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#1
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Study Suggests Math Teachers Scrap Balls and Slices
http://www.nytimes.com/2008/04/25/science/25math.html
Study Suggests Math Teachers Scrap Balls and Slices By Kenneth Chang New York Times, April 25, 2008 'One train leaves Station A at 6 p.m. traveling at 40 miles per hour toward Station B. A second train leaves Station B at 7 p.m. traveling on parallel tracks at 50 m.p.h. toward Station A. The stations are 400 miles apart. When do the trains pass each other? Entranced, perhaps, by those infamous hypothetical trains, many educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn. That idea may be wrong, if researchers at Ohio State University are correct. An experiment by the researchers suggests that it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations, in this case 40 (t + 1) = 400 - 50t, where t is the travel time in hours of the second train. (The answer is below.)' rest at site If the study is correct, I wonder which math curricula are most consistent with it. It appears to contradict the philosophy of Everyday Mathematics (EM), which our public school use. The EM site http://everydaymath.uchicago.edu/about.shtml#curriculum says "Students acquire knowledge and skills, and develop an understanding of mathematics from their own experience. Mathematics is more meaningful when it is rooted in real life contexts and situations, and when children are given the opportunity to become actively involved in learning. Teachers and other adults play a very important role in providing children with rich and meaningful mathematical experiences." |
#2
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Study Suggests Math Teachers Scrap Balls and Slices
On Apr 25, 2:58*am, Beliavsky wrote:
http://www.nytimes.com/2008/04/25/science/25math.html Study Suggests Math Teachers Scrap Balls and Slices By Kenneth Chang New York Times, April 25, 2008 'One train leaves Station A at 6 p.m. traveling at 40 miles per hour toward Station B. A second train leaves Station B at 7 p.m. traveling on parallel tracks at 50 m.p.h. toward Station A. The stations are 400 miles apart. When do the trains pass each other? Entranced, perhaps, by those infamous hypothetical trains, many educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn. That idea may be wrong, if researchers at Ohio State University are correct. An experiment by the researchers suggests that it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations, in this case 40 (t + 1) = 400 - 50t, where t is the travel time in hours of the second train. (The answer is below.)' This claim is ridiculous. Learning how to translate a verbal statement of the problem into equations is far more important than the mindless manipulations used to solve (in this case) linear equations. The latter only involves application of an algorithm. |
#3
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Study Suggests Math Teachers Scrap Balls and Slices
Beliavsky wrote:
http://www.nytimes.com/2008/04/25/science/25math.html Study Suggests Math Teachers Scrap Balls and Slices By Kenneth Chang New York Times, April 25, 2008 'One train leaves Station A at 6 p.m. traveling at 40 miles per hour toward Station B. A second train leaves Station B at 7 p.m. traveling on parallel tracks at 50 m.p.h. toward Station A. The stations are 400 miles apart. When do the trains pass each other? Entranced, perhaps, by those infamous hypothetical trains, many educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn. Actually, that isn't the idea that I've seen. The idea is that making math more relevant makes kids more willing to learn, and provides at least some hope that they'll have some use for the math once they walk away from the classroom. The average person never sees an abstract equation in real life after graduation (other than possibly e=mc^2 where the equation is iconic rather than being an equation to be understood). lojbab |
#4
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Study Suggests Math Teachers Scrap Balls and Slices
In article , Bob LeChevalier says...
Beliavsky wrote: http://www.nytimes.com/2008/04/25/science/25math.html Study Suggests Math Teachers Scrap Balls and Slices By Kenneth Chang New York Times, April 25, 2008 'One train leaves Station A at 6 p.m. traveling at 40 miles per hour toward Station B. A second train leaves Station B at 7 p.m. traveling on parallel tracks at 50 m.p.h. toward Station A. The stations are 400 miles apart. When do the trains pass each other? Entranced, perhaps, by those infamous hypothetical trains, many educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn. Actually, that isn't the idea that I've seen. The idea is that making math more relevant makes kids more willing to learn, and provides at least some hope that they'll have some use for the math once they walk away from the classroom. From my physics and engineering training, math is exactly *about* describing the real world. Indeed, aspects of the real world can only be approached mathematically (relativity, quantum physics). So the idea of not invoking the real world in teaching mathematics makes absolutely no sense. I haven't read the article, but I suspect it's not invoking real world examples that have hobbled math education. Rather, it's the reliance on expressing math oin *verbal* terms in the kind of examples that elementary schools have favored lately (see many threads in misc.kids about that). Banty |
#5
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Study Suggests Math Teachers Scrap Balls and Slices
Beliavsky wrote:
