Given this background, it seems reasonable to propose that schools should also see the need to train learners to become effective in collaborative learning settings. Therefore, instruction that promotes collaborative skills of learners ought to be designed. In this article a collaborative or group approach refers to a classroom arrangement in which learners sit together to discuss and solve mathematics tasks or problems (see Dhlamini, 2012). Our premise is that a classroom arrangement that incorporates group learning activities provides learners with ‘effective tools to reinforce their problem solving system’ (Dhlamini, 2012, p. 241). This is possible because the processes that occur during group discussion include verbalising explanations, justifications and reflections (Beers, Boshuizen & Kirschner, 2007; Kirschner, Beers, Boshuizen & Gijselaers, 2008), giving mutual support (Van Boxtel, Van der Linden & Kanselaar, 2000) and developing arguments about complex problems (Munneke, Andriessen, Kanselaar & Kirschner, 2007). In the same vein, Dhlamini (2012) emphasises three elements of group learning activities: discussion, argumentation and reflection. According to Van Boxtel et al. (2000), group learning activities can allow learners to provide explanations of their understanding, which can help them to elaborate and reorganise their knowledge. Lai (2011) notes that group learning activities, such as providing elaborated explanations to group members, improve learner comprehension of conceptual knowledge.

In this study, we constructed group approach learning environments in experimental schools. Control schools mostly followed a didactic teaching approach in which teaching is associated with transmission of knowledge by the teacher, and learning associated with passive receiving of knowledge. The didactic teaching approach primarily involves lecturing and is essentially teacher-centred, limiting learner participation and reflection (Johnes, 2006). This approach is usually associated with traditional approaches to teaching. In almost all control schools teachers employed traditional methods of teaching that were primarily presented in a non-group teaching mode. Therefore, this study investigated the effects of the two teaching approaches on the performance of Grade 10 mathematics learners. Our primary research question was: What is the effect of using a group approach on the performance of learners when teaching certain topics in Grade 10 financial mathematics?

In light of the national search for teaching approaches that can improve mathematics achievement of learners in South Africa, we believe that our outcome variable, Grade 10 performance in financial mathematics, is particularly timely. Indeed, our results may be of importance for those interested in empowering teachers to meet the challenges of the new curriculum.

In contrast, long-term memory has the capacity to permanently store huge amounts of knowledge and chunks of domain-specific skills in a form of cognitive schemas or schemata (Dhlamini & Mogari, 2011). A schema can hold a huge amount of information which can be treated as a single unit when it is processed in the working memory (Kirschner, 2002). Given that a schema can be treated as a unit, it may be processed effectively in the working memory. Cognitive load is reduced when information is treated as a unit in the working memory. For this reason cognitive load theory has focused on designing teaching methods to counter the effect of the limitations of human working memory which, when not managed properly, may increase cognitive load (for examples, see Dhlamini, 2012; Dhlamini & Mogari, 2012a).

Although cognitive load theory generated lessons are meant to manage individual working memory load (cognitive load) of individuals (Kirschner, Paas, Kirschner & Janssen, 2011), Kirschner, Paas and Kirschner (2009) have emphasised an alternative technique of effectively dealing with individual working memory limitations by making use of the multiple working memories of individuals in group approach learning environments. From a cognitive load theory perspective, it is argued that dividing the processing of information in the working memory across individuals in a group approach environment is useful because this technique allows information to be divided across a larger reservoir of cognitive capacity, thus increasing the working memory capacity (Kirschner et al., 2011). According to Dhlamini (2012), in a group approach learning environment, which represents a huge working memory system, the limitations of individual working memories are not exposed because individual working memories are not subjected to processing each piece of problem-solving information. Therefore the risk of overloading each group member is lowered, and an individual’s working memory capacity is freed up whilst the group’s collective working memory capacity is expanded, and the cognitive load may be reduced (Kirschner, 2009).

