more negative and positives stuff
[email protected]
Here is another example - I'm not sure which one is more clear - they are
different and its good to fool with more than one way to do it.
******************************
If you have 3 kids standing in a line, all facing east, and they each take
two forward (toes first) steps, then you'll have a total of 6 steps to the
east.
If you have 3 kids standing in a line, all facing east, and the each take two
backward steps (heals first), then you'll have a total of 6 steps to the
west.
If you have 3 kids standing in a line, all facing west, and they each take
two forward steps (toes first), then you'll have a total of 6 steps to the
west.
If you have 3 kids standing in a line, all facing west, and they each take
two backward steps (heels first), then you'll have a total of 6 steps to the
east.
Try it first before going farther.
If you have three people - stand up and actually follow the directions. If
you don't have 3 people, draw it - be sure you draw it so you can tell which
way they step - heels first or toes first.
******************************************
Okay - now to put some math notation to it.
Let's let the sign of the first number stand for the direction they're
facing. If they're facing east, we'll make the number positive and if they're
facing west, we'll make it negative.
The value of the first number is how many kids there are standing in the
line.
Now let the sign of the second number be whether they walk forward (toes
first) or backward (heels first). If they walk forward (toes first) we'll
give it a positive sign and if they walk backward (heels first) we'll give it
a negative sign.
The value of the second number will be how many steps each takes.
We'll let the sign of the answer be positive if they end up to the east of
where they started and negative if they end up to the west of where they
started.
The value of the answer is the total number of steps taken by all kids.
If you have 3 kids standing in a line, all facing east, then the first number
is a +3.
As they each take two forward (toes first) steps, the second number is +2.
Since they are facing east and walking forward, they'll move east. So the
sign of the answer will be positive.
Since there are three of them, each taking 2 steps, you'll have a total of 6
steps to the east and the answer is +6.
We have: +3 X +2 = +6
**********************
If you have 3 kids standing in a line, all facing east, then the first number
is a +3.
As they each take two backward (heels first) steps, the second number is -2.
Since they are facing east and walking backward, they'll move to the west. So
the sign of the answer will be negative.
Since there are three of them, each taking 2 steps, you'll have a total of 6
steps to the east and the answer is -6.
We have: +3 X -2 = -6
******************
If you have 3 kids standing in a line, all facing west, then the first number
is a -3.
As they each take two forward (toes first) steps, the second number is +2.
Since they are facing west and walking forward, they'll move to the west. So
the sign of the answer will be negative.
Since there are three of them, each taking 2 steps, you'll have a total of 6
steps to the east and the answer is -6.
We have: -3 X +2 = -6
*****************
If you have 3 kids standing in a line, all facing west, then the first number
is a -3.
As they each take two backward (heels first) steps, the second number is -2.
Since they are facing west and walking backward, they'll move to the east. So
the sign of the answer will be positive. (They'll end up east of where they
started.)
Since there are three of them, each taking 2 steps, you'll have a total of 6
steps to the east and the answer is +6.
We have: -3 X -2 = +6
*********************
(Note - i could have made the meanings of the signs be reversed - let the
sign of the first number be walking forward or backward and the sign of the
second number be which direction they are facing, east or west. It doesn't
matter. Sometimes, for some real-world situations, one way makes more sense.
But it doesn't matter here.)
--pamS
different and its good to fool with more than one way to do it.
******************************
If you have 3 kids standing in a line, all facing east, and they each take
two forward (toes first) steps, then you'll have a total of 6 steps to the
east.
If you have 3 kids standing in a line, all facing east, and the each take two
backward steps (heals first), then you'll have a total of 6 steps to the
west.
If you have 3 kids standing in a line, all facing west, and they each take
two forward steps (toes first), then you'll have a total of 6 steps to the
west.
If you have 3 kids standing in a line, all facing west, and they each take
two backward steps (heels first), then you'll have a total of 6 steps to the
east.
Try it first before going farther.
If you have three people - stand up and actually follow the directions. If
you don't have 3 people, draw it - be sure you draw it so you can tell which
way they step - heels first or toes first.
******************************************
Okay - now to put some math notation to it.
Let's let the sign of the first number stand for the direction they're
facing. If they're facing east, we'll make the number positive and if they're
facing west, we'll make it negative.
The value of the first number is how many kids there are standing in the
line.
Now let the sign of the second number be whether they walk forward (toes
first) or backward (heels first). If they walk forward (toes first) we'll
give it a positive sign and if they walk backward (heels first) we'll give it
a negative sign.
The value of the second number will be how many steps each takes.
We'll let the sign of the answer be positive if they end up to the east of
where they started and negative if they end up to the west of where they
started.
The value of the answer is the total number of steps taken by all kids.
If you have 3 kids standing in a line, all facing east, then the first number
is a +3.
As they each take two forward (toes first) steps, the second number is +2.
Since they are facing east and walking forward, they'll move east. So the
sign of the answer will be positive.
Since there are three of them, each taking 2 steps, you'll have a total of 6
steps to the east and the answer is +6.
We have: +3 X +2 = +6
**********************
If you have 3 kids standing in a line, all facing east, then the first number
is a +3.
As they each take two backward (heels first) steps, the second number is -2.
Since they are facing east and walking backward, they'll move to the west. So
the sign of the answer will be negative.
Since there are three of them, each taking 2 steps, you'll have a total of 6
steps to the east and the answer is -6.
We have: +3 X -2 = -6
******************
If you have 3 kids standing in a line, all facing west, then the first number
is a -3.
As they each take two forward (toes first) steps, the second number is +2.
Since they are facing west and walking forward, they'll move to the west. So
the sign of the answer will be negative.
Since there are three of them, each taking 2 steps, you'll have a total of 6
steps to the east and the answer is -6.
We have: -3 X +2 = -6
*****************
If you have 3 kids standing in a line, all facing west, then the first number
is a -3.
As they each take two backward (heels first) steps, the second number is -2.
Since they are facing west and walking backward, they'll move to the east. So
the sign of the answer will be positive. (They'll end up east of where they
started.)
