In 1976 Skemp published an important discussion paper spelling out the differences between relational and instrumental understanding as they apply to mathematical teaching and learning. Skemp highlights two faux amis, the first is understanding. Skemp defines understanding in two ways: 1) instrumental understanding and 2) relational understanding. The second faux amis is the word mathematics which he describes as two different subjects being taught. I have considered Skemp’s article in four sections.
1. Faux amis
2. Instrumental and relational understanding
3. The mismatch
4. Implication for mathematics teaching
Key terms: Schema; faux amis; Instrumental understanding; relational understanding; mathematics.
Setting the Scene
It is extremely difficult to define understanding. Skemp attempts to assimilate it into some form of an appropriate or inappropriate schema that is dependent upon many variables such as language, environment, belief, tradition and culture. Could understanding be an abstract thing, brain pattern or rule? Skemp uses the term ‘faux amis’ to mean that language can have different meanings to different people even though the root origins of words are the same. He looks at French and English and identifies what he calls a ‘mismatch’. He uses analogies and understandings based on his own experience and others in his community of practice (Mellin-Olsen, 1981). This mismatch, he believed, is the root of many difficulties in mathematics education including the word mathematics itself. This assignment attempts to appraise his arguments in relation to other literature and my own personal experience.
A schema is a mental structure we use to organize and simplify our knowledge of the world around us. We have schemas about ourselves, other people, mechanical devices, food, and in fact almost everything. Skemp uses the notion that different mathematical ideas are seen as objects and processes for an individual to encapsulate into some form of schema. It is argued that schema theory lacks a consistent definition and it has its roots in idealist epistemology and mixed empirical studies (Sadoski, 1991).
Instrumental understanding is where you apply an appropriate remembered rule to the solution of a problem without knowing why the rule worked. Instrumental understanding can be seen in terms of tacit content knowledge of the methods of mathematics. Tacit meaning it is difficult to transfer the knowledge by means of writing it down and is therefore communicated somehow to the student. Gregory Chaitin, a mathematician and computer scientist argued that understanding something means being able to figure out a simple set of rules that explains it (Chaitin, 2006). In teaching directed numbers and rationalising the denominator as a remembered rule is much easier than trying to get students to identify with the concept relationally. In my experience as a teacher certain pupil’s go through their mathematics education at...