5. Music, Spatial Reasoning and Mathematical Performance

Historically, there has long been interest in the relationship between acquired musical skills and performance in mathematics. It has been assumed that there is a strong connection between music and mathematics, as many musicians play from notation and are constantly required to adopt quasi-mathematical processes to subdivide beats and turn rhythmic notation into sound. More recently, there has been interest in the relationship between music and spatial-temporal reasoning, which contributes to some areas of mathematical understanding. Spatial-temporal reasoning is the ability to transform mental images in the absence of a physical model (Rauscher et al., 1997; Shaw, 2000). It involves the ability to manipulate and understand complex shapes through mental imagery, as the individual develops and evaluates patterns which change in space and time. Cooper (2000) viewed spatial-temporal reasoning as an abstract model of cognition consisting of several elements, including pattern-seeking, recognition, retention and recall; visualising imagery; perceiving figures as wholes; generating a whole image from a fragment; grasping the whole of a problem; understanding spatial relationships from multi-perspectives and among internal movement of parts; maintaining orientation within space; and mentally manipulating shapes within two- or three-dimensional space. The key features used in spatial-temporal reasoning include the transforming and relating of mental images in space and time, the use of symmetries to compare physical and mental images, and temporal sequencing (Grandin et al., 1998). These skills are high-level mathematical abilities which are useful in learning proportional reasoning (Grandin et al., 1998; Shaw, 2000) and induce advanced understanding of mathematical concepts such as fractions, proportions, symmetry and other arithmetic operations (Tran et al., 2012). Developing spatial-temporal thinking may also be related to geometrical skills.

Early mathematical skills tend to be one of two types: number knowledge or number operation (Griffin, 2004). The latter is linked with the formulation of mental number lines, which enable children to understand magnitudes, relations between them and arithmetic operations (Jordan et al., 2008; Gunderson, 2012). Mental number lines are linked with spatial-temporal reasoning. The development of a mental number line is fundamental for mathematical understanding and facilitates performance, especially in arithmetic (Ramani and Siegler, 2008; Booth and Siegler, 2008; Van Nes and Doorman, 2011; Gunderson et al., 2012). Spatial skills have also been linked with spatial structuring, which is important in determining quantities, as well as comparing and calculating them (Butterworth, 1999; Mulligan and Mitchelmore, 2009). Undertaking such tasks early in development occurs unitarily. This takes time and can also lead to errors. Most children gradually learn to organise objects in ways that enable them to count more accurately and efficiently. This helps them to understand the decimal system. Spatial awareness contributes to the development of patterning, while the temporal element might be used in structuring and strategy choice.

Explanations for the relationship between music and spatial-temporal reasoning have been sought in neuroscience. Two main approaches have developed. The first concerns connectivity, and proposes that the processing of music and spatial tasks is underpinned by overlap in brain functions (Fiske, 1996). In contrast, near-transfer theory suggests that music and spatial-temporal reasoning share some processes, and the development of one leads to the development of the other (Rauscher, 2009; Schellenberg, 2004). Explanations for the links between music and spatial-temporal reasoning relate to connectionism—the development of neural connections (Sporns, 2011)—and modular theory, which is related to near-transfer (Jordan-DeCarbo and Nelson, 2002). The connectivity proposal has been supported by Shaw (2000), who suggested that musical and spatial processing overlap in the brain and, as a result of these cortical connections, the development of certain kinds of musical and spatial abilities (especially spatial-temporal abilities) is intertwined. Near-transfer suggests that several kinds of thinking are required in order to learn and make music. Both are multi-dimensional processes. A range of spatial skills might be improved because of the practice required in making music (Jordan-DeCarbo and Nelson, 2002).

Particular interest in the relationship between music and spatial reasoning skills developed following a study by Rauscher and colleagues (1993), who claimed that after listening to Mozart’s ‘Sonata for Two Pianos (K448)’ for ten minutes, adult participants showed significantly better spatial reasoning skills than after periods of listening to relaxation instructions designed to lower blood pressure, or listening to silence. The mean spatial reasoning scores were eight and nine points higher after listening to the music than in the other two conditions. However, the effect only lasted for ten to fifteen minutes. Early attempts to replicate the phenomenon were unsuccessful (Chabris, 1999; Steele et al., 1999). Using a logical rather than spatial reasoning task, Newman and colleagues (1995) tested 114 students before and after listening to either eight minutes of Mozart’s music, relaxation instructions or silence, and found that all participants showed a practice effect with no particular enhancement in the music group. Similarly, Rideout and Laubach (1996) tested four female and four male undergraduates on two equivalent spatial tests, following either the presentation of Mozart’s ‘Sonata for Two Pianos in D Major’ or a non-musical activity. EEG was recorded during, at baseline and at two task performance periods. Correlations were generated between task performance and EEG variables. Performance improved significantly following the presentation of the music. In a later study, Rideout and colleagues (1998) studied 16 participants who showed reliable improvement on a paper-folding and cutting task after listening to Mozart’s ‘Sonata for Two Pianos in D Major’. The enhanced performance was also noted for 16 other participants after listening to a contemporary musical selection with similar musical characteristics. In both cases, the control procedure included ten minutes of listening to a relaxation tape. Similarly, Wilson and Brown (1997) examined the effect of Mozart’s music on 22 college undergraduates who had listened to a selection of Mozart’s music. Each participant performed a pencil and paper maze task after a ten-minute presentation of each of three listening conditions: a piano concerto by Mozart, repetitive relaxation music and silence. Limited support for the previously obtained enhancing effect of listening to Mozart’s music was revealed in measures of performance accuracy on this spatial task, whereas no effect was found for either the number of maze recursions or the overall quality of maze solutions. Hetland (2000b) carried out two meta-analyses and found that music significantly enhanced performance on a variety of spatial tasks, but that music other than Mozart also enhanced spatial-temporal performance over a short period of time.