According to Kirschner (2009), within a group setting information processing is characterised by active and conscious sharing (i.e. retrieving and explicating information), discussing (i.e. encoding and elaborating the information) and remembering (i.e. personalising and storing the information) valuable task-relevant information and knowledge held by each group member. For a group to perform a mathematics task, it is not necessary that all group members be highly knowledgeable in task-related information or be able to process all available information by themselves and at the same time (Johnson, Johnson & Stanne, 2001). As long as there is communication and coordination between group members, the information elements within the task and the associated cognitive load can be shared amongst group members (Kirschner, 2009).

Given that there is a paucity of studies to test whether or not learners in group approach settings perform better than learners in non-group approach settings (Kirschner et al., 2009), we decided to investigate this observation. The aim of this study therefore was to investigate the comparative effects of a group approach and a non-group approach on the performance of Grade 10 mathematics learners. In a sense, this study was conducted within the ‘effect paradigm’. Studies conducted within the effect paradigm examine outcomes of collaboration rather than the collaborative process itself. In effect the comparison is between a group performance and a non-group or individual performance (Lai, 2011). Some of the results from these studies suggest that a group approach can have powerful effects on learner performance (Lai, 2011).

The intervention in each school lasted for two weeks. During the intervention the pairing of schools was such that each experimental school was paired with a control school. Whilst implementing a group approach in experimental schools the first author visited each control school once to observe a comparative mathematics lesson being implemented (see Figure 1). Control schools were visited on days on which the first author had little teaching time at corresponding experimental schools. The fifth experimental school was not paired with a control school, but this did not affect the results because results in each group were aggregated.

FIGURE 1: Intervention program for paired schools in the study.

4. Your brother wins a LOTTO competition and decides to invest R50 000 now. He secures an interest rate of 9% p.a. compounded annually. The inflation rate is currently running at 12% p.a.
4.1 What will the future value of your brother’s money be in 15 years from now?
4.2 Due to inflation, what money will have the same buying power as R50 000 in 15 years’ time?
4.3 By how much will the buying power of your brother’s money have declined after 15 years?

An example of a compound interest test item was:

2.3 Calculate the compound interest on a loan of R800 at 7% p.a. if the interest is compounded half yearly.

The FMAT had a total mark score of 60. The same test was written by all learners in both conditions before and after the investigation. The pre-test determined learners’ initial performance status before intervention. A post-test was given at the end of a two-week intervention to examine the effects of a group approach (experimental group) and teachers’ conventional teaching (control group) on the performance of learners.

In addition, the first author administered an instrument to investigate the influence of a group learning approach on the cognitive load of learners in experimental schools. The researchers adopted a self-reporting instrument developed by Paas, Van Merriënboer and Adams (1994). This instrument uses a post-test questionnaire in which test takers are asked to report the amount of mental effort (cognitive load) invested in performing problem-solving tasks in a test; there are nine choices. Each of the choices was presented in the learners’ answer booklets or script immediately following each session of the FMAT (pre-test and post-test). The first measure of learners’ cognitive load served as a baseline measure before intervention. The second measure served as an indication of the influence of a group approach on learners’ cognitive load, and related problem-solving performance.

Finally, to gain access to learners’ conceptions of whether or not working in groups, and also in non-group learning environments in control schools, enhanced their performance the first author did post-intervention semi-structured interviews. Two learners in each of the nine participating schools (N = 18) were purposively sampled for the interviews to provide their opinions on the style of teaching they followed in their respective schools. The following sampling criteria were established to select learners from both groups for interviews: (1) learners’ performances in post-test (all performance categories were represented) and (2) quality of involvement during lessons (learners who were observed to participate minimally or maximally were equally selected). Interviews enabled us to elicit first-hand, in-depth primary data from learners. For the study of classroom practice and forms of classroom interactions that characterise approaches to FMAT-related worksheets, we asked all four teachers in control schools to allow the first author to observe their lessons only once. The task of the first author was to observe whether teachers’ conventional teaching approaches followed a model of learner-centeredness or teacher-centredness. Using the descriptions presented earlier, a group approach was classified as a learner-centred approach, and teacher-dominated lesson was classified as a teacher-centred approach.

