Drive learning

The teacher uses the learners’ ideas as starting points for discussion. The teacher responds to learners’ contributions in order to make relevant instructional decisions. In particular, learner responses drive the learning process. The establishment of learners’ prior knowledge is one of the criteria that can be used to guide the facilitation of lessons. Schoenfeld (1985:12) refers to the concept of ‘resources’ available to an individual, which consists of a set of relevant facts available to the problem solver, algorithmic procedures known by the learner, and routine procedures and procedural knowledge about the agreed-upon rules for working in the domain. In this case, one of the questions asked during the interview was: “What is your opinion regarding the statement “a major goal of mathematics instruction is to help learners develop the belief that they have the power to control their own success in mathematics”?

Teacher A

Teacher A responded: “Obviously, my opinion would be that it is correct, but as teachers we

need to create a situation whereby learners take charge of their own learning [. . .] we need to allow learners to take charge of their own learning”. Learners’ responses should ‘drive’

92

instructional decision should be based on learners’ responses and abilities. He gives the impression that he is the one responsible for creating such opportunities. Teacher A’s approach seems to be in line with the claim that teachers should create opportunities and strategies that evoke learners’ mathematical thinking and explanations and know what to do with what they heard in order to make instructional decisions (Franke & Kazemi, 2001:104).

However, during the classroom observation carried out, Teacher A could not follow up on some of the comments the learners made during his lesson. From Table 4.16 regarding the discussion of better loan repayments options, the conversation ended without the teacher responding to one of the learner’s comments that monthly repayments will depend on affordability. Instead, Teacher A maintained his opinion that he will recommend that people pay higher monthly payments so that they can pay off the loan quickly and save money on interest. However, Teacher A could in fact have followed up on the learner’s response and found out why the learner said he would prefer to pay low monthly repayments, paying a higher final amount. Teacher A’s actions appear to contradict the views of Stigler and Hiebert (1998: 2) that mathematics teachers should elicit learners’ mathematical thinking and anticipate multiple strategies for solving problems, as mathematics is not just a set of procedures or algorithms to be followed.

Teacher B

Teacher B said: “I must first see what they know . . . Initially, I believed that for the success

of the learners it depends upon the teacher [. . .] so we were not giving them enough chance to explore on their own”. These comments from Teacher B seem to confirm that he needs to

explore what learners know before he can provide his own ideas. From Teacher B’s comments one can conclude that he believes learners ‘drive’ the facilitation process. However, from Table 4.17 it appears that the learners’ responses do not guide teaching and learning. Teacher B accepts the learners’ one-word responses without allowing them to explore their answers. Hence, one needs to note that the traditional way of accepting answers only is inadequate. Furthermore, he continues with the presentation without requesting the learners’ responses or comments. The facilitation process in Teacher B’s case looks similar to that of Teacher A. Both Teacher A and Teacher B are occasionally tempted to proceed with the lesson, ignoring learners’ contributions.

93

Teacher B does not elicit learners’ mathematical thinking and explanations, as suggested by Stigler and Hiebert (1998:2). In one of the conversations Teacher B had with the Grade 10 learners he asked “In the equation ymxc which variable represents the gradient?”

One of the learners’ response was mx . But Teacher B failed to provide an explanation or ask the learner why he thinks mx represent the gradient and not the variable m . He only

said m represents the slope of the function y mxc. He continued his lesson by saying “Now today I want us to look at how to draw the graph using the intercepts”. This is an indication that the learners’ responses do not drive the flow of the lesson.

Teacher C

Teacher C said: “Using problem solving, learners have to come up with what they know

before we can tell them what to do”. Teacher C’s comments seems to be in line with

Schoenfeld’s (1985: 12) view that any mathematical problem-solving activity is built on a foundation of basic mathematical knowledge, which is called ‘resources’. However, as is the case with teachers A and B, Teacher C seems to agree that she would build her facilitation of lessons on learner responses in order to enhance their problem-solving skills. However, Teacher C’s lesson did not proceed in accordance with her claims.

From Table 4.13, three learners gave a wrong answer when asked the value of the gradient in the function y = 2. Despite these wrong answers given by the learners, Teacher C ignored the learners’ misconceptions about the value of the gradient when a function is given. Instead, she proceeded with the lesson by introducing the concept of the y-intercept. She does not seem to be able to switch between her presentation in order to explore the concept of the gradient, but proceeded with the objectives of her lesson, which was to draw the graphs and then identify the effect of a using the graph. Yet, according to Stigler and Hiebert (1998:3), struggling and making mistakes and then seeing why learners are making mistakes are essential parts of the learning process.

Teacher D

Teacher D noted: “Learners should be guided in their own learning to strengthen their own

mathematical learning. When learners do problems themselves they will not forget easily and will have deeper understanding of concepts so that they can apply the new knowledge themselves. If I stand in front and then facilitate they might forget mathematical concepts

94

easily”. From Table 4.14, Teacher D did not present his lesson in accordance with this view.

He stood in front and demonstrated to the learners how to calculate the gradients of the two lines, PT and RS.

In his presentation, the learners only gave the ‘Yes’ answers, without exploration. He proceeded to the next stage of his presentation and never explored the learners’ understanding, as he claimed to do. In the case of Teacher D, as in the case with teachers B and C, it appears that the learners’ answers do not drive learning. Artzt et al. (2008: 9) mention that learning with understanding enhances learners’ remembering strategies and assists learners to relate new ideas of mathematics to what they already know and can do, to use their previous knowledge and skills to construct new meaning and to apply their learning to new contexts. According to Lau et al. (2009:309), the assumption is that the acquisition of skills by a learner is an activity in which the readily relevant skills are combined to meet new, more complex task requirements.