Some research has explored the so-called Mozart effect on children. For instance, as part of the BBC programme Tomorrow’s World, a replication of Rauscher’s study was undertaken with over 6,000 ten- and eleven-year-old children (Hallam, 2001). They were tested after they listened simultaneously to either contemporary pop music by Blur or Oasis, the same piece of music by Mozart that was used in Raucher’s study, or a talk given about experiments. After being assigned at random to one of the three listening experiences, each child completed two tests of spatial abilities. No statistically significant differences were found between the performance of the three groups on the two tests of spatial reasoning. A reanalysis of the data using a different statistical approach by Schellenberg and Hallam (2006) showed that performance on one of the tests—square completion—did not differ as a function of the listening experience, but performance on the paper-folding test was superior for children who listened to the popular music compared to the other two groups. This was interpreted in terms of the arousing effects of the popular music, which the children also enjoyed, leading to an increase in their motivation.

This mixed, although mainly negative, evidence relating to the Mozart effect led to research focusing on the role of music when it was played alongside the completion of a range of different intellectual tasks. The impact of background music is considered in more depth in Chapter 11.

Comparisons between Musicians and Non-Musicians, and Correlation Studies

One strand of research has compared the performance of musicians with non-musicians on spatial-temporal reasoning tasks, while a further strand has compared performance on mathematical tasks. Neuroscientific research into brain structures has confirmed that the areas of the brain where spatial reasoning occurs are more pronounced in adult musicians, and that the processing of music and spatial-temporal tasks activates similar neural structures. Sluming and colleagues (2002) found that musicians achieved better results than controls on a line orientation test and were better in finding the middle of a line (Patston et al., 2006). Skills used in these two tasks may be related to the ability to manipulate the mental number line. Taking account of the importance of the concept of mental number line for the development of mathematical thinking, it is possible that active engagement with music enhances this process (Siegler and Booth, 2005; Ramani and Siegler, 2008).

There is considerable evidence from research with professional musicians or those training to become professional musicians that they have better spatial-temporal reasoning abilities than non-musicians, including mental rotation. Pietsch and Jansen (2012) compared students of music, sports and education, and demonstrated better performance on mental rotation tasks among the first two groups, while Sluming and colleagues (2007) found that members of orchestras outperformed controls in mental rotation tasks. They suggested that this was linked with more pronounced development of Broca’s area in the brains of the musicians, while Mark (2002) showed that the areas of the brain which are activated whilst performing music and spatial-temporal tasks are proximate.

Musicians are better at a range of visuospatial search tasks. Patston and Tippett (2011) administered a language comprehension task and a visuospatial search task to 36 expert musicians and 36 matched non-musicians in conditions of silence and correct or incorrect piano music playing in the background. Musicians performed more poorly on the language comprehension task in the presence of the background music compared to silence, but there was no effect of background music on the musicians’ performance on the visuospatial task. In contrast, the performance of non-musicians was not affected by the music on either task. This suggests that, when musicians process music, they recruit a network that overlaps with the network used in language processing. Musicians have better reaction times to selective and divided visual attention tasks (Rodrigues et al., 2013). They are better at matching a set of coloured blocks to a visual image (Stoesz et al., 2007), have better memory for line drawings (Jakobson et al., 2008), and are more accurate when asked to mark the centre of a horizontal line (Patston et al., 2006) and when asked to judge the orientation of a line (Patston et al., 2007). These findings might be particularly important in linking music with mathematics, as the ability to visualise a horizontal line and localise a middle and proportional distance on it is closely related to the notion of the mental line used for a variety of mathematical operations (Gunderson et al., 2012). However, Helmbold and colleagues (2005) compared 70 adult musicians and 70 non-musicians matched for age, sex and level of education on their performance on different aspects of primary mental abilities including verbal comprehension, word fluency, space, flexibility of closure, perceptual speed, reasoning, number and memory—they found no significant differences except for flexibility of closure and perceptual speed, where the musicians performed reliably better than non-musicians.

Pannenborg and Pannenborg (1915) compared individuals with varying degrees of musical talent and found only a slightly higher level of mathematical ability in those with high levels of musical ability. In contrast, Haecker and Ziehen (1922) administered a self-report questionnaire via the internet to 227 musical and 72 unmusical male participants, who were doctoral-level members of the American Mathematical Association or the Modern Language Association. The questionnaire assessed musicality, music perception, music memory and musicianship, music performance and music creation. The mathematics group did not exhibit higher levels of either musicality or musicianship. The mathematicians reporting high-level music performance ability did not report significantly greater musicality than did the literature or language scholars. Similarly, Haimson and colleagues (2011a; 2011b) recruited participants from the online membership of the American Mathematical Society and the Modern Language Association and presented them with a questionnaire assessing skills in musicality and musicianship. Members of both groups reported relatively low levels of musicality with no statistically significant differences between them. Revesz (1954) also found that reported levels of interest or aptitude for mathematics in musicians were low.

