This qualitative study seeks to explore Mathematics teachers’ constructions and their enactments of leaner-centred practices. Of particular interest in the study is an in-depth understanding of teachers’ conceptions of learner-centred practices and the extent to which they enabled meaningful learning as they engaged learners in their teaching of Mathematics. Vygotsky’s (1978) socio- cultural theory of learning and Ausubel et al. (1978) notion of meaningful learning theory both provide key theoretical bases that underline the study.
Di Napoli (2004) argues that the learner-centred philosophy is a paradigm that reflects views about teaching, learning, and knowledge acquisition. On the same note of learner-centred philosophy, Novak and Gowin (1984) state that learner- centred philosophy provides a framework for a variety of teaching methods
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geared to improve learning. This means that when teaching within the learner- centred philosophy, different types of teaching methods are used and each chosen method must be consistently informed by the learner-centred philosophy. When teaching within the learner-centred philosophy, the teacher acts as a facilitator of the learning process and learners work in collaboration with one another to pursue conceptual understanding (Blumberg, 2008). The teacher can as well be viewed as a manager of the learning process wherein s/he provides guidance and support to learners in their plight to seek meaning within a learner-centred setting. According to Vavrus et al. (2011), learner-centred philosophy places an emphasis on the learner who is doing the learning and the teacher’s role is to help learners access and process knowledge. The authors are of the view that learner-centred philosophy,
Is an approach that informs the practices of teaching based on the assumption that people learn best by actively constructing and assimilating knowledge rather than through the passive addition of discrete facts to an existing store of knowledge (p.
27).
This means that within the learner-centred philosophy, learners actively construct knowledge whilst the teacher facilitates the learning process rather than present knowledge; hence the teacher no longer assumes the traditional role of knowledge transmitter. Furthermore, Vavrus et al. (2011) report that learner- centred philosophy draws deeply upon constructivism theory of learning with the notion that learning takes place when learners are actively engaged in a learning process. Basically learner-centred philosophy has some relations with the social constructivist view on learning, which emphasizes learners’ active participation in the learning process in learner-centred practices. This means that the learner-centred philosophy is based on a constructivist theory of learning where the teacher engages learners in the learning process. Learners actively participate during the lesson in their plight to seek conceptual understanding. In the process they construct their own knowledge and use it as means to build new ones. It can be argued that the knowledge that they use to build new ones can be interpreted as the knowledge that learners already have in relation to the new ones. When learners construct new knowledge, the pre-
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existing knowledge is important for the formation of a connection between the pre-existing knowledge and the new one.
According to Vygotsky’s (1978) theory of socio-cultural development, learning occurs in and through socially mediated activities and language plays a key role in mediation. He argues that knowledge is constructed by learners within a social setting. This means that learners construct meaning individually or socially when they are in the learning process. Vygotsky (1978) placed emphasis on social and linguistic influences on learning, and in particular on the role of the teacher in the education process. He advanced a concept, known as
‘the Zone of Proximal Development’ (ZPD) to provide some measure of learners’ cognitive development related to instruction. Vygotsky’s notion of ZPD is significant in understanding how learners actively construct knowledge with assistance by the teacher or an adult or a capable peer during learner- centred practices. He defines the ZPD as:
The distance between the actual development level as determined by independent problem solving and the level of spontaneous development as determined through problem solving under adult guidance or in collaboration with more cable peers (Vygotsky, 1978:86).
Here, Vygotsky implied that with appropriate teaching, there may be potential for a learner to reach higher conceptual understanding within the zone itself than s/he would be able to achieve without instruction. In the zone therefore, the learner will actively construct his/her own mathematical knowledge as the teacher assists him/her during the learning process. According to Vygotsky (1978), there is no single zone for each learner; rather a zone is created for every concept. This means that for every new mathematical concept being introduced, there is a zone such that the learner has to integrate new knowledge with his/her relevant developed cognitive structure. The teacher is supposed to create classroom situations in learner-centred practices through which a learner could construct mathematical knowledge. Furthermore, the teacher has to help the learner to bridge the gap between what s/he already knows and what is to be learned. Vygotsky contends that if the teacher teaches to current developmental levels, s/he condemns a learner to remain at these levels. This suggests that if the teacher teaches a learner what s/he already knows, it is unlikely that the
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learner will construct ‘new’ mathematical knowledge. For the learner to develop cognitively, the teacher needs to be constantly in advance of his/her development, leading and directing it. So basically in a learner-centred classroom setting, learning will occur with the support of the teacher or competent peer.
