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# Teaching and caring are inextricably linked. But: only one of them is difficult.

Eh... You should watch it. He's not really talking about the kind of "applied problem solving" that been thrown at students for the past 20 years. He's more into "patient problem solving": requiring the students to develop the structure necessary to solve a loosely structured problem. Not bad, really.

But also not math. The stuff he's doing is more introductory physics. Mitrovarr is right: math needs abstraction.

posted by mr_roboto at 9:51 PM on May 14, 2010

I don't entirely disagree, but I'm solidly in the camp that there definitely needs to be a lot more practice at applied problem solving. One of my strongest memories from student teaching was going through one of the story problem sections that tend to follow the rest of a largely abstract unit with one of my Algebra 2 classes. They were so stymied by the homework that I wanted to stop and spend a week on it. The followup reaction when I told the class was fearful and vehement -- these students were dead scared of having to try to face word problems. And my cooperating teachers really didn't want to linger more than an extra day. One of a handful of issues that made me wonder what in the world I was doing there and what we're doing teaching math in general if we're not getting our students over that hurdle.

I think you have to give students practice in answering concrete questions, whether it's the trajectory of a body in motion or "What's more economical -- a Corolla or a Prius?", that you have to

Lots of time to sell people on the power of abstraction once they've got the foundation in place for a bridge connecting it to reality.

posted by weston at 10:32 PM on May 14, 2010

The thing is, I think the bridge is planted on abstraction, not reality. My strongest memory of math is when I trained-wrecked out at the calculus level. The problem, in the end, was that I had no foundation. I had been pretty good at math before that, and even gotten awards when I was younger. However, when it became really difficult, I realized that even when I knew what to do, I didn't truly understand what I was doing - and that I never really had.

I didn't know (and still don't) the fundamental theoretical underpinnings of math itself that allow you to say 'oh, you're allowed to do this, but not this other operation.' I just made progress as long as I did because I was able to memorize rules and apply math as a methodological solution to problems, not truly understand it. That kind of thing, I think, is a great way to learn how to answer questions in the short term but is very destructive to long-term success - it is easier, so you get used to it, but it's only one-way. You can't use a learned methodological ability to do math to do anything novel. Also, I think, the complexity involved in using it as such usually stops you before you get too far in. I got through calc 3 and probably could have kept going for at least a while, but it would have been the same thing - not understanding, just learning and applying tools. I think that is fundamentally the wrong way to do things.

posted by Mitrovarr at 11:42 PM on May 14, 2010

You're misrepresenting Peano. 2 is defined to be the natural number that comes after 1. Addition is defined recursively, so that m + n is the number that comes after m + (n-1), and m + 1 is the number that comes after m. Thus, 1 + 1 must be 2.

These don't actually exist, by definition.

That's because most geometry courses are based on Euclid's Elements in no small part, and understanding the content requires using proofs. Ideally, the people who were teaching these students would have a good understanding of what a proof actually is, but unfortunately (IME) most math education majors are highly unqualified to teach anyone about math.

Triangle congruences underly a baffling number of geometric proofs, unless you think that the whole of Euclidean geometry was created for instructional purposes.

posted by TypographicalError at 2:30 AM on May 15, 2010 [2 favorites]

tigger, did you read/watch the links? What's being described by Dan has certainly not been taught in grade school for the past 20 years.

Constructivism is not a method, though many methods are consist with it. If you're going to pull authority, throw us some citations.

posted by mathtime! at 11:17 AM on May 15, 2010

Post

# Teaching and caring are inextricably linked. But: only one of them is difficult.

May 14, 2010 9:13 PM Subscribe

Dan Meyer is a high school math teacher with a clever idea: make math about the real world. On his blog, he writes about classroom management, the real skills of teaching, labels, information design, and assessment.

