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One way to find out what students understand is to ask them true/false questions, but have them justify their answers. These justifications bring out confusions about the concepts, but are also the beginning, for calculus students, of writing mathematical proofs.
Background and Purpose
My initial experimentation with modified true/false questions was prompted by frustration with my students' performance in the first-year calculus courses which I was responsible for teaching as a graduate assistant at the University of Colorado, Boulder. Although my students' ability to perform the various algorithms of the course was largely acceptable, the proofs which they were producing (especially on tests) left much to be desired. As a replacement for these "proofs," I began to use true/false questions addressing course concepts on exams; to avoid mere guessing, I also required students to write a short explanation of their response. These written explanations proved to be extremely valuable as a method of identifying student misconceptions and learning difficulties, thereby allowing me to modify my instruction to address those problems.
After joining the faculty at the University of Southern Colorado, I began to use modified true/false questions both for on-going assessment and on exams in a variety of lower division courses, including calculus, college algebra, liberal arts mathematics courses, and content courses for pre-service elementary teachers. I have also used them as one means to assess conceptual understanding on upper division course exams. A regional state institution with moderately open enrollment, USC limits class size in mathematics to 45 students, and in many classes the size is closer to 30. This allows me to use modified true/false and other writing assignments as often as once a week so that students become familiar with both the format and my expectations of it. Regular use of these questions provides me with instructional feedback, and serves as a mechanism to encourage student reflection on concepts.
Method
The statements themselves are designed to test students' understanding of specific theoretical points, including definitions, properties, theorems, and relations among concepts. (See below for examples.) By casting these statements in the form of implications, it is possible to draw attention to the logical form of mathematical theorems, such as the distinction between the hypothesis and the conclusion, and the relation between a statement, its contrapositve and its converse. The form of the question is thus conducive to a wide variety of concepts, making it readily adaptable to different topics. The critical step in designing a good question is to identify those aspects of the concept are (i) most important and (ii) most likely to be misunderstood. The purpose should not be to trip students up on subtle technical points, but to call their attention (and reflection) to critical features of the concepts. Some examples:
Determine if each of the following is true or false, and give a complete written argument for your response. Your explanation should be addressed to a fellow student whom you are trying to convince of your reply. Use diagrams, examples, or counterexamples as appropriate to supplement your written response.
If 6 | n and 4 | n, then 24 | n.
If a is a real number, then .
If f is differentiable at a, then 
exists.
If f(c) is a local maximum of f, then f ´(c)=0.
If the series 
converges, then 
.
A typical true/false writing assignment consists of two - four questions addressing the same basic concept or relation; sample instructions for the students are shown above. The stipulation that students write their explanation to a fellow student is made to encourage students to provide a convincing, logical argument consistent with their current level of formal understanding. (In upper division courses, more formalism and rigor is expected.) The goal is to have students communicate their understanding as precisely and clearly as possible, but to articulate this understanding in their own terms, rather than using formal terms or conventions which they don't understand. When employing writing assignments as a part of the course grade, the role of this audience is explained and emphasized to the students at the beginning of and throughout the semester. Students are allowed three to five days to complete the assignment, and are encouraged to discuss the questions with each other and with me during this time. Students who wish to discuss the assignments with me must show that they've already made a significant attempt, and they find that I am not likely to give them "the answer."
In evaluating individual responses, credit is given primarily for the explanations, rather than the true/false response. In fact, a weak explanation for a correct response may earn less credit than a "good" explanation for a wrong response. Both composition and content are considered, with the emphasis placed on content. Typically, I read each response twice: the first time to make comments and place papers into rough piles based on either a three or five point scale, the second time to verify consistency within the piles and assign the score. When using true/false assignments as part of the course grade, a generic version of the scoring guide is shared with students early in the semester. This generic guide is tailored to specific assignments by setting specific content benchmarks for each score (for two types of such scales, see the article by Emenaker in this volume, p. 116). The scores from these assignments can then be incorporated into the final course grade in a variety of ways depending on overall course assessment structure and objectives. Methods that I have used include computing an overall writing assignment grade, incorporating the writing assignment scores into the homework grade, and incorporating the writing assignment scores into a course portfolio grade.
Findings
Students tend to find these assignments challenging, but useful in building their understanding. Their value in building this understanding has led me to use writing increasingly as an instructional device, especially with difficult concepts. The primary disadvantage of this assignment is the fact that any type of writing requires more time for both the instructor (mostly for grading) and the student (both for the physical act of writing and the intellectual challenge of conceptual questions). I have found that modified true/false questions require less time for me to evaluate than most writing assignments (as little as one hour for three questions in a class of 25). Because these questions focus on a specific aspect of the concept in question, most students are also able to respond fairly briefly to them, so that including them on exams is not unreasonable provided some extra time is allowed.
Use of Findings
The instructional feedback I've obtained from student
responses to these questions has led me to modify my
teaching of particular concepts in subsequent courses (see
[1]). For instance, the responses I initially received to
questions addressing the role of derivatives in locating extrema
suggested that students were confused, rather than
enlightened, by class presentations of "exceptional" cases (e.g., an
inflection point that is also an extremum). Accordingly, I
now use true/false assignments (e.g., "If
(c , f(c)) is an inflection point, then
f(c) is not a local maximum of
f.") as a means to prompt the students to discover such examples
themselves, rather than including them in class presentations. This
worked well in calculus, where students have enough
mathematical maturity for such explorations, but was less successful
in college algebra. In response, I began to modify the
question format itself in order to force the students to confront
conceptual difficulties. For example, rather than ask
whether for all reals a, I may tell the students this
statement is false, and ask them to explain why.
A more global instructional change I made as a result of a true/false assignment occurred when one of the best students in a first semester calculus course repeatedly gave excellent counterexamples in response to false statements, while insisting that the statements themselves were not false since he knew of other examples for which the conditions of the statement did hold. I now regularly explore the distinction between universally and existentially quantified statements in introductory courses, whereas I had previously not considered it an issue.
Success Factors
Early in a semester, one should expect the quality of responses to be rather poor, both in terms of content and composition. Quality improves dramatically after the first few assignments as students become familiar with its require ments. To facilitate this process, it is important to provide each student with specific written comments on their work, and to encourage revisions of unacceptable work. I have also found that sharing samples of high quality student responses helps students learn to distinguish poor, good, and outstanding work. For some students (non-native English speakers, learning disabled, etc.), the writing requirement (and the fact that composition does count) has posed problems not solved by these measures. Since the number of these students is small at USC, I have always been able to arrange an alternative for them, such as an oral interview or a transcriber.
References
[1] Barnett, J. "Assessing Student Understanding
Through Writing." PRIMUS (6:1), 1996. pp. 77-86.
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