They aren't buzzwords, they are simple ideas and don't have to be referred to with those terms if they are triggering you. The OP just wants interactive, flexible systems to learn math from, which I wholeheartedly agree with.

Okay, my bad, then. Can you tell me what exactly is meant by a "hackable" math learning environment? And how will a REPL (however it is realized) make learning mathematics better?

Imagine if all the proofs in your linear algebra textbook were done in Isabelle, Idris or something similar to that. Then you'd be able to interactively explore the proof, it would be unambiguous, and you would know exactly like it worked. If you don't understand how a C program generates its result, you could start a debugger and single-step through it. With a paper proof, how can you convince yourself that there is nothing missing in the proof? Maybe you think you understood, but you didn't.

Excel, for all its warts, is hackable. Programming languages are hackable.

Fill inn these boxes and get the solution is not hackable (unless the programmer seriously failed at validating input.)

The great tragedy of maths education is that people come out the other side and think "fill in the box" is maths. No, that is doing you sums, not living in the world of mathematics.

The filling in of the boxes is most fairly compared to finding the typo in the complicated regex, or the file with incorrect permissions on the web server

"Fill in the box" is not referring to sums. It's referring to extremely rigid homework where you fill out each step in a process to finding out a result very precisely. It coincides with the horrific bubble tests where you select 1 of 5 answers and there is no partial credit.

This style of learning is what dominates US schools now from basic arithmetic all of the way up through advanced calculus.

Then replace the domain specific terminology with something more palatable to educators.

The point made in the parent comment is that teaching math often fails in its intent, to provide students with the rewards of mathematical insight and ability to reason.

Instead basic math education results in a rift between those who "aren't good with numbers" and, well, masochists.

What is it about math education that so abhors references to applications?

What is so crucial about references to applications? Sometimes the applications are so distant that it is not imaginable for high school teacher to dig into them.

People learn about mammals in biology and don't ask for references to applications. Why is it different in math?

What equivalence do you see between math and biology? The pedagogy seems to begin from opposite directions. What axioms of biology are comprised of "mammals"?

This thread began with a parent article on matrix algebra, which has many, many, exciting real-world applications.