Oliver Grillmeyer, Ph. D., U. C. Berkeley
Sarah Chance, Ph. D., U.C. Santa Barbara
What is the best way to teach math to high school and college students? Conventional classroom teaching has included explanation of the subject matter accompanied by a limited number of examples, after which students are assigned unsolved problems to work out. Recent research has investigated the effect of increasing the use of "worked out" examples of math problems, and the results have revealed that this teaching method produced more effective results (Carroll, 1992 & 1994; Ward, & Sweller, 1990; Pass & Van Merrienboer, 1994). This manuscript includes (1) a brief review of research findings suggesting that providing students with worked out examples is more effective than the conventional math instruction method (2) a discussion of the probable reasons for this effect (3) comments on what this body of research suggests for teaching today's math students, with specific consideration of a recently developed math-learning resource, hotmath.com.
Providing students with worked out examples of math problems has been found to be more effective than simply assigning the same problems for the students to work out on their own. In one experiment (Carroll, 1994), 40 high school students were instructed in how to solve linear equations. In an "acquisition phase" the students were divided into two groups and their instruction differed in the following way: in the "conventional learning" group, students were assigned 44 unsolved problems to work out (in the classroom and at home homework), and in the "worked examples" group students were provided with the same problems, but half of the problems were accompanied by correct solutions. After completion of the assigned problems, both groups were tested on 12 related problems, 10 of which were very similar to the linear equations presented in the acquisition phase, and 2 of which were word problems, used to test whether students could transfer and extend their knowledge to a new context. No worked out examples were available during the test. The test results revealed that students in the "worked examples" group outperformed students in the "conventional learning" group on both types of the test problems. A second experiment, employed a similar methodology but focused on "low achieving" students (students with a history of failure in mathematics, and students identified as learning disabled). Here, the data revealed that students in the "worked examples" group required less acquisition time, needed less direct instruction, made fewer errors, and made fewer types of errors than students in the "conventional learning" group.
Related research (Pass & Van Merrienboer, 1994) sheds light on the cognitive underpinnings of the effects described above. In this study, 60 college-aged students were instructed in geometry concepts. As in the Carroll experiments, students were assigned un-worked problems to solve or worked out examples to review (unlike the Carroll study, the "worked examples" group was assigned no un-worked problems to solve). In this study, the researchers manipulated the nature of the problems presented to the students: within each group, some students received problems which were all similar to each other, while others received a more varied problem set. Furthermore, the researchers measured the "cognitive load" experienced by the students. This research revealed that while students in the worked examples group completed their work more quickly, they perceived the work as less demanding and displayed better transfer performance at test. The effect was most pronounced for the students given highly-variable problems. The researchers suggest that the reduced cognitive load associated with the worked examples enabled students to "take advantage of" the variability in problems by using the available cognitive resources to process the underlying similarity in the problems (i.e., the mathematical concepts being taught), and to integrate the current problem with existing knowledge (Linn, 2000).
hotmath.com is a new approach for mathematics homework assistance for high school and college students. It is a Web site that provides students with a means of learning mathematics by giving step-by-step solutions to the textbook homework problems assigned by their teachers. Students are given a complete path from the problem statement to the solution showing each step to take to reach that solution, along with hints, and explanations of each step. Students can work at their own pace through the solutions and move back to replay steps. Hotmath asks questions during the problem solving process as well. This encourages more thought on the part of the students and ensures that the process is interactive.
Hotmath provides solutions to relevant and pertinent problems-the ones that students are given by their teachers to solve. Teachers know that students are focusing on the exercises that are important to students, and students are getting help with their homework while still learning the important material.
The challenge is to create instruction that is effective to the challenged students. Hotmath does this by providing accessible, worked solutions to problems. Students using Hotmath focus on the correct series of steps needed to reach the solution. The theory of knowledge integration as described by Linn (2000) and others is one such technique. There are four tenets in knowledge integration: make science accessible, make thinking visible, promote autonomy, and encourage social supports. Hotmath's instructional approach helps to make science accessible by presenting relevant problems to students to promote integration of the ideas obtained while learning how to solve these problems with the students' existing knowledge and ideas. Hotmath helps to make the expert's problem solving thought process more visible by clearly showing the sequence of steps taken and providing explanations, discussions of alternative paths and approaches, and showing a complete history of the steps taken to reach the solution. Hotmath promotes autonomy by prompting students with questions that encourage them to reflect on the information presented. This promotes an inquiry process that will help students to learn autonomously.
Hotmath uses graphs and diagrams to explain the solutions when applicable and necessary. These additional sources of information are integrated within the HotMath instructional environment to reduce cognitive load that would be imposed by having to integrate them (Ward & Sweller, 1990). The use of multimedia to provide alternative forms of representation has been shown to be effective when properly presented (Mayer, 1997). HotMath is looking into providing narrations of the explanations to provide instruction that is easier to follow.
Hotmath uses ideas founded in principles of constructivism and knowledge integration, and that leverage off of the effectiveness of worked examples and multimedia. The goal is to provide instruction that reduces cognitive load and results in knowledge that is integrated with students' existing ideas such that it can be more effectively used in a wide range of contexts.
Carroll, W. M. (1992). The use of worked examples in teaching algebra. ED353130. Apr. 1992.
Carroll, W. M. (1994). Using worked examples as an instructional support in the algebra classroom. Journal of Educational Psychology. v86, n3, 360-367.
Linn, M. C. (2000). Designing the knowledge integration environment. International Journal of Science Education. Vol. 22, No. 8, 781-796.
Mayer, R. E. (1997). Multimedia learning: Are we asking the right questions? Educational Psychologist, 32(1), 1997, 1-19.
Paas, F. G. W. C. & Van Merrienboer, J. J. G. (1994). Variability of worked examples and transfer of geometrical problem-solving skills: A cognitive-load approach. Journal of Educational Psychology. v86, n1, 122-133.
Papert, S. (1980). LOGO's roots: Piaget and AI. In Mindstorms: Children, computers, and powerful ideas. New York: Basic Books.
Piaget, J., & Inhelder, B. (1971). The Psychology of the Child. New York: Basic Books.
Ward, M. & Sweller, J. (1990). Structuring effective worked examples. Cognition and Instruction, 1990, 7(1), 1-39.