Sometimes I hear that one subject is harder than another, but never scientifically investigated. For example, in high school, it's commonly thought that mathematics is the hardest subject. But I think that this definition is poorly built. I remember when I was in high school, and mathematics wasn't hard, but I really had no interest, and I ignored it and obtained poor grades. Today I've started to like mathematics, and it's really not that hard. I can solve problems and prove propositions, and the only thing I feel that changed in my mind is the addition of interest.

It seems that sometimes, the subject is not hard, and the student is under the effect of something that compels him to not do it, such as mathematical anxiety or the lack of interest as I pointed out before. I believe that the concept of difficulty per se is a poor measure of skill needed for doing something and that it interacts with other aspects. So, I got curious: are there studies on the definition of difficulty?

I have found this question after making my question, and it seems to address some of the questions I've made, but it doesn't mention the effects of willingness or lack of interest in obtaining knowledge.

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    $\begingroup$ There are studies, mostly focusing on one area or learning, such as foreign vocabulary, text understanding, maths, etc., not comparing them to each other. Here is a journal article that attempts to theoretically classify learning problems according to their difficulty. I'm too stupid to understand what it says, but maybe it is of help to you: citeseerx.ist.psu.edu/viewdoc/… If you do understand it, maybe you'd like to summarize it in an answer to your question. $\endgroup$
    – user3116
    Feb 1, 2014 at 20:13
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    $\begingroup$ I found that article through a Google Scholar search. You can see if there's more or optimize the query: scholar.google.com/scholar?q=measuring+difficulty+learning $\endgroup$
    – user3116
    Feb 1, 2014 at 20:16
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    $\begingroup$ Willingness and interest are characteristics of the learner, not of the learning matter, and therefore not aspects of its difficulty. $\endgroup$
    – user3116
    Feb 2, 2014 at 0:08
  • $\begingroup$ Those things are very individual. One is really capable for math and other for literature. How can you expect to measure it? In addition, it also depends how and where you are raised. For example, you you are born and raised in China/Japan you will struggle with recognizing Caucasian people. The same is visa versa. But when a group of Korean children of age ~10 were adopted in Europe, at the age of 30 yo they recognized Caucasian faces better than Asian. So, how will you measure such a difficulty? $\endgroup$ Feb 3, 2014 at 21:17

2 Answers 2


First, let me start by saying your topic is extremely broad. There are many reasons why something may be difficult to learn. However, the exact "difficulty associated with learning something" is known by many different terms in the scientific literature, and in particular, I have found cognitive load theory to be a particularly useful description of this. According to Sweller, Cognitive Load falls into three distinct domains:

  1. Intrinsic cognitive load is that difficulty that is associated with the material itself, independent of the learner.
  2. Extraneous cognitive load is associated with the instructional methods or materials (poor instructional techniques increase this load).
  3. Germane cognitive load is load that the learner requires to create associations and connections among the concepts, and depends on the learner's knowledge base.

The basic premise is that subjects are difficult to learn if they require a great number of connections to be formed (a mental model or schema). The greater the connections required, the greater the difficulty. Complicating this, certain topics are inherently difficult, such as mathematical computation and critical thinking, and learners bring their own pre-conceived mental notions to the table. Cognitive Load theory attempts to provide a framework for thinking about the interactions between these variables.

I have studied the difficulty with teaching/learning engineering subjects, such as statistics, variability, critical thinking, etc. using controlled experiments that attempt to hone in on the second and third aspects of CLT. Here is a link to a conference proceeding that I published on the subject; I have another paper in-work that I can't share until it is published- but so far, my results very much support CLT.

Interest can be a factor, because attention given to a learning subject will take away from the students' ability to process other cognitive loads. Likewise, anxiety can also be a factor. Lots of research left to do in this field if you ask me, though - so expect these theories to develop substantially in the next 10-20 years.


