Abstract
School algebra serves as the language of mathematics and a foundational subject for learning advanced mathematics courses. This makes developing learners’ proficiency in algebra the most desirable instructional goal of school mathematics. Despite having such importance emphasis, however, studies indicate that the vast majority of learners are characterised by inadequate mathematics proficiency levels in general and in the algebra syllabus topics in particular. Consequently, this quasiexperimental study attempted to investigate the efficacy of using discoursebased instruction as an instructional approach to developing proficiency in algebra unit topics. One hundred and six (N = 106) Grade 11 learners participated in the study and were randomly grouped into an experimental group (n = 52) and a control group (n = 54). Using a test instrument that consisted of 24 Raschvalidated items, both pretest and posttest data were collected from both groups under similar conditions. The MannWhitney U statistical analysis of the pretest data revealed no significant difference between the control and experimental groups. The MannWhitney U analysis performed on the posttest data demonstrated that the experimental group scored significantly higher in the posttest scores when compared to the control group after the intervention. The study findings provided evidence of the efficacy of discoursebased instruction over teachercentred instruction for developing learners’ algebra proficiency.
Contribution: The study has contributed to the conceptual and practical understanding of how discoursebased instruction can be used to concretise learners’ proficiency in basic algebra.
Keywords: Classroom discourse; discoursebased approach; mathematical proficiency; experimental study; teaching method.
Introduction
This manuscript reports the findings of a doctoral study (Legesse, 2022) that investigated the effectiveness of discoursebased instruction of algebra and function syllabus topics. In this dynamically changing world, mathematics plays an essential role in scientific investigation, technological progress, and every walk of life (Suh & Seshaiyer, 2013). Mathematics educators and researchers advocate the teaching of mathematics for developing mathematically proficient citizens (Suh, 2007; Suh & Seshaiyer, 2013). For a learner to be mathematically proficient in a domain of mathematics, the learner must demonstrate conceptual understanding (comprehension of mathematical concepts, operations, and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently, and appropriately), strategic competence (the ability to formulate, represent, and solve mathematical problems), adaptive reasoning (the capacity for logical thought, reflection, explanation, and justification), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile), coupled with a belief in diligence and one’s efficacy (Kilpatrick et al., 2001, p. 116).
Like many other countries around the world, Ethiopia has envisioned the teaching and learning of mathematics to enable all learners to be mathematically competent (Ethiopian Ministry of Education [MOE], 2010a). Despite this vision, the findings of the Ethiopian National Assessment and Examination Agency (ENAEA, 2010) showed that the majority of Grade 10 and Grade 12 learners scored below the average passing mark of 50% in the mathematics assessment examination. The assessment results further revealed that inadequate mathematical proficiency across different content domains of school mathematics has remained a persistent problem for the vast majority of learners (Legesse et al., 2020).
To alleviate this problem, research literature in mathematics education recommends different forms of instructional approach for enabling learners to develop mathematical understanding, problemsolving, and thinking (Bennett, 2014; Bradford, 2007; Cross, 2009; Manouchehri & Enderson, 1999; National Council of Teachers of Mathematics [NCTM], 1991, 2014). Among the recommendations was the use of classroom discourse orchestrated around mathematical tasks that foster learners’ understanding of mathematical ideas, communication, and problemsolving skills (Bennett, 2014; Cross, 2009; Legesse et al., 2020; NCTM, 1991; Smith & Stein, 2011; Walshaw & Anthony, 2008). Moreover, the literature suggests the engagement of learners in mathematical discursive practices of explaining, justifying, listening to, sharing, comparing, evaluating, and interpreting each other’s mathematical ideas and reasons and constructing convincing arguments (Bennett, 2014; Rumsey & Langrall, 2016; Smith, 2018; Smith & Stein, 2011) to enhance their mathematical proficiency and achievement results (Anthony & Hunter, 2017; Bennett, 2014; Bradford, 2007; Smith, 2018). However, evaluation studies of the effectiveness of discoursebased mathematics teaching and learning on targeted academic outcomes in different cultural contexts are scant (Bradford, 2007; Rumsey & Langrall, 2016). In particular, there is limited practical effort in using mathematical discourse as a teaching strategy in Ethiopian school classrooms.
