mathematics
Pam Sorooshian
Linda Wyatt posted this on Facebook and it is a good explanation of
what math really is:
<http://www.fordham.edu/academics/programs_at_fordham_/mathematics_departme/what_math/index.asp>
The first analogy is "Scaffolding" and I like it a lot. The second
analogy is a comparison to training for sports and I don't like it much.
I'll copy the first one and briefly comment, then the second, and then
my short comment on why I don't like the second. There is more to the
article that is worth reading.
******
Scaffolding.
When a new building is made, a skeleton of steel struts called the
scaffolding is put up first. The workers walk on the scaffolding and use
it to hold equipment as they begin the real task of constructing the
building. The scaffolding has no use by itself. It would be absurd to
just build the scaffolding and then walk away, thinking that something
of value has been accomplished.
Yet this is what seems to occur in all too many mathematics classes in
high schools. Students learn formulas and how to plug into them. They
learn mechanical techniques for solving certain equations or taking
derivatives. But all of these things are just the scaffolding. They are
necessary and useful, sure, but by themselves they are useless. Doing
only the superficial and then thinking that something important has
happened is like building only the scaffolding.
The real "building" in the mathematics sense is the true mathematical
understanding, the true ability to think, perceive, and analyze
mathematically.
*******************
Just saying - those construction workers who are building scaffolding
know the purpose of it and know how it will be used during construction
of the building. Students in math classes seldom know the purpose of the
"scaffolding" they are supposed to be building and almost never see any
point to most of it and, for MOST of those students, there is no point
because they are not going to be using it (most of what is taught in an
algebra class, for example) at all for the rest of their lives.
------------------------------------------------------------------------
************************
Ready for the big play.
Professional athletes spend hours in gyms working out on equipment of
all sorts. Special trainers are hired to advise them on workout
schedules. They spend hours running on treadmills. Why do they do that?
Are they learning skills necessary for playing their sport, say basketball?
Imagine there're three seconds left in the seventh game of the NBA
championship. The score is tied. Time out. The pressure is intense. The
coach is huddling with his star players. He says to one, "OK Michael,
this is it. You know what to do." And Michael says, "Right coach. Bring
in my treadmill!"
Duh! Of course not! But then what was all that treadmill time for? If
the treadmill is not seen during the actual game, was it just a waste to
use it? Were all those trainers wasting their time? Of course not. It
produced (if it was done right!) something of value, namely stamina and
aerobic capacity. Those capacities are of enormous value even if they
cannot be seen in any immediate sense. So too does mathematics education
produce something of value, true mental capacity and the ability to think.
********************
My problem with the sports comparison is in the conclusion - that math
education produces "true mental capacity and the ability to think."
First - math "education" doesn't often produce that - more likely the
opposite. Math education almost always produces LESS ability to think
and more mental confusion and reliance on rote memorization. Second,
there are many other ways to develop mental capacity and the ability to
think - math is not the only (or necessarily the best) way and certainly
not even a good way for everybody. Third, and most importantly, even if
learning about mathematics helps people think more logically, there is
much more to "true mental capacity and the ability to think," than that.
Mathematics does not, for example, help people think more empathically,
more globally, more creatively, or more independently.
-pam
[Non-text portions of this message have been removed]
what math really is:
<http://www.fordham.edu/academics/programs_at_fordham_/mathematics_departme/what_math/index.asp>
The first analogy is "Scaffolding" and I like it a lot. The second
analogy is a comparison to training for sports and I don't like it much.
I'll copy the first one and briefly comment, then the second, and then
my short comment on why I don't like the second. There is more to the
article that is worth reading.
******
Scaffolding.
When a new building is made, a skeleton of steel struts called the
scaffolding is put up first. The workers walk on the scaffolding and use
it to hold equipment as they begin the real task of constructing the
building. The scaffolding has no use by itself. It would be absurd to
just build the scaffolding and then walk away, thinking that something
of value has been accomplished.
Yet this is what seems to occur in all too many mathematics classes in
high schools. Students learn formulas and how to plug into them. They
learn mechanical techniques for solving certain equations or taking
derivatives. But all of these things are just the scaffolding. They are
necessary and useful, sure, but by themselves they are useless. Doing
only the superficial and then thinking that something important has
happened is like building only the scaffolding.
The real "building" in the mathematics sense is the true mathematical
understanding, the true ability to think, perceive, and analyze
mathematically.
*******************
Just saying - those construction workers who are building scaffolding
know the purpose of it and know how it will be used during construction
of the building. Students in math classes seldom know the purpose of the
"scaffolding" they are supposed to be building and almost never see any
point to most of it and, for MOST of those students, there is no point
because they are not going to be using it (most of what is taught in an
algebra class, for example) at all for the rest of their lives.
------------------------------------------------------------------------
************************
Ready for the big play.
