A multi-digit number writing model based on a hierarchical representation of the number syntax
Read the full dissertation (in Hebrew)…
My thesis examined the cognitive processes involved in writing numbers. Understanding these processes is important to extend the theoretical knowledge in this area, and to help develop tools and methods that may improve how we teach mathematics, especially to those who experience math difficulties. The existing knowledge about the number writing processes is limited: we know there are separate processes for handling the number structure and for handling individual digits or number words, but the exact nature of these processes is still unclear. In this study I focused on the processing of the number’s structure.
I dictated numbers with 3-6 digits to 21 adults with no learning disorders. The participants heard each number and wrote it on a digital tablet that recorded the pen movement. I analyzed the time gaps between adjacent digits. In doing so I relied on previous studies that have identified that the syntactic structure of the verbal number affects how people write it as a sequence of digits, and specifically affects the processing time before writing each digit.
I found a hierarchical pattern of time gaps: the gap between the triplets was longer than the gaps before the decade digit in each triplet.; in the right triplet, the gap before the decade digit was longer than the gap before the unit digit. This hierarchical pattern was also observed also when controlling for eacj number word’s phonological length. This pattern supports the idea that there is a tree-like hierarchical representation of the syntactic structure of the number. I found different patterns of gaps when the same numbers were dictated as a series of digits (“one, two, three”), i.e. when there was no syntactic structure, compared to when they were dictated as a structured multi-digit number (“one hundred and twenty-three”). This difference further supports the conclusion that the hierarchical representation reflects the syntactic structure of the multi-digit verbal number rather than some features of the digit sequence. Furthermore, the gap between the triplets was different when the number was dictated as a pair of triplets (without the word “thousand” between them) and when the same number was dictated as a 6-digit number. I concluded that the hierarchical representation merges the two triplets.
In the analysis of the time gaps before the decade and unit digits in the right triplet, and before the hundred digit in 4-digit numbers, I found shorter time gaps before the digit 0 than before other digits in the same positions. In contrast, in 5- and 6-digit numbers, the time gap before the hundred digit was longer when the hundred digit was 0 than when it was another digit. I concluded that the numbers are processed in chunks: the processing of 0 is faster than the processing of other digits when the 0 appears in the middle of a chunk, but slower when it appears at the beginning of the chunk. This is probably because the digit 0 at the beginning of the chunk creates an irregular structure of the chunk. In any case, these findings show unequivocally that the effect of 0 on writing times originates in its syntactic status and not in the grapho-motor properties of writing the digit 0.
The effect of the number structure was reflected not only in the digit 0 but also in the digit 1: the gap before the units digit was shorter when the decade digit was 1 than when it was 2-9. I concluded that a teen numbers (11-19) is processed as a single unit.
The patterns of time gaps in the left triplet were different from the right triplet. The hierarchical pattern – a longer gap before the decade digit than before the units – was not observed in the left triplet. Furthermore, the effect of the digit 0 was not found within the left triplet, and in 5-6 digit numbers with an irregular structure (0 in the hundred digit), within-triplet time gaps were longer in the left triplet than in the right one. I concluded that the processing of the left triplet is different from the right triplet. I propose that the participants started processing the number’s structure while they were still hearing it, before they started writing it on paper.
Overall, this study indicates that writing multi-digit numbers involves a hierarchical representation of the syntactic structure of the number. The number’s syntactic structure is processed “in real time”, as it is being heard, and is affected by the digits 0 and 1. The number writing model proposed here is more detailed than any previous models. This model describes the writing process in adults without learning disorders, and can form the basis for analyzing writing patterns also in children or adults with learning disorders.