# Assessing Higher Order Thinking in Mathematics

By Gerald Kulm | Go to book overview

way to explore such empirical results is to collect samples of qualitative data at the same time as the item scores are recorded. This could be as straightforward as collecting a sample of the students' answer sheets (especially if they were encouraged to "show their work"). A more formal strategy would be to interview a sample of the students taking the test.

An interesting and informative way to look at SOLO data of the kind described here is to analyze both the dichotomous question- level data and the polytomous SOLO super- items using IRT models, then compare the two ( Wilson, 1988; Wilson & Iventosch, 1988). Such a comparison could not be done in this case because the only data available for this analysis were the superitem scores.

Comparing the SOLO data with the Doing Mathematics variable seemed to raise some doubts about the relational level. Looking more closely at some of the relational questions within the items (e.g., items 1, 2, 3, and 4) leads one to speculate whether the relational level has been well-realized by these items. Certainly the relational question within each of these items would be expected to be more difficult, but that is not sufficient for it to be considered as indicating a higher level within the SOLO taxonomy. For example, in item 2, the relational question asks the student to place the use of a railway timetable into the broader context of a real-life problem where one has to consider time taken to get to and from the railway station. This is adding an extra variable to the problem, but is it addressing the mathematical relations among the components of the timetable? What is needed is a strongly mathematical idea of how to apply SOLO. One potential source for this is the van Hiele ( 1986) mathematics learning sequence. If one compares the SOLO idea, which is a general approach, to the van Hiele approach, one realizes that the van Hiele levels constitute successive relational levels that could be used in a SOLO framework. The interesting complication is that SOLO provides a framework for assessing within the van Hiele levels, and van Hiele levels provide a framework for linking between SOLO items at different levels.

Acknowledgments

This study was sponsored by the National Center for Research in Mathematical Sciences Education, School of Education, University of Wisconsin- Madison. The author would like to thank Tom Romberg, Director of the Center, for providing the data on which this study is based and for his encouragement and support.

References

Aitchison, J., & Silvey, S. D. ( 1957). "The generalization of probit analysis to the case of multiple responses". Biometrika, 44, 131-140.

Andrich, D. ( 1985). A latent-trait model for items with response dependencies: Implications for test construction and analysis. In S. E. Embret son (Ed.), Test design: Developments in psychology and psychometrics. Orlando, FL: Academic Press.

Biggs, J. B., & Collis, K. F. ( 1982). Evaluating the quality of learning. The SOLO Taxonomy. New York: Academic Press.

Carpenter, T. P., & Moser, J. M. ( 1984). "The acquisition of addition and subtraction concepts in grades one through three". Journal for Research in Mathematics Education, 15, 179-202.

Collis, K. ( 1983). "Development of a group test of mathematical understanding using superitem SOLO technique". Journal of Science and Mathematics Education in South East Asia, 6( 1), 5-14.

Collis, ICF., & Davey, H. A. ( 1986). "A technique for evaluating skills in high school science". Journal of Research in Science Teaching, 23( 7), 651-663.

Cureton, E. E. ( 1965). "Reliability and validity: Basic assumptions and experimental designs". Educational and Psychological Measurement, 25, 326- 346.

Edwards, A. L., & Thurstone, L. L. ( 1952). "An internal consistency check for scale values determined by the method of successive integers". Psychometrika, 17, 169-180.

Glaser, R., Lesgold, A., & Lajoie, S. ( 1987). Toward a cognitive theory for the measurement of achievement. In R. Ronning, J. Glover, J. Con oley , & J. Witt (Eds.), The influence of cognitive psychology on testing. Hillsdale, NJ: Lawrence Erlbaum Associates.

Guttman, L. ( 1941) "The quantification of a class of attributes: A theory and method for scale construction". In P. Horst (Ed.), The Predication of Personal Adjustment, pp. 319-348. New York: Social Science Research Council.

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