Abstract
Background: This study explored the impact and effectiveness of using erroneous examples for teaching Grade 9 algebraic linear equations.
Aim: The strategy was designed to allow learners to learn from their own mistakes and, therefore, help them close the gap(s) between their prior knowledge and the new knowledge.
Setting: This study was conducted with two Grade 9 classes from two secondary schools in the Gauteng Central district of South Africa.
Methods: A quasi-experimental design in the form of a pre-test–intervention–post-test was used with the experimental group (n = 31 in the pre-test and n = 22 in the post-test), while the pre-test and post-test with regular teaching were administered with the control group (n = 28 and n = 24). Both schools are public schools. The experimental school is in the former disadvantaged African township and the control school is in the semi-urban Indian township.
Results: The results revealed a significant difference between the pre-test and the posttest. The control group performed better than the experimental group, with the mean scores of x = 23.00000 and x = 18.83333, respectively. However, the experimental group’s mean scores have improved significantly in the post-test, from x = 1.12903 to x = 18.83333.
Conclusion: The findings revealed that the experimental group’s mean score increased better than the control group’s after the intervention.
Contribution: The study suggests that using erroneous examples effectively improves learners’ performance in solving algebraic linear equations.
Keywords: algebraic linear equations; errors; erroneous examples; mathematics education; learner performance.
Introduction
Mathematics is an essential school subject, potentially stimulating national economic and personal development (Capuno et al. 2019). However, algebra, one of the branches of mathematics, is crucial in understanding mathematical concepts and other science-related subjects. Mathematics teachers acknowledge that algebra is challenging to many learners. As a result, algebra plays a ‘gatekeeper’ role to learning more advanced mathematics. A conceptual understanding of algebraic linear equations at the Grade 9 level is essential for progress at higher levels of learning (Strauss 2012, as cited in Lee & Mao 2021). Algebra constitutes the most significant percentage (35%) of the mathematics content at the Grade 9 level (Department of Basic Education [DBE] 2022). Learning algebra at the Grade 9 level presents many learners with challenges, resulting in misconceptions, other associated errors and poor performance in National Senior Certificate Examinations (DBE 2022).
The majority of Grade 9 learners in South Africa perform below expectations in international and regional mathematics and science tests (Bowie et al. 2022); for instance, Trends in International of Mathematics and Science Studies (TIMSS) 2019 tests and Southern and Eastern African Consortium for Monitoring Educational Quality (SACMEQ) 2017 tests. Out of 39 countries that participated in the TIMSS 2019 test, South Africa (SA) came second from last with a very low score of 389 out of TIMSS’s average of 500. Moreover, in the SACMEQ 2017 test, South Africa took position 6 out of 15 participants but performed below the SACMEQ’s benchmark of 500. Even at the national level, performance in mathematics is unsatisfactory (DBE 2021–2024). Table 1 provides a summary of National Senior Certificate (NSC) performance from 2021 to 2024.
| TABLE 1: National senior certificate performance from 2021 to 2024. |
The DBE (2022) suggests learners’ poor performance in the final national mathematics examinations is because of their lack of conceptual understanding of essential algebraic skills they should have mastered by Grade 9. The DBE (2021–2024) reports that most learners fail to factorise simple algebraic expressions and misapply the distributive rules, indicating insufficient understanding of essential mathematics concepts. Mastering essential mathematics skills is crucial for understanding more advanced concepts in the future (Hall, Meyer & Rose 2012). Teachers’ teaching strategies could contribute towards learners’ poor performance.
Teaching mathematics concepts has been characterised by traditional teacher-centred approaches, which focus on drill and practice (Zhao & Acosta-Tello 2023). As a result, learners only know the mathematical rules without understanding why they work. A different teaching approach should be adopted to teach conceptual understanding.
While erroneous examples have been used in mathematics as an intervention strategy in the United States of America and Europe, little is known about the effectiveness of using erroneous examples as a teaching strategy in South Africa. This study used erroneous examples to teach Grade 9 algebraic linear equations and explore their impact on improving learners’ performance. To achieve this, the following hypotheses were assumed:
H0a: Using erroneous examples to teach Grade 9 algebraic linear equations does not have a significant effect on learner performance between the experimental and control groups.