Since there are three of them, each taking 2 steps, you'll have a total of 6
steps to the east and the answer is +6.
We have: -3 X -2 = +6
*********************
(Note - i could have made the meanings of the signs be reversed - let the
sign of the first number be walking forward or backward and the sign of the
second number be which direction they are facing, east or west. It doesn't
matter. Sometimes, for some real-world situations, one way makes more sense.
But it doesn't matter here.)
--pamS
Karin
Non-mathie here! : : : : : : : waving : : : : : : :
I understand this example better than the car one. The car and east/west/mph
etc. was confusing to me and I got lost about 1/2 way through. This "3 kids
in line" seems more basic and easier for me to understand.
But it's just a formula for helping one to remember how to do these kinds of
problems, right?
What would really help me understand is if there is ever a time IRL when
there is a need to multiply 2 negative-signed numbers, or a postive times a
negative, etc.. Is there such an example? Or do these problems exist
exclusively on paper?
I got pretty good grades in math when I was in school - even algebra,
because I could usually understand how to apply the learned formula to the
problem. But all that has left me a LONG time ago. Seems I'm back to "basic"
math knowledge and I can help myself in most cases to figure out whatever
real-life problems that come up. Beyond that - I'd need a refresher course
or something, or learning formula's on an unschooling e-mail list. ;-)
Karin
I understand this example better than the car one. The car and east/west/mph
etc. was confusing to me and I got lost about 1/2 way through. This "3 kids
in line" seems more basic and easier for me to understand.
But it's just a formula for helping one to remember how to do these kinds of
problems, right?
What would really help me understand is if there is ever a time IRL when
there is a need to multiply 2 negative-signed numbers, or a postive times a
negative, etc.. Is there such an example? Or do these problems exist
exclusively on paper?
I got pretty good grades in math when I was in school - even algebra,
because I could usually understand how to apply the learned formula to the
problem. But all that has left me a LONG time ago. Seems I'm back to "basic"
math knowledge and I can help myself in most cases to figure out whatever
real-life problems that come up. Beyond that - I'd need a refresher course
or something, or learning formula's on an unschooling e-mail list. ;-)
Karin
>two
>
> Here is another example - I'm not sure which one is more clear - they are
> different and its good to fool with more than one way to do it.
>
> ******************************
> If you have 3 kids standing in a line, all facing east, and they each take
> two forward (toes first) steps, then you'll have a total of 6 steps to the
> east.
>
> If you have 3 kids standing in a line, all facing east, and the each take
>the
> backward steps (heals first), then you'll have a total of 6 steps to the
> west.
>
> If you have 3 kids standing in a line, all facing west, and they each take
> two forward steps (toes first), then you'll have a total of 6 steps to the
> west.
>
> If you have 3 kids standing in a line, all facing west, and they each take
> two backward steps (heels first), then you'll have a total of 6 steps to
> east.which
>
> Try it first before going farther.
>
> If you have three people - stand up and actually follow the directions. If
> you don't have 3 people, draw it - be sure you draw it so you can tell
> way they step - heels first or toes first.they're
> ******************************************
> Okay - now to put some math notation to it.
>
> Let's let the sign of the first number stand for the direction they're
> facing. If they're facing east, we'll make the number positive and if
>it
> facing west, we'll make it negative.
>
> The value of the first number is how many kids there are standing in the
> line.
>
> Now let the sign of the second number be whether they walk forward (toes
> first) or backward (heels first). If they walk forward (toes first) we'll
> give it a positive sign and if they walk backward (heels first) we'll give
>number
> a negative sign.
>
> The value of the second number will be how many steps each takes.
>
> We'll let the sign of the answer be positive if they end up to the east of
> where they started and negative if they end up to the west of where they
> started.
>
> The value of the answer is the total number of steps taken by all kids.
>
> If you have 3 kids standing in a line, all facing east, then the first
>6
> is a +3.
>
> As they each take two forward (toes first) steps, the second number is +2.
>
> Since they are facing east and walking forward, they'll move east. So the
> sign of the answer will be positive.
>
> Since there are three of them, each taking 2 steps, you'll have a total of
> steps to the east and the answer is +6.number
>
> We have: +3 X +2 = +6
>
>
> **********************
>
> If you have 3 kids standing in a line, all facing east, then the first
>is -2.
> is a +3.
>
> As they each take two backward (heels first) steps, the second number
>So
> Since they are facing east and walking backward, they'll move to the west.
>6
> the sign of the answer will be negative.
>
> Since there are three of them, each taking 2 steps, you'll have a total of
> steps to the east and the answer is -6.number
>
> We have: +3 X -2 = -6
>
> ******************
> If you have 3 kids standing in a line, all facing west, then the first
>So
> is a -3.
>
> As they each take two forward (toes first) steps, the second number is +2.
>
> Since they are facing west and walking forward, they'll move to the west.
> the sign of the answer will be negative.6
>
> Since there are three of them, each taking 2 steps, you'll have a total of
> steps to the east and the answer is -6.number
>
> We have: -3 X +2 = -6
>
> *****************
>
> If you have 3 kids standing in a line, all facing west, then the first
>is -2.
> is a -3.
>
> As they each take two backward (heels first) steps, the second number
>So
> Since they are facing west and walking backward, they'll move to the east.
>they
> the sign of the answer will be positive. (They'll end up east of where
> started.)6
>
> Since there are three of them, each taking 2 steps, you'll have a total of
> steps to the east and the answer is +6.the
>
> We have: -3 X -2 = +6
>
> *********************
> (Note - i could have made the meanings of the signs be reversed - let the
> sign of the first number be walking forward or backward and the sign of
> second number be which direction they are facing, east or west. It doesn'tsense.
> matter. Sometimes, for some real-world situations, one way makes more
> But it doesn't matter here.)
>
> --pamS
>
>
>
> ~~~ Don't forget! If you change the topic, change the subject line! ~~~
>
> To unsubscribe from this group, send an email to:
> [email protected]
>
> Visit the Unschooling website:
> http://www.unschooling.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
Dena Lambert
Thanks Pam! These made much more sense to me than the first ones, probably because it was more visual for me. This is interesting, thanks for clarifying integers for another non-mathie!
