An Attempt to Understand Students' Understanding of Basic Algebra

This paper reports the results obtained with a group of 24 14-year-old students when presented with a set of algebra tasks by the Leeds Modelling System, LMS. These same students were given a comparable paper-and-pencil test and detailed interviews some four months later. The latter studies uncovered several kinds of student misunderstandings that LMS had not detected. Some students had profound misunderstandings of algebraic notation: Others used strategies such as substituting numbers for variables until the equation balanced. Additionally, it appears that the student errors fall into several distinct classes: namely, manipulative, parsing, clerical, and “random.” LMS and its rule database have been enhanced as the result of this experiment, and LMS is now able to diagnose the majority of the errors encountered in this experiment. Finally, the paper gives a process-oriented explanation for student errors, and re-examines related work in cognitive modelling in the light of the types of student errors reported in this experiment. Misgeneralization is a mechanism suggested to explain some of the mal-rules noted in this study.

[1]  William J. Clancey,et al.  Tutoring rules for guiding a case method dialogue , 1979 .

[2]  Derek H. Sleeman.  Basic Algebra Revisited: A Study with 14-Year-Olds , 1985, Int. J. Man Mach. Stud..

[3]  D. Norman.  Categorization of action slips. , 1981 .

[4]  Herbert P. Ginsburg,et al.  Children's arithmetic: The learning process , 1977 .

[5]  Richard M. Young,et al.  Errors in Children's Subtraction , 1981, Cogn. Sci..

[6]  Mikhail V. Matz.  A process model for high school algebra errors , 1982 .

[7]  John Seely Brown,et al.  Diagnostic Models for Procedural Bugs in Basic Mathematical Skills , 1978, Cogn. Sci..

[8]  John Seely Brown,et al.  An Investigation of Computer Coaching for Informal Learning Activities. , 1978 .

[9]  Derek Sleeman,et al.  ACE: A system which analyses complex explanations , 1979 .

[10]  J. S. Brown,et al.  Pedagogical, natural language, and knowledge engineering techniques in SOPHIE-I, II and III , 1982 .

[11]  J. R. Hartley,et al.  Towards more intelligent teaching systems , 1973 .

[12]  R. Tennant.  Algebra , 1941, Nature.

[13]  EDWINA RISSLAND MICHENER,et al.  Understanding Understanding Mathematics , 1978, Cogn. Sci..

[14]  Derek H. Sleeman,et al.  Modelling Student's Problem Solving , 1981, Artif. Intell..

[15]  Ira P. Goldstein,et al.  The genetic graph: a representation for the evolution of procedural knowledge , 1979 .

[16]  Clayton Lewis.  Kinds of Knowledge in Algebra. , 1980 .

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

[18]  Kurt VanLehn,et al.  Felicity conditions for human skill acquisition: validating an ai-based theory , 1983 .