Developing Fluency in Mathematics

Mary T. Treanor, Valparaiso University

The ability to discuss mathematics is crucial for mathematics majors and for students from across the campus whom we meet in service courses. This talk will address the concerns of both sets of students; there are higher expectations for majors, but many of the same strategies can be used or easily modified for science, business, or engineering majors. While most students will readily agree that the ability to communicate with colleagues, others in their profession, and clients will be crucial to their success in the future, fewer students expect to develop and exercise these skills now in their mathematics classes.

Before I can expect students to discuss mathematics orally or in writing, I need to encourage them to read their text and to provide other selected materials as well; no one is comfortable expressing ideas—or asking questions--when the terms, idioms, and conventions of a topic are foreign. Mathematics is a language, but too often majors lack confidence in their ability to speak clearly. Students from other majors often seem to assume that mathematics will always be foreign, a language that they will never know. They often concentrate only on the “boxed” portions of their text; they seem surprised by my expectation that they will understand if they read. And initially all students may be puzzled when I ask them to explain, to use words, to do more than use an algorithm to solve a problem, and to care about understanding the solution instead of focusing only on a correct answer. (Certainly, English professors expect that students listen to, read, and analyze good literature in order to improve their ability to express themselves well, and I think that the analogy holds as well in mathematics.)

Small collaborative groups can be a good setting for speaking about mathematical ideas and for thrashing out an explanation of a solution that will be presented either orally or in written form. In small groups, the discussion will be lively. Errors and questions can be more easily raised and addressed by either students or by the instructor in this non-judgmental setting, and students experience control and build confidence in their ability to speak and to listen in mathematically accurate ways.

The materials for these groups can be very varied: questions about text that they have already read, a problem to solve with directions that the solution be written up specifically to be helpful to a student who was absent for the introduction of the topic, a problem whose solution requires them to make finer distinctions about material with which they are already somewhat familiar, their creation of an original problem and its solution, True-False questions. There is always a written report sheet, unless each group will report orally. Almost all of the materials are also suitable for students to use individually.

I find that students often develop confidence in their ability to articulate their thinking when they are responding to well-formulated True-False questions. Students are directed to label each statement correctly, and, if the statement is False, they are to give a counterexample and to explain why their example is appropriate. Deciding which statements are True or False and then, for a False statement, responding to each of the other two parts will often stimulate discussion, raise clarifying questions, and lead to a careful articulation of the group’s insights. Explanations for true statements are optional, but students often tackle the challenges that such explanations pose.

Students’ questions, whether posed by an individual or arising in a group who can’t resolve the issue for themselves, sometimes trigger my ideas for new materials that will stimulate student discussion and writing. Problems that lead students to make clear distinctions between ideas that are easily confused are particularly valuable.

It is my experience that an emphasis on articulating mathematical understanding encourages everyone in class to listen more thoughtfully, to speak and write more clearly, and to understand more thoroughly—and that includes the instructor. In this talk, I will share strategies and materials that I have found helpful in my classes; examples from courses for majors and from service courses will be included.