I am thinking really deep about this topic with no answer. For example while learning set theory first time you find it very difficult to get your head around concepts like Union, Intersection etc but after some practice it becomes quite obvious, but as we go further there comes concepts like Group theory, Abstract algebra etc which even after reading 100 times doesn't makes any sense. On the other side you see some people playing with those concepts very easily.

Has anyone any idea on this topic?

  • $\begingroup$ Abstract concepts are more difficult to understand. Also, domains with many concepts connected with many complex relations. $\endgroup$ Oct 26, 2015 at 16:19
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    $\begingroup$ @DesignerAnalyst your comment seems the most abstract comment to me :) $\endgroup$
    – CodeYogi
    Oct 26, 2015 at 16:44
  • $\begingroup$ The complete answer probably involves some reasonably complex Person X Stimulus type X Context interactions (i.e., it depends). $\endgroup$
    – mrt
    Oct 28, 2015 at 2:08
  • $\begingroup$ @mrt Please explain Stimulus and Context. $\endgroup$
    – CodeYogi
    Oct 29, 2015 at 3:11
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    $\begingroup$ This question is pretty broad as it covers a bunch of different domains of cognitive psychology. Are you asking about mathematical concepts specifically? In what domain of mathematics? Are you talking about crossing between domains of mathematics? Is the question specific to adults or to developing children? $\endgroup$
    – Seanny123
    Oct 31, 2015 at 19:37

2 Answers 2


Firstly lets define criteria for what difficult learning material is. My interpretation of the matter states that the task of developing mastery or competency in understanding and applying difficult content:

  • Takes an increased amount of time
  • or Demands increased cognitive effort

It's relevant to keep in mind The 2 systems that occur in the brain

The brain is known to have a limited amount of cognitive capacity on the amount of calculations or work able to be performed, at once. This capacity is constant, unless brain development due to certain hard activities occur, in which, overtime the capacity of the brain will increase, but this happens over a long period of time(weeks months etc).

Thinking Fast and Slow by Daniel Kahneman references an experiment where demand was found to correlate to pupil dilation:

We recorded strings of digits on beats of the metronome and instructed the subject to repeat or transform the digits one by one, maintaining the same rhythm. We soon discovered that the size of the pupil varied second by second, reflecting the changing demands of the task.

At a certain point though, the capacity of the brain is met:

reaches an almost intolerable peak as you rush to produce a transformed string during and immediately after the pause, and relaxes gradually as you “unload” your short-term memory. The pupil data corresponded precisely to subjective experience: longer strings reliably caused larger dilations, the transformation task compounded the effort, and the peak of pupil size coincided with maximum effort.

The evidence provided strongly suggests that the theory put forward has some accuracy

According to Thinking Fast and Slow there are several factors which use up this cognitive capacity/energy, but the most important ones to consider are:

  • Multitasking Mentally computing a series of tasks, while keeping the state of other tasks in memory. For example when multiplying 15 * 34 without a calculator, you must keep the value of 5 * 34 in your head while doing 10* 34
  • Resisting subconscious processes For example unlearning habits, or the Stroop effect
  • Memory how much memory you are actively storing

These factors, which tax cognitive capacity, are ultimately the root cause of what determines the difficulty of concept, the more the learning material provokes any of these tasks, the more difficult the content is to learn.

These factors are mainly affected by two elements of knowledge: the complexity of the knowledge, and how well it relates to information currently in your head.

There are many theories on what knowledge is, and how it is interpreted by humans, including Bloom's taxonomy but the most relevant one for this question is Concept Learning which I will briefly state the 3 levels of concepts:

Not a concept simple recall or discrimination of knowledge

Concrete or Perceptual Concepts knowledge that bears physical similarity to one another from the learner.

Defined, or relational and associated concepts where the stimuli bears no obvious physical similarities, but cohere due to functional properties.

from not a concept -> concrete or perceptual concepts, concepts basically grow in amount of similar or distinct properties, and abstractions; distinct physical properties, but similar functional properties.

The more properties equates to more demand for memory. The more distinct and similar properties present, the more demand for multitasking, which therefore makes complex concepts harder to understand.

In Make it Stick, a book on learning strategies suggests that humans learn content through understanding it in short term memory, but after practice, the information goes through a process of:

consolidation, in which memory traces (the brain’s representations of the new learning) are strengthened, given meaning, and connected to prior knowledge

This point relates to Resisting Subconscious Processes which is described in "thinking fast and slow" tax cognitive capacity.

Also the more foreign the information is to you;the information already stored in your head, the harder consolidation will be, taking up more mental capacity.

Another consideration is elaboration. Knowledge is suggested to trigger the thought of other knowledge when the brain understands the relationship between knowledge. If concepts are not understood completely, and instead its information is just memorized, then this process will not occur and competency and mastery in understanding and applying the concept will be reduced. This is discussed in A Mind Of Numbers. This process not triggering will then require the logical part of the mind to figure out the connection, where the subconscious mind does not supply it, and demand more mental capacity.

So basically, the more complex information is, and the more foreign the information is, the more taxing the information will be in learning it, which makes it harder to understand.

  • $\begingroup$ Well written! but I would need couple to readings more to understand it completely since its a foreign knowledge to me :) $\endgroup$
    – CodeYogi
    Oct 28, 2015 at 4:24
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    $\begingroup$ That said! I think only hard work doesn't pay off completely and you feel cheated, lose motivation and neglected to work harder next time. It feels that your mind resists you to think further, you just need to finish the study material as soon as possible. I know setting possible goals might help, but in this case your mind becomes your own enemy. There have been countless times when given a puzzle / problem I find that my mind is thinking nothing I am just gazing over it sweating, feel my heart pumped up and fell headache. Feels like my mind has become too lazy to do anything. $\endgroup$
    – CodeYogi
    Oct 28, 2015 at 4:30
  • $\begingroup$ Also, I forgot to mention that I even feel too distracted to read long posts like these in one sitting. Really I find it too disturbing like I mentioned in my first comment. While reading your post I switched tab many times, changed songs, had an urge to watch some cool video. Is this normal? $\endgroup$
    – CodeYogi
    Oct 28, 2015 at 4:34
  • $\begingroup$ Its completely normal, I am probably worse than you!!, I've got finals tomorrow and haven't studied all day. Its called procrastination, though I do not believe your problem directly relates to difficult concepts, it is more to do with your emotional intelligence, how well you can control your thoughts. Its basically procrastination, as the work seems to hard or boring to do. I suggest you apply the stuff in this video:youtube.com/watch?v=wmx_35rQIRg $\endgroup$
    – tristo
    Oct 28, 2015 at 6:14

Part of it has to do with the ability to construct a model of reality. Every developer can relate to that. Visible, concrete things are easy to deconstruct in pragmatic terms. But once you get into abstract concepts, it gets harder and harder to bind it with your existing view of the world.

Some concepts are also very difficult to grasp if you can't connect parts of the concept with things you already understand. This process is called chunking, and it can be understood as dividing the problem until each part already makes sense to you, or is small enough to tackle the question on your own.

On top of this, you have to know what you're really looking for in order to ask the right questions. Some students just ask "How does this work? I don't understand", but the teacher can't answer something so vague. Richard Feynman make a very good statement about this problem in this interview.

  • $\begingroup$ Ya, for example I tried to visualize tree while learning counting but it becomes too cumbersome when you start too difficult problems. $\endgroup$
    – CodeYogi
    Oct 28, 2015 at 4:12
  • $\begingroup$ Please check my second comment on @tristo's answer. $\endgroup$
    – CodeYogi
    Oct 28, 2015 at 4:31

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