Proofs and Counterexamples about Continuous Functions

This study focused on reasoning and proof from undergraduate math majors in calculus. Proofs of truths and counterexamples of false facts have been shown to help students understand them. Not only do they help explain why a statement is true or false, but students will also be able to see how. However, reading proofs and constructing them are different processes. In my personal experience, constructing proofs is very difficult especially when I don't know what direction in which I need to head.

Before constructing proofs/counterexamples, students and teachers need to decide which one to produce (to prove a truth or to disprove a falsity?), which means they must decide on the validity of the proposition (is it true or false?), which means they must have a deep understanding of the concepts involved. In fact, many students and teachers have difficulty deciding whether a proposition is true or false because of their inadequate understanding of the mathematical content. Still, many mathematicians often struggle with formal proofs. Thus more effective ways of teaching these are needed, at least in the undergraduate math major domain.

Ko and Knuth (2009, p. 69) pose two research questions for this study:

1. How well do undergraduates construct proofs and generate counterexamples in the domain of continuous functions?
2. What problems appear in the proofs students construct or the counterexamples they generate?

One thing that the authors stated that I found striking was the fact that proof serves as a means of communication among mathematicians. This makes sense, because I noticed when learners can explain something in each others' terms, they gain a better understanding of the idea. Informally, they are proving it to one another.

We usually think of counterexamples, as disputes to generalizing statements. For example, the number $2$ would be a counterexample (in fact, the only one) to the false proposition, All prime numbers are odd. Some counterexamples can actually be ‘examples’, if we think of them as a proof to the generalizing statement's negation (e.g., $2$ is an example that proves the conjecture, There exists an even prime number).

The authors also discussed different types of proof schemes and counterexamples and the properties of each. They created a scale or spectrum of proofs, and one for counterexamples, that ranged by robustness. After receiving calculus students' work on a problem set, they rated it using their own spectra.

Since constructing a proof requires sufficient background knowledge and the decision of whether to prove or disprove, Ko and Knuth (2009) suggest that math teachers should provide a learning environment in which students can easily decide the veracity of statements. There are several methods of doing this. Especially in secondary school, formal proof is not even in the question. Therefore teachers must foster students' informal proof and reasoning as often as possible. Fishbein (1982) states, Ignoring students' intuitive representations of proof does not promote their mathematical understanding (as cited in Ko and Knuth, 2009, p. 75). In addition to drawing attention to students' intuitive thinking, teachers should also explicitly identify students' logical flaws and misconceptions so that they will have a stronger understanding of the processes in constructing (even informal) proofs.

References

• Ko, Y., & Knuth, E. (2009). Undergraduate mathematics majors' writing performance producing proofs and counterexamples about continuous functions. Journal of Mathematical Behavior, 28, 68–77.

How Students View Assessment

Assessment seems to have a completely different effect on students' attitudes based on whether they frequently pass or frequently fail.

Students who demonstrate winning streaks are confident and hopeful, demonstrate continual evidence of success, and are excited to learn. These students are likely to seek more feedback and more challenges. On the other hand, students who demonstrate losing streaks are frequently hopeless and give up easily. They do not feel safe at school and feel that they are always being evaluated. They see feedback as criticism and do not seek challenges or new ideas. Stiggins tries to eliminate this gap.

Rather than using assessment to sort students into winners and losers based on performance, educators should use assessment to help student learn. Stiggins offers an alternative notion, Assessment For Learning (as opposed to assessment of learning). This alternative encourages teachers to turn assessment into a process that involves sharing goals and targets with students, provide frequent and continual feedback in student-friendly language, and provide examples of outstanding student work. This causes students to self-assess and notice trends in their own achievement. Students become more aware of their academic progress. They begin to understand what is expected and make decisions on how to become better. They also generate their own feedback and set their own goals. The hope, Stiggins says, is not to eliminate failure but to eliminate losing streaks. This helps boost student confidence and motivates them to try more.

References

• Stiggins, R. (2007). Assessment through the student's eyes. Educational Leadership, 64(8), 22–26.