Investigations in Number, Data, and Space

Transcription

1 Investigations in Number, Data, and Space First Edition Impact Study MODES OF TEACHING AND WAYS OF THINKING Anne Goodrow TERC/Tufts University Abstract From drawing a set or writing numerals to represent quantities or a problem and its solution, children express their mathematical thinking. Their ways vary in sophistication and complexity, both as they develop their understanding of number and as they are exposed to school instruction and conventional mathematical representation. This study examined (a) the development of number sense and number representation by children in traditional 1, transitional 2, and constructivist 3 second grade mathematics classrooms and (b) how different teaching approaches influence the way children deal with computation exercises. Results to tasks designed to explore number sense and number representation show that children in constructivist classrooms used a wider variety of representations to express their mathematical thinking. Their ways included the literal recording of one's counting to solve a problem, the invention of number strings (numbers broken apart and recombined in a different order, e.g = ), and the use of negative numbers. When asked to solve addition and subtraction computation problems presented in vertical form and in horizontal form, children in traditional classrooms nearly always chose to use algorithm procedures taught at school; some of these children found it necessary to rewrite in vertical form problems presented horizontally. In contrast, children in constructivist classrooms, who had not learned algorithmic procedures for addition and subtraction but, instead, relied on their own number sense, presented a higher number of correct responses through use of varied strategies which revealed understanding of number relations and of the properties of the decimal system. Introduction Number is central to primary mathematics, regardless of the pedagogical theory applied. The traditional mathematics curriculum focuses on the acquisition of numerical skills such as number order, counting on, addition and subtraction facts, place value, and addition and subtraction algorithms. In the constructivist mathematics curriculum, which is grounded in Piaget's theory of child development, sense-making about number is a primary concern. Children's development of number sense is grounded in meaningful experiences, including solving real-world problems as well as decontextualized mathematics problems. These two approaches are compared and contrasted in Table 1. A third type of classroom, which might be described as "mixed" or "transitional" is also a 1

2 commonly found educational practice. In these classrooms, teachers use a constructivist curriculum, however, they teach mathematics in a traditional manner. This study examined (a) the development of number sense and number representation by children in traditional, transitional, and constructivist second grade mathematics classrooms and (b) how different teaching approaches influence the way children deal with computation exercises. Table 1: A Look at School Environments Method A total of 30 children, 10 from each of the classroom types, participated in the study. Children were interviewed individually and asked to do oral and written tasks involving addition, subtraction, number composition, and the properties of the decimal system. All interviews were untimed and audiotaped. This paper analyzes children's responses to two of these tasks: the Number of the Day and the Two-digit computation worksheet. These tasks are described in Table 2. 2

3 Table 2: Computation Tasks Requiring Written or Verbal Explanation Results Children's responses were coded as correct or incorrect, and their written and verbal responses were coded for type of solution strategy used. Non-parametric statistical tests (the Kruskal-Wallis and Mann Whitney U) were used to test the significance of differences. The Number of the Day Children in constructivist and Mixed classrooms performed better at this task than those in the Traditional group. Children in the constructivist group created significantly more correct number sentences than those in the Traditional group (p =.0015). Most children in the Traditional group used addition only; all used two terms and one operation. Children in the Mixed and Constructivist groups often used several terms and more than one operation. 3

4 Two-Digit Addition Problems All groups successfully solved at least ten of the twelve addition problems. There was no significant difference between the groups in addition, even when regrouping was necessary. The standard addition algorithm taught in US schools constituted 93.6% of the addition strategies used by the Traditional group, and 60% of those used by the Mixed group. In the Constructivist group, the standard addition algorithm was documented only 6% of the time. In the Constructivist group, the most widely used addition strategies were adding tens first and recomposing. The Traditional group used only 2 different strategies, while 7 different strategies, reflecting different ways of thinking about combining numbers were used by children in the Constructivist group. Example of children's written work and verbal explanations of addition are shown in Figures 3 and 4. 4

