(new) unschooling dad SOS!
amari28may
hey, all!
writing from the tropical paradise of Toronto, ON (!!!).....
i am a brand-new unschooling father of two amazing boys (7 and 3). my boys read - totally unprompted - for HOURS every single day. and if they aren't nose deep in the latest Geronimo Stilson, Amulet or Bone book,they are running around the house, screaming and wreaking havoc...or they are outside running around, investigating creepy-crawlies, theorizing about how they catch insects, etc...or they are watching 'Avatar (the GREATEST cartoon EVER!)', nature vids on YouTube...or they are practicing their martial arts.
although the older has been comfortably reading at a Grade 7-8 level for about a year now(i am trained as a Literacy Specialist and discovered this when he asked to complete an assessment that i was using for my middle school students) and can easily complete comprehension exercises written for children much older than him, he has as yet displayed NO interest in doing so. neither has he shown a yen to pick up a workbook and do math sums just for the heck of it.
now, having scoured Joyce's beautifully-wrought responses to many of the queries that paranoid newbies like Yours Truly inevitably have, i realize - intellectually - that his learning is inextricably linked to his interests. which is EXACTLY as it should be.
BUT - the visceral reality is that i worry that my sons will 'fall behind (i know, i know - that's my 'schooling' whispering in my ear! but still.....)' if they haven't mastered, say, multiplication and division by age x.
i'm trying mightily to cross the bridge separating my head from my gut...y'all help me out....PLEASE?????
T-Dot Daddy
writing from the tropical paradise of Toronto, ON (!!!).....
i am a brand-new unschooling father of two amazing boys (7 and 3). my boys read - totally unprompted - for HOURS every single day. and if they aren't nose deep in the latest Geronimo Stilson, Amulet or Bone book,they are running around the house, screaming and wreaking havoc...or they are outside running around, investigating creepy-crawlies, theorizing about how they catch insects, etc...or they are watching 'Avatar (the GREATEST cartoon EVER!)', nature vids on YouTube...or they are practicing their martial arts.
although the older has been comfortably reading at a Grade 7-8 level for about a year now(i am trained as a Literacy Specialist and discovered this when he asked to complete an assessment that i was using for my middle school students) and can easily complete comprehension exercises written for children much older than him, he has as yet displayed NO interest in doing so. neither has he shown a yen to pick up a workbook and do math sums just for the heck of it.
now, having scoured Joyce's beautifully-wrought responses to many of the queries that paranoid newbies like Yours Truly inevitably have, i realize - intellectually - that his learning is inextricably linked to his interests. which is EXACTLY as it should be.
BUT - the visceral reality is that i worry that my sons will 'fall behind (i know, i know - that's my 'schooling' whispering in my ear! but still.....)' if they haven't mastered, say, multiplication and division by age x.
i'm trying mightily to cross the bridge separating my head from my gut...y'all help me out....PLEASE?????
T-Dot Daddy
Joyce Fetteroll
On Oct 7, 2011, at 1:01 AM, amari28may wrote:
are still based on years and years of experience that has gathered a
very different picture.
We're all immersed in a society that believes what children need to
learn is too hard for them to learn from living. (All parents can see
their kids learning from life. But all they see is trivial,
unimportant stuff. Not the "hard" important stuff they teach in
schools.)
What reinforces that belief are several things:
Schools base instruction on memorization. While humans can memorize,
while it will put a chunk of information into someone's head faster
than absorbing a holistic understanding, it's difficult for us. It
needs gone over and over to keep in in our heads. (It's especially
hard when someone doesn't care to get it into their heads!)
Memorization doesn't lead to understanding. Some people manage to see
patterns in what they've memorized and pull understanding from that,
but the memorization can happen without understanding. (It wasn't
until after college that I realized that multiplication was repeated
addition. It wasn't until well after college that I got an intuitive
grasp of what calculus was and why someone might use it to get
information they wanted from a situation. But I could do the problems
fine because I'd memorized the types of problems and what processes I
was supposed to use with each type.)