In an Ethiopian context, poor performance in mathematics is often attributed to teacherdominated classroom teaching practices (Dhoj & Verspoor, 2013). Deressa (2004) characterised teacherdominated instruction as being associated with poor social interactions and communication between teachers and learners and among learners over the learning content. One approach to improving learners’ mathematics proficiency learning outcomes could be exploring the potential of researchinformed teaching approaches, namely the discoursebased approach. Consequently, the purpose of this study was to investigate the effectiveness of discoursebased instruction of algebra and function topics on Grade 11 learners’ mathematical proficiency compared to traditional lecturebased instruction.
School algebra serves as the language of mathematics and a foundational subject for learning higher mathematics, science subjects, and engineering courses (Grønmo, 2018). Improving the classroom instruction of algebra enables learners to pursue learning advanced mathematics; hence, proficiency in algebra plays an important role in college and university mathematics courses (Grønmo, 2018; Kilpatrick et al., 2001). The present study, therefore, attempted to determine if discoursebased instruction enables learners to enhance their proficiency in algebra unit topics based on the Ethiopian Grade 11 mathematics syllabus. The research question was: Are there differences in mathematical proficiency scores between Grade 11 learners who were taught algebra unit topics using discoursebased instruction and those learners who were taught the same content using the traditional lecture method before and after the treatment?
This study used the term discourse to refer to dialogic talk that promotes the teaching and learning of mathematics through engaging in sociocultural practices, such as questioning, reasoning, listening, sharing, explaining, and justifying (Alexander, 2008; Brown & Hirst, 2007; Steele, 2001). A discourse counts as mathematical if it is about mathematical objects and involves the use of symbols, notations, representations, and definitions (Moschkovich, 2007; Sfard, 2008). Mathematical discourse includes the communication of definitions, rules, procedures, ideas, theorems, and proofs, verbal explanations of problemsolving strategies, and justification of reasons (Shilo & Kramarski, 2018).
Theoretical background and literature review
The present study is guided by a sociocultural perspective. A sociocultural perspective underlines the role of dialogic talk in the construction of knowledge in a social setting (Alexander, 2008). Through engaging in dialogic talk that promotes social interactions, learners develop mathematical understanding and construct meanings by explaining how mathematical tasks can be accomplished or how procedures work, and by challenging each other’s ideas, comparing different solution strategies, sharing ideas and reasoning (Bradford, 2007; Gravemeijer & Cobb, 2006; Hiebert & Wearne, 1993; Lampert, 1990; Steele, 2002).
Mathematical discourse as a vehicle for learning promotes the construction of mathematical understanding by
helping learners to clarify and organize their thoughts, facilitating personal and collective sensemaking, supporting building connections between representations and multiple strategies, enabling learners to use others as a resource of ideas to challenge and broaden understanding, and helping learners learn the mathematical language. (Anthony & Hunter, 2017, p. 101)
Hoyles (1985) describes mathematical understanding as the ability to
form a view of the mathematical idea, step back and reflect upon it, use it appropriately and flexibly, communicate it effectively to another, reflect on another’s perspective of the idea, and incorporate another’s perspective into one’s own [schema] or challenge and logically reject this alternative view. (p. 212)
This also involves asking clarification questions, sharing ideas and reasoning, agreeing and disagreeing with each other’s ideas, and respecting and listening to each other’s ideas to promote the construction of mathematical knowledge (Chapin et al., 2003).
Engaging learners in collaborative classroom discourse that incorporates mathematical discursive activities such as agreeing and disagreeing with others’ ideas, explaining their reasoning and thinking, and discussing and comparing solution methods promotes the development of the understanding of mathematical topics (Anthony & Hunter, 2017; Kazemi, 2008). To create a classroom environment that invites all learners to participate, the teacher should build confidence in learners that everyone can contribute to the classroom lesson by informing all learners to respect each other’s responses, share, and exchange their ideas, listen to each other’s responses, appreciate making mistakes as an opportunity for learning, and to question each other (Cobb, 1994; Maguire & Neill, 2006; Yackel & Cobb, 1996).
