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"Problem solving, which is preferred, occurs when students are not able to apply a mechanical, memorized response, but rather have to figure out what to do in a new situation."Kids who know math can look at a problem and apply a straightforward, mathematical solution. For kids who know squat, EVERYTHING is a problem that has to be figured out. That is not a good thing. The route to complex problem solving does not bypass mastery of the basics.
Discovery learning in math (pdf file)
memory lane
This reminds me of the parent uprising that took place here a few years back.
Kids in the accelerated math course at the middle school were doing so badly on their tests, and the tests were so different from what had been taught in class, that the district was finally forced to hold a public meeting to discuss the situation.
The then math chair, by way of explanation, told us that the reason kids were struggling was that the course goal wasn't just to teach math. "I want your children to be able to solve problems," she said.
She elaborated on this point: there was something about astronauts, as I recall. She wanted our kids to be able to use math to solve the kinds of problems an astronaut might have to solve. Something out of the ordinary.
That was the goal.
Ergo: hard course, 6th grade children weeping over their homework, and parents hiring tutors. Specifically: parents hiring math tutors already employed by the district as math teachers.
Later, a parent standing in the back of the room raised his hand and when called on observed that students hadn't done so well on measurement in the last round of state tests.
The department chair said emphatically, a hint of triumph in her voice, "Your children can't measure!" She seemed to relish this factoid. Your children!
A friend of mine heard from the administration just the other day that kids did badly on the measurement scale again this year.
So I guess the 4 intervening years of Math Trailblazers hasn't done the trick.
21 comments:
I would like to propose a Murray Gell-Mann corollary.
Those who really understand a subject area can't even begin to have a discussion with those who don't.
How do you explain to someone that they are completely wrong? In a nice way? What if you are currently sitting in a little chair during a math open house?
"...the course goal wasn't just to teach math. 'I want your children to be able to solve problems,' she said."
This is not just in math. I have had many situations where the difference between what I want and what the school is doing is so great that I have been at a loss for words. How do you tell a school that they got it backwards?
Those who really understand a subject area can't even begin to have a discussion with those who don't.
That's my problem with the "Writer's Day" the middle school will have next year, the one where professional writers will talk to the students, then the students will engage in a "hands on" writing project that will "emulate" the professional.
How does an 11-year old emulate writing a **book** with a famous expert on animal behavior who has autism?
As for talking to one's district about these things, I've recently learned that it may take as long as 20 years for writers to be able to read themselves as others read them.
In other words, it takes two decades for writers to be able to edit their own prose well.
(Another article to get posted...)
If that's the case, the LAST thing writing instruction should be centered around is "process" and "editing."
If I get the energy, I may simply start telling people here that it takes 20 years for writers to become expert self-editors SO they should be focusing on teaching kids to write fairly well on the first pass.
That might have some appeal since having kids edit & rewrite is so fantastically labor-intensive.
btw, I'm not sure about all this --- but I do know, that if I were currently teaching composition, I definitely wouldn't be focused on a "writing is rewriting" approach.
I'd be using sentence combining, sentence composing (Killgallon), and text reconstruction - with students writing essays that mimic the structure of the text-reconstructed essays.
I would also teach 5-paragraph essays using Kerrigan & The Shape of College Writing.
(For college kids & possibly advanced h.s. kids, I'd use Graff's book They Say, I Say.)
Then lots and lots and lots of reading & lots and lots of analysis of the reading.
To learn how to solve "all sorts of new problems" one has to start with basic problem solving techniques that are generalizable to a variety of different situations. What the ed school gurus were advising was what happened in Catherine's school as she described. Giving students problems for which they did not have the prior knowledge or skills to solve them.
If you throw a kid who doesn't know how to swim and/or is afraid of water in the deep end of a swimming pool with the instruction that he/she is to swim to the other side, one result is the kid may drown. The other result is the kid may do a combination of thrashings to keep himself afloat and somehow make it to the other side. If you asked the kid how he did it, a likely response may be "I have no idea what I did, but I never want to do that again!"
The wacky part is that with all this emphasis on writing and re-writing and "modeling professional behavior", the college students I teach often have no idea of how to produce a draft. They are so used to recomposing on the fly (which computers help with) that they don't really have the steps down at all. It is a slight problem when we do journal-type reports, but that isn't too bad since they have a well-defined format. Even then, I require that they give me a rough draft that I read over to make sure that they have the right things in the right sections.
It is a huge problem in senior seminar, when they have to write a twenty page review article. I actually started requiring that they give me an outline which they discuss with me individually, as too many of them just wanted to sit down start writing without any ideas about organization. I discovered I then had to add a class period on how to outline, and this is for college seniors!
Catherine: "In other words, it takes two decades for writers to be able to edit their own prose well."
