In my earlier post on the meaning of understanding, I describe understanding of mathematics as making connections: To understand is to make connections. These connections are not done in random. Concepts are linked with other concepts in order to create a richer image for the new concept that is being learned. To understand therefore is to form concept image. And a concept image is not formed by defining the concept. The definition of a concept is different from the concept image. Let me share with you a an excerpt from my paper which discusses this idea. You can view the references here.
Understanding the definition does not imply understanding the concept. In order to understand a concept one must have a concept image for it. One’s concept image includes all the non-verbal entities, visual representations, impressions and experiences that are created in our mind by a mention of a concept name (Vinner, 1992). Vinner stressed that the concept definition is not the first thing that is learned in understanding a concept but the experiences associated with it, which becomes part of one’s concept image. Vinner believes that in carrying out cognitive tasks, the mind consults the concept image rather than the concept definition.
Vergnaud (1997) also noted, “it is misleading, even in mathematics [despite its precision in defining], to consider that the properties of a concept are self-contained in its definition” (p. 5). To study and understand how mathematical concepts are develop in students’ minds through their experience both in and outside school, Vergnaud proposed that one needs to consider a concept C as a three-uple of three sets:
C = (S, I, R)
S: the set of situations that make the concept useful and meaningful.
I: the set of operational invariants that can be used by individuals to deal with these situations.
R: the set of symbolic representations, linguistic, graphic or gestural that can be used to represent invariants, situations and procedures (p. 6)
What are these telling us mathematics teachers? Well, firstly, that it is not a good practice to start with concept definitions and secondly that it is important to design learning experiences that will create and enrich the concept image because it is then and only then when students can relate to the definition. Definitions are already an abstraction of the concept. For example, if you give the child the definition of a dog, do you think he’d recognize one if he sees one? Kidding. In order to be suitable to learners , a definition must consist of concepts known to the learner (Cobb). It should rely, as far as possible, on the intuition of the student (Fischbein). It should be within the grasp of the learner (inside the learner’s zone of proximal development — Vygotsky).
All the lessons and tasks I present in this blog target the creation of concept images. For example, on the teaching of integers, the following lessons provides the ground work for understanding integers.
- A problem solving approach for introducing integers
- Sorting number expression
- Introducing the integers via number line with a twist