# What research has modelled the difficulty of mental mathematical calculation?

I posted this also on mathoverflow.

What research has modelled the difficulty of evaluating a formula mentally (for your average, numerate, person, not a trained mental calculator)?

For instance, evaluating an arithmetic expression requires in general requires three registers, but some only require two and are considerably easier to deal with. Additions is easier than subtraction which is easier than multiplication which is easier than division. Multiplication by small integer number is easy, as are multiplications and divisions by 2,5,10...

Many rule of thumbs are intuitive, but I'm looking for a more comprehensive work on the topic.

• @Arthur Just checking; are you limiting your question to addition, subtraction, multiplication and division? – Jeromy Anglim Jun 30 '12 at 4:57
• @JeromyAnglim no, not really, no particular limit. Though generally, mental techniques to do anything else break down to addition, subtraction, multiplication of digits and decimal shifts. – Arthur B. Jul 3 '12 at 14:18

There is a huge amount of research on this topic, particularly in developmental psychology.

Siegler: As @Joel has noted, Robert S. Siegler provides a great entry point into this literature (see his list of selected publications with PDFs). He has done a lot of research on the strategies that children use to solve mathematical problems. His research includes both empirical and computational modelling approaches. He also has a number of studies that compare performance across features of arithmetic problems. For example, you could look at Siegler and Lemaire (1997) which shows in an adult sample the differences in speed and accuracy for different kinds of multiplication problems (e.g., multiples of 10 or not, single versus double digits). As another example, Siegler and Shrager (1984) show empirical error rates and strategy use in children for all possible combinations of single digit addition.

Act-R: There are also a number of publications using the ACT-R framework on mathematical problem solving (see this list from the ACT-R website). Perhaps check out Ritter et al (2007) for a review.

Defining difficulty: In general, there are questions about how you would define difficulty. Typical indicators would include task completion time and accuracy. Also, when talking about the difficulty of a mathematical operation, you need to consider the strategy that an individual is using. In particular, there is general shift with practice from using an algorithmic strategy to using a retrieval strategy (for a review, see Delaney et al, 1998). Thus, at first children use strategies like repeated addition to solve multiplication problems whereas with practice, they typically learn to retrieve the answer from memory. The ability to retrieve is broadly related to the number of times the specific problem has been encountered with the correct answer available, along with forgetting effects related to periods where there is no exposure.

### References

• Delaney, P.F., Reder, L.M., Staszewski, J.J. & Ritter, F.E. (1998). The strategy-specific nature of improvement: The power law applies by strategy within task. Psychological Science, 9, 1-7. PDF
• Siegler, R. S., & Lemaire, P. (1997). Older and younger adults' strategy choices in multiplication: Testing predictions of ASCM via the choice/no choice method. Journal of Experimental Psychology: General, 126, 71-92. PDF
• Siegler, R. S., & Shrager, J. (1984). Strategy choices in addition and subtraction: How do children know what to do? In C. Sophian (Ed.), The origins of cognitive skills (pp. 229-293). Hillsdale, NJ: Erlbaum. PDF1 PDF2
• Ritter, S., Anderson, J.R., Koedinger, K.R. & Corbett, A. (2007). Cognitive Tutor: Applied research in mathematics education. Psychonomic bulletin & review, 14, 249-255. PDF