Activating Inappropriate Existing Knowledge
McNeil, in a much more recent work with other researchers, highlights the importance of understanding math equivalence, that is, appreciating what an equal sign really entails. To assess how well a student understand math equivalence, questions like the one shown below can be used:
|"If we do not think spirally, why do we have to learn or teach spirally? And this is what makes it so difficult. There are many right words in the quote and the right intentions. But, as long as the fundamentals are wrong, the pedagogy behind the spiral approach is not viable.
Concepts - classes - are not organized in a single hierarchy. Rather there is a web of related classes in which a variety of hierarchies can be distinguished. Recognizing individual hierarchies is as important as discerning relationships between the classes. But spirals?"
Figure and caption taken from http://www.cut-the-knot.org/Mset99/examples.shtml
Gamoran, A. (2001). Beyond curriculum wars: Content and understanding in mathematics. In T. Loveless, Ed., The Great Curriculum Debate, pp. 134-162. Washington, D.C.: Brookings Institution Press.