From what I can tell, the author of the article wanted her students to engage in philosophy rather than mathematics, but maybe didn't make this clear to them.
Mathematics is indeed all about rigor, proof, and finding "correct answers". Philosophy, on the other hand, takes a more lenient approach, with emphasis on the process of thinking, rather than the final result.
Perhaps the children in this class merely thought that a more mathematics-oriented approach was what the teacher desired. That's reasonable, if that's how their previous classes had been done.
Apparently the students themselves adjusted quickly enough, once the teacher made her expectations clear. As the article states, "Things went so much better the second time around with not one student asking me if their answer was “right”".
So this could very well come down to a simple misunderstanding between the teacher and her students, in this rather isolated case.
I disagree with your claim that what the teacher wanted the students to do was not really math. Instead she was claiming that they (and apparently you) have a narrow conception of what doing math involves.
The "right answer" consisting of a valid proof is very different from the "right answer" to a numerical computation. The thinking and communication process is surely more relevant to the former than the latter. And if anything, as you say, the former is what math is really about.
Yes, "mathematics" is a relatively narrow concept, even if it embodies a great deal of knowledge.
When it comes to mathematics, the exact form of the outcome isn't very important. Maybe it's a valid proof in one case, maybe it's a specific number in another case.
It's the degree of rigor that's important. Rigor is what separates mathematics from other fields of study.
Thinking and reasoning about abstract concepts aren't without value, of course. But if this is done without an extreme degree of rigor, it probably should not be considered to be mathematics.
Something tells me you've never sat in on an algebraic topology class where the teacher draws a donut and a pair of pants on the board, along with a line or two, and calls it a proof. A key characteristic of mathematics is that it can be made rigorous, but we certainly don't work in full rigor day to day. A look at Principia Mathematica should demonstrate simply why that would be infeasible.
Fundamentally they both have only one hole through them, so under the right sort of deformations you can turn one shape into the other: They have the same topology.
As a practicing mathematician, I disagree completely. Much mathematics is done with the understanding that the ideas can be made rigorous, and the point of educating people in mathematics is to allow them to craft arguments and revise when they find holes. One can and should introduce rigor gradually over time, and get the benefits of learning to reason and doing mathematics.
When it comes to mathematics, the exact form of the outcome is extremely important. Proofs that are aesthetically pleasing are better than proofs that are not. Proofs that glean insight are better than proofs that do not. In fact, most people don't care about whether a theorem is true unless the proof provides sufficient insights.
If you want mathematics to be it's own little walled garden free from ambiguity, that's fine. But don't you think it's kind of important that people be able to translate between problems in the messy non-mathematical universe and mathematics.
Mathematics is indeed all about rigor, proof, and finding "correct answers". Philosophy, on the other hand, takes a more lenient approach, with emphasis on the process of thinking, rather than the final result.
Perhaps the children in this class merely thought that a more mathematics-oriented approach was what the teacher desired. That's reasonable, if that's how their previous classes had been done.
Apparently the students themselves adjusted quickly enough, once the teacher made her expectations clear. As the article states, "Things went so much better the second time around with not one student asking me if their answer was “right”".
So this could very well come down to a simple misunderstanding between the teacher and her students, in this rather isolated case.