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I had a six hour conversation today with my husband on proofs vs. calculations. How can I be so sure that this proof approach is the right thing to do? While I personally think it's entertaining, how do I know that the kid will do okay on the SAT or won't be counting on his fingers later on?
ooooo....
This sounds fun:
I've exhausted my interest in the topic of Fuzzy Math and I'm now interested in Junk Geometry.
Drat These Greeks
Birkhoff's Geometry, Singapore NEM (doesn't cover quadratic equation until 9th grade), Frank Allen's axiomatic algebra instruction, Apostol Calculus, and more.
Go read.
9 comments:
Junk Geometry? What qualifies?
I propose my high school son's entire year of public school geometry based on the Prentice Hall series be the benchmark for junk geometry. There's probably worse out there. The fact that they are still only on chapter 6 and 3/4 of the way through the year is an indication of how little progress they are making.
Moreover, he has no idea how to set up any of the geometry problems to solve them. The entire course seems to revolve around teaching the kids the order to push their buttons on their graphing calculator.
He has never written a single proof.
He has never had an assignment that didn't use the calculator. Expect for the writing assignments when he writes about geometry (actually he writes about some real world problem that may have some tangential relationship to geometry).
It is a complete joke to call this a geometry class.
He has never had an assignment that didn't use the calculator. Expect for the writing assignments when he writes about geometry (actually he writes about some real world problem that may have some tangential relationship to geometry).
This is a nightmare.
What is the textbook?
I've said before that I think Geometric proofs are one of the six or so things that should absolutely be taught in secondary schools. Frankly, I take the lack of support from the usual constructivist suspects for teaching this subject as evidence that they are lying about wanting to "teach kids to think".
Proof-based geometry is entirely about teaching a new way to think.
Doug,
Here's what I'm not getting about how the word "conceptual understanding" is used. In K-6 it means something along the lines of demonstrating with manipulatives whatever arithmetic problem you are working on. What does it mean in high school math?
"Conceptual Understanding" of geometry to me means being able to prove the theorem that you used to calculate the value of angles. I am learning this vague phrase really can mean anything to anyone.
Frankly, I take the lack of support from the usual constructivist suspects for teaching this subject as evidence that they are lying about wanting to "teach kids to think".
good point
in fact, EXCELLENT point
Conceptual understanding is knowing how to set up the problem.
Procedural fluency is being able to finish it.
FWIW, my son's school uses the Prentice Hall Geometry book. An earlier edition was reviewed on Mathematically Correct, but it is essentially the same book.
The book is really made worse by the teaching, however. I mean, it is a mediocre book with only a superficial treatment of geometry at best. But if the teacher fails to assign the bulk of the problems or skip the sequencing, then it just makes the problem worse.
I did the calculus taught at UW Madison mentioned by that blog.
Apostol is 100 percent inappropriate to learn calculus if it is your first time and you are anything short of extraordinarily gifted at math.
If you are just good at math, or want to be, here's what you do to learn calculus
1- buy an old copy of stewart and work every single problem in it from the first cover to the last cover. This will take 9-12 months.
2 - Grab Apostol 1 and 2 and work them. Almost cover to cover, thought you can skip the ODEs if you want. The linear algebra in Apostol 2 is excellent.
2b - Alternatively, grab Spivak's Calculus. It's just as good as Apostol 1.
3 - Now baby Rudin. Prepare to cry. A lot. Also be prepared to spend, at least in my case, 30-75 minutes reading each and every page in order to fully understand, on a line by line basis, just what Rudin said.
Note that Stewart teaches calculus; Apostol and Spivak bridge calculus and analysis; and Rudin is analysis.
A hard book done through homeschooling at the student's pace is a completely different experience than the same book done on a schedule in some sort of a formal classroom setting. The fact is that learning calculus heuristically -- without the proofs -- is no way to learn it at all. I would rather "spin my wheels" for 2 years and have my child really internalize the formal definition of a limit than to cover all of a typical college freshman calculus sequence.
I guess in the end it all comes down to what you are trying to accomplish. Inasmuch as you are trying to actually impart knowledge on your student, you will teach the proofs of the theorems as much as if not more than their mere statement and application. If you are just trying to cover as much material as you can for their upcoming career as an engineer or something, then maybe knowledge isn't what you're after so much as the memorization of facts.
I might also add that Baby Rudin, itself, is actually just a bridge to the real Real Analysis of Rudin's other book or Royden. There is no doubt about it that Baby Rudin is terse and difficult. One reviewer on Amazon says "Thank you Dr. Rudin for your wonderfull book on analysis. You made a man of me." But, it still isn't even there yet! And, the reason it exists at all is because we don't get started teaching real math until way too late in the game. In my own opinion, this has more to do with bad math ed in K-12 than anything -- the bad math ed in college undergraduate programs. And, why do we do it this way? So we can have a shitstorm of civil engineers, apparently. God help us if we don't have enough engineers.
At any rate, Apostol appears to be a Baby, Baby Rudin which is precisely what Calculus really ought to be. And, if it just takes longer to really cover integration and differentiation in a single variable, then so be it. It takes what it takes -- you can't redefine what constitutes knowledge or what Calculus is just so that physics majors can get a BS degree in 4 years or so that Engineering majors can become "qualified" on some purely aritificial timeline. Apostol is really how colleges should be treating the subject rather than this false half-assed stuff they try to do in half the time it really takes. The only reason Apostol seems like it is only for the super gifted is just because students are expected to do it on the same timeline as the fake calculus.
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