# Hotmath

Oliver Grillmeyer, Ph. D., U. C. Berkeley

Sarah Chance, Ph. D., U.C. Santa Barbara

## Introduction

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## Research

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).

## Why do worked examples work better than conventional methods?

- Reduced cognitive overload. Solving problems imposes a high degree of cognitive load on the learner. Students lose track of the key issues to be learned as the process of solving problems is so cognitively demanding; this interferes with learning of the desired information.
- Reduced frustration. Students often feel frustrated as they struggle trying to reach the solution. Providing step-by-step solutions can create a "scaffolding" or schema around which to organize challenging concepts, making them more manageable and less overwhelming.
- Opportunity for great problem variability. A greater number and range of examples can be presented, which facilitates abstraction, especially for students with low achievement.
- Shifted responsibility. Worked examples encourage active mental participation on the part of the students by providing the student with the tools needed to reach a solution on their own without needing to rely on others such as a parents or tutors, resources that may not always be available.
- Immediate correction. Correctly worked examples reduce the risk of "faulty learning" or practicing incorrect solutions and the learning of incorrect associations (Siegler, 1988).
- Greater feeling of accomplishment and confidence. Knowing that it will be possible to complete schoolwork correctly makes learning math less daunting. Knowing that work has been completed correctly contributes to confidence and the desire to continue.

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## References

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.

June, 2001