Vaughn (2000) meta-analysed studies comparing mathematics achievement in students with and without self-selected music study, and only reported a very small positive association between mathematics and musical engagement. Working with fourth-grade children, Haley (2001) investigated the effects of participating in an instrumental music programme, band or orchestra on their academic achievement. The children were placed into three groups. The first consisted of children who had studied an instrument prior to the introduction of band and orchestra in fourth grade, the second consisted of children just beginning to study an instrument and the third consisted of children with no experience of instrumental instruction. The findings showed that students who had studied an instrument prior to fourth grade had higher scores in mathematics achievement than did students in the other groups.

Comparing performance in reading and mathematics in two schools with different levels of music education, one with an outstanding music programme and the other with no music programme, Deere (2010) carried out a survey. Students experiencing high-quality music education had higher Tennessee Comprehensive Assessment Program (TCAP) reading and mathematics scores in the fourth grade. There was also a high correlation between music education and TCAP scores in reading and mathematics. In the eighth grade, where musical education was of high quality, students also reported higher TCAP reading and mathematics scores.

In a study with young children, Williams and colleagues (2015) investigated parent-child home music activities in a sample of 3031 Australian children participating in Growing Up in Australia: The Longitudinal Study of Australian Children. Frequency of shared home music activities was reported by parents when children were two to three years old. A range of social, emotional and cognitive outcomes were assessed by parent and teacher report and direct testing two years later, when the children were four to five years old. A series of regression analyses found that frequency of shared home music activities had a small significant partial association with measures of children’s numeracy. The findings suggested that there may be a role for parent-child home music activities in supporting children’s mathematical development.

Catterall and colleagues (2000), using the NELS:88 data, studied low socioeconomic status students who exhibited high mathematics proficiency in twelfth grade and found that 33 percent were involved in instrumental music compared with 15 percent who were not involved. Miksza (2010) extended this research, examining the potential relationship between participation in high-school music ensembles and extra musical educational outcomes, including achievement in mathematics, using data from the Education Longitudinal Study of 2002. The sample of 12,160 students was representative of white and minority high-school students from 603 rural, suburban and urban schools across the United States. The students who belonged to school music ensembles had higher scores in standardised mathematics tests. The study controlled for socioeconomic status but not mathematical performance prior to any music training. Similarly, Bergee and Weingarten (2020) used multi-level mixed modelling to test the extent to which students’ music achievement scores were related to their reading and mathematics achievement scores. Of the four levels examined—individual students, classrooms, schools and districts—only individuals and districts accounted for a significant portion of the total variance in achievement scores. There was a strong relationship between music scores and reading/mathematics achievement. In higher education, Barroso and colleagues (2019) aimed to identify the cognitive and affective factors related to mathematics and music theory that best explained undergraduate music theory achievement. The findings suggested that mathematic scores and music theory confidence were important predictors of grades in undergraduate music theory examinations.

Musical Interventions and Spatial-Temporal Reasoning

While the research comparing musicians and non-musicians, and that showing relationships between music, spatial-temporal reasoning and mathematics is important, it is not able to demonstrate causality. To demonstrate causality, it is necessary to carry out experimental intervention studies where the impact of musical engagement is compared with the impact of other activities or no activity. Rhythm may be particularly important, as infants engage in significantly more rhythmic movement to music and other rhythmically regular sounds than to speech, and also to some extent exhibit tempo flexibility (Zentner and Eerola, 2010).

General music instruction—including singing, movement and playing percussion instruments—has been shown to assist four- to six-year-old children in the development of spatial ability (Bilhartz et al., 1999). Zafranas (2004) studied 61 kindergarten children who received two piano or keyboard lessons weekly during one school year. Following piano or keyboard instruction, participants improved significantly in hand movement, gestalt closure, triangles, spatial memory and arithmetic, but not in matrix analogies. Similarly, Gromko and Poorman (1998) investigated the effect of music training on 15 preschoolers’ performance on subtests of the Wechsler Preschool and Primary Intelligence Scale. For the three-year-olds in the study, this musically intellectually stimulating environment resulted in an increase in the ability to perform spatial-temporal tasks.

Rauscher and colleagues (1997) assigned 78 students from three preschools to music, computer or no instruction groups. The instruction groups received training in one of the following: piano or keyboard (either individually or coupled with group singing lessons), group singing lessons only or computer instruction. The children were pre- and post-tested using one spatial-temporal reasoning task, object assembly, and three spatial recognition tasks (geometric design, block design and animal pegs). There were no differences between groups in pre-test scores, but after instruction the children in the piano group scored significantly higher on the spatial reasoning task compared to children in the other conditions. There were no differences amongst the groups on the spatial recognition tasks. The computer group, singing and no-instruction groups did not improve significantly over time on any of the tests. Later studies (Rauscher, 2002; Rauscher and Zupan, 2000)—which were undertaken over three years with upper-middle-income children who were provided with eight months of weekly 40-minute keyboard instruction in groups of eight to ten beginning in either kindergarten, aged five, or first grade, aged six—scored higher on two spatial-temporal tasks, puzzle-solving and block-building compared to children who did not receive music instruction. No enhancement was found for a pictorial memory task. However, these effects were not maintained when music instruction was terminated, although when lessons resumed in second grade the same children’s scores increased again, surpassing the levels that they had reached before the lessons were terminated. The children who received instruction over a period of three years scored higher on the spatial-temporal tasks compared to children who had not received instruction. While the scores of the keyboard group improved every year, although not significantly, after kindergarten the scores of children who began instruction in the second grade did not improve, suggesting that it was important that the training began early. Rauscher and La Mieux (2003) also reported that children who received keyboard lessons, singing training or rhythmic instruction scored higher than controls on spatial reasoning tasks. Further studies examined the effects of musical instruction on spatial-temporal reasoning in middle-income elementary-school children (Rauscher and Hinton, 2011). Two groups—a music group and an animated reading group—received 40 minutes of lessons in groups of eight to ten for nine months. At the end of the study, the children who received the keyboard lessons scored significantly higher on spatial-temporal reasoning tasks than those who received the animated reading lessons, although the improvement for the keyboard group was only for the girls.