Sociomathematical norms refer to what counts as an acceptable mathematical explanation and justification of the ‘why’ and ‘how’ aspects of teachers’ and learners’ activities that are specific to the mathematical discussion (Cobb, 1994; Maguire & Neill, 2006; Stephan, 2014; Yackel & Cobb, 1996). The teacher and learners may set criteria for deciding what is regarded as an acceptable explanation, or a different solution method (Stephan, 2014). The teacher is responsible for establishing a ‘psychological safety’ learning environment in which learners feel free to respectfully express their thinking and reasoning during group discussions of mathematical ideas (Cobb et al., 1990). The teacher should create a classroom culture in which learners are expected to explain and justify their solution strategies and reasoning (Cobb et al., 1990).
Regardless of the method of instruction, sociomathematical norms are present in all classrooms; however, what counts as an acceptable mathematical explanation and solution in traditional classrooms may not be a mathematically elegant and sophisticated explanation and solution in learnercentred classrooms (Stephan, 2014; Yackel & Cobb, 1996). For instance, just describing the solution procedure for solving a quadratic equation might be counted as an acceptable explanation in traditional classrooms (Stephan, 2014). Setting the use of accurate mathematical terms, symbols, representations, syntax, rules, and notations as sociomathematical norms can help learners develop mathematical language proficiency (Pourdavood & Wachira, 2015; Summers, 2012). Questions that can be used to establish sociomathematical norms include: How can you prove that your answer is right? Can you prove it in more than one way? How is your solution strategy different from that of another learner’s solution strategy? Do you agree or disagree with another learner’s solution? Why? Why does strategy A work? Why does strategy B not work? (Kazemi, 2008, p. 414).
This study conceived discoursebased mathematics instruction as the teaching and learning of mathematics through engaging in dialogic talk orchestrated around carefully designed tasks in which the topics to be learned are embedded. Discoursebased instruction can be characterised by the engagement of learners in dialogueelicited tasks wherein the learnerteacher, learnerlearner, and learnergroup interactions are anchored on discourse practices of challenging each other’s ideas, sharing ideas, agreeing and disagreeing with solution strategies and ideas, comparing solution procedures, and explaining problemsolving strategies (Legesse et al., 2020, 2021). In such classroom learning environments, mathematics learning is more than ‘appending’ new knowledge to existing knowledge; it involves the reconstruction of understanding and building a web of interconnected conceptual understanding that fosters the transfer of learning to new contexts (Hiebert et al., 1997).
Some evaluation studies indicate that discourseoriented forms of instruction positively influenced mathematics learning outcomes (Bradford, 2007; Cross, 2009; Legesse et al., 2020; Sepeng & Webb, 2012; Smith, 2018). Bradford (2007) found that discourseoriented instruction in prealgebra classes helped lowachieving learners improve their mathematics achievement and problemsolving skills. A quasiexperimental study by Legesse et al. (2020) examined the effects of discoursebased mathematics instruction on Grade 11 learners’ acquisition of conceptual and procedural knowledge of probability and statistics topics in an Ethiopian secondary school. Legesse et al. found that discoursebased instruction of probability and statistics increased learners’ knowledge of concepts and procedures. Moreover, the discussionbased teaching strategy improved the experimental group of Grade 9 learners’ word problemsolving performance when compared to the control group taught with the traditional lecture method (Sepeng & Webb, 2012).
Research material and methods
The present study employed a quasiexperimental design with a pretest and posttest control group (Shadish et al., 2002). A quasiexperimental design with a control group is an appropriate method of inquiry when the primary intent of the study is to evaluate the effectiveness of an instructional intervention (Goodwin, 2009). Such a quasiexperimental design is also appropriate when the randomisation of each participant is impossible for practical reasons (Gall et al., 2003). Thus, a quasiexperimental study with a pretest and posttest control group established a causeeffect relationship between discoursebased instruction and proficiency in algebra unit topics.
Participant selection
This study was conducted with Grade 11 learners in a randomly selected public school in Bahir Dar, Ethiopia. The school principal provided permission to conduct the study with the intended groups of learners. The study lasted for about 10 class weeks. During each class week, there were five lessons and each lesson was 42 minutes.
