What does it mean to understand mathematics?

I’m sharing in this post some of the “theories” underpinning the lessons, learning tasks, and math teaching ideas that I blog here. I thought, rather than talking about them in each post I make I will just refer the reader to this and to some other similar “theories” I will write about some other time. This is part of the literature review of my dissertation titled “A Framework of Growth Points in Students Developing Understanding of Function”. If you are a researcher and wants to see the references, you may download the dissertation here.

What is understanding?

Good ideas are networks

To understand is to make connection. The stronger the connection, the more powerful the knowledge. The more connected a concept is to other concepts, the easier it is to retrieve from memory and the wider the applications. In mathematics, this idea implies that one understands when he or she can make connections between ideas, facts or procedures (Hiebert & Wearne, 1991). In making connections, one not only links new mathematical knowledge to prior knowledge but also creates and integrates knowledge structures (Carpenter & Lehrer, 1999). Thus, the process of understanding is like building a network. Networks are built as new information is linked to existing networks or as new relationships are constructed (Hiebert & Carpenter, 1992). If one imagines a weblike structure, the mental representations constructed in the process of understanding can be thought of as nodes. These nodes are themselves “networks”. These smaller networks resemble what is called a schema in cognitive psychology which is a network of well-connected ideas, skills and strategies an individual uses in working with a particular task (Marshall, 1990).

The importance of the acquisition of cognitive structures (schemas) has been shown in studies of people who have developed expertise in areas such as mathematics, physics, chess, etc.  Bransford, Brown & Cocking (1999) summarised the key principles of experts’ knowledge. Some of these are:

1. Experts notice features and meaningful patterns of information that are not noticed by novices.

2. Experts have acquired a great deal of content knowledge that is organised in ways that reflect a deep understanding of their subject matter.

3. Experts’ knowledge cannot be reduced to sets of isolated facts or propositions but, instead, reflect contexts of applicability: that is, the knowledge is “conditionalized” on a set of circumstances.

4. Experts are able to flexibly retrieve important aspects of their knowledge with little attentional effort (p. 19).

Von Glasersfeld (1987) described understanding as a “never-ending process of consistent organization” (p. 5). It is not an all or none phenomenon hence “it is more appropriate to think of understanding as emerging or developing rather than presuming that someone either does or does not understand a given topic, idea, or process” (Carpenter & Lehrer, 1999, p. 20).

What is Conceptual and Procedural Knowledge?

Related to the notion of understanding is knowledge of concepts and procedures. Conceptual knowledge in mathematics is “knowledge of those facts and properties of mathematics that are recognized as being related in some way” (Hiebert & Wearne, 1991, p. 200).  It is “knowledge that is understood, … a knowledge that is rich in relationships. … A unit of conceptual knowledge is not stored as an isolated piece of information; it is conceptual knowledge only if it is a part of a network” (Hiebert & Carpenter, 1992, p. 78) [italics, mine]. This implies that the quality of conceptual knowledge is a function of the strength of the connection or relationships between the concepts involved.

Hiebert and Carpenter defined procedural knowledge as a sequence of actions and as such, the connection between concepts involved is minimal. An example of procedural knowledge is knowledge of standard computation algorithms, which consist of a step-by-step sequence of procedures of symbol manipulation. Hiebert and Carpenter argued that procedural knowledge could contribute to mathematical expertise only if it is related to conceptual knowledge: “From the expert’s point of view, procedures in mathematics always depend upon principles represented conceptually” (p. 78).



The image is from the post Good Ideas are Networks in  Slow Muse by Deborah Barlow.

A Framework of Growth Points in Students’ Developing Understanding of Function – PhD Thesis by the Author

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