| Topic: |
Sociology > Education |
| User: |
"Dom" |
| Date: |
01 Jun 2007 09:02:46 PM |
| Object: |
Teaching "algebra" in second grade |
Now I have a much better grasp of why so many college students are
placed in remedial algebra classes. I must tell our director of
placement testing to purchase "Base 10 blocks, number lines, etc.,"
and allow students to use them on the placement test. I will also put
in an order for these items for use in my remedial classes.
=================
http://blog.seattlepi.nwsource.com/educatingmom/archives/115426.asp
Posted by unregistered user at 5/16/07 8:38 a.m.
My 2nd grader was telling me this morning about how her class was
STRATEGIZING the math problem 100-X=85. They were allowed to come up
with their own ways of finding the answer using Base 10 blocks, number
lines, etc. One girl took the 85 and broke it down into 80+5, then
added 20 to the 80 to get 100, then subtracted 5 from the 20 and got
the answer of 15.
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| User: "met00" |
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| Title: Re: Teaching "algebra" in second grade |
06 Jun 2007 05:57:18 AM |
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Dom wrote:
Now I have a much better grasp of why so many college students are
placed in remedial algebra classes. I must tell our director of
placement testing to purchase "Base 10 blocks, number lines, etc.,"
and allow students to use them on the placement test. I will also put
in an order for these items for use in my remedial classes.
Every day I present my child with a problem. He has all day to use any
strategy they want to solve it. He can ask any question and I'll explain
the theory behind it with examples.
Here are Monday and Tuesdays problems:
Monday. Introduction to Order of Operations
(2*2(2)) + 3(3) - 15 + 22 + ((3*6)+2) * 2 - 41 =
Tuesday: Improper Fractions and Reduction
3 7/5 + 2 8/10 =
Now, for a fifth grader one would expect these to be easy to resolve.
But my child is a first grader who just loves to play with numbers and
learn how they work (if only my other child had that interest).
For the whole school year I attempted to "hold my child back" so that he
wouldn't get too far ahead of the other students because I know what
happens when a kid gets bored in class because they see the work in one
subject as easy... they lose interest in learning - even in the subjects
that they are in grade level on. But, you can't stop a rushing river,
and he loves playing with numbers and learning what you can do with
them, so I finally have given in and am letting him learn at his pace.
Will this lead to problems when he enters grade 2 and is doing math at
grade 5 levels? That is what I'll be dealing with next year. But in the
mean time, consider this. When you don't tell a child HOW to resolve a
problem, but give them the rules and tools to find the answer
themselves, they can be exceptionally creative in not only finding the
answer, but determining rules that you had not taught them by seeing
what failed to work in their experimentation. These intuitive leaps are
things of beauty.
When my child heard about square roots he asked me what they were. So I
explained what a square root was and provided examples with the squares
of numbers between 1-12. The next day he came to me and asked if there
was a way to show a square roots "root". I showed him the way to express
exponents and explained what it was and what it meant. The next day he
had written out a problem he wanted to solve and in the middle of the
problem was + 3(3). So I asked him what that was and he said... "well if
3(2) is 6, then 3(3) is 24." I pointed out that computers used "2" as
the base. Less than an hour later he has written down every exponent
from 2(1) through 2(10). Not as an assignment, but to see what the
answers were...
My point is, if a child is presented math as a series of "rules" and
allowed to explore the concepts with whatever methods work for them,
math is interesting to a number of them, and we shouldn't be surprised
to see some of the blossom and far exceed what is expected of them.
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| User: "Pubkeybreaker" |
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| Title: Re: Teaching "algebra" in second grade |
06 Jun 2007 09:57:00 AM |
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met00 wrote:
(2*2(2)) + 3(3) - 15 + 22 + ((3*6)+2) * 2 - 41 =
Tuesday: Improper Fractions and Reduction
The artificial distinction that is placed between "proper" and
"improper"
fractions is a lot of flim-flam taught by people who do not understand
that
the distinction itself DETRACTS from understanding. There are no
"improper"
fractions. There are just rational numbers. An integer is a fraction
with a
denominator of 1. A so-called "improper" fraction is just the sum of
an
integer and a non-integral rational number. Toss in the distributive
law and
that division is just multiplication by the reciprocal and now you
have
EVERYTHING needed to do arithmetic with fractions.
