KTM User Pages
16 Mar 2006 - 00:03
While visiting Hung Hsi-Wu's website yesterday, I found an article he published last year on The Role of Open-Ended Problems in Mathematics Education. I'm still not at the level where I can easily read his work, though I think I could muscle my way through. (I should put this proposition to the test, shouldn't I?) That said, I assume Hung Hsi-Wu is talking about the kind of problem I think of as Problems The Kids Can't Do. We call them Extended Response here in Irvington; they have various names elsewhere. Possibly the most famous such problem is the haybaler problem from IMP, which Barry Garelick posted awhile back. Google "haybaler problem" and you get 619 hits. Here is Hung Hsi-Wu:
Open-ended problems have become a popular educational tool in mathematics education in recent years. Since mathematical research is nothing but a daily confrontation with open-ended problems, the introduction of this type of problems to the classroom brings mathematical education one step closer to real mathematics. The appearance of these problems in secondary education is therefore a welcome sight from a mathematical standpoint. More than this is true, however. While these problems may represent something of a pedagogical innovation to the professional educators, the fact is that many mathematicians have made use of them in their teaching all along and do not regard their presence in the classroom as any kind of departure in educational philosophy. For example, I myself have often given such problems in my homework assignments and exams.2 Nevertheless, I have chosen to take up this topic for discussion here because, after having reviewed a limited amount of curricular materials for mathematics in the schools, I could not help but notice that they pose certains hazards in practice. These hazards include the possibility of misinforming the students about the very nature of mathematics itself.
Two things: a) open-ended problems are not confined to high school mathematics. and b) I'm going to dive in and take it as a given that open-ended problems for 9-year olds and misinformation about the very nature of mathematics go hand-in-hand. But in fact, I don't know. What do you think of these problems? Eggs for 9 year olds (pdf file)
multiples of 4 that end in 4 for 13 year olds (pdf file)
the Million Dollar Job a group problem for 8th graders (pdf file)
Sources of Mathematics Open-Response Items
World Class Arena
it's always worse than you think I've now skimmed enough of Hung Hsi Wu's article to see that I was right. His subject is Problems The Kids Can't Do. And, yes, it's always worse than you think:
...in discussing these three problems with some teachers, I was astounded to be told by one and all that they considered the first part of Problem I (“WHAT MIGHT ITS AREA BE??”) to be a good problem because it allows the students to make up their own questions and answers, but that they thought the second part (“WHAT WOULD THE LARGEST AREA BE??”) was bad because it pins down the students to a single correct answer. Since a good part of mathematics, pure or applied, is pre-occupied precisely with such maximization problems, we have here an example of an educational philosophy that has distorted the way a group of teachers think about the subject they are supposed to teach.5 This should be a matter of grave concern.I'll say.
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
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Wu has hit the nail on the head with that piece. Yes, open ended problems are important but there's a time and place. There's also a difference between open-ended problems (problems that have more than one right answer, or more precisely a solution set which contains more than one element) and problems kids can't do (like the Haybaler Problem). I was keeping this a dirty secret, but I'm in ed school right now, taking my first evening class. What I've learned is that the education profession LOVES problems with more than one answer. Questions that can be answered in more than one way are called "divergent" questions. For reasons I don't understand and don't want to understand, the education profession feels that convergent answers are narrow, confining and mind numbing, while divergent answers open children's minds. Yes, I believe it's bullshit also and I hear unanimous consent, so I shall move on. As for open-ended problems, there's are limits to their value. Yes, you can give a problem like "You are starting a bike messenger service and need to purchase a bicycle. How much money should you spend?" (Thanks to John Hoven, an economist, for this example). The problem is one of "it depends" and solving it would involve exploring what those things are. The question is, how much class or homework time would that take, and could students be making better use of their time learning skills and concepts that