An Alternative Theory: Deep Understanding of Mathematics
Secondary and Middle Grades Education
Comments on an article by Stephen G. Schilling and Heather C. Hill. This series of articles presents a systematic approach to judging the validity of a test within a framework recommended by Kane. The author's approach effectively integrates quantitative and qualitative tools, showing the role of psychometric models in validation, and emphasizes the importance of clearly explaining the theory underlying the development of the test. Indeed, validation of the test is commensurate with validation of the theory underlying its development. In this commentary, I'd like to address the difficulties the authors encountered in connecting the empirical evidence with the theory, and suggest an alternative theory that might account for the difficulties. I propose that a single construct underlies all the measures described in this series of studies, and that construct is deep knowledge of mathematics, not a mathematics different from that known by mathematicians or mathematically competent adults, and not a superficial understanding of algorithms that is often the outcome of students' learning in the current system of mathematics education.