Number notation effect
Most of the time when we write numbers and do some calculations we use the Indo-Arabic number notation. Numbers can be denoted in many ways, for example, Roman notation is quite well known in our culture. Actually the two notation mentioned here belongs to two families of number notations: place-value notation (like the Indo-Arabic notation in which the power is denoted by the position of the digit) and sign-value notation (like the Roman notation in which the digits simply should be added up to get the number it denotes).
We wanted to know whether notation type has an effect on how we process numbers. There are quite a few studies that investigate the effect of modality of the input (e.g. visual vs. auditory), or the existence of notation independent number representation, or whether numbers are denoted with digits or letters, but we wanted to learn about the effect of the notation structure, whether sign-value and place-value notations are processed differently. We had quite a few motivations to do that, but none of those reasons came from psychology:
- First, while most of the mathematicians and historians say that calculation with place-value notations is efficient, a minority of scholars highlight the efficiency of sign-value notations. What are the cognitive advantages and drawbacks of these notation types? Computational considerations show that some features of the sign-value notation is misunderstood, and actually, sign-value notation can be used more simply than place-value systems.
- Second, most of the high cultures originally used a sign-value notation to denote numbers, and place-value notation was a much later development. We can observe many times that if something is complex for the human mind, then it will appear late in the history of culture. If this is the case here, then sign-value notation should be easy to handle for humans. In our study, because Indo-Arabic and Roman notation can not be used to investigate this question for several reasons, we introduced two artificial number notation: a sign-value notation and a place-value notation. Participants had to solve simple numerical tasks (comparison and addition). (The construction of the numbers, and the whole design is quite complicated, because we wanted the notation structure to be the only thing that differs between the two number systems and that can account for the results. So be prepared that reading those parts of our paper requires a considerable effort.) We consistently found that sign-value notations are easier to use than place-value notations. This is surprising in the light of the stressed efficiency of place-value notation, but it is quite consistent with out historical and computational considerations.
Finally, when we wanted to find a cognitive model for this number notation effect, we found that none of the known models can explain our results. Thus, we introduced an object based number representation. In this model numbers are represented as objects and groups. For example, 325 can be represented like 5 apples, 2 baskets (all including 10 apples each), and 3 sacks (all including 100 apples each). Sign-value notation can be easier because its notation form or structure is more similar to this object based representation than the place-value notation structure. (A similar explanation was used to account for how preschoolers understand zero.)
Krajcsi, A., & Szabó, E. (2012). The role of number notation: sign-value notation number processing is easier than place-value. Frontiers in Psychology, 3(463). https://doi.org/10.3389/fpsyg.2012.00463
Krajcsi, A., Szabó, E., & Dorfberger, S. (2011). The role of number notation in numerical processing. Presented at the Typical and Atypical Neurocognitive Aspects of Numerical Processing, Beer Sheva, Haifa, Israel.
(Download our posters presenting some part of the project. Note that in some cases the graphs do not show the final published data.)
Krajcsi, A., & Szabó, E. (2011). Representation of multi-power numbers in preschool children. Presented at the The developing brain: Perspectives from typical and atypical development, Granada, Spain.
See a short presentation on some parts of this project. Note that in some cases the graphs do not show the final published data.
Description of the project in Hungarian.