Recently I ran across a study of college students' understanding of
math. They were faced with this problem (imagine the / is the division
symbol):
2 x 50 / 50
Half -- half! -- went through the problem left to right, doing all the
calculations. They'd memorized a process to get such problems over and
done with. They couldn't see the whole problem. They didn't care to.
The strategy that had worked for *their* needs for 12 years -- to
experience as little pain as possible while meeting the teacher's
requirements -- was to just plow through.
Schools use the methods they do because initially public school was to
get the masses at least reading and doing arithmetic so they'd be
employable. And then later when schools needed to show kids were
learning, the methods suited that well. It's very hard to test the
quality of someone's understanding. But it's very easy to test what
someone has memorized.
If someone's knowledge of Spanish is tested based on how many words
they had memorized from this week's list in the text book, a 90% would
say they'd done well. But could they understand a normal daily
conversation and join in? Could they ask someone what aisle the potato
chips are in? Isn't that a big part of what language is for? After 2
years of Spanish I had some vocabulary, grammar, the basics of
declention memorized. It was only useful for passing Spanish tests.
After 2 years of professional instruction, just about any 2 year old
Hispanic child was leaps and bounds ahead of me in understanding
Spanish.
The above is more head knowledge but your feelings still have all
those years of building up what right math learning looks and feels
like. Just as an Italian feels in his being the right way to greet a
certain level of acquaintance is to give them a big hug. But if they
moved to England and knew hugs weren't as freely given, how long would
it take for his feelings when he restrained himself from hugging to
match his head?
What your feelings know is the right way to do *school* math. What
will probably continue to confuse your feelings is that school math
seems way more complex than what kids might use in daily life. So it
seems like kids can't possibly get that same amount of understanding
from pursuing interests.
But what they will get from math that's used for personally meaningful
reasons is a holistic understanding of how and why you'd want to break
something into chunks, compare the chunks, look at them from different
angles, regroup them, expand them, contract them. The numbers are
irrelevant to the process. Whether you're adding 2+3 or
2734.952+34103.14, the underlying meaning is the same. What's
important is understanding which process will manipulate the
information you have to get you the information you want. A kid who
has combined and divided up armies and cash and points and cookies is
not going to be confused about which process he needs to use when.
He'll know which one *feels* right. The numbers are merely details.
And calculators can handle the numbers. A calculator can't tell you
which process to use.
Joyce
[Non-text portions of this message have been removed]
> BUT - the visceral reality is that i worry that my sons will 'fallFeelings tend to lag behind the head. And that's because your feelings
> behind (i know, i know - that's my 'schooling' whispering in my ear!
> but still.....)' if they haven't mastered, say, multiplication and
> division by age x.
are still based on years and years of experience that has gathered a
very different picture.
We're all immersed in a society that believes what children need to
learn is too hard for them to learn from living. (All parents can see
their kids learning from life. But all they see is trivial,
unimportant stuff. Not the "hard" important stuff they teach in
schools.)
What reinforces that belief are several things:
Schools base instruction on memorization. While humans can memorize,
while it will put a chunk of information into someone's head faster
than absorbing a holistic understanding, it's difficult for us. It
needs gone over and over to keep in in our heads. (It's especially
hard when someone doesn't care to get it into their heads!)
Memorization doesn't lead to understanding. Some people manage to see
patterns in what they've memorized and pull understanding from that,
but the memorization can happen without understanding. (It wasn't
until after college that I realized that multiplication was repeated
addition. It wasn't until well after college that I got an intuitive
grasp of what calculus was and why someone might use it to get
information they wanted from a situation. But I could do the problems
fine because I'd memorized the types of problems and what processes I
was supposed to use with each type.)
Recently I ran across a study of college students' understanding of
math. They were faced with this problem (imagine the / is the division
symbol):
2 x 50 / 50
Half -- half! -- went through the problem left to right, doing all the
calculations. They'd memorized a process to get such problems over and
done with. They couldn't see the whole problem. They didn't care to.