H0a: Using erroneous examples to teach Grade 9 algebraic linear equations has no significant effect in the pre-test and post-test.
The researcher argues that using erroneous examples to teach Grade 9 linear equations can help learners recognise and correct common mistakes, leading to improved performance. An erroneous example is a worked example in which one or more steps are deliberately made incorrect. Using erroneous examples for teaching involves assigning learners into small groups of at most five learners, who work as detectives, identifying errors in erroneous examples and helping each other find correct ways of solving the problems.
Researchers (Zhao & Acosta-Tello 2023) have shown that when learners study erroneous examples, they develop critical thinking.
Rushton’s (2018) study on erroneous examples reveals that when learners study errors committed by fictitious learners, they reflect on their errors, which helps them reconstruct and correct them. In concurring, Metcalfe (2018) argues that using learners’ erroneous examples during lessons enables learners to identify their knowledge gap and consequently assists in closing it.
However, many Mathematics teachers are reluctant to discuss erroneous examples in class, fearing learners may be confused and reproduce errors when solving problems in the future. These teachers remember Skinner’s (1968) and other behaviourists’ studies, which showed that undesirable behaviour would be repeated if not punished (Metcalfe 2018). Nevertheless, constructivists believe individuals are responsible for constructing their knowledge from their mistakes (Nesher 1987). As such, the researcher argues that enabling learners to exhibit their errors and discussing erroneous examples allows learners to fill the gap(s) between what they already know and the new knowledge they are learning. Thus, using erroneous examples should be a teaching strategy that enables learners to develop a conceptual understanding of solving algebraic linear equations, forming a firm foundation for learning further concepts in mathematics and other related subjects.
Several studies have been carried out using erroneous examples for teaching and learning with positive results (e.g. Rushton 2018; Zhao, Dua & Singh 2017). However, despite positive results of using erroneous examples for teaching and learning, there is little research on using this strategy to teach Grade 9 algebraic linear equations in South Africa.
This study sought to explore the effectiveness of using erroneous examples for teaching Grade 9 algebraic linear equations. In exploring the effectiveness of using erroneous examples, the researchers utilised a quasi-experimental design to measure the significant effect between the pre-test and post-test and then the experimental and control groups.
Theoretical perspectives
This study is underpinned by Vygotsky’s zone of proximal development (ZPD), scaffolding, and Bandura’s observational theory of learning. Qin (2022) asserts that the ZPD is the difference between the child’s level of actual development, which is defined by what tasks they can solve independently, and the level of possible development, defined by the activities they can solve under the guidance of adults or in cooperation with more intelligent peers. However, Zaretsky (2021) states that the ZPD is defined by what a child can accomplish while collaborating with adults. The argument is that a learner may be able to solve a problem while being assisted by an adult today, and tomorrow, the learner can solve the problem alone (Zaretsky 2021). Furthermore, when teachers create appropriate learning conditions, learners’ problem-solving capabilities develop. A written test was used before implementing an intervention strategy to ascertain the learners’ ZPD in this study.
Scaffolding is a process where the teacher creates a situation that makes problem-solving easy for learners (Qin 2022). As the learner shows the capability to manage the situation, the teacher gradually withdraws support. Kusmaryono, Jupriyanto and Kusumaningsih (2021) describe scaffolding as those actions by teachers and peers meant to support, facilitate, assist and accelerate learners’ learning tasks. According to Lei et al. (2020), scaffolding is the transitory support given to learners while solving problems. In this study, the researcher used erroneous examples for scaffolding.