In addition, in experimental schools the potential of more robust interaction was exploited with several worked-out problem examples in financial mathematics that were given to groups in worksheets (e.g. see Figure 2). Worked-out examples are a set of problem-related examples that presents an instructional step-by-step guideline on how to solve a problem (Dhlamini, 2011; Dhlamini & Mogari, 2011). Because worksheets were developed in collaboration with the four control schools teachers who participated in the study, both experimental schools and control schools were exposed to identical worksheet tasks. However, the mode of presenting the worksheet tasks varied between the two groups. In experimental schools, whilst group members studied steps in worked-out examples the first author walked from one group to another to ensure that learners understood the task assigned to them and adhered to the roles that the first auther assigned to them. Essentially, the first author also wanted to determine whether the group learning dynamics of listening, writing, answering, questioning and critically assessing contributions were taking place. According to Lai (2011), when teachers walk and circulate amongst groups, learners are more engaged and discussions are more fruitful.

FIGURE 2: A sample of a worked-out problem-solving example.

When solving new tasks the first author encouraged learners to work independently in order to apply some of the ideas they discussed in groups. They were advised to seek assistance from group members only when they encountered challenges. Whilst they interacted in groups the first author asked questions such as: ‘What do you think about what your fellow learner said?’; ‘Do you agree with your fellow group member?’; ‘Do you think you can learn from other learners?’ Responses to these questions were considered to indicate learners’ views about a group approach to mathematics. On the last day both the experimental group and control group wrote a post-test to measure the comparative effects of the instructional methods on learners’ performance. The post-test had exactly the same questions as the pre-test.

Based on the total score obtainable, learners were designated as low-performing (below 24), average-performing (between 24 and 42), and high-performing (above 42). The pre-test mean score of experimental schools was 20.9 whilst the mean score of the control schools was 22.0. The differential between these two means was 1.1 marks. The small differential suggests baseline equivalence of the two groups before intervention, thus confirming the earlier classification of learners as low-performers based on the Grade 12 results of their schools.

Post-test results suggested greater improvement in experimental schools, where a group approach was implemented. Here the mean score of the experimental schools was 33.3 (SD = 4.21; N = 378), whilst the mean score of control schools was 25.8 (SD = 4.10; N = 328). Because of the non-random sampling techniques employed in the study a one-way analysis of covariance (ANCOVA) was computed in order to test for statistical difference amongst all the variables and covariates. Pre-test scores were entered as the covariate and post-test scores as the dependent variable.

The phrase between-subjects (or within-learners) in Table 3 represents a measure of how much the FMAT scores of each learner tended to change (or vary) over the period of two weeks; this is measured by comparing the scores of the pre-test and post-test. The results in Table 3 show that when the effect of pre-test scores is removed, the effect of the group approach becomes significant, as confirmed by F(1.703) = 558.7, p < 0.05. This means that the group approach, as implemented in the experimental schools, is superior to the conventional teaching approaches implemented in the control schools in substantially improving learners’ performance in certain topics in Grade 10 financial mathematics.

Learner 1: Eh … this method is good because in my class we are not talking [implying that the teacher explained everything during the lesson], but here we are talking to our friends for answers. It is good because I get it better from my fellow learners.
Learner 2: A group approach is better than our school method because we also talk as learners. I like it when we sit and discuss our views in maths. This is the best method to teach maths.
Learner 3: We must do these types of problems in my class. Our teacher must listen to us as students because these are our problems.

In particular, learners emphasised the importance of verbal interaction between learners when solving mathematics tasks. In fact, Learner 1 emphasised that she ‘gets it better’ when the explanation emerged from a fellow group member. When participants were asked about the difference between an explanation from the teacher and one from a fellow learner, typical responses were:

Learner 4: I think it is easier to challenge my friend than a teacher.
Learner 5: Sometimes it is easier when we use our language as students because we understand each other.
Learner 6: I think because we are all not perfect, if we discuss our mistakes as students we can improve.

Responses suggest that learners preferred a learner-centred approach (group approach) in which they engage in opportunities to share and discuss their mathematics ideas. A different picture emerged from learners in control schools. Learners from control schools revealed that they were never given an opportunity to discuss mathematics ideas amongst themselves. Most of the learners emphasised that in their schools they were subjected to a didactic teaching approach. The term didactic teaching approach is used to represent a popular view by learners from control schools that teaching in control schools was teacher-dominated. In another round of questions learners were asked to describe aspects of a group approach and traditional teaching approaches that they thought accounted for each instruction’s beneficial effect. Responses from experimental schools pointed to worked-out examples as an effective learning tool.