The strategy that had worked for *their* needs for 12 years -- to
experience as little pain as possible while meeting the teacher's
requirements -- was to just plow through.
Schools use the methods they do because initially public school was to
get the masses at least reading and doing arithmetic so they'd be
employable. And then later when schools needed to show kids were
learning, the methods suited that well. It's very hard to test the
quality of someone's understanding. But it's very easy to test what
someone has memorized.
If someone's knowledge of Spanish is tested based on how many words
they had memorized from this week's list in the text book, a 90% would
say they'd done well. But could they understand a normal daily
conversation and join in? Could they ask someone what aisle the potato
chips are in? Isn't that a big part of what language is for? After 2
years of Spanish I had some vocabulary, grammar, the basics of
declention memorized. It was only useful for passing Spanish tests.
After 2 years of professional instruction, just about any 2 year old
Hispanic child was leaps and bounds ahead of me in understanding
Spanish.
The above is more head knowledge but your feelings still have all
those years of building up what right math learning looks and feels
like. Just as an Italian feels in his being the right way to greet a
certain level of acquaintance is to give them a big hug. But if they
moved to England and knew hugs weren't as freely given, how long would
it take for his feelings when he restrained himself from hugging to
match his head?
What your feelings know is the right way to do *school* math. What
will probably continue to confuse your feelings is that school math
seems way more complex than what kids might use in daily life. So it
seems like kids can't possibly get that same amount of understanding
from pursuing interests.
But what they will get from math that's used for personally meaningful
reasons is a holistic understanding of how and why you'd want to break
something into chunks, compare the chunks, look at them from different
angles, regroup them, expand them, contract them. The numbers are
irrelevant to the process. Whether you're adding 2+3 or
2734.952+34103.14, the underlying meaning is the same. What's
important is understanding which process will manipulate the
information you have to get you the information you want. A kid who
has combined and divided up armies and cash and points and cookies is
not going to be confused about which process he needs to use when.
He'll know which one *feels* right. The numbers are merely details.
And calculators can handle the numbers. A calculator can't tell you
which process to use.
Joyce
[Non-text portions of this message have been removed]
Meredith
"amari28may" <juniorburchall@...> wrote:
Good for him!
Really think about what you wrote up there. How many adults sit and do "comprehension exercises" or math problems for the heck of it? Why would you even bother?
I minored in French in college and now and then like to read a bit in French to renew my acquaintance with the language. I don't need to do comprehension exercises to know if I'm understanding what I'm reading! It's pretty obvious to me. So why should a child do "comprehension exercises" for Any other reason than to prove to Someone Else what his or her skills are? They know if they're getting the information they want out of a text. Hopefully, your son knows he can ask for help when he doesn't get what he wants - that's far more important than how well he understands, the confidence to go looking when he doesn't.
I've done math problems for fun - do them pretty regularly - but they don't look Anything like sitting down with a book or worksheet. Working a "math" problem looks like figuring something out - how much mortar mix to buy to re-tile the area around my stove? And then, since the instructions on the bag are for mixing up the whole fifty pounds, how much water do I need if I only want ten pounds? That's a real math problem in my real life this week. I can't say I "enjoyed" the math - it was just part of the process - but I love the way my kitchen looks right now ;)
I also make quilts, though, and I can honestly say I enjoy the math problems I pose myself on a regular basis. I play with a lot of algebra and geometry for the fun of it (I work with a lot of odd angles and curves) - but the idea of sitting down with a worksheet gives me the heebie jeebies. Why would your son want to do such a thing? I bet he'd rather do Real math to get a project done - in fact, I bet he already does. Math is everywhere.