Observational learning is when someone gains new responses by observing others (Greer, Dudek-Singer & Gautreaux 2020), and it takes place when someone learns something by observing others doing it. According to Ahn, Hu and Vega (2020), observational learning has four main tenets: attention, retention, replication and motivation. They contend that paying attention is crucial in observational learning because simply exposing learners to the modelling situation does not guarantee that the individual will learn something (Ahn et al. 2020). Thus, for learners to benefit from observation, they should pay attention during lessons. In support of the above, He (2022) argues that selective attention is critical in observational learning as it determines what learners observe and what they will learn. Ahn et al. (2020) further argue that for observational learning to be effective, the model being observed should possess attractive attributes that lure observers to pay attention to the cues. He (2022) argues that the characteristics of the object of observation are also crucial in that learners may have an interest in observing novel characteristics or those similar to theirs. In this study, learners’ errors were used as models in erroneous examples, and this prompted learners’ eagerness to find out where they went wrong and why. The second tenet, retention, is crucial because the learners should retain what they observe to reproduce the same in the future. Retention is the process of actively transforming information from modelled events into codes that can be used to generate new information (Ahn et al. 2020). In this study, retention was enabled through group discussions on the possible causes of errors in erroneous examples so that errors can then be avoided in future situations. Replication is the learner’s ability to transform the coded observed behaviour into actions (Ahn et al. 2020). Motivation is also crucial in observational learning. Learners should be motivated to want to mimic what they observe. This motivation enables learners to pay attention during observation, and by paying attention, learners can code and retain information for future replication. Learners were also motivated to learn when they discussed erroneous examples and could see that they were not the only ones making errors when solving equations.
Observational learning played a crucial role in this study and helped the teacher and learners at the experimental school to observe the researcher, demonstrating how to use erroneous examples to help learners overcome their errors when solving algebraic linear equations. When studying erroneous examples, learners identified their errors and managed to control them in subsequent exercises.
Literature review
This study’s literature review focuses on learners’ errors and erroneous examples.
Errors in mathematics
Mathematics teachers know learners may write wrong answers in class exercises, homework or tests. Learners exhibit different types of errors for different reasons. According to Pan et al. (2020), an error is a mismatched fact or process for any standard. Meanwhile, Mulungye, O’Connor and Ndethiu (2016) state that an error is a deviation or mistake from the truth when applying a procedure to solving a problem. Mathaba and Bayaga (2019) state that when learners fail to see the connection between their prior knowledge and what they are learning, they commit many errors. According to Aliustaoglu, Tuna and Bider (2018), most learners overgeneralise their arithmetic knowledge when studying algebra, and this causes them to commit errors. Overgeneralisation can also be seen when learners solve equations such as (x + 4)(x + 7) = 30 and learners proceed to write x + 4 = 30 or x + 7 = 30, leading to getting wrong answers of x = 26 or x = 23. The learner has overgeneralised the rule used when the right-hand side equals zero. For instance, (x + 4)(x + 7) = 0, of which the answers are x = –4 or x = –7.
Meanwhile, Makonye and Khanyile’s (2015) study grouped errors into three categories: careless errors, errors and random errors. They (Makonye & Khanyile 2015) argue that learners commit systematic errors when they lack a complete understanding of the essential concept information. Implicitly, they (Makonye & Khanyile 2015) contend that systematic errors are learned errors. This is because learners commit the same type of errors in similar situations. However, Luneta and Makonye (2010) describe systematic errors as misconceptions and further state that they indicate learners’ flawed line of thinking, which is revealed in their written or spoken works.
Luneta and Makonye (2010) argue that learners commit systematic errors when they attempt to solve the problem ‘6x – 12x and get the answer as 6x instead of –6x. In lower grades, learners would use their prior knowledge, where they learned that more significant numbers could not be subtracted from smaller ones. Because borrowing is impossible, the learner subtracted the smaller number from the bigger one but could not use the correct sign in the final answer.
Careless errors are wrong answers, which learners can quickly correct when they revise their work; for instance, when learners simplify an expression like 15x – 3x and get the answer as 2x. Parwati and Suharta (2020) contend that when learners commit calculation (careless) errors, they can easily correct themselves when they recheck their work. They (Parwati & Suharta 2020) further argue that learners exhibit substantial errors when they have insufficient understanding of symbols, incorrectly apply processes and even fail to plan how to solve the problem.
