Diana Wearne

Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics

We argue that reform in curriculum and instruction should be based on allowing students to problematize the subject. Rather than mastering skills and applying them, students should be engaged in resolving problems. In mathematics, this principle fits under the umbrella of problem solving, but our interpretation is different from many problem-solving approaches. We first note that the history of problem solving in the curriculum has been infused with a distinction between acquiring knowledge and applying it. We then propose our alternative principle by building on John Dewey’s idea of “reflective inquiry,” argue that such an approach would facilitate students’ understanding, and compare our proposal with other views on the role of problem solving in the curriculum. We close by considering several common dichotomies that take on a different meaning from this perspective

Instructional Tasks, Classroom Discourse, and Students’ Learning in Second-Grade Arithmetic

To investigate relationships between teaching and learning mathematics, the six second-grade classrooms in one school were observed regularly during the 12 weeks of instruction on place value and multidigit addition and subtraction. Two classrooms implemented an alternative to the more conventional textbook approach. The alternative approach emphasized constructing relationships between place value and computation strategies rather than practicing prescribed procedures. Students were assessed at the beginning and the end of the year on place value understanding, routine computation, and novel computation. Students in the alternative classrooms, compared with their more traditionally taught peers, received fewer problems and spent more time with each problem, were asked more questions requesting them to describe and explain alternative strategies, talked more using longer responses, and showed higher levels of performance or gained more by the end of the year on most types of items. The results suggest that relationships between teaching and learning are a function of the instructional environment; different relationships emerged in the alternative classrooms than those that have been reported for more traditional classrooms.

Children's Conceptual Structures for Multidigit Numbers and Methods of Multidigit Addition and Subtraction.

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 2digit 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.

Mathematics Teaching in the United States Today (and Tomorrow): Results From the TIMSS 1999 Video Study:

The Third International Mathematics and Science Study (TIMSS) 1999 Video Study examined eighth-grade mathematics teaching in the United States and six higher-achieving countries. A range of teaching systems were found across higher-achieving countries that balanced attention to challenging content, procedural skill, and conceptual understanding in different ways. The United States displayed a unique system of teaching, not because of any particular feature but because of a constellation of features that reinforced attention to lower-level mathematics skills. The authors argue that these results are relevant for policy (mathematics) debates in the United States because they provide a current account of what actually is happening inside U.S. classrooms and because they demonstrate that current debates often pose overly simple choices. The authors suggest ways to learn from examining teaching systems that are not alien to U.S. teachers but that balance a skill emphasis with attention to challenging mathematics and conceptual development.

Links between teaching and learning place value with understanding in first grade.

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.

Understanding and Improving Mathematics Teaching: Highlights from the TIMSS 1999 Video Study

portion of the TIMSS 1999 Video Study included Australia, the Czech Republic, Hong Kong SAR, Japan, the Netherlands, Switzerland, and the United States. In this article, we focus on the mathematics lessons; the science results will be available at a later date. Stimulated by a summary article that appeared in the Kappan and by other reports, interest in the TIMSS 1995 Video Study focused on its novel methodology and the striking differences in teaching found in the participating countries. In particular, the sample of eighth-grade Understanding and Improving Mathematics Teaching: Highlights from the TIMSS 1999 Video Study

Making Mathematics Problematic: A Rejoinder to Prawat and Smith

Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H., Olivier, A., & Wearne, D. (1996). Problem solving as a basis for reform in curriculum and instruction: The case of mathematics. Educational Researcher, 25(4), 12-21. Smith, J. P. & Thompson, P. W. (in press). Additive quantitative reasoning and the development of algebraic reasoning. In J. Kaput (Ed.), Employing childrens natural powers to build algebraic reasoning in the context of elementary mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates. Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25,165-208.

Acquiring meaning for decimal fraction symbols: A one year follow-up

Fourth graders with differing achievement records participated in a specially designed two week unit on decimal fractions. Students were encouraged to connect meaningful referents with decimal fraction symbols and use these meanings to develop procedures for adding and subtracting decimal numbers. One year later these students and a matched set of fifth graders were interviewed and given paper-and-pencil tests. Three questions were of interest: (1) Do short term changes in the processes students use to solve problems remain stable over time; (2) Do students who have been instructed in conceptually-based processes exhibit a higher level of performance one year later than their conventionally taught peers; and (3) What is the relationship between entry achievement level and the year-long effects of conceptually-based instruction? The results suggest that: (1) If students used the meanings of written symbols as a basis for solving problems immediately after instruction, they used these processes to solve problems one year later, regardless of entering achievement; (2) Compared to their conventionally taught peers, students in the lower achievement group benefitted relatively more from the conceptually-based instruction than students in the higher achievement group; (3) However, higher achieving students were more likely to exhibit use of conceptually-oriented processes one year later than the lower achieving students.