How we perform carry operation, why is it so difficult, and what can be learned from it about algorithms?
Read the full dissertation (in Hebrew) or the article based on Shibolet’s dissertation
One of the central mathematical abilities is the ability to perform calculations orally. This is a basic skill that has a wide range of applications: managing household finances, effective activity in the workplace, and more. However, mental calculation poses a challenge for both children and adults, and it is affected by various factors such as exercise characteristics, calculation strategies, individual memory capacity, etc. For example, mental calculation involving carry operation (e.g., 27+36) is more difficult than calculations without carry operation (e.g., 21+32). Nevertheless, to date, only a few studies have systematically examined the precise origin of difficulty in such calculations.
To examine these factors, we asked 31 participants to perform 2-digit addition exercises with decade-crossing using two specific strategies: right-to-left calculation (units first, then decades) and left-to-right calculation (decades first, then units). As response, the participants were instructed to say each intermediate result (decade sum, unit sum) and then the final answer. For example, for 27+36 with the right-to-left strategy, the participants were to say “13, 50, 63”. By measuring the duration of each step in the calculation (units addition, tens addition, merging intermediate results into the final answer), we could examine the calculation process step by step, not just as a whole.
Two findings demonstrated the significant role of working memory in the calculation process. First, calculation was slower in the stages in which the working-memory load was lower. Second, these stages were particularly sensitive to the participant’s working memory capacity. Our findings indicate that the working memory load was not constant but changed dynamically during the calculation: it was the highest in the first calculation stage (either units or decades addition), lower in the second stage, and yet lower in the merge stage. We conclude that working memory undergoes updates during the calculation, such that no-longer-necessary information, namely the already-added operands, is removed from memory to make room for the intermediate results. This memory management scheme reduces the load on working memory and promotes efficient and flexible memory resource management in the calculation process.
In the merge stage (and only in this stage), the presence of a decade crossing created more difficulty when the carry was at the unit position (e.g., 27+36) than when it was at the decade position (e.g., 72+63). An explanation for this pattern is that the decade crossing position affects the specific calculation algorithm invoked during merge: when the crossing was in the unit position, a more complex sub-algorithm must be used to merge the intermediate results (e.g., 50+13 compared to 130+5).
The second calculation step (either units or decades addition) was more difficult in right-to-left calculation than in left-to-right calculation. This may arise from load imposed on the syntactic representation of the number during calculation: when calculating from right to left, i.e., when starting from the units addition, the first interim result is a ones/teens word, and in the second addition step the participant may have to update the structural representation of the number into a 2-digit number with a tens word. Presumably, this update consumes time and cognitive resources.
A similar pattern was found in the merge step: it was slower in right-to-left calculation than in left-to-right calculation. This effect may stem from the order of the previously memorized intermediate results in working memory: in left-to-right calculation, the memorized order (130+5) aligns with the order of words in Hebrew verbal numbers, whereas in right-to-left calculation the memorized order (5+130) does not. A need to reverse the order of words of the intermediate results may consume time and cognitive resources.
Overall, the research demonstrates that multidigit calculation is a complex process, which involves several types of difficulty stemming from various factors. A precise assessment of performance in high granularity, i.e., at the level of each calculation step, can reveal these different sources. Further examination of the difficulty factors in oral calculation will allow understanding in detail the relations between individual abilities and exercise characteristics and could promote success in complex mathematical operations.