I've worked several different jobs over the years which set me up to do a good bit of math and I'll tell you something they don't teach in schools: most people can't do multiplication past 6x7 reliably in their heads and almost no-one remembers how to add fractions. Nevermind algebra - most people faked their way through that to begin with, but adding fractions is an actually useful skills and most people can't do it. They convert to decimals and do the math on a calculator. The only people who can consistently add fractions are people who do it all day long for their jobs and most of them learned it On the job, having forgotten what they learned in school. The guy training them showed them how to use a tape-measure as a number line and after six weeks of counting, they remembered what you get when you add 11/16 to 3/8. Go to a construction site, some time - you can tell who's got a new guy from the math problems scrawled all over the drywall and subfloor.
So what's the worst case? Your kids will be no worse off than folks who went through school, passed all their classes, and promptly forgot most of it after the test. But chances are, they'll do better than that because they won't have the angst and baggage around math and reading. If they need to learn something, they'll do it. If they need to take a "remedial class" it will be no big deal, just a stepping stone to what they want, and the rest of the class will all be *repeating* the subject for the nth time, having wasted all that time doing math worksheets they don't remember when they could have been reading Bone or watching Avatar (yes! it Is the best cartoon ever and I'm off to look up Geronimo Stilson since my daughter already loves the other stuff you listed).
---Meredith
>> although the older has been comfortably reading at a Grade 7-8 level for about a year now...*****************
> and can easily complete comprehension exercises written for children much older than him, he has as yet displayed NO interest in doing so. neither has he shown a yen to pick up a workbook and do math sums just for the heck of it.
Good for him!
Really think about what you wrote up there. How many adults sit and do "comprehension exercises" or math problems for the heck of it? Why would you even bother?
I minored in French in college and now and then like to read a bit in French to renew my acquaintance with the language. I don't need to do comprehension exercises to know if I'm understanding what I'm reading! It's pretty obvious to me. So why should a child do "comprehension exercises" for Any other reason than to prove to Someone Else what his or her skills are? They know if they're getting the information they want out of a text. Hopefully, your son knows he can ask for help when he doesn't get what he wants - that's far more important than how well he understands, the confidence to go looking when he doesn't.
I've done math problems for fun - do them pretty regularly - but they don't look Anything like sitting down with a book or worksheet. Working a "math" problem looks like figuring something out - how much mortar mix to buy to re-tile the area around my stove? And then, since the instructions on the bag are for mixing up the whole fifty pounds, how much water do I need if I only want ten pounds? That's a real math problem in my real life this week. I can't say I "enjoyed" the math - it was just part of the process - but I love the way my kitchen looks right now ;)
I also make quilts, though, and I can honestly say I enjoy the math problems I pose myself on a regular basis. I play with a lot of algebra and geometry for the fun of it (I work with a lot of odd angles and curves) - but the idea of sitting down with a worksheet gives me the heebie jeebies. Why would your son want to do such a thing? I bet he'd rather do Real math to get a project done - in fact, I bet he already does. Math is everywhere.
>if they haven't mastered, say, multiplication and division by age x.Psst... that's what calculators are for!
I've worked several different jobs over the years which set me up to do a good bit of math and I'll tell you something they don't teach in schools: most people can't do multiplication past 6x7 reliably in their heads and almost no-one remembers how to add fractions. Nevermind algebra - most people faked their way through that to begin with, but adding fractions is an actually useful skills and most people can't do it. They convert to decimals and do the math on a calculator. The only people who can consistently add fractions are people who do it all day long for their jobs and most of them learned it On the job, having forgotten what they learned in school. The guy training them showed them how to use a tape-measure as a number line and after six weeks of counting, they remembered what you get when you add 11/16 to 3/8. Go to a construction site, some time - you can tell who's got a new guy from the math problems scrawled all over the drywall and subfloor.
So what's the worst case? Your kids will be no worse off than folks who went through school, passed all their classes, and promptly forgot most of it after the test. But chances are, they'll do better than that because they won't have the angst and baggage around math and reading. If they need to learn something, they'll do it. If they need to take a "remedial class" it will be no big deal, just a stepping stone to what they want, and the rest of the class will all be *repeating* the subject for the nth time, having wasted all that time doing math worksheets they don't remember when they could have been reading Bone or watching Avatar (yes! it Is the best cartoon ever and I'm off to look up Geronimo Stilson since my daughter already loves the other stuff you listed).
