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."

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.

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

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

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

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

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

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/

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

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