The best argument for this is the amazing amount people who struggle to compute a 15% or even 20% tip on a dinner bill. This skill should be required for graduation from 8th grade.

posted by thorny at 9:34 PM on May 14, 2010

posted by thorny at 9:34 PM on May 14, 2010

I don't know. If you emphasize concrete math over abstract math, it seems like you're setting up your students for a trainwreck later when you get into the fields in which abstract math is absolutely necessary - advanced algebra, calculus, and such. That kind of emphasis always seems to result in people who can

I bet it does great on standardized tests, though, so it's probably going to be the next teaching paradigm.

posted by Mitrovarr at 9:48 PM on May 14, 2010 [1 favorite]

*do*math, but don't understand math - and there is only so far that attitude will take you.I bet it does great on standardized tests, though, so it's probably going to be the next teaching paradigm.

posted by Mitrovarr at 9:48 PM on May 14, 2010 [1 favorite]

*Isn't this always the clever idea?*

Eh... You should watch it. He's not really talking about the kind of "applied problem solving" that been thrown at students for the past 20 years. He's more into "patient problem solving": requiring the students to develop the structure necessary to solve a loosely structured problem. Not bad, really.

But also not math. The stuff he's doing is more introductory physics. Mitrovarr is right: math needs abstraction.

posted by mr_roboto at 9:51 PM on May 14, 2010

Yeah... Now that I've watched it all the way through... It looks like this guy is doing an awesome introduction to physics class. Not much math, though.

posted by mr_roboto at 9:53 PM on May 14, 2010

posted by mr_roboto at 9:53 PM on May 14, 2010

This is indeed an old idea, "make math about the real world," but Meyer has approached it with the emphasis on problem-solving.

Right now I'm trying to help my son who's desperately far behind in math, depressed by it, and afraid of the consequences. What I found potentially very useful is the emphasis on having the child 'own' the math problems that they solve. Here are some of the key suggestions he makes in the TED video:

1. Use multimedia - as in, bring the real world into it. If your problem is, 'how long will it take to fill this rain barrel,' show a barrel being filled.

2. Encourage student intuition - let them define the way they want to solve the problem.

3. Ask the shortest question you can - Eliminate the detailed steps of how you want them to use formulas for area, rate, etc. and just ask the key question.

4. Let students build the problem - They defined it; let them build the steps to solve it

5. Be less helpful - Modern textbooks do too much handholding. Encourage the student's problem-solving, esp. now that you've let them define the problem and its steps.

The idea of letting my math-averse son define what word problems are and what math means for him is, to me, exciting, liberating, and full of potential.

posted by Hardcore Poser at 9:58 PM on May 14, 2010 [6 favorites]

Right now I'm trying to help my son who's desperately far behind in math, depressed by it, and afraid of the consequences. What I found potentially very useful is the emphasis on having the child 'own' the math problems that they solve. Here are some of the key suggestions he makes in the TED video:

1. Use multimedia - as in, bring the real world into it. If your problem is, 'how long will it take to fill this rain barrel,' show a barrel being filled.

2. Encourage student intuition - let them define the way they want to solve the problem.

3. Ask the shortest question you can - Eliminate the detailed steps of how you want them to use formulas for area, rate, etc. and just ask the key question.

4. Let students build the problem - They defined it; let them build the steps to solve it

5. Be less helpful - Modern textbooks do too much handholding. Encourage the student's problem-solving, esp. now that you've let them define the problem and its steps.

The idea of letting my math-averse son define what word problems are and what math means for him is, to me, exciting, liberating, and full of potential.

posted by Hardcore Poser at 9:58 PM on May 14, 2010 [6 favorites]

He's not getting rid of the abstract part of math. The problem he is addressing is that a typical math course skips the part that comes before, defining the question and finding the right abstraction. I remember a math class, must have been in 7th or 8th grade, where we joked (half-joked? 10%-joked?) that the right answer when you were called on was always "plug it in," meaning follow the obvious formula.

posted by domnit at 10:05 PM on May 14, 2010

posted by domnit at 10:05 PM on May 14, 2010

Clever idea? People have been trying to do that for decades.

Frankly, I always found "story problems" tedious and boring when I was in school. Obviously you have to learn how to apply math to the real world, but it's not actually any more

posted by delmoi at 10:10 PM on May 14, 2010

Frankly, I always found "story problems" tedious and boring when I was in school. Obviously you have to learn how to apply math to the real world, but it's not actually any more

*fun*posted by delmoi at 10:10 PM on May 14, 2010

Mitrovarr: "

In both the above links and the video, Meyer talks pretty much exclusively about teaching Algebra I. You might appreciate his discussion in the video of teaching a slope problem: where a traditional textbook lumps together abstract model with concrete picture, Meyer begins with a picture as a means of demonstrating the efficacy of using abstract models. In other words, Meyer's job is at east in part cultivating the interest in and relatability of math.

