6.4 ANSWERS TO RESEARCH QUESTION 3
6.4.1 The teachers displayed narrow conceptions of meaningful learning
The teachers had acknowledged the value of using learner-centred teaching in the teaching of Mathematics. Hence, they conducted the observed lessons within their personal enactment of “learner-centred” practices of which it emerged that they believed that the role of the teacher was that of a guide. In the study, the literature review has revealed that learner-centred practices have a strong link with meaningful learning because that is where the latter occurs.
This would imply that the teachers’ conceptions of meaningful learning basically inform the way they enable effective learning in their “learner-centred”
practices. And in the literature review chapter, I have established that the constructs meaningful learning and effective learning mean the same thing. In this section, therefore, I will provide a discussion of the teachers’ conceptions of meaningful in their personal enactments of “learner-centred” practices.
In education literature, meaningful learning can be viewed as learners’
experiences that have particular meaning to them (Kostiainen et al., 2018;
Wong, 2015) e.g. the teacher links theory and practice, or learners’ engagement in a task situation, or learners’ success in solving a mathematical problem, or learners working collaboratively. Whilst in cognitive development meaningful learning is viewed as a situation where learners make connections between new ideas and their own existing related ideas (Ausubel et al., 1978). In the study, the teachers differed in the way they viewed meaningful learning in their
“learner-centred” practices.
Milton held two contrasting views about meaningful learning. On one hand, he believed that meaningful learning was the learner’s ability to solve a given
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problem and on the other hand he believed it was the learner’s ability to follow steps as they attempted to work on a problem. Apparently Milton based his first description of the meaning of meaningful learning on learners being able to solve a problem (Kostiainen et al., 2018). Despite some of his learners not getting the task that he had given them to do on constructing a triangle correct, Milton believed that meaningful learning took place because learners attempted the problem.
On the second account, Milton’s conception of meaningful learning seems to be associating it with the application of mathematical procedures to problems with no concern about concepts connections which resonates with Skemp’s (1976) construct of instrumental understanding and Star and Stylianides’ (2013) notion of procedural knowledge. In my literature review chapter, I have pointed out that instrumental understanding and procedural knowledge do not lead to meaningful learning because, here, the learners do not make some effort to link some mathematical ideas. However, there is need for learners to make conceptual connections in order to enable effective learning among them.
In constructing the triangle, the learners were in fact trying to follow steps that Milton had demonstrated to them. It is important to note that during his demonstration, Milton did not explain his steps to the learners. It was possible that they got the construction wrong because the sharp point of the compass was not placed at the zero mark on the ruler as the start off point of the radii of the arcs. As such his learners did not make a connection between radii of their arcs and the zero mark of the ruler as used in the measurement of line segments.
Hence, they seemed not to make meaning of the steps he used during his demonstration.
Milton’s demonstration of constructing a triangle did not enable effective learning. Without being able to follow the steps, there was no basis for them to make sense of the actual construction procedure. Furthermore, without understanding the steps they would not be able to reflect about how and why the construction procedure resulted in the triangle with the required dimensions.
Neither would they be able to make connections between the steps and the concepts. Learners need to understand the reasons for doing things and to make
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connections between the concepts just like in Skemp’s (1976) construct of relational understanding and Star & Stylianides’ (2013) notion of conceptual knowledge.
Themba’s conception of meaningful learning was based on connecting Mathematics to real life situations (Wong, 2015), for example linking meaningful learning to future carriers. Themba believed that meaningful learning is achieved if learners can associate mathematical concepts with future carriers. He maintained that if learners could apply the concepts that they learnt in class to everyday life then there was meaningful learning. To him, meaningful learning is informed by learners’ application of scientific concepts to real life.
Themba explained in his interview that if you teach learners Geometry, then there will be meaningful learning because some of them will end up being carpenters or builders. This view of the importance of linking mathematics concepts to real life applications is supported in the literature, however at no point in Milton’s lesson did he pointed out the links or make the links explicit.
He seems to believe doing such mathematics concepts was his task and that it was up to his learners to recognise or make up the connections to real life.
Just like Milton, Sabelo held two conceptions about meaningful learning. His first conception is that it occurs when learners can express their ideas while they are working in groups. Sabelo believed that group work enables effective learning because in their groups learners discuss and share mathematical ideas which is shared by the researchers (Mtika & Gates, 2010; Webb et al., 2009;
Wong, 2015). As the learners discuss in their groups learning opportunities are generated because they would be bringing in various experiences to the discussions. However, Sabelo could not utilize group work effectively to enable meaningful learning.
Sabelo’s second conception about meaningful learning was that learners should be kept busy by the teacher discussing a problem and showing their understanding. This belief he held about meaningful learning resonates with Kostiainen et al.’s (2018) assertion that learners should be seen engaged in a
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task situation that they have been given by the teacher so that there is effective learning.
Indeed, after Sabelo had given his learners a task to work in small groups, he went around observing what learners were doing in their groups without any interference with what they were doing in their groups. He made sure that learners were seated in groups and seemed to be working on something. His interventions fell short of ensuring that the learners were working together meaningfully while being engaged in the task. All he did was check if the answers were correct and provided the correct answers when necessary.
In the study, the teachers displayed narrow conceptions of meaningful learning in their personal enactments of “learner-centred” practices. Their conceptions of meaningful learning were rooted on what the learners were capable of doing in a classroom environment. Despite the teachers’ emphasis on the role of prior knowledge in their observed lessons, they did not associate it with it view in cognitive development. In cognitive development, meaningful learning is viewed in terms of learners’ association of new knowledge with what they already know.
6.4.2 The teachers tried to emphasize the role of prior knowledge in their