Vygotsky’s theory of socio-cultural development within his notion of ZPD recognizes knowledge resulting from the teacher’s role of mediation and negotiation. It can therefore be argued that Vygotsky’s concept of ZPD and the social construction of mathematical ideas are of value in viewing learner-centred practices, constructions, and personal enactments of Mathematics teachers in their teaching practices. Now the question is: What relation does learner-centred practices have with the notion of meaningful learning?
The main objective of education is to engage learners in meaningful learning which occurs when learners are making meaning of what they are being taught (Wong, 2015). Classroom teachers play an important role in facilitating making meaning by learners. They support learners in their attempt to achieving meaningful learning. According to Kostiainen et al. (2018), meaningful learning is a concept describing experiences that learners believe to have a particular meaning to them. Such experiences may incorporate for example situations where:
The teacher links theory and practice during practical activities. In the context of education, theory here involves knowledge of the concepts that include principles and conceptual definitions i.e. the content knowledge that is being taught, whilst practice is a variety of learning experiences that are either meaningful or meaningless to learners.
Learners’ engagement in task situations that have been given to them by the teacher.
Learners’ being successful in solving mathematical problems, which according to them have special meaning.
Nevertheless, this does not mean that all experiences may be meaningful to learners. Whilst some learners may be successful in problem solving, others may not, hence they would perceive the problem solving situation as
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meaningless to them. Kostiainen et al.’s (2018) basic claim about learning experiences is that they become meaningful to learners only when learners give meaning to them. In other words, meaningful means some knowledge that is understandable to the learner. It can be argued therefore that for meaningful learning to occur, learners must be engaged into some learning processes so that they feel some experiences.
On the same note of meaningful learning, Wong (2015) identified a combination of five attributes that would generate meaningful learning as depicted in the diagram below:
Figure 3.1 Five attributes for generating meaningful learning (adopted, Wong, 2015: p.182)
According to Wong (2015), meaningful learning incorporates combinations of active learning, constructive learning, intentional learning, authentic learning, and cooperative learning. In active learning, learners explore and manipulate objects, and observe the results of their actions. In that way they construct their own meaning as they manipulate and interpret the environment. Hence, Wong points out that meaningful learning requires learners’ engagement in task situations in which they are hands-on and seeing the results of their actions.
However, learners’ manipulation of task situations and observations are not sufficient for meaningful learning (Wong, 2015). It is for this reason that Wong points out that learners need to also articulate their hands-on activities and observations and he refers to this as constructive learning. During constructive learning, learners integrate their new experiences with what they already know
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(Wong, 2015). The next attribute that generates meaningful learning that was identified by Wong is intentional learning. This attribute is about learners associating what they are learning to fulfil their career aspirations or trying to achieve a cognitive goal, for example understanding how to add fractions with different denominators. Wong also identified authentic learning wherein mathematical problems are situated in real-life contexts. In that way, it is assumed that learners can better understand the problem hence they would be in a position to transfer their understanding to new situations. The last attribute to meaningful learning is cooperative learning. Essentially, this is about collaborative learning where learners engage on a task or problem situation in small groups in order to accomplish a common understanding. The above attributes, which according to Wong generate meaningful learning, do not happen in isolation; rather they have a connection with one another.
However, in cognitive development, meaningful learning is viewed in terms of the acquisition of knowledge. According to Novak (2002), for meaningful learning to take place the knowledge learned must be relevant to the existing knowledge. The new knowledge being introduced to the learners must be conceptually related to what learners have learnt within the same knowledge base. For example, when teaching addition of mixed fractions/numbers, learners must have learnt addition of common fractions and must have been introduced into the concept of mixed fractions.
