Thinking Digitally: Individual Differences in Mental Representations of Number and Computational Algorithms Associated with Exact Quantities

The present research explored mental representations of number, the computational algorithms used to solve multi-digit arithmetic tasks, and how mental manipulation of exact quantities are modulated by individual differences in working memory capacity and method of instruction through six experiments. The first three experiments focused on holistic or decomposed mental representation of 2-digit numbers in Australian university students using a magnitude judgement and number bisection task. Since holistic or decomposed magnitude representations are associated with different multi-digit addition algorithms, Experiment 4 investigated individual differences in orthographic precision and strategies for multi-digit addition using a same-different mental arithmetic task. Experiment 5 explored the relationship between working memory and construction of novel strategies in a mathematical problem solving task using Diophantine equations. Individual differences in mental representations were further explored in Experiment 6 by comparing Australian university students with Japanese children who were skilled users of a ‘mental abacus’. Although the Japanese children’s performance was superior to the university students on virtually all tasks, both groups appeared to use qualitatively similar strategies of computing the total of 2-digit numbers by decomposing each of the operands according to unit and decade positions. Why are some people better at maths than others? The results suggest that computational speed and accuracy were associated with larger working memory capacity, stronger decomposed mental representations of number, and an educational approach that involved plenty of practice. All of these attributes facilitate application of exact multi-digit algorithms. Aside from the theoretical implications of these findings for the field of numerical cognition, the results are also practically important for teachers and policy makers