Professional athletes spend hours in gyms working out on equipment of
all sorts. Special trainers are hired to advise them on workout
schedules. They spend hours running on treadmills. Why do they do that?
Are they learning skills necessary for playing their sport, say basketball?
Imagine there're three seconds left in the seventh game of the NBA
championship. The score is tied. Time out. The pressure is intense. The
coach is huddling with his star players. He says to one, "OK Michael,
this is it. You know what to do." And Michael says, "Right coach. Bring
in my treadmill!"
Duh! Of course not! But then what was all that treadmill time for? If
the treadmill is not seen during the actual game, was it just a waste to
use it? Were all those trainers wasting their time? Of course not. It
produced (if it was done right!) something of value, namely stamina and
aerobic capacity. Those capacities are of enormous value even if they
cannot be seen in any immediate sense. So too does mathematics education
produce something of value, true mental capacity and the ability to think.
********************
My problem with the sports comparison is in the conclusion - that math
education produces "true mental capacity and the ability to think."
First - math "education" doesn't often produce that - more likely the
opposite. Math education almost always produces LESS ability to think
and more mental confusion and reliance on rote memorization. Second,
there are many other ways to develop mental capacity and the ability to
think - math is not the only (or necessarily the best) way and certainly
not even a good way for everybody. Third, and most importantly, even if
learning about mathematics helps people think more logically, there is
much more to "true mental capacity and the ability to think," than that.
Mathematics does not, for example, help people think more empathically,
more globally, more creatively, or more independently.
-pam
[Non-text portions of this message have been removed]
dola dasgupta-banerji
Hi Pam,
I liked the bit about "scaffolding" too. Wouldn't it be interesting to have
math lesson which read more like a story than simply boring formulae and
figures. Sometime back I had read about a math teacher who told his pupils
stories of great discoveries, great architecture and exploring that used
math as a tool. Among them where the building of the pyramids, the mapping
and scaling of the entire Himalayan range and how the height of Everest was
calculated without actually climbing it with a tape and measure and the most
interesting one I found was the secret behind why one is one and two is two
and three is three and so on.
One is one because in the Arabic numerals one is written in such a way that
it has one angle, two has two, three has three and guess what zero is zero
since it has no angles. I wish someone had taught me math that way.
Math is probably one of the most creative area of the human mind but
unfortunately for most it has been reduced to common market place......
The intermingling of art, culture, music, math and science is and surely
must be the future of education for holistic growth of the human heart and
mind. Does any math teacher ever tell us how math plays a important role in
deciding the meter of a tune or how many beats each Indian classical Raga
has. I learn Kathak, a classical dance form, and the amount of math that
goes into each composition is intricate and amazing. And most of the older
gurus of dance have never been to any 'school' or 'college'!
Thanks for this lovely post.
Dola
On Thu, Dec 23, 2010 at 12:45 AM, Pam Sorooshian
<pamsoroosh@...>wrote:
I liked the bit about "scaffolding" too. Wouldn't it be interesting to have
math lesson which read more like a story than simply boring formulae and
figures. Sometime back I had read about a math teacher who told his pupils
stories of great discoveries, great architecture and exploring that used
math as a tool. Among them where the building of the pyramids, the mapping
and scaling of the entire Himalayan range and how the height of Everest was
calculated without actually climbing it with a tape and measure and the most
interesting one I found was the secret behind why one is one and two is two
and three is three and so on.
One is one because in the Arabic numerals one is written in such a way that
it has one angle, two has two, three has three and guess what zero is zero
since it has no angles. I wish someone had taught me math that way.
Math is probably one of the most creative area of the human mind but
unfortunately for most it has been reduced to common market place......
The intermingling of art, culture, music, math and science is and surely
must be the future of education for holistic growth of the human heart and
mind. Does any math teacher ever tell us how math plays a important role in
deciding the meter of a tune or how many beats each Indian classical Raga
has. I learn Kathak, a classical dance form, and the amount of math that
goes into each composition is intricate and amazing. And most of the older
gurus of dance have never been to any 'school' or 'college'!
Thanks for this lovely post.
Dola
On Thu, Dec 23, 2010 at 12:45 AM, Pam Sorooshian
<pamsoroosh@...>wrote:
>[Non-text portions of this message have been removed]
>
> Linda Wyatt posted this on Facebook and it is a good explanation of
> what math really is:
> <
> http://www.fordham.edu/academics/programs_at_fordham_/mathematics_departme/what_math/index.asp
> >
>
> The first analogy is "Scaffolding" and I like it a lot. The second
> analogy is a comparison to training for sports and I don't like it much.
> I'll copy the first one and briefly comment, then the second, and then
> my short comment on why I don't like the second. There is more to the
> article that is worth reading.
>
> ******
> Scaffolding.
