In continuing the argument for spatial reasoning and its relationship to both mathematical and musical problem solving, it should be helpful to consider Gestalt theory. Original proponents of this theory, Wertheimer (1922), Koffka (1922) and Köhler (1929), elucidated the tendency for humans and nonhumans to group stimuli together and to perceive the whole versus the parts.
This “grouping” predisposition causes one to structure and interpret visual, auditory or conceptual phenomena based on proximity, similarity, closure and or simplicity according to
symmetry, regularity and smoothness. These factors were referenced as the laws of organisation and explained in the context of perception and problem solving. Wertheimer (1922, 1959) was
particularly interested in the problem-solving processes of geniuses such as Einstein as well as children; he saw the essence of successful problem-solving behaviour as the ability to see the overall structure of a problem. Luchins & Luchins were two of many Gestalt psychologists who applied Gestalt theory to mathematical pedagogy (Wertheimer, 1922, 1959; Luchins & Luchins, 1985, 1988, 1997; Arcavi, 2003). Additionally, this theory may explain why one normally hears a musical tone as a whole rather than in terms of separate overtone fragments; this allows insight regarding suggested writing rules and cognitive results of musical composition as well (Lerdahl & Jackendoff, 1983; Levitin, 2006, pp. 76-82).
Richardson (2004) illustrated the existence of Gestalt theory in an illustration of children’s use of spatial logic in her preschool classroom. Originally, she had asked her students to count the number of dots on cards (similar to dot arrangements on dice). Yet instead of counting, they made shapes to match what they had seen, such as an “X” shape to show the five-dot configuration and a square shape to show the nine-dot configuration. This example supports the argument by van Nes & de Lange (2007, p. 219) of the propensity of spatial logic to exist prior to and in support of numeracy.
As an illustration of how music can enhance spatial reasoning and therefore logical thought, a case study of a blind learner will be summarised. Clearly, a case study of one person should not be generalised, yet it may provide insight into the potential that active and evaluative listening may hold in encouraging logical thinking. This student, in his early 20s, exhibited certain cognitive difficulties apparently deriving from an episodic rather than holistic, Gestaltian perception of reality. His spatial reasoning and ability to make connections and distinguish relevancies was lacking possibly due to the scarcity of stable systems of reference by which to organise space. Based on the cognitive
intervention programme, Instrumental Enrichment (Feuerstein, Rand, Hoffman & Miller, 1980), an intervention was undertaken, which utilised tactile and analytical musical methodologies. This was
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developed by Gouzman (2000) in order to address cognitive deficiencies that are relatively common for the blind learner.
Before listening to numerous musical selections, three compositional techniques of musical form were discussed: repetition, used to create a sense of familiarity, balance, unity and symmetry,
contrast and variation, the former of the two also used for symmetry and balance and both for
novelty as well as to keep some elements of a musical thought while changing others. Aided by this vocabulary, the student was then guided through numerous listening sessions that encouraged him to analyse the structure of each piece, including how differing elements make up the whole. In order to activate abstract and yet focused thinking, he was asked to aurally follow the pathways of the music and to locate goals and points of articulation throughout. Listening to and locating specific aspects of the music is analogous to visually or kinesthetically locating multiple points and angles on a
geometric shape or orienting oneself within a space.
After only three sessions, cognitive skills such as strategic focusing, comparative and hypothetical thinking, communication of mental processes and Gestaltian perception, aided by identification of musical points of reference and overall organisation had been stimulated in this learner, as revealed by the transformed quality of his discussions. The first session had been primarily analytical, the second, which included improvisation and composition, encouraged creative problem solving and syntactical reasoning, building a whole from parts. The third session revealed an
emerging ability to hierarchically identify, classify and evaluate several sources of information. This case study suggests the potential that active listening and analysis of music holds for the development of organised and abstract reasoning (Portowitz, 2001). Again, this is a case study of solely one person and the results should not be generalised to an entire population; it is nonetheless reasonable to acknowledge that certain modes of thought could arise from such analytical work. My own experience helping children and adult learners to clarify their thoughts and develop coherent
reasoning via listening, evaluating, performing and discussing musical principles resonates with this example.
Additionally, music compositional and improvisational skills encourage both divergent thinking (associated with creativity) and convergent thinking (usually associated with problem solving). As with composition and improvisation, higher-order mathematical problems and calculations have been found to benefit from both forms of thinking also (Dunn, 1975; Haylock, 1987; Toshihiro, 2000; Cropley, 2006).
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Nes and de Lange (2007) point out the advantage of applying spatial structures to mathematical problems such as comparing numbers of objects, continuing patterns, building block constructions and identifying quantity using structure and grouping, for example, by seeing six as groups of three and three. Certain musical pieces are particularly effective for encouraging spatial-temporal
reasoning as well as counting, accurate quantity discernment, categorisation, pattern recognition and even cognitive task switching8. The song America, for instance (Bernstein & Sondheim, 1957) is in a mixed metre, with alternating 6/8 and 3/4 time signatures or rhythm patterns. It provides a good opportunity to teach numerous skills such as those mentioned above and to alert students to the role that rhythm can play in structure. See Figure 4 below.
1 2 3 4 5 6 1 & 2 & 3 &
1 2 3 1 2 3 1 2 1 2 1 2
Figure 4 Different methods of notation for the rhythm in “America” (Bernstein & Sondheim, 1957).
Beneath the standard method of notation shown at the top of Figure 4 above are two other ways to count the song “America” from the musical West Side Story (Bernstein & Sondheim, 1957). Though the underlying tempo and pulse remain constant, the accents (in bold) fall on different beats as reflected by the alternating time signature of 6/8 3/4 and are therefore grouped and counted accordingly. The top row is a “proper” way to count in line with the underlying eighth note (or quaver) pulses within this alternating time signature. The first measure (bar) could also be counted as two large beats, each containing three small beats within (compound duple metre, or compound time) followed by the different emphases in the second measure, in which three large beats are each divided into halves. If considered in that way, one could count, “1 ee uh, 2 ee uh, 1&, 2&, 3&.” The bottom row shows an alternate way to count, which points out the accents and subsequent grouping structure and can be readily understood by learners whether or not they have had notational training. This musical piece provides an effective opportunity to teach multi-layered skills including counting, the
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retention of a steady beat while accentuating alternating ones, recognition of rhythmic patterns and structures as well as the flexibility of cognitive task switching. The top method of counting is most appropriate for students who are learning more complex standard rhythmic notations.
Mathematical abilities such as ordering, comparing, generalising and classifying are supported by an ability to grasp spatial structure (National Council of Teachers of Mathematics, 2000; Waters, 2004). More formal, complex operations such as addition, subtraction, multiplication and use of algebraic variables also benefit from a solid foundation in spatial reasoning (Kieran, 2004; van Nes & de Lange, 2007). Research has shown that children with serious mathematical difficulties tend to use minimal levels of structure if at all (Mulligan, Mitchelmore & Prescott, 2005). Therefore, it seems clear that improving spatial and spatial-temporal reasoning is important for later mathematical development.
As shown and discussed in multiple ways, music education may provide a method for assisting growth in that area. This suggests possibilities for a comprehensive education that includes an inter- disciplinary approach for the enhancement of spatial, structural and comprehensive mathematical skills. It would not be appropriate to simply substitute music training for spatial and structural awareness guidance in the classroom, but it appears to be potentially helpful as a supplement.