I've been brushing up on a lot of basic arithmetic, algebra, and logic as I work towards a review of calculus (and beyond), and I keep noticing that in order to fully understand many principles in the abstract, I really have to include some practice with concrete examples.

I might struggle through a rigorous proof and manage to understand each logical step, but when I step back to view the whole, it's often too much to hold at once. On the other hand, I could learn an algorithm and manage to execute it properly without understanding why it works.

But if I do my best to follow the proof, and then apply the result on some practice problems, I often find that if I revisit the proof, I understand it fully. Tracking the variables and their behavior takes less mental effort once I've seen some examples, and often a statement that was difficult to parse in abstract language is straight-forward, and even intuitive, once I've worked through some concrete examples. I might have to go back and forth between proof and practice a few times to really nail it down.

I'm sure this is nothing new, but I'm constantly astonished at how mixing these two approaches can get me over hurdles that once seemed insurmountable. Is this the conventional wisdom in math education?

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    $\begingroup$ Not sure if it is a legitimate question or a rant masked a question. +1 anyway. $\endgroup$
    – Git Gud
    Mar 13, 2013 at 19:32
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    $\begingroup$ I wonder if what you call "practice" I call "examples". When I'm stuck on a proof, I often examine some specific situations in which the proof applies and others when it doesn't apply (counter-examples, in a way). If that doesn't get me unstuck, then I'm really in trouble. $\endgroup$ Mar 13, 2013 at 19:35
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    $\begingroup$ It would be very nice if all (or even many) students understood that the abstractions deal, ultimately, with concrete objects. $\endgroup$ Mar 13, 2013 at 19:39
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    $\begingroup$ @GitGud Yeah, that crossed my mind as I was posting, but I though I'd put it out there. I was momentarily overcome by my own enthusiasm :) $\endgroup$
    – ivan
    Mar 13, 2013 at 19:41
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    $\begingroup$ My impression is that your experience is pretty common. People seem to fall on a spectrum: some of us $-$ I’m one $-$ generally do best by starting with the abstraction and then looking at examples, but I think that we’re in a minority. Others generally do best by looking at a number of examples before ever seeing the abstraction. Most probably fall somewhere between the two extremes. But no matter where one falls, one eventually needs a fund of good examples on which to test ideas. $\endgroup$ Mar 13, 2013 at 21:17

1 Answer 1


"In order to fully understand many principles in the abstract, I really have to include some practice with concrete examples."

I'll say, "yes you are on the right path." That's how it works for everyone who has the desire and passion to learn math. One needs to struggle with concepts in a variety of ways before one can climb up (or over) to a new (or different) level of abstraction.

I like how Sam Alexander relates it to weight training: "Like weight training, to train your mathematurity you need to struggle with concepts and exercises which really challenge you." (How to Train Your Mathematical Maturity by Sam Alexander http://www.xamuel.com/mathematical-maturity). [Emphasis added by me.]

However, "proofs and problems" is not the only dimension of mathematics. There is the "creative math muscle" to be developed. And there's relating math with philosophy, religion, reason, logic, physics, history, etc. all of which are fascinating and can help one to even better understand math beyond what you think you know about it.

"In order to fully understand many principles in the abstract..."

When you fully understand any principle in the abstract, please teach that to me :) Math can be a rabbit hole when you dig down the point of understanding principles.

On the other hand, it's not the "wisdom of conventional education" because conventional education is based on "teaching" and what you are doing is something very different. You are learning.

Finally, note the biological reason for your experience. When you endeavor to learn new ways of thinking (or any new skill for that matter), you are creating a new neural net of pathways in your brain. This takes both effort and time for those neural pathways to grow.


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