AOSIS OpenJournalshttp://www.openjournals.netinfo@openjournals.nethttp://www.pythagoras.org.za1012-23462223-7895<article_title>Exploring mathematical discussion in word problem-solving
</article_title>Percy Sepeng1,2
1Marang Centre for Mathematics and Science Education, University of the Witwatersrand, South Africa
2Centre for Educational Research, Technology and Innovation, Nelson Mandela Metropolitan University, South Africa
Paul Webb2
2Centre for Educational Research, Technology and Innovation, Nelson Mandela Metropolitan University, South Africa
This study explored whether discussion as a teaching strategy in mathematics classrooms could have positive gains in improving learners’ problem-solving performance, as well as their ability to make sense of real wor(l)d problems. This article discusses the partial findings of a bigger study that used a pre-test intervention or post-test mixed-method design and utilised both quantitative and qualitative data. Analysis of the data generated from both pre-tests and post-tests suggests that the intervention strategy significantly improved the experimental group’s problem-solving skills and sense-making performance. The statistical results illustrate that the experimental group performed significantly better than the comparison group in the post-test. The main finding of this study is that in classrooms of experimental schools in which discussion technique was successfully implemented, there was a statistically significant improvement in the learners’ competence in solving word problems.
60331http://dx.doi.org/10.4102/pythagoras.v33i1.60
Percy Sepeng
percy.sepeng@wits.ac.za
Private Bag X3, WITS 2050, South Africa
10 Oct. 201122 June 201215 Aug. 2012Sepeng, P., & Webb, P. (2012). Exploring mathematical discussion in word problem-solving. Pythagoras, 33(1), Art. #60, 8 pages. http://dx.doi.org/10.4102/pythagoras.v33i1.60
© 2012. The Authors. Licensee: AOSIS OpenJournals. This work is licensed under the Creative Commons Attribution License.
IntroductionProblem solving and integrated assessment are seen as the cornerstones of school mathematics and the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 2000) called for mathematics instruction and assessment to focus more on conceptual understanding than on procedural knowledge or rule-driven computation (Hamilton, 2004; Kilpatrick, Swafford & Findell, 2001). Major arguments for including word problems in the school mathematics curriculum have always been their potential ability to promote realistic mathematical modelling and problem-solving. Solving word problems also help learners to develop the skill of knowing when and how to apply classroom mathematical knowledge as well as everyday life-knowledge when solving problems. In this article we therefore argue that there are multiple benefits and good outcomes when learners participate in mathematical discussion in the classroom.
This article begins with a discussion of the current debates on using discussion as a strategy for the teaching and learning of mathematics in general. The study draws largely on the results of a research project conducted in six schools in the Eastern Cape Province of South Africa (Sepeng, 2010). Much of the work in this study relies both theoretically and methodologically on notions of classroom mathematics discourse and mathematical modelling. The main argument in this article is that careful use of discussion as a teaching strategy in mathematics classrooms appears to have positive gains in improving learners’ problem solving performance in word problems.
The hypothesis for the study was that the introduction of mathematical discussion in the teaching and learning of word problems would improve and enhance learners’ problem solving performance and the ability to make sense of real wor(l)d problem-solving. The purpose of this article is therefore to demonstrate how the results of the study were found to support this hypothesis.
Solving word problems
Word problems have been defined differently in different studies. For the purpose of this study, the definition provided by Verschaffel, Greer and De Corte (2000) is used. These researchers define word problems as ‘textual descriptions of situations assumed to be comprehensive to the reader, within which mathematical questions can be contextualised’ (p. ix). They also stress that word problems ‘provide, in convenient form, a possible link between the abstractions of pure mathematics and its applications to the real-world phenomena’ (p. ix). According to Palm (2009) mathematical word problems include pure mathematical tasks ‘dressed up’ in a real-world situation that require students to ‘undress’ these tasks and solve them (p. 60).
