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Discrete versus continuous
We’ve been looking in detail at the early experiences of number. One aspect that has ramifications at multiple levels is the tension and interplay between discrete and continuous. We are not simply referring to the metaphysical paradoxes that Zeno (Dantzig, 1954) arrived at when representing distance/time either as discrete quanta or as an infinitely divisible continuum but considerations that directly relate to the learning, teaching and use of mathematics. Below are three instances.
PART A: Parallel Individuation versus Approximate Number System
Neuroscientific research (Izard et al, 2008), (Leibovich et al, 2016) suggests that the infant brain has two distinct ‘systems’ for processing number – one that handles small discrete numerosities (up to 3?) and the other that uses a continuous/analogue mechanism for estimating larger numerosities. There are studies showing that the strength of these systems is correlated with mathematical achievement but there isn’t consensus on how these systems relate to exact calculations. A critical question is how these dual approaches could be leveraged to better support a child’s conception and use of number.
PART B: Counting first or measuring first? Arithmetic first or algebra first?
Given that our neurological ‘number line’ represents magnitudes approximately and ‘the extent to which two numbers can be discriminated depends on their ratio’ (Izard et al, 2008), there are some who believe that early teaching of number should foreground estimation and multiplicative reasoning rather than working additively. Separately, Davydov and others have suggested that starting with comparison and measurement of continuous quantities could provide a more secure foundation for mathematics. In particular, it would not create an artificial divide between whole and fractional numbers. Related references can be found in Keith Devlin’s blog, Should Children Learn Math by Starting with Counting?
This also ties in with whether we develop arithmetic first or algebra first, whether we progress from the specific to the general or the general to the specific (Mason 2016).
Of course, we need not see these as either/or questions. But developing these themes concurrently raises subtle questions on ordering particular exposures, how connections are made and what learners and teachers need to attend to. A key feature is the choice and use of units, which enable the distilling of magnitudes to number. This could afford an early and natural introduction to counting, division and remainders, and multiple representations (counting in 2s, 5s, 10s, 12s, ... can also be seen as a choice of units).
PART C: The number line
The paradox and power of the interplay of discrete and continuous come to the fore in the model of the number line. The Dutch, and others, have made good use of the empty number line (Anghileri, 2001) in supporting children’s arithmetic calculations. Yet Sarama and Clements (2009), state that ‘there is much to be cautious about in considering the use of the number line as a representation for beginning arithmetic’. Though they also report that children in New Zealand do better with the number line. How then should learners be supported to develop a holistic sense of the number line, incorporating whole and non-whole numbers, especially given later development of the density of the rationals, and the existence of the irrationals. Would alternate representations of number (numbers as areas, arrays, graphs, tangles, games, ...) be helpful?
Anghileri, J. (ed.) (2001) Principles and practices in arithmetic teaching: innovative approaches for the primary classroom, Open University Press.
Dantzig, T. (1954) Number: The Language of Science. 4th edition, Macmillan.
Devlin, K. (2009) Should Children Learn Math by Starting with Counting? Devlin’s Angle (blog) http://www.maa.org/external_archive/devlin/devlin_01_09.html
Izard, V., Pica P., Spelke, E., and Dehaene, S. (2008) Exact equality and successor function: two key concepts on the path towards understanding exact numbers. Philosophical Psychology 21(4), pp. 491-505. http://nrs.harvard.edu/urn-3:HUL.InstRepos:10139289
Leibovich, T., Katzin, N., Harel, M. and Henik, A. (2016) ‘From “sense of number” to “sense of magnitude” – The role of continuous magnitudes in numerical cognition’, Behavioral and Brain Sciences, pp. 1–62. http://dx.doi.org/10.1017/S0140525X16000960
Mason, J. (2016) Overcoming the Algebra Barrier: being particular about the general, and generally looking beyond the particular, in homage to Mary Boole. In And the Rest is Just Algebra, S. Stewart (Ed.), Springer pp. 97-117. http://link.springer.com/chapter/10.1007/978-3-319-45053-7_6
Sarama, J. & Clements, D. (2009) Early childhood mathematics education research: Learning trajectories for young children. Routledge.