Learners who fail to observe the rules of operations might commit transformation errors (Hansen et al. 2020). For instance, when learners incorrectly apply the order of operations (BODMAS) when simplifying expressions involving two or more operations. Makonye and Hantibi’s (2014) study observed that most learners are confused about multiplication and subtraction when working with negative numbers. For example, –7 × –8 = –15, and they fail to consider the order of operations and get answers like 3 – 3 × 4 = 0.
Another source of learner errors is when new knowledge interferes with what is being learned. For instance, when learners study algebraic expressions immediately after learning the laws of exponents (Gumpo 2014). Learners may misuse the laws of operations when simplifying expressions like 3x + 5x and get 8x2 instead of 8x.
Erroneous examples
Erroneous examples are worked-out examples in which a single or multiple steps in the work are deliberately made incorrect. A study by Richey et al. (2019) observed that learners’ cognition develops when they compare correct and erroneous examples. Exposing learners to erroneous examples may motivate learners who may not be willing to participate during lessons for fear of making mistakes. When learners become aware that making mistakes in mathematics is expected, they may feel that they are not the only ones struggling with mathematics.
Rushton (2018) observed that when learners study the hypothetical errors committed by other learners, they reflect on their errors, which enables them to reconstruct and correct them. Additionally, when learners study erroneous examples, they might think about their own thinking (metacognition) when analysing why erroneous examples might be incorrect. According to Richey et al. (2019), studying erroneous examples might assist learners to revise and correct their understanding of concepts. Moreover, Durkin and Rittle-Johnson (2015) observed that learners’ misconceptions were reduced and their conceptual and procedural understanding of decimal magnitude was improved when they studied erroneous examples. Therefore, this study investigated the effectiveness of erroneous examples for teaching algebraic linear equations.
Methods
This study investigated the effectiveness of erroneous examples for teaching Grade 9 algebraic linear equations using a quasi-experimental design in a quantitative research approach. Pre-tests and post-tests were used to measure the difference in performance between the experimental and control groups in solving algebraic linear equations. According to Creswell (2014), through quantitative methods, researchers can analyse theories by collecting data that may confirm or refute the theories.
To explore the effectiveness of erroneous examples for teaching Grade 9 algebraic linear equations, the researcher established two study groups: the experimental group and the control group. Both study groups wrote the pre-test and post-test. However, the experimental group received the treatment in the form of erroneous examples to test the hypotheses: null hypothesis, which states that using erroneous examples to teach Grade 9 algebraic linear equations does not have a significant effect on learner performance between the experimental and control groups, and the alternative hypothesis, which states that using erroneous examples to teach Grade 9 algebraic linear equations has a significant effect on learner performance between the experimental and control groups.
The quantitative approach used in this study warranted a positivist paradigm because this allowed the researcher to appraise and evaluate the theory and establish the relationship between the chosen variables (Creswell & Creswell 2018).
The learners and teachers used in this study were purposively selected because their characteristics were relevant to the study (Andrade 2021). The learners at the experimental school were not performing well, while learners at the control school were performing well. Moreover, the experimental school was located in the former disadvantaged township, and the control school was located in the semi-urban township. Both groups of learners had different sociocultural backgrounds and languages. Although learners had different sociocultural and language backgrounds, the language of teaching and learning at both schools is English. However, to cater for the language diversity among learners, clear and simple language was used in word problem questions 7 and 8. Learners were also allowed to ask for an explanation when they did not understand the questions. Most learners at the two study schools used public transport to and from school. This mode of transport affects learner attendance when there is a taxi strike.
The intervention process for this study was implemented as described below.
Initially, direct instruction was used to teach the EG to solve algebraic linear equations by applying inverse operations and the distributive rule. The teacher walked around, taking note of learners’ incorrect solutions in preparation for class discussions on using erroneous examples for learning algebraic linear equations. The teacher writes learners’ incorrect solutions on the board and leads class discussions during which learners identify mistakes and the possible reasons for the mistakes. Learners then solved the problems correctly and were given two to three problems as homework to reinforce the concepts.
