[Numeracy 636] Re: That old thinking style thing
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Wed Dec 22 11:04:45 EST 2010
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Hi Dorothea,
Thanks for your response--I've spent a good half hour reading it and thinking about representing those equations in words and in pictures or demonstrations.
I also went back to look at your article again--it was a great issue of Focus on Basics, and I was very pleased to also have an article in it. (Here's the link to the pdf of the issue: http://www.ncsall.net/index.php?id=1195 )
My experience certainly echoes your research in that many adult learners don't have the part/whole concept, and I like your idea of using diagrams to show various numbers in a problem as either parts or wholes. I won't attempt to draw diagrams here, but list readers can look at your article for some examples. I find such a part/whole diagram especially useful when working on per cent problems.
An interesting wrinkle comes when we start to talk about problems such as "Marco has three times as many books as Antonio. If Marco has 12 books, how many does Antonio have?" or an equation like 3A = 12. Here the three is not a part of the whole, but rather the number of (equal) parts we're looking at.
I know that some of the pleasure I find in math comes in being able to think and write paragraphs like the previous one. And some of the pleasure I find in teaching math comes as learners are able to show their understanding, not the results of rote memory. From my reading of this list, I think many of us are in agreement there.
So I continue to search for ways to demonstrate math concepts before or while I talk about them, because, unlike me, most of my adult learners do better when they can see and do, rather than listen and talk.
Kate Nonesuch
Victoria, BC
kate.nonesuch at viu.ca
________________________________
From: numeracy-bounces at lincs.ed.gov on behalf of ROBERT G STEINKE
Sent: Tue 12/21/2010 11:43 AM
To: The Math and Numeracy Discussion List
Subject: [Numeracy 634] Re: That old thinking style thing
Kate Nonesuch - I'm replying to your question about "showing equations of
the type X - 3 = 2 and X/2 = 5".
Here is where the idea of "only 2 kinds of problems" comes in (see the May
2008 Focus of Basics article I wrote).
In addition and multiplication equations (3 + 2 = x or 5 * 2 = x) the WHOLE
is by itself on one side of the equal sign.
In equations using subtraction and division signs, (X - 3 = 2 and X/2 =
5) the WHOLE is the first number in the equation.
When you are using the two mats that you describe, the WHOLE is in one
location, the PARTS are in the other, regardless of the operation.
You don't solve X - 3 = 2 by subtraction; you solve it by addition because
it is a "find the WHOLE" problem.
You don't solve X/2 = 5 by division; you solve it by multiplication because
it is a "find the WHOLE" problem.
Dorothea Steinke
Dorothea at numberworks4all.com
-----Original Message-----
From: numeracy-bounces at lincs.ed.gov
[mailto:numeracy-bounces at lincs.ed.gov]On Behalf Of Susan Jones
Sent: Tuesday, December 21, 2010 9:41 AM
To: The Math and Numeracy Discussion List
Subject: [Numeracy 633] Re: That old thinking style thing
I'm hoping this discussion doesn't get lost in the holidays and that people
will chime in when they come back if they're festivating...
One of our Math teachers does something really similar to this (I don't
think he uses mats -- but I'll suggest it, since visual boundaries have a
peculiarly positive effect on many), and this is the kind of project I may
try in a workshop setting.
I think that once the idea is grounded, with dedicated explicit practice in
translating the manipulative experience into "regular person" language as
well as "math language," and then a few more times practicing... then we
could show it visually ... say with something put together on the ol'
computer... maybe a Flash exercise. (Which I have yet to learn. Maybe
start this break :)) Or, draw it on the board and make an "old fashioned"
video of it.
Subtraction... let's see... that idea that something *was taken away* is
hard to convey, but I like the six pack with the empty places idea. And,
I think, it's a grand place to work in the "normal people language" and make
up situations to learn that "x - 2 = 3" is a different way of saying "what
is three two less than?" and working to make that statement *mean*
something. "Three is two less than what?" they'd probably get... show 'em
that we're "just" turning some words around...
