Learner’s everyday life abounds with mathematical objects and concepts. An awareness of the mathematics in the environment is a function of learners’ ability to interpret the relevance of both the formal and informal mathematics they have been experiencing in the classroom. Regardless of the context in which learners engage with mathematics, it is generally agreed that interconnections provide teachers insight into developments in learner's understanding of the extent of their own understandings. These understandings are built on learners’ personal experiences, intuitions and formal knowledge taught in the classroom.
Mathematics educators and teachers have invested considerable effort in exploring instructional strategies that would help learners to develop a better grasp of mathematical concepts. One stream of inquiry about teaching approaches has focused on teaching practices that aid in the construction of a powerful and meaningful understanding of mathematics and its utility. Developments in cognitive psychology and domain expertise have yielded significant lines of inquiry about what we mean by powerful understandings and these might be investigated.
Mathematics curriculum reforms call for teaching and learning experiences that optimize the development of a substantive understanding of mathematics (National Council of Teachers of Mathematics 2000). The question remains as to how we can characterize this type of understanding? Discussions about understandings and the manifestation of understandings within knowledge rich areas such as mathematics have focused on the structure of knowledge that is constructed by learners and its impact on meaning making and vice verse. The notion of structure implies the existence of links among strands of knowledge. Prawat (1989) argued that the building of well organised knowledge is indicative of a sophisticated understanding of a subject. It is argued, that this type of knowledge is qualitatively superior to one that is less well organized in terms of access to and use of that knowledge during problem solving.
Perspectives about how learners come to know and make sense of mathematics have contributed to the emergence of socio-constructivist views. According to this framework, mathematics learning is seen as an individual, as well as, a shared activity during which learners should be encouraged to investigate, argue, justify and test conjectures (Cobb 1995). Such activities need a classroom-learning environment that supports learners to question what teachers and peers say about a particular mathematics concept.
The approach towards teaching needs to move away from the transmission mode towards one that fosters free and open inquiry and debate. The social context in which learning takes place is as important as the concepts themselves in the growth of the type of understanding that is constructed by the learners (Lee 1998). The social and participative nature of learning reflects a view of knowledge that involves personal constructs as opposed to ‘received wisdom.’ Pedagogies that reflect the communal nature of learning mathematics would also practice different norms in the classroom. In these mathematics classrooms the rules of learner behaviour and mathematical discourse are negotiable. Learners are offered multiple opportunities to engage constructively and critically with mathematical ideas. Teachers who subscribe to such an open classroom environment need to display an understanding of how individual students would work in that environment, including their background and beliefs about what it means ‘to do mathematics.’ Thus, the scaffolding of learning has emerged to be a priority for teachers.
Arguments about shared mathematical ‘meaning making’ by learners places premium on the language that is used in this process. During the course of negotiated understanding, learners need to communicate their own ideas with peers and others in the community. Wertsch (1998) suggested that language is a cultural artefact or tool that mediates individual’s cognition in mathematics. The mathematical language that is used by members of the mathematical community includes special words, symbols, diagrams and representations with meanings that are different from those occurring in everyday language. The vocabulary of mathematics that is used by mathematicians, mathematics textbooks and, to some extent, mathematics teachers, tends to be at variance and conflict with that used by learners.
While mathematicians may understand each other, learners often experience difficulties with the language of mathematics and every-day language (Sfard and Kieran 2001). One of the difficulties here is that learners attempt to extract the meaning that is embedded in what teachers and textbooks say in their own mathematical language. For example, the notion of function and roots as used by the mathematical community has a special meaning in comparison to the every-day meaning of function. Through a process of acculturation learners come to discriminate the duality of meanings that is associated with these two terms. This process entails the learner engaging mathematics concepts in activities that make sense and aids in resolving the apparent tensions. The adoption of such activities to foster debate and discussion about mathematics terminology and the way these are used constitutes an important strategy for assisting learners with different cultural backgrounds to engage in knowledge building (White 2003).
The failure of pedagogies to recognize learning mathematics as a problem concerning language and communication exacerbates the situation for certain groups of learners. Learners might form the view that they have not grasped what is being taught when in fact the problem could be their difficulty to communicate their understandings and intuitions into mathematically precise statements. Many learners’ problems with mathematical understandings that are related to the solution of word problems can be attributed to problems of language. The above tension between mathematical language and the learner’s own language is impeding, if not preventing, the construction of important connections that are necessary for the growth of deeper levels of understanding. The gap in communication between teachers and learners is further complicated by the multiple and idiosyncratic interpretations that learners construct about a particular mathematical concept and their own linguistic backgrounds (Clarkson 1992).
The interest in knowledge organization and performance has generated a number of theoretical frameworks including connectedness. Mayer (1975) examined the notion of connectedness as involving the accumulation of new information in long-term memory, adding new nodes to memory and connecting the new nodes with components of the existing knowledge network of the learner. He identified two types of connectedness: internal and external. Internal connectedness refers to the degree to which new nodes of information are connected with one another to form a single well-defined whole or schema. Schemas refer to clusters or chunks of knowledge about a particular mathematical concept. They represent objects, contexts and prior experiences of the learner with that concept. Beyond the above elements, schemas also contain a network of interrelations. For example, a student might have developed a schema about whole numbers. This whole number schema could include information about size of numbers and operations involving this class of numbers. The nature of schemas and how these may be used to account for learning has been investigated in a number of areas, such as, composing stories (Bereiter and Scardamalia 1986), chess (Chase and Simon 1973), analogical problem solving (Gick and Holyoak 1983) and mathematics (Skemp 1987).
Connectedness refers to both the presence of nodes related to a schema and the quality of the network of interrelationships established among those nodes. The broad notion of quality of knowledge here can be related, in part, to what Anderson (2000) refers to as ‘strength’ of connections. In this sense the stronger the connections among the nodes in a particular schema, the better the quality of that well-defined structure. Mayer (1975) visualized external connectedness as the degree to which newly established knowledge structures are connected with structures already existing in the learner’s knowledge base. Let us examine these ideas in the context of a schema for proportion. A learner might be expected to relate a schema for proportion with schemas for ratio or fraction. These external connections between proportion and ratio or fraction will have a certain quality that would impact on learner’s ability to use them in order to solve problems or provide alternative representations. In such a case, the new schema for proportion will have both a certain quality in its internal structure (internal connectedness) and a certain quality in its connections to related schemas (external connectedness). This analysis of connectedness was used to support the argument that the linking of the different pieces of knowledge of geometry and algebra reflect deeper and richer understandings (Chinnappan 1998; Chinnappan and Thomas 2003).
In classrooms where teachers support students to talk, the higher level of input from learners during their critical evaluation of mathematical concepts would help them reflect and reconstruct new understandings. The ensuing debate and problem solving that adumbrates examination of a focus concept can be expected to aid in learners building the elements that are necessary for the growth of external connectedness. For example, young learners working on the fractions will be motivated to explore meaningful contexts where part-whole relations that are embedded in the fraction number are given concrete and richer representations. Mayer’s (1975) analysis of connectedness, while useful for the exploration of organizational features of conceptual aspects of mathematical understanding is somewhat limited in that it does not explicitly consider idiosyncrasies of the individual learner and the contexts in which learning takes place. Authenticity of learning tasks contribute directly to the quality of schemas that can be constructed but this feature of the task is embedded is the context that moulds one’s thinking.
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