---Meredith
[email protected]
Thank you for sharing this brilliant insight! It has actually helped me better understand my own relationship with math.
I have an honours degree in math, but I actually feel ashamed of it. When people know about my degree, they invariably express that they are impressed, and I feel that they accord me greater respect and admiration. I think this happens because of the discouraging, confounding experience most people have had with school math. I must be really smart to have "mastered" such an inaccessible, baffling, and eventually frightening subject.
But the truth is, I feel no more competent in math than they do, and my elevation in their esteem is undeserved. In high school, I had a knack for memorizing the formulas and plugging in the numbers. I gained absolutely no understanding. My lack of understanding exposed me as mediocre to poor in university math, which was no longer restricted to mere rote. But my other school skills pulled me through, just barely.
If I had a "re-do," and could choose to either gain a more authentic understanding of the math that is all around us in everyday life, or to gain the benefits of my math degree through schoolish abilities like memorization, I would take the deeper, non-institutionally recognized understanding of math in a heartbeat. If nothing else, through the unschooling approach I would at least know whether or not I had any real interest and/or talent in math. If not, I could spend my time and energy pursuing something more fulfilling for me. Either way I would feel more comfortable in my own skin.
Finally! I can see why, with my externally conferred status as outstanding in the subject of math, I feel as scared and ill-equipped as everybody else to help my kids with it. Your post has helped clarify and shift my thinking about math. Previously I was pushing it fearfully to the back of my mind, which was relatively easy since my children are young. I can now envision that I will be able to let go of the idea that my children must somehow learn math as *I* have experienced it, that I will be able to let go of school math and discover something else richer, more exciting and more engaging (whatever it is, wherever it leads) along with my children.
Thank you, Joyce. This one is a keeper for me,
Mairi.
Joyce Fetteroll wrote: << Schools base instruction on memorization. While humans can memorize, while it will put a chunk of information into someone's head faster than absorbing a holistic understanding, it's difficult for us. It needs gone over and over to keep in in our heads. (It's especially hard when someone doesn't care to get it into their heads!)
Memorization doesn't lead to understanding. Some people manage to see patterns in what they've memorized and pull understanding from that, but the memorization can happen without understanding. (It wasn't until after college that I realized that multiplication was repeated addition. It wasn't until well after college that I got an intuitive grasp of what calculus was and why someone might use it to get information they wanted from a situation. But I could do the problems fine because I'd memorized the types of problems and what processes I was supposed to use with each type.) >>
I have an honours degree in math, but I actually feel ashamed of it. When people know about my degree, they invariably express that they are impressed, and I feel that they accord me greater respect and admiration. I think this happens because of the discouraging, confounding experience most people have had with school math. I must be really smart to have "mastered" such an inaccessible, baffling, and eventually frightening subject.
But the truth is, I feel no more competent in math than they do, and my elevation in their esteem is undeserved. In high school, I had a knack for memorizing the formulas and plugging in the numbers. I gained absolutely no understanding. My lack of understanding exposed me as mediocre to poor in university math, which was no longer restricted to mere rote. But my other school skills pulled me through, just barely.
If I had a "re-do," and could choose to either gain a more authentic understanding of the math that is all around us in everyday life, or to gain the benefits of my math degree through schoolish abilities like memorization, I would take the deeper, non-institutionally recognized understanding of math in a heartbeat. If nothing else, through the unschooling approach I would at least know whether or not I had any real interest and/or talent in math. If not, I could spend my time and energy pursuing something more fulfilling for me. Either way I would feel more comfortable in my own skin.