Had you said something like, "edit their own prose and get anything useful out of the process", I might agree. I don't think that most writers ever get to the point where they can "edit their own prose well." The best that you can expect is that if you give your work a week or two before you look at it with an eye to editing you can sometimes see poor writing of various sorts.
Oh, you might be able to do a good job of basic copyediting. But I think that significant writing is too closely tied to each writer's idiosyncratic thinking process for the writer to notice flaws of logic, organization, and flow as often as you really need to. And absent those things, I don't think you can say that the writer can edit his own work "well".
Frankly, finding even a professional editor that can do a good job of structural editing of the work of others is a non-trivial task. Even good editors are often lured down the alleyways of comma placement, spelling, and a random selection editorial peeves of dubious validity.
IME, a good reader who is willing to say, "I don't follow this part", is as useful as (sometimes more useful than) most editors. If you can tell me where you're having a problem, I can often see all or part of the problem and rewrite to correct it. If you can articulate why you don't understand the part, you're invaluable.
But good readers are hard to find, too.
On outlining: In many ways, the formal outlining taught in the 70s was worse than useless. Until I managed to get past the cumbersome process involved in the sort I was taught, I actively avoided outlining at all, to the great detriment to my writing.
Now, I always outline any writing longer than a page or two, but I haven't used a formal outline since the last time it was required in high school. I make notes, draw lines, and rewrite the parts I can no longer read for all the scribbling. At the end, I have a plan that is functionally an outline, but it bears little resemblance to the roman numeral, capital letter, lower-case roman numeral (etc.) outline taught in some writing classes.
Barry: An interesting anecdote about the deep end of the pool...
When I was a little guy (don't remember how old) I was sent to swimming lessons/camp. The camp founder was a retired Olympic swimming coach and his solution for kids who insisted on keeping their face dry while 'swimming' was to carry them to the diving board and drop them off. No instructions were given nor required.
I guess you would say he was 'old school'. It occurs to me now that his 'instruction' wasn't about swimming it was about overcoming fear. I think some of my early English teachers assigned work with similar designs.
Paul B,
Thanks. Sounds like in the case of your coach, the kids already knew how to swim, at least enough to get them to safety. Did it work in your opinion?
My husband had a similar experience with a swimming instructor. He was tossed into deep water and told to swim to the other side. The result? A 40 year old who can barely swim and doesn't ever voluntarily go into the water.
Bear cubs learn to swim by the mother bear throwing the cubs into water. So why doesn't it work with humans? Cargo cult science at work?
"What if you are currently sitting in a little chair during a math open house?"
That was me last night. I had to leave halfway through because my baby started fussing but I almost choked when the presenter (a textbook author) claimed that "if kids have to memorize a formula, that's not mathematics". Another gem: "calculating is not mathematics, 'flexible thinking' is mathematics".
"if kids have to memorize a formula, that's not mathematics". Another gem: "calculating is not mathematics, 'flexible thinking' is mathematics".
Do you happen to know the name of the textbook author and/or the name of the textbook. I'm collecting all these words of wisdom in one place so that if I am put on the spot and asked to define "idiot" I can have examples at the ready.
"Bear cubs learn to swim by the mother bear throwing the cubs into water. So why doesn't it work with humans?"
Because if a human tries to throw a bear cub into the water, the mother bear will kill the human. :-) Or maybe I don't understand the question :-)
-Mark Roulo
Textbook authors may have their own ideas, but publishers know the score. If you look at their web sites, you will see two classes of pre-algebra and algebra books; rigorous and fuzzy. Only the fuzzy ones exist for K-6 because a rigorous series is too tough to sell. Unfortunately, ignorance drives K-6 math education. These people feel perfectly capable and willing to select curricula without content-expert help.
Or maybe I don't understand the question :-)
Apparently not.
The quote was from Phil Gonsalves, Alameda County Office of Education math coordinator and professor at Cal State East Bay Hayward. His books include "Math Around the World" and he's listed as a co-author on the MacMillan/McGraw Hill elementary series.
Okay, so I glanced through the sample of the 3rd grade math book since that's the level at which my DD is currently working. It's not as bad as the infamous EDM or TERC but it's still pretty fuzzy. The book is very big on having the students use manipulatives and/or draw pictures to solve problems and pretty light on memorization of basic facts & standard algorithms. Typical problem: "Use the math tool chest to model each multiplication sentence..." Huh?
Forgot to include the link for the 3rd grade book sample: http://advancetracker.com/Advance_eBook/Main.html?bookId=760124
Crimson Wife,
Wow. Math tool chest. You better keep on with Singapore.
That sounds a lot like Math Trailblazers.
SusanS
My DD has been working through a test prep book since she'll be taking the CA Achievement Test to placate my DH.
This morning she came to me all confused, asking "What's a 'number sentence'?"
When I explained to her that the book meant an "equation", she replied: "Why didn't they just say that???????" ROTFL!
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