Children's Conceptual Structures for Multidigit Numbers and Methods of Multidigit Addition and Subtraction
Karen C. Fuson, Diana Wearne, James C. Hiebert, Hanlie G. Murray, Pieter G. Human, Alwyn I. Olivier, Thomas P. Carpenter, 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.
Deepening the Analysis: Longitudinal Assessment of a Problem-Centered
Terry Wood, Patricia Sellers
Longitudinal analyses of the mathematical achievement and beliefs of 3 groups of elementary pupils are presented. The groups consist of those students who had received 2 years of problem-centered mathematics instruction, those who had received 1 year, and those who had received textbook instruction. Comparisons are made for the groups using a standardized norm-referenced achievement test from first through fourth grade. Next, student comparisons are made using instruments developed to measure conceptual understanding of arithmetic and beliefs and motivation for learning mathematics. The results of the analyses indicate that after 2 years in problem-centered classes, students have significantly higher achievement on standardized achievement measures, better conceptual understanding, and more task-oriented beliefs for learning mathematics than do those in textbook instruction. In addition, these differences remain after problem-centered students return to classes using textbook instruction. Comparisons of pupils in problem-centered classes for only 1 year reveal that after returning to textbook instruction, these students' mathematical achievement and beliefs are more similar to the textbook group. Also included are exploratory analyses of the pedagogical beliefs held by teachers before and after teaching in problem-centered classes, and those held by teachers in textbook classes. The results of the student and teacher analyses are interpreted in light of research on children's construction of nonstandard algorithms and the nature of classroom environments.
Assessment and Grading in High School Mathematics Classrooms
Sharon L. Senk, Charlene E. Beckmann, Denisse R. Thompson
The assessment and grading practices in 19 mathematics classes in 5 high schools in 3 states were studied. In each class the most frequently used assessment tools were tests and quizzes, with these determining about 77% of students' grades. In 12 classes other forms of assessment, such as written projects or interviews with students, were also used, with performance on such instruments counting for about 7% of students' grades averaged across all 19 classes. Test items generally were low level, were stated without reference to a realistic context, involved very little reasoning, and were almost never open-ended. Most test items were either neutral or inactive with respect to technology. Written projects usually involved more complex analyses or applications than tests did. The teachers' knowledge and beliefs, as well as the content and textbook of the course, influenced the characteristics of test items and other assessment instruments. Only in geometry classes did standardized tests appear to influence assessment.
Developing Ratio and Proportion Schemes: A Story of a Fifth Grader
Jane Jane Lo, Tad Watanabe
There is a growing theoretical consensus that the concepts of ratio and proportion do not develop in isolation. Rather, they are part of the individual's multiplicative conceptual field, which includes other concepts such as multiplication, division, and rational numbers. The current study attempted to clarify the beginning of this developmental process. One fifth-grade student, Bruce, was encouraged to schematize his trial-and-error-based method, which was effective in solving so-called missing-value tasks. This study describes several advancements Bruce made during the teaching experiment and analyzes the challenges Bruce faced in attempting to schematize his method. Finally, the mathematical knowledge Bruce needed to further develop his ratio and proportion concepts is identified. The findings provide additional support for the view that the development of ratio and proportion concepts is embedded within the development of the multiplicative conceptual field.
Brief Report: Results of Third-Grade Students in a Reform Curriculum on the Illinois State Mathematics Test
William M. Carroll
Results of Third-Grade Students in a Reform Curriculum on the Illinois State Mathematics Test William M. Carroll, University of Chicago School Mathematics Project Over the past decade, there has been a call for major reforms in mathematics educattion from classrooms where students memorize facts and practice.