Working with elementary-school students, Johnson and Davis (2016) investigated the effects of a programme combining musical ensembles in residence with regular classroom music instruction on students’ auditory discrimination and spatial intelligence. In combination with regular, sequential general music classes, participants in the programme received two half-hour lessons each week from musical ensembles in residence, lasting for four consecutive years. The chamber ensembles provided aural models for reinforcing fundamental concepts. Data were collected from a stratified, random sample of students in grades two and four to five receiving the experimental programme, and from demographically similar comparison schools which did not receive any regular music instruction. A total of 684 elementary students participated in the study. Children participating in the programme with the chamber music ensembles showed consistent and statistically significantly greater scores in both auditory discrimination and spatial intelligence measures.

Holmes and Hallam (2017) examined the potential of active music-making to improve mathematics achievement in primary-school pupils. In a quasi-experimental design, 60 children aged five or six participated in the music programme, while the same number of pupils from parallel classes made up two control groups. Lessons contained a variety of musical, predominantly rhythmical activities, based on popular nursery rhymes. Spatial-temporal skills were tested at the beginning and the end of the study. Throughout the intervention, pupils were assessed on musical skills, as well as general and specific mathematical skills. A strong relationship between musical and spatial-temporal skills was found in both age groups. The younger group scored higher than their peers on a picture test and a puzzle test. The results for the older children were also higher for the music group in both spatial-temporal tests. Some enhancement in mathematics in the intervention group was found, although there was no significant contribution of spatial-temporal abilities to general mathematics achievement.

One strand of research has focused on preschool children from deprived backgrounds participating in Head Start programmes (Rauscher, 2003, Rauscher et al., 2005). In the first study, 87 Head Start children were randomly assigned to one of three groups—piano, computer or no instruction—for 48 weeks over two years. At the end of the intervention, the children who received music instruction scored significantly higher than control groups on visual and auditory tasks that required spatial and temporal skills. Performance on an arithmetic task also improved following music instruction. A second study focused on whether different types of music instruction had different effects. Over 100 Head Start children of mixed ethnicity were assigned randomly to one of four conditions: piano, singing, rhythm or no instruction. All of the children in the music groups received weekly individual instruction for a period of 48 weeks over two years. The data from the three music groups replicated the data from the first study. The children in the music groups scored significantly higher at post-test on tasks requiring spatial and temporal skills. The rhythm group scored significantly higher than the piano and singing groups on temporal and arithmetic tasks. A third study was conducted to determine whether the effects endured after instruction stopped. The scores of the Head Start children who received lessons in the first and second studies were compared with three groups of grade-matched children participating in Head Start who did not receive music instruction, at-risk children not involved in Head Start, and middle-income children who did not receive music instruction. The children who had received music instruction in the first study continued to score higher than all of the other groups of children, with the exception of the age-matched middle-income children, on three of the four tests two years after instruction had ended. The data from the children who participated in the second study when they progressed to kindergarten showed that the singing, piano and rhythm groups scored higher than the Head Start and at-risk children on five of the tests. In addition, the rhythm group scored higher than the singing and piano groups on an arithmetic subtest, and scored significantly higher than the middle-income children on the temporal, arithmetic, mathematical reasoning and numeracy tasks. These findings suggest that rhythm instruction has the strongest impact on a range of mathematically related tasks. Rauscher and Hinton (2011) summarised the results from several of these studies and showed that music groups had higher scores on arithmetic and spatial abilities following musical interventions, although they were equivalent initially (Rauscher, 2014).

Several research projects have been undertaken within the context of the El Sistema approach to musical engagement, a structured extracurricular orchestral programme. For instance, Osborne and colleagues (2015) studied pupils from a low-income neighbourhood participating in El Sistema and showed that they had greater improvement in spatial reasoning, verbal and mathematical skills than comparison groups. Further evidence for music being responsible for enhanced spatial reasoning in at-risk children comes from an Israeli study, in which a two-year music training intervention of two to three hours per week was introduced in some after-school centres for at-risk children, but not in other centres (Portowitz et al., 2007). Children participating in the intervention showed larger improvements in remembering and reproducing a complex line drawing.

The most effective music interventions for enhancing spatial temporal reasoning in all children seem to be based on rhythm (Hetland, 2000a; Holmes, 2017; Holmes and Hallam, 2017; Rauscher and Le Mieux, 2003). Children in the early years of primary school seem to benefit the most from such interventions (Costa-Giomi, 2004; 2013; Graziano et al., 1999; Holmes, 2017; Holmes and Hallam, 2017; Rauscher, 2002, Rauscher and La Mieux, 2003; Rauscher and Zupan, 2000; Schellenberg, 2004). The optimal length of interventions that is required for there to be a sustainable impact has not been conclusively established. Rauscher and Zupan (2002) showed improvement in spatial-temporal skills which continued throughout a four-year programme, whilst Rauscher (2000) suggested that there was a need for programmes to last for at least two years to achieve lasting change. The underpinnings of such accelerated progression in disadvantaged and other pupils are not yet clear, and these enhancements might be mediated by the development of general cognitive abilities. It is also possible that these programmes raise participants’ motivation, self-efficacy, and perseverance. Overall, the majority of studies have shown that spatial-temporal skills can be improved by musical training. Interestingly, when other related cognitive abilities have been assessed—for instance, pictorial memory (Rauscher and Zupan, 2000), spatial recognition (Rauscher 1994; 1997), and number recall (Rauscher and La Mieux, 2003)—there has been no significant improvement related to musical engagement.

