A Developmental Model of Algebra Problem Solving: Trade-offs between Grounded and Abstract Representations.
Kenneth R. Koedinger, Martha Wagner Alibali, Mitchell J. Nathan · 1999
This paper presents a developmental model of students' acquisition of competence in quantitative and algebraic problem solving. A key notion underlying the developmental model is a distinction between grounded and abstract representations. Grounded representations, like problems, are more concrete and familiar, closer to physical objects and everyday events. Abstract representations, like symbolic equations, are concise and easy to manipulate, but are distanced from any physical objects of reference. The complementary computational characteristics of grounded and abstract representations lead to hypotheses about the order of skill acquisition. In prior research, the authors demonstrated that early in the development of algebraic competence, the advantages of grounded representations outweigh those of abstract representations--for simpler problems, students are better at than the analogous equations. This paper presents two studies that test the hypothesis that later in algebra development, the advantages of abstract representations emerge--for more complex problems, students are better at equations than the analogous problems. Includes 6 tables, 7 figures, and 16 references. (Author/WRM) ******************************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. ******************************************************************************** Algebra Problem Solving Development 1 A Developmental Model of Algebra Problem Solving: Trade-offs Between Grounded and Abstract Representations Kenneth R. Koedinger M :BLE 1 PERMISSION TO REPRODUCE AND DISSEMINATE THIS MATERIAL HAS BEEN GRANTED BY ktiadj TO THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) U.S. DEPARTMENT OF EDUCATION Office of Educational Research and Improvement EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) This document has been reproduced as ived from the person or organization originating it. Minor changes have been made to improve reproduction quality. Points of view or opinions stated in this document do not necessarily represent official OERI position cr policy. Koedinger, K. R., Alibali, M. W., Nathan, M. J. (1999). A developmental model of algebra problem solving: Trade-offs between grounded and abstract representations. Paper prepared for the annual meeting of the American Educational Research Association, Montreal. Algebra Problem Solving Development 2 ABSTRACT We present a developmental model of students' acquisition of competence in quantitative and algebraic problem solving. A key notion underlying our developmental model is a distinction between grounded and abstract representations. Grounded representations, like problems, are more concrete and familiar, closer to physical objects and everyday events. Abstract representations, like symbolic equations, are concise and easy to manipulate, but are distanced from any physical objects of reference. The complementary computational characteristics of grounded and abstract representations lead to hypotheses about the order of skill acquisition. In prior research (Koedinger & Nathan, 1999), we demonstrated that early in the development of algebraic competence, the advantages of grounded representations outweigh those of abstract representations for simpler problems, students are better at than the analogous equations. This paper presents two studies that test the hypothesis that later in algebra development, the advantages of abstract representations emerge for more complex problems, students are better at equations than the analogous problems.We present a developmental model of students' acquisition of competence in quantitative and algebraic problem solving. A key notion underlying our developmental model is a distinction between grounded and abstract representations. Grounded representations, like problems, are more concrete and familiar, closer to physical objects and everyday events. Abstract representations, like symbolic equations, are concise and easy to manipulate, but are distanced from any physical objects of reference. The complementary computational characteristics of grounded and abstract representations lead to hypotheses about the order of skill acquisition. In prior research (Koedinger & Nathan, 1999), we demonstrated that early in the development of algebraic competence, the advantages of grounded representations outweigh those of abstract representations for simpler problems, students are better at than the analogous equations. This paper presents two studies that test the hypothesis that later in algebra development, the advantages of abstract representations emerge for more complex problems, students are better at equations than the analogous problems. Algebra Problem Solving Development 3 INTRODUCTION This paper presents a developmental model of students at different levels of competence in quantitative and algebraic problem solving. In prior research, we employed the difficulty factors assessment (DFA) methodology to explore early algebra problem solving, and identified effects that contradict common beliefs and current instructional practices (Koedinger & Tabachneck, 1995; Koedinger & Mac Laren, 1997; Koedinger & Nathan, 1999). In this paper, we review these results and present new results about student problem solving at higher levels of competence. Together these results on difficulty factors in algebra problem solving provide a picture of student development from arithmetic competence through various distinguishable levels of algebraic competence. A key notion underlying our developmental model is a distinction between grounded and abstract representations. Grounded representations are ones that are more concrete, closer to physical objects and everyday events. In the context of quantitative reasoning, real world problem situations or story problems are more grounded than symbolic equations because they use familiar and refer to familiar physical objects and events'. For example, consider the following problem. Ted works as a waiter. He worked 6 hours in one day and also got $66 in tips. If he made $81.90 that day, how much per hour does Ted make? Given some experience with money and waiters, the words, objects and events described in this problem are relatively familiar to students. Students' understanding of the quantitative relationships described is thus grounded in these familiar terms. 1 How grounded a particular problem is for a particular student depends on that student's past experiences with the particular objects and events in a problem as well as the or symbols used to refer to them. To the Algebra Problem Solving Development 4 Abstract representations, in contrast, are short and concise, leaving out any direct indication of the physical objects and events being referred to. Consider the following algebraic representation of the problem above.representations, in contrast, are short and concise, leaving out any direct indication of the physical objects and events being referred to. Consider the following algebraic representation of the problem above. x * 6 + 66 = 81.90 The equation is clearly shorter and more concise than the problem above. Besides the numbers common to both (6, 66, 81.90) there are only four other characters in the equation (x, *, +, .) whereas there are 100 other characters in the story. What is left out, however, is any reference to the familiar objects and events like hourly wages and tips. Furthermore, the terminology is different. The words in the algebraic sentence (i.e., x, *, +, =) are less familiar than the phrases expressing analogous meanings in the (i.e., how much, hours in one day, also got, he made). The difference between grounded and abstract representations is not as discrete as these examples might indicate. There are multiple levels of intermediate groundedness or abstractness. Consider the following word problem, which is devoid of situational content: Starting with some number, if I multiply it by 6 and then add 66, I get 81.90. What number did I start with? It is intermediate in abstractness between the and the equation, having 78 characters besides the numbers. It is also intermediate in its groundedness. While it is missing familiar references to money and waiters, it contains that are more familiar (i.e., some number, multiply, add, get) than the characters expressing analogous meaning in the equation (i.e., x,