A further methodological issue, which socio-cultural approaches have yet to address satisfactorily, arises from the increasingly multicultural nature of mathematics classrooms. Students’ interpretations of mathematics classroom interaction relate in part to their different social, cultural and linguistic backgrounds. When classroom interaction is analysed, therefore, some way of taking account of this diversity needs to be found, otherwise there is the risk that a single cultural perspective, that of the researcher, will be imposed. Discursive psychology has the potential to address some of the above-mentioned issues.
Discussion in mathematics classrooms
The use of discussion as a tool to increase reasoning has gained emphasis in classrooms worldwide, as earlier reports had predicted (Yore, Bisanz & Hand, 2003). Discussion, however, requires scaffolding and structure in order to support learning (Norris & Phillips, 2003). Wood, Williams and McNeal (2006) found variations in students’ ways of seeing and reasoning, and these were attributed to the particular differences established in classrooms early in the year pertaining to when and how to contribute to mathematical discussions and what to do as a listener. Their conclusions are consistent with the findings reported by several other researchers (e.g. Dekker & Elshout-Mohr, 2004; Ding, Li, Piccolo & Kulm, 2007; Gillies & Boyle, 2006; Webb, Nemer & Ing, 2006), who also suggested that participation obligations put boundaries around the opportunities for students to share their ideas and to engage in mathematical practices. When they make a difference through classroom discourse, teachers shift students’ cognitive attention toward making sense of their mathematical experiences, rather than limiting their focus to procedural rules. In doing so, students become less engaged in solutions to problems than in the reasoning and thinking that lead to those solutions
(Yackel & Cobb, 1996).
Students develop a mathematical disposition through the patterns of interaction and discourse created in the classroom and the process of ascribing meaningfulness to one another’s attempts to make sense of the world. Learning about other ways to think about ideas, to reflect, and to clarify and modify thinking is fundamental to moving learning forward. Carpenter, Franke and Levi (2003) maintain that the very nature of mathematics presupposes that students cannot learn mathematics with understanding and without engaging in discussion. However, more talk in the classrooms does not necessarily enhance student understanding. Better understanding is dependent on particular pedagogical approaches, purposefully focused on developing a discourse culture that elicits clarification and produces consensus within the classroom community.
A variety of situations may arise in which the outcomes are not fully realised. For example, a number of studies have reported that some students appear to thrive more than others in whole-class discussions. In their respective research, Baxter, Woodward and Olson (2001) and Ball (1993) found that highly articulate students tend to dominate classroom discussions. Low academic achievers usually remain passive, and when they do participate visibly, their contributions are comparatively weaker, and their ideas sometimes muddled. Nevertheless, pedagogical practices that create opportunities for students to explain their thinking and to engage fully in dialogue have been reported in research undertaken by Steinberg, Empson and Carpenter (2004). In a study which was part of their Cognitively Guided Instruction Project, they found that classroom discussion was central to a sustained change in students’ conceptual understanding.
MethodSample in this study
The sample consisted of Grade 9 learners (n = 176) and their teachers in six township secondary schools, four of which were experimental schools (n = 107) and two of which were comparison schools (n = 69) (where no intervention took place). The six schools chosen were a convenient sample of a cluster of similar schools in Port Elizabeth. All the schools were functional (as opposed to dysfunctional – which is the case in many instances in South Africa), had similar characteristics in their approach to teaching and learning contexts and were public and previously marginalised schools. The schools drew their learners from communities of low economic status. The two comparison schools were chosen randomly within the group identified.
Research design
In this study we used a pre-test – intervention – post-test design. We investigated the situation in terms of the problem-solving performance of Grade 9 learners with English as their second language, using a pre-test to establish how the learners solved mathematics word problems and what problems they might have mathematically. Then we wanted to find out if introducing discussion into the classroom as part of the learning experience had any effect on learners’ problem-solving performances and/or strategies. A post-test, which was exactly the same as the pre-tests, investigated if there were any changes. We also used this test to find explanations or reasons for any changes in learners’ sense-making and solutions of real wor(l)d problems. In the next section we present the intervention strategy used to promote discussion in the teaching and learning of word problems.
