In accordance with the vision of mathematics teaching advocated by the National Council of Teachers of Mathematics in the Professional Standards for Teaching Mathematics, I feel that the role of a teacher is to be an evaluator, a guide, and a facilitator. A teacher must evaluate students' present understandings and determine if misunderstandings exist. Misunderstandings, if not diagnosed, can develop into obstacles which stymie the development of understanding of related concepts. Therefore, a teacher must evaluate students' present understandings and then guide students to higher levels of understanding.
The process of guiding students toward higher levels of understanding can be done in many different ways but generally involves placing students in either individual or collaborative explorations which challenge their present understandings and force the students to build new mental connections to overcome the cognitive dissonance. Beyond taking into account the students' prior knowledge, these explorations should be designed to focus on either particular mathematical content areas or the development of general problem-solving skills. Such a schema should build upon students' prior understandings and aid them in internalizing the concepts better than learning passively. In order to accomplish this, I believe that collaborative experiences, such as group quizzes and group problem-solving activities, are good ways to challenge the students to clearly communicate their understandings and these in turn create forums for students to grapple with the understandings of others and compare them against their own. The result of this process is revised understandings that are more complex and better aligned with accepted mathematical conceptions.
In addition to being a guide, the teacher should be a facilitator of reflection in the students. Many students tend to be merely concerned with following steps and are not concerned with being reflective on their learning or with the outcomes of their work beyond the grade. As part of the calculus course I taught this past Fall semester, I initiated two methods to induce student reflection. The first one took the form of journal entries which the students handed in each week. The journal entries were a forum for the students to summarize what had been discussed in class the previous week, to discuss their difficulties, and to propose a plan to address those difficulties. Beyond allowing me to keep in closer touch with the ninety students in my lecture section, the journals, according to many of the students, aided them in assessing what they had learned and in considering if they needed help in a more timely fashion rather than waiting right before a test. The second reflection activity was that students were given a sheet to be filled out which asked the students to reexamine their mistakes on a test, hypothesize why those mistakes were made, and provide a plan to refrain from those mistakes again. In addition, the students were required to correct all the mistakes and hand in a 100% correct test. In order to aid the students, a complete solution for each of the test problems was available and the students were encouraged to speak to each other because they were not to just hand in the solution but to add a discussion which delineated why the particular steps were taken.
In addition to the facilitation of reflection, I feel that the teacher must facilitate the construction of connections between concepts and topics. For example, when trying to distinguish the connection between the world of precalculus mathematics to that of calculus, one could characterize it as the study of static relationships versus the study of dynamic relationships. Algebra and trigonometry can be used to examine static "snapshots" of a dynamic situation but the analysis of the dynamic situation requires the use of calculus techniques. An extremely useful tool for building connections is technology. Calculator and computer developed presentations can take a situation and bring it alive through modeling relationships. The use of graphical, tabular, and algebraic representations allows the students to see the same relationship through multiple perspectives. For example, the examination of a limit no longer needs to be restricted to the study of calculus. With the increasing availability of powerful but relatively inexpensive graphing calculators, students can develop an intuitive sense of limits by simply examining graphical and tabular examples without discussing the e-d ideas. Even in calculus, the e-d ideas can be bolstered in the students' minds by connecting them with a graphing device's viewing window. The connection of e and d can be illustrated by manipulating the range settings of the viewing window so that for a particular (x, f(x)) and a given epsilon (i.e., YMAX = f(x) + epsilon and YMIN = f(x) - epsilon), the graph cuts obliquely across the screen. This type of exploration allows students to visualize why the delta is chosen as the minimum distance and how any xi in the delta-neighborhood forms a resultant which satisfies the inequalities, YMAX >= f(xi) >= YMIN. Such technological interactions have the potential to incite student-developed questions as well as to change the nature of the curriculum.
The increasing availability of technology forces mathematicians and math educators to reexamine our thinking about mathematics as a discipline. We need to rethink what skills need to be honed and how to develop the connections between concepts so that students might have a smooth transition through the undergraduate mathematics sequence. In addition, we need to attend to the particular needs and learning styles of the students. One particular methodology generally does not work for all. The use of combinations of tools, i.e., assessments, technology, collaborative work, explorations, reflective activities, and discussions can accomplish more than any single tool on its own. Therefore, I believe the tools should be looked upon as pieces to a puzzle that display the path for students to develop higher levels of understanding; they are all needed and should be applied at the appropriate times.