Data collection
This study utilised a pre-test–intervention–post-test design in which the pre-test was administered to assess the learners’ baseline knowledge before the intervention was applied to the experimental group, and the post-test was used to assess whether the strategy resulted in learners gaining knowledge on the content that was taught (Alam 2019). The data for this study were collected through a pre-test and a post-test. After administering the intervention to the experimental group, both groups wrote the post-test. The experimental group consisted of 31 learners (N = 31, 10 boys and 21 girls), and the control group consisted of 28 learners (N = 28, 13 boys and 15 girls) who participated in the pre-test. Because of transport problems on the day the post-test was administered, the experimental group comprised only 12 learners (N = 12, five boys and seven girls), while the control group consisted of 24 learners (N = 24, 10 boys and 14 girls). The learners were tested in solving Grade 9 algebraic linear equations. Immediately after administering the pre-test, the researcher marked the learners’ scripts to understand how the two study groups performed. After the pre-test, the researcher spent 6 weeks with the teacher of the experimental group, implementing twelve intervention lessons.
The teacher of the experimental group and the researcher designed worksheets using erroneous examples, which learners used during group discussions. The study was conducted in three phases. In the first phase, learners were put into small groups of five by picking cards numbered from one to five, and all learners who picked the same number belonged to the same group. However, it was observed that during the first phase, learners of almost the same ability had been assigned to the same group, and group discussions were almost impossible in groups with less competent learners. During the second phase, learners were regrouped into mixed-ability groups. However, it was also observed that most learners had difficulties explaining errors created by fictitious learners. During the third and last phase, the teacher of the experimental group and the researcher designed worksheets using learners’ own errors. While still working in mixed-ability groups, it was observed that learners were able to explain their erroneous reasoning during group discussions, which helped them correct their understanding. High-ability learners were also helped by explaining how to solve some of the problems of low-ability learners (Pozas, Letzel & Schneider 2020).
After the intervention, a 2-week washout period was allowed before the post-test was administered to minimise the effects of the intervention in the post-intervention test (Verma 2021). After the washout period for the experimental group, a post-test was administered to both groups on the same day, simultaneously. The researcher requested the teachers of the respective groups to be invigilators. The post-test was administered to gauge the intervention’s effectiveness and to compare the performance of the two groups. The results of the pre- and post-tests for the experimental group were compared to those of the control group.
Furthermore, the test items were piloted before being used in the main study. In the pilot study, it was noted that the initial time allocated was not enough because most learners could not finish the test. Additionally, it was observed that questions 7 and 8 were very difficult for most learners, and these were replaced with relatively easier ones in the main study. After marking the scripts, the researcher gave the scripts to the teachers to check for marking accuracy. The two teachers were also given the statistical analysis to approve and confirm the explanations.
Data analysis
Microsoft Excel was used to collect, collate and manage the raw data, while IBM SPSS version 28 was used for statistical analysis to understand the learners’ performance in the pre-test and post-test. The researcher used a t-test to determine if there was a statistically significant difference between the mean scores for the experimental and control groups. Additionally, a t-test was used to determine if the average scores of the two groups depicted that the sample used in this study did not originate from the same group of participants. Mean scores were used to explain and interpret the outcomes of the two groups at the 95% confidence limit (2-sided). Consequently, the outcomes were declared statistically significant for all p-values less than 0.05.
A t-test was used in this study because the data satisfy the conditions of normality, homogeneity and independence (Liang, Fu, & Wang 2019:21). The data were independent because they came from two different groups of the same type. A t-test was used because there is a small difference between the standard deviations and the means.
Ethical considerations
The researcher applied for and obtained the ethical clearance certificate (reference no: 2022/07/06/57344574/22/AM) from the UNISA College of Education Ethics Review Committee and obtained the research permission letter from the Gauteng Department of Education. The researcher also obtained approval letters from the principals of the two schools that participated in this study. The teachers and their learners were requested to sign consent and assent forms, respectively, as evidence of agreement. The learners’ parents also signed consent letters agreeing that they allowed their children to participate in this study.
Results
The pre-test and post-test data were analysed descriptively in the section below. Microsoft Excel was used for data collection and organisation, while IBM SPSS version 28 was used to analyse the information statistically. The experimental and control groups were compared using a t-test. The results were interpreted as significant if the p-value was less than 0.05 at the 95% confidence limit.
