A theory of algebra-word-problem comprehension and its implications for the design of learning environments.

A tutoring approach is derived from a model of problem comprehension, based on the van Dijk and Kintsch (1983; Kintsch, 1988) theory of discourse processing. A problem statement is regarded as a text from which the student must glean propositional and situational information and make critical inferences. The competent student must coordinate this information with known problem models so that formal (i.e., algebraic) operations can be applied and exact solutions can be obtained. We argue that this task is a highly reading-oriented one in which poor text comprehension and an inability to access relevant long-term knowledge lead to serious errors. In particular, poor students often omit from their solutions or misspecify necessary mathematical constraints that are based on reading inferences needed to describe fully the problem situation. Furthermore, formal algebraic expressions are so abstract that their meaning is often elusive; this contributes to mistranslations and misinterpretations. The competent app...

[1]  Daniel G. Bobrow,et al.  Natural Language Input for a Computer Problem Solving System , 1964 .

[2]  Allen Newell,et al.  Human Problem Solving. , 1973 .

[3]  Eva Teubal,et al.  Verbal cues as an interfering factor in verbal problem solving , 1975 .

[4]  H. Simon,et al.  Models of Thought , 1979 .

[5]  Kurt VanLehn,et al.  Repair Theory: A Generative Theory of Bugs in Procedural Skills , 1980, Cogn. Sci..

[6]  Robert E. Reys,et al.  Solving Verbal Problems: Results and Implications from National Assessment. , 1980 .

[7]  Seymour Papert,et al.  Mindstorms: Children, Computers, and Powerful Ideas , 1981 .

[8]  A. Caramazza,et al.  Naive beliefs in “sophisticated” subjects: misconceptions about trajectories of objects , 1981, Cognition.

[9]  Richard E. Mayer,et al.  Frequency norms and structural analysis of algebra story problems into families, categories, and templates , 1981 .

[10]  Laura Gould,et al.  A study of TRIP: a computer system for animating time—rate—distance problems* , 1982 .

[11]  Richard E. Mayer,et al.  Memory for algebra story problems. , 1982 .

[12]  John R. Anderson.  Acquisition of cognitive skill. , 1982 .

[13]  W. Kintsch,et al.  Strategies of discourse comprehension , 1986 .

[14]  C SchankRoger,et al.  Dynamic Memory: A Theory of Reminding and Learning in Computers and People , 1983 .

[15]  John R. Anderson,et al.  Cognitive principles in the design of computer tutors , 1984 .

[16]  W Kintsch,et al.  Understanding and solving word arithmetic problems. , 1985, Psychological review.

[17]  Alan H. Schoenfeld,et al.  Mathematical Problem Solving , 1985 .

[18]  Betty Collis.  Reflections on Inequities in Computer Education: Do the Rich Get Richer? , 1985 .

[19]  John R. Anderson,et al.  The Geometry Tutor , 1985, IJCAI.

[20]  John Seely Brown,et al.  Process versus Product: A Perspective on Tools for Communal and Informal Electronic Learning , 1985 .

[21]  Douglas H. Clements,et al.  Effects of Logo and CAI Environments on Cognition and Creativity , 1986 .

[22]  Douglas H. Clements,et al.  The effects of Logo and CAI problem-solving environments on problem-solving abilities and mathematics achievement , 1986 .

[23]  Etienne Wenger,et al.  Artificial Intelligence and Tutoring Systems: Computational and Cognitive Approaches to the Communication of Knowledge , 1987 .

[24]  Nancy Pennington,et al.  Comprehension strategies in programming , 1987 .

[25]  Susan E. Newman,et al.  Cognitive Apprenticeship: Teaching the Craft of Reading, Writing, and Mathematics. Technical Report No. 403. , 1987 .

[26]  Marlene Scardamalia,et al.  Computer-Supported Intentional Learning Environments , 1989 .

[27]  R. Mayer,et al.  Students' miscomprehension of relational statements in arithmetic word problems. , 1987 .

[28]  Stephen K. Reed,et al.  A structure-mapping model for word problems. , 1987 .

[29]  Claudius Magnus Kessler.  Transfer of programming skills in novice lisp learners , 1988 .

[30]  W. Kintsch.  The role of knowledge in discourse comprehension: a construction-integration model. , 1988, Psychological review.

[31]  K. Fuson,et al.  Teaching Children to Use Schematic Drawings to Solve Addition and Subtraction Word Problems. , 1988 .

[32]  W. Kintsch,et al.  The role of understanding in solving word problems , 1988, Cognitive Psychology.

[33]  L. Resnick.  Treating Mathematics as an Ill-Structured Discipline. , 1988 .

[34]  D. Kibler,et al.  Exploring the Episodic Structure of Algebra Story Problem Solving , 1989 .

[35]  R. Glaser,et al.  Learning Theory and the Study of Instruction , 1989 .

[36]  A. Lewis.  Training students to represent arithmetic word problems , 1989 .

[37]  John R. Anderson,et al.  The Transfer of Cognitive Skill , 1989 .

[38]  G. Vining,et al.  Data Analysis: A Model-Comparison Approach , 1989 .

[39]  John R. Anderson,et al.  Skill Acquisition and the LISP Tutor , 1989, Cogn. Sci..

[40]  Ramanathan V. Guha,et al.  Building Large Knowledge-Based Systems: Representation and Inference in the Cyc Project , 1990 .

[41]  Kurt Reusser,et al.  From text to situation to equation: cognitive simulation of understanding and solving mathematical word problems , 1990 .

[42]  Walter Kintsch,et al.  Routine Computing Tasks: Planning as Understanding , 1991, Cogn. Sci..

[43]  W. Kintsch,et al.  Enhancing students' comprehension of the conceptual structure of algebra word problems. , 1992 .