Design type and the intervention
In this study an intervention strategy was used to investigate the effect of discussion on the problem-solving performance of Grade 9 second language learners. The intervention took place over a period of six weeks and focused on the use of writing and mathematical discussion to solve word problems. The intervention was implemented in the experimental group and its aim was to assist learners not only to identify problem situations that were problematic from a realistic point of view, but also to consider the (in)appropriateness of applying mathematical operations directly as their solutions. Moreover, the teachers of the experimental schools were introduced to a typical collaborative learning context(s) where the teacher posed a task (or a problem) and the learners, after sufficient time to complete the task, engaged in a discussion of the solution methods and/or strategies that they had developed in small groups. The whole class discussion continued for, at most, 15 minutes before another short segment (three to five minutes) of group task. This cycle was repeated five to six times in a double period of about
90 minutes. During the small group interaction learners had to develop reasons to support their thinking and/or think about some relevant issue(s) or question(s) instead of solving a specific mathematics question. Because of the continual emphasis on both justification and reasoning, whole classroom discussions resulted in the emergence of key concepts in wor(l)d problem solving within the contexts of realistic considerations in particular, and sense-making in general. As a consequence, the teaching approach appeared to have considerable potential for in-depth conceptual development growing out of the learners’ discursive activity.
Although learners had some time to explore their reasoning with one another other during the limited time allocated to small group discussion, the interruptions brought about by the teachers in the classroom discussion did not give them much time to pursue their own ideas. However, learners were expected to accept the obligation and engage in thinking about the issues at hand and in sharing their thinking within their smaller groups. As such and since the discussion inevitably focused on their reasons, learners were in a good frame of mind to compare and contrast their reasons with those of others. Consequently, their thoughts and discussions formed a basis for engaging meaningfully in the subsequent classroom discussion. In some instances, concept cartoons in mathematics were used as a stimulus or trigger for discussion while learners were solving problems in their small groups. The purpose of introducing discussion was to help learners seek, share and construct knowledge when engaging in solving word problems. The discussions took the form of dialogue and talk (formal and informal) in both English and the learners’ home language (isiXhosa). In promoting discussion, learners were expected to engage critically with problems and build positively on what others had said.
The observations during and after the intervention were done with the aim of measuring teachers’ implementation of the strategies that they had learnt during teacher workshops. These workshops gave teachers from experimental schools the opportunity to be trained on how to get learners discussing, arguing and writing about their views and experiences when they solved mathematics word problems. The aim of promoting these strategies in their teaching was to develop and improve their approaches to the teaching and learning of word problem solving in their classrooms. These teachers were introduced to and trained in strategies to improve their pedagogical content knowledge and their ability to promote the teaching and learning of mathematics when solving word problems.
The intervention also focused on the language of mathematics embedded within word problems. Simple translations were provided for phrases that are often used in mathematical word problems to simplify the meaning of these problem statements. Table 1 shows some examples of the
translations provided.
The problem-solving tasks
The pre-test and post-test consisted of the following three modelling problems adapted from Verschaffel, Greer and Van Dooren (2009. Learners were expected to complete these problem-solving (PS) tasks and give a written explanation of how they arrived at their answers:
PS1: 100 children are transported by minibuses to a
summer camp at the seaside. Each minibus can
hold a maximum of 8 children. How many
minibuses are needed?
PS2: Two boys, Sibusiso and Vukile, are going to help
So nwabo rake leaves on his plot of land. The plot
is 1200 square meters. Sibusiso rakes 700 square
meters in four hours and Vukile does 500 square
meters in two hours. They get 180 rand (R) for
their work. How are the boys going to divide the
money so that it is fair?
PS3: John’s best time to run 100 meters is 17 seconds.
How long will it take him to run 1 kilometre?
Coding of the tests
The three PS tasks were coded using a schema that was an elaboration of the classification schema developed by Verschaffel, De Corte and Lasure (1994). The classification schema comprised fourteen categories, which were reduced to three general categories:
Realistic reaction (RR): All cases where a learner either gave the (most) correct numerical solution that also took into account the real-world aspects of the problem context, as well as cases where there was a clear indication that the learner tried to take into account those real-world aspects, without giving the mathematically and situationally (most) accurate numerical answer.
