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Ms. K rarely assigns word problems. What problems she has assigned have been, frequently, far above the kids' skill level. (examples here: scroll down for the entire list) The parents do these problems at home, then the kids turn them in. This is an open secret. My favorite Extended Response moment happened last year, before Christopher had moved to Phase 4. One weekend parents all over the soccer grounds were grabbing each other & asking whether anyone knew how to do the latest Extended Response. These were all highly educated Westchester parents with important jobs requiring advanced training. And they're running around the soccer games accosting people about the latest Challenge Problem their kids have to hand in. This year there's one student in one of the Phase 4 classes who, last semester, was getting 60s & 70s on his tests and had straight '10s' — the highest score possible — on the Extended Response problems.
NOTE to IMS Math Department:
Look for a pattern!
Cs and Ds on tests, A+ on Extended Response problems — what does this suggest?
The Extended Response problems are assigned because, in the beginning, the Phase 4 students were supposed to be mathematically gifted, and Irvington's pedagogical philosophy where the mathematically gifted are concerned can be summed up in two words: Math Olympiads.
gifted and talented according to Math Olympiads The MATH OLYMPIADS approach to educating the gifted and talented, as far as I can determine, is the following:
I have the Challenge Philosophy in writing. The Assistant Superintendent for Curriculum said, in a letter to me, that Phase 4 kids 'need to be challenged' — although he agreed that kids shouldn't be given problems so challenging their parents would have to do them. He may be right about kids who really are GATE in math, although I can't imagine GATE kids don't need instruction. And my leaning where GATE kids are concerned is towards acceleration over enrichment. I may be wrong about GATE kids. But I'm right about the high achievers. Kids who do well in math because they're high-achieving don't need Math Olympiad problems. In fact, I'll go for the Strong Form here: Kids who do well in math because they're high-achieving are harmed when they spend time on Math Olympiad 'challenge problems' instead of word problems pitched to their level and embodying the concepts they are currently trying to master. Ms. K assigns Challenge problems, not Instructional problems. As a result, virtually all of the word problems Christopher has done this year fall under the heading of lost instructional time. What Christopher needs are brilliant instructional word problems of the kind provided by Action Math.
has Math Olympiads become a national curriculum? I always saw the Extended Response problems in the accelerated class as an 'add-on.' The mathematically talented kids were taught math like everyone else, only they had to do Extended Response problems, too. Now I'm wondering whether in fact the 'challenge' approach is simply another manifestation of constructivist math. Instead of being taught how to do word problems, kids are handed a problem and told to figure it out on their own. Here's what I see in Christopher's class:
I'm not privy to Ms. K's thinking, but I know exactly what the effect of her approach to word problems has been on Christopher (and I know he's not the only one):
I'm guessing that this approach is the result of the constructivist pedagogy Ms. K, who is very young, would have been taught in ed school — whether she's aware of it or not. She teaches the 'basics' in class, the kids memorize what she's put on the board, then the kids discover how to apply the basics to word problems on their own.
In fact, it's probably worse than that, since Ms. K. told a friend of mine that she teaches the concepts the day after the kids have done homework on those concepts. My friend said Ms. K told her this in a 'DUH!' tone, as if it should just be obvious a teacher wouldn't teach a new concept before assigning homework on the concept. I'm wondering whether this is an ed school truism at this point. Do ed schools teach future math teachers to have the students discover everything first, including rules, properties, and procedures, and then "go over it" later after the kids have discovered whatever they're going to discover? I don't know.
this is where bar models come in Here is 1 of the 5 problems Christopher's class was assigned for Wednesday night:
None of the kids has been given any instruction whatsoever in how to set up such a problem algebraically. Nor have they been given any instruction in the Official Prentice-Hall Problem Solving Strategies:
Wednesday afternoon I was working on these problems with Christopher and his friend M. Needless to say, neither boy Looked for a Pattern, Guessed and Tested, Simplified the Problem, Made an Organized List, Worked Backwards, Accounted for All Possibilities, Made a Table, Wrote an Equation, Solved by Graphing, Drew a Diagram, Made a Model, Solved Another Way, or Simulated the Problem. No. Instead, both boys, working independently, subtracted 280 from 2870 and then stopped. They knew they weren't done, but they didn't know what to do next. They didn't know why they'd subtracted 280 from 2870, either. I pointed out to M. that one runner went 280 m further than the other. Unfortunately I can't remember what he did with this information. I do know that he ended up with answers that were 280 m different, but added up to a whopping big batch of meters, far more than the original 2870. Then he started getting upset, and insisting his answer was right. I decided it was bar model time. In hindsight, this was the wrong decision where M. was concerned. Both of the kids do know how to translate English words into an equation, and M. might have been able to think the problem through using x to stand for one runner's distance. He was flat-out unwilling even to look at a bar model. 'I don't understand anything about this problem,' he said, and that was that. He was done. My mistake. Christopher was game. He knew he was getting nowhere doing what he was doing, and he'd had enough experience with bar models to take it on faith that a bar model would work. Which reminds me: I must stress to Christopher that the point of the bar model isn't to solve the problem, but to show you which operations you need to do in what sequence. Bar models are a way of setting up the problem. He didn't know exactly where to start, though he did know he should draw two bars, one for each runner. When I started talking him through he caught on quickly and he was able to label everything correctly and quickly on his own, without prompting. Here's what he drew (this is my version):
Seeing the problem laid out this way, Christopher again subtracted the 280 from the 2870. Then he got stuck again, in the same place he'd been stuck before. We've got work to do. However, when I started walking him through it, asking what we had left now that we'd subtracted the 280, he was able to say that we had the two other segments left, and he was able to say, with prompting, that these two segments were equal. The instant he said they were equal he realized he needed to divide the difference by 2.
