Difficult Numbers?

Some numbers appear to be more difficult than others. It will come as no surprise to teachers, that children find single-digit multiplication with operands 7 and 8 more difficult than multiplications with other operands. So far difficulty has been analysed through correctness of performed operations. How the difficulty of numbers is perceived irrespective of the individuals’ ability to perform a calculation correctly, has not been systematically investigated. This could have implications for how we construct teaching materials and tests for educational as well as for research purposes. We present here a first exploratory study investigating the research questions: Are some numbers perceived as more difficult than others? If so, is this dependent on general mathematical ability?

In a survey conducted on Facebook, 691 adults (age > 18; 515 women, 174 men) answered a short questionnaire on number preferences and motivation to perform multiplications and calculations in context. Informants were asked to state on a scale from 1 to 5 if they believed they would be successful and if they would prefer to do the calculation or not.

Difficulty of numbers was scored from 1 to 6 based on the presence of digits: the easiest operands (e) were 1, 2, 5 and 10, intermediate difficult (i) were 3, 4, 6, 9, and most difficult (d) were 7 and 8. Difficulty level 1: ee or eee, 2: ei or eei, 3: ed or eed, 4: ii or eii, 5: id or eid, and 6: dd or edd. General mathematical ability was self-reported (self-evaluation on a scale from 1(low) to 5 (high)).

Overall, men reported all numbers as easier than women even after correcting for achievement level and difficulty of numbers. More interestingly, low achievement groups perceived difficult numbers, that is numbers including digits 7 and 8, as relatively more difficult than high achievement groups did.

Whether or not some numbers are perceived as more difficult than others by some achievement groups have implications for teaching as well as for research. If a person perceives a task as difficult from the presence of specific digits, that could influence performance and ability to engage in for example learning activities. As one participant stated: “I am a math teacher, but even so it feels like a shock through my body if I have to do arithmetic with 7 and 8 and the result exceeds 10.”