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"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Look for patterns may be a bit out of date.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
7 comments:
Lockhart's Lament should be called "Lockhart's Whine". Much more accurate.
There is a definition problem with toolkit, and most people use math as a toolkit in life, even mathematicians and engineers. Few are able to create completely new things in math or anything else. Most amazing things are evolution and not revolution. Then again, I get lots of paradigm shifting ideas about a lot of things, but I don't have the knowledge or wherewithal to do anything about them. Creativity is easy.
In my math toolkit I have things like explicit, implicit and parametric functions. I have vectors and matrices. I have dot products, cross products, divergence and curl. I have Stokes' Theorem. I just explained to my son how to transform parametric functions defined as vectors with homogeneous coordinates using 4X4 matrices. You transform functions, not numbers. This is part of a toolkit of knowledge.
I use my toolkit to come up with new solutions to problems. The math is old, but the solutions are new and could be paradigm-shifting. Then again, paradigm shifts are more of a market issue than a math issue. Creating the Apple I or TRS-80 didn't require a paradigm shift. They were computers with a CPU, machine language, memory, storage, and an I/O device. I remember coveting the guts of an old PDP-44 computer in 1976 - for $5000. The shift wasn't even clear with the advent of VisiCalc or Word. Even the internet came slowly from things like the Merit Network and ARPANET.
Then again, what's the problem here? It's not about having a subject matter difference of opinion with math content experts. I had no issues with my son's high school math teachers. I did, however, have an issue when my son's first grade teacher lectured us parents about the benefits of MathLand while we sat in little chairs in the school library.
The "finding interesting patterns" definition sounds like a definition of data mining, not mathematics!!
Take a look at the most recent gasstationwithoutpumps post for an in depth look at the different kinds of thinking, including critical thinking and mathematical thinking
http://gasstationwithoutpumps.wordpress.com/2014/10/26/critical-thinking/
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
That two trains problem always reminds of the amusing story about John Von Neumann and a similar problem, about a fly flying back and forth between two cyclists. It's a fun anecdote; if you don't know it, check it out here
Interesting post on what mathematics grad students do:
http://www.quora.com/What-do-grad-students-in-math-do-all-day
This isn't how traditional arithmetic is taught. This isn't how constructivist arithmetic is taught.
I don't think this is how we *WANT* most K-12 students taught.
-Mark R.
I have to say, the first time I rad Lockhart's essay, lots of it resonated with me. In particular, I don't like the way many geometry classes de-emphasize proofs or sanitize them with the statement-reason format. And there are is such a thing as more trigonometry than you need.
On the other hand, if his essay is used to justify a complete switch to "discovery" learning, that's a problem. I believe that there should be SOME time for SOME of the curriculum to be "discovered". But those discoveries are hard to make. they take time and they take teachers with MORE proficiency than those who directly instruct, not less. You can be the sage on the stage...but you better also be the sage on the side or you're going to waste a lot of class time.
I agree with what I think Lockhart was saying, that we don't want math to be delivered from on high as some received wisdom. Students should occasionally get to see the messy process that leads us to these understandings. If you can guide you students so that of what you teach, maybe 5% is "discovered", you are doing really well. Even that 5% may be too high a fraction.
Phil
Lockhart's "interesting patterns" are interesting to a mathematician patterns. Historically this means things like creating the quadratic formula from noticing patterns in how you can solve visually solve a quadratic using squares. A more hands on version: I'm currently working with a 7th grader who is a budding mathematician (he notices patterns that are interesting to a mathematician). A few weeks ago he came in with a pattern that he had found in the tens (and hundreds) digits of perfect squares, and we spent a couple of weeks writing out different ways to represent perfect squares to see if any of them would give us insight into whether his pattern was universally true, and why it worked (it's universally true, and it's a bit awkward and unlikely to get us anywhere new). A mathematician's interesting patterns rarely have much to do with real world applications. As a way to learn and teach, I think it really works best (only?) with students who have a lot of mathematical curiosity.
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