By drawing concept maps, students strengthen their understanding of how a new concept is related to others they already know.
Background and Purpose
Concept maps are drawings or diagrams showing connections between major concepts in a course or section of a course. I have used concept maps in precalculus and calculus classes at Surry Community College, a medium sized member of the North Carolina Community College System. About one-third of our students transfer to senior institutions. This includes most of the students who enroll in precalculus and nearly all of those who take calculus.
Concepts maps can be a useful tool for formative assessment. They can also provide an added benefit of helping students organize their knowledge in a way that facilitates further learning. I first learned about concept maps from Novak . Further insight about concept maps and their uses in the classroom can be found in  and .
Concept maps are essentially drawings or diagrams showing mental connections that students make between a major concept the instructor focuses on and other concepts that they have learned . First, the teacher puts a schematic before the students (on a transparency or individual worksheet) along with a collection of terms related to the root concept. Students are to fill in the empty ovals with appropriate terms from the given collection maintaining the relationships given along branches. From the kinds of questions and errors that emerge, I determine the nature and extent of review needed, especially as related to the individual concepts names that should be familiar to each student. I also give some prompts that enable students to proceed while being careful not to give too much information. This is followed by allowing the students to use their text and small group discussions as an aid. The idea at this stage is to let the students complete the construction of the map. Eventually they all produce a correct concept map. Once this occurs, I discuss the concept map, enrich it with other examples and extensions in the form of other related concepts.
This strategy seems to work particularly well when dealing with several related concepts, especially when some concepts are actually subclasses of others. I have used this procedure in precalculus in order to introduce the concept of a transcendental number. Students chose from the list: integers, real, 3/5, rational, irrational, e, algebraic, p, transcendental,, and natural to complete the concept map on the top of page 90.
As students proceeded individually, I walked around the classroom and monitored their work, noting different areas of difficulty. Some students posed questions that revealed various levels of understanding. Among the questions were "Don't these belong together?" "Does every card have to be used?" and "Haven't some groups of numbers been left out?"
I have also used this procedure in calculus to introduce the gamma function. I wanted to introduce the gamma function by first approaching the idea of a non-elementary function. This prompted me to question what students understood about the classifications of other kinds of functions. I presented a schematic and gave each student a set of index cards. On each card was one of the following terms: polynomial, radical, non-elementary, trigonometric, inverse trigonometric, hyperbolic, and gamma. The students were told to arrange the cards in a way that would fit the schematic.
Use of Findings
The primary value of using concept maps this way is that it helps me learn what needs review. They provide a means for detecting students misconceptions and lack of knowledge of the prerequisite concepts necessary for learning new mathematics.
The primary caveats in this approach are that the teacher should be careful not to let the students work in groups too quickly and not to give away too much information in the way of prompts or hints. One alternative approach is to have the students, early in their study of a topic, do a concept map of the ideas involved without giving them a preset schematic. Then you gather the concept maps drawn and discuss the relations found: what's good about each, or what's missing. You can also have students draw a concept map by brainstorming to find related concepts. Then students draw lines between concepts, noting for each line what the relationship is.
 Angelo, T. A. and Cross, K. P. Classroom Assessment Techniques, A Handbook for College Teachers (2nd ed., p. 197). Jossey-Bass, San Francisco, 1993.
 Jonassen, D.H., Beissneer K., and Yacci, M.A. Struc-tural Knowledge: Techniques for Conveying, Assessing, and Acquiring Structural Knowledge. Lawrence Erlbaum Associates, Hillsdale, NJ, 1993.
 Novak, J.D. "Clarify with Concept Maps: A tool
for students and teachers alike," The Science
Teacher, 58 (7), 1991, pp. 45-49.