This paper describes an emerging approach to the design of task sequences to promote reflective abstraction. The approach aims at promoting particular mathematical understandings. Central to this approach is the identification of available student activities from which students can abstract the intended ideas. The approach differs from a…

Linear functions are an important part of the middle school mathematics curriculum. Students in the middle grades gain fluency by working with linear functions in a variety of representations (NCTM 2001). Presented in this article is an activity that was used with five eighth-grade classes at three different schools. The activity contains 15 cards…

This study considers tertiary calculus students' and instructors' conceptualizations of slope. Qualitative techniques were employed to classify responses to 5 items using conceptualizations of slope identified across various research settings. Students' responses suggest that they rely on procedurally based conceptualizations of…

Activity-based instructional strategies promote physical interaction with manipulatives to represent abstract concepts. As a means to improve student achievement in mathematics, educational researchers and practitioners often recommend instructional strategies that capitalize on the assumed benefits of manipulatives. Recent research has indicated…

Looking for, recognizing, and using underlying mathematical structure is an important aspect of mathematical reasoning. We explore the use of mathematical structure in children's integer strategies by developing and exemplifying the construct of logical necessity. Students in our study used logical necessity to approach and use numbers in a…

This paper addresses how secondary students might reason about amounts of change in covarying quantities. Two empirically based forms of covariational reasoning are distinguished. The first form-- reasoning about quantities as varying simultaneously and independently--supports tandem comparison of amounts of change. The second form--coordination…

This case study examines the salient features of two individuals' reasoning when confronted with a task concerning the cardinality and associated cardinal number of equinumerous infinite sets. The APOS Theory was used as a framework to interpret their efforts to resolve the "infinite balls paradox" and one of its variants. These cases…

We present a synthesis of findings from constructivist teaching experiments regarding six schemes children construct for reasoning multiplicatively and tasks to promote them. We provide a task-generating platform game, depictions of each scheme, and supporting tasks. Tasks must be distinguished from children's thinking, and learning situations…

The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) identifies the strategic use of appropriate tools as one of the mathematical practices and emphasizes the use of pictures and diagrams as reasoning tools. Starting with the early elementary grades, CCSSM discusses students' solving of problems "by drawing." In later…

In the authors' examination of various instructional programs, they observed that most provide all the necessary data to solve proportion problems, employ compatible numbers that are usually unrealistic, present numbers (data) in the order in which they are to be manipulated, discuss contexts that cannot be easily replicated, and present…

Proportional reasoning is important for informed decisions in proportional problem situations. This paper reports on mathematical content knowledge related to proportional reasoning of second-year, pre-service teachers. Responses to two ratio items provide insights into their correct method of solutions and common misconceptions. Anchor Points…

Successful outcomes for a "Transition Course in Mathematics" have resulted from two unique design features. The first is to run the course as a "survey course" in mathematics, introducing sophomore-level students to a broad set of mathematical fields. In this single mathematics course, undergraduates benefit from an introduction of proof…

The following is a report of an exploration of what mathematical reasoning might look like in classrooms. Focusing on just one lesson in one classroom, data are presented that indicate that upper primary students are willing and able to reason for themselves, especially in classrooms in which the culture for such reasoning has been established. It…

The papers in the monograph address different topics related to mathematics teaching and learning processes which are of great interest to both students and prospective teachers. Some papers open new research questions, some show examples of good practice and others provide more information about earlier findings. The monograph consists of six…

Kaminski, Sloutsky, and Heckler (2008a) published in "Science" a study on "The advantage of abstract examples in learning math," in which they claim that students may benefit more from learning mathematics through a single abstract, symbolic representation than from multiple concrete examples. This publication elicited both enthusiastic and…