In collaboration with the head of the mathematics department at the participating school, two comparable Grade 11 mathematics teachers, in terms of professional and academic qualifications, who were willing to be involved in the study were recruited. The learner population in the participating school has similar ethnicity and socioeconomic status. All Grade 11 learners enrolled in the natural science stream were allocated into 14 sections. After this, each section was assigned a unique natural number. Fourteen equalsize pieces of paper were prepared and each piece was numbered 1 through 14. Then each piece was folded and placed in a carton. After shaking the box and mixing the pieces well, an independent person randomly picked two pieces of paper one after the other without replacement, representing two sections of Grade 11 learners. The two sections were assigned to either the control group (n = 54) or the experimental group (n = 52). One hundred and six Grade 11 learners (N = 106) participated in this study. After being briefed about the study, all learners in the selected classes offered their verbal consent for participation. Names and details of participants were kept confidential while analysing the data and reporting the results.
Based on prior studies in a similar context (e.g. Legesse et al., 2020), there was an anticipation that the data would violate the assumptions for the parametric tests. Accordingly, the MannWhitney U test was chosen for analysing and comparing differences in the pretest and posttest scores between the control and experimental groups. Using the evidence from the literature (e.g. Happ et al., 2019; WMWssp, n.d.), a sample size calculation based on the expected medium effect size (r = 0.349) for the MannWhitney U test (Mangiafico, 2016), power (1–β):0.80, and twosided type I error (α: 0.05) produced an estimated total sample size of 112 participants. This power analysis indicated that the sample size of 106 was sufficient to detect a statistically significant difference between the control and experimental groups on the continuous dependent variable.
Locating the control and experimental groups in the same school controls differences in the physical conditions (Fraenkel et al., 2012) and avoids possible threats due to differential selection (Gall et al., 2003). On the other hand, it might cause treatment contamination. Danga and Korb (2014) examined the effect of treatment diffusion using an educational experimental design in a Nigerian secondary school where learners had maximum opportunities to exchange information and talk about their classroom teaching and teach each other. The study found that placing the control group and the experimental group in the same school did not result in a potential treatment diffusion to affect learners in the control group (Danga & Korb, 2014). In this study, the authors made efforts to prevent or reduce potential treatment diffusion (Shadish et al., 2002) between the control group and the experimental group by making the groups blind to one another and by taking the groups to different classes in physically separated buildings (Rhoads, 2011). Under such circumstances, it was most unlikely there would be information leakage about the nature of treatment between the control and experimental groups.
Implementation of discoursebased mathematics instruction
Two comparable mathematics teachers (both male) in terms of academic profiles and professional qualifications were both chosen from the Grade 11 classes involved in the study. After recruitment, the two teachers received two different training sessions of four hours each on how to design and implement discoursebased lessons. The training involved the design of discourseelicited tasks, crafting different questioning strategies, facilitation strategies of learners’ classroom discourse, and setting up and maintaining social and sociomathematical norms. During the training session, the teachers were maintained blind to the hypotheses of the study (Rhoads, 2011). The training was offered in such a way that the teachers did not disclose information to their learners about the new method of teaching.
After the training, using a coin toss, the teachers were assigned to teach either the experimental group or the control group. Both the control and experimental groups were taught the same unit topics outlined in the first two chapters of the Grade 11 mathematics syllabus (MOE, 2009). The unit topics include relation and function, inverse relation, graphs of relation and inverse relation, even and odd functions, onetoone and onto functions, absolute value and signum functions, inverse functions and their graphs, simplification of rational expressions, arithmetic operations on rational expressions, rational functions and their graphs, solving rational equations and inequalities, and some applications of rational equations and inequalities as word problems.
The experimenter teacher was provided with a lesson plan format, prototype lessons, and an intervention guide, that assist in discoursebased lesson preparation and implementation while the other teacher was strictly informed not to implement discoursebased instruction in the control group but to use the traditional lecture method. The classroom discourse in the experimental class was structured in individual work, smallgroup discourse, and wholeclass discourse. The experimental group underwent discoursebased mathematics instruction where learners actively engaged with mathematics learning through explaining, justifying, conjecturing, comparing, sharing, and questioning.