The following conditions were considered:
- If the p-value < 0.05, then the outcomes were considered significant.
- If the p-value ≥ 0.05, then the outcomes were considered insignificant.
The mean ( ) scores were used to test the statistically significant effect between the experimental and control groups’ performance in the pre-test and post-test, and these are discussed below on a question-by-question basis from question 1 to question 8.
Table 2 shows how EG and CG performed before and after intervention from Q1 to Q8. The results revealed different mean values for Q1 before and after intervention. The outcome showed that the EG performed significantly differently before and after intervention in Q1 (p-value < 0.001), which is less than (p-value = 0.05) at the 95% confidence limit. The results showed that the learners in the EG improved significantly in Q1 from a mean ( = 0.00000) before intervention to a mean ( = 4.58333) after intervention. The significant performance improvement in Q1 for the EG suggested the intervention’s positive impact on learners’ performance. However, the analysis of the results also indicated that the control group did not perform significantly differently with the mean score ( = 2.50000) on the pre-test, and the mean score ( = 3.45833) in the post-test for Q1 (p-value = 0.134), which is greater than (p-value = 0.05) at the 95% confidence limit.
| TABLE 2: Pre-test and post-test (Q1 to Q8 test results). |
The analysis of results for Q2 indicated that the experimental and control groups performed significantly differently, recording a p-value < 0.001 on both the pre-test and post-test. Both groups of learners showed improved performance in solving algebraic equations with variables on the two sides of the equal sign. Moreover, the results analysis also indicated that the experimental group outperformed the control group, suggesting that the intervention impacted learners’ performance positively.
Data for Q3 indicated that the experimental group performed significantly differently in the post-test after intervention (p-value < 0.001), which is below (p-value = 0.05) the 95% confidence limit. After the intervention, most learners in the experimental group could solve algebraic linear equations involving brackets. Most learners were able to apply the distributive laws to remove brackets and group like terms correctly. Similarly, learners in the control group also performed significantly better in the post-test versus performance in the pre-test (p-value < 0.001) which is less than (p-value = 0.05) at the 95% confidence limit. Although both study groups improved significantly in Q3, the experimental group recorded a better improvement than the control group, suggesting that using erroneous examples as an intervention strategy positively affected learners’ performance.
The results analysis for Q4 showed that the experimental group performed significantly differently in both the pre-test and post-test (p-value = 0.007), which is less than (p-value = 0.05) at the 95% confidence limit, suggesting that the intervention was effective in improving learners’ performance in solving algebraic linear equations on each side of the equal sign. Moreover, the control group also performed significantly differently in the pre-test and post-test (p-value < 0.001), which is less than (p-value = 0.05)at the 95% confidence limit (2-sided).
The analysis of results for Q5 showed that the experimental group performed significantly differently in the pre-test and post-test (p-value = 0.016), which is lower than (p-value = 0.05) at the 95% confidence limit. The experimental group recorded a higher mean score ( = 1.25000) in the post-test versus the mean score ( = 0.12903) in the pre-test, indicating that the experimental group improved performance after the intervention. The better performance by the experimental group in the post-test suggested that the strategy effectively enhanced learners’ performance in solving algebraic linear equations with fractions. However, the analysis of the results for Q5 depicted that the control group performed insignificantly differently in the post-test (p-value = 0.971), which is higher than (p-value = 0.05) at the 95% confidence limit.
The analysis of results for Q6 showed that the experimental group performed significantly differently in the pre-test and post-test (p-value = 0.005), which is less than (p-value 0.05) at the 95% confidence limit. Correspondingly, the analysis of results for Q6 revealed that the control group also performed significantly differently on the two tests (p-value = 0.002), which is less than (p-value = 0.05) at the 95% confidence limit. However, although both study groups performed significantly differently in the two tests, the experimental group was outperformed by the control group, showing a better improvement with a post-test mean ( = 4.12500) compared to the post-test mean ( = 1.83333) for the experimental group. Moreover, the higher mean score ( = 1.83333) for the experimental group after the intervention compared to the mean ( = 0.16129) before the intervention suggests that the intervention positively impacted learners’ performance.
The results analysis for Q7 showed that the experimental group did not perform significantly differently in the two tests (p-value = 0.179), which is greater than (p-value = 0.05) at the 95% confidence limit. Equally, the analysis of results showed that at a 95% confidence limit, the control group performed insignificantly differently in the pre-test and post-test (p-value = 0.272), which is greater than (p-value = 0.05) at the 95% confidence limit. Although the experimental group performed insignificantly differently in the two tests, there was a performance improvement after the intervention, showing a mean score ( = 0.58333), which is higher compared to the mean score ( = 0.09677) before the intervention. The higher mean score after intervention recorded by the EG suggested that the strategy positively affected learners’ performance. The results analysis for Q7 also showed a decline in performance for the control group from a mean score ( = 1.03571) on the pre-test to a mean score ( = 0.54167) in the post-test. The low mean scores in the results analysis for Q7 revealed that learners in the two study groups still had difficulties solving word problems involving algebraic linear equations.
The results analysis for Q8 indicated that the experimental group performed insignificantly differently in the pre-test and post-test (p-value = 0.858), which is greater than (p-value = 0.05) at the 95% confidence limit. However, the experimental group showed a decline in performance, mean score ( = 0.16667) after intervention against the mean score ( = 0.19355) before the intervention, suggesting that the strategy did not positively affect learners’ performance in Q8. Although the control group performed insignificantly differently in the two tests, there was an improvement from the mean ( = 0.64286) in the pre-test to mean ( = 1.20833) in the post-test, suggesting that learners gained knowledge from traditional teaching strategies.
Comparison between experimental and control groups in the pre-test and post-test
The total mean score between the experimental and control groups was used to prepare the pre-test and post-test summary outcomes, and Table 3 shows the findings.
| TABLE 3: Pre-test and post-test summary. |
The analysis of results, as shown in Table 3, indicated that the experimental and control groups performed significantly differently in the pre-test (t = – 4.370; p-value < 0.001), which is less than (p-value = 0.05) at the 95% confidence limit. The control group recorded a higher mean score ( = 12.82143) against the mean score ( = 1.12903) for the experimental group. However, the two study groups performed insignificantly differently post intervention (t = –1.329; p-value = 0.195), which is greater than (p-value = 0.05) at the 95% confidence limit. Although the two study groups did not perform significantly differently after the intervention, the control group outperformed the experimental group. The more significant p-value (p = 0.195) that is greater than (p < 0.05) at the 95% confidence limit after the intervention suggests that there is no significant difference between the pre-test and post-test when teaching Grade 9 algebraic linear equations using erroneous examples (H0). Despite there being no significant difference in performance between the experimental and control groups in the post-test, the experimental group significantly improved performance in the post-test ( = 1.12903) in the pre-test to ( = 18.83333) on the post-test, increasing by ( = 17.7043) versus the control group’s improvement of (, from ( = 12.82143) on the pre-test to ( = 23.00000) in the post-test indicated that the intervention improved learners’ performance in solving algebraic linear equations. The results also revealed that learners in the control group gained knowledge from the traditional teaching approaches. However, the experimental group gained more knowledge from the intervention strategy.
Discussion of findings
The study’s outcomes indicated that the experimental and control groups performed significantly differently in solving Grade 9 algebraic linear equations in the pre-test (p-value < 0.001), which is less than 0.05 at the 95% confidence limit. This study ascertained the effectiveness of erroneous examples in the teaching and learning of algebraic equations. Barbieri, Miller-Cotto and Booth (2019) observed that using erroneous examples draws learners’ attention to the underlying factors that make the answer incorrect and refines the same when solving problems. The experimental group recorded a dismal mean score of ( = 1.12903) compared to the control group’s mean score of ( = 12.82143). The findings also indicated that both study groups lacked the essential knowledge and skills Makonye and Khanyile (2014) for solving Grade 9 algebraic linear equations, as indicated by the low mean scores. No learner in the experimental group could answer question 1 before the intervention, and learners’ performance in the rest of the questions was far below expectations. The DBE, in several of their National Senior Certificate diagnostic reports, has echoed a lack of understanding of the essential knowledge and skills as the reason for poor performance (DBE 2022).
