Evidence from the international comparative Trends in Mathematics and Science Study (TIMSS) supports the claims that many teachers find teaching mathematics challenging and that many South African learners face obstacles learning mathematics. The 2015 TIMSS report, for example, indicated that 61% of Grade 5 learners in South Africa could not do basic mathematics, ranking these learners as the lowest performing of the participating countries (Mlachila & Moeletsi 2019; Spaull 2018). For our own province (Eastern Cape), the 2015 TIMSS indicated that the percentage of Grade 5 learners who were unable to demonstrate basic mathematics knowledge and skills was even higher (74%) (Reddy et al. 2016). The TIMSS 2019 results, released on 08 December 2020, indicated minimal change in the Grade 5 learners’ achievement average, which remains below the basic threshold of 400 TIMSS points (Reddy et al. 2020).

South Africa’s matriculation results for mathematics are also a cause for concern. Only 58% of learners taking mathematics in 2018 achieved the over 30% level (Department of Basic Education [DBE] 2019:132). In 2019, only 54.6% of matric learners achieved the over 30% level (DBE 2020:5). These statistics say nothing, however, of the substantial attrition rate in the number of learners actually taking mathematics in the years preceding matric (Adler & Pillay 2017), and Shay (2020) notes that the situation is not improving because of a further increase in attrition of 17.9% from 2018 to 2019.

Our focus for this article is on the use of music in the teaching of fractions at Grade 5 level. As literature indicates, fractions form an integral part of foundational knowledge for mathematics achievement, especially the development of fractional reasoning (Cortina, Visnovska & Zuniga 2014; Dole 2010; Hilton et al. 2016; Tzur 2016); yet, developing a sound understanding of fractional knowledge is identified as being particularly challenging (Tzur 2016). Literature demonstrates many connections between music and mathematics as well as the benefits using these links can have for education (Civil 2007; Edelson & Johnson 2003; Geist, Geist & Kuznik 2012; Song et al. 2016).

A number of previous studies have suggested possible ways in which musical activities can be used to support mathematics teaching and learning. For example, research by Bresler (1995) identified four different ways in which arts were integrated into other primary school subjects, depending on how the teacher used the various arts forms. She named the different styles of arts integration as the subservient approach, the affective style, the social integration style and the co-equal, cognitive style. The first three styles of integration involve using the arts to illustrate content, to enhance a particular atmosphere within the classroom or to create meaningful links to the community surrounding the school, respectively. The co-equal, cognitive style (which, for the remainder of this article, we refer to as the ‘co-equal style’ as per Bresler’s abbreviation in her own work) was identified as the most rigorous, but least prevalent style, because it requires that teachers have the necessary knowledge and skills of the arts form (Bresler 1995; Grumet, Randolph & Stanley 2014). In a similar study, Song et al. (2016:20) identified and categorised three ways in which teachers integrated music into the mathematics curriculum, namely mathematics as a tool during music creation, music as a catalyst for mathematical cognition and music as a pedagogical approach to mathematics. They note, however, that research exploring the third integration strategy is limited and called for further research aimed at identifying practical strategies that teachers can employ when teaching mathematics at the primary school level (Song et al. 2016).

Both Bresler (1995) and Song et al. (2016) identify the integration of mathematics and music at a pedagogical level to be the most demanding way for teachers to use the two subjects in which it requires that teachers have knowledge, skills and confidence in both subjects, as well as enough time to plan and effectively prepare integrated lessons. Such difficulty notwithstanding, South Africa’s Curriculum and Assessment Policy Statement (CAPS) highlights the importance of integration between school subjects (DBE 2011a).

The present study shares elements from the first author’s response to this call to begin addressing the gap that exists in research about the integration of mathematics and music at a pedagogical level. The research reported in this article derives from the first author’s master of education case study research (Lovemore 2020).

At the time when this research was carried out, the first author taught Grade 5 mathematics. She had also taught class music and violin lessons before the study. Having noticed the mathematics found in music, she recognised an opportunity to explore how music and mathematics could be integrated to support her own teaching of fractions. Subsequently, using an action research case study, she designed various learning support materials (LSMs) and trialled a range of practical strategies with her Grade 5 mathematics class across eight weeks of teaching.