Not all of the research has shown an impact of music on spatial reasoning. For instance, Hanson (2003) investigated the effects of a sequenced Kodály literacy-based music programme on the spatial reasoning skills of kindergarten students. Fifty-four kindergarten children participated. One group of children received Kodály music instruction, a second group computer instruction and a third group no intervention. The programme lasted for seven months. Spatial-temporal reasoning, spatial reasoning and a nonspatial measure were assessed. The analysis revealed no statistically significant differences in pre-, post- or gain scores for any of the measures.

The Relationships between Spatial Skills and Mathematics

Children engage with arithmetic long before they experience formal mathematics education. Some number processing is present prior to the development of language. Preschool children understand estimation and comparison of quantities often before they can count or use number terminology. They have a sense of ordinality (Kaufmann, 2008) and use and develop strategies and procedures in solving problems (Bisanz et al., 2005). Very young children can discriminate between small groups of items containing different numbers of objects. Understanding increasing quantity by adding objects and decreasing quantity by removing them depends on observing ordinal relations among numbers (Bisanz et al., 2005). This skill is related to addition and develops earlier than subtraction. Children gradually develop greater accuracy until they can provide exact solutions to arithmetic problems. This is usually achieved by four to five years old. They also begin to develop rules and concepts that inform and constrain their growing ability to manipulate numbers (Bisanz et al., 2005). Krajewski and Schneider (2009) developed a three-phase model of this process: basic numeric skills, quantity number concepts and number relationships. At the third level, visual-spatial skills play a vital role, while non-verbal representations of magnitudes are essential for problem-solving (Rasmussen and Bisanz, 2005). This model supports a strong relationship between spatial skills (Cheng and Mix, 2014), the visual–spatial components of working memory and the development of mathematical abilities.

Alternatively, Spelke (2008) proposes a broader model which outlines three main systems which support young children’s mathematical learning: a system for representing small exact numbers of objects, up to three; a system for representing large approximate numerical magnitudes—for example, about 20—and a system for representing geometric properties and relationships. Each system is malleable and relatively independent in young children, but as basic concepts and mathematical operations develop, children learn to connect the three systems. Linking representations of numbers with representations of space helps in creating mental number lines, which are central to understanding relationships between numbers and calculations.

Spatial structuring is essential for many mathematical activities of a numerical or geometrical nature. Van Nes and de Lange (2007) propose that the ability to imagine a spatial structure relates to a specific magnitude, and to mentally manipulate it helps in understanding quantities and the process of counting and also speeds up that process. Van Nes and Dorman (2011) describe the mathematical skills which rely on spatial structures as composing and decomposing of quantities; counting and grouping; part-whole knowledge in addition, multiplication and division; comparing a number of objects; patterning; building a construction of blocks; ordering, generalising and classifying; and more sophisticated mathematical operations; for instance, algebra, proving, predicting, and mental rotation of structures.

Booth and Siegler (2008) examined whether the quality of numerical magnitude representations of first-grade children with a mean age of 7.2 years was correlated with, predictive of and causally related to their learning of arithmetic. The children’s pre-test numerical magnitude representations were correlated with their pre-test arithmetic knowledge, and were predictive of their learning of answers to unfamiliar arithmetic problems. The relation to learning to solve unfamiliar problems remained after controlling for prior arithmetic knowledge, short-term memory for numbers and mathematics achievement test scores. In addition, presenting randomly chosen children with accurate visual representations of the magnitudes of addends and sums improved their learning of the answers to problems. Representations of numerical magnitude are both correlationally and causally related to arithmetic learning. These abilities are engaged not only in geometry, but also in number sense, comparing and calculating quantities, and effectively using strategies to solve problems.

Similarly, Gunderson and colleagues (2012), using two longitudinal data sets, found that children’s spatial skills and mental transformation ability, at the beginning of first and second grades, were a predictor of improvement in linear number-line knowledge over the course of the school year. Spatial skill at age five predicted performance on an approximate symbolic calculation task at age eight. This relationship was mediated by children’s linear number-line knowledge at age six. Similarly, working with 760 preadolescent college students and high- and low-ability college bound youths, Casey and colleagues (1995) found that spatial skill (as measured by the Vandenberg Mental Rotation Test) was highly related to success in mathematics. For all of the female samples, mental rotation predicted mathematics aptitude even when verbal aptitude scores were entered into the regression first. For the male samples, the relationship varied as a function of the ability of the sample. Overall, spatial skills are widely used in many levels of mathematical thinking and their development is considered a strong predictor of achievement in mathematics at primary school and other stages of education.

The Relationships between Music, Spatial Skills and Mathematics

Another strand of research has studied the relationships between music, spatial skills and mathematics. Understanding ratio enables children to calculate fractions, divisions and proportions, while pattern recognition is used in spatial-temporal tasks and in a broad variety of mathematical tasks. Schlaug and colleagues (2005) suggested a link between these skills and using rhythmic notation, while Gordon (1993) saw the link as being through the processing of structures of sound. Geist and colleagues (2012) argue that music is children’s first patterning experience and helps engage them in mathematics even though they do not recognise this.

