[Math] How to Garner Mathematical Intuition

Question

Motivated by Why Is Intuition so Important to Mathematicians but Missing from Mathematics Education? $^{1}$ by Leona Burton, I would like to learn about specific ideas or strategies to attain mathematical intuition, which I exemplify in my Answer below. Many textbooks only present definitions, theorems, and proofs, without any explanation of the intuition or motivation behind a definition or theorem, or the idea of the proof. I illustrate by examples the meaning of "intuition":

● $\{\emptyset\} $ can be intuited as a box containing nothing a box containing an empty box.

● The Fundamenal Theorem of Calculus can be intuited, via a geometric picture, by interpreting the interpretation of $g(x) = \int_a^{x} f(t) \, dt$ as the area under the graph of $f$ and $g(x)$ as the "area so far" function from $a$ to $x$

● The Jacobian Determinant for a transformation can be intuited by approximating the image region of the new variable with a parallelogram determined by secant vectors.

Despite the quote by Henri Poincaré ("It is by logic that we prove, but by intuition that we discover.") on the averred significance of intuition in math, the aforesaid article astounds me:

"None of the mathematicians talked about working on their intuitions
to improve their frequency or reliability…"

"Intuition, insight or instinct was seen by most of the seventy mathematicians whom I interviewed as a necessary component for developing knowing. Yet none of them
offered any comments on whether, and how, they themselves had had their intuitions nurtured as part of their learning process."

"These practising research mathematicians speak with such enthusiasm and joy of their practices. However, with the notable exception of the work of Fischbein, accounts of the deliberate nurturing of intuition and insight is absent from
the mathematics education literature, even from process-based research, and, despite the claim for the centrality of it to mathematical work, it is equally absent from practices
with students…"

1. http://www.jstor.org/stable/40248307?seq=1&uid=3738176&uid=2&uid=4&sid=21102540224927

Answer 1

TL;DR

My opinion is that "intuitive" is well approximated by "compatible with our internal model of reality" and I try to argue that such treatment might be useful in gaining or teaching intuition, that is, in my words, updating someone's model to match the reality.

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Disclaimer 1:

I am not sure what really is the question about and what are the answers the OP wishes for, however, as this topic seems to be of interest of many, I can take a guess. Most of it is probably well known to you, but I suspect the point is to spell it out. I can only hope the opinions provided would be useful to some.

Disclaimer 2:

All the things below are my own opinions based on my (rather small compared to others) experience with intuition in teaching, doing research, discussions, etc. However, it is worth noting, that quite large part of any success I had was due to gaining intuition either by me or people I communicated with. (I wouldn't cite here the testimonials of students I taught, but you are welcome to search through my answers, especially those that contains pictures, e.g. 1, 2, 3, 4, 5 or 6. Also, this is not an (failed) attempt at self-promotion.)

Disclaimer 3:

I find the cited article of Leona Burton misleading and not useful, in particular, I don't agree with meanings of intuition cited by her in the context provided by the paper. Although the aforementioned points are correlated with adjective "intuitive", in my opinion the relation is much weaker than implied by the author.

• Intuitive is the opposite of rigorous. There are many proofs that are intuitive and rigorous, as well as proofs that are informal and not intuitive.
• Intuitive means visual. Although surely visual learners would find visual arguments more intuitive, this feature is orthogonal to intuition, kinesthetic or auditory learners would find other approaches intuitive and the fact that most of us are visual learners is not a reason enough to connect those two terms.
• Intuitive means plausible, or convincing in the absence of proof. It's actually hard to decide here. The meaning of this sentence depends on meaning of "plausible", that is, among others, how "reasonable" or "probable" a thing has to be, to be qualified as "plausible". On the other hand, in a situation where we have hardly any intuition, we can still call some things "plausible", "reasonable" or "probable". Besides, in context of human behavior, we can say that "somebody acted on intuition instead of reason" which would mean that an act does not have to be "reasonable" to be "intuitive". Once again, we are here buried within natural-language semantics.
• Intuitive means incomplete. This is plain false; there are proofs that are intuitive and complete, and there are proofs that are incomplete and aren't intuitive.
• Intuitive means based on a physical model or on special examples. This is close to `heuristic'. There are physical models, special examples and heuristics that are counter-intuitive, also there are intuitive approaches that are general and purely theoretical.
• Intuitive means holistic or integrative as opposed to detailed or analytic. I can agree that intuition often regards the global perspective, but it's not not enough to be intuitive. Moreover, I cannot recollect right now, but there was a theorem where a single detail inside the proof made the whole thing intuitive, it was one of the cases such that when solved, you would know how to solve all the other cases.

