Transcoding and math

By Prof Chris Donlan

 

Why is the game of Bingo so popular? Perhaps because there’s a simple pleasure in matching a spoken number, e.g. “sixty six”, to its Arabic numeral form ‘66’. Spoken numbers and Arabic numerals are complementary codes. Each pair of items is a unique match. Translating from one to the other is called ‘transcoding’, and it’s a feature of everyday life. If someone asks you the time, you might look your phone and, without thinking, transcode 09:10 to “nine ten” or “ten past nine”. Transcoding becomes effortless, for most of us. But it has to be learned, and that learning starts early in childhood.

 A recent US study found that 70% of a sample of pre-schoolers were able to choose reliably between numerals ‘36’ and ‘306’ when hearing the spoken number “thirty-six”.

Recent longitudinal studies have confirmed the essential role of early transcoding skills in the development of later arithmetic competence. In order to use the number system effectively, there is a need to understand place value, the principle whereby the position of a digit registers its value. Not only is ‘306’ greater than ‘36’, but also ‘21’ is greater than ‘12’, and ‘321’ is greater than ‘123’, etcetera. A recent Canadian study found that most children were able to apply the place-value system by the age of 6. Importantly, those who were able to read multi-digit numerals aloud (i.e. to transcode them) were most likely to apply the place value rule. 

The linkage between spoken numbers and Arabic numerals is seen clearly in children’s early efforts to write multidigit numbers to dictation. For “five hundred and sixty-two” they may write ‘500602’. Here the regularities of the spoken English forms are evident in the child’s production. At this stage of learning each element of the spoken form is fully expressed. The ordering of the spoken form corresponds to the left-right structure of multi-digits and provides a scaffold for learning about place value.  However, the English number system, while transparent for transcoding hundreds (“five hundred” represents ‘5 x 100’), and semi-transparent for decades (“sixty” represents ‘6 x 10’),  presents major challenges in teen numbers where spoken forms reverse the order of their Arabic equivalents, e.g. “fourteen” for ‘14’, and sound confusingly similar to the decade forms, e.g. “forty”.

Spoken number systems vary widely between languages. Portuguese, for example, assimilates the sounds of the hundreds. The spoken form of ‘500’ is expressed as “quinhentos” instead of “cinco cem”, with consequent challenges for the learner. Many Asian languages, in contrast provide a fully transparent correspondence between spoken and Arabic forms. Some researchers view this systematicity as a significant advantage in mathematical development. 

Children across the world acquire transcoding skills, many learning through informal exposure (like through the Maths@Home games). However, some individuals struggle, especially if they have learning difficulties and if the number systems to which they are exposed are not transparent. These learners may require structured exposure, building on regularities to provide a firm basis for the development of number knowledge. 

Further Reading

Gilmore, C., Goebel S.M., Inglis M. (2018). An introduction to mathematical cognition. Abingdon: Routledge. Chapter 3, Symbolic Number, pp. 29-49. 

Yuan, L., Prather, R., Mix, K.S., Smith, L.B. (2019). Preschoolers and multi-digit numbers: A path to mathematics through the symbols themselves. Cognition, 189, 89-104. 

https://doi.org/10.1016/j.cognition.2019.03.013

Habermann, S., Donlan, C., Göbel, S.M.,Hulme, C. (2020) The critical role of Arabic numeral knowledge as a longitudinal predictor of arithmetic development. Journal of Experimental Child Psychology, 193, 104794. https://doi.org/10.1016/j.jecp.2019.104794

Cheung, P., Ansari, D. (2021). Cracking the Code of Place Value: The Relationship Between Place and Value Takes Years to Master. Developmental Psychology, 57, 227-240. https://doi.org/10.1037/dev0001145