Take care,
Dena
Take care,
Dena
----- Original Message -----
From: PSoroosh@...
Sent: Monday, May 06, 2002 8:00 PM
To: [email protected]
Subject: [Unschooling-dotcom] more negative and positives stuff
Here is another example - I'm not sure which one is more clear - they are
different and its good to fool with more than one way to do it.
******************************
If you have 3 kids standing in a line, all facing east, and they each take
two forward (toes first) steps, then you'll have a total of 6 steps to the
east.
Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service. Get more from the Web. FREE MSN Explorer download : http://explorer.msn.com
[Non-text portions of this message have been removed]
Fetteroll
on 5/6/02 8:49 PM, Karin at curtkar@... wrote:
owing money. But the positive and negative can represent *anything* that's
opposite. In Pam's example, they represent opposite cardinal directions.
They can also represent charges on a particle, whether something is
increasing or decreasing, coming after or before or above or below some
chosen point.
It's from being taught how to do arithmetic before we've had lots of
experience with numbers that has cemented the idea that positives are
something and negatives are less than nothing.
Multiplying 2 negatives numbers is just multiplying 2 qualities that come in
opposite flavors to their 2 positive counterparts. The positives and
negatives sometimes make sense like up and rising being positive and down
and falling being negative. Sometimes they're just chosen so everyone is
using negative the same way. (The "negative" charge of an electron isn't
really negative. It's just opposite the "positive" charge of the proton.)
The numbers can tell you *how big* something is. The positives and negatives
can tell you *where* it is in relation to some (sometimes arbitrarily
chosen) reference (zero) point.
Life came first. Then we invented math as a way to describe life. Teaching
math out of the context of what it's describing is like teaching a foreign
grammar and vocabulary without ever hearing or using the language.
Math makes sense in context. Out of context it's just memorization.
Joyce
> But it's just a formula for helping one to remember how to do these kinds ofNo, it's real stuff!
> problems, right?
> What would really help me understand is if there is ever a time IRL whenUsually people explain positive and negative numbers to kids as having and
> there is a need to multiply 2 negative-signed numbers, or a postive times a
> negative, etc.. Is there such an example? Or do these problems exist
> exclusively on paper?
owing money. But the positive and negative can represent *anything* that's
opposite. In Pam's example, they represent opposite cardinal directions.
They can also represent charges on a particle, whether something is
increasing or decreasing, coming after or before or above or below some
chosen point.
It's from being taught how to do arithmetic before we've had lots of
experience with numbers that has cemented the idea that positives are
something and negatives are less than nothing.
Multiplying 2 negatives numbers is just multiplying 2 qualities that come in
opposite flavors to their 2 positive counterparts. The positives and
negatives sometimes make sense like up and rising being positive and down
and falling being negative. Sometimes they're just chosen so everyone is
using negative the same way. (The "negative" charge of an electron isn't
really negative. It's just opposite the "positive" charge of the proton.)
The numbers can tell you *how big* something is. The positives and negatives
can tell you *where* it is in relation to some (sometimes arbitrarily
chosen) reference (zero) point.
Life came first. Then we invented math as a way to describe life. Teaching
math out of the context of what it's describing is like teaching a foreign
grammar and vocabulary without ever hearing or using the language.
Math makes sense in context. Out of context it's just memorization.
Joyce
[email protected]
In a message dated 5/6/2002 5:30:21 PM Pacific Daylight Time, curtkar@...
writes:
won't have to memorize a formula or so that, at the least, the formula will
have some sense behind it. I searched math books - dozens of the most popular
of them - almost every one just gave the formula, none gave understandable
examples. So that's what I'm trying to do - write up more understandable
examples.
It is an example of the positive and negative signs having an actual meaning,
instead of just a formula.
The "formula" says: "When multiplying two numbers, if the signs of the two
numbers are the same, the product is positive and if the signs are different,
the product is positive."
That is just a formula - no meaning and hard to remember since there is no
"reason" or "explanation" for it - just a rule to follow.
One thing I've become very aware of (and this might help some of you) is that
people who consider themselves "nonmathies" often don't really read EACH
little bit of an explanation. My mom and my sisters are all very math anxious
- when I explain something to them, they don't "hear" it all. Their attention
wanders, they block bits, they are too used to the kind of reading where they
can skim a little and still follow the action. Whatever the reason - the fact
is that these kinds of explanations require someone to focus on one bit at a
time and make sure that each bit is understood before moving on to the next.
It is SLOW reading, sometimes painstaking. But - the understanding you can
get from doing it that way is very very empowering.
It is hard, in writing, to make this clear, because I can't watch your eye
movement, facial expression, body language, etc., and so I can't (gently <G>)
bring you back to the step where you got lost (it almost always is because of
lack of enough attention to the step right before that one).
--pamS
[Non-text portions of this message have been removed]
writes:
> But it's just a formula for helping one to remember how to do these kinds ofNo - not a formula. It is a way to give some MEANING to the signs so that you
> problems, right?
won't have to memorize a formula or so that, at the least, the formula will
have some sense behind it. I searched math books - dozens of the most popular
of them - almost every one just gave the formula, none gave understandable
examples. So that's what I'm trying to do - write up more understandable
examples.
It is an example of the positive and negative signs having an actual meaning,
instead of just a formula.
The "formula" says: "When multiplying two numbers, if the signs of the two
numbers are the same, the product is positive and if the signs are different,
the product is positive."
That is just a formula - no meaning and hard to remember since there is no
"reason" or "explanation" for it - just a rule to follow.
One thing I've become very aware of (and this might help some of you) is that
people who consider themselves "nonmathies" often don't really read EACH
little bit of an explanation. My mom and my sisters are all very math anxious
- when I explain something to them, they don't "hear" it all. Their attention
wanders, they block bits, they are too used to the kind of reading where they
can skim a little and still follow the action. Whatever the reason - the fact
is that these kinds of explanations require someone to focus on one bit at a
time and make sure that each bit is understood before moving on to the next.
