementary education curriculum. Such endorsements require careful research. This paper presents the general results of a longitudinal classroom investigation of children's thinking and representations over two and a half…