This book demonstrates how K-2 students deepen their mathematical ideas when encouraged to represent and communicate their thinking. Numerous vignettes from actual classrooms and reflective comments from teachers provide a model for how show and tell can enhance classroom teaching and improve children's learning. Examples of students' work and…

This paper consists of a learning unit for college algebra in which the problem presents three different payment plans for a tuition plan to pay for college expenses. The details of each payment plan are presented, each followed by several exercises so that comparisons of the results can be made across payment plans. (MM)

This study investigates whether the National Council of Teachers of Mathematics (NCTM) professional standards are appropriate for a two-year college developmental mathematics courses. The NCTM professional standards of worthwhile mathematical tasks, classroom discourse, and teacher reflection were all part of this preparatory mathematics course.…

Our teaching conception acknowledges the teacher's central role as a decision maker, influenced by knowledge, beliefs, and emotions. We believe that teachers' education must be focused on teachers' awareness of the complexity of the teaching process, of the incidence of these factors in it, and of the importance of looking at theory as a strong…

As a result of questions arising from differences in performance on Algebra 1 final examinations by school during the 1999-2000 school year, students' middle school mathematics experiences have received renewed focus. Specific questions have arisen regarding whether all students are sufficiently prepared to handle algebraic content by grade 8 or…

This paper describes "The Migration Method," a new technique for graphing a quadratic function (one of the major foundations of undergraduate mathematics) and finding its zeroes. The method promotes a higher degree of conceptualization, reinforces the concept of symmetry, encourages mental visualization, and strengthens the ability to…

This paper examines the question of whether the introduction and use of the function machine representation as a scaffolding device helps undergraduates enrolled in a developmental algebra course to form a rich, foundational concept of function. It describes students' developing understanding of function as an input/output process and as an…

The major focus of this study is to trace the cognitive development of students throughout a mathematics course and to seek the qualitative differences between those of different levels of achievement. The aspect of the project described here concerns the use of concept maps constructed by the students at intervals during the course. From these…

This book consists of a series of activities aimed at providing a problem solving, hands-on approach so that students can experience concepts in algebra. Topics include ratio and proportion, patterns and formulas, integers, polynomials, linear equations, graphs, and probability. The activities come in the form of reproducible blackline masters…

This book was written as a resource for middle and high school teachers hoping to ease their students' transition to algebra. Drawing on experiences with three professional development programs, key "habits of thinking" that characterize the successful learning and use of algebra are outlined. Strategies teachers can use to cultivate…

This paper presents a developmental model of students' acquisition of competence in quantitative and algebraic problem solving. A key notion underlying the developmental model is a distinction between grounded and abstract representations. Grounded representations, like story problems, are more concrete and familiar, closer to physical objects and…

Recent reports have pointed to the need for teachers to have "deep understanding" of concepts and procedures in school mathematics, particularly those related to functions. However, previous research has demonstrated that many students, including preservice teachers, have limited understanding of functions. This paper describes a set of nonroutine…

Which manual calculation skills are still needed when students use graphic/symbolic calculators or computers with computer algebra systems (CAS)? What should students be able to do manually, i.e. just using paper and pencil? This text is the outcome of a two-day discussion on these questions, held by the four authors. Our answers and proposals are…

The purpose of this study was to determine if the use of Learning Logic, an integrated-learning system, by high school students in Algebra I had a significant effect on the performance of these students when they subsequently enrolled in a traditionally taught Algebra II course. Four high schools were selected in which Learning Logic was used for…