For better or worse, you don't need advanced algebra or calculus to get into or even get through most colleges and universities in the US. Meyer doesn't say so, but he seems to be working in and among low-college-attendance populations. In that case, any math skills at all are pretty much an improvement on the status quo.

Teaching math is full of perverse incentives. People who are good at or curious about math are encouraged to teach the populations which have the most skills with math and, as a result, the youngest and lowest-performing kids are often stuck with teachers who had little to no math preparation in college.

The standardized test part of your comment I just don't get. If abstract mathematical reasoning is what we're getting at here, how other than a test do you propose to assess it? Meyer's actually got some pretty radical ideas about testing, including throwing out static unit tests for assessments at the end of each concept which allow students to skip those concepts they've already mastered. Find a teacher who parrots the textbook, he's accused of teaching to the test. Find a teacher who makes his own rich, multimedia driven teaching portfolio mapped to his own style and the needs of his students, he's accused of teaching to the test.

posted by l33tpolicywonk at 10:13 PM on May 14, 2010 [2 favorites]

*I don't know. If you emphasize concrete math over abstract math, it seems like you're setting up your students for a trainwreck later when you get into the fields in which abstract math is absolutely necessary - advanced algebra, calculus, and such. That kind of emphasis always seems to result in people who can do math, but don't understand math - and there is only so far that attitude will take you.*

I bet it does great on standardized tests, though, so it's probably going to be the next teaching paradigm."I bet it does great on standardized tests, though, so it's probably going to be the next teaching paradigm.

In both the above links and the video, Meyer talks pretty much exclusively about teaching Algebra I. You might appreciate his discussion in the video of teaching a slope problem: where a traditional textbook lumps together abstract model with concrete picture, Meyer begins with a picture as a means of demonstrating the efficacy of using abstract models. In other words, Meyer's job is at east in part cultivating the interest in and relatability of math.

For better or worse, you don't need advanced algebra or calculus to get into or even get through most colleges and universities in the US. Meyer doesn't say so, but he seems to be working in and among low-college-attendance populations. In that case, any math skills at all are pretty much an improvement on the status quo.

Teaching math is full of perverse incentives. People who are good at or curious about math are encouraged to teach the populations which have the most skills with math and, as a result, the youngest and lowest-performing kids are often stuck with teachers who had little to no math preparation in college.

The standardized test part of your comment I just don't get. If abstract mathematical reasoning is what we're getting at here, how other than a test do you propose to assess it? Meyer's actually got some pretty radical ideas about testing, including throwing out static unit tests for assessments at the end of each concept which allow students to skip those concepts they've already mastered. Find a teacher who parrots the textbook, he's accused of teaching to the test. Find a teacher who makes his own rich, multimedia driven teaching portfolio mapped to his own style and the needs of his students, he's accused of teaching to the test.

posted by l33tpolicywonk at 10:13 PM on May 14, 2010 [2 favorites]

*math needs abstraction.*

I don't entirely disagree, but I'm solidly in the camp that there definitely needs to be a lot more practice at applied problem solving. One of my strongest memories from student teaching was going through one of the story problem sections that tend to follow the rest of a largely abstract unit with one of my Algebra 2 classes. They were so stymied by the homework that I wanted to stop and spend a week on it. The followup reaction when I told the class was fearful and vehement -- these students were dead scared of having to try to face word problems. And my cooperating teachers really didn't want to linger more than an extra day. One of a handful of issues that made me wonder what in the world I was doing there and what we're doing teaching math in general if we're not getting our students over that hurdle.

I think you have to give students practice in answering concrete questions, whether it's the trajectory of a body in motion or "What's more economical -- a Corolla or a Prius?", that you have to

*start*with questions like that in order to make all the symbolic manipulation skills and even some of the higher reasoning skills worthwhile.

