The national assessment of Educational Progress (NAEP) is used by the federal government and by states to gauge achievement in several subject areas, including mathematics. The results of the NAEP tests in mathematics at the eighth grade are used here to help us explore students' mathematics achievement over the decade from 1990 to 2000. In particular, we use these data to counteract the media portrayal of students' achievement in mathematics as steadily declining.

# Search Results

### James Hiebert and Diana Wearne

Conceptually based instruction on place value and two-digit addition and subtraction without regrouping was provided in four first-grade classrooms, and more conventional textbook-based instruction was provided in two first-grade classrooms. An observer compiled extensive notes of 20 lessons in each kind of classroom. Students who received conceptually based instruction performed significantly better on items measuring understanding of place value and two-digit addition and subtraction with regrouping and used strategies more often that exploited the tens and ones structure of the number system. Content and pedagogical differences between the instruction lessons are linked to the learning differences and are used to explain between-group differences in levels of performance and understanding. Observations are offered on the complex interactions between instruction, understanding, and performance.

### Diana Wearne and James Hiebert

A theory is proposed for how students develop competence with the written symbols of decimal fractions. The theory views competence as the cumulative and sequential mastery of four separate cognitive processes in working with written symbols. The first two processes develop for the user the semantics of the symbol system, and the second two elaborate its syntax. Initial predictions from the theory were tested by instructing small groups of students for about 2 weeks. As predicted, (a) most students were able to acquire the first two processes, (b) these processes generated correct performance and transferred to novel tasks, and (c) it was more difficult for students to acquire the semantic-based processes once they had routinized syntactic processes. The theory and data are interpreted in the context of some persistent issues in mathematics learning.

### Diana Wearne and James Hiebert

### Edited by Patricia F. Campbell

Students who get the correct answer may differ in their understanding .

### Jennifer E. Jacobs, James Hiebert, Karen Bogard Givvin, Hilary Hollingsworth, Helen Garnier and Diana Wearne

Debates about the future of school mathematics in the United States often center on whether standards-based instruction is improving or undermining students' achievement. Critical for making progress in these debates is information about the actual nature of classroom practice in U.S. classrooms. This article focuses on one key element of classroom practice—teaching—and presents the results of two studies of randomly selected, nationally representative U.S. eighth-grade mathematics lessons that were videotaped as part of the TIMSS 1995 and 1999 Video Studies. Analyses compare features of teaching found in these lessons with pedagogical recommendations for middle school teachers in the *Principles and Standards for School Mathematics* (*Principles and Standards*) in order to examine the extent to which teaching in U.S. eight-grade classrooms is standards-based. Results show that typical mathematics teaching, in both 1995 and 1999, is more like the kind of traditional teaching reported for most of the past century (Cuban, 1993; Fey, 1979; Weiss, Pasley, Smith, Banilower, & Heck, 2003; Welch, 1978) than the kind of teaching promoted in *Principles and Standards*.

### Karen C. Fuson, Diana Wearne, James C. Hiebert, Hanlie G. Murray, Pieter G. Human, Alwyn I. Olivier, Thomas P. Carpenter and Elizabeth Fennema

Researchers from 4 projects with a problem-solving approach to teaching and learning multidigit number concepts and operations describe (a) a common framework of conceptual structures children construct for multidigit numbers and (b) categories of methods children devise for multidigit addition and subtraction. For each of the quantitative conceptual structures for 2-digit numbers, a somewhat different triad of relations is established between the number words, written 2-digit marks, and quantities. The conceptions are unitary, decade and ones, sequence-tens and ones, separate-tens and ones, and integrated sequence-separate conceptions. Conceptual supports used within each of the 4 projects are described and linked to multidigit addition and subtraction methods used by project children. Typical errors that may arise with each method are identified. We identify as crucial across all projects sustained opportunities for children to (a) construct triad conceptual structures that relate ten-structured quantities to number words and written 2-digit numerals and (b) use these triads in solving multidigit addition and subtraction situations.