Finally! I can see why, with my externally conferred status as outstanding in the subject of math, I feel as scared and ill-equipped as everybody else to help my kids with it. Your post has helped clarify and shift my thinking about math. Previously I was pushing it fearfully to the back of my mind, which was relatively easy since my children are young. I can now envision that I will be able to let go of the idea that my children must somehow learn math as *I* have experienced it, that I will be able to let go of school math and discover something else richer, more exciting and more engaging (whatever it is, wherever it leads) along with my children.
Thank you, Joyce. This one is a keeper for me,
Mairi.
Joyce Fetteroll wrote: << Schools base instruction on memorization. While humans can memorize, while it will put a chunk of information into someone's head faster than absorbing a holistic understanding, it's difficult for us. It needs gone over and over to keep in in our heads. (It's especially hard when someone doesn't care to get it into their heads!)
Memorization doesn't lead to understanding. Some people manage to see patterns in what they've memorized and pull understanding from that, but the memorization can happen without understanding. (It wasn't until after college that I realized that multiplication was repeated addition. It wasn't until well after college that I got an intuitive grasp of what calculus was and why someone might use it to get information they wanted from a situation. But I could do the problems fine because I'd memorized the types of problems and what processes I was supposed to use with each type.) >>
lylaw
I am not at all surprised about the college students’ approach. there’s a good article (albeit not about unschooling) by alfie kohn about how learning the "”steps” to solve math problems actually poisons the well of understanding, when compared to just fiddling around with ideas and coming to one’s own approach. it’s an article that is an answer to “why not allow both approaches” (instruction and fiddling). here it is:
http://www.alfiekohn.org/teaching/edweek/rotten.htm
we weren’t always unschoolers, but we always chose alternative ed environments for our kids. when my son was in 1st grade (age 6), his class used to have an activity called “number of the day” or something along those lines. the kids would all be gathered with the teacher on the carpet and there would be a number written on the board. kids would raise their hands to call out “number sentences” that equaled the number of the day. early in the year, the number was 10. kids called out 5x2, and 4+6, and even 10 divided by 1. then my son raised his hand (I was there, volunteering), and said “5 divided by a half”
the teacher was absolutely convinced that he was wrong. she didn’t say that but asked if he could explain how he got that.
it took more than 24 hours, and her asking her chemist husband about it (because I guess she knew she wasn’t so great at fractions), for her to come around and apologize and tell him that she realized he was correct.
I presume she learned the steps for solving fraction problems, and didn’t really *understand* fractions, which my son clearly did without ever having had a single lesson or conversation about it. he just saw the world that way, and it made sense to him.
lyla
[Non-text portions of this message have been removed]
http://www.alfiekohn.org/teaching/edweek/rotten.htm
we weren’t always unschoolers, but we always chose alternative ed environments for our kids. when my son was in 1st grade (age 6), his class used to have an activity called “number of the day” or something along those lines. the kids would all be gathered with the teacher on the carpet and there would be a number written on the board. kids would raise their hands to call out “number sentences” that equaled the number of the day. early in the year, the number was 10. kids called out 5x2, and 4+6, and even 10 divided by 1. then my son raised his hand (I was there, volunteering), and said “5 divided by a half”
the teacher was absolutely convinced that he was wrong. she didn’t say that but asked if he could explain how he got that.
it took more than 24 hours, and her asking her chemist husband about it (because I guess she knew she wasn’t so great at fractions), for her to come around and apologize and tell him that she realized he was correct.
I presume she learned the steps for solving fraction problems, and didn’t really *understand* fractions, which my son clearly did without ever having had a single lesson or conversation about it. he just saw the world that way, and it made sense to him.
lyla
[Non-text portions of this message have been removed]
BRIAN POLIKOWSKY
A Mathematician’s Lament
by Paul Lockhart
http://www.maa.org/devlin/LockhartsLament.pdf
A very good read that kinda makes proves they way they teach math in school
is really really not helping children understand math!
A must read if you think your kids need to know this or that by the age of x !!!!
Alex Polikowsky
[Non-text portions of this message have been removed]
by Paul Lockhart
http://www.maa.org/devlin/LockhartsLament.pdf
A very good read that kinda makes proves they way they teach math in school
is really really not helping children understand math!