Lots of time to sell people on the power of abstraction once they've got the foundation in place for a bridge connecting it to reality.

posted by weston at 10:32 PM on May 14, 2010

I actually think this might help kids do "abstract math". That is, by removing all of the step-by-step nudging that a text book gives a reader, and forcing them to create these steps on their own, patterning will emerge....and this is what I consider to be a great use of abstraction: taking that line of reasoning which, through considerable work, allowed me to answer a problem and applying it to other problems. A process is an abstraction, even if it appears to be mere calculation.

posted by klausman at 10:49 PM on May 14, 2010 [2 favorites]

posted by klausman at 10:49 PM on May 14, 2010 [2 favorites]

people are bad at algebra because they never mastered arithmetic, and flunk calculus because they can't do algebra.

in order to get someone who is good at calculus when they are 18 you need to start training them when they are 6. is anyone is the U.S. planning twelve years ahead? we don't like to make that kind of investment, especially not in a person.

posted by ennui.bz at 10:55 PM on May 14, 2010

in order to get someone who is good at calculus when they are 18 you need to start training them when they are 6. is anyone is the U.S. planning twelve years ahead? we don't like to make that kind of investment, especially not in a person.

posted by ennui.bz at 10:55 PM on May 14, 2010

**weston:**

*Lots of time to sell people on the power of abstraction once they've got the foundation in place for a bridge connecting it to reality.*

The thing is, I think the bridge is planted on abstraction, not reality. My strongest memory of math is when I trained-wrecked out at the calculus level. The problem, in the end, was that I had no foundation. I had been pretty good at math before that, and even gotten awards when I was younger. However, when it became really difficult, I realized that even when I knew what to do, I didn't truly understand what I was doing - and that I never really had.

I didn't know (and still don't) the fundamental theoretical underpinnings of math itself that allow you to say 'oh, you're allowed to do this, but not this other operation.' I just made progress as long as I did because I was able to memorize rules and apply math as a methodological solution to problems, not truly understand it. That kind of thing, I think, is a great way to learn how to answer questions in the short term but is very destructive to long-term success - it is easier, so you get used to it, but it's only one-way. You can't use a learned methodological ability to do math to do anything novel. Also, I think, the complexity involved in using it as such usually stops you before you get too far in. I got through calc 3 and probably could have kept going for at least a while, but it would have been the same thing - not understanding, just learning and applying tools. I think that is fundamentally the wrong way to do things.

posted by Mitrovarr at 11:42 PM on May 14, 2010

Mitrovarr: "

Von Neumann once told a student "in mathematics you don't understand things. You just get used to them." Just as I'd go crazy if I had to write software in java bytecode, I think people would go nuts if they went into calculus armed only with addition and multiplication. Exponents and other short hands allow us to conveniently forget expansion of terms.

And once you study number theory and Peano axioms, ultimately things like 1+1=2 are unprovable. It's a social convention. There's no bridge from logic and fundamental axioms to 1+1=2. And that's to say nothing of things like unprovable theorems. Finally, I'm not sure children have the reasoning capacity to intuitively understand things like implication (or stop snickering at the phrase

Of course, it doesn't get better with age =( We do teach students proofs, and they hate it with a passion. Every time you hear someone complain about geometry, what they really mean is they hate proofs, because for some reason we decided geometry would be the one course we teach students proofs in. I'm pretty sure all the variations of side-angle-side were crafted for instructional purposes only. Perhaps a mathematician will prove me wrong by citing a recent publication involving it's application, but for the moment I feel no threat of contradiction.

The worst part is that proofs don't have to be illustrative or constructive. I mean, proof by contradiction is powerful but all the reader really learns about is things that don't and can't exist. I actually don't remember using proof by contradiction in geometry, but it came up all the time in Programming Logic and Algorithm Analysis. Which isn't so bad, except ideally you'd like the proof to hint at how to write the damn program.

So don't worry too much about your theoretical footing; it's counterproductive and even the smartest folks in the room sometimes loose those links.

posted by pwnguin at 1:04 AM on May 15, 2010

*I didn't truly understand what I was doing - and that I never really had.*

I didn't know (and still don't) the fundamental theoretical underpinnings of math itself that allow you to say 'oh, you're allowed to do this, but not this other operation.'"I didn't know (and still don't) the fundamental theoretical underpinnings of math itself that allow you to say 'oh, you're allowed to do this, but not this other operation.'

Von Neumann once told a student "in mathematics you don't understand things. You just get used to them." Just as I'd go crazy if I had to write software in java bytecode, I think people would go nuts if they went into calculus armed only with addition and multiplication. Exponents and other short hands allow us to conveniently forget expansion of terms.