I think that stuff coudl be conveyed with some creative graphics. If we can
get that "whole to part" concept, then we'll be conveying "that X in 'x -
3'? That's the whole thing that you *started* with, before you yanked
three away." (I'm trying to think on behalf of the language-oriented
learners as well as the visually-oriented ones.)
http://www.hewlett.org/programs/education-program/open-educational-resources
says that in Mid-December they'll accept letters of inquiry for the
development of Open Source Materials. Their wording talks about fairly big
attempts -- but other things I've been reading suggest that they would
really like to do some focus on "deeper learning" in basic math for people
at community colleges. I wonder if some collaborative work could be
funded... some lessons in Flash or getting somebody who already knows to
put things together...
Susan Jones
Academic Development Specialist
Center for Academic Success
Parkland College
Champaign, IL 61821
217-353-2056
sujones at parkland.edu
Webmastress,
http://www.resourceroom.net <http://www.resourceroom.net/>
http://www.bicycleuc.wordpress.com <http://www.bicycleuc.wordpress.com/>
>>> "Kate Nonesuch" <Kate.Nonesuch at viu.ca> 12/20/2010 2:26 PM >>>
Hi Susan,
I'm intrigued by your ideas of making videos--I should make a video of the
following idea--it would be easier than writing it out.
Here's a visual and kinaesthetic way I've used to demonstrate solving
equations involving addition and multiplication:
You need two mats, or two spaces, one for each side of the equals sign.
On the left, a closed envelope represents X (or E), with some visible
counters as necessary. On the right are some counters. The total number of
counters on each mat is the same.
For example: To show X + 3 = 5, on the left hand side, the envelope has 2
counters inside it and there are 3 counters lying in the open. The right
hand side has 5 counters and no envelope.
The question to learners is: The number of counters on each mat is the same.
How many counters in the envelope? How do you know? What operation did you
do to find out?
Example 2: To show 3X = 12, on the left hand side there are three
envelopes, each with four counters inside. On the right hand side are 12
counters lying in the open.
The question to learners is: The number of counters on each mat is the
same. Each envelope has the same number of counters inside. How many
counters in each envelope? How do you know? What operation did you do to
find out?
I encourage students to play with the counters on the right hand side to
help them figure it out, e.g., in example 1, to separate out three counters
on the right hand side to isolate the number hidden in the envelope.
And, as you said in your post, the thinking is the important part--after
I've done a few, I ask students to make up some problems for the other
students. Sometimes they set it up with the envelope first, then have to
write the equation based on what they have set up; sometimes they write an
equation first, and then have to figure out how to represent it with
envelope and counters. Either way, lots of thinking about what an equation
represents, and the difference between 2X and X+2 comes through loud and
clear!
However, this method is not so useful in showing equations of the type X - 3
= 2 and X/2 = 5. Anybody have ideas for that?
Kate Nonesuch
Victoria, BC
kate.nonesuch at viu.ca
________________________________
From: numeracy-bounces at lincs.ed.gov on behalf of Susan Jones
Sent: Fri 12/17/2010 1:19 PM
To: kabeall at comcast.net; The Math and Numeracy Discussion List
Subject: [Numeracy 627] That old thinking style thing
... take all that stuff with the x's and show LOTS of examples of putting
numbers in there for the x's. For students who do have some number sense,
plugging in numbers and *testing* adding 3x + 2x and discovering that it's
5x no matter what you call x... as long as you call it the same thing...
could really help. (I'm also trying to figure out ways to do this that
*aren't* entirely depending on language and discussion... some of my guys
get *so* much smarter when they can see things...)
..........
I'm curious -- what are other folks doing to engage students in that
elusive comprehension aspect?
Susan Jones
Academic Development Specialist
Center for Academic Success
Parkland College
Champaign, IL 61821
217-353-2056
sujones at parkland.edu
Webmastress,
http://www.resourceroom.net <http://www.resourceroom.net/> <http://www.resourceroom.net/>
http://www.bicycleuc.wordpress.com <http://www.bicycleuc.wordpress.com/> <http://www.bicycleuc.wordpress.com/>
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