This study investigated the development of length measurement ideas in students from prekindergarten through 2nd grade. The main purpose was to evaluate and elaborate the developmental progression, or levels of thinking, of a hypothesized learning trajectory for length measurement to ensure that the sequence of levels of thinking is consistent…

This article reports on students' learning through conjecturing, by drawing on a semester-long teaching experiment with 6 sixth-grade students. It focuses on 1 of the students, Josh, who developed especially powerful ways of operating over the course of the teaching experiment. Through a fine-grained analysis of Josh's actions, this article…

Functions are all around us, disguised as actions on concrete objects. Composition of functions, too, is all around us, because these actions can be performed in succession, the output of one serving as the input for the next. In terms of Gray and Tall's (2001) "embodied objects" or Lakoff and Nunez's (2000) "mathematical idea…

A study designed to investigate a method examining some of the processes involved in students' forming of mathematical concepts in an experimental mathematics course. Results suggest that the card sorting task offers a useful tool for use in evaluation programs. (FL)

Uses situated abstraction to understand nurses' conceptions of intensive quantity of drug concentration. Explores nurses' conceptions to undertake a pointed examination of the degree of situatedness of nurses' knowledge and reasoning. Demonstrates that nurses' conceptions were abstracted within their practice when they coordinated mathematical…

Data were collected on tactics used when making a sequence of set comparisons in terms of more or equal number. (MP)

The effects of negative instances on the acquisition of the mathematical concepts of commutativity and associativity were examined. Also investigated were possible transfer effects that might result from the use of negative instances. For 64 ninth-grade subjects, results favored the treatments containing mixed instances and supported the transfer…

A study asked 24 first grade children from 3 different classes to write number sentences and select appropriate alternative number sentences for addition and subtraction word problems. Results indicated that children could be characterized into five clusters according to their flexibility in accepting alternative number sentences and that…

A theory is proposed for how students develop competence with the written symbols of decimal fractions. Competence is viewed as the cumulative and sequential mastery of four separate cognitive processes. The theory was tested with small groups of students, most of whom were able to acquire the first two processes. (MNS)

Discusses duality between process and concept in mathematics, ambiguity of symbolic notation, and flexibility in thought processes of successful students. Content analysis and empirical evidence suggest a qualitatively different kind of mathematical thought in more able compared to less able elementary students (n=72) in arithmetic. (Contains 42…

Reports on a study designed to investigate aspects of young children's knowledge of fractions and partitions of quantities. Findings suggest that although children did not have a deep understanding of the concept of one half, a significant number demonstrated a powerful systematic procedure leading to the creation of accurate equal fractional…

A teaching experiment with six third graders to analyze their ways of operating while solving fraction tasks. Children's quantitative reasoning with fractions was based on their quantitative reasoning with natural numbers. Presents the constructive itinerary of one of the most advanced children in the group. (Contains 44 references.) (Author/MDH)

In an attempt to bridge the perceived distinction between internalized conceptions of mathematics and their external representations, this discussion treats mathematics both as an individual, constructive activity and as a communal, social practice. It is argued that this approach might transcend the tripartite scheme of teacher, student, and…

Investigates issues of conceptual understanding in teaching and learning mathematics provided conceptually based instruction on place value and two-digit addition and subtraction without regrouping in four first grade classrooms. Conventional textbook-based instruction was provided in two first grade classrooms. Experimental-group students scored…