http://www.nytimes.com/2008/04/25/science/25math.html Study Suggests Math Teachers Scrap Balls and Slices By Kenneth Chang New York Times, April 25, 2008 'One train leaves Station A at 6 p.m. traveling at 40 miles per hour toward Station B. A second train leaves Station B at 7 p.m. traveling on parallel tracks at 50 m.p.h. toward Station A. The stations are 400 miles apart. When do the trains pass each other? Entranced, perhaps, by those infamous hypothetical trains, many educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn. That idea may be wrong, if researchers at Ohio State University are correct. An experiment by the researchers suggests that it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations, in this case 40 (t + 1) = 400 - 50t, where t is the travel time in hours of the second train. (The answer is below.)' rest at site If the study is correct, I wonder which math curricula are most consistent with it. It appears to contradict the philosophy of Everyday Mathematics (EM), which our public school use. The EM site http://everydaymath.uchicago.edu/about.shtml#curriculum says "Students acquire knowledge and skills, and develop an understanding of mathematics from their own experience. Mathematics is more meaningful when it is rooted in real life contexts and situations, and when children are given the opportunity to become actively involved in learning. Teachers and other adults play a very important role in providing children with rich and meaningful mathematical experiences." I see all sorts of possible issues with this. First, I'm not sure I buy the notion that you can generalize easily from college students to elementary students. Developmentally they're in different places when it comes to abstract thinking. That, however, is obviously an empirical question that could be tested. Second, I believe there is other evidence on the table that some students learn better inductively and some better deductively, and that some have more need of concrete representations than others in attempting to understand concepts. That said, I do think that this notion that everything is best *taught* through a consistent emphasis on concrete examples is hogwash, and I'm not at all surprised by the results of the experiment. The task they measured the students on required the students to think more theoretically--they had to get the theory to be able to apply to the specific situation. It seems likely to me that the students who were specifically taught the theory would do better than those who had to induce the theory from the specific examples. Third, while it's important for the kids to know the theory, for the overwhelming majority of them the importance of knowing the theory is so that they can apply it. Therefore, the "two trains" problem is still an important component because a student who has mastered the material should be able to apply the theory to figure out the answer to the problem. That's a separate issue from how best to get students to learn and understand the theory. Best wishes, Ericka |
#6
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Study Suggests Math Teachers Scrap Balls and Slices
In article , Banty says...
In article , Bob LeChevalier says... Beliavsky wrote: http://www.nytimes.com/2008/04/25/science/25math.html Study Suggests Math Teachers Scrap Balls and Slices By Kenneth Chang New York Times, April 25, 2008 'One train leaves Station A at 6 p.m. traveling at 40 miles per hour toward Station B. A second train leaves Station B at 7 p.m. traveling on parallel tracks at 50 m.p.h. toward Station A. The stations are 400 miles apart. When do the trains pass each other? Entranced, perhaps, by those infamous hypothetical trains, many educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn. Actually, that isn't the idea that I've seen. The idea is that making math more relevant makes kids more willing to learn, and provides at least some hope that they'll have some use for the math once they walk away from the classroom. From my physics and engineering training, math is exactly *about* describing the real world. Indeed, aspects of the real world can only be approached mathematically (relativity, quantum physics). So the idea of not invoking the real world in teaching mathematics makes absolutely no sense. I haven't read the article, but I suspect it's not invoking real world examples that have hobbled math education. Rather, it's the reliance on expressing math oin *verbal* terms in the kind of examples that elementary schools have favored lately (see many threads in misc.kids about that). I just read the article. I don't know if it's more in the headlining, or the usual news media oversimplification int his article, but from this study teachers would *not* be "scapping" balls and slices. (Or trains, as in the first paragraph.) Balls and trains have been discussed in elementary phsyics and algebra forever. From the article: ________________________ In the experiment, the college students learned a simple but unfamiliar mathematical system, essentially a set of rules. Some learned the system through purely abstract symbols, and others learned it through concrete examples like combining liquids in measuring cups and tennis balls in a container. ________________________ This is about the learning-by-doing active learning model, not about relating real-world concrete examples to mathematics. Learning *through* pouring stuff and counting stuff, rather than having the abstract *related to* volumes, speeds, etc. And, yes, I'm not surprised it doesn't work well for mathematics. Kids need to learn the abstraction (it *is* abstractions after all) AND have the world world application. But the abstract representation is basic. Banty |
#7
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Study Suggests Math Teachers Scrap Balls and Slices
Banty wrote:
[snip] I haven't read the article, but I suspect it's not invoking real world examples that have hobbled math education. Rather, it's the reliance on expressing math oin *verbal* terms in the kind of examples that elementary schools have favored lately (see many threads in misc.kids about that). There seems to be a fuller press release he http://www.eurekalert.org/pub_releas...-ced042108.php It says that teaching *college* students with abstract examples enabled them to grasp the underlying maths more easily then when they learnt a system using concrete examples. The NYT article also discussed some experiments with 11yo children (which aren't in the press release). I don't think that you can necessatily extrapolate further to children of the age to be using ths Everyday maths, which seems to be Kindergarten to Grade 6. -- Penny Gaines UK mum to three |
#8
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Study Suggests Math Teachers Scrap Balls and Slices
In article
