tember 2002 Workshop). Some content analyses claimed that materials had too many errors (Braams, 2003a). Virtually no one disputes that curricular materials should be free from errors; all authors and publishers indicated that errors should be quickly corrected, especially in subsequent versions. It appears that not all content analyses paid appropriate attention to the accuracy issue. For example, Richard Askey, University of Wisconsin, in commenting on the Department of Education reviews to the committee during his September 17, 2002, testimony, pointed out that “In these 48 reviews, no mention of any mathematical errors was made. While a program could be promising and maybe even exemplary and contain a significant number of mathematical errors, the fact that no errors were mentioned strongly suggests that these reviews were superficial at best.”

It surfaced over time that some of the debate over the quality of the materials focused on the relative importance of different types of mathematical activity. To assist in deliberations, we chose to stipulate a distinction between mathematical inquiry and mathematical reasoning. Mathematical inquiry, as used in the report, refers to the elements of intuition necessary to create insight into the genesis and evolution of mathematical ideas, to make conjectures, to identify and develop mathematical patterns, and to conduct and study simulations. Mathematical reasoning refers to formalization, definition, and proof, often based on deductive reasoning, formal use of induction, and other methods of establishing the correctness, rigor, and precise meaning of ideas and patterns found through mathematical inquiry. Both are viewed as essential elements of mathematical thought, and often interact. Making too strong a distinction between these two elements is artificial.

Frequent debates revolve around the balance between mathematical inquiry and mathematical reasoning. For example, when content material has weak or poor explanations, does not establish or is not based on appropriate prerequisites, or fails to be developed to a high level of rigor or lacks practice in effective choices of examples, issues of mathematical reasoning are often cited as missing. At the same time, rather than focusing solely on the treatment of a particular topic at one particular point in the material, it is essential to follow the entire trajectory of conceptual development of an idea, beginning with inquiry activities, and ensuring that the subsequent necessary formalization and mathematical reasoning are provided. Moreover, one must determine in a content analysis whether a balance between the two is achieved so that the material both invites students’ entry and exploration of the origin and evolution of the ideas and builds intuition, and ensures their development of disciplined forms of evidence and proof.

To illustrate this tension and the need for careful communication and exchange around these issues, we report here two viewpoints, one pre-



The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement