In this section, I present a development of three key ideas which underpin this study. I first discuss some skills needed for mathematical literacy, before moving on to the theory of constructivism and then linking these to some complexities involved in the contextualisation of tasks.
In PISA’s definition (OECD, 2003, p.24):
Mathematical literacy is an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded mathematical judgements and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen.
The above definition conveys three important ideas (Steen & Turner, 2007, p.286):
mathematical literacy is much more than arithmetic or basic skills.
mathematical literacy requires something quite different from traditional school mathematics, and
mathematical literacy is inseparable from its contexts.
The third idea is of particular importance and is emphasised in the Department of Education’s Subject Assessment Guideline for Mathematical Literacy (2008):
On the one hand, mathematical content is needed to make sense of real life contexts;
on the other hand, contexts determine the content that is needed (DoE, 2008, p.7).
It further states:
When teaching and assessing Mathematical Literacy, teachers should avoid teaching and assessing mathematical content in the absence of context. At the same time teachers must also concentrate on identifying in and extracting from the contexts the underlying mathematics or ‘content’. Assessment in Mathematical Literacy needs to reflect this interplay between content and context. Learners should use mathematical content to solve problems that are contextually based (DoE, 2008, p.7).
This suggests that assessment tasks should be contextually based, that is, based in real-life contexts and use real-life data, and should require learners to select and use appropriate mathematical content in order to complete the task. Competency in mathematical modelling helps one to tackle problems from the “real world”. Steen & Turner (2007, p. 286) use the phrase “mathematics acting in the daily lives of citizens”. The National Curriculum
Statement (DoE, 2006, p.42) requires the subject to be “rooted in the lives of the learners”.
This view is synonymous with the theory of Realistic Mathematics Education (RME) developed by the Freudenthal Institute in the Netherlands. Freudenthal made two important points, namely that RME must be close to children and be relevant to everyday life situations (Freudenthal, 1991).
The approach required to develop Mathematical Literacy is “to engage with contexts rather than applying mathematics already learned to the context” (DoE, 2003, p.42). RME claims that this can be achieved when mathematics is taught as a human activity instead of transmitting mathematics as a pre-determined system constructed by others. In RME, as with mathematical literacy, contextual problems are the basis for progressive mathematisation, and through mathematising, the students develop informal context-specific solutions strategies from experientially realistic situations (Gravemeijer & Doorman, 1999). The emphasis in the preceding sentence is on “experientially realistic situations”. The context used in mathematical literacy has to be close to the learners experience for them to be meaningful.
The concept of mathematisation was extended by Treffers (1987, cited in Gravemeijer &
Doorman, 1999, p.116), who made a distinction between horizontal and vertical mathematisation. In horizontal mathematisation, the students come up with mathematical tools which can help to organise and solve a problem located in a real-life situation. Vertical mathematisation is the process of reorganisation within the mathematical system itself, for instance, finding shortcuts and discovering connections between concepts and strategies and then applying these discoveries. In short, “horizontal mathematisation involves going from the world of life into the world of symbols, while vertical mathematisation means moving within the world of symbols” (Freudenthal, 1991, p.42).
In RME, applying mathematics is very difficult if taught ‘vertically’, that is, various subjects are taught separately (Lange, 1996). Curriculum 2005 also emphasised an integration of Learning Areas so that “learners experience the Learning Areas as linked and related” (DoE, 2002a, p.13). RME share many similarities with the theory of constructivism in Mathematical Literacy. Constructivism is a theory of learning which posits that students learn by actively constructing their own knowledge, knowledge is created not passively received and views learning as a social process (Clements & Battista, 1990).
Vygotsky (1962) promoted the view of learning based on both individual and social construction, and showed the importance of social interaction and language in supporting and extending learning. Initially, a child’s new knowledge is interpsychological, meaning it is learned through interaction with others, on the social level. Later, this same knowledge becomes intrapsychological, meaning inside the child, and the new knowledge or skill is mastered on an individual level. The Mathematical Literacy classroom also emphasises the need to “work collaborately in teams and groups to enhance mathematical understanding”
(DoE, 2003. p.10).
Constructivism argues that all knowledge is constructed by individuals rather than transferred directly by an expert, such as a teacher, parent or book, to the learner. The constructivist perspective argues that all knowledge is actively constructed by the learner. This means that that the learner plays the primary role in organising input from outside into meaningful knowledge. According to Epstein (2002), constructivism emphasises the importance of the knowledge, beliefs, and skills an individual brings to the experience of learning. It recognises the construction of new understanding as a combination of prior learning, new information, and readiness to learn.
Epstein (2002, p.3) offers a deeper insight into constructivism by identifying nine general principles of learning that are derived from constructivism. Of these nine principles, four are directly related to this study, and will be used to derive some of the categories that will be used in the analysis in Chapter Five.
Learning involves language: the language that we use influences our learning.
Learning is a social activity: our learning is intimately associated with our connection with other human beings, our teacher, our peers, our family, as well as casual acquaintances.
Learning is contextual: we learn in relationship to what else we know, what we believe, our prejudices and our fears.