The post-test results indicated that the two groups did not perform significantly differently (p-value = 0.195) at the 95% confidence limit greater than 0.05, showing that the experimental group managed to reduce the performance gap with the control group. The more significant p-value (p = 0.195) indicated no significant difference between the two groups’ means, implying that they performed the same on the post-test after the experimental group received some treatment. The results indicated that using erroneous examples as an intervention strategy for teaching Grade 9 algebraic linear equations effectively enhanced learners’ performance. Although the experimental group’s performance improved post-intervention, the control group also improved performance without the intervention. However, the experimental group’s performance improvement was better because it received some treatment. The experimental group improved from a mean score of = 1.12903 in the pre-test to a mean score of = 18.83333, an increase of = 17.7043 compared to the control group, which improved from the pre-test mean score of = 12.82143 to a post-test mean score of = 23.00000, increasing by = 10.17857. The improvement in performance by the experimental group suggests that learners might have been motivated to pay attention during lessons, which allowed them to retain what they had been taught (He 2022).
The more significant experimental group’s mean score post the intervention indicates the intervention strategy’s effectiveness in helping learners gain skills and knowledge to solve algebraic linear equations. Using erroneous examples in small groups may have helped learners identify mistakes and develop correct strategies collaboratively, boosting their confidence and problem-solving performance (Crisianita & Mandasari 2022).
Limitations of the study
The findings of this study could be limited to the study’s sample because only two schools out of 68 schools, two teachers, and their learners participated in the study. Moreover, not all learners who wrote the pre-test wrote the post-test because of transport problems on the day of writing the post-test. Had all learners written the post-test, different results could have been reached. Moreover, this study’s focus was on algebraic linear equations. Had other areas been used, a different result could have been reached.
Suggestions for future research
The results demonstrated that when learners use erroneous examples, they take responsibility and learn effectively from their mistakes. However, only two teachers and two schools out of 68 schools took part in this study. Future studies could be carried out with more teachers and more learners. Additionally, future research could be done to investigate the impact of erroneous examples in other subjects.
Conclusion
The findings showed that erroneous examples help learners take responsibility and learn effectively from their mistakes. In small mixed-ability groups, learners act like detectives, identifying errors and building collaboration skills by helping each other solve problems. Moreover, teachers should ascertain learners’ prior knowledge to enable mixed-ability grouping. Additionally, the findings showed that learner participation and teacher support for struggling students could improve performance in solving algebraic linear equations. Erroneous examples improved learners’ algebraic performance, raising the mean score from = 1.13, with a total score of 16 out of 60, to = 18.83, with 33 out of 60 after the intervention.
Acknowledgements
The authors acknowldge the participants of this study who shared data to complete the research. This article is partially based on Julius Gwenzi’s thesis titled ‘Using erroneous examples for teaching Grade 9 Algebraic Linear Equations at a School in Johannesburg Central District’ towards the degree of Master of Education at the University of South Africa, South Africa, May 2024, with supervisor Tšhegofatšo P. Makgakga. It is available here: https://uir.unisa.ac.za/server/api/core/bitstreams/b95d519a-9868-44f7-a454-231445821f88/content.
Competing interests
The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.
Authors’ contributions
J.G. contributed to the conceptualisation, methodology, analysis, investigation and writing of the original draft. T.P.M. contributed to the conceptualisation, methodology, analysis, investigation, writing and review of the original draft in a supervisory capacity.
Funding information
This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.
Data availability
The data that support the findings of this study are not openly available and are available from the corresponding author, T.P.M., upon reasonable request.
Disclaimer
The views and opinions expressed in this article are those of the authors and are the product of professional research. They do not necessarily reflect the official policy or position of any affiliated institution, funder, agency or that of the publisher. The authors are responsible for this article’s results, findings and content.
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