Research has provided evidence for the relationships between music, spatial-temporal reasoning and mathematics. For instance, McDonel (2015) found strong correlations between musical aptitude, rhythm achievement and scores in numeracy tests. However, the sample size was very small, so the findings have to be interpreted cautiously. Spelke (2008) compared performance in tasks measuring performance on the three main systems supporting young children’s mathematical learning: representing small exact numbers of objects, large approximate numerical magnitudes, and representing geometric properties and relationships in students aged five to seventeen with no music training, with sports training, with training in other art forms and with music training which was considered on three levels of intensity: moderate, intense and highly intense. The first experiment, with children who had low levels of music training, did not show that such instruction enhanced any core mathematical skills. The second experiment included students with mixed levels of music training. Here, the children with intense music instruction outperformed the others in all tests related to spatial awareness. In the third experiment, students with extensive music training achieved higher scores in tests of sensitivity to geometry, including a task which assessed children’s ability to relate numerical and spatial magnitudes, and involved operations on a mental number line. Researching these relationships is complex, because musical training may be associated with some aspects of mathematics but not others. For instance, Bahna-James (1991) found that high-school students’ music theory grades correlated with their grades in algebra, geometry and pre-calculus, but not with grades on an advanced mathematics course on logic. Similarly, Bahr and Christensen (2000) reported that performance on a mathematics test and a musicianship rating scale correlated in areas where music and mathematics shared structural overlap in pattern recognition and symbol usage, but not for other areas of mathematics, where there was no overlap. However, not all of the research supports this. For instance, Helmbold and colleagues (2005) failed to demonstrate any advantage for musicians in pattern recognition.

Holmes and Hallam (2017), working with primary-school children showed correlations between music and only some, rather than all, mathematical skills related to spatial reasoning, while changes in mathematical skills reliant on memory were much smaller. This finding suggests that the development of spatial skills may act as a moderator between rhythmic instruction and attainment in mathematics. There were correlations between spatial reasoning scores and music performance. These were high for a picture test and a puzzle test score. Correlations between music score, the two puzzle tests and various mathematical performances showed strong correlations with some but not all mathematical tests. The strongest correlations were with two- and three-dimensional shapes. There were lower or no correlations with addition, subtraction, counting and number recognition.

Cranmore and Tunks (2015) adopted a qualitative approach asking 24 high school students to share their direct experiences with music and mathematics, as well as their perceptions of how the two fields were related. Participants were divided into four groups based on school music participation and level of achievement in mathematics. Most of the students saw mathematics as a foundation for musical ability, suggesting a different direction to most previous studies. Rhythm was perceived to have the most connections with mathematics.

Musical Engagement and Mathematical Performance

Some studies have concentrated on the impact that learning music might have on the development of specific cognitive skills which are considered useful in acquiring mathematical understanding; for instance, notions of proportions, fractions and patterns. Gardiner and colleagues (1996) showed that children participating in an arts programme—which included seven months of supplementary music lessons with a lower score on mathematics at baseline—outperformed controls in terms of mathematics achievement. Those participating for the longest period of time had the highest scores overall. As all of the groups participated in music and other arts, it was not possible to conclude that it was the music element that produced the effect.

Whitehead (2001) examined the effect of Orff Schulwerk music instruction on the mathematical scores of middle- and high-school students. Subjects were randomly placed into three groups: a full treatment group which received music instruction for 50 minutes five times each week, a limited treatment group which received 50 minutes of instruction once a week and a no treatment group which received no music instruction. After 20 weeks, the full treatment group showed higher significant gains in mathematics than the other two groups. The limited treatment group showed limited mathematics improvement and the no treatment group showed the lowest gain.

Ribeiro and Santos (2017) aimed to verify the efficacy of non-instrumental musical training on numerical cognition in children with low achievement in mathematics. Using cluster analysis, they examined whether children with low scores on numerical cognition would be grouped in the same cluster pre- and post-musical training. Primary-school children were divided into two groups according to their scores on an arithmetic test. Testing with a battery of numerical cognition tests revealed improvements for the children with low achievement in mathematics, especially for number production capacity, compared to normative data. The number of children with low scores in numerical cognition decreased after the intervention.

Neville and colleagues (2008) examined the differences in results between four groups of preschoolers who received music training; attention training; no training and general teaching delivered in a small group; and no training and general teaching in a large class. Music instruction was delivered daily and included listening to music, making music, moving to music and singing. The intervention lasted for eight weeks. A statistically significant change was recorded in numeracy and visual cognition for the music group and the attention group. Children from the music group performed especially well in verbal counting and estimating magnitudes. Similarly, Geoghegan and Mitchelmore (1996) investigated the impact of a weekly early-childhood music programme on the mathematics achievement of preschool children aged four to five. The group of children involved in musical activities scored higher on a mathematics achievement test than the control group, although home musical background may have been a confounding factor. The children who listened more frequently to adults singing and to their own music collection at home performed better than other children.

Cheek and Smith (1999) examined whether the type of music training was related to the mathematics achievement levels of eighth-grade students. Data were collected from the Iowa Academic Achievement Tests of Basic Skills and through a survey on participants’ music background, including type of musical instrument, number of years of school music lessons, number of years of private lessons and demographics. No significant difference was found between the mathematics scores of students who did and did not receive private music lessons. However, students with two or more years of private lessons had a significantly higher mean mathematics score than students with no private lessons. Furthermore, students who had keyboard lessons had significantly higher mathematics scores than students who had music lessons on other instruments.