But teachers with inadequate understanding will continue to make
arithmetic
with fractions much more mysterious than it should be.
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| User: "Herman Rubin" |
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| Title: Re: Teaching "algebra" in second grade |
07 Jun 2007 02:08:51 PM |
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In article <1181141820.622443.129440@q66g2000hsg.googlegroups.com>,
Pubkeybreaker <pubkeybreaker@aol.com> wrote:
met00 wrote:
(2*2(2)) + 3(3) - 15 + 22 + ((3*6)+2) * 2 - 41 =
Tuesday: Improper Fractions and Reduction
The artificial distinction that is placed between "proper" and
"improper"
fractions is a lot of flim-flam taught by people who do not understand
that
the distinction itself DETRACTS from understanding. There are no
"improper"
fractions. There are just rational numbers. An integer is a fraction
with a
denominator of 1. A so-called "improper" fraction is just the sum of
an
integer and a non-integral rational number. Toss in the distributive
law and
that division is just multiplication by the reciprocal and now you
have
EVERYTHING needed to do arithmetic with fractions.
But teachers with inadequate understanding will continue to make
arithmetic
with fractions much more mysterious than it should be.
Definitely. The fraction m/n, with both positive, is
taking the NUMBER m of parts of size 1/n, n the NAME
of the part. This is why m is called the NUMERATOR
and n the DENOMINATOR. It is easy to see by breaking
parts up that m/n = km/kn, k an integer not 0. This
can be reversed, so fractions can be put in lowest terms.
Also, m_1/n + m_2/n = (m_1 + m_2)/n; from this, we can
see how to add fractions with any common denominator.
The least common denominator makes the numbers smallest,
but has no other effect.
The other properties of fractions can be deduced from
these, especially with a little algebraic notation.
The use of variables is linguistic, and not at all
difficult if one knows it.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
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| User: "toto" |
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| Title: Re: Teaching "algebra" in second grade |
06 Jun 2007 10:50:02 PM |
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On Wed, 06 Jun 2007 07:57:00 -0700, Pubkeybreaker
<pubkeybreaker@aol.com> wrote:
met00 wrote:
(2*2(2)) + 3(3) - 15 + 22 + ((3*6)+2) * 2 - 41 =
Tuesday: Improper Fractions and Reduction
The artificial distinction that is placed between "proper" and
"improper" fractions is a lot of flim-flam taught by people who
do not understand that the distinction itself DETRACTS from
understanding. There are no "improper" fractions. There are
just rational numbers. An integer is a fraction with a
denominator of 1. A so-called "improper" fraction is just the
sum of an integer and a non-integral rational number. Toss
in the distributive law and that division is just multiplication by
the reciprocal and now you have EVERYTHING needed to
do arithmetic with fractions.
I couldn't agree more. When I taught hs algebra, this was the
hardest thing for the kids to unlearn. There is NO need to change
the answers into mixed numbers and doing so gives the impression
that a fraction with a larger numerator than denominator is not
acceptable.
But teachers with inadequate understanding will continue to
make arithmetic with fractions much more mysterious than it
should be.
There are many things that kids thing are mysterious in math which
are really quite straightforward.
Another really annoying one from algebra is *like terms.* For some
reason, we no longer seem to teach conversions of units which is
actually an introduction to like terms in some ways. I'm not sure
when that happened, but I know that in the late 80s, high school
students seemed not to have the concept of converting from one
unit to another when you wanted to combine things.
--
Dorothy
There is no sound, no cry in all the world
that can be heard unless someone listens ..
The Outer Limits
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| User: "toto" |
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| Title: Re: Teaching "algebra" in second grade |
01 Jun 2007 09:45:58 PM |
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On Fri, 01 Jun 2007 19:02:46 -0700, Dom <DRosa@teikyopost.edu> wrote:
Now I have a much better grasp of why so many college students are
placed in remedial algebra classes. I must tell our director of
placement testing to purchase "Base 10 blocks, number lines, etc.,"
and allow students to use them on the placement test. I will also put
in an order for these items for use in my remedial classes.
