"The essence of mathematics is recognizing interesting patterns in interesting abstractions of reality and finding properties of those patterns and abstractions."This strikes me as completely wrong, not that I have much confidence in my intuitions concerning the essence of math. So take this as a confession, not an argument.
Everything About The Way We Teach Math Is Wrong
"Recognizing interesting patterns" -- even "recognizing interesting patterns in interesting abstractions of reality" -- strikes me as the toolkit approach to math.
I personally -- another confession, not an argument -- can't stand the toolkit approach to math. Blech!
I have no interest -- none! -- in endless iterations of function problems designed to determine how much profit the guitar teacher will make teaching x number of students while paying y rent on the studio for 20 weeks, or whether Jim should buy the monthly contract or the annual, as useful and important as those questions are in daily living.
Nor do I relish the thought of encountering yet another roller coaster depicted as yet another instance of geometry, or another textbook with a nautilus shell splashed across the cover. Enough with the nautili.
If you want to make me hate math, real-world math will do the job.
That reminds me.
I did not, as a child, solve two trains leaving the station problems, and I wish I had. I was utterly charmed by the two-trains problems when I encountered them as an adult, working my way through "Russian Math."*
I don't remember whether we were asked to solve bathtub problems (I think we were), and I was charmed by those problems, too.
(Speaking of the real world, the bathtub problem is all you need to know to understand why fiscal stimulus doesn't work when the Federal Reserve targets inflation. One party is putting money into the system; the other party is taking it back out.)
To be fair, I'm not at all sure that "toolkit math" is what Lockhart actually means: "That’s what math is — wondering, playing, amusing yourself with your imagination."
Nevertheless, inside K-12, the toolkit approach is what look-for-patterns turns into.
Look for patterns may be a bit out of date.
Today, with Common Core, at least here in NY, we've moved beyond look-for-patterns to modeling, for pete's sake. In my district, the entire high school mathematics curriculum, the entire rationale for the entire mathematics curriculum, is modeling.
Lots and lots of function problems modeling stuff nobody cares about.
This is always the conundrum with constructivism and progressive education.
Progressive educators think the real world is fun and motivating.
* Mathematics 6 by Enn Nurk and Aksel Telgmaa