26
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
27
words, in order to facilitate learners’ constructions of mathematical knowledge, the teacher needs to ask them appropriate and judiciously selected questions that are relevant to the task at hand. The teacher needs to develop his/her questioning skills in order to enhance learners’ achievement. According to Marzano, Pickering, and Pollock (2001), the teacher’s classroom practices that involves questioning is more effective than one without questioning. As the teacher asks learners some questions, their mathematical thinking and participation are stimulated hence enabling learning. In a way, this implies that effective questioning can lead to meaningful learning. Now the question is: What kind of questions should the teacher ask leaners in order to foster effective learning?
Badham (1994) identified four main categories of questioning that classroom teachers can use to promote effective learning. The first category of questioning Badham identified is: Starter questions. These questions basically direct learners’ thinking to the new knowledge and they seek multiple responses from learners in order to initiate a discussion (Badham, 1994). For example if the topic for the day is “Addition of fractions with different denominators”, one of the questions that the teacher may ask learners would be: Give me an example of a pair of some fractions which have different denominators. This question would give learners a starting point so that they begin to think about new knowledge that they are about to learn. To respond to this question it is expected that learners identify such fractions. Starter questions take the form of ‘pivotal’
questions wherein learners have to think and focus their attention to the new topic.
The second category of questioning that was identified by Badham is: Questions to stimulate mathematical thinking. According to Badham (1994), these questions help learners to make connections between previous knowledge and experiences with the new knowledge. For example, in following up on the question about a pair of fractions with different denominators, one could ask about how these fractions could be represented as fractions with the same denominator. This would get them ready to relate the new knowledge to an already known fact of adding fractions with like denominators. Such questions help learners to see patterns and relationships between what they already know and what is new to them. Basically the teacher asks learners this type of question
28
in order to find out what learners already know and to help them make links to what they know (Ausubel, 1978). The teacher is expected to spend some time at this stage of the lesson, making sure that learners’ prior knowledge is confirmed.
The third category of questioning is: Assessing questions. In essence these are follow-up or probing or leading questions where the teacher perceives learners’
responses as inadequate or inappropriate. A teacher asks such questions when learners are engaged into a task or problem situation. Examples of assessing questions are: How did you….? Why do you think…? What if…? What about…, etc. These type of questions allow the teacher to get learners’
clarification, elaboration, to see what they understand and to stimulate their thinking (Badham, 1994). Such questions involve cognitive manipulation of information in order to support an idea or a solution to a problem. The teacher may ask probing questions to an individual learner or group of learners or the entire class to get more information or think and express their ideas in-depth.
As learners respond to the teacher’s assessing questions, it is important for the teacher to give timely feedback which could be simple comments such as right, or correct, or more corrective ones as a way of moderating their responses.
The last category of questioning is: Final discussion questions. According to Badham (1994), such questions allow learners to share and compare their solutions, and the methods they used to arrive at the solution. At this stage of the lesson, the effort of the class is drawn together to share meaning. Learners think about their peers’ mathematical ideas and methods which in itself is key to effective learning. Examples of final discussion questions are: Which groups have the same solution? Which group has a different solution? Are your results the same? (Why/why not?); Is there another strategy of finding the solution?
During all the above discussed categories of questioning, wait time is essential in stimulating learners’ thinking after the teacher has paused a question (Shahrill, 2013). In other words, the teacher should give learners enough wait time to allow them to think before responding to a question that s/he has posed.
This would lead to learners’ active participation and giving thoughtful responses.
29
From the discussion thus far, it can be argued that questioning is a fundamental instrument of enabling meaningful learning. The teacher should be competent with content knowledge and have good questioning skills. The contention is that having conceptual understanding of the subject Mathematics would facilitate the teacher’ good questioning skills. This means that for the Mathematics classroom teacher to ask learners appropriate questions during his/her lessons, s/he must have full knowledge of the subject matter. The teacher must be an expert in the area of Mathematics otherwise it will not be possible for him/her to ask learners good relevant questions as the lesson progresses. However, a teacher may have inadequate or no training in questioning techniques during his/her pre-service training. This would affect learners’ classroom participation and academic achievement as the teacher will not be competent enough on the art of questioning techniques. His/her empowerment at pre-service training on asking learners some questions during a lesson would improve his/her practice hence potentially enable meaningful learning.
2.4.1 The use of manipulatives
One of the innovative teaching strategies used by some classroom teachers is manipulatives. The teachers use manipulatives when mediating mathematical concepts. Their belief is that the use of manipulatives would help learners cope with the abstractness of mathematical concepts (Tall, 2008). In other words, by using manipulatives they believe that abstract mathematical concepts would be more accessible to the learners. Hence during the teaching of Mathematics, the use of manipulatives is associated with effective teaching.
Manipulatives is a word that is used when educators refer to concrete objects such as Dienes blocks, geoboards and rubber bands, and Cuisenaire rods that can be used in the teaching and learning of Mathematics (Clements & McMillen, 1996). Clements and McMillen mention that though learners who use manipulatives in their Mathematics class usually do better than those who do not use them, but this is only true for certain topics. According to them, manipulatives do not guarantee success in the learning of Mathematics. They acknowledge the idea that manipulatives have an important place in learning Mathematics, but they point out that manipulatives do not carry the meaning of
30
the mathematical idea. Their argument is that learners sometimes learn to use manipulatives only in a rote manner. However, in rote learning, the current knowledge is not linked to pre-existing knowledge hence there is no meaningful learning. Clements and McMillen (1996) explain that at times physical actions with certain manipulatives may suggest mental actions different from those that teachers wish learners to learn. For example, when using a number line as a manipulative to find the sum of 7 and 5, learners locate 7 on the number line and start to count 1, 2, 3, 4, 5 and read the answer as shown below.
Figure 2.1 Number line for finding the sum of 7 and 5
Such a procedure does not help them to solve the problem mentally hence are not using the number line as a tool. The expected procedure would be to count 8, 9, 10, 11, 12 or rather 8 is 1, 9 is 2, 10 is 3, 11 is 4 and 12 is 5 so that it matches the mental activity intended by the teacher. Clements and McMillen’s (1996) arguments imply that not all manipulatives are sufficient to guarantee meaningful learning. In other words, it does not mean that if learners use manipulatives in their Mathematics class, meaningful learning is guaranteed rather learners should make connections between manipulative models and real life situations, and mathematical concepts in order to attain conceptual understanding. On the same note of manipulatives, Clements and Battista (1990) point out that teachers use them as a vehicle to get to the abstract, symbolic and established Mathematics.