The persistent poor performance of learners in the NSC mathematics examinations prompts the investigation of the factors contributing to the crisis. Practice reveals that pre-service teachers of mathematics still operate from rote understanding rather than conceptual coherence. One suspects that this approach to mathematics is perpetuated through conceptually starved teaching practices. Metacognition as mathematical soliloquy seeks to promote critical consciousness in mathematics teaching and learning.
Metacognition, first coined by Flavell (1976), the active personal regulation and monitoring of one’s cognitive processes is inherently part of the fabric of doing, talking about, and understanding mathematics. Metacognition is a private endeavour, and as such it aligns with a radical constructivist notion of knowledge acquisition. The mathematical soliloquy, the personal interrogation and conversation that characterises the thought processes associated with dealing with mathematical objects reflects the understanding that underlies action. Schoenfeld (1985, 1987, 1988, 1992) relates metacognition to problem solving. In this practice based research metacognition is not specifically related to problem solving but rather a state of mind that is able to interrogate mathematical objects and thickly and rigorously describe them. It is from this level of cognitive accomplishment that teaching and learning could occur effectively.
In terms of conceptual frameworks, even though not advertently metacognitive, the following authors promote the tenets of metamathematics. Kaput (1985) advocates that learning is enhanced through being able to represent mathematical objects. This in itself is a metacognitive skill. In this research the verbal representation (description) of a mathematical object is an indicator of metacognitive activity. Skemp (1976, 1986,1993) and Tall (1989) work with the notions of instrumental and relational understanding, where the former refers to a rote epistemology as opposed to the conceptual depth of the latter. Dreyfus (1990) and Vinner (1992) relate the concept image and concept definition of a mathematical entity possibly as two extremes of a continuum where the former is concrete and the latter the rigorous translation of that entity into its abstract algebraic equivalent.
Pimm (1987) speaks about what elicits exact responses in interaction with learners. He claims it is learners’ responses that embody meanings they attach to mathematics concepts. Pimm’s perspective enters the cognitive domain that separates the stimulus and response. Brown (1997) also claims that ‘Mathematics can only be shared in discourse and the act of realising mathematics in discourse brings to it much beyond the bare symbols.’ Brown (1997) introduces the hermeneutic in analysing learners’ understanding of mathematics. Pimm and Brown enter the realm of metamathematics in probing cognitive content through descriptions offered by the mathematics of learners.
The project promotes conceptual understanding amongst teachers. It forms part of the third and fourth year methodology courses, and provides research opportunities for Honours, Masters and PhD studies.