It is SLOW reading, sometimes painstaking. But - the understanding you can
get from doing it that way is very very empowering.
It is hard, in writing, to make this clear, because I can't watch your eye
movement, facial expression, body language, etc., and so I can't (gently <G>)
bring you back to the step where you got lost (it almost always is because of
lack of enough attention to the step right before that one).
--pamS
[Non-text portions of this message have been removed]
[email protected]
In a message dated 5/7/2002 3:48:12 AM Pacific Daylight Time,
fetteroll@... writes:
TOO early and too much emphasis on arithmetic is looking to me like the REAL
cause of all the math anxiety and math ignorance in this country.
I'm going to start doing a little research about other countries. The PROBLEM
is that comparisons are based on standardized tests and you don't have to
understand much, just follow formulas, to do well on standardized tests. Kids
in other countries might just be better "drilled" on formulas and do better
on standardized tests. So I'm not sure how I'm going to compare countries -
but I'm interested in it.
--pamS
[Non-text portions of this message have been removed]
fetteroll@... writes:
> It's from being taught how to do arithmetic before we've had lots ofSAY IT AGAIN, JOYCE!!!
> experience with numbers that has cemented the idea that positives are
> something and negatives are less than nothing.
TOO early and too much emphasis on arithmetic is looking to me like the REAL
cause of all the math anxiety and math ignorance in this country.
I'm going to start doing a little research about other countries. The PROBLEM
is that comparisons are based on standardized tests and you don't have to
understand much, just follow formulas, to do well on standardized tests. Kids
in other countries might just be better "drilled" on formulas and do better
on standardized tests. So I'm not sure how I'm going to compare countries -
but I'm interested in it.
--pamS
[Non-text portions of this message have been removed]
[email protected]
In a message dated 5/7/02 10:47:26 AM, PSoroosh@... writes:
<< It is hard, in writing, to make this clear, because I can't watch your eye
movement, facial expression, body language, etc., and so I can't (gently <G>)
bring you back to the step where you got lost (it almost always is because of
lack of enough attention to the step right before that one). >>
You also couldn't see that I glazed over five words into it and didn't read
any of it. <g>
Good for other people for being your test subjects, because I would have
cried.
Sandra
<< It is hard, in writing, to make this clear, because I can't watch your eye
movement, facial expression, body language, etc., and so I can't (gently <G>)
bring you back to the step where you got lost (it almost always is because of
lack of enough attention to the step right before that one). >>
You also couldn't see that I glazed over five words into it and didn't read
any of it. <g>
Good for other people for being your test subjects, because I would have
cried.
Sandra
[email protected]
In a message dated 5/7/2002 10:23:11 AM Pacific Daylight Time,
SandraDodd@... writes:
So - another request <G>.
If anybody really did find that reading these examples was uncomfortable - or
if you keep intending to do it, but just can quite get around to it, or if
you don't have time (but have time for all the other mail) or have any other
reason for not reading it --- YOU are the people I really want to hear from
about your own math backgrounds.
I also want to hear, especially from that above group, how you handle math
stuff with your kids -- how do you make sure you don't pass on your math
anxiety to them? How do you create an environment in which math concepts get
bumped into and expanded on - like you would for other stuff?
--pamS
[Non-text portions of this message have been removed]
SandraDodd@... writes:
> You also couldn't see that I glazed over five words into it and didn't readI still really want to hear those math stories, though.
> any of it. <g>
>
> Good for other people for being your test subjects, because I would have
> cried.
So - another request <G>.
If anybody really did find that reading these examples was uncomfortable - or
if you keep intending to do it, but just can quite get around to it, or if
you don't have time (but have time for all the other mail) or have any other
reason for not reading it --- YOU are the people I really want to hear from
about your own math backgrounds.
I also want to hear, especially from that above group, how you handle math
stuff with your kids -- how do you make sure you don't pass on your math
anxiety to them? How do you create an environment in which math concepts get
bumped into and expanded on - like you would for other stuff?
--pamS
[Non-text portions of this message have been removed]
[email protected]
In a message dated 5/7/02 11:49:40 AM, PSoroosh@... writes:
<< I also want to hear, especially from that above group, how you handle math
stuff with your kids -- how do you make sure you don't pass on your math
anxiety to them? How do you create an environment in which math concepts get
bumped into and expanded on - like you would for other stuff? >>
Games, puzzles, building sets, computer games with math elements (I'm really
liking this new Math Arena), poker chips, cash, card games, dominos (spinner,
where all scores are multiples of five), measuring, sewing, menus. Just from
not calling anything "math" but having lots of opportunities to see patterns
and figure out odds and percentages and keeping scores on games.
And totally without me, they have math exposure from video games, D&D, Magic,
Pokemon, pooling money with friends to get snacks while playing games,
figuring out how many weeks' allowance it will take to pay for a concert
ticket or video game,...
I didn't call history "history," didn't call rocks, bugs and stars "science,"
didn't call reading poetry "English" or "Language Arts," and along with that
never called anything "math," but rather called it Games, puzzles, building
sets, computer games, poker chips, cash, card games, dominos ...
Sandra
<< I also want to hear, especially from that above group, how you handle math
stuff with your kids -- how do you make sure you don't pass on your math
anxiety to them? How do you create an environment in which math concepts get
bumped into and expanded on - like you would for other stuff? >>
Games, puzzles, building sets, computer games with math elements (I'm really
liking this new Math Arena), poker chips, cash, card games, dominos (spinner,
where all scores are multiples of five), measuring, sewing, menus. Just from
not calling anything "math" but having lots of opportunities to see patterns
and figure out odds and percentages and keeping scores on games.
And totally without me, they have math exposure from video games, D&D, Magic,
Pokemon, pooling money with friends to get snacks while playing games,
figuring out how many weeks' allowance it will take to pay for a concert
ticket or video game,...