@JeromyAnglim's answer to your linked question says most of what I'd have wanted to say in response to your original (title) question. Difficulty of any given question in terms of item response theory is generally defined as the probability of answering it correctly at a given level of a (often latent) relevant skill. This definition of difficulty can be applied to an aggregate of questions as well (e.g., the probability of getting an "A" as one's final grade in a math class).

Mathematics may have a reputation for being a relatively "hard" subject because it relies more directly on quantitative, spatial, and logical reasoning than on verbal, social, or emotional reasoning. This may not be the case for many other subjects. These kinds of reasoning and the aptitudes associated with them aren't completely independent, but they are somewhat independent (i.e., intelligence as a whole is partly multidimensional, though the extent to which this is true is debated hotly), and may certainly be practiced somewhat independently. The latter set of reasoning skills (those not-so-relevant to math) may be practiced more frequently in everyday experience, especially during adolescence, and to a probably lesser extent, youth in general. Furthermore, those with special strengths in verbal, social, or emotional reasoning (and without corresponding strengths in math-relevant aptitudes, however rare that may be) may have more influence over the reputation of the subject. Thus the popular reputation may disproportionately reflect the opinions of those who are used to feeling stronger in other subjects, even though they may still be relatively strong in math as well.

A second major factor in both willingness to learn and the experience of interest is stereotype threat. This is one of probably many causes of academic anxiety in topics such as math—in fact, stereotype threat in math may be one of the most commonly studied varieties of the general social phenomenon. This may be due to Western society's many demographic stereotypes and prejudices with respect to quantitative reasoning, most notably ethnicity and gender. Stereotype threat research describes a complex process beginning with incidental membership in a group with a relevant stereotype. When stereotyped individuals are aware of a negative stereotype (e.g., their group is bad at math), they often fear that they will confirm the stereotype through their behaviors (e.g., earning a bad grade in math). This fear interferes with performance in often already high-pressure testing environments, often making the feared outcomes harder to avoid (an example of a self-fulfilling prophecy). Fear also reduces the capacity to enjoy the challenge of a test or other learning opportunity; enjoyment and self-confidence are crucial to the experience of intrinsic motivation.

Loss of interest and motivation, especially when due to fear, may often motivate use of cognitive coping mechanisms in defense of one's ego. One such mechanism is disparagement of the source of anxiety. In the all-too-common case of a person who experiences stereotype threat regarding his or (more likely) her mathematical ability, one may choose (not necessarily consciously) to see mathematics classes as inherently uninteresting, unfair, and difficult, even though none of these are necessarily true (for instance, imagine taking a middle school math class at your age). Naturally, one who believes these things is more likely to voice these things, particularly in as much as doing so may afford opportunities for ego compensation, or may be necessary to further offset others' negative judgments.

Stereotype threat theory has received some pretty serious criticism via empirical study (e.g., it's not the only cause of differences; (Sackett, Hardison, & Cullen, 2004). Nonetheless, it seems to have survived the test of meta-analysis reasonably well (Nadler & Clark, 2011). These analyses have revealed several moderators, including strength of identification with mathematics among women (Nguyen & Ryan, 2008).


- Nadler, J. T., & Clark, M. H. (2011). Stereotype threat: A meta‐analysis comparing African Americans to Hispanic Americans. Journal of Applied Social Psychology, 41(4), 872–890.
- Nguyen, H. H. D., & Ryan, A. M. (2008). Does stereotype threat affect test performance of minorities and women? A meta-analysis of experimental evidence. Journal of Applied Psychology, 93(6), 1314–1334. Retrieved from http://www.researchgate.net/publication/23489223_Does_stereotype_threat_affect_test_performance_of_minorities_and_women_A_meta-analysis_of_experimental_evidence/file/60b7d51a8da2aeacb0.pdf.
- Sackett, P. R., Hardison, C. M., & Cullen, M. J. (2004). On interpreting stereotype threat as accounting for African American-White differences on cognitive tests. American Psychologist, 59(1), 7–13. Retrieved from http://www.asc.upenn.edu/usr/ogandy/C45405%20resources/Sackett%20et%20al%20stereotype%20threat.pdf.


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