Schoenfeld’s problem solving theory in a student controlled learning environment

Abstract

This paper evaluates the effectiveness of a student controlled computer program for high school mathematics based on instruction principles derived from Schoenfeld’s theory of problem solving. The computer program allows students to choose problems and to make use of hints during different episodes of solving problems. Crucial episodes are: analyzing the problem, selecting appropriate mathematical knowledge, making a plan, carrying it out, and checking the answer against the question asked.

The effectiveness of the computer program was evaluated by means of a pre-test–post-test quasi experimental design study. Four classes worked with the computer program in three periods of two consecutive weeks each, whereas five classes received only traditional mathematics education. These classes served as a control group. The results show evidence of intervention effectiveness. The students who worked with the computer program showed increased problem-solving ability compared to the students in traditional mathematics instruction. The use of hints could explain an essential part of the increase in students’ problem solving skills.

Introduction

In most classes of secondary education mathematics, teachers teach students to solve mathematics problems by having them copy standard solution methods provided by the textbook. Little time is devoted to teaching students how to solve problems (Foshay & Kirkley, 2003). The effect is that students have great difficulty in solving non-standard problems that require the application of domain knowledge and routines. Researchers (Garofalo and Lester, 1985, Schoenfeld, 1987, Van Streun, 2000) indicate that students’ problem solving failures are often not the result from a lack of mathematical knowledge but from the ineffective use of their knowledge. Schoenfeld (1992) stated that students need to learn to define goals and to self-regulate their problem solving behaviour in order to improve solving of non-standard mathematics problems.

Schoenfeld observed that during problem solving, students display distinct categories of behaviour. Schoenfeld calls these categories ‘episodes’. Crucial episodes are: analyzing the problem, selecting appropriate mathematical knowledge, making a plan, carrying it out, and checking the answer against the question asked. Schoenfeld observed that experts spend more time analyzing problems and verifying the adequacy of possible answers to a problem than novices do. Novices (new learners), on the other hand, spend more time on familiar procedures (e.g. calculations) without making sure they followed a correct solution plan. They spend less time reading a problem text and thinking about the appropriateness of the procedures they use.

Dreyfuss and Eisenberg, 1996, Goos, 2002, Teong, 2003 confirm the findings of Schoenfeld. They conclude that strong problem solvers are capable of representing problem situations in many forms: a sketch, a graph, a table or a numerical example. During the solution plan episode, strong problem solvers often use more formal representations and algorithms. Strong problem solvers also are flexible in their approach and usually monitor their solution process. Novices on the other hand spend much time carrying out procedures (e.g. calculations) without questioning the adequacy of their solution plan. They spend little time reading a problem text and thinking about the appropriateness of the procedures they use. Schoenfeld (1992) shows how novices can learn to apply episodes in their solution process if you let them think about the way they try to find a solution. For instance one may ask the students what they want to find out, which steps they have taken so far and which steps they will take next in order to solve the problem. The teacher can try to improve self-regulatory behaviour and instruct students in order to reflect on the episodes needed to solve their problem successfully. The teacher stimulates students to use different solution methods and choose the most suitable method when working at a problem.

Implementing instruction in problem-solving skills may profit from the use of new technologies that enable the design of interactive computer software. Interactive computer programs may be designed to provide feedback and hints to assist students in becoming aware of their solution process. The problem here is how to design a computer program that is both interactive and adaptive to students’ needs for help (Jonassen, 1996). The program therefore should allow students to obtain hints according to their needs. This is no trivial matter. Wood and Wood (1999) e.g. report that students with low prior knowledge tend to seek less help after an incorrect response in an adaptive algebra tutoring program than students with high prior knowledge. However, the finding that weak students seek less help may be due to Wood and Wood’s algebra tutoring program. This program provided students help by means of a single problem solving method. Perhaps, if different solution methods were offered weak students might have sought help more frequently.

In this paper we report on the development and evaluation of a computer program based on Schoenfeld’s theory of problem solving and his ideas about adaptive hints to help students solve a mathematics problem. The program offers a student controlled learning environment with instructional hints for each episode in problem solving and with different solution methods to choose from within each hint. By offering different hints based on informal and formal solution methods (see next paragraph) all students can choose a level of instruction that can support them effectively. There are three research questions:

• Do students actually use hints on different episodes during problem solving?

• During which episodes are these hints most effective?

• Does a computer program according to Schoenfeld’s theory of episodes help students to become better problem solvers?