# NSF CAREER — Teaching Mathematics Well in Community Colleges: Understanding the Impact of Reform-Based Instructional Resources

## Vilma Mesa

Verification Strategies in Selected Examples in Textbooks Used in the Classes Observed

The purpose of the textbook analysis was to determine the extent to which examples, presumably the most important resource for students and instructors, make explicit verification strategies as part of the solution. We used the same methodology described in Mesa (in press) to describe the verification strategies present in all examples in sections in the textbooks that corresponded to one of the lessons for each class we observed. The analytical framework comes from Balacheff’s theory of conceptions (Balacheff & Gaudin, in press).

Verification strategies can be (1) procedural (the student repeats the procedure and if the answer obtained is the same the problem is correct), (2) determined by the didactical contract (solutions are chosen because they are presented in the section of the textbook), (3) based in the mathematical content (they require the use of theorems or definitions). Mesa had documented that for the most part, examples in calculus textbooks in sections devoted to differential equations do not take advantage of the richness of the content to determine what solution method has to be used, how to determine that an answer has been found and how to determine that the answer is correct. With few exceptions, examples only present the steps with little justification, using themes of the corresponding section, and with no discussion about the plausibility or correctness of the solution.

While strategies based on procedures are easy to identify, strategies based on the didactical contract or the content require an analysis of the notions available to students. For the purpose of determining the content used, we constructed concept maps of the sections from which the lessons came from and analyzed all the examples in those sections. We added another section from each textbook for comparison purposes. In total we examined 93 examples across the three textbooks, College Algebra by Cynthia Y. Young, Analytic Trigonometry with Applications by Raymond A. Barnett, Michael R. Ziegler, and Karl E. Byleen, and Precalculus: A Concise Course by Ron Larson and Robert P. Hostetler.

## Key Findings:

• The concept maps reveal the high complexity of the notions that are introduced to students in each lesson.
• Like in the Mesa study, 85% (79 of the examples) show the steps to solving the problem, 10 (11%) of the examples showed how to decide what do to solve the problem and 22 (24%) of the examples elaborated on the solution. This suggests that the verification strategies are process based.
• Fifty-one (55%) of the examples explicitly stated the answer, that is, it stated that an answer has been found.
• Only 12 (13%) of the examples verified that the solution found was indeed correct.

## Next Steps:

• Additional modification of the survey based on feedback regarding self-handicapping behavior questions and desire to capture SES.
• Continued analyses of subgroups?
• Administer the PALS surveys to students and instructors in the entire department at the community college.