I didn't call history "history," didn't call rocks, bugs and stars "science,"
didn't call reading poetry "English" or "Language Arts," and along with that
never called anything "math," but rather called it Games, puzzles, building
sets, computer games, poker chips, cash, card games, dominos ...
Sandra
[email protected]
In a message dated 5/7/02 12:20:05 PM, curtkar@... writes:
<< What I
don't understand is WHY multiplication would be applied to negative numbers
in the first place. If I could understand this, I think I would *get it*.
Maybe I'm asking for a word problem?? >>
I agree.
Plots on graphs, fine--that's already a mathematical construct. Tell us of a
practical everyday real-life need for multiplying negatives by negatives.
Like what? Unpaid car payments? What real thing?
I'm coming into the old-enough-to-die-of-old-age range soon, and except for
debts (bookkeeping stuff), I don't deal with negative numbers that I know of.
Sandra
<< What I
don't understand is WHY multiplication would be applied to negative numbers
in the first place. If I could understand this, I think I would *get it*.
Maybe I'm asking for a word problem?? >>
I agree.
Plots on graphs, fine--that's already a mathematical construct. Tell us of a
practical everyday real-life need for multiplying negatives by negatives.
Like what? Unpaid car payments? What real thing?
I'm coming into the old-enough-to-die-of-old-age range soon, and except for
debts (bookkeeping stuff), I don't deal with negative numbers that I know of.
Sandra
Karin
>or
> So - another request <G>.
>
> If anybody really did find that reading these examples was uncomfortable -
> if you keep intending to do it, but just can quite get around to it, or ifother
> you don't have time (but have time for all the other mail) or have any
> reason for not reading it --- YOU are the people I really want to hearfrom
> about your own math backgrounds.Call me thick-headed, slow-to-get-it, what have you.
>
I am still NOT getting this.
Even though Joyce and you tried to explain to me how these positive/negative
numbers come up and apply to real life situations, and I don't doubt that
they do, I have never come across the need to multiply negative and positive
numbers, or two negatives numbers. I think that is my main stumbling block
for not understanding this whole thing.
>> But it's just a formula for helping one to remember how to do these kindsof
>> problems, right?you
>No - not a formula. It is a way to give some MEANING to the signs so that
>won't have to memorize a formula or so that, at the least, the formula willpopular
>have some sense behind it. I searched math books - dozens of the most
>of them - almost every one just gave the formula, none gave understandablemeaning,
>examples. So that's what I'm trying to do - write up more understandable
>examples.
>It is an example of the positive and negative signs having an actual
>instead of just a formula.Okay, maybe formula is a wrong word to use, on my part. Your example with
the cars or people standing in line is helpful to understand how the numbers
move in the negative and positive direction when multiplication is applied,
and how and why the right answer is determined. I do understand this. What I
don't understand is WHY multiplication would be applied to negative numbers
in the first place. If I could understand this, I think I would *get it*.
Maybe I'm asking for a word problem?? (What am I DOING? I HATE word
problems! <g>)
>I also want to hear, especially from that above group, how you handle mathget
>stuff with your kids -- how do you make sure you don't pass on your math
>anxiety to them? How do you create an environment in which math concepts
>bumped into and expanded on - like you would for other stuff?Since my kids are just 9 & 11, we haven't yet gotten to harder math conepts.
I'd also be very interested how to create a natural and easy math learning
environment without math anxiety. I'm so glad this subject is being
discussed. Thank-you, Pam!
Karin
[email protected]
In a message dated 5/7/2002 11:07:28 AM Pacific Daylight Time,
SandraDodd@... writes:
mean I think it should be "taught" as a separate subject to kids. I haven't
done that with my own kids.
In fact, what I'm in the process of doing is gathering anecdotal evidence,
really, about what happens when math IS taught to children in a formal,
progressive, systematic, programmed way and what happens when it is not. My
theorem is that bad things often happen when arithmetic computational skills
are taught BEFORE concepts have been absorbed and integrated. A corollary is
that concepts can be absorbed in daily life, through fun and games, through
conversation and argument and by observation.
"Math" is something. Just like "reading" is something. But that doesn't mean
that "math" has to be learned separately in a series of out-of-context steps
any more than it means "reading" has to be learned separately in a series of
out-of-context steps. And neither has to be "taught" in the usual meaning of
the word.
However, only a few unschooling parents have "reading anxiety" and I doubt
very many at all have serious "reading phobia" - where they zone out, feel
faint, get sweaty, feel like throwing up, or black out when faced with
reading. So - we unschoolers spend a fair amount of time talking about how
our kids learn to read, yet that seems to me to be a sort of "no-brainer" <G>
compared to the question of how our kids learn mathematics, especially given
that the vast majority of the population, including unschooling parents,
suffer from moderate to severe math anxiety or phobia.
--pamS
[Non-text portions of this message have been removed]
SandraDodd@... writes:
> I didn't call history "history," didn't call rocks, bugs and starsWhich reminds me to reiterate - because I'm talking about math it doesn't
> "science,"
> didn't call reading poetry "English" or "Language Arts," and along with
> that
> never called anything "math," but rather called it Games, puzzles, building
>
> sets, computer games, poker chips, cash, card games, dominos ...
mean I think it should be "taught" as a separate subject to kids. I haven't
done that with my own kids.
In fact, what I'm in the process of doing is gathering anecdotal evidence,
really, about what happens when math IS taught to children in a formal,
progressive, systematic, programmed way and what happens when it is not. My
theorem is that bad things often happen when arithmetic computational skills
are taught BEFORE concepts have been absorbed and integrated. A corollary is
that concepts can be absorbed in daily life, through fun and games, through
conversation and argument and by observation.
"Math" is something. Just like "reading" is something. But that doesn't mean
that "math" has to be learned separately in a series of out-of-context steps
any more than it means "reading" has to be learned separately in a series of
out-of-context steps. And neither has to be "taught" in the usual meaning of
the word.