A must read if you think your kids need to know this or that by the age of x !!!!
Alex Polikowsky
[Non-text portions of this message have been removed]
Stacey
>>when they could have been reading Bone or watching Avatar (yes! it Is the best cartoon ever and I'm off to look up Geronimo Stilson since my daughter already loves the other stuff you listed).Yes! We LOVE the last air bender series - have watched it several times. I hear they are thinking of a new series and the avatar is a girl! Right now we are eating up anything by Studio Ghibli. We all enjoyed the Matrix movies and read the brothers got their inspiration from "Spirited Away". Sigh. I love homeschooling.
Currently "math" in our house is my husband, 8 year old and 6 year old figuring their way through the beta version of the new Star Wars game, Knights Of The Old Republic, because my husband went into a lottery to be a tester...and was picked! It is Thanks Giving up here in Canada this weekend and it will be celebrated as a couple of Jedi Knights and a Sith. Plus Turkey. And pumpkin pie....but I will divide the pie into equal 5ths and eat every last bite!
Stacey
Stacey
That was brilliant, beautiful actually! Thanks for sharing Alex! I have already shared it with many who need to read it...a few times.
Stacey
Stacey
--- In [email protected], BRIAN POLIKOWSKY <polykowholsteins@...> wrote:
>
> A Mathematicianâs Lament
> by Paul Lockhart
>
> http://www.maa.org/devlin/LockhartsLament.pdf
>
> A very good read that kinda makes proves they way they teach math in school
> is really really not helping children understand math!
> A must read if you think your kids need to know this or that by the age of x !!!!
>
> Â
> Alex Polikowsky
>
> [Non-text portions of this message have been removed]
>
Tova
A great read! Thank you for posting it.
I like this part:
"The main problem with school mathematics is that there are no problems. Oh, I know what passes for problems in math classes, these insipid “exercises.” “Here is a type of problem. Here is how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.” What a sad way to learn mathematics: to be a trained chimpanzee.
But a problem, a genuine honest-to-goodness natural human question— that’s another thing. How long is the diagonal of a cube? Do prime numbers keep going on forever? Is infinity a number? How many ways can I symmetrically tile a surface? The history of mathematics is the history of [human]kind’s engagement with questions like these, not the mindless regurgitation of formulas and algorithms (together with contrived exercises designed to make use of them).
A good problem is something you don’t know how to solve. That’s what makes it a good puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions."
I like this part:
"The main problem with school mathematics is that there are no problems. Oh, I know what passes for problems in math classes, these insipid “exercises.” “Here is a type of problem. Here is how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.” What a sad way to learn mathematics: to be a trained chimpanzee.
But a problem, a genuine honest-to-goodness natural human question— that’s another thing. How long is the diagonal of a cube? Do prime numbers keep going on forever? Is infinity a number? How many ways can I symmetrically tile a surface? The history of mathematics is the history of [human]kind’s engagement with questions like these, not the mindless regurgitation of formulas and algorithms (together with contrived exercises designed to make use of them).
A good problem is something you don’t know how to solve. That’s what makes it a good puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions."
--- On Sat, 10/8/11, BRIAN POLIKOWSKY <polykowholsteins@...> wrote:
From: BRIAN POLIKOWSKY <polykowholsteins@...>
Subject: Re: [unschoolingbasics] Re: (new) unschooling dad SOS!
To: "[email protected]" <[email protected]>
Date: Saturday, October 8, 2011, 10:35 AM
A Mathematician’s Lament
by Paul Lockhart
http://www.maa.org/devlin/LockhartsLament.pdf
A very good read that kinda makes proves they way they teach math in school
is really really not helping children understand math!
A must read if you think your kids need to know this or that by the age of x !!!!
