If we consider the educational aspects of the possible positive effects of integrating mathematics and music, then we can observe on one side that in pre-schools and children’s educational TV shows, a new knowledge or story usually is presented through integrating verbal language, visual models, music and dance, which all are different intelligences in Gardner’s (1983) Multiple Intelligences theory. Conversely, this type of integrated mode of teaching by using a range of combined human intelligences, or by integration of different subjects, appears to be rarely used in schools and colleges, especially as the “efforts to integrate mathematics with music are rare” (Kleiman, 1991, p. 1). Regarding mathematics education, this means that in schools and colleges a new mathematical concept, that is, the concept of function, is usually introduced through using mainly the logical-mathematical intelligence channel. Other intelligence channels, excluding the verbal-linguistic and the visual- spatial ones, are rarely used. Because students possess different mixtures of skills, integrated teaching approaches can be employed to empower the understanding of mathematics by activating different intelligences. Students’ individual learning preferences can be addressed by a range of relevant teaching methods, and in the development of these methods Gardner's (1983) Multiple Intelligences theory is highly applicable and useful also in practice (Gardner, 1993).
Because students learn in different ways, using different intelligences, often more intelligences simultaneously, Munro (1994) suggests the followings for the teaching of mathematics:
Present mathematics ideas in a range of ways. When teachers are teaching
an idea, they can vary how they present it; in words either visually or auditorily, in pictures or as a series of actions. These things are under the control of teachers; they can select the input format.
Cueing students to think about the idea in different ways. Teachers can't
control how students think about the ideas being learnt. Even if ideas are presented visually, this does not mean that the students will visualize images.
Switching ways of thinking about ideas. Building ideas in one
representational format doesn't suit all students. Many students can be helped to learn an idea when they are cued to think of it in different ways.
Encouraging students to monitor how they learn best and to understand their preferred ways of learning. Students can learn to understand and
value their own approach to mathematics learning, to see that they can learn successfully.
(Munro, 1994, p. 9) My study has considered Gardner’s (1983) Multiple Intelligences theory, and Munro's (1994) suggestions above, mainly the first. The study has been designed in an effort to integrate mathematics with music, present the concept of function in four different ways (embodiments), and use multiple intelligences such as mathematical- logical (through algebraic and tabular representations of functions), musical (through audible representations of different aspects of functions by relevant author-designed musical analogies) and visual-spatial (through graphical representations of functions).
Wahl (1998) presents 14 activities and over 90 sub-activities of teaching mathematics up to school year level 10 through the original seven basic intelligences of Gardner's (1983) Multiple Intelligences theory. These activities are well presented, adaptable to students' various knowledge levels, empower both individual and group work, and are mathematically meaningful up to school year level 10. These activities, as models, were a source of inspiration for my study, which was designed to be well
presented, adaptable, empowering both individual and group work, and mathematically meaningful at Year 11 mathematics level.
I believe that this research study has expanded the theoretical and practical knowledge accumulated in Wahl's (1998) publication by targeting a higher school year level, Year 11 mathematics, and the concept of function through the active integration of mathematics and music. This study also addressed the mathematical- logical, visual-graphical, verbal and inter-personal intelligences explicitly, which together with the musical intelligence made a total of five out of the original seven basic intelligences of Gardner's (1983) theory. The two not explicitly addressed intelligences being the kinaesthetic and the intra-personal intelligences.
My study has considered the power of the relationships between mathematics and music in a unique, particular way, through using nine relevant musical analogies in order to describe in a non-traditional, musical-auditory way the six different aspects of the function concept explicitly requested by the Year 11 Mathematical Methods Australian Curriculum 2014.
The Hungarian born world-famous mathematician and cognitive psychologist Zoltán P. Dienes (1916-2014), who created the field of Psychomathematics, the psychology of mathematics learning, worked around the globe, in England, France, Germany, Italy, Australia (mathematics education professor at The University of Adelaide in the late 1950s and early 1960s), New Guinea, USA, Chile, Brazil, Argentina and Canada, has had an important contribution to the 20th century's mathematics education. Dienes (1987) has developed a sequence of different structured activities, named lessons, integrating music and literature to teach both explicit- and hidden mathematics, confirming that learning real mathematics has to be an enjoyable process, and has used innovative games, songs, poetry and dance to teach abstract mathematical concepts. I was inspired by this approach to mathematics teaching, being curious how these ideas might apply in practice. I was inclined towards finding answers to questions such as how does it work, how effective it is using games, music and dance during mathematics lessons?
In order to have a deeper inside look into these type of questions, to check how this works in my mathematics teaching practice, I initiated, designed and coordinated a non-conventional study, titled “Dancematics” (Nagy, 2008), involving a small group of four Year 11 mathematics and dance students, aiming for the integration of mathematics and dance. The “Dancematics” project was presented by a student at the South Australian Mathematics Talent Quest 2008 (SAMTQ 2008), organised by The Mathematical Association of South Australia in 2008, and has been awarded a “Judges Commendation for Independent Initiative in Exploring Maths in Dance”. At that time I was not prepared to perform a research study by posing an appropriate research question and collecting and analysing relevant data. Despite the fact that there are a range of auditable documents which have been prepared by the “Dancematics” project, there are recorded videos of the mathematics presentation by the student on a DVD, and the success of the project at SAMTQ 2008 was published in the Term 4 Newsletter 2008 of The Mathematical Association of South Australia, I, as the coordinator of the project, did not consider it a proper research study.