Learners in the experimental group were grouped into heterogeneous small groups (Webb, 1991) of four or five learners. The experimenter teacher was responsible for establishing classroom rules that equally entertain and respect all learners’ ideas and contributions (Bennett, 2014) and creating discourseelicited mathematical tasks (Lampert, 1990). The teacher briefed his learners on how to respectfully listen to and share ideas, communicate, agree or disagree with each other’s ideas, work in groups, and maintain classroom norms (Bennett, 2014; Legesse et al., 2021). The experimenter teacher followed the implementation process illustrated in Table 1.
TABLE 1: The process of implementing discoursebased mathematics instruction (Legesse et al., 2020). 
As outlined in Table 1, the design and implementation of discoursebased instruction goes through planning and creating of tasks that embed the learning topics in the stages of task presentation: individual work (each learner thinks about the given task), smallgroup discussion, wholeclass discussion, and reflection. The task implementation requires the active involvement of learners in individual work (give learners the task and allow each learner to think about it individually for some minutes), smallgroup discourse (encourage learners to work together in small groups to talk about the task, discuss, and listen to each other’s ideas), wholeclass discourse (allow learners to present and share their ideas and strategies), and reflection (allow learners to reflect on the task through openended questions, such as ‘What solution strategies did you find very useful for solving rational equations?’ ‘What challenges have you faced in today’s lesson?’ ‘What concepts have you understood well and what concepts do still you want to understand?’).
The teacher is responsible for establishing classroom rules that equally entertain and respect all learners’ ideas and contributions (Bennett, 2014) and creating discourseelicited mathematical tasks (Lampert, 1990). Learners should respectfully challenge, listen to, and share each others’ ideas, work in groups, and be accountable for their learning (Bennett, 2014).
By applying the five practices model – anticipating, monitoring, selecting, sequencing, and connecting (Smith & Stein, 2011, p. 8) – the experimenter teacher (A) facilitated and guided learners’ discourse of conversations about the given tasks. The experimental group of learners engaged in discursive activities that include generating examples and nonexamples, agreeing and disagreeing, comparing equationsolving strategies, exploring different simplification strategies for rational expressions, explaining rational equationsolving strategies, describing relations using multiple representations, and evaluating mathematical statements. From the leading author’s classroom observations, a sample lesson on identifying functions graphically in the experimental class is presented below.
Planning for instruction: The teacher articulated the goal of the lesson and asked learners to open their mathematics textbooks. Then the teacher wrote the task on the chalkboard.
Task choice or design: The teacher read and presented the task shown in Figure 1. The questions asked were ‘Which of the following graphs represent functions?’ and ‘Which graph does not represent a function? Explain your answer.’
Individual work: The teacher allowed learners to think about the task independently for some minutes.
Small group discourse: The teacher encouraged learners to discuss and exchange their ideas in small groups for about six minutes. After ensuring all groups completed the task, the teacher asked the class to share their ideas and explanations.
Wholeclass discourse: The teacher selected a learner from a group. The learner answered ‘the graphs in (a) and (b) represent functions’. The teacher posed the question ‘How did you know that?’ The same learner verbally explained that ‘if we draw a vertical line, it crosses exactly at one point. So, the graphs represent functions by a vertical line test’. The teacher asked the class ‘Do you all agree with this explanation?’ The class approved the explanation by saying loudly ‘yes’. The teacher asked, ‘What about the graphs in (c) and (d)?’. Some other selected learners answered that the graphs in (c) and (d) do not represent a function. The teacher asked the class ‘Who would like to explain why these graphs do not represent functions?’ A learner explained to the class by saying ‘a vertical line crosses the graphs at two points’. The teacher asked ‘do you agree?’ The class agreed with the learners’ explanations. The teacher said ‘But what does it mean when the vertical line crosses the graph at two points?’ A learner explained by saying that ‘two numbers in the domain are related to the same number in the range’.

FIGURE 1: A discourseelicited task presented to the experimental class: By a vertical line test, Blocks (a) and (b) represent functions, whereas Blocks (c) and (d) do not. 