Alex Polikowsky
[Non-text portions of this message have been removed]
[Non-text portions of this message have been removed]
jo70mo
And sometimes that personally meaningful reason will not be something as concrete as dividing up cookies or money and be a curiosity about numbers and how they work. It is amazing to me how inherently fascinating numbers are to my children. Especially when it is almost ingrained that people won't like maths and won't do it unless they have too.
My 8 year old son is fascinated by the concept of infinity and keeps coming up with trick questions for us and his friends such as what is 1 plus the square root of infinity. I cannot recall how either the concept of square roots or infinty came onto his radar and would probably guess at tv programmes such as Pineas and Ferb.
When he was 5 or 6 he was in our bed and at some late hour turned to us and said " you know 4 x 3 is the same as 2 x 6"
We don't quiz our children but I think DH had a very sleepy lapse and said and what is that. DS just shrugged his shoulders and carried on with whatever his mind had now moved on to.
It was fascinating to me to see that he hadn't used the end product or even thought about it but must have been looking at the way changing one number in the equation would affect the other by halving and doubling etc. I have no idea what internal process he was using to compute it but it wasn't about what the answer to the sum was but about how manipulating one number affected the other. That really blew me away at the time.
That sort of internal number play - be it for concrete practical reasons or just fascination will surely lead to a much deeper understanding of how maths works.
My 8 year old son is fascinated by the concept of infinity and keeps coming up with trick questions for us and his friends such as what is 1 plus the square root of infinity. I cannot recall how either the concept of square roots or infinty came onto his radar and would probably guess at tv programmes such as Pineas and Ferb.
When he was 5 or 6 he was in our bed and at some late hour turned to us and said " you know 4 x 3 is the same as 2 x 6"
We don't quiz our children but I think DH had a very sleepy lapse and said and what is that. DS just shrugged his shoulders and carried on with whatever his mind had now moved on to.
It was fascinating to me to see that he hadn't used the end product or even thought about it but must have been looking at the way changing one number in the equation would affect the other by halving and doubling etc. I have no idea what internal process he was using to compute it but it wasn't about what the answer to the sum was but about how manipulating one number affected the other. That really blew me away at the time.
That sort of internal number play - be it for concrete practical reasons or just fascination will surely lead to a much deeper understanding of how maths works.
>
> But what they will get from math that's used for personally meaningful
> reasons is a holistic understanding of how and why you'd want to break
> something into chunks, compare the chunks, look at them from different
> angles, regroup them, expand them, contract them. The numbers are
> irrelevant to the process. Whether you're adding 2+3 or
> 2734.952+34103.14, the underlying meaning is the same. What's
> important is understanding which process will manipulate the
> information you have to get you the information you want. A kid who
> has combined and divided up armies and cash and points and cookies is
> not going to be confused about which process he needs to use when.
> He'll know which one *feels* right. The numbers are merely details.
> And calculators can handle the numbers. A calculator can't tell you
> which process to use.
>
> Joyce
>
> [Non-text portions of this message have been removed]
>
Meredith
"jo70mo" <jo70mo@...> wrote:
Yes! and sometimes it won't even "look like" math... if you're thinking in terms of school math. Patterns. Colors. Fashion design. Music. Building. Poetry. Running errands. Hosting a party. It's eeevvvery where! Most of the time we miss it because we're so heavily indoctrinated to think of "math" in terms of certain kinds of number problems we miss all the ways we use values and relationships to solve problems.
---Meredith
>***********************
> And sometimes that personally meaningful reason will not be something as concrete as dividing up cookies or money and be a curiosity about numbers and how they work. It is amazing to me how inherently fascinating numbers are to my children. Especially when it is almost ingrained that people won't like maths and won't do it unless they have too.
Yes! and sometimes it won't even "look like" math... if you're thinking in terms of school math. Patterns. Colors. Fashion design. Music. Building. Poetry. Running errands. Hosting a party. It's eeevvvery where! Most of the time we miss it because we're so heavily indoctrinated to think of "math" in terms of certain kinds of number problems we miss all the ways we use values and relationships to solve problems.
---Meredith