will help them better define and construct solutions to such problems? Defenders of such problems claim that they are "real world" and therefore prepare students better than the typical "drill and kill" problems. Well, actually, disciplines like economics, accounting, business management, physics, chemistry, etc. prepare students later on for the complexities of solving problems, but the basic skills and concepts of math are there in all of them. But that brings us to the next favorite thing of the education profession: Interdisciplinary learning. A math class just isn't a math class unless you can bring in elements of economics, literature, social studies, you name it. We don't want to narrow and dull students minds by just teaching them what they ought to be learning in a math class, now do we? -- BarryGarelick - 16 Mar 2006
I think you once linked to a post by a maths teacher who had learnt to give surface area problems not as: "How paint you would need to put two coats onto a room that's 8 feet high, 10 feet wide and 12 feet long, with one door that's 2 feet wide and 7 feet high?" But as: "Calculate how much paint you would need if you were painting your room." That strikes me as a good open-ended question. Though it does involve a bit more work in checking the answers. And I did have teachers at school who would start off a subject by giving us a problem and letting us struggle with it for a bit, and then showing us the new technique that would solve it (all in class). For example before we started on calculus we were given a "calculate the maximum area given so many metres of fencing" problem. But I agree giving open-ended problems and then not showing students how they can be solved is cruel. And confusing. -- TracyW - 16 Mar 2006
Barry! Congratulations!!! Yay, you! Boy, the teaching profession needs you. I say we all INVADE. you're SO right about interdisciplinary. Check out The Jason Project (we just came back from our Jason Project 'Museum': JASON At-Home Edition is Integrated and Multidisciplinary:
Life, Physical, and Earth Sciences
-- CatherineJohnson - 16 Mar 2006
This is supposed to be a huge selling point TO HOMESCHOOLERS, for pete's sake. Then there's this: To enhance your JASON At-Home experience, take an Online Expedition Workshop and purchase the Carolina Biological Disappearing Wetlands materials kit. A KIT! FOR THE DISAPPEARING WETLANDS! FORGET ERECTOR SETS! YOU CAN GET A DISAPPEARING WETLANDS KIT! -- CatherineJohnson - 16 Mar 2006
But I diverge........ -- CatherineJohnson - 16 Mar 2006
Tracy I have no idea when & where open-ended problems become important.....(Saxon has one 'Problem-Solving' problem at the beginning of each and every Lesson....but as usual, they do have a solution, the solution is presented and explained, and the problems are included because you're on the Approach Path to learning that concept. Still, he has one in each lesson. I like them!) But I agree giving open-ended problems and then not showing students how they can be solved is cruel. And confusing. THis i what's always amazing to me. EVERYTHING is topsy-turvy. The open ended problem is supposed to have more than one answer — oops! Just one answer! It's supposed to be real-world — oops! Antopolis is weird and not fun! It's supposed to Increase Appreciation for Mathematics — oops! No one wants to do problems they can't solve EVER. -- CatherineJohnson - 16 Mar 2006
Barry, wow, congratulations! Are you keeping your day job? (I presume so since you are taking an evening class). I hope ed school doesn't turn you off. -- CarolynJohnston - 16 Mar 2006
"Barry, wow, congratulations! Are you keeping your day job? (I presume so since you are taking an evening class). I hope ed school doesn't turn you off." Thanks for the congrats from Johnson and Johnston. Yes, I am keeping the day job. I am eligible to retire in 5 years after which I should have my teaching credential, and then I will try to get a job teaching math in high school. It hasn't turned me off as yet. I'm taking a genral introduction to secondary education course. (Not a math teaching class yet). I happen to like the teacher and he is open to opinions, and his philosophy is that you should teach in the manner in which you are comfortable. In other words, don't be something you're not. There's a bit more to it than that, but that's the essence. So if I say I don't like pure discovery, and ill-posed problems are a waste of everyone's time, he values the opinion. I'm taking the opportunity to express my opinions now, because my advisor is an NCTM adherent despite her masters in math. She teaches the math teaching methods class, so the fun is over once I hit those. She and I exchanged some emails about calculators and it was evident where she was coming from. Her line of thinking is that "math evolves" and that we have now "evolved" into using calculators so go with the flow. The "math evolves" business brings to mind Steve H's quip: "Have you checked what 8 x 7 is lately?" which I ALMOST put in an email back to her, but then thought better of it. I'll wait til I get the certificate first. Catherine: A disappearing wetlands kit? What about a disappearing math knowledge kit? That would be for social studies classes. -- BarryGarelick - 16 Mar 2006
