Evidence addressing development of natural number conceptualization comes from improvements in math found related to musical pitch skill learning in first and second graders (Gardiner Reference Gardiner2000; Reference Gardiner, Vrobel, Rössler and Marks-Tarlow2008a; Gardiner et al. Reference Gardiner, Fox, Knowles and Jeffrey1996). The specificity of connection could not be explained fully by improved general learning. But since both these areas of skill profit at this stage from mental engagement (Gardiner Reference Gardiner, Vrobel, Rössler and Marks-Tarlow2008a) involving sequential representation, even if in different ways, this similarity may well account for the pitch–math connection (Gardiner Reference Gardiner2000; Reference Gardiner, Lerner, Schaimberg, Anderson and Miller2003; Reference Gardiner, Vrobel, Rössler and Marks-Tarlow2008a). This and further evidence to be discussed supports Rips et al.'s proposal in the target article, that innate processing capability has an essential role in the development of natural number concept; but it implies that innate capability concerning sequence is still more foundational than the capability at math-related abstraction that Rips et al. propose.
To briefly review the learning evidence: First-grade students receiving arts skill “test arts” training (1 hr 40 min/week) accelerated significantly at math, but not similarly at reading, passing controls who received arts teaching that emphasized appreciation rather than skill (50 min/week) (Gardiner Reference Gardiner2000; Gardiner et al. Reference Gardiner, Fox, Knowles and Jeffrey1996); this occurred whether the skill training students were at the bottom, middle, or top of the class leaving kindergarten. At the end of second grade, those with two years of test arts did best, those with one year next best, and those with no test arts most poorly at math, but again not at reading. The test arts component that developed arts skill most substantially was the Kodaly method music training (Chosky Reference Chosky1981).
Studies with Kodaly alone again found math improvements related to the extent of the Kodaly training (Gardiner Reference Gardiner2000). Students receiving Kodaly and Kodaly-related training since at least the first grade have now been found to perform significantly better on math at the end of the third grade, compared to all other students in their district as a group (79% vs. 48% at grade level, p<.001), and 17 – 34% better, regardless of whether or not they had diagnosed learning disability, or familial poverty (Gardiner Reference Gardiner2008b).
First- and second-grade math progress correlated significantly with Kodaly-trained capability involving musical pitch but not rhythm (Gardiner Reference Gardiner2000). This may seem surprising, as musical rhythm is often taught as involving counting. But Kodaly method avoids rhythm counting, instead emphasizing discrimination between rhythm pattern types.
I have proposed (Gardiner Reference Gardiner1998; Reference Gardiner2000; Reference Gardiner, Vrobel, Rössler and Marks-Tarlow2008a) that the positive effects of pitch and math learning on one another reflect an interaction that moves mental engagements advantageously toward incorporating sequence, away from less favorable options. In math, engagement with the sequential nature of the natural numbers provides a foundation for developing engagement, with addition and subtraction as operations both involving movement, but in opposite directions along the natural number sequence. In pitch engagement, the sequential representation of ordering of discrete pitches from lower to higher according to vibration frequency within the musical scale, underlies the developing sense of upward and downward movement in perceived melody and helps the young learners remember and sing accurately many different melodies built from the same scale.
The beginning math student has other options for engaging with arithmetic; for example, memorizing production rules like “1+2 gives 3.” The student learning melodies may develop and memorize the sequence of vocal gestures producing each melody separately. But such strategies are ultimately limiting. Further development in both mathematics and music builds advantageously upon the sequential conceptualizations of natural numbers and musical scales, respectively.
As Rips et al. discuss, conceptualization of the natural numbers must capture the basic Dedekind-Piano axioms (Dedekind Reference Dedekind1888/1963) of first element, successor, predecessor, and inductive continuation. Note to what extent the notion of sequence is at the heart of these particular axioms. Besides the possibility of infinite continuation, the elements of a musical scale also satisfy these axioms – even though the choice of musical pitches for a scale has no relationship to set cardinality, but rather, has to do with frequency interrelationships of interest to music. Indeed, most scales rise in pitch not only through “whole,” but also through “half” steps.
Our data thus suggest that innate “Basic Sequencing Capability (BSC),” adapted in so many different ways in skill learning (Gardiner Reference Gardiner2007), is also foundational to the development of the natural number concept. Like advances in counting and the arithmetic operations, development of further abstractions concerning the natural numbers can then profit from mental representation capturing the sequential nature of natural numbers.
That representation concerning sequential properties of natural number is especially foundational, is further implied by the variety of engagement applications involving the ordering of natural numbers that must be developed. Natural numbers are directly related to set cardinality when discrete objects are counted; however, ordinal applications (first, second, third, etc.) concern position in a counting sequence but not set cardinality, and applications concerning measurement require a further bridge, again to sequence (Gardiner Reference Gardiner2007).
If innate capability concerning sequence is critical to natural number development, as I suggest, then the extensive development and practice by young children at counting (Gelman & Gallistel Reference Gelman and Gallistel1978; cf. Gelman & Butterworth Reference Gelman and Butterworth2005), which already connects sequential operation to progression through natural numbers, may prepare for further development. The next step could then be to internalize the mechanisms involved in counting out loud, substituting for sequential auditory verbalization purely mental acts concerning sequential position and movement along sequence (Gardiner Reference Gardiner2007). Our evidence then could imply that experience with musical-scale engagement that also involves sequence, even if differently, can accelerate the development of a sequence based mental internalization of natural number engagement (Gardiner Reference Gardiner, Vrobel, Rössler and Marks-Tarlow2008a).
Rips et al. have comprehensively reviewed research showing that young learners have means other than counting for differentiating magnitudes and for establishing cardinality of small sets of objects. Such capabilities are important in their own ways, still exist beyond childhood, and can be disrupted along with capabilities based on natural numbers by injury or by experimental methods causing localized temporary disruption of brain processing capability (Cappelleti et al Reference Cappelleti, Barth, Fregni, Spelke and Pascual Leone2007). This can imply brain architecture that places processing involving natural number near other forms of quantitative processing, perhaps to facilitate interaction, but not necessarily that the quantification built on natural numbers is developed from these other engagement forms.
Further potentially valuable opportunities for mental interaction with the arts as learners advance in math capability is discussed elsewhere (Gardiner Reference Gardiner2007; Reference Gardiner2008c).