, Beliavsky wrote: That idea may be wrong, if researchers at Ohio State University are correct. An experiment by the researchers suggests that it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations, in this case 40 (t + 1) = 400 - 50t, where t is the travel time in hours of the second train. (The answer is below.)' rest at site If the study is correct, I wonder which math curricula are most consistent with it. It appears to contradict the philosophy of Everyday Mathematics (EM), which our public school use. The EM site http://everydaymath.uchicago.edu/about.shtml#curriculum says "Students acquire knowledge and skills, and develop an understanding of mathematics from their own experience. Mathematics is more meaningful when it is rooted in real life contexts and situations, and when children are given the opportunity to become actively involved in learning. Teachers and other adults play a very important role in providing children with rich and meaningful mathematical experiences." As someone else mentioned, the content and method of methods for teaching mathematics to adults are different from those for children. I just wanted to note that not all children come to school with the same level of numeracy (which is obvious) BUT they don't come with the same learning style OR at the same developmental level. A good teacher has to cope with all these things. -- Chookie -- Sydney, Australia (Replace "foulspambegone" with "optushome" to reply) http://chookiesbackyard.blogspot.com/ |
#9
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Study Suggests Math Teachers Scrap Balls and Slices
In article ,
Beliavsky wrote: http://www.nytimes.com/2008/04/25/science/25math.html Study Suggests Math Teachers Scrap Balls and Slices By Kenneth Chang New York Times, April 25, 2008 'One train leaves Station A at 6 p.m. traveling at 40 miles per hour toward Station B. A second train leaves Station B at 7 p.m. traveling on parallel tracks at 50 m.p.h. toward Station A. The stations are 400 miles apart. When do the trains pass each other? Entranced, perhaps, by those infamous hypothetical trains, many educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn. That idea may be wrong, if researchers at Ohio State University are correct. An experiment by the researchers suggests that it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations, in this case 40 (t + 1) = 400 - 50t, where t is the travel time in hours of the second train. (The answer is below.)' The most important part of algebra for the non-mathematician is formulation. The position of the train leaving station A at time t ( =6), relevant to station A, is 40*(t-6). For the train leaving station B, the position, relative to station A is 400 - 50*(t-7), assuming t = 7. One sets these two equal and solves for t. Then comes the solution; showing how to do it deserves most of the credit even if the arithmetic is bad. The most important part is formulation. If one can formulate the problem correctly, it can be fed into a sufficiently advanced calculator and get solved. If it is not formulated correctly, who cares if the student knows how to do the rest. The solution for this problem, and for most, follows the rule of equality, which states that the same operation done on equal entities gives equal results. It this is used, any calculator can be used to get the results. If it is not known how to use it, the process of getting from the formulation to the use of arithmetic is likely to produce wrong answers. Finally comes the arithmetic. Knowing how to do the arithmetic was not even that important BC (before computers) and is of little importance now; it may be useful, and I see no reason not to teach it. A student (as distinguished from a warm body occupying a space in a classroom) will try to learn the useful materical, and should be discouraged from overdoing it. If the study is correct, I wonder which math curricula are most consistent with it. It appears to contradict the philosophy of Everyday Mathematics (EM), which our public school use. The EM site http://everydaymath.uchicago.edu/about.shtml#curriculum says "Students acquire knowledge and skills, and develop an understanding of mathematics from their own experience. Mathematics is more meaningful when it is rooted in real life contexts and situations, and when children are given the opportunity to become actively involved in learning. Teachers and other adults play a very important role in providing children with rich and meaningful mathematical experiences." : -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University Phone: (765)494-6054 FAX: (765)494-0558 |
#10
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Study Suggests Math Teachers Scrap Balls and Slices
In article , Penny Gaines wrote:
Banty wrote: [snip] I haven't read the article, but I suspect it's not invoking real world examples that have hobbled math education. Rather, it's the reliance on expressing math oin *verbal* terms in the kind of examples that elementary schools have favored lately (see many threads in misc.kids about that). There seems to be a fuller press release he http://www.eurekalert.org/pub_releas...-ced042108.php It says that teaching *college* students with abstract examples enabled them to grasp the underlying maths more easily then when they learnt a system using concrete examples. The NYT article also discussed some experiments with 11yo children (which aren't in the press release). I don't think that you can necessatily extrapolate further to children of the age to be using ths Everyday maths, which seems to be Kindergarten to Grade 6. -- Penny Gaines UK mum to three From my experience with my children, and also from seeing what happens to college students, including PhD students in mathematics and statistics, the biggest problem with starting with concrete examples and special cases is that the excess baggage from the special cases is hard to discard. This also holds for people with my abilities. Children can understand abstract ideas, up to a reasonable level of complexity. Much of mathematics has ideas which are not that complex, and are much simpler than what is now taught. The axioms for the development of the integers, especially the ordinal ones which are complete, are much simpler than the use of decimal notation. The complete rules for formal logic can be done on a couple of pages. The use of symbols for entities, the key part of algebra, can be taught with beginning reading; the entities can be anything, including words and phrases. As the children grow older, and get taught mainly by memorization and routine, this ability gets smothered, and it is difficult to reawaken later. Children understand abstract ideas; it is worse than pulling teeth to try to get this though the heads of most adults. Getting it by just being shown examples requires research ability. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University Phone: (765)494-6054 FAX: (765)494-0558 |
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