However, only a few unschooling parents have "reading anxiety" and I doubt
very many at all have serious "reading phobia" - where they zone out, feel
faint, get sweaty, feel like throwing up, or black out when faced with
reading. So - we unschoolers spend a fair amount of time talking about how
our kids learn to read, yet that seems to me to be a sort of "no-brainer" <G>
compared to the question of how our kids learn mathematics, especially given
that the vast majority of the population, including unschooling parents,
suffer from moderate to severe math anxiety or phobia.
--pamS
[Non-text portions of this message have been removed]
[email protected]
In a message dated 5/7/2002 11:20:04 AM Pacific Daylight Time,
curtkar@... writes:
uses for it.
Negative and positive numbers are just a tool - something useful IF you know
how to use it. I don't find a lot of uses for soldering iron in my life ---
but I have a friend who does. Why? Because he knows how to use one and, when
something comes up that he wants to do, he thinks of it as part of his tool
kit. If I am faced with the same situation, a soldering iron won't be in my
tool kit and so I won't even see the situation as one that requires a
soldering iron. I'll see it as needing glue or tape or maybe as needing a
friend who might have more tools <G>. So - with negative and positive numbers
- if you don't have a facility with them, you won't notice all the times they
could have been useful.
That is the case with algebra. People often say they never got algebra and it
never hurt them because they never had any use for it anyway. But, I use
algebra frequently in my regular day-to-day life (not counting using it
professionally in statistics and economics). I use it because I have a
facility with it and so I use it to figure things out that other people might
have figured out (with more difficulty, less flexibility, and more likelihood
of error) some other way or might have just ignored or might have guessed at
or might have taken someone else's word for.
So - this isn't an argument that everybody SHOULD learn algebra or how to use
positive and negative numbers. No more than that everyone SHOULD learn to use
a soldering iron. But it is just to explain why you don't always notice all
the possible "uses" for all these mathematical tools in your daily life.
However, I'm also uncomfortable leaving the impression that I might think
understanding math is only worthwhile because it is useful in daily life.
Reading is good for more than understanding assembly instructions, filling
out job applications, reading traffic citations <G>, and other "practical"
applications - it is a way to communicate and to understand the world
(reading literature, poetry, interesting information, biographies, history,
etc.). Mathematics is another way to communicate and understand the world,
too, and goes far beyond day-to-day utilitarian purposes.
--pamS
[Non-text portions of this message have been removed]
curtkar@... writes:
> Even though Joyce and you tried to explain to me how these positive/negativeOne general problem is that if you don't know how to do it, you won't see
> numbers come up and apply to real life situations, and I don't doubt that
> they do, I have never come across the need to multiply negative and
> positive
> numbers, or two negatives numbers. I think that is my main stumbling block
> for not understanding this whole thing.
uses for it.
Negative and positive numbers are just a tool - something useful IF you know
how to use it. I don't find a lot of uses for soldering iron in my life ---
but I have a friend who does. Why? Because he knows how to use one and, when
something comes up that he wants to do, he thinks of it as part of his tool
kit. If I am faced with the same situation, a soldering iron won't be in my
tool kit and so I won't even see the situation as one that requires a
soldering iron. I'll see it as needing glue or tape or maybe as needing a
friend who might have more tools <G>. So - with negative and positive numbers
- if you don't have a facility with them, you won't notice all the times they
could have been useful.
That is the case with algebra. People often say they never got algebra and it
never hurt them because they never had any use for it anyway. But, I use
algebra frequently in my regular day-to-day life (not counting using it
professionally in statistics and economics). I use it because I have a
facility with it and so I use it to figure things out that other people might
have figured out (with more difficulty, less flexibility, and more likelihood
of error) some other way or might have just ignored or might have guessed at
or might have taken someone else's word for.
So - this isn't an argument that everybody SHOULD learn algebra or how to use
positive and negative numbers. No more than that everyone SHOULD learn to use
a soldering iron. But it is just to explain why you don't always notice all
the possible "uses" for all these mathematical tools in your daily life.
However, I'm also uncomfortable leaving the impression that I might think
understanding math is only worthwhile because it is useful in daily life.
Reading is good for more than understanding assembly instructions, filling
out job applications, reading traffic citations <G>, and other "practical"
applications - it is a way to communicate and to understand the world
(reading literature, poetry, interesting information, biographies, history,
etc.). Mathematics is another way to communicate and understand the world,
too, and goes far beyond day-to-day utilitarian purposes.
--pamS
[Non-text portions of this message have been removed]
Jocelyn Vilter
> From: PSoroosh@...This is me, raising my hand over here. I got it last night, skimmed it,
> Reply-To: [email protected]
> Date: Tue, 7 May 2002 13:38:32 EDT
> To: [email protected]
> Subject: Re: [Unschooling-dotcom] more negative and positives stuff
>
> If anybody really did find that reading these examples was uncomfortable - or
> if you keep intending to do it, but just can quite get around to it, or if
> you don't have time (but have time for all the other mail) or have any other
> reason for not reading it --- YOU are the people I really want to hear from
> about your own math backgrounds.
saved it to read when I wasn't so pooped. Skimmed it again this morning,
thought "I'll save it to really read when Matthew gets up and I can kill two
birds with one stone - and he can explain it to me."
Hmmm, my math background? I have choppy memories of school as it is. I
recall pretty early on having to fill in grid papers with numbers. 1 through
as far as we could go, apparently. I also remember stuffing them into the
black hole that was my desk and "losing" them, I think out of sheer boredom.
Looking back now that seems a Sisyphusian challenge and to what end? Skip
ahead to memorizing multiplication facts with my dad in the kitchen. I
wasn't getting them in school, so we drilled and drilled on them at home. I
remember some tears and oddly enough, a few good times too. Skip ahead to
jr. high and me taking and only squeaking by in pre-algebra. I repeated the
class, even though no one was going to force me to do so. Got slightly
better grades that next year, and took no more math after that. Ever. Got
an AA and a BA and an MFA, all without taking any more math.