And once you study number theory and Peano axioms, ultimately things like 1+1=2 are unprovable. It's a social convention. There's no bridge from logic and fundamental axioms to 1+1=2. And that's to say nothing of things like unprovable theorems. Finally, I'm not sure children have the reasoning capacity to intuitively understand things like implication (or stop snickering at the phrase

*modus ponens*...). So teach them order and arithmetic young.Of course, it doesn't get better with age =( We do teach students proofs, and they hate it with a passion. Every time you hear someone complain about geometry, what they really mean is they hate proofs, because for some reason we decided geometry would be the one course we teach students proofs in. I'm pretty sure all the variations of side-angle-side were crafted for instructional purposes only. Perhaps a mathematician will prove me wrong by citing a recent publication involving it's application, but for the moment I feel no threat of contradiction.

The worst part is that proofs don't have to be illustrative or constructive. I mean, proof by contradiction is powerful but all the reader really learns about is things that don't and can't exist. I actually don't remember using proof by contradiction in geometry, but it came up all the time in Programming Logic and Algorithm Analysis. Which isn't so bad, except ideally you'd like the proof to hint at how to write the damn program.

So don't worry too much about your theoretical footing; it's counterproductive and even the smartest folks in the room sometimes loose those links.

posted by pwnguin at 1:04 AM on May 15, 2010

I can't put my finger on it but, something about teacher with a blog just doesn't sit right with me.

posted by wcfields at 2:28 AM on May 15, 2010

posted by wcfields at 2:28 AM on May 15, 2010

*And once you study number theory and Peano axioms, ultimately things like 1+1=2 are unprovable. It's a social convention.*

You're misrepresenting Peano. 2 is defined to be the natural number that comes after 1. Addition is defined recursively, so that m + n is the number that comes after m + (n-1), and m + 1 is the number that comes after m. Thus, 1 + 1 must be 2.

*unprovable theorems*

These don't actually exist, by definition.

*for some reason we decided geometry would be the one course we teach students proofs in.*

That's because most geometry courses are based on Euclid's Elements in no small part, and understanding the content requires using proofs. Ideally, the people who were teaching these students would have a good understanding of what a proof actually is, but unfortunately (IME) most math education majors are highly unqualified to teach anyone about math.

*I'm pretty sure all the variations of side-angle-side were crafted for instructional purposes only.*

Triangle congruences underly a baffling number of geometric proofs, unless you think that the whole of Euclidean geometry was created for instructional purposes.

posted by TypographicalError at 2:30 AM on May 15, 2010 [2 favorites]

I used to tell my (community college, lots of folks who literally can't add anything above 5+5 in their heads) that they shouldn't hate word problems. With word problems, they can at least know if their answer makes sense. In a regular problem, you get an answer, and you can't apply basic intuition to see if the answer makes sense. In a word problem, if you're figuring out the volume of a swimming pool and you get 17 gallons, you know you fucked up.

posted by notsnot at 5:03 AM on May 15, 2010

posted by notsnot at 5:03 AM on May 15, 2010

"Skill practice and conceptual development are both essential. I have no interest in any war between them, nor in anyone who suggests they're enemies."

TRUTH BOMB

posted by escabeche at 6:08 AM on May 15, 2010

TRUTH BOMB

posted by escabeche at 6:08 AM on May 15, 2010

Nice! Sounds like he's a UC Davis graduate, and he has a nice post about the guy in the office next door to mine, which is kind of cool.

posted by kaibutsu at 6:16 AM on May 15, 2010

posted by kaibutsu at 6:16 AM on May 15, 2010

Link shortners in posts are frowned upon.

posted by blue_beetle at 6:27 AM on May 15, 2010

posted by blue_beetle at 6:27 AM on May 15, 2010

Hardcore Poser, what you've described is called constructivism. It's the way math, and most other subjects, have been taught it grade school for the past 20 years or so. Lately there's been some movement away from this method. Current research in cognitive science doesn't support it. But teachers (and I speak as one) are slow to accept and implement empirical evidence from outside teaching. Hence the emphasis on untested constructivist practices such as differentiated instruction. To help your son in math take a look at jump math. It pretty much argues against everything you just described, but it's amazingly effective. It was also created by an actual math professor, unlike most math curricula.