"Calculate how much paint you would need if you were painting your room." I wonder how many kid's rooms would suddenly become 10' x 10' x 10', given this problem. -- TerriWheeler - 16 Mar 2006
I used to have a Disappearing Wetlands Kit, but it disappeared. -- GoogleMaster - 16 Mar 2006
I am eligible to retire in 5 years after which I should have my teaching credential, and then I will try to get a job teaching math in high school. CAN WE GET YOU TO MOVE TO IRVINGTON????!!!! (I should add that I don't know anything about the high school math teachers here; they may be great. I certainly haven't heard anything bad about them....) EVEN SO, CAN WE GET YOU TO MOVE TO IRVINGTON??? -- CatherineJohnson - 16 Mar 2006
TERRI LOL!! -- CatherineJohnson - 16 Mar 2006
Regarding the problems attached at the end, are these supposed to be examples of open-ended problems? I personally like the 13-year-old and 8th grade problems. The 13-year-old problem is totally solvable and has one right answer. My classmates and I did a variation of the 8th grade problem--I think it's pretty common. Of course, it was followed up with instruction in powers and such. I think the use of problems like this capture kids' interest and show them that more math knowlege can be a time-saving tool for "real life" problems, even if they don't expect to encounter a situation exactly like this one. I'm not sure about the 9-year-old problem. I think it is solvable by a 9-year-old, but I'm not sure what its point is. It seems like it might help them practice reasoning skills, reading and comprehending problems, and learning to organize data in columns or a table. I'm not sure how it would/should match up with the curriculum at that age. -- AndyJoy - 16 Mar 2006
Hey, the Million Dollar Job question is trying to meet "KENTUCKY LEANER GOALS" (sic). I wonder if that's like a Tennessee Walker, only not quite. -- GoogleMaster - 16 Mar 2006
Holy Moses, I think too much. The egg question:
Ambiguities:Mrs Newman has five children. Three of them are girls. Two of them are boys. The children buy chocolate eggs to give to each other. Each girl gives each boy a red egg. Each boy gives each girl a yellow egg. Each girl gives each of the other girls a blue egg. Each boy gives each of the other boys a green egg. How many eggs of each colour do the children buy?
"This is an open-ended question for sure." It is an ambiguous question(as is the bicycle question). Open-ended questions are not necessarily ambiguous, and they are not necessarily problems that kids can't do. These distinctions are important if you want to understand the paper. -- KtmGuest - 16 Mar 2006
"CAN WE GET YOU TO MOVE TO IRVINGTON??? " Well, our plans are to move to California, so I'm afraid Irvington is out, much as I'd like to be your neighbor. High school math teachers, at least in the area where I live, are pretty good. I've been observing some math classes (which I have to do for my class in ed. school) and many of the teachers have degrees in math and are no-nonsense type of people. Actually one teacher, not a math major, told me prior to the class, that he didn't believe in "that discovery crap" and he doesn't have kids work in groups. The math "lounge" for the faculty contains many different math texts that teachers can use to supplement the "official" bad texts. Among them are the Doliciani algebra series which are made use of. Also Moise-Downs "Geometry", but not too many teachers make use of that, unfortunately. Proofs are still not a mainstay of geometry classes, except honors geometry and even then (I saw such a class today), students consider proofs to be the hardest part of geometry. It used to be that that's what geometry classes were all about. Now it's a "sometimes" thing, but the honors classes have a good bit of them. The regular geometry classes don't have any. Interestingly, the teacher I observed today for a class in "analysis" (sort of a non-honors version of pre-calculus) told me that the last few years students are having a hard time with beginning algebra. They can't do as much math as kids in previous years. He blamed the SOL tests for that. There's a lot of preparation for SOL's which because the questions are so dumb, necessary leaves out a lot of math if you are gearing kids up to do well on that. -- BarryGarelick - 16 Mar 2006
SOL tests meaning Student-Out of-Luck tests? -- SmartestTractor - 16 Mar 2006
The official meaning of SOL is Standards of Learning, but your definition is by far the more accurate one. -- BarryGarelick - 16 Mar 2006