The creator of the Six-stage Theory of Learning Mathematics, Dienes (2004) identified an analogy between interpreting music and teaching mathematics, and also posed a question regarding finding other ways to teach the beauties of mathematics, stating that:
The concert pianist makes gigantic efforts to interpret the amazing complexity of a Beethoven sonata to the lay public by decoding an abstruse symbol system and transforming it into an enjoyable artistic experience. Why could we not find ways to interpret the beauties of mathematics to the lay public through the methods more accessible to the general public?
(Dienes, 2004, p. 12)
An earlier dated, but a similar statement is in Dienes (1966): “Just as music is accepted as a valuable part of the general education of all, with the few excelling, so in mathematics everyone can reach enjoyment, receiving its long-valued stimulation and satisfaction” (p. 14).
Dienes observed that “the reason mathematics is boring in schools is because no real mathematics is taught in schools” (Dienes, 25/04/2006, as cited in Sriraman & Lesh, 2007, p. 67).
Bálint (2015) summarised Dienes' work and states that the “new mathematics” is the mathematics of structures. Teaching the new mathematics is not about teaching a sequence of knowledge but to help students to develop their authentic mathematical mode of thinking, to form mathematical structures mentally. Dienes has developed games which made possible to observe, experiment, understand and represent these structures. Representing these structures is possible through multiple embodiments of the abstract relationships.
I made an effort in this study to find a way to interpret the beauties of mathematics, to teach “real mathematics” through a method which I believe was really appreciated by the participating students: using structural analogies between mathematics and music. The present study was inspired by Dienes’ (1960, 1966, 1987, 2000a, 2000b, 2001, 2004) general work, including the above mentioned analogy between interpreting music and teaching mathematics and the possibility to reach students' enjoyment in mathematics. Instead of developing a game, which would have been a 3D model, I have developed analogies which show similar structures between function and music. My approach was one of many possible embodiments, using music. I have embodied the multiple facets of a mathematical abstract concept in music. I believe that this study fully confirms Dienes’ (2004) findings and statements in a particular way.
The American cognitive psychologist Jerome S. Bruner (1915–2016), who made fundamental contributions to cognitive learning theory in educational psychology in the 20th century, emphasises the role and importance of both the teacher and the
The teacher is not only a communicator but a model. Somebody who does not see anything beautiful or powerful about mathematics is not likely to ignite others with a sense of the intrinsic excitement of the subject. A teacher who will not or cannot give play to his own intuitiveness is not likely to be effective in encouraging intuition in his students.
(Bruner, 1960, p. 90) I have seen and appreciate the beauty and power of mathematics since decades and I think that I have ignited many of my students with a sense of excitement of mathematics and have encouraged the intuition of many of my students effectively. Bruner (1960) states that “The effort to give visible embodiment to ideas in mathematics is of the same order as the laboratory work” (p. 81). I made an effort to give both visible and audible embodiments of different aspects of function in this study and I confirm Bruner's statement. The effort to give visible and audible embodiments was a very pleasant and exciting work, similar to a pleasant and exciting laboratory work.
Learning mathematics is a long, complex process which can be aided, in part, by some non-mathematical approaches. One of the possible non-mathematical aids is using different skill transfer effects between music and mathematics (McKeachie, 1987). My study has been designed to measure, firstly, the change (increase or decrease) of the cognitive skills of understanding of function, and secondly, the change of non-cognitive skills of beliefs and attitudes towards mathematics at senior secondary level, when the process is aided by skill transfer effects between music and mathematics. In order to perform this action research, I designed and presented nine musical analogies – musical embodiments – of different aspects of function using “a multiple embodiment approach – introducing the function concept in a variety of settings with the hope of effecting transfer of learning” (Eisenberg, 2002, p. 141).
Regarding the possible positive effects of music on mathematics achievement, Kelstrom (1998) stated that these have been poorly employed in the curriculum. Rauscher (2003) concludes that “music may act as a catalyst for cognitive abilities in other disciplines” (p. 4). I believe that my study confirms Rauscher's (2003) statement in a particular way, with regards to senior secondary mathematics students' cognitive abilities concerning the mathematical concept of function.
In their study about common misperceptions in learning mathematics and music and the interactions of these disciplines, Malone, Leong and Lamb (2001, p. 6) state that “music provides the reality and physical experiences corresponding to mathematical abstractions”. I believe that my study totally confirms Malone et al.'s (2001) observation because my study did show through analogies that music provides the reality and physical audio experiences corresponding to mathematical abstractions of different aspects of function.
Using analogies in teaching can be beneficial in observing inter-concept similarities, understanding new concepts and in improving attitudes (Treagust, 2001). I believe that, by targeting the concept of function, using analogies and observing inter- concept similarities between different aspects of the mathematical concept of function and the similar aspects of the musical concept of melody, my study also confirms and fully supports Treagust’s (2001) findings.