No reaction (NR): All those cases where there is no indication that the learner was aware of the realistic modelling difficulty, for example, mathematically correct but situationally inaccurate and/or incorrect or inappropriate responses, computational errors, etc. This category also provides a measure of the problem-solving performance of the learner.
Other reaction (OR): All cases where a learner did not provide a numerical response and did not give any written comment that indicated that the learner was aware of the realistic modelling difficulty that prevented him or her from answering the problem, as well as instances where the learner generated incorrect responses with mathematical (or computational) errors.
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Overview of reliability and validity in this study
The primary strategy utilised in this study to ensure external validity was the provision of thick, rich and detailed descriptions so that anyone interested in transferability would have a solid framework for comparison (Merriam, 1988). Nixon and Power (2007) point out that warranting of claims must fulfill the criteria of trustworthiness, soundness, coherence, plausibility and fruitfulness. Trustworthiness refers to the quality of qualitative data collected (Anastas, 2004); and in the sensof neutrality in the findings or decisions of the study (Guba & Lincoln, 2005).
Reliability is the degree to which the instrument consistently measures whatever it is measuring (Ary, Jacobs & Razavier, 1990; Best & Kahn, 2003). According to Silverman (1999), reliability refers to the degree of consistency with which instances are assigned to the same category by different observers or by the same observer on different occasions. Neuman (2003) suggests reliability has to do with the issue of dependability. Dependability of the data in this study was established by capturing the observations on a tape and video recorder, and transcribing them both manually in writing and with computer software. Attempts were made to reproduce the interview scripts as accurately as possible to eliminate possible threats to the reliability of the instruments used in this study. Creswell (2005) defines threats as the problems that threaten our ability to draw correct cause-and-effect inferences that arise because of the experimental procedures or the experiences of participants.
Ethical considerations
The Education, Research Technology and Innovation Committee gave prior permission to conduct this research as part of the Integrated School Development and Improvement project offered by the Centre for Educational Research, Technology and Innovation at the Nelson Mandela Metropolitan University. After obtaining ethics clearance, the first author approached the principals and teachers of the participating schools, where their roles as participants and their rights to choose to be participants and to participate or not in the study were explained to them. Both the teachers and parents (on behalf of the learners) gave their informed consent. They were assured of confidentiality and also that participation was voluntary. They were given a guarantee that they could withdraw from the study at any time and that no personal details would be disclosed. They were promised confidentiality of information collected in the schools and were assured that no portion of the data collection would be used for any purpose other than this research. Learners were also assured that the test results would not influence their school marks.
Statistical analysis of data
The quantitative statistical data generated from the pre-tests and post-tests were captured in a Microsoft Office Excel spreadsheet and subjected to repeated measure ANOVA techniques (using the STATISTICA Software package) which simultaneously accounted for pre-test and post-test data of the experimental and comparison groups in order to provide both descriptive and inferential statistics. ANOVA techniques were used to determine the statistical (non)significance of the results, based on mean differences between experimental and comparison groups before and after the intervention. Where necessary, the statistical technique of matched-pairs t-tests was computed to compare the mean scores of the comparison and experimental groups.
Cohen’s d statistics were calculated to determine whether statistically significant (p < 0.0005) pair-wise differences were practically significant. A small practical significance is noted where 0.2 < d < 0.5; a moderate practical significance is noted if 0.5 < d < 0.8; and a large practical difference is recorded if d > 0.8. Expressed differently, an effect size of less than 0.2 is considered to be insignificant, an effect size between 0.2 and 0.5 is considered to be of small significance; an effect size between 0.5 and 0.8 is considered as being moderately significant, while an effect size of 0.8 and greater is considered to be highly significant. Effect size as expressed by the Cohen’s d statistics is defined as the difference in means divided by the pooled standard deviation and is a measure of magnitude (or significance) of the differences between the pre-test and post-test scores (Gravetter & Walnau, 2008).