On the same note of meaningful learning Ausubel (1962) argues that in any educational classroom setting, meaningful learning takes place when new connections are made between the learner’s own relevant cognitive structure and the new knowledge to be learnt. His argument seems to resonate with Novak’s (2002) notion of meaningful learning in education. Novak views meaningful learning as learners’ attempt to link concepts they have learnt and to those that are relatively new to them. Hence, for meaningful learning to occur they must be an integration of new concepts with the learner’s relevant pre- existing concept ideas. However, Agra et al. (2019) found an extended meaning of meaningful learning.
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According to Agra et al. (2019), meaningful learning is
a learning process, in which the learner is motivated to learn, understands, reflects and attributes new concepts, starting from previous knowledge and experiences, modifying the existing meanings, by means of the organization and integration in the cognitive structure of the previous and new concepts, making them meaningful, which, necessarily, are transferred to other situations that experience (p. 248-255).
In addition to relating new knowledge with pre-existing knowledge as articulated by Ausubel (1962) and Novak (2002) about meaningful learning, Agra et al. (2019) mention that a learner must be motivated to learn. By saying that the learner must be motivated what they mean is that s/he must be influenced to relate new ideas with pre-existing ideas during the learning process. Another addition by Agra et al. (2019) to the meaning of meaningful learning is about the learner being able to transfer what s/he has learnt to new situations according to its usefulness for example, to his/her everyday life or to other concepts within Mathematics or other subject disciplines. This means that when the learner has learnt meaningfully, s/he must be able to use that knowledge to new situations.
Though Ausubel indicates that meaningful learning applies to reception (expository) learning in classroom settings, his theory is helpful in the current study because it provides a framework within which one can view and describe teachers’ conceptions of meaningful learning in their teaching practices. Thus Ausubel’s notion of meaningful learning will shed some light on how Mathematics teachers will enable meaningful learning.
It is somewhat difficult to distinguish between the two phrases ‘meaningful learning’ and ‘learner-centred practices’, however, I will briefly try to disentangle the meanings. In essence, meaningful learning is an outcome that can be achieved as a result of the teaching of mathematical concepts within learner-centred practices. In a way learner-centred practices are the context at which meaningful learning takes place. During his/her teaching, therefore the teacher has to operate within learner-centred practices.
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In retrospect, there is a strong link between learner-centred practices and meaningful learning. The link is brought about by learners’ pre-existing knowledge which can be considered as a necessary requirement for learning new concepts. Within the learner-centred framework, the teacher negotiates learners’ constructions of new knowledge which they are supposed to associate with pre-existing knowledge hence enabling meaningful learning (Ausubel, 1962), and learner-centred practices form context at which meaningful learning occurs.
According to Ausubel (1962), the teacher should ensure that a learner already possesses appropriate knowledge in his/her cognitive structure that s/he can assimilate the new knowledge with. What this implies is that in learner-centred practices, the teacher should confirm and emphasize the learner’s relevant prior knowledge. For example, when teaching place value to the thousands, one’s appropriate prior knowledge would be place value to hundreds. Here, the teacher has to make sure that learners have a firm understanding of place value to hundreds. This would enable them to assimilate hundreds to place value of thousands.
What emerges in the foregoing is that meaningful learning can involve active learning, constructive learning, intentional learning, authentic learning, and cooperative learning within learner-centred practices. Meaningful learning can as well be viewed as relating prior ideas with pre-existing ideas within learner- centred practices. And in turn, learner-centred practices would involve learners collaborating with one another as the teacher facilitates the learning process.
Learner-centred practices and meaningful learning, therefore, cannot be divorced from one another because learner-centred practices provide a context at which meaningful learning occurs. Thus, in the current study more understanding is needed on how Mathematics teachers make their meaning of learner-centred practices and the extent to which they enable meaningful learning. The diagram below illustrates the framework that draws together learner-centred practices and meaningful learning.
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Figure 3.2 Framework for learner-centred practices and meaningful learning The above framework indicates that the teacher plays the role of facilitator during learner-centred practices. In other words, s/he is the manager of the learning process. As the teacher facilitates the learning process, he/she draws in learners’ prior knowledge whilst learners collaborate with one another. Thus, as learners are engaged in small group activities, in their plight to achieve conceptual understanding, meaningful learning is enabled.