### Helen M. Gerlach, Ph.D.

##### Advisor: Dr. Jerome Frieman

##### Dissertation Title

Individual schema development in solving mathematical word problems

##### Dissertation Abstract

Prior research has shown a continuum of ability (Novice, Transitional, Expert) on which individuals were able to solve increasingly more complex mathematical word problems involving proportional reasoning. Given problems X, Y, and Z (of increasing difficulty), a Novice could correctly solve only X, a Transitional X and Y, and an Expert all three. Three experiments explored the problem solving differences between subjects at each of the three ability levels.

A larger percentage of subjects whose major was engineering and who had taken a computer science course in high school solved the most difficult problem correctly compared with other students enrolled in an introductory computer science course.

Performance was not affected when the word problem was presented in a more directive format. However, there was a practice effect. A significantly larger percentage of students correctly solved the Z-level problem when it was presented within a set of six problems, each of increasing difficulty, where all of the problems were written in a more directive format.

An integration task of memory and a problem sorting task tapped schema differences. Novices did not make connections between problem types. Transitionals focused on the Z-level problem. Experts viewed all problems as related.

Problem solving strategy varied across problems and groups of subjects. Novice subjects used less refined strategies both to solve the problem and verify their answers. Transitional subjects employed a ratio strategy to solve X-level problems and static strategies (e.g., word order matching) for both the Y and Z-level problems. However, a simple verification strategy ("does that make sense?") enabled Transitional subjects to correctly solve the Y problems. Expert subjects employed a ratio strategy to solve both the X and Z-level problems and an active strategy (e.g., balance the equation) for the Y-level problems. Experts were more likely to verify their answers in a mathematical fashion.

Thus, performance on proportional reasoning word problems is enhanced with increased experience in solving problems in a highly structured environment, where subjects are able to see the progression from one problem to the next, and when subjects are capable of employing mathematical verification strategies.

##### Education

**Ph.D.,** Psychology, Kansas State University, 1986