=================
The student below is a second grader, not a high school student.
Younger students often need visuals and concrete representations to
help them understand what they are doing.
Note that the student below understood how to break up the numbers and
how to use the properties of numbers to come up with a solution (and
her solution was correct). The fact that she did not do this using
the standard algebraic notation doesn't seem to me to be a problem.
Certainly she understands her own procedure and it works to break down
problems into easier problems. That doesn't mean that she should not
later be taught the standard way of doing such problems.
Now, in high school, presumably, she would get beyond using
manipulatives to help and she would probably quickly learn the
procedure for adding equals to both sides of an equation to solve it.
One of the problems that I've had with high school algebra students
involves the fact that they have trouble understanding adding anything
but numbers to both sides of an equation. OTOH, I had a student tell
her mother that algebra was like *cheating* because when you do all
the steps the answer just falls out.
http://blog.seattlepi.nwsource.com/educatingmom/archives/115426.asp
Posted by unregistered user at 5/16/07 8:38 a.m.
My 2nd grader was telling me this morning about how her class was
STRATEGIZING the math problem 100-X=85. They were allowed to come up
with their own ways of finding the answer using Base 10 blocks, number
lines, etc. One girl took the 85 and broke it down into 80+5, then
added 20 to the 80 to get 100, then subtracted 5 from the 20 and got
the answer of 15.
Just because she didn't use algebraic notation does not mean she
wasn't using algebraic reasoning even if she used more steps than most
people would. Her reasoning was correct.
Algebraically, she did this:
100 - x = 85
100 - x + x = 85 + x
100 = 80 + 5 + x
100 - 80 = 5 + x
20 = 5 + x
20 - 5 = 5 - 5 + x
15 = x
That's pretty sophisticated for 2nd grade, don't you think?
--
Dorothy
There is no sound, no cry in all the world
that can be heard unless someone listens ..
The Outer Limits
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| User: "Dom" |
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| Title: Re: Teaching "algebra" in second grade |
02 Jun 2007 04:08:39 AM |
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On Jun 1, 10:45 pm, toto <scarec...@wicked.witch> wrote:
[snip]
http://blog.seattlepi.nwsource.com/educatingmom/archives/115426.asp
Posted by unregistered user at 5/16/07 8:38 a.m.
My 2nd grader was telling me this morning about how her class was
STRATEGIZING the math problem 100-X=85. They were allowed to come up
with their own ways of finding the answer using Base 10 blocks, number
lines, etc. One girl took the 85 and broke it down into 80+5, then
added 20 to the 80 to get 100, then subtracted 5 from the 20 and got
the answer of 15.
Just because she didn't use algebraic notation does not mean she
wasn't using algebraic reasoning even if she used more steps than most
people would. Her reasoning was correct.
Algebraically, she did this:
100 - x = 85
100 - x + x = 85 + x
100 = 80 + 5 + x
100 - 80 = 5 + x
20 = 5 + x
20 - 5 = 5 - 5 + x
15 = x
That's pretty sophisticated for 2nd grade, don't you think?
In my opinion, the minds of students would be much less stunted if
they did this type of problem mentally (e.g 100=80+10+5=80+15) instead
of using blocks, etc.
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| User: "Don K" |
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| Title: Re: Teaching "algebra" in second grade |
02 Jun 2007 09:19:01 AM |
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"Dom" <DRosa@teikyopost.edu> wrote in message
news:1180775319.102834.142960@r19g2000prf.googlegroups.com...
On Jun 1, 10:45 pm, toto <scarec...@wicked.witch> wrote:
[snip]
http://blog.seattlepi.nwsource.com/educatingmom/archives/115426.asp
Posted by unregistered user at 5/16/07 8:38 a.m.
My 2nd grader was telling me this morning about how her class was
STRATEGIZING the math problem 100-X=85. They were allowed to come up
with their own ways of finding the answer using Base 10 blocks, number
lines, etc. One girl took the 85 and broke it down into 80+5, then
added 20 to the 80 to get 100, then subtracted 5 from the 20 and got
the answer of 15.
