Hermeneutic and empirical analyses of graphically inspired metamathematics that reflect critical consciousness within perspectives of personal and social justice - Van Jaarsveld, Pieter Paul (Rhodes University, Faculty of Education, Department of Education, 2007)
My involvement with mathematics education amongst township educators and learners over the past seven years has highlighted the absence of sustained meaning and meaning making of mathematical concepts. It appears though that this instrumental rather than relational understanding of FET mathematics is not unique to township learners but is encountered amongst learners of all socio-economic classes and is representative of many FET mathematics learners. Given that the language of learning and teaching is a major contributory factor in the South African education system, it appears that the language of mathematics itself is a greater exacerbating factor for many learners of mathematics. The exclusive algorithmic approach to classroom mathematics further seems to alienate many learners from the essence of the meaning of mathematical tasks. This research undertakes to determine whether metateaching and metalearning as forerunners to metacognition facilitates the acquisition of the sustained meaning of mathematical concepts. Metateaching and metalearning refer to the acute and deliberate awareness by educator and learner as to what constitutes concepts. Teaching and learning therefore presupposes the deconstruction of concepts into its subsumed derivative roots. It also assumes an awareness of the tacit degrees of abstraction that characterise tasks and the content of tasks. This in turn has implications for the educator's adopted sequence of topics for instruction. Metacognition implies awareness on the part of the learner (and educator) as to how material is learned and a further awareness as to how that learning can be sustained. Whether we ascribe meaningful learning to radical or social constructivism, or to associationist didactive approaches, or a combination of these, we are making assumptions about how learners acquire and sustain mathematical meaning because mathematics is, by and large a symbolic language often devoid of affective connotation. Furthermore our assessments of learners' tasks amount to clinical corrections of austere formulae wrapped in algorithmic procedures which manifest nothing of a learner's experience of mathematics or the deeper understanding (or misunderstandings) which characterise a learning and/or assessment episode. To this end the research design of this interpretive case study requires learners to expound in textual accounts their thoughts as they describe the evolution of a mathematical process as they approach a solution and eventually interpret it. The textual account exposes the concept definition for what it really is in a learner's understanding of it and it is the expressiveness of language that indicates whether the understanding of a learner is approaching the concept image. The textual accounts vary in richness in terms of mathematical register and this in turn reflects the conceptual depth. The mechanism which seems to promote the conversion from concept definition to concept image is the graphical representation of the mathematical task or procedure, possibly because of its greater concreteness as opposed to the abstraction of its algebraic form.
Keywords: Mathematics -- Study and teaching -- South Africa; Mathematics -- Study and teaching -- Social aspects -- South Africa; Algorithms -- Study and teaching -- South Africa; Critical pedagogy; Metacognition; Hermeneutics