The concept of curriculum integration and Bresler’s styles of arts integration guided the first author through the broader study. Bresler’s styles of arts integration (1995) provided the theoretical framing of the study. The author located her research within the co-equal style. This captured the essence of her research goal, namely, to demonstrate possible strategies whereby music note values could be used in creative ways in her teaching of fractions to her Grade 5 learners. We believe her research story may be valuable to other teachers interested in exploring the use of curriculum integration, in particular, integration of music and mathematics. In our concluding section, we discuss some implications for further research in this area.

The topic of fractions forms an important part of curricula internationally. It is widely accepted as a difficult topic requiring a great deal of time not only for teaching, but also for addressing misconceptions (Cortina et al. 2019). Many learners, for example, may think of a fraction as consisting of two numbers separated by a line as opposed to seeing it as representing a single number with a single position on a number line (Musser, Burger & Peterson 2011). In responding to challenges posed by such misconceptions, and noting that the concept of a fraction is ‘a complex, multifaceted idea involving a variety of meanings that relate to each other in an intricate manner’ (Thompson & Saldanha 2003 in Cortina et al. 2019:20), Cortina et al. (2019) advocate a shift from teaching fractions through rote procedures to teaching for more conceptual understanding and for developing proportional forms of reasoning.

The term ‘fraction’, as Musser et al. (2011:218) explain, is used in two different ways in primary school mathematics: as a numeral (the ‘part-to-whole model’ indicating the number of parts of a whole) or as a number where the focus is on a relative amount. We see music as constituting an ideal vehicle through which an integrated teaching approach could be employed to create potentially richer fraction-learning opportunities. Music is made up of notes, which vary in duration, known as the note value. Note values are expressed as ‘relative lengths to one another’ and their ‘fraction-like names imply, the relative values’ (Kornfeld 2005:10). Hence, as shown in Figure 1, a whole note is equivalent to two half notes or four quarter notes or eight eighth notes.

The study was conducted in an independent school in the Eastern Cape. Reddy et al. (2016:5) contrast independent schools to public schools in their report on the 2015 TIMSS study, noting that a significantly higher percentage of learners (60%) in these schools achieved scores which demonstrated basic knowledge in mathematics. Reddy et al. (2016) attributed this to greater access to resources. A further factor making it possible for the first author to conduct this study at the school was that the management team actively encouraged teachers to experiment with teaching strategies, including taking advantage of opportunities for integration across the curriculum. Thus, it paved the way for her to trial strategies for linking the learning outcomes of music and mathematics.

The Grade 5 class comprised of 16 learners, eight boys and eight girls, all within the 11-12 year age range. This group of learners had diverse learning needs and socio-economic and cultural backgrounds. Because of the first author’s prior experiences of working with this group of learners, she was aware of their different learning needs. The school’s language of learning and teaching is English. Ten learners had English as their first language, and six faced the challenge of English being only their second or third language. Some other factors contributing to the diverse needs of the group were diagnoses by educational psychologists indicating that one learner had dyslexia, two learners had attention-deficit hyperactivity disorder and one was experiencing ‘mathematics anxiety’.

Before beginning the intervention, the first author needed to ensure that her research design and conduct observed the ethical requirements both of the university and of research ethics more generally. Following the submission of a carefully considered ethics application, the university’s ethical standards committee granted ethical clearance for this study. Applying for this clearance further stimulated the first author, as the action researcher, to critically reflect upon, and question, various aspects of the study from multiple perspectives. She carefully considered influences, which her personal involvement and her dual role as the researcher and research participant may have over the other participants in the study. Core ethical aspects addressed included obtaining gatekeeper permission from the school and ensuring clear explanation of all aspects of the study to critical peers, learners’ guardians and the learners themselves. Thus, written, informed consent was secured from all participants. The first author emphasised to all participants that the focus of the action research design intentions was on her teaching strategies and LSMs, not the learners. She indicated that all participants had the right to withdraw from the study at any stage without prejudice. Learners’, colleagues’ and the school’s anonymity was preserved by using pseudonyms. Finally, she ensured that there was no disruption to the school’s normal programme. The mathematics lessons took place during school hours, according to the set timetable, and the first author remained mindful of the importance of maintaining focus on her regular teaching responsibilities.