FindingsThe word problems in this study, which are examples of a central part of mathematics learning, can be seen as attempts to connect mathematical reasoning to learners’ everyday life experiences and/or knowledge (Sepeng, 2011). In other words, these problem-solving tasks can be viewed as a manifestation of the notion that mathematics is or should be part of mundane practices in everyday life (Verschaffel et al., 2009).
Effect of intervention on problem solving
Table 2 depicts a summary of the statistical analysis of learners’ word problem solving achievements as well as their realistic reactions. Analysis of pre-test results indicate a statistically significant difference between the experimental and comparison groups, with the experimental group’s performance being worse than that of the comparison group (p < 0.0005). After the intervention, the experimental group’s performance was statistically significantly better (p < 0.0005) than the comparison group’s, with a mean difference (Δχ) of 29.14. In Table 2 a positive mean difference implies that the mean score of the experimental group was more than that of the comparison group in the post-test.
Practical significance of the differences
As p < 0.0005 in all cases, Cohen’s d was calculated in order to gauge the effect size of the practical significance of the differences in the experimental and the comparison groups. A large practical significance (d = 1.56) was noted. When the effect sizes on the RR difference between the experimental and the comparison groups were compared, a moderate practical significance (d = 0.57) was calculated (see Table 2).
Sense-making (or realistic considerations) of word problem solving (PS tasks)
Table 3 shows the number and percentage of learners who succeeded in producing three, two, one, and zero RRs to the PS tasks. It illustrates that learners performed rather poorly in these tasks, which required not only computational skills, but realistic sense-making as well. Only 1% of learners in the experimental schools produced three situationally accurate answers or reacted three times in a way that showed attention to the realistic modelling complexity of the problems. A closer look at the post-test results for the experimental schools shows a 10% improvement in the production of 2 RRs compared to a drop of 6% in the comparison schools.
The mean difference (difference between the mean scores) shows [Δχ = 0.47] a statistically significant (p < 0.0005)
difference between the experimental and the comparison groups for the RR after the intervention. The positive mean score shows that, although the comparison group had a tendency to consider reality and sense-making when solving the word problems before the intervention (Δχ = -0.02), its performance
was well below the experimental group’s after the intervention.
Matched-pairs t-tests
Table 4 shows the results of a matched-pairs t-test that was used to test whether there was a significant mean difference between experimental and comparison groups before and after the intervention (or pre-test and post-test). In addition to this, data in Table 4 depict mean scores of the experimental and comparison groups for word problem solving and sense-making (RRs) in this study.
At the p < 0.0005 significance level, the study gives overwhelming evidence that the problem-solving scores of the experimental group improved by 17.08 after the intervention, whilst the practical significance calculated for the experimental group is moderate. Although the comparison group’s mean score was higher than that of the experimental group before the intervention (pre-test), a negative mean difference (Δχ = -26.67) suggests that the comparison group not only scored well below the experimental group after the intervention, but that their scores were significantly lower than the experimental group’s.
A practical non-significance was also calculated for the comparison group. Despite a small practical significance found for the experimental group’s RRs, the experimental group did significantly better than the comparison group with a marginally significant improvement in sense-making scores (Δχ = 0.23) in the experimental group after the intervention.
Table 5 provides a brief summary of findings based on learners’ solving and sense-making of word problems.
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DiscussionProblem solving
Analysis of the data obtained from pre-tests and post-tests revealed that learners’ problem-solving performance in the post-tests improved over time after the intervention. Statistical results illustrate that there was a statistically significant difference between the experimental and comparison groups before the intervention (pre-tests), with the comparison group performing significantly better than the experimental group. However, the experimental group performed statistically significantly better than the comparison group after the intervention. In other words, it appeared that the intervention strategy in this study (see introduction of the discussion) had a positive influence on learners’ skills in solving word problems. As a result, a large practical significance was also noted. The experimental group improved significantly from the pre-test to the post-test.
