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Here is a common process I witness in office hours:


  1. Student struggles with a difficult concept.
  2. Student gains understanding of the concept.
  3. Student, now bewildered since the concept is mastered, declares "Oh! That's it??."

They not only learned the concept, but they have declared the concept to be easy!!

The student then maintains the following incorrect perception: "I only understand easy math."


I hear this all the time, at every level.

  • Some percentage of students in Calculus II think that Calc I was easy, which is why they got an A, but now Calc II is hard, so they are not succeeding.
  • Some percentage of students in algebra think that arithmetic was easy, which is why they got an A, but now algebra is hard.
  • To be honest, even I personally have the perception that everything before Hartshorne's Algebraic Geometry is relatively easier, which is why I did well in it!

How many of these people are correct? Is there any research or profound wisdom on this issue out there? I'd prefer research but I'm not sure how it would even be conducted.

  • All math is approximately the same difficulty, from arithmetic to algebraic geometry. People slip up because of a bad semester somewhere along the way and that is why anyone ever gets stuck in math, or
  • There are "hard spots" in math where you can get stuck; the hard spots become more and more dense as you progress further in math, or
  • Something else??
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I would say there are definitely hard spots and what they are depends on orders you teach it and how it is taught. –  ruler501 Apr 10 at 16:58
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One possible way to answer your question (about concept mastery?) is to look to the literature on expertise. My knowledge of this area of research begins around the time of studies on expert vs novice paradigms in chessboard perception. A fair amount of this has to do with chunking information; perhaps the differences you are observing are w.r.t. how fine/coarse one's understanding is. Math classes are often ordered so that only the next class gives you a broad enough view to chunk the previous one(s) properly (e.g. Calc I, II, III, Analysis, Topology). Just a comment for now... –  Benjamin Dickman Apr 10 at 21:05
    
@BenjaminDickman, I hope you can find a way to expand this a little into an answer. I think it is the kind of answer that I would want to accept (I have never heard about the chunking concept). –  Chris Cunningham Apr 11 at 19:13
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@ChrisCunningham Maybe you could check out a couple of the relevant papers and put up the answer? One of the early papers can be found here: bura.brunel.ac.uk/bitstream/2438/1339/3/Multiple_boards.pdf.txt Check also its citations, as well as the articles that cited it; the latter can be found at: scholar.google.com/… –  Benjamin Dickman Apr 11 at 20:45
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2 Answers 2

It seems to me that you're wondering about two different issues:

  1. are some topics or areas in mathematics more or less difficult to understand?

  2. the nature of developing an understanding.

I can't speak to the first point, but the second point... there is something about arriving at an understanding such that you cannot ever imagine what it was like to NOT understand. The insight is so great, the sense of knowing is strong, and so you almost immediately forget that you didn't know, feel like it was weird that you never knew it. It's not because the topic was inherently easy that you understood -- but that once you understand something, it seems "easy!" It's an attribution error to assume that if you understand something, then it must be easy.

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For 1. there actually is something to integral vs derivative calc. Derivative calc is a straightforward plug 'n' play algorithm where as antiderivatives/integrals are more heuristic. (There is a generalized antiderivative algorithm to match the derivative one, but it's intractable for non-computers). So it can be fair to say that, while it may not be "more difficult", Calc II is somewhat characteristically different from Calc I. –  Jsor Apr 10 at 20:13
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Perhaps because all math is simple once you understand it? ;-)

On a bit of reflection, you'll see that each area of human endeavour advances until the going gets though for the geniuses in the area, who struggle at the forefront. They are followed by bright people who struggle to understand them, and sometimes try to make the genius' findings understandable to the unwashed masses. Thus in the end for us integration by parts is routine, but it was a very complicated procedure to Leibnitz (IIRC, Dunham in one of his books tells the story). The going gets harder as you approach the bleeding edge.

On there being specific tough spots, I've heard it said that abstract algebra "separates men from boys," and in my own experience (personal and as a teacher) many people have a very hard time wrapping their brain around abstract concepts with no direct connection to everyday experience (thus struggle with group theory, for example, or in my case in Computer Science with automata and languages). Once the hurdle is overcome, it is smooth sailing.

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