One needs knowledge to learn: it is not possible to absorb new knowledge without having some structure developed from previous knowledge to build on.
The focus of this study is the learners’ responses to contextualised test items in a Mathematical Literacy provincial examination. Language plays four important roles here.
Firstly the written context is a textual representation of a real life context, and cannot capture all the details of the context. Thus it may convey a limited representation of the real context.
Secondly, putting questions into context inevitably involves using additional words to ask the question. If learners have to negotiate more textual information in order to answer questions, then their reading ability is being tested as well as their understanding of mathematics (Ahmed & Pollitt, 2001). Language can thus be a barrier to particularly second language English learners when dealing with contextually based questions. Thirdly, the language used in the assessment tool influences the ways in which the learners respond to the items.
Sometimes learners may misinterpret instructions which are not clearly and unambiguously framed. They may provide a response different to what the examiner expects because their interpretation of the question was different from what the examiner intended. Fourthly, language also plays a role in the assessment of the learners’ responses. Learners who struggle with writing or speaking English may be disadvantaged by questions which require much explanations, reflections or reasons.
The principle that learning is a social activity has applicability to this study because learners learn from their peers as well as their family and teachers. All these encounters contribute to the ways in which learners’ knowledge develops.
The third principle that learning is contextual has particular significance to this study. The context here has various interpretations. It refers to the context in which learning takes place, such as the classroom, the home or environment. However, we can also interpret it as the context that is used as a setting for instruction or assessment. In the case of Mathematical Literacy, the DoE advise that context and context should be intertwined:
When teaching and assessing Mathematical Literacy, teachers should avoid teaching and assessing mathematical content in the absence of context. At the same time teachers must also concentrate on identifying in and extracting from the contexts the underlying mathematics or ‘content’. Assessment in Mathematical Literacy needs to reflect this interplay between content and context. Learners should use mathematical content to solve problems that are contextually based (DoE, 2008, p7).
Bernstein (1996) argues that the process of recontextualisation whereby school mathematics is being recontextualised into another field – in this case the everyday – creates a new set of
demands previously not recognised. Both teachers and learners will need to acquire new recognition rules and new realisation rules. Recognition rules are the means by which individuals are able to recognise the speciality of the contexts in which they are. Realisations rules determine how we put meanings together and how we make them public i.e. allowing the production of ‘legitimate text’.
Many teachers of Mathematical Literacy comment that learners with a good command of the English language, greater exposure to everyday experiences and from schools situated in higher socio-economic suburbs generally perform better than, for example, English second language learners, coming from low socio-economic areas with limited everyday experiences.
This may be related to the way learners interpret questions in mathematical literacy, where learners from lower socio-economic backgrounds using real life experiences in making sense of the questions. Bernstein’s theory would attribute this phenomenon to such learners not acquiring the recognition and realisation rules. The current study will look for such patterns when examining learners’ responses to context-based questions.
Real-life context based tasks often come with their own specialised language and complexities (Bansilal & Khan, 2009). The terminology of the context may involve vocabulary relating to adult concepts. As mentioned, examples of these are ‘200 free kilometres’ in car hire scenarios and ‘base occupancy’ in room bookings, or in the case of this provincial examination, ‘infant mortality rate’ in the context of infant deaths (Q 4.2 of the examination paper) and ‘win by a margin of…’ in the context of soccer goals (Q2.2).
Assessment using contexualised items often requires learners to recognise and distinguish between context information and crucial information, where crucial information refers to information without which the task cannot be solved, while context information refers to facts or information about the context, which may be disregarded in the solution of the task (Bansilal & Wallace, 2008). Sometimes there are context-specific rules, bound to the context that needs to be interpreted by the learner, for example, working out the transfer duty of a house. Examples in this study are working out the costs of various transactions from a bank from Q3.3, as well as the FIFA and fanatics scoring systems presented in Q 2.2 of the examination paper.
The fourth principle that learning is built on previous knowledge is also relevant to the study.
One implication of this is that all learners’ previous encounters with particular contexts will
affect their perception and understanding of contexts that are used for instructional or assessment purposes. Thus it will be of value to investigate learners’ perceptions and experiences of contexts used in the examination paper to identify ways in which their perceptions may influence their responses.
An important consideration in learning new knowledge is the organisation of knowledge and making connections with other concepts. Mathematical proficiency described by Kilpatrick, Swafford & Findell (2001) is made up of five interwoven strands. Of interest in this study is the strand of Conceptual Understanding, which refers “to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organised their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know”
(Kilpatrick et al., 2001, p.118).
The fourth principle also helps explains what may happen when learners’ new knowledge does not fit in with accepted mathematical traditions. As part of the normal process of constructing knowledge, learners often develop misconceptions. Brodie & Berger (2010, p.169) cite theories by Confrey (1990); Nesher (1987) and Smith, DiSessa & Roschelle (1993) to elaborate that a misconception is a “conceptual structure by the learner, which makes sense in relation to her/his current knowledge, but is not aligned with conventional mathematical knowledge.”