The control group was taught the same unit topics with the traditional lecture method and attended the same class hours per week in similar shifts as the experimental group. The duration of each lesson was 42 minutes. The traditional lecture method can be characterised as the ‘chalk and talk’ method where the teacher mainly dominates the classroom talk, and the majority of learners are passive listeners to the teacher’s lecture with a lack of opportunities for challenging ideas, or discussing with peers to construct an understanding of concepts (Rosenthal, 1995). The lesson observations showed that the classroom instruction in the control group was dominated by the teacher’s demonstration of rules and procedures for arithmetic operations of rational expressions, simplification strategies, equation solving, and graphing procedures for rational functions. Learners often engaged with procedural and computation tasks that did not provide opportunities for learners to be actively engaged in discourse practices of explaining, justifying, questioning and answering, agreeing and disagreeing, and comparing solution strategies. Learners copied notes and worked on examples and were passive listeners to what was explained by the teacher.
Findings from the observations were consistent with the traditional teachercentred forms of teaching practices (Deressa, 2004; Lampert, 1990). In an Ethiopian context, the traditional teaching method was found to manifest poor social interactions and communication between teacher and learners and among learners over the learning content (Deressa, 2004). The implementation of the tasks is based on teacherled demonstration where learners are accustomed to rehearsing facts and mimicking formulas, solution procedures, and rules to do similar tasks (Hsu, 2013). The instruction focuses on teaching procedures and computational skills. The teacherdominated way of teaching mathematics restricts opportunities for learners to explain, share, and discuss their ideas and thinking (Hsu, 2013). For instance, the teaching of simplification of rational expressions in a teacherdominated classroom focuses on enabling learners to memorise and practise procedural rules by assigning a set of routine tasks (Kooloos et al., 2019).
Data collection instrument
The variables involved in the present study were methods of teaching as the independent variable and mathematical proficiency scores as the dependent variable. Test scores as quantitative and classroom observations as qualitative data were collected. The total test score (total sum of scores in each strand) was used as a measure of participants’ proficiency in algebra. The maximum possible score assumed was 48 while the minimum was 0. During the progression of the experiment, the researcher as a neutral observer made unannounced classroom observations twice per week in both groups. Accordingly, the researcher observed and documented teachers’ actions, learners’ participation, and classroom activities.
Anchoring to the characterisations of the four strands of mathematical proficiency (see Table 2), the authors developed 24 test items from reviewing different literature sources including national mathematics examination papers. The constructs to be measured were operationalised through the test items that covered a variety of algebra and function topics outlined in the Grade 11 mathematics syllabus (MOE, 2009). More specifically, the Rasch partial credit model (Wilson, 2005) was applied to validate the test items using Winsteps software (version 4.4.2). The Rasch analysis provided psychometric evidence of item reliability (0.98), person reliability (0.79), item separation (7.66), and person separation (1.97), item fit, item difficulty, item polarity, local independence, and unidimensionality that made up the test instrument. Measures of item difficulty were found to be within the range from –1.96 logits to 1.92 logits; which is located inside the normal range from –3 logits to 3 logits (Timofte & Siminiciuc, 2018). Items with difficulty measures below –2.0 or above 2.0 are described as easy and hard items (De Ayala, 2009).
TABLE 2: Characterisations of the four strands of mathematics proficiency (Legesse, 2022). 
Before the start of the experiment, pretest data were collected using Raschvalidated test items (see Table 3) on the dependent variable from both groups. The time permitted to complete the test was 2 hours. After the completion of the implementation phase, using the same test instrument, posttest data were collected from both groups under similar conditions. The same classroom teachers and two doctoral students invigilated the administration of the pre test and post test. The researcher attended the invigilation processes as a neutral observer. Learners’ test performance was marked using an itemspecific scoring rubric (0/1/2). An initial format of the scoring rubric was created after analysing each item for its consistency with the content and characterisations of the proficiency strands. It was then reviewed by doctoral students and senior mathematics teachers. The modified version of the rubric was used for scoring learners’ test performance.
Data analysis results
Statistical data analyses were performed using the Statistical Package for the Social Sciences (SPSS, version 23). The normality of the test data was examined by considering the absolute sizes of the Zratios for skewness and kurtosis (Orcan, 2020). The Zratio is determined by dividing the value of skewness and kurtosis by their standard errors (Orcan, 2020). For a small sample size, when the absolute sizes of the Zratios for skewness and kurtosis are greater than 1.96, the data would be nonnormally distributed (Orcan, 2020).
