The learning of general mathematical strategies : a developmental study of process attainments in mathematics, including the construction and investigation of a process-oriented curriculum for the first secondary year
Consideration of the relative place of content and process in the mathematics curriculum leads to the following questions: 1. What is the nature of the mathematical process and how does it relate to the content? 2. Does the process comprise learnable strategies; if so, what are feasible learning objectives for different ages? 3. Can content and process be learned simultaneously or are there incompatibilities between effective teaching methods? A theoretical study shows that the content of mathematics - structures, symbol-systems and models - arises directly from the application of the basic processes of generalisation and abstraction, symbolisation and modelling, to the objects of experience. Experimental studies based on (a) the development of a process-enriched curriculum for the early secondary years, and (b) age and ability cross-sectional studies of pupils' proof activity show that: i. the awareness that proof requires consideration of all cases is generally weak among secondary pupils, but is relatively easily taught, ii. with a process-enriched curriculum, 11 year olds can acquire strategies of experimenting, making generalisations and constructing complete (finite) sets but still have little sense of deducing one result from another, iii. the main types of deficiency in proof-explanations are (a) fragmentary arguments, (b) non-explanatory re-statements of the data, (c) unawareness of suitable starting assumptions. Strategies for improving proof activity are inferred from pupils' responses, and are shown to be effective in a sixth form teaching experiment. An informal study shows that students entering university mathematics departments possess generalisation skills and logical awareness to a much higher degree than 15 year olds, but still have only vague ideas of the nature of axiom systems. On question 3 the evidence suggests that there need be no substantial loss of content learning in the process-enriched curriculum, and both in this and in the teaching experiment an improvement in general understanding and involvement was observed.