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My take on this is that kids come in normal distributions, i.e. they come to your classroom with a range of capabilities. In low SES districts the spread, the standard deviation, is quite large. I've had classrooms with kids ranging from 1 year above grade to 5 years below grade.
If you are in a failing district, like mine, you are blanketed with consultants, coaches (of which I am one), tight curriculum maps, walk throughs, and on and on. The sum total of this is that you're asking teachers to teach to a really narrow portion of a theoretical distribution while, at the same time, you are 'delivered' children with a 5-6 year span in abilities.
This means that if you follow the rules, and it's perilous not to, you are by definition throwing 80% of your class under the bus. Teachers adjust the curriculum to try to push as many of the distribution as possible through the eye of the needle. They teach to their median. So by definition 40% of your class is bored and 40% don't get it.
My district retention policy is "We don't have one!" Even if they had one there is no remediation program so the miniscule portion of students who are retained are put back into the very same classroom that failed them with the expectation that the second time through will be magic.
"Just stand right there in the middle of center field, Stan, and I'll hit you a few balls," says Ollie, as he proceeds to spray 60 balls at poor Stan; all at once and all over the outfield!
Friday, May 23, 2008
under the bus
from Paul:
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When middle schools don't retain failing students they are in effect setting them up for extreme failure in the high school.
Most of the data indicates that there are serious problems developing between the 4th and 8th grades. If we could ensure that students entered high school prepared to learn at that level, maybe more students would graduate with a diploma that actually meant something.
It gets worse when you factor in kids with ed plans. These kids 'work' with SPED teachers who generally have less mathematics background than the already low background of the classroom teacher the child is immersed with. With these kids you have this ludicrous recipe where the ones with the biggest need for expertise get handed off to the assistant chef.
I developed a method for division that allowed kids with minimal multiplication skill to be successful with two digit divisors, something that middle school kids really struggle with (even when they can multiply). Actually I just documented a method I found on the internet (don't remember where). Anyway, I offered to teach it to every sped teacher on staff at my school. I had no takers.
When I get a chance to teach this method to a classroom I get incredible smiles and joy out of kids who thought this to be an insurmountable task. Once, I had a sped teacher in the room during one of these sessions. She said "I don't get it so I'm just going to teach this the standard way." Essentially her approach was to continue down the path that wasn't working rather than spend the time to learn another path. Even if it doesn't float your personal boat, teachers, especially sped teachers, should be looking for multiple representations, eh?
Concerned -- absolutely
Paul -- can you teach us your method??
Interesting comment, which sparks a thought: I wonder whether in high-performing districts you get extremely good SPED teachers?
It's not that there aren't zillions of problems in SPED everywhere; there are.
But at least in my district I absolutely don't see any indication that SPED teachers are less capable or well-educated than non-SPED teachers.
I wasn't trying to flame SPED teachers in my post, not in any way. They have a real challenge that I would not sign up for.
It's just that when it comes to modifications in my district, they don't have content expertise so their modifications tend to focus on templating, more friendly numbers, multi-step problem breakdown, etc. These are all generic kinds of modifications which I suspect work for most any content.
There is nothing like, say, let's use bar models for this word problem, or let's try a different representation of the math. There is no content specific modification. When we have PD opportunities, math teachers get math PD and SPED teachers never get to go to these. I'm not sure where they go but I suspect they're off developing generic sorts of strategies.
I have a slide show on this method which is not ready for prime time yet. Here is the essence:
For 1563 divided by 17.
Take a single piece of paper and make a big T chart on it allowing a few inches at the top.
Write the problem 1563 divided by 17 in the top left.
Find the 1st,2nd and 5th multiples of the divisor: 17,34, and 85. Write this work on the top right so that you have the multiples and multipliers listed next to each other. This is the end of multiplication.
Then realize that this also provides you with the 10th, 20th, and 50th or 100th, 200th, 500th, by just adding zeros.
On the left side, write the divisor and then take successive multiples (or some power of 10 of a multiple) out of the divisor by subtraction while keeping track of the multipliers for same on the right.
You just work down the page on the left until there's nothing left to take out, then sum the right side to get the answer.
This is a horrible explanation. I'll post a slide when I get a chance. It's power is (especially for struggling multipliers) minimal multiplication and no frustrating guess and check. Best of all, if you do subtract something less than optimal it doesn't make the problem blow up. You just keep subtracting. So more steps maybe, but no dead ends requiring 43 pounds of erasers.
Thank you for your explanation. It made sense to me once I pulled out a piece of paper and tried it. I showed the method to my husband and he said that it's very similar to the way he was introduced to division in school.
I have a copy of a book first published almost 30 years ago that suggests that blind students use Paul's method. Two advantages over the standard long division algorithm are that it doesn't rely so heavily on getting a complex spatial arrangement right and it doesn't require repeatedly moving up and back down the page.
This book refers to this method by the terms "serial subtraction," "subtractive method," and "T-method."
It is really quite easy: differentiate!!!!
The method of division PaulB describes is called the partial quotients method and is the one taught in Everyday Math. It makes up for lack of multiplication skills, which students of EM surely do not have. It could be a nice stepping stone to the traditional method but EM simply leaves it as is.
Since the developers of EM and similar atrocities are always talking about the importance of number sense and estimation, it seems that mastery of the standard long division method would go a long way toward building such skills in students.
Barry:
Very true about the transition. I wait until kids are proficient with this and what happens is pretty cool. Many kids will invent variations on their own like instead of using multiples of 1,2, and 5 they use 1,2,4. Sometimes they'll invent something on the fly like using a multiple of 8 because they discover that it gets them where they want to go quicker.
At this point I know kids are ready for the next step, a transition to partial quotients in the standard algorithm. I have kids draw the standard division 'box' but instead of using just significant figures I make them fill out the partials all the way to the ones place with zeros. Then as they go through the algorithm I have them 'add' above the box just like they added in the initial treatment.
This lets them see, after a time, that the standard algorithm is just taking advantage of all those zeros they're generating. Some kids get to this point and finally 'see' the standard for the first time. The beauty of this phase is that they can still eliminate guess and check and succeed with imperfect multiplication knowledge. Estimating too low does not require backing up, you just keep going.
Sadly, EM and Investigations both have some great techniques buried in their presentation. The problem is that their 'discovery' bias prevents them from every bringing a method 'home'. It's like they're afraid to admit the end game is the standard algortithm so they just sort of present everything and hope for the best.
I've had 6th graders that divide by using tick marks because it has equal weight to anything else they learned in Investigations.
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