As shown in Table 4, the absolute size of the Z ratio for the skewness of pretest scores in the experimental group and that of posttest scores in the control group are greater than 1.96. The values of skewness and kurtosis indicated that the pretest scores and posttest scores were nonnormally distributed for each sample (Orcan, 2020). Results of the ShapiroWilk test supported that the raw scores on the dependent variable failed to meet the normality assumption for each sample.
TABLE 4: Descriptive statistics for pretest scores and posttest scores. 
Davison (1999) recommended nonparametric tests when the violation of the assumptions for parametric tests becomes evident. Consequently, the authors performed MannWhitney U test analyses on rankconverted pretest and posttest raw scores (Feys, 2016; Pallant, 2016).
Pretest and posttest data analysis
To determine if there was a significant difference in the pretest scores between the control group and the experimental group (see Table 5) before the treatment, the pretest scores were analysed using the MannWhitney U test.
TABLE 5: Normality test for pretest scores and posttest scores. 
The result of the MannWhitney U test indicated no statistically significant difference in ranked pretest scores between the control group and the experimental group (U = 1281, z = –0.786, p = 0.432) (see Table 6), which showed that both groups were comparable at the start of the experiment.
TABLE 6: Results of the MannWhitney U test for the pretest scores. 
The result of the MannWhitney test on the posttest scores showed that the ranked posttest scores in the experimental group were significantly higher compared to the ranked posttest scores in the control group (U = 1068.50, Z = –2.124, p = 0.034) after the intervention (see Table 7), which revealed that the experimental group scored statistically higher in the posttest scores than the control group.
TABLE 7: Results of the MannWhitney U test on the posttest scores. 
Calculation of the effect size for the MannWhitney U test (r = 0.21) using the formula , where Z refers to the absolute standardised test statistic Z and N is the total number of participants (Pallant, 2016; Tomczak & Tomczak, 2014), showed that the degree to which the experimental group had posttest scores with higher ranks than the control group was small (Mangiafico, 2016). This value of effect size might suggest that learners should be exposed to discoursebased instruction for a relatively longer period of intervention for having a bigger outcome difference.
Description of discourse types observed in the experimental class
The analysis of discourse types focused on capturing and describing the types of discourse practices enacted in the experimental class. Using the discourse observation protocol (Weaver et al., 2005), learners’ actions and teachers’ actions were observed for the potential discursive indicators during classroom instruction. A total of eight classroom lesson observations were conducted during the implementation phase, apart from the first and last weeks of the experiment period. Themes of potential discursive indicators were identified by reviewing the literature (e.g. Moschkovich, 2007; Weaver et al., 2005). The discourse of conversation utterances made between learners and teacher and among learners along the segments of each observed lesson were coded and categorised for the following discourse types: sharing of ideas, solution procedures and strategies (S), conceptual explanation (E), exemplifying (EX), questioning (Q), justification (J), comparison and contrast (CC), agreeing and disagreeing (AD), and generalising (G) (Weaver et al., 2005). The discourse types extracted from the observed lessons are summarised and presented in Table 8.
TABLE 8: Discourse types extracted from lessons observed in the experimental class with discursive activities (Legesse, 2022). 
Descriptive analysis of classroom instruction in the control group
A total of six classroom observations were made for the control group. The classroom instruction predominantly focused on the teacher’s demonstration of rules and procedures for arithmetic operations of rational expressions, simplification strategies, rational equation solving, and graphing procedures for rational functions. Learners often engaged with procedural and computation tasks that did not provide opportunities for learners to foster the construction of mathematical ideas and concepts through engaging in discourse practices of explaining, justifying, questioning and answering, agreeing and disagreeing, and comparing solutions strategies. Descriptive analysis of the observed lessons revealed that learners in the control group lacked opportunities to compare and explain different solution strategies and graphing procedures and to express their thoughts in the process of simplifying rational expressions and solving equations and inequalities.
The teacher often interrogated learners’ understanding by gathering information about their mastery of the rules, definitions, and algorithms, using questions such as ‘What did you recall about yesterday’s lesson?’, ‘What is the definition of relation?’, and ‘What are the graphing procedures for rational functions?’. For instance, the teaching of simplification of rational expressions focuses on enabling learners to memorise and practise procedural rules by assigning a set of routine tasks (Kooloos et al., 2019). Such proceduraloriented mathematics teaching restricts opportunities for learners to explain, share, and discuss their ideas and thinking (Hsu, 2013).