2870 - 280 = 2590 2590 ÷ 2 = 1295
At that point I asked him where the 1295 belonged on the bar model, and he knew. Then he knew that Runner 1's distance was 1295 + 280 meters while Runner 2's distance was 1295 meters.
So I'm thinking.... a) is this happening in other school districts? are math teachers taking a discovery approach to word problems? and — b) the best defense is a good offense
If I had little ones I'd be teaching them bar models just so they have a way to tackle all the discovery word problems they're not going to be taught how to do in the years to come.
My mom used to always tell us to Drive Defensively. Same thing here. Teach Defensively.
extended response problem from IL state test
extended response problem 1
extended response problem 2
extended response problem 6
extended response problems 7, 8, 9
direct instruction & the rigor conundrum
Dan's daughter reacts to extended response problem
defensive teaching of Singapore bar models
open-ended problems in math ed
problems that teach - "Action Math"
email to the principal
-- CatherineJohnson - 10 Mar 2006 Back to: Main Page.
Ms. K rarely assigns word problems. What problems she has assigned have been, frequently, far above the kids' skill level. (examples here: scroll down for the entire list) The parents do these problems at home, then the kids turn them in. This is an open secret. My favorite Extended Response moment happened last year, before Christopher had moved to Phase 4. One weekend parents all over the soccer grounds were grabbing each other & asking whether anyone knew how to do the latest Extended Response. These were all highly educated Westchester parents with important jobs requiring advanced training. And they're running around the soccer games accosting people about the latest Challenge Problem their kids have to hand in. This year there's one student in one of the Phase 4 classes who, last semester, was getting 60s & 70s on his tests and had straight '10s' — the highest score possible — on the Extended Response problems.
The Extended Response problems are assigned because, in the beginning, the Phase 4 students were supposed to be mathematically gifted, and Irvington's pedagogical philosophy where the mathematically gifted are concerned can be summed up in two words: Math Olympiads.
gifted and talented according to Math Olympiads The MATH OLYMPIADS approach to educating the gifted and talented, as far as I can determine, is the following:
- The mathematically gifted child needs, above all, to be challenged.
- The best way to challenge the mathematically gifted child is to give him super-hard problems he's never been taught how to do & send him off to grapple with them on his own.
I have the Challenge Philosophy in writing. The Assistant Superintendent for Curriculum said, in a letter to me, that Phase 4 kids 'need to be challenged' — although he agreed that kids shouldn't be given problems so challenging their parents would have to do them. He may be right about kids who really are GATE in math, although I can't imagine GATE kids don't need instruction. And my leaning where GATE kids are concerned is towards acceleration over enrichment. I may be wrong about GATE kids. But I'm right about the high achievers. Kids who do well in math because they're high-achieving don't need Math Olympiad problems. In fact, I'll go for the Strong Form here: Kids who do well in math because they're high-achieving are harmed when they spend time on Math Olympiad 'challenge problems' instead of word problems pitched to their level and embodying the concepts they are currently trying to master. Ms. K assigns Challenge problems, not Instructional problems. As a result, virtually all of the word problems Christopher has done this year fall under the heading of lost instructional time. What Christopher needs are brilliant instructional word problems of the kind provided by Action Math.
has Math Olympiads become a national curriculum? I always saw the Extended Response problems in the accelerated class as an 'add-on.' The mathematically talented kids were taught math like everyone else, only they had to do Extended Response problems, too. Now I'm wondering whether in fact the 'challenge' approach is simply another manifestation of constructivist math. Instead of being taught how to do word problems, kids are handed a problem and told to figure it out on their own. Here's what I see in Christopher's class:
number one: The kids have been given virtually no 'normal' word problems — normal meaning do the odd problems for homework-type problems — all year. number two: They've been given 9 Extended Response problems, only 1 or 2 of which they could plausibly solve on their own. number three: To my knowledge, they've been given little-or-no direct instruction in the kinds of word problems that will appear on the state test. number four: This week, when Ms. K. finally did assign a page of word problems for homework, she gave them no instruction whatsoever on how to do them. number five: Having read all of the sample problems for the state test, I would be stunned if any problems like the ones Ms. K. assigned this week will appear on the state test next week. number five: My guess is she didn't demonstrate how to do the problems in class the next day, either, unless the students asked her to. (I'll ask Christopher.) I don't see how she could have. There were 5 problems in all, each requiring a different approach the kids have not been taught, and they spent at least 10 minutes writing in their math journals. That doesn't leave a lot of time for demonstrating and explaining five different word problems in one class period. update: As I suspected, Ms. K went over 'the problems kids had trouble with,' which means that it was up to the kids to a) know they needed help and b) say so in front of a class filled with peers who, at lunchtime, are going to be calling them 'fat,' 'gay,' and/or 'stupid.' The only problem Christopher remembers her going over in class was 'the runner problem.' (This is two days ago, we're talking.) He has no memory, none whatsoever, of what she actually said about how to do the runner problem. I'm sure the runner problem came up because no one in the class could do it, so there was no shame in admitting defeat. Almost certainly most of the kids solved problems 8, 11, & 12 through guess-and-check, and that was that. It's unlikely that any of the 11-year olds Christopher knows would say, "I got the right answer, but I'm wondering whether there's a more elegant and efficient way to go approach this problem."
