There are various strategies that are used by Mathematics teachers in order to enable conceptual understanding during their day-to-day classroom practices.
Umugiraneza, Bansilal, and North (2017) found that in South Africa, although teachers were familiar with progressive teaching strategies such as classroom discussion, group work and experiments, they focused widely on strategies such as expository, chalk and talk, question and answer. However, research has indicated that teaching strategies that involve active teaching methods like investigation, discovery method, problem solving, and collaborative learning are more effective than traditional strategies in the teaching of Mathematics, because they enable learners to deepen their conceptual understanding. These strategies are rooted in the Vygotskian socio-cultural notion that views learning as taking place through social interactions (Firmender, Gavin, and McCoach (2014) wherein effective learning is facilitated. According to Vygotsky (1978), mental operations are initiated in an individual through active social interaction with more competent peers and adults. Effective learning is therefore necessitated by an effective teacher.
An effective Mathematics teacher is one who can stimulate a learner to learn the concepts of Mathematics (Clements & Battista, 1990). His/her teaching strategies should enable learning amongst learners in a constructivist classroom.
According to Clements and Battista, a constructivist classroom is where learners are actively involved in the sharing of knowledge as they socially interact with one another whilst the constructivist teacher guides and supports them. The constructivist teacher should offer meaningful and appropriate tasks, and enable opportunities for discussion among learners (Clements & Battista, 1990). On the other hand, learners must be communicating about Mathematics with one
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another and making sense of the Mathematics. In this study, effective learning has the same meaning as meaningful learning because both constructs focus on conceptual understanding.
2.3.1 Qualities of an effective teacher
Hattie (2003) identified five major qualities of excellence of an effective teacher. According to Hattie, an effective teacher is one whose teaching strategies has positive effects on the learner’s learning. During his/her teaching, an effective teacher enables a powerful achievement of meaningful learning amongst his/her learners. Below are Hattie’s five major effective qualities that underpin an effective teacher which he refers to as dimensions of an expert teacher:
Identifies essential representations of their subject
Guides learning through classroom interactions
Monitors learning and provides feedback
Attends to affective attributes
Influences learners’ outcomes (Hattie, 2003, p. 6)
A teacher who identifies essential representations of his/her learners is one who makes use of prior knowledge with regards to the learner. Such knowledge may be within Mathematics itself, or learners’ everyday knowledge, or knowledge drawn from other subject disciplines. All these types of knowledge would serve as basis for understanding new knowledge. In short, this type of teacher possesses a cohesive form of knowledge that s/he is able combine with new content according to the needs of learners to enable meaningful learning among them. The challenge with this type of teacher is that his/her learners may be coming from different backgrounds such that the prior knowledge s/he has decided to use may not be all of the learners. Hattie also identified an effective teacher whose quality is guiding learning through classroom interactions. This type of a teacher allows class engagement such that there is effective learner- learner and learner-teacher interactions. Here, learners are at liberty to question and their errors are appreciated by the teacher thereby allowing for appropriate feedback. One of the most important quality of a teacher identified by Hattie is monitoring and providing feedback. According to Hattie, a teacher who is able
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to monitor his/her learners is the one who can identify learners’ difficulties during a lesson and their level of understanding. S/he is skilful in observing and assessing the level of understanding of learners during the lesson. Through monitoring learners’ problems as they engage with a problem, the teacher would provide prompt and appropriate feedback in order to enable understanding. In any classroom situation, feedback is essential to foster effective learning;
basically corrective feedback is the most powerful type of feedback where clarification, elaboration, and learners’ evidence is sought by the teacher.
Hattie (2003) mentions that feedback is the most powerful single moderator that enables understanding. For a teacher to enhance understanding among his/her learners there must be relevant and proper feedback. In the same vein, Hattie and Timperley (2007) mention that feedback is one of the most powerful influences on learning and achievement. According to them feedback can be used in a classroom situation as a corrective measure to improve and enhance learning. It is most powerful when learners are engaged in a problem or task situation. Hattie and Timperly mention that for feedback to be more effective the learner’s response must be faulty. In other words, its main focus is on errors that are made by learners during a lesson otherwise feedback is not effective when there is a complete misunderstanding or lack of information among them.
Furthermore, Hattie and Timperly point out that feedback can as well be understood as information provided by a teacher or learner regarding one’s performance or understanding. This implies that on one hand a teacher can provide a corrective information to a learner and on the other hand a learner can provide an alternative strategy to solving a problem to his/her peers as they engage in a task situation during whole class discussion or small group discussion. In essence, feedback provides information that bridges the gap between what the learner attempts to understand and what s/he already knows.
Feedback can take the form of verbal comment or probing questions on individual learners or a group of learners. However, giving feedback needs the teacher’s high level of skill. Furthermore, the classroom teacher may not provide effective feedback to every learner in the class particularly if there is a large number of learners (McDonough, 2004).
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Another quality identified by Hattie (2003) is attending to affective attributes wherein the teacher treats learners equally, respects and cares for them during lessons. This is about creating a conducive atmosphere for learning in class.
Here, the teacher takes special care about his/her learners’ successes and failures. However, learners may take advantage of a teacher who is too much respectful to them and begin to misbehave which may affect their concentrations. The last quality of an effective teacher identified by Hattie is about influencing learners’ outcomes. This is where the teacher constructs an appropriate and challenging task, and gives it to his/her learners to work out.
The task is aimed at meeting his/her instructional objectives for the topic. As learners work on the given task, the teacher monitors the proceedings. Here, s/he is not supposed to dominate the lesson rather learners actively participate the lesson progresses. The task s/he gives to learners is not just to keep learners busy, rather to involve them in the lesson meaningfully in order to facilitate learning. It can be pointed out that constructing and designing a task oriented problem can pause a reasonable amount of challenge to a teacher.
All the above dimensions of an effective teacher are significant in enabling meaningful learning. However, Hattie (2003) argues that too few of the dimensions have been put into practice by classroom teachers. Despite all of Hattie’s dimensions of an effect teacher, he still has to possess pedagogical content knowledge. According to Shulman (1987) pedagogical content knowledge refers to the teacher’s competency on the knowledge of the subject matter and knowledge of instructional practice. So for the teacher to facilitate learners’ conceptual understanding of Mathematics, s/he needs to have knowledge of the subject and effective teaching strategies.
2.4 TEACHERS ENACTING LEARNER-CENTRED PRACTICES