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6 Two-Digit Subtraction Problems All children were equally successful in solving subtraction problems without regrouping. As problems became more difficult - requiring regrouping, and/or subtracting with a 0 in the ones place - children in the Constructivist group were more successful than either the Traditional or Mixed groups. Significant differences emerged between the Constructivist and Mixed groups in: 1) the total correct (see Figure 5); 2) the number of correct problems with 1 regrouping; and 3) the number of problems in the horizontal form solved correctly. The standard algorithm taught in US schools constituted 85% of the subtraction strategies used by the Traditional group, 48.3% of those used by the Mixed group, but only 3.7% of those used by the Constructivist group. Subtracting tens first and recomposing accounted for more than 60% of the strategies used in the Constructivist group. Examples of children's written work and verbal explanations of their thinking about subtraction are shown in Figures 6 to 9. 6

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8 Discussion and Summary The children in the Constructivist group were encouraged to solve addition and subtraction problems by using what they knew about number to combine and break apart numbers in ways that made sense to them. The Number of the Day results show that these children were more flexible in thinking about number than the children in the Traditional class. As a result, although they did not receive instruction in the use of the standard algorithms, the children in the Constructivist were the most successful at both two-digit addition and subtraction. In summary: Children in the Traditional group almost exclusively used the standard addition and subtraction algorithms. Many children in the Mixed group also used the standard algorithms. Use of the algorithms was procedural. Children in the Mixed group also sometimes used strategies that suggested a basic understanding of part-whole relationships, such as counting on or using the commutative property. The Constructivist group relied on number sense strategies which demonstrated greater understanding of part-whole relationships. The standard algorithms were rarely used. When regrouping was required, the Constructivist group performed better than either of the other two groups, and the Mixed group performed the least well. The data from this study support the view that children are more successful at computation when they rely on their own thinking about number rather than on taught procedures. In contrast, when children rely on procedural knowledge of the standard algorithm, their errors suggest over generalization of rules (Resnick and Omanson, 1987); and, due to column by column focus of the procedure, they lose a sense of the whole quantities with which they are working. As Kamii (1989) has suggested children combine and separate numbers in ways that make sense to them, they develop part-whole relationships and further their understanding of place value. These understandings are reflected in children's written representations of computation problems and verbal explanations of their thinking. References Brooks, J. and Brooks, M. (1993). In search of understanding: The case for constructivist classrooms. Alexandria, VA: Association for Supervision and Curriculum Development. Kamii, C. (1989). Young children reinvent arithmetic: Implications of Piaget's theory. New York: Teachers College Press. 8

9 Resnick, L. and Omanson, S. (1987). Learning to understand arithmetic. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 3, 41-95). Hillsdale, NJ: Lawrence Erlbaum Associates. Footnotes 1. The Traditional classroom emphasized basic skills, used a basal textbook, and emphasized individual work on conventional algorithms for addition and subtraction. (Goodrow, Anne M., Children's Construction of Number Sense in Traditional, Contructivist, and Mixed Classrooms. Unpublished dissertation for the degree of Doctor of Philosophy in Child Development. Medford, MA: Tufts University, May Pp ) 2. The Transitional classrooms used Investigations in Number, Data, and Space. Although a constructivist curriculum, the teaching was characterized by more traditional mathematics pedagogy. For example, discussions were fairly didactic and contained little talk about children's mathematical thinking. (Goodrow, pp ) 3. The Constructivist classrooms also used the Investigations in Number, Data, and Space curriculum. These teachers encouraged students to generate their own strategies for solving addition and subtraction problems, and emphasized discussion and demonstration of these strategies. (Goodrow, p. 70.) Goodrow, Anne M. Modes of Teaching and Ways of Thinking. Paper presented at the International Society for the Study of Behavioral Development. Bern, Switzerland, July, (Summary of: Children's Construction of Number Sense in Traditional Constructivist, and Mixed Classrooms. Unpublished dissertation for the degree of Doctor of Philosophy in Child Development. Medford, MA: Tufts University, May 1998). Reprinted with permission from the author. All rights reserved. 9