In an innovative study, Kvet (1985) investigated whether significant differences existed in sixth-grade reading, language and mathematics achievement between students who were excused from regular classroom activities for the study of instrumental music and students not studying instrumental music. Over 2000 sixth-grade students participated. The analyses showed that there was no significant difference in sixth-grade reading, language and mathematics achievement between those who were excused from regular classroom activities for the study of instrumental music and those not studying instrumental music.

Focusing on emotions related to mathematics as well as achievement, An and colleagues (2014) studied 56 third-grade elementary students in a pre-post-test control group design, which was utilised to examine changes between two groups of participating students in mathematics achievement and dispositions, including beliefs about success, attitude, confidence, motivation and usefulness. The students in the music group received music-mathematics integrated lessons, while the students in the control group received traditional lecture- and textbook-based mathematics instruction. Analysis of the results demonstrated that, despite statistically equivalent pre-test scores prior to the intervention, after the intervention the music group students had statistically significantly higher positive mathematics disposition scores than their non-music-group peers. This suggests that there are advantages for teachers in utilising music-themed activities as a context for offering students the opportunity to learn mathematics in a challenging yet enjoyable learning environment.

While the evidence for the impact of musical activity on mathematics performance is mixed, some authors have proposed that there may be a link between the use of fractions and proportions in rhythm, and point out that the processing of these requires mathematic specific skills (Shaw, 2000; Schlaug et al., 2005; Jones, 2011). For instance, Courey and colleagues (2012) examined the effects of an academic music intervention on conceptual understanding of music notation, fraction symbols, fraction size and equivalency in third-graders from a multicultural, mixed socioeconomic public-school setting. Sixty-seven students were assigned in their class to their general education mathematics programme or to academic music instruction for 45 minutes, twice a week for six weeks. The academic music students used their conceptual understanding of music and fraction concepts to inform their solutions to fraction computation problems. Statistical analysis revealed significant differences between experimental and comparison students’ music and fraction concepts, and fraction computation following the intervention, with large effect sizes. Students who began instruction with less fraction knowledge responded well to the intervention and produced post-test scores similar to their higher achieving peers. Similarly, Azaryahu and colleagues (2019) examined the effect of two integrated intervention programs representing holistic versus acoustic approaches to teaching fraction knowledge. Three classes of fourth-grade children attended 12 lessons on fractions. One class attended the MusiMath holistic programme focusing on rhythm within the melody, while the second class attended the academic music acoustic programme (Courey et al., 2012) which used rhythm only. The third class of children received regular mathematical lessons on fractions. Students in both music programmes learned to write musical notation and perform rhythmic patterns through clapping and drumming as part of their fraction lessons. They worked toward adding musical notes to produce a number fraction, and created addition–subtraction problems with musical notes. The music programme used a 4/4 time signature with crotchets, quavers and semiquavers. In the mathematics lessons, the students learned the analogy between musical durations and half, quarter and eighth fractions, but also practised other fractions. Music and mathematics skills were assessed before, immediately following, and three and six months after the intervention. The analysis indicated that only the MusiMath group showed greater transfer to intervention trained and untrained fractions than the comparison group. The academic music group showed a positive trend on trained fractions. Despite this, both music groups outperformed the comparison group three and six months after the intervention on the trained fractions. Only the MusiMath group demonstrated greater gains in untrained fractions. Similarly, Hamilton and colleagues (2018) describe a pilot study which aimed to determine whether understanding in mathematics, and specifically, fractions, equivalence, ordinance and division improved when music and musical rhythm were used in lessons. The preliminary data suggested that students responded positively to this novel method of teaching in terms of engagement but also test performance.

Focusing on piano keyboard skills, Johnson and Edelson (2003) developed an activity for teaching children aspects of mathematics through musical concepts, including the use of musical instruments and musical symbols, to expand the concepts of serial order, fractions, sorting, classification and ratios. They concluded that music had the potential to assist in developing mathematical skills. Also using piano keyboard lessons but combined with a video game, Graziano and colleagues (1999) demonstrated that preschool children given six months of piano keyboard lessons improved dramatically on spatial-temporal reasoning, while children in appropriate control groups did not improve. The researchers also developed a Spatia1 maths video game which was designed to teach fractions and proportional mathematics. It was extremely successful in a study involving 237 second-grade children, aged six to eight years old. The children participating in the piano keyboard training as well as the maths video game scored significantly higher on proportional mathematics and fractions than children who experienced non-musical training along with the maths video game. Lim and colleagues (2018) investigated future teachers’ experiences and perceptions of using a virtual reality game for elementary maths education. The virtual reality game was designed and developed to integrate a musical activity, beat-making, into the learning of fractions. The mathematics education students who participated perceived that the concept of fractions was effectively represented via beat-making in the virtual reality game.

Wentworth (2019) explored the effectiveness of an integrated approach to music and mathematics in high school. Four lessons were taught to an intervention and control class to determine how mathematically motivated music instruction affected students’ understanding of operations of functions, composition of functions, inverse functions, domain and range. A pre-post-test design was used to determine the effect on achievement of the integrated lessons; a questionnaire was also given out, to identify differences in students’ mathematical perceptions, self-efficacy and determination. The intervention group demonstrated significantly greater gains overall. Three major differences were identified between the groups—the intervention group used function notation more frequently than the control group; the control group demonstrated confusion between composition of functions and inverse functions, while the intervention group did not; and the intervention group showed more mathematical work for the applications portion of the test than the control group. The integrated instruction led to comparable and, in some cases, significantly better mathematics outcomes than the control group, giving students an increased willingness to work with mathematical applications both on the post-test and moving forward.

