Earlier quoted context omitted.
The author of the linked article is a math professor. Additionally, not all math is writing proofs. It's also applying their results (a student doesn't need to prove 2+2=4 to use 2+2=4 in some computation) and developing intuitions about the nature of mathematical objects (which can then be used to guide future development by helping to discover new hypotheses to prove).
Fair enough. I didn't see the last line. I retract what I said earlier. Still I am not very convinced that it is a good idea to teach STEM students that way. The author of the article works at a liberal arts college.
Take it from a math PhD with quite some teaching experience that we do need more of what this professor is teaching, and less of number-mangling and rule-memorization, especially for STEM people. Understanding of what is going on is far more important than the formalism, which always comes later.
Also understand that the contents of a course aren't well condensed into a short article about it, and you won't get much out of the latter.
In the end, some complex mathematical notions have very hands-on representations that are faithful. The beauty comes from realizing that they are the same. Some examples:
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1a)The limit of the iterated dynamical system (f_1(z) = (1+i)z/2, f_2(z) = 1 - (1-i)z/2) in the complex plane
1b)The shape you get if you fold a paper over many times, and unfold keeping the angles at 90 degrees [1]
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2a)The algebraic field resulting from adjoining the roots of the polynomial x^2 + 1 = 0 to the real numbers
2b)All the ways you can move, rotate, and scale a flat shape on a desk [2]
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3a)The problem of classifications of embeddings of S^1 into R^3 up to ambient isotopy (a whole field of mathematics whose primary problem has remained open for over 100 years, and is connected to many others)
3b)Can you come up with a way to tell if you and I are tying our shoelaces the same way? [3]
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4a)The study of the following class: a set S with an associative operation ×, under which it is closed, such that every element is invertible, up to mappings that preserve × (that is, maps F such that F(g × h) = F(g) × F(h)).
4b)Study of reversible operations on an object that don't change the nature of it[4]. Like shuffling a deck of cards[5], spinning a globe on gimbals[6], or maybe swapping left and right children of some nodes in a binary tree here and there[7].
(The last example is more abstract, but hey, I made a thesis out of things like that!).
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The notions described in a) and b) are exactly the same. The way mathematics is taught is often you don't see b) while looking straight at it! And yet the formalisms in a) are much better understood when you know that they really are b).
If you have seen any definitions in part a), but part b) comes as a surprise - it's a problem. And yet that's the state of affairs.
That's the disaster that this professor is trying to fix.
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[1] https://en.wikipedia.org/wiki/Dragon_curve
[2] https://en.wikipedia.org/wiki/Complex_plane
[3] https://en.wikipedia.org/wiki/Knot_theory
[4] https://en.wikipedia.org/wiki/Group_(mathematics)
[5] https://en.wikipedia.org/wiki/Symmetric_group