Sense-making
The overall results of this study illustrate that learners’ performance changes dramatically when discussion is introduced into the mathematics classroom as a teaching strategy. When word problems are taught, teachers should consider carefully how to model the situation (or context), and also whether the information provided is relevant and sufficient for solving the problem (Säljö, Riesbeck & Wyndham, 2009). The results of this study showed that before the intervention (pre-test) learners had a tendency to respond to the problems even if the information given was irrelevant to the information needed to answer the given question. It is interesting that intercultural comparison studies show similar findings (Säljö et al., 2009; Verschaffel et al., 2000; Xin, 2009; Xin, Lin, Zhang & Yan, 2007).
The statistical results revealed a significant difference between the experimental group and comparison group. The experimental group appeared to show a tendency to consider reality marginally better than the comparison group. A large significant practical difference between the experimental group and the comparison group was also noted after the intervention.
The results of the study demonstrate that the introduction of discussion in the teaching and learning of word problems in mathematics not only had a positive effect on learners’ problem-solving performance, but also on their ability to consider reality when they had to solve word problems. The data generated in this study also suggest that whole-class discussion and problem-based approaches to the teaching of word problems can be applied appropriately and successfully (to certain degree) in second language teaching and learning settings, and can assist both mathematics teachers and learners to improve their knowledge of the real wor(l)d effects of mathematics problems.
ConclusionIn this study discussion as a strategy to improve second language learners’ word problem solving and sense-making skills was explored. The literature suggests, and initial observations appeared to confirm, that the difficulties in solving word-problems are related to effective pedagogical strategies that advance problem-based and whole-class discussion approaches to the teaching and learning of word problems in mathematics (Verschaffel et al., 2000, 2009). The results of this study seem to illustrate and substantiate that what teachers do serves as a fundamental component to raising learning outcomes (Douglas, 2009). In the study teachers’ pedagogical practices during the intervention resulted in lessons which involved group interaction and communication. The lessons were organised in such a way that there was some form of taking turns, where each member of a small group had to make their talk comprehensible to all (Heap, 1990). In this way the lessons in the mathematics classrooms came to be dominated by cooperative learning discussions where talk within teacher-learner interactions and/or learner-to-leaner(s) interactions were of high quality.
Statistical analysis of variance showed statistically significant and large practically significant evidence that the introduction of discussion in the teaching and learning of word problems in mathematics increases the problem-solving and sense-making performance of second language learners. The level at which the threshold of p was set in the study was 0.0005, which meant that there was a 0.05% chance that the results were accidental. The large practical significance noted in the study implies a research result that should be viewed as important for teaching practice in mathematics classrooms. The findings of this study suggest that when discussion was introduced into the mathematics classroom, better connections between classroom mathematics and out-of-school mathematics were made and that there was better integration between the learners’ formal written mathematical language and their informal spoken mathematical language. In fact, learners not only generated more computationally correct responses, but also produced more situationally accurate and appropriate solutions to real-wor(l)d problems in the post-test (or after the intervention).
Although the intervention of the study targeted only a limited number of teachers and schools in Port Elizabeth townships, and the conclusions drawn from the study cannot be generalised, the findings provide sufficient insights from which tentative recommendations for mathematics teacher development can be drawn. Analysis of the quantitative data suggests that promoting the introduction of discussion techniques in mathematics classrooms had benefits and in all probability promoted the participating learners’ problem-solving performance and significantly increased the likelihood of realistic consideration of word problem solving. However, to successfully implement such a strategy, teachers need appropriate fundamental skills and the necessary knowledge of managing and maintaining classroom discourses that allow the development of formal written mathematical language and the skills necessary for the negotiation of meaning within informal spoken mathematical language.
Acknowledgements
This research is supported by the Centre for Educational Research, Technology and Innovation at the Nelson Mandela Metropolitan University. Any opinions, findings and conclusions or recommendations expressed in this study are those of the authors and do not necessarily reflect the views of the Centre for Educational Research, Technology and Innovation.
Competing interests
We declare that we have no institutional and/or personal or financial relationships which may have inappropriately influenced us in writing this article.
Authors’ contributions
P.S. (University of the Witwatersrand) conducted all the research and wrote the manuscript. P.W. (Nelson Mandela Metropolitan University) was the promoter of the bigger study and made conceptual contributions to the bigger study and to the manuscript.
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