Not all of the research has shown that music has a positive effect on learning mathematics. For instance, Costa-Giomi (2004) worked with nine- to ten-year-old children from low-income families, who were involved over three years with weekly individual piano lessons. All of the children who participated in the study were given an instrument so that they could practise at home. Self-esteem and musical understanding were enhanced for the music group, but their academic achievement in mathematics and English was no different from a control group.

The quality of the musical input is crucial in any transfer of skills. This was illustrated in a three-year study to explore whether group music instruction could improve the test scores of economically disadvantaged elementary-school children (including almost 600 kindergarten to fifth-grade students from four elementary schools). One school provided 30 minutes of keyboard lessons per week, another a 40-minute lesson every six days, while the remaining two schools acted as controls. All lessons were in groups of 20 to 25 pupils. Participants were pre-tested with two subtests measuring verbal abilities, two measuring quantitative abilities and one measuring spatial-temporal abilities. Tests were then repeated at 9, 18 and 27 months. During the first two years of the study, there were difficulties in the implementation of the music programme, and it was only at the end of the study when the children had received one year of high-quality tuition that there were any gains (Rauscher, 2005).

Similarly, Yang and colleagues (2014) examined the relationship between long-term music training and child development based on 250 Chinese elementary-school students’ academic development of first language, second language and mathematics. They found that the musician children outperformed the non-musician children only on musical achievement and second language development. Although music training appeared to be correlated with children’s final academic development of first- and second-language learning and mathematics, it did not independently contribute to the development of the first language, nor all mathematical skills. The authors argued that other variables might be important; for example parents’ level of education.

Two experiments by Rickard and colleagues (2012) also revealed inconclusive results. The first was based on an already existing music programme for ten- to thirteen-year-olds. Comparing participation in drama, art or music groups, there was some improvement in the music group on a non-verbal IQ test but not in academic achievement. A second musical intervention, provided externally over six months, included playing music with percussion instruments, composing, improvising, playing in a group, singing, active listening and analysis of a wide range of styles. As the programme was introduced in a private school, all of the students were of middle or high economic status. Three groups participated in music, drama or an additional activity. Students from the music group achieved better results in mathematics but this result was also in evidence in the drama group. The authors argued that the age of the children may have been a factor in the outcomes of the research.

Cox and Stephens (2006) compared high-school students with different numbers of music credits in relation to their mean mathematics grade point averages, or their mean cumulative grade point averages. Students were then separated into two groups based on the number of music credits. Those who had earned at least two music credits per grade level were placed into Group A. This included ninth-graders with two or more music credits, tenth graders with four or more music credits, eleventh graders with six or more music credits, and twelfth graders with eight or more music credits. The remaining students were placed into Group B. The group A students performed better than the group B students but the differences were not statistically significant, although there was a slight upward trend in grade point average as the number of music credits increased. Lower grade point averages were non-existent as music credits increased.

Overview

There have been a number of reviews of the impact of music on cognitive skills, including spatial reasoning and mathematics. In relation to spatial reasoning, Hetland (2000a) reviewed 15 studies and found a strong and reliable relationship and concluded that music instruction led to dramatic improvement in performance on spatial temporal measures. She commented on the consistency of the effects and likened them to differences of one inch in height or about 84 points on the SAT (p. 221). She showed that the effects were likely to be stronger among younger children, three to five years than those aged six to twelve years. Similarly, Črnčec and colleagues (2006) reviewed the evidence on the impact of music teaching on spatio-temporal reasoning skills and found that there was a consistent effect, although improvements in associated academic domains, such as arithmetic, had not been reliably shown. More recent research( as reviewed above) generally supports a positive role for music in developing spatial temporal reasoning skills, the consistency of the findings suggesting a near transfer, automated effect. Where spatial temporal skills are well developed the wider and more appropriate the choice of strategies and the more efficient and less erroneous mathematical operations. Rhythm-based instruction seems to be the most conducive for the improvement of spatial temporal reasoning skills, followed by learning to play the piano. Singing leads to smaller changes.

Focusing on the impact of engaging with music on a broader range of academic skills and educational attainment, including mathematics, Hodges and O’Connell (2007) suggest that a moderate position needs to be taken. At one extreme, the data support the contention that music improves academic performance, but at the other extreme there is no basis for saying that music instruction has no effect on academic achievement. Hodges and O’Connell argue that human learning is so complex that any simplistic explanations must be rejected. They suggest that some music experiences have a positive impact on academic performance under certain circumstances. What is neglected is the impact that an individual teacher can have. Excellent teachers who are enthusiastic and who relate well to students may make a greater difference to educational outcomes than particular methods used, although if the overall quality of tuition is poor it can have a negative impact. A more recent systematic review also suggests that the findings are inconclusive and contradictory (Jaschke et al., 2013). Jaschke and colleagues attribute this to differences in research design, the analytical methods used, the nature of the musical interventions and differences in neural activation during the processing of these tasks. Recent meta-analyses have come to similar conclusions about the challenges faced by research and some have concluded that overall, music does not have an impact on children’s cognitive skills including mathematics (Sala and Gobet, 2020).

To conclude, the evidence suggests that active engagement in musical activities enhances a range of spatial processing skills, particularly in young children. These skills may support the development of some simple mathematical skills, mediated by line number knowledge, but do not transfer to all mathematical skills. There remain many questions about the type of training that may be effective, that involving rhythm seems a likely candidate; how long training needs to be sustained, the type of mathematical activities which may be most influenced by musical activity, and the overall quality of the training on offer.