Just because she didn't use algebraic notation does not mean she
wasn't using algebraic reasoning even if she used more steps than most
people would. Her reasoning was correct.
Algebraically, she did this:
100 - x = 85
100 - x + x = 85 + x
100 = 80 + 5 + x
100 - 80 = 5 + x
20 = 5 + x
20 - 5 = 5 - 5 + x
15 = x
That's pretty sophisticated for 2nd grade, don't you think?
In my opinion, the minds of students would be much less stunted if
they did this type of problem mentally (e.g 100=80+10+5=80+15) instead
of using blocks, etc.
The point is there are many ways to visualize a problem and come up with a solution.
By being allowed to play around with manipulatives, some students may be more
likely to visualize the relationship between the various parameters.
Once the students have an intuitive understanding of the relationships and
what is going on, then you can show them efficient ways to solve these
problems mentally or using abstract symbols.
People will forget the tricks and shortcuts if they aren't used frequently, but
if they understand the fundamental concepts, they can always go back and
derive the more efficient solution method for themselves.
Don
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| User: "Herman Rubin" |
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| Title: Re: Teaching "algebra" in second grade |
02 Jun 2007 09:25:54 PM |
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In article <1180775319.102834.142960@r19g2000prf.googlegroups.com>,
Dom <DRosa@teikyopost.edu> wrote:
On Jun 1, 10:45 pm, toto <scarec...@wicked.witch> wrote:
[snip]
http://blog.seattlepi.nwsource.com/educatingmom/archives/115426.asp
Posted by unregistered user at 5/16/07 8:38 a.m.
................
Just because she didn't use algebraic notation does not mean she
wasn't using algebraic reasoning even if she used more steps than most
people would. Her reasoning was correct.
Algebraically, she did this:
100 - x = 85
100 - x + x = 85 + x
100 = 80 + 5 + x
100 - 80 = 5 + x
20 = 5 + x
20 - 5 = 5 - 5 + x
15 = x
That's pretty sophisticated for 2nd grade, don't you think?
Not if the MAJOR rule for solving problems has been taught,
which it should be in first grade at the latest. Instead
of teaching a large number of rules,
The same operation performed on equals gets equal results.
In my opinion, the minds of students would be much less stunted if
they did this type of problem mentally (e.g 100=80+10+5=80+15) instead
of using blocks, etc.
They should not do it with blocks, but not mentally either.
Not everyone can keep track of that much mentally.
The limitations on my arithmetic ability, which few can
match, is my lack of the ability to keep a sufficient
number of digits straight, and erase when done. Gauss
was supposed to have used seven-point interpolation to
get the logarithms he needed for astronomical calculation;
I would not even try.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
.
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| User: "Dom" |
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| Title: Re: Teaching "algebra" in second grade |
02 Jun 2007 10:16:35 AM |
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On Jun 2, 5:08 am, Dom <D...@teikyopost.edu> wrote:
On Jun 1, 10:45 pm, toto <scarec...@wicked.witch> wrote:
[snip]
http://blog.seattlepi.nwsource.com/educatingmom/archives/115426.asp
Posted by unregistered user at 5/16/07 8:38 a.m.
My 2nd grader was telling me this morning about how her class was
STRATEGIZING the math problem 100-X=85. They were allowed to come up
with their own ways of finding the answer using Base 10 blocks, number
lines, etc. One girl took the 85 and broke it down into 80+5, then
added 20 to the 80 to get 100, then subtracted 5 from the 20 and got
the answer of 15.
Just because she didn't use algebraic notation does not mean she
wasn't using algebraic reasoning even if she used more steps than most
people would. Her reasoning was correct.
Algebraically, she did this:
100 - x = 85
100 - x + x = 85 + x
100 = 80 + 5 + x
100 - 80 = 5 + x
20 = 5 + x
20 - 5 = 5 - 5 + x
15 = x
That's pretty sophisticated for 2nd grade, don't you think?
In my opinion, the minds of students would be much less stunted if
they did this type of problem mentally (e.g 100=80+10+5=80+15) instead
of using blocks, etc.
Typo: The 80 should have been 85.
.
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