The question guiding the first author’s action research case study was as follows:

The study was qualitative in nature, influenced by an interpretivist paradigm acknowledging that individuals’ lived experiences are shaped by their context and their perceptions about knowledge are socially constructed (Maxwell 2012; Thomas 2013; Willis 2007). It focussed on the design and implementation of a set of 10 Grade 5 mathematics lessons on fractions into which music was integrated, in ways that targeted skills in both subjects and in ways that allowed the first author to integrate the curriculum of both subject areas equally. The research question lent itself well to reflective processes of action research. Cycles of implementation of intervention strategies, reflection, adaptation and further implementation occurred (Ferrance 2000; Friedman, Gray & Aragon 2018; McCallister 2018), guiding the first author in critically reflecting on her chosen strategies, the LSMs she developed and the ways in which this appeared to support the teaching of fractions to Grade 5s. Her critical reflections also assisted her in keeping the lessons at the co-equal style of integration with equal focus on both mathematics and music.

Given that the research question focussed on ways in which the first author could enrich her teaching of fractions, she acted as both the researcher and research participant in this action research case study, reflecting throughout each of the intervention cycles on her own teaching. Her learners were also research participants, and they provided feedback based on their experiences of the lessons. In addition, two of the first author’s Grade 5 colleagues became research participants, having agreed to contribute by acting as critical peers. In this role, they viewed video recordings of critical moments from the lessons, identified by the first author, and discussed their insights on these with her. The study sample was thus both purposeful and convenient (Maxwell 2003), and the participants were ideally placed to provide the first author with rich feedback, understanding the dynamics and expectations present within their specific educational setting.

The reflective nature of action research (Ferrance 2000; Friedman et al. 2018; McCallister 2018) guided the first author to critically evaluate whether her lesson goals were achieved at the end of each lesson and to adjust her teaching strategies and LSMs across the course of the research cycle. Feedback from learners and critical peers contributed further to improvements for the following lesson of each cycle.

The 10 intervention lessons took place over an 8-week period. As Table 1 shows, the main focus in the first four lessons was on music note values, thereby providing a necessary foundation for learners’ transitioning across into explicit use of such knowledge in solving fraction problems (Lesson 5 onwards).

Multiple data collection tools from multiple sources were used to gain rich insight into this 10-lesson unit. Data sources included the first author’s reflective research journal, video and audio recordings of intervention lessons, photographs, examples of learners’ work, the learners’ self-assessment feedback forms and informal oral feedback during the lessons as well as the voice-recorded semi-structured interviews with critical peers (Mrs H and Mr K are both Grade 5 teachers. Mrs H is in her early fifties, and Mr K is in his mid-twenties). Triangulating across sources reduced the risk of bias deriving from any single source (Bush 2002; Ferrance 2000; Willis 2007), and, as Maxwell (2003:245) noted, ‘cross-checking’ across sources served to strengthen the credibility of a study.

After every second lesson, learners were asked to complete a self-assessment feedback form which asked them to indicate on a continuum on how they felt about the lesson in general, their level of understanding of the lesson content and their level of participation. Learners also had the opportunity to complete the written prompt, ‘I want my teacher to know …’. Three informal oral feedback sessions were held and voice recorded (at the beginning, middle and end of the 10-lesson intervention), where the first author invited her learners to discuss their general feelings about and experiences of the various lesson activities and the LSMs used.

Similarly, after every two lessons, the first author identified critical moments within the video recordings of the lessons and met with her Grade 5 critical peers to watch through the relevant sections with her and share their ideas about the successes and challenges in relation to the teaching strategies and LSMs. All nine (four individual and one combined) of these semi-structured interview sessions were recorded and subsequently transcribed. Through member checking (Brenner 2006; Willis 2007), the first author sought her critical peers’ further feedback regarding the accuracy of her transcripts of these interviews.

The first author’s own reflections (on lessons, on the feedback from the research participants and on examples of learners’ work and photographs) recorded in her reflective research journal played a crucial role. Journal entries followed a template originally designed by the University of Birmingham’s Academic Skills Centre (2015) based on Schön’s principles of being a reflective practitioner (Schön 1983, 1987). The first author’s analyses and reflections on the strengths and areas for improvement after each lesson informed adjustments for subsequent lessons. Figure 2 depicts the broader action research cycle which she followed in her design of the lessons. Figure 3 shows some of the ways in which critical reflections informed subsequent lesson planning and implementation.

Some of the ways in which critical reflections informed the first author’s subsequent lesson planning and implementation are shown in Figure 3.

Lessons 5–10 similarly included reflections and subsequent changes to lessons that followed. Examples of these reflections and changes are summarised in Table 2.

