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?
What makes some things difficult or easy to understand?
learning;mathematical psychology;mathematical ability
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 effortIt'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 accuracyAccording 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* 34Resisting subconscious processes For example unlearning habits, or the Stroop effect Memory how much memory you are actively storingThese 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 knowledgeConcrete 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 brains representations of the new learning) are strengthened, given meaning, and connected to prior knowledgeThis 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.