Search

This report describes a research project that investigated how to design instruction to help children take the step to generalizing and formalizing their knowledge into powerful abstract systems for representing and operating mathematical ideas. The studies build on existing research on cognitively guided instruction (CGI). The goal of this paper is to begin to understand how to provide support for children to reflect on their procedures in order to form generalizations from them and construct notations for representing their procedures and generalizations abstractly. The conclusion is that overall it appears that students in the primary grades can engage in formulating, representing, and justifying conjectures even though their justifications might not always be sufficient to validate all of the conjectures they are capable of identifying. (Contains 20 references.) (MM)

This paper describes a research-based teacher professional development program for elementary school mathematics and includes an overview of cognitively guided instruction (CGI). Also described are the CGI professional development program and the research base for CGI with regard to children's thinking; teachers' knowledge and beliefs about children's thinking and the relation of teachers' knowledge and beliefs to their student's achievement; the effect of the CGI professional development program on teachers' knowledge, beliefs, and practice; and the achievement of students in CGI classes. (Contains 27 references.) (MM)

This study investigated: (1) the relationship between the development of information processing capacity and certain Piagetian logical reasoning abilities; and (2) how the development of these cognitive abilities related to acquisition of certain measurement concepts. Forty first-grade children were individually administered tests of conservation of length and number, transitivity of length, information processing capacity, and basic length measurement concepts. The Piagetian measures of logical reasoning were positively correlated with information processing capacity, but the measures of information processing capacity failed to account for much of the variability of performance on the logical reasoning tasks. Some children at the highest levels of processing capacity failed the logical reasoning tasks and some at the lowest level passed them, suggesting that the logical reasoning tasks are not simply measures of information processing capacity. Furthermore, information processing measures accounted for 25% of the variance in performance on the linear measurement tasks, and length conservation accounted for an additional 23%. Although these two measures accounted for almost half of the variance, it is not clear that they represent prerequisites for learning basic length measurement concepts. Some children at low levels on both measures successfully completed the measurement tasks. (Author/MNS)

The purpose of this paper is to identify how the theory and techniques of research in cognitive development can be applied to the study of learning and instruction in mathematics. Five basic research paradigms are characterized; major directions of research on number, measurement, geometry, and adolescent reasoning are identified. Most of the research was built upon the work of Piaget and his associates. However, recently translated Soviet research and information processing techniques are also discussed. Potential education applications discussed include: content and sequencing of mathematical topics, matching instruction to appropriate levels of development, and choosing instructional strategies. (Author/MK)

Forty-three first-grade children, who had received no formal instruction in addition and subtraction, were individually administered 20 problems that could be solved using addition or subtraction. The problems were selected to represent the following semantic types: joining, separating, part-part-whole, comparison, and equalizing. Responses were coded in terms of appropriateness of strategy, correct or incorrect answer, type of error, mode of representation, and solution strategy. For every problem but the two addition comparison problems, over 70% of the subjects chose a correct strategy. There were very few systematic errors. Only 15 of the 860 responses involved the wrong choice of operation. The majority of solutions involved the use of concrete objects, but a significant number used fingers or did not use any physical representation. Contrary to previous analysis of children's solution processes for addition and subtraction problems, these results suggest that children do not transform problems so they can apply a single strategy. Rather, they have a rich repertoire of strategies which they apply directly to a problem based on its semantic structure. The results also suggest that verbal problems may be an appropriate context to introduce addition and subtraction operations. (Author/MK)

This paper describes the research program of the Mathematics Work Group of the Wisconsin Research and Development Center for Individualized Schooling. The major interest is in the development of children's abilities to solve verbal addition and subtraction problems and particularly in the processes and strategies used by children. Three factors are considered: (1) problem structure, (2) student characteristics, and (3) the nature of instruction. An analysis of verbal problems is presented. This analysis includes a discussion of various types of problem entities: discrete sets, continuous attributes, and actions or transformations. Problem structure is also analyzed along three dimensions: action vs. static relationships, set inclusion, and order (larger vs. smaller). Examples of various problem types are given. Results from the first year of an ongoing (1978-1981) longitudinal study of primary age students are included. These results are mainly gathered from individual problem-solving interviews with about 150 subjects. Various strategies employed and their change over time are examined. The paper concludes with a discussion of the implications of the research, both present and contemplated, for instruction. (Author/MK)

This study investigated the effect of initial instruction on the processes children use to solve basic addition and subtraction verbal problems. Prior to instruction and following a 2-month introductory unit on addition and subtraction, 43 first-grade children were individually tested on verbal problems representing different models of addition and subtraction. Prior to instruction, children's solution processes modeled the action or relationships described in the problem. Following instruction, they generally used a separating strategy for all subtraction problems. Although they could solve the problems, few children could coordinate their solutions with the arithmetic sentence they wrote representing the problem. (Author/MK)

The focus is on the problem-solving behaviors of primary age children on one-step verbal or story problems involving addition and subtraction. When children are given a simple word problem for which they have not learned the necessary algorithms, they are often able to derive a solution on their own. This report focuses on the child-invented processes pupils use on word problems that would normally be solved through use of an addition or subtraction algorithm. Much of the data comes from a three-year longitudinal study of about 100 first-grade pupils from one public and one parochial school in the Madison, Wisconsin area serving middle to upper-middle class neighborhoods. It was felt that invention was a rearrangement of elements into similar structures. The data suggested that instruction has a bearing upon changes in invented behavior. Thus, any characterization of children's formal or invented mathematics concepts and procedures needs to consider the role of instruction. (MP)

This material is designed to examine the research on how children acquire basic addition and subtraction concepts and skills. Two major lines of theories of the development of basic number concepts, called logical concept and quantification skill approaches, are identified. Major recurring issues in the development of early number concepts are also discussed. This is followed by examination of studies specifically related to addition and subtraction. It is noted that by kindergarten most children understand the rudiments of these operations, but a complete understanding develops over a protracted span of years. The direction of initial and ongoing research is discussed, and word problem studies and investigations on children's solutions of symbolic addition and subtraction problems are viewed to be concerned with both problem difficulty and solution processes. The review suggests that a great deal is known both about children's knowledge of addition and subtraction and how they solve addition and subtraction problems. It also indicates that there is not yet a clear picture of how to apply such new information to design more effective instruction. (MP)

This study investigated children's representation of addition and subtraction problems with canonical and noncanonical number sentences, to see whether children would directly represent the structure of a problem if both types of sentences were available. Subjects were 22 first graders and 41 second graders randomly assigned either to a Canonical group or a Noncanonical group. Each group received two 30-minute periods of instruction on writing and solving number sentences and writing number sentences to represent word problems. The Noncanonical group was introduced to all six basic open sentence types, while the Canonical group was introduced only to the one for addition and the one for subtraction in which the unknown is to the right of the equals sign. Children were then given two 12-item tests, on one of which they were instructed to write a number sentence for the problem and to solve the problem; on the second test, they only had to write a number sentence for the problem. Results suggest that most first graders are limited to direct symbolic representations of word problems. Both first and second graders can learn to write noncanonical number sentences and use them to represent word problems. Implications for instruction are discussed. (MNS)