Analysis was an iterative and ongoing process throughout the study (Maxwell 2003). Because of the rich data collected from the intervention lessons and interviews, the first author needed to systematically analyse, interpret and present the ideas of the participants as authentically as possible (Friedman et al. 2018). Although it was time-consuming, she transcribed all the lessons and interviews herself, to develop a deeper understanding of and to make meaning from the rich data. Subsequently, two analytical frameworks based on analysing video recordings were chosen to guide the analysis process. These were Adler and Ronda’s (2015) framework for analysing mathematics discourse in instruction and the viewing, investigating and discussing environments of learning mathematics framework by Karsenty and Arcavi (2017). From these two frameworks, a colour-coded template (identifying the various sources of data) was developed, in which quotations, reflections and critical elements from the lessons could be recorded. For example, the research looked to answer questions such as: ‘How did the LSMs add value to the lesson? How can they be improved?’ and ‘What links were established between mathematics and music? Did the musical activities support or hinder the teaching and learning of mathematics?’

This combination of analytical strategies and multiple data sources facilitated a clear and holistic interpretation of the broader study’s findings. By systematically and iteratively working through the data, major elements and emerging themes were identified. For a full discussion of these findings, see Lovemore (2020). The ‘thick description’ (Geertz 1973) of the intervention lesson cycles shared there will, we hope, allow readers to immerse themselves sufficiently in the context of the study to make their own assessment of the overall credibility and relatability (Creswell & Miller 2000:129) of the study’s findings.

From the process of data collection, presentation and analysis, the first author derived findings, which she grouped into five categories, based on repeated key phrases, as well as key elements within each category. A summary of these findings is shown in Table 3.

The focus for the present article is, as was indicated, the value of integrating music into mathematics lessons on fractions. In the discussion which follows, we highlight three aspects of this:

Because this discussion section focusses so exclusively on action research and classroom-based data, we now replace the term ‘first author’ with ‘teacher’. Whilst the research took place within the Grade 5 classroom of the teacher (in her mid-twenties), fractions are introduced in the South African curriculum in the foundation phase. The focus of the activities and the LSMs on using multiple and concrete representations invoking learners to employ multiple senses (auditory, verbal, visual, tactile and kinaesthetic) is particularly relevant to foundation phase learners. We, therefore, argue that the following themes that emerged from the research would likely similarly apply to the teaching of fractions through integration with music in foundation phase classrooms.

As the foregoing discussion has shown, the integration of music into mathematics provided a way in which fractions could be represented other than the traditional representation of pizza or chocolate bars so commonly used in primary school lessons. Such integration, we argue, is beneficial in that it offers teachers an additional strategy to draw on in representing fractions.

This action research study was not about identifying generalised findings, but rather hoping that the findings spark interest and recognition by other teachers in other settings of the value of curriculum integration. We acknowledge that a potential limitation of this research could be that the first author was able to use her strong musical knowledge in the design and implementation of her fraction lessons. This limitation notwithstanding, we argue that integration of mathematics and music holds much potential for supporting student conceptual understanding of fractions and developing productive mathematical learning dispositions.

The question then would be to what extent these findings might be seen as useful to teachers without the necessary musical knowledge. Further research would be needed to find out whether other, non-musically trained, teachers could benefit from the use of these strategies and LSMs; for example, the use of the American note-naming convention may make it a manageable strategy for such teachers. We believe that collaborative working between music and mathematics teachers could overcome challenges that emerge for those teachers who do not have specialised knowledge of teaching both. Research can also be conducted on the use of fewer note values and lower number ranges of fractions to adapt the strategies for younger grade learners.

A key contribution of this article is that it shows curriculum integration as a means of achieving the broader curriculum aims of, for example, developing ‘an appreciation for the beauty and elegance of mathematics’ and recognising ‘that mathematics is a creative part of human activity’ (DBE 2011a:8). These aims, which hold for the mathematics curriculum at all levels, have tended to be overlooked despite their surviving the various curriculum revisions. In this action research case study, the teacher was able, and encouraged, to explore the creative link between music and mathematics, and the resulting integration provided an opportune space for the teacher, her learners and her critical peers to experience the beauty and creativity in mathematics. Our hope is that this article may inspire other teachers in developing their own integrated strategies and pedagogical innovations, through drawing on their talents, interests and competencies, to enrich their teaching of mathematics. Finally, we hope also that this article will provide some useful insights and suggestions for teachers interested in undertaking action research in their own classroom-based settings, and who, through systematic cycles of reflection and trialling, set out to explore and strengthen aspects of their own innovative teaching practices.