>I knew early on that I didn't want to do to Matthew what had been done to
> I also want to hear, especially from that above group, how you handle math
> stuff with your kids -- how do you make sure you don't pass on your math
> anxiety to them? How do you create an environment in which math concepts get
> bumped into and expanded on - like you would for other stuff?
me, mathwise. Like most kids, he seemed interested in numbers early and
used to love to count things and play around with them. He liked to have us
ask him math questions in the car, so I used to say what's 5 + 6 and he'd
think a minute and answer 11, so I'd say what's 8 +3 and he'd think a minute
and answer 11. We'd do several of these in a row, and what I always found
interesting, was that he'd take the time to think it through, even though he
saw what I was doing - he was delighted with the idea that there was more
than one way to get the same answer. We noodled around with other number
things too. Multiplication for example - he'd want me to make up problems
for him, so I'd say what's 3 x 5, then I'd rephrase it what are three fives.
I find that that subtle difference in wording really helps me to actually
SEE what it is I'm supposed to be doing.
He's always had access to computer games and board games and card games and
puzzles and real life use of money as well.
We've never drilled him on memorizing tables or spent much time on writing
"problems" down, most of this stuff is all verbal. I find he gets concepts
very easily and is able to jump to the next level in stuff pretty
intuitively. I'm just barely hanging on by my fingernails, but I think
we're pretty safely over the hump of me having to worry about having passed
on math phobia to him.
It helps a lot that his dad is not nearly so much a mathphob as me.
It also helps a LOT that we know Pam in real life, so that when he has a
question about some concept that has me baffled, he can talk to her about
it.<g>
Jocelyn Vilter
[email protected]
In a message dated 5/7/2002 11:20:04 AM Pacific Daylight Time,
curtkar@... writes:
situations where someone found it convenient to multiply negative/positive
numbers - rather than my made-up examples.
I'll have to think about it and dig some up. Maybe other people have some?
They will usually occur when using algebra --- I'm trying to think if there
are other REAL times I can think of that don't involve algebra.
--pamS
[Non-text portions of this message have been removed]
curtkar@... writes:
> Okay, maybe formula is a wrong word to use, on my part. Your example withOkay - I think I know what you're asking for. You're asking for some REAL
> the cars or people standing in line is helpful to understand how the
> numbers
> move in the negative and positive direction when multiplication is applied,
> and how and why the right answer is determined. I do understand this. What
> I
> don't understand is WHY multiplication would be applied to negative numbers
> in the first place. If I could understand this, I think I would *get it*.
> Maybe I'm asking for a word problem?? (What am I DOING? I HATE word
> problems! <g>)
situations where someone found it convenient to multiply negative/positive
numbers - rather than my made-up examples.
I'll have to think about it and dig some up. Maybe other people have some?
They will usually occur when using algebra --- I'm trying to think if there
are other REAL times I can think of that don't involve algebra.
--pamS
[Non-text portions of this message have been removed]
Karin
>YES! That's what I'm asking for.
> Okay - I think I know what you're asking for. You're asking for some REAL
> situations where someone found it convenient to multiply negative/positive
> numbers - rather than my made-up examples.
But, just to clarify, your made-up examples were helpful, too.
In fact, I'm printing them out and saving them for future use.
Karin
zenmomma *
>>Usually people explain positive and negative numbers to kids as having andOkay, now I'm getting it too. Cool. I never had trouble with flipping the
>>owing money. But the positive and negative can represent *anything* that's
>>opposite. In Pam's example, they represent opposite cardinal directions.
>>They can also represent charges on a particle, whether something is
>>increasing or decreasing, coming after or before or above or below some
>>chosen point.>>
numbers around in school, but figuring out where they fit into real
life...that's another story. Guess I'm now another unschooling success
story. ;-)
Life is good.
~Mary
_________________________________________________________________
Join the world�s largest e-mail service with MSN Hotmail.
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Karin
> >>Usually people explain positive and negative numbers to kids as havingand
> >>owing money. But the positive and negative can represent *anything*that's
> >>opposite. In Pam's example, they represent opposite cardinal directions.Thanks for pointing this out Mary! (and Joyce)
> >>They can also represent charges on a particle, whether something is
> >>increasing or decreasing, coming after or before or above or below some
> >>chosen point.>>
>
> Okay, now I'm getting it too. Cool. I never had trouble with flipping the
> numbers around in school, but figuring out where they fit into real
> life...that's another story. Guess I'm now another unschooling success
> story. ;-)
>
> Life is good.
> ~Mary
Even though I already read it once, I think I glazed over it and missed the
significance of *this* part.
I think I have an involuntary brain-block when it comes to reading mathy
stuff. <g>
I'd really love to get past that, though, for my own sake and for my kids.
Karin
[email protected]
Tonight I was telling some people about how an increase in government
spending or a decrease in taxes works to stimulate the economy and how the
impact is a multiple of the original change. I wrote it down, to explain it
better, and, lo and behold, discovered myself using a negative number to
indicate a reduction in taxes. I won't bore you with the rest of the details.
But this was an example where there was a sensible meaning to using a
negative number (which did get used further in multiplication) so I just
thought I'd mention it.
--pamS
spending or a decrease in taxes works to stimulate the economy and how the
impact is a multiple of the original change. I wrote it down, to explain it
better, and, lo and behold, discovered myself using a negative number to
indicate a reduction in taxes. I won't bore you with the rest of the details.
But this was an example where there was a sensible meaning to using a
negative number (which did get used further in multiplication) so I just
thought I'd mention it.
--pamS
rumpleteasermom
--- In Unschooling-dotcom@y..., SandraDodd@a... wrote:
the Ohio Historical Society's sites. We call stuff about stars
"astronomy" when we go to the astonomy club's events and meeting. But
knowing that the world has categories to help us learn and find things
is a far cry from saying everything has to be kept in it's own
category. The girls are learning lots of 'math' at their HAM radio
classes. They learned some at chemistry too - I think we need to add
a chem lab here so they don't have to take the class anymore, but
that's another thought.
Other things my girls 'study' would include library science, botany,
biology and I suppose english. That does not mean they sit down and
focus on one thing at a time. It merely means they know how the
categories are set up - maybe that comes from using the library so
much.
