Analysis of responses from both groups

For each of the questions 1, 2 and 6, learners’ mean scores in both groups were greater than 1. As a result these questions were grouped together in the following discussions. The mean scores ranged between 1,09 and 2,09 out of a possible score of 5 (See Table 5). On the other hand for questions 3, 4, 5, 7, and 8, learners’ mean scores are less than 1 (almost 0) as such these will also be discussed together.

Mean score per question in a pre-test

0 0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 Questions M e an sc o re s Control group Experimental group

Question 1

The total score for this question was 5 and the mean scores were 1.84 and 2.09 for control and experimental groups respectively.

Most learners recognized the pattern; however, a majority of these learners were not able to answer the question correctly. Several learners’ responses were as follows:

Other learners did not complete the pattern to the 10th stop as indicated in the example below:

Eighteen learners in the experimental group and 12 learners in the control group obtained four marks and higher.

Question 2

The total score for this question was 5 and the mean scores were 1.09 and 1.18 for control and experimental groups respectively.

Most learners wrote down the answer without showing how it was found. The researcher believed that these learners might have found the answer by trial and error. On the other hand it is possible that learners did not think that the solution process was important but only the answer as it is usually emphasized in traditional classrooms. This confirmed what was pointed out by Wilson et al (1993:61), that traditional models of problem solving lead to an emphasis on getting the answer.

Question 6

The mean scores for this question were 1.16 and 1.40 for control and experimental groups respectively. The maximum score for this question was 5.

During discussions with learners it became evident that this was one of the questions that was clearly understood by most learners since they were involved in sports competitions in one way or the other. It is important then as indicated by the Department of Education (2003:42) that tasks selected should be related to learners’

experiences so that they become meaningful and learners build on these experiences. Learners were however, not able to solve the problem, but at least most attempted this

Several learners divided 8 by 2 and said 4 games will be played.

It became obvious that learners lacked adequate skills or strategies to solve this problem. The best way to find the solution to question 6 was to draw a diagram showing different teams. However an interesting solution by one of the learners is as follows:

Questions 3, 4, 5, 7 and 8

As indicated earlier, the mean score for each of these questions was smaller than 1 out of a possible score of 5.

In questions three, four and five learners either did not write anything or copied some of the numbers appearing in the questions without any attempt to solve them. This reaction may be caused by the fact that question three involved fractions, question four involved percent and question five involved ratio. These concepts are

related, though they are not portrayed as such in most classrooms. This seems with the argument that isolating topics gives an impression that there are many concepts to be learned and learners do not easily see the relationship among topics thus they may take longer to understand (Cangelosi 1996). A learner who does not understand one of these concepts may also have difficulty working with the others. Therefore the researcher felt that learners were not competent in working with these concepts.

In question seven, it was evident that although learners understood this question, they did not know what to do with the given information. An observation here was that some learners gave their answers in rands instead of days indicating that they never looked back at the problem to see whether their answer made sense or not. The following responses from learners’ work reveal the above observation.

One learner wrote:

Another learner wrote:

During problem solving one may lose sight of the original problem and looking back strategies force one to check the answer against the requirements of the problem (Ohio Department of Education 1980b:20). Reflection is one of the major aspects of problem solving and in this case learners did not reflect on their solutions. This is one of the issues that needs attention in teaching by problem-based approach.

In question eight, learners wrote the formulae for the area of the rectangle and/or formulae for the area of a circle but did not know what to do with these formulae. Below are examples of what some learners wrote:

In most mathematics classrooms, the mistake made by educators is to do an example on the board using the formulae after which learners are given problems to practice using the formulae. If there is no example done for these learners, then it is difficult for them to use these formulae in solving problems even though they know the formulae. Hiebert et al (1996:17) believe these learners possess knowledge that they do not use to inform their procedures, thus resulting in memorization and execution of procedures without understanding.

The common observation made while marking and during informal interviews with learners was that they believed the answer was more important than the solution method leading to the answer. It could be that this practice was based on traditional instruction which contributes to this belief. Kaput (1999:133) points out that in these classrooms, learners are graded not on understanding of the mathematical concepts and reasoning, but on their ability to produce the right symbol string- answers. It was not surprising therefore that in some instances learners wrote only the answer and in others, where they were uncertain they did not write anything. Lester points (1985:47) out that the right direction might be for teachers to focus more attention on solution attempts and less on correct answers.

4.3 The post-test