Learner 7: I like this method of teaching because you have taught us with examples.
Learner 8: Eh … I liked this method. It is good and train students’ minds. It is good because we were given examples that helped us to work alone. We will pass maths now.
Learner 9: I think a group method was good because it works with things that we know and good examples which make us to be familiar to the problems.

The responses provided by Learner 8 and Learner 9 emphasise the importance of retention and familiarity in learning. To enhance the positive benefits of a group approach learners sat in groups to discuss worked-out examples. Although identical tasks were tackled in control schools, the main mode of teaching used by teachers did not replicate the performance observed in experimental schools.

More recently Kirschner et al. (2009) observed that learning by an individual is less effective and efficient than learning by a group of individuals during mathematics tasks. In terms of cognitive load theory, arranging learners in groups for learning purposes provides an opportunity to deal effectively with the limitations of working memory at individual level. This is important because within a group approach environment members have more processing capacity and so the construction of task-related schemas is promoted (Kirschner et al., 2009).

We acknowledge that the design of our investigation permitted the emergence of two independent variables within our study, namely a group learning approach and a worked-out examples approach. Hence, one of the challenges in interpreting the results of our investigation is separating the influence of these variables on the outcome of our experiment. To address this issue, and further provide a justification to moderate the confounding influence of the worked-out examples approach on learners’ performance, we explored the following considerations: (1) a relatively larger component of the methodological setup of our investigation was hinged on the group approach rather than on the worked-out example approach and (2) there is a considerable empirical evidence to support the claim that a group learning approach promotes learner performance and long-term retention of the studied material, essentially in learning environments that put exclusive emphasis on this approach (see Lai, 2011; Whicker, Bol & Nunnery, 1997). Given these considerations, it may be reasonable to conclude that when learners in experimental schools discussed worksheet-related examples within a group approach context, they gained greater understanding of mathematics tasks, and were able to retain this knowledge and understanding longer.

Another confounding issue relates to the time span within which this investigation was conducted. It may be argued that a two-week period may not suffice to influence learners’ problem-solving performance. The instructional design that was implemented in this study, which largely embraced aspects of cognitive load theory, yielded the outcome that successfully overcame the posing constraints of time. The results of this study provide substantial evidence to suggest that learners’ problem-solving performance could be positively influenced over a short period of time (see Table 4). However, this study shows that this is possible when instructional design is aligned to learners’ cognitive architecture to moderate the effect of extraneous problem-solving demands (cognitive load) on learners’ performance (see also Dhlamini, 2012). Within group approach learning environments that were systematically constructed in experimental schools, learners’ cognitive load was therefore reduced within a relatively short time span. Hence, this study shows that group approach instruction that is aligned to the principles of cognitive load theory may speedily accelerate learners’ problem-solving performance.

We therefore emphasise that the outcome of our investigation was largely influenced by the type of instructional modes that were employed in both groups during the experiment. One of the learners commented: ‘Working in a group also helped me to understand the example steps in worksheets better’. In our study worked-out examples were presented as mathematics tasks, not as a form of instruction. Worked-out examples worksheets served as springboards to group discussions. Therefore, the accelerated examples-related performance that is also reported by the learner in previous sentences could be linked to the fact that learners in experimental schools worked in groups. Despite these observations, we acknowledge that a further investigation with a modified methodological design should be conducted to explicitly demarcate the effects of a group learning approach and a worked-out examples approach on the mathematics performance of participants.

In conclusion, we acknowledge that the teaching of mathematics has often been viewed as an isolated, individualistic or competitive activity in which one sits and struggles alone to solve problems. Hence, at the beginning of this study we asked: What is the effect of using a group approach on the performance of learners when teaching certain topics in Grade 10 financial mathematics? The results of this study, and those of other related studies, provide some hope that group learning may still have a place in mathematics teaching. Specifically, the results of this study provide some evidence to suggest that a group approach may be an effective way to teach certain topics in Grade 10 mathematics.

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