Classroom observations in the control group revealed that the instruction was characterised by the absence of questions that probed learners’ justification and reflection. The teacher questioning strategy mainly manifested a characteristic of teacher askinglearner salientteacher answering for definitions, procedures, and rules. The nature of classroom discourse was dominated by the unidirectional flow of ideas from the teacher to learners, less frequent interactions among learners and between the teacher and learners, asking closed questions, lack of opportunities for learners to conjecture, discuss, and share their ideas and solution strategies. The teacher was the sole responsible person to approve learners’ responses and to state rules and procedures to be followed without soliciting learners’ thinking (Lampert, 1990).
Limitations of the study
Unwanted researcherrelated bias and teacherrelated variables such as teaching experience, teaching style, difference in their views about mathematics and its teaching, lack of discourseoriented learning materials, and relatively large class size might influence the implementation of discoursebased teaching as was witnessed by Bradford (2007). The study did not provide enough information on which strands of proficiency either group might perform better relative to the other one. A separate analysis of groups’ scores in easy and difficult items was also interesting. Furthermore, the study did not examine the interaction of the mathematical tasks, questions, and discourse types to influence learners’ learning and this limitation was also observed in other similar studies (Hiebert & Wearne, 1993). After the completion of the experiment, the researcher planned to treat the control group with discoursebased instruction. However, about two months after the completion of the experiment, it was impossible to expose the control group to the treatment in a similar manner to the experimental group because schools were closed due to the COVID19 pandemic. To circumvent this unexpected situation, the control group learners received classroom activities that were enacted in the experimental class in the form of handouts and group assignments via their mathematics teacher after schools reopened.
Conclusion and recommendation
The study examined the effectiveness of discoursebased mathematics instruction on Grade 11 learners’ proficiency in the syllabus topics of algebra and function. Results of the data analyses showed that learners who engaged in discoursebased instruction of algebra topics demonstrated better performance in mathematics proficiency than those who were taught the same syllabus topics through teachercentred instruction. It appeared that discoursebased instruction was more effective in developing learners’ proficiency than the teachercentred method for teaching mathematics. That is, learners in the experimental group gained a better understanding of mathematical concepts and procedures than those learners in the control group. This was most likely the result of their regular engagement in the discoursebased learning activities that promote individual thinking and discussions of mathematical ideas where learners compared solution methods, explained and posed questions for clarification, and shared and listened to one another’s ideas and reasoning. The findings in this study were consistent with theoretical and empirical literature (e.g. Bradford, 2007; Cross, 2009; Franke et al., 2015; Legesse et al., 2020; Sepeng & Webb, 2012; Smith, 2018; Stein, 2007). For instance, a sociocultural learning perspective contends that a classroom environment that creates a learning platform for social interactions and cultural contexts fosters the construction of knowledge and understanding.
Overall, the study findings provided innovative ideas on how school mathematics should be taught in Ethiopian classroom contexts and that assist mathematics teachers in being aware of the potential benefits of discoursebased mathematics teaching for enhanced conceptual learning outcomes. Based on these findings, the study recommended discoursebased instruction for teaching mathematics at upper secondary and university levels. It proposed discoursebased education as a component of mathematics teachers’ professional development programme. For these findings to be highly informative, replication studies should be conducted across different grade levels and other mathematical content domains with similar or mixed research designs.
Acknowledgements
This manuscript is based on a doctoral thesis submitted to the University of Johannesburg for which the university provided funding.
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
K.L., conceived the research and conceptualised the research design and methodology as well as the initial analysis of the data. M.Y.L., collected the data and developed the data analysis tools, finalised the analysis of the data, and the initial writing of the manuscript.
Ethical considerations
Ethical clearance to conduct this study was obtained from the University of Johannessburg Faculty of Education Research Ethics Committee (no. Sem 22019031).
Funding information
This research received no specific grant from any funding agency in the public, commercial, or notforprofit sectors.
Data availability
Data sharing is not applicable to this article as the participants did not authorise the authors to share the data collected and analysed in this study.
Disclaimer
The opinions, findings, conclusions, and recommendations communicated in this manuscript are those of the authors and do not necessarily reflect the views of other persons or any institution.
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