**Introduction**

Assessing Higher Order

Mathematical Thinking: What We

Need to Know and Be Able to Do

Assessing Higher Order

Mathematical Thinking: What We

Need to Know and Be Able to Do

GERALD KULM

Most mathematics teachers are able to construct a test for a specific class or course in order to assess students' attainment of a set of objectives, perhaps including objectives that could be identified as higher order thinking. Furthermore, many teachers would probably use, more or less informally, a framework such as cognitive level by mathematical content to establish the breadth and depth of items included in the test. That is, the teacher would use his or her own knowledge about the abilities of the students in the class to construct or choose a set of items that include a few easy ones, a majority that are moderately difficult, and a few challenging ones. The teacher would also make sure that all of the mathematical content that is deemed important is covered by the test items.

Unless our notions about assessment are likely to change significantly, this type of a procedure will continue to be used by teachers and test developers. More importantly, this approach to testing is likely to remain as the primary criterion that mathematics teachers, school administrators, and the general public use to define what mathematics is and how well students have learned. In order to make changes in school mathematics, we must be able to specify the parameters that determine how much and what part of a subject has been learned by whom. One approach is to modify the traditional cognitive-level-by-content framework so that it reflects new conceptions of mathematics thinking and learning. Another approach -- embraced by those who believe that a two-dimensional framework cannot possibly reflect the way mathematics is learned -- is to start anew, building upon more recent psychological models of learning and more modern ways of looking at the content of mathematics.

A second fundamental issue for assessment concerns the question: What is mathematics? In assessing higher order thinking, we must be able to identify the parameters that characterize mathematical thinking processes. In the case of applications and other problem- based contexts for doing mathematics, we need to be able to identify essential mathematical content that is embedded and to have some idea about how the context and content interact with performance.

Mathematics has often been chosen as an example subject area for experiments by educational psychologists and test developers, partly because it appears to be so readily amenable to breaking into nice, simple, linear pieces. It might be argued that the behavioral objectives craze and the focus on learning hierarchies were largely responsible for widening the gap between school mathematics and "real" mathematics. Anything that could not be stated and measured behaviorally gradually disappeared from the curriculum. Is

-1-