Please understand, that I don't say that intuition is not connected with the above, those points are related, but do not cover what I think we would like "intuition" to describe. (Obviously, meanings of words and sentences are not independent, but, nevertheless, try to think about what would happen if we were to perform some kind of orthogonalization on the meaning of the above).

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On intuition:

My most useful approximation of "intuition" is

$$\color{blue}{\text{Intuitive means compatible with our model of reality.}}$$

In other words, things we find counter-intuitive behave differently as we would expect them to behave, i.e. our internal model of reality is not consistent with "experimental results" we observe. This might seem like a truism, but it is still an useful approximation.

There is hardly anything we can do about the reality, so to gain intuition we need to update our internal map (like a chart, not a function). There are many approaches. (In fact, we learn if and only if we change our behavior, if some activity does not change our reality model/map/whatever, then we are not learning).

• Test model on simpler cases. For example, many geometrical theorems still work where some segments degenerate to points, e.g. the formula for area of cyclic quadrilateral becomes the Heron's formula.
• Maybe there is an important factor that was not taken into account by our model. This happened for me with the Monty Hall problem, the missing piece of information was that some events are not independent.
• Use different model. This happened for me in linear algebra $$(U + V) \cap W \neq (U \cap W) + (V \cap W),$$ but even better example would be the 3D proof (instead of 2D) of the Desargues' theorem (if you know the proof, this is also a great example for the next point).
• Compose a few already known models. The Einstein-Pythagoras' theorem is a classic example: $$E = m (a^2 + b^2).$$
• Model the model. This might feel weird, but meta-modelling is still a nice technique. It might be useful when you don't know integrals and you want to deduce $\sum_{k=0}^{n}k^\alpha$ for some $\alpha \in \mathbb{N}$. It is easy to intuitively guess sums for $\sum_k 1$ or $\sum_k k$, and then for $\sum_k k^2$. Modelling the model is exactly the process which allows us to guess that $\sum_{k = 0}^{n}k^\alpha$ is $\Theta(n^{\alpha+1})$. Similar generalization might happen with quadrature of higher order, or generalization of Brianchon's theorem to other conics. I'm not sure if I'm overdoing it, but I would say that the category theory is the essence of this trope.

When everything fails, we can still:

• Chart the territory until the pattern arrives. Actually, this is helpful if we have no idea which model would be applicable to considered part of reality. This corresponds to enumerating a few first examples in order to familiarize ourselves with the problem we are solving.
• Produce a completely new model and internalize it. This is a process similar to one where a child with a questionable sense of balance learns how to ride a bicycle, especially making turns; the parent provides the model (lean left/right) and the child tries to make it work.

On teaching intuition:

My experience with teaching intuition is that people think differently, have different reality maps/models and successful "intuition transfer" happens when they updated their models enough to be compatible with presented theorems/data/etc. Some useful techniques:

• Ensure you are "going through", i.e. use words that students understand, pictures with visual thinkers, etc.
• Chart the territory for those who have no models at all, i.e. show basic examples.
• Consider simpler cases first, so that students can decide which of their already-learned models could apply.
• Pinpoint issues that are frequently missed (you need some experience to do this), e.g. show counterexamples for some common fallacies.
• Present different perspectives, maybe from different domains if possible.
• Divide the problems into subproblems so that previous intuitions might be used.
• Give some intuitive (but not necessarily simple) special cases that could be later generalized.
• Let the students think, i.e. allow enough "spare" time for internalization.

Conclusion:

As I said before, all this techniques are mostly known, and my little experience is not enough to attest it works or not. I also doubt the above lists to be complete. Nevertheless, the OP displayed such an eagerness in this topic that I hope he would still find this post useful.

$$\ddot\smile$$