posted by trigger at 7:07 AM on May 15, 2010

posted by trigger at 7:07 AM on May 15, 2010

All the innovative ideas in the world aren't going to help in a society where it's accepted that there are some people who just "can't do" math and that's okay because nobody needs it anyway.

posted by Legomancer at 7:09 AM on May 15, 2010

posted by Legomancer at 7:09 AM on May 15, 2010

The biggest point in getting it for students that I tutored was to make them really explain it. Most HS students have a math workflow (even on long-answer word problems) which goes something like this:

posted by a robot made out of meat at 7:24 AM on May 15, 2010

- Problem
- Noodles
- Answer

posted by a robot made out of meat at 7:24 AM on May 15, 2010

After suffering through the swirling vortex of stupid that was Saxon math, believe me, this (and just about anything else, really, when it comes to math instruction) looks awfully good to me.

I wish I knew more about math. I had awful teachers, a frankly idiotic curriculum (that got scrapped shortly after I made it out of school -- sigh), and began to associate math with bamboo-under-fingernails-level torture. I'm perfectly good at practical math -- fractions, calculating tips, figuring out the best way to structure something mathematically in a piece of knitwear (designing knitted stuff involves a decent amount of geometry and algebra, believe it or not). So I'm happy to see anyone taking an approach that might actually help students not only get it, but also not hate it as much as I did for years.

posted by bitter-girl.com at 8:12 AM on May 15, 2010

I wish I knew more about math. I had awful teachers, a frankly idiotic curriculum (that got scrapped shortly after I made it out of school -- sigh), and began to associate math with bamboo-under-fingernails-level torture. I'm perfectly good at practical math -- fractions, calculating tips, figuring out the best way to structure something mathematically in a piece of knitwear (designing knitted stuff involves a decent amount of geometry and algebra, believe it or not). So I'm happy to see anyone taking an approach that might actually help students not only get it, but also not hate it as much as I did for years.

posted by bitter-girl.com at 8:12 AM on May 15, 2010

The problem with math instruction is not that math is hard. It isn't that math is obscure or abstract. The problem is that

How do you get over the fear? The same way you get over any fear: Exposure. Practice. Confidence. If necessary, one-on-one focused, patient assistance.

posted by DU at 8:25 AM on May 15, 2010

**students are afraid of it**. Why are they afraid of it? Because we keep telling them, directly and indirectly, that it's hard, obscure and abstract.How do you get over the fear? The same way you get over any fear: Exposure. Practice. Confidence. If necessary, one-on-one focused, patient assistance.

posted by DU at 8:25 AM on May 15, 2010

*Hardcore Poser, what you've described is called constructivism. It's the way math, and most other subjects, have been taught it grade school for the past 20 years or so.*

tigger, did you read/watch the links? What's being described by Dan has certainly not been taught in grade school for the past 20 years.

*Lately there's been some movement away from this method. Current research in cognitive science doesn't support it.*

Constructivism is not a method, though many methods are consist with it. If you're going to pull authority, throw us some citations.

posted by mathtime! at 11:17 AM on May 15, 2010

Apologies for the shortened link: entirely the result of poster error. For the record, it links to an Ezra Klein post w/ an embedded TED video of the subject of the OP. If the mods care to, please fix it.

posted by l33tpolicywonk at 11:30 AM on May 15, 2010

posted by l33tpolicywonk at 11:30 AM on May 15, 2010

I fixed it, carry on.

posted by jessamyn at 12:49 PM on May 15, 2010 [1 favorite]

posted by jessamyn at 12:49 PM on May 15, 2010 [1 favorite]

Abstraction isn't an end to itself, there always needs to be a reason. We shouldn't be abstracting the abstraction away at an earlier age, we should be constantly jumping between abstraction and real life. Here's the abstract concept, here is how we use it.

If someone had told me in sophomore year that all that crap about arctangents and sin and all that was about

posted by gjc at 1:22 PM on May 15, 2010

If someone had told me in sophomore year that all that crap about arctangents and sin and all that was about

*circles*and*motion*rather than a different kind of algebra, I might have understood it. Or even remembered it. I had a (seemingly) high powered math education in HS, such that I was able to sleep through Calc1 at university and pass fairly handily, but I didn't understand a lick of it and remember even less. Turns out, I was taught pure abstraction and memorization.posted by gjc at 1:22 PM on May 15, 2010

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