High school students’ use of patterns and generalisations

Generalisation is an important aspect in mathematics that permeates all branches of the subject and is a feature highlighted in the teaching of the subject at practically all levels. For example, in Arithmetic a child may generalise that multiplication of a whole number by 5 gives a product ending in 0 or 5. As a statement that is true for all members of some set of elements, theorems in geometry can be considered as generalisations (Mason, 1996). On the other hand, in algebra, we commonly use variables, which Schoenfeld and Arcavi (1988) described as general tools in the service of generalisation. So, what do we mean by generalisation? Several attempts have been made to explain the term generalisation. Kaput (1999) claimed that generalisation involves deliberately extending the range of reasoning or communication beyond the case or cases considered by explicitly identifying and exposing commonality across the case or the cases. He added that this resulted in lifting the reasoning or communication to a level where the focus is no longer on the cases or situations themselves but rather on the patterns, procedures, structures, and relations across and among them, which in turn become new, higher-level objects of reasoning or communication. This hierarchical aspect is similar to what Sfard (1991) proposed in her theory of reification, in which processes at one level become the new objects at another level. The idea of creating new objects for subsequent actions was also used by Davidov (1972/1990) who described generalisation as “inseparably linked to the operation of abstracting” (p. 13). The link between generalisation and abstraction was also highlighted by Dreyfus (1991). However, Dreyfus used the term generalisation as the recognition of some general characteristics in a set of mental objects and claimed that generalisation involves the expansion of an individual’s knowledge structure. Regarding cognitive activities involved in generalising, Harel and Tall (1991) made a distinction between three types of generalisations, (a) expansive generalisation – one that extends the students’ existing structure without requiring changes in current ideas; (b) reconstructive generalisation – one that requires the reconstruction of the existing cognitive structure; and (c) disjunctive generalisation – one which adjoins the new particular as an extra case or generates a new structure distinct from the others. On the other hand, Radford (1996) claimed that a goal in generalising geometricnumeric patterns is to obtain a new result. This new result depends on the observer’s conceptualisation of the mathematical objects and the relations involved. Radford added that accordingly, generalisation is not a concept but rather a procedure and that a generalisation procedure g arrives at a conclusion α, starting from a sequence of “observed facts”, a1, a2,. , an, which can be written as: a1, a2, . , an → α (α is derived from a1, a2, ., an). The most significant aspect of the generalisation is its logical nature that makes possible the conclusion α. Mathematics: Essential Research, Essential Practice — Volume 1