I'm not privy to Ms. K's thinking, but I know exactly what the effect of her approach to word problems has been on Christopher (and I know he's not the only one):
a) properties, rules, and procedures are learned by rote and b) all word problems, including simple, beginning problems in algebra, become Challenge Problems
I'm guessing that this approach is the result of the constructivist pedagogy Ms. K, who is very young, would have been taught in ed school — whether she's aware of it or not. She teaches the 'basics' in class, the kids memorize what she's put on the board, then the kids discover how to apply the basics to word problems on their own.
In fact, it's probably worse than that, since Ms. K. told a friend of mine that she teaches the concepts the day after the kids have done homework on those concepts. My friend said Ms. K told her this in a 'DUH!' tone, as if it should just be obvious a teacher wouldn't teach a new concept before assigning homework on the concept. I'm wondering whether this is an ed school truism at this point. Do ed schools teach future math teachers to have the students discover everything first, including rules, properties, and procedures, and then "go over it" later after the kids have discovered whatever they're going to discover? I don't know.
this is where bar models come in Here is 1 of the 5 problems Christopher's class was assigned for Wednesday night:
None of the kids has been given any instruction whatsoever in how to set up such a problem algebraically. Nor have they been given any instruction in the Official Prentice-Hall Problem Solving Strategies:
Wednesday afternoon I was working on these problems with Christopher and his friend M. Needless to say, neither boy Looked for a Pattern, Guessed and Tested, Simplified the Problem, Made an Organized List, Worked Backwards, Accounted for All Possibilities, Made a Table, Wrote an Equation, Solved by Graphing, Drew a Diagram, Made a Model, Solved Another Way, or Simulated the Problem. No. Instead, both boys, working independently, subtracted 280 from 2870 and then stopped. They knew they weren't done, but they didn't know what to do next. They didn't know why they'd subtracted 280 from 2870, either. I pointed out to M. that one runner went 280 m further than the other. Unfortunately I can't remember what he did with this information. I do know that he ended up with answers that were 280 m different, but added up to a whopping big batch of meters, far more than the original 2870. Then he started getting upset, and insisting his answer was right. I decided it was bar model time. In hindsight, this was the wrong decision where M. was concerned. Both of the kids do know how to translate English words into an equation, and M. might have been able to think the problem through using x to stand for one runner's distance. He was flat-out unwilling even to look at a bar model. 'I don't understand anything about this problem,' he said, and that was that. He was done. My mistake. Christopher was game. He knew he was getting nowhere doing what he was doing, and he'd had enough experience with bar models to take it on faith that a bar model would work. Which reminds me: I must stress to Christopher that the point of the bar model isn't to solve the problem, but to show you which operations you need to do in what sequence. Bar models are a way of setting up the problem. He didn't know exactly where to start, though he did know he should draw two bars, one for each runner. When I started talking him through he caught on quickly and he was able to label everything correctly and quickly on his own, without prompting. Here's what he drew (this is my version):
Seeing the problem laid out this way, Christopher again subtracted the 280 from the 2870. Then he got stuck again, in the same place he'd been stuck before. We've got work to do. However, when I started walking him through it, asking what we had left now that we'd subtracted the 280, he was able to say that we had the two other segments left, and he was able to say, with prompting, that these two segments were equal. The instant he said they were equal he realized he needed to divide the difference by 2.
2870 - 280 = 2590 2590 ÷ 2 = 1295
At that point I asked him where the 1295 belonged on the bar model, and he knew. Then he knew that Runner 1's distance was 1295 + 280 meters while Runner 2's distance was 1295 meters.
So I'm thinking.... a) is this happening in other school districts? are math teachers taking a discovery approach to word problems? and — b) the best defense is a good offense
If I had little ones I'd be teaching them bar models just so they have a way to tackle all the discovery word problems they're not going to be taught how to do in the years to come.
My mom used to always tell us to Drive Defensively. Same thing here. Teach Defensively.
extended response problem from IL state test
extended response problem 1
extended response problem 2
extended response problem 6
extended response problems 7, 8, 9
direct instruction & the rigor conundrum
Dan's daughter reacts to extended response problem
defensive teaching of Singapore bar models
open-ended problems in math ed
problems that teach - "Action Math"
email to the principal
-- CatherineJohnson - 10 Mar 2006 Back to: Main Page.