>
> When a new building is made, a skeleton of steel struts called the
> scaffolding is put up first. The workers walk on the scaffolding and use
> it to hold equipment as they begin the real task of constructing the
> building. The scaffolding has no use by itself. It would be absurd to
> just build the scaffolding and then walk away, thinking that something
> of value has been accomplished.
>
> Yet this is what seems to occur in all too many mathematics classes in
> high schools. Students learn formulas and how to plug into them. They
> learn mechanical techniques for solving certain equations or taking
> derivatives. But all of these things are just the scaffolding. They are
> necessary and useful, sure, but by themselves they are useless. Doing
> only the superficial and then thinking that something important has
> happened is like building only the scaffolding.
>
> The real "building" in the mathematics sense is the true mathematical
> understanding, the true ability to think, perceive, and analyze
> mathematically.
> *******************
>
> Just saying - those construction workers who are building scaffolding
> know the purpose of it and know how it will be used during construction
> of the building. Students in math classes seldom know the purpose of the
> "scaffolding" they are supposed to be building and almost never see any
> point to most of it and, for MOST of those students, there is no point
> because they are not going to be using it (most of what is taught in an
> algebra class, for example) at all for the rest of their lives.
> ----------------------------------------------------------
> ************************
> Ready for the big play.
>
> Professional athletes spend hours in gyms working out on equipment of
> all sorts. Special trainers are hired to advise them on workout
> schedules. They spend hours running on treadmills. Why do they do that?
> Are they learning skills necessary for playing their sport, say basketball?
>
> Imagine there're three seconds left in the seventh game of the NBA
> championship. The score is tied. Time out. The pressure is intense. The
> coach is huddling with his star players. He says to one, "OK Michael,
> this is it. You know what to do." And Michael says, "Right coach. Bring
> in my treadmill!"
>
> Duh! Of course not! But then what was all that treadmill time for? If
> the treadmill is not seen during the actual game, was it just a waste to
> use it? Were all those trainers wasting their time? Of course not. It
> produced (if it was done right!) something of value, namely stamina and
> aerobic capacity. Those capacities are of enormous value even if they
> cannot be seen in any immediate sense. So too does mathematics education
> produce something of value, true mental capacity and the ability to think.
> ********************
>
> My problem with the sports comparison is in the conclusion - that math
> education produces "true mental capacity and the ability to think."
> First - math "education" doesn't often produce that - more likely the
> opposite. Math education almost always produces LESS ability to think
> and more mental confusion and reliance on rote memorization. Second,
> there are many other ways to develop mental capacity and the ability to
> think - math is not the only (or necessarily the best) way and certainly
> not even a good way for everybody. Third, and most importantly, even if
> learning about mathematics helps people think more logically, there is
> much more to "true mental capacity and the ability to think," than that.
> Mathematics does not, for example, help people think more empathically,
> more globally, more creatively, or more independently.
>
> -pam
>
> [Non-text portions of this message have been removed]
>
>
>
plaidpanties666
dola dasgupta-banerji <doladg@...> wrote:
Reading (or watching or hearing) stories in general, fact or fiction, involves math if you think of "math" in terms of patterns and relationships rather than "formulae and figures". That's the biggest problem most people have understanding "math" and one of the ways reading about the history of mathematics can help. Its not just random numbers, its an attempt to describe the wonders of the universe.
For some people that "common market place" is a wonderful universe in and of itself. It's always amazing to me how many people are in any kind of market as much for the experience as anything else. People play at shopping, socialize, connect, swirl around and learn new things - whether at high-end auctions or at street vendors or in actual shops. And all that swirling contains patterns and relationships, the underpinnings of mathematics.
---Meredith
>Wouldn't it be interesting to haveThere are teaching methods that do things like that - teach basic math via history or mythology or music or science. Not in public schools, for the most part these days. And there have been college level "physics for poets" and "math for artists" classes that use similar strategies. Looking into those sorts of things can be helpful for parents looking at natural learning and thinking "but there's No Math" - not so good for natural learning if parents try to push "The History of Zero" (fascinating book, if you like that sort of thing) with an agenda of "trying to promote math".
> math lesson which read more like a story than simply boring formulae and
> figures
Reading (or watching or hearing) stories in general, fact or fiction, involves math if you think of "math" in terms of patterns and relationships rather than "formulae and figures". That's the biggest problem most people have understanding "math" and one of the ways reading about the history of mathematics can help. Its not just random numbers, its an attempt to describe the wonders of the universe.
> unfortunately for most it has been reduced to common market place......***************
For some people that "common market place" is a wonderful universe in and of itself. It's always amazing to me how many people are in any kind of market as much for the experience as anything else. People play at shopping, socialize, connect, swirl around and learn new things - whether at high-end auctions or at street vendors or in actual shops. And all that swirling contains patterns and relationships, the underpinnings of mathematics.
---Meredith