As for me, I don't *tell* Rachel to go study botany. Or Jenni to go
work on her library science. They just do and we all know where each
fits into the world.
Bridget
>"science,"
> I didn't call history "history," didn't call rocks, bugs and stars
> didn't call reading poetry "English" or "Language Arts," and alongwith that
> never called anything "math," but rather called it Games, puzzles,building
> sets, computer games, poker chips, cash, card games, dominos ...We DO call history "history" here - especially when we are going to
>
the Ohio Historical Society's sites. We call stuff about stars
"astronomy" when we go to the astonomy club's events and meeting. But
knowing that the world has categories to help us learn and find things
is a far cry from saying everything has to be kept in it's own
category. The girls are learning lots of 'math' at their HAM radio
classes. They learned some at chemistry too - I think we need to add
a chem lab here so they don't have to take the class anymore, but
that's another thought.
Other things my girls 'study' would include library science, botany,
biology and I suppose english. That does not mean they sit down and
focus on one thing at a time. It merely means they know how the
categories are set up - maybe that comes from using the library so
much.
As for me, I don't *tell* Rachel to go study botany. Or Jenni to go
work on her library science. They just do and we all know where each
fits into the world.
Bridget
Fetteroll
on 5/8/02 9:11 AM, rumpleteasermom at rumpleteasermom@... wrote:
We're *trained* to do that from the moment we enter school.
The point of talking about not categorizing things by subjects is to *help*
new (and old!) unschoolers break out of a way of thinking that can hamper
their getting or being better at unschooling. It's not a way to get people
to stop categorizing things into subjects. It's an exercise to *help* people
free their minds.
*Anyone* can categorize an historical museum as history. But how many people
can see English and math and science and geography and architecuture and
sociology and so on there? The point is most people coming to unschooling
will see the historical museum as *just* history when in reality it drew on
all those tools (subjects) to be what it is.
And even that thinking is a trap but it's a step towards unschooling. It
hopefully helps people to get to two places:
1) where they can be comfortable that helping a child figure out their game
score is math in a much bigger sense than seemingly having done one practice
problem that schooled kids would be doing 100 times more of and
2) that if everything is in everything then they can start to ease the
worries about whether their kid has done any math or history or writing
recently.
Joyce
> We DO call history "history" here - especially when we are going to*Everyone* who has been to school categorizings things into subject areas.
> the Ohio Historical Society's sites.
We're *trained* to do that from the moment we enter school.
The point of talking about not categorizing things by subjects is to *help*
new (and old!) unschoolers break out of a way of thinking that can hamper
their getting or being better at unschooling. It's not a way to get people
to stop categorizing things into subjects. It's an exercise to *help* people
free their minds.
*Anyone* can categorize an historical museum as history. But how many people
can see English and math and science and geography and architecuture and
sociology and so on there? The point is most people coming to unschooling
will see the historical museum as *just* history when in reality it drew on
all those tools (subjects) to be what it is.
And even that thinking is a trap but it's a step towards unschooling. It
hopefully helps people to get to two places:
1) where they can be comfortable that helping a child figure out their game
score is math in a much bigger sense than seemingly having done one practice
problem that schooled kids would be doing 100 times more of and
2) that if everything is in everything then they can start to ease the
worries about whether their kid has done any math or history or writing
recently.
Joyce
Tia Leschke
>I found the first one hard to stay with, but I did stay on and understood
>
>If anybody really did find that reading these examples was uncomfortable - or
>if you keep intending to do it, but just can quite get around to it, or if
>you don't have time (but have time for all the other mail) or have any other
>reason for not reading it --- YOU are the people I really want to hear from
>about your own math backgrounds.
it. Then the second one arrived, and it just seemed like to much work. I
put it off, and now I have 20 or 30 math messages to read that I
skipped. (I did finally read your second example and found it easier than
the first.)
I don't remember anything at all about my math (arithmetic) experiences
before grade 4. I think I didn't have any problems with it. In grade 4, I
bogged down on long division. I probably stayed after school every day
that year trying to finish my work. It just seemed to take so long with
each problem to figure out how many times a number would "go into" another
one. I'd try one number, and it would be too big. Then I'd try another,
and it would be too small, etc. etc. etc. From that time on, I was in the
lowest math group. In grade 5, we didn't finish the book (by about
half). In grade 6 we finished even less (because of having to finish the
previous one) and then I got to grade 7, where we were tracked. I got put
into the top (college bound) track because of my language arts skills. We
were expected to do grade 7 and 8 math in that year and go on to algebra in
grade 8 instead of 9. Remember I hadn't come close to finishing the grade
6 book. I got through because we had a teacher who would raise our grade a
little bit for each oral report we did about math. I did tons of them,
even though I hated public speaking, and managed to get through.
Algebra completely stumped me, not because I couldn't get the concepts but
because I took so long to do the arithmetic and got behind. Maybe I would
have been ok if we could have used calculators for the arithmetic, but that
was before calculators. And we weren't allowed to use slide rules, even if
I had known how. On to geometry after barely managing to get a C in
algebra. I did great until we hit square roots . . .
In college, I had to take a math course and took "math for liberal arts
students". It was symbolic logic, and I *loved* it. I even took a more
advanced course and was told I should go into programming. (I did but
didn't finish before moving to Canada. Here, you had to have a whole pile
of math and science to even get in, even though I had been getting straight
As in programming in California.)
So there you have my math experiences.
>I also want to hear, especially from that above group, how you handle mathI'd love to know the answers to those questions. I fear I haven't
>stuff with your kids -- how do you make sure you don't pass on your math
>anxiety to them? How do you create an environment in which math concepts get
>bumped into and expanded on - like you would for other stuff?
succeeded well at all in that area. My son was actually "ahead" of his
schooled friends in the early years. He managed to learn most of the
practical arithmetic up through percentages and some fraction stuff, mostly
through his interest in money. After that, he's had no need for the stuff
they learn in school and so hasn't learned it.
Tia
No one can make you feel inferior without your consent.
Eleanor Roosevelt
*********************************************
Tia Leschke
leschke@...
On Vancouver Island