The key question in establishing students’ conceptions or perceptions of mathematics is
‘What is mathematics?’ Marton (1981:178) notes that “any answers to questions that seek to solicit people’s understanding, meanings and interpretations of social phenomena are statements about people’s conception of reality”. The variation in meanings attached to this question is thus interpreted as perceptions and conceptions. This is because internalised knowledge and meanings attached to social phenomena are expressed in speech or utterances.
In Bourdieu’s language, the embodied immanent structures of the world are expressed through a variety of activities including speech and walk among others (Bourdieu 1990:70).
Such is the basis upon which inferences about students’ history of interactions and the meanings attributed to social contexts and aspects of their mathematics reality is drawn on as conceptions and perceptions.
Douglas (1996:12) identified three major themes upon which students’ conceptualise mathematics. These themes have been found to relate to the kind of social object or social
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activity or ‘affordance’26 that mathematics is to individual students. The author argues that the way students conceive mathematics ranges from the nature of mathematical knowledge, the character of mathematical activity and the essence of learning mathematics. Such is the continuum within which Douglas (1996) has found students’ meanings attached to mathematics to range. Outlining the various aspects of how students conceptualise mathematics is thus a guide in our understanding and interpretation of students' meanings and interpretations of social situations in their dynamic interactions with the socio-cultural context in which they are embedded (Volte, 1999:628). Douglas has found students to conceive or perceive mathematics as follows.
3.5.1 The nature of mathematical knowledge
This is where mathematics knowledge is conceptualised in terms of its composition, structure or its status. Three sub-themes are identified by Douglas (1996:12) under the conception of the nature of mathematical knowledge and these are:
i) Composition of mathematical knowledge - includes the representation of mathematics knowledge, formulas and algorithms. This conception according to Reid et al (2003) is where students view mathematical activity such as numbers and calculation and make no essential advance beyond elementary arithmetic.
ii) Structure of mathematical knowledge - entails interpreting mathematics either as consisting of interlinked content or as a collection of a variety of content that is unrelated. This is where mathematics is conceived as consisting of methods and applications that are not related and as having no meaningful use in everyday life (Reid, Wood, Smith and Petocs, 2003).
iii) Status of mathematical knowledge - entails students defining mathematics either as a rigid entity or as a dynamic field. This is where students view mathematics as ‘rule- like’ and as having no room for creativity such as applying the law of gravity.
26Refers to what the social environment offers the individual student. This includes everything that is at the individual student’s disposal in the social environment such as social meanings, knowledge and know-how, values, beliefs, culture, social artefacts etc (McGrenere and Ho, 2000:1).
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Douglas (1996) states that students also conceptualise mathematics as an activity; either as
“doing mathematics or validating established mathematical ideas”. Doing mathematics is a process that involves one embarking on a journey of finding things out or discovery through trial and error. Dossey (1992:45) argues that learning mathematics actively by ‘doing’
encourages learning the skills of the discipline through a deep approach to learning. He argues that learning by doing encourages students to see mathematics as having a worth in their lives. This has been substantiated by a study conducted by Liljedahl (2005:220) which maintains that there is a change in perception from negative perceptions to positive perceptions about mathematics that comes with making an effort to engage with mathematical operations. Liljedahl’s (2005:220) states that the negative public narrative about mathematics being hard is challenged as students discover solutions to mathematical operations by aggressively engaging themselves. He defines it as an ‘AHA! moment’, or a moment of self discovery, an eye opener and beliefs alteration about their mathematical abilities. Lerman (1996: 137) argues that know-how or any skill is acquired through practice and not imparted or transmitted. Likewise mathematical knowhow comes about with practice.
Validating ideas is where students conceive of mathematics as a process that was discovered by the ‘knowledgeable others’ of the field so that the role of the student is just to reproduce the solutions that were discovered. Douglas (1996:14) argues that a student who conceives of mathematics as validating ideas looks forward to being assisted by others with mathematical applications and makes no effort to discover solutions through independent trial and error.
This confirms Kloosterman, Raymond and Emenakor (1996:40) findings that most “students believe that the goal of mathematics is to get the right answers and that the role of the student is to receive the mathematical knowledge and teachers are transmitters of the knowledge”.
Haggis (2003:99) defends the portrayal of validating ideas as completely negative, arguing that students have varying academic developmental capabilities and validating is a starting point.
3.5.3 The essence of learning mathematics
This theme according to Douglas is where mathematics learning is perceived as constructing knowledge so that knowledge develops from memorising to application to construction of
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new knowledge. The essence of learning mathematics has been outlined as a developmental pattern in which the knowledge construction and knowledge memorisation are stages in the mathematics knowledge development continuum which students have to demonstrate as they progress through their educational trajectory. This is what Haggis (1996:98) refers to as
‘stage theory’ because it perceives the development of knowledge to be in developmental stages.
3.5.4 Life conception
This pertains to students making a connection between the mathematics learned in class and their day to day tasks activities (Reid, 2003). It is where mathematics is viewed as a way of life and a way of thinking and in life conception, the learners make strong personal connections between mathematics and their own lives and view mathematics as useful.
Douglas (1996:15) argues that the perceived usefulness or value of mathematics is a key determinant of whether students pursue mathematics or not. The value of mathematics relates to the kind of ‘affordance’ that mathematics is to the individual students as highlighted in the theoretical framework. This viewpoint is supported by Dossey (1992:45) who states that the relevance of a subject for a student is its utility value or perceived usefulness in everyday life and future. Dossey (1992) asserts that the utility value of a task is a significant factor in why someone would want to stay attached to that task. This statement is affirmed by Kloosterman, Raymond and Emenakor (1996:40) who state that for students to put an effort into what they are learning, they would have to realise its value. The emphasis of students’ advancement in their conceptualisation of mathematics to a level where mathematics is conceived as a way of life (Reid 2003), is an imprint of Entwistle’s (2004) conception of the ‘stage theory’.
Does Douglas’s (1996) analysis of how students conceive and perceive mathematics imply that human behaviour is predicable? McLeod (1992:581) cautions against the conceptualisation of the meanings that students attach to mathematics according to Douglas’s (1996) suggestion as a blue-print approach, suggesting that the meanings that students attach to mathematics are not one dimensional. McLeod (1992) argues that “there are many different kinds of mathematics as well as a variety of conceptions about each type of mathematics”. He further argues that meanings that people attach to social contexts arise from two different backdrops. One is as a result of invoking a repeated negative or positive experience which was perceived negatively or positively. Second is the attribution of an
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already existing conception or perception to a new but related task. For instance; a student who has a negative conception or perception to mathematics may attribute the same meaning to statistics or any form of quantitative research. In other words, Douglas’s framework on students’ conceptions and perceptions is a guide in our understanding but not cast and stone.