[Numeracy 635] Re: That old thinking style thing
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Tue Dec 21 23:14:24 EST 2010
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I teach solving equations by telling students that the person who made up the equation already knew the answer (certainly when I make up an equation, that is so). What they are to do is to undo what the person who made the equation did. (There is a lot more to my explanation, but this discussion made me stop and think...) Perhaps it would help if students were assigned a number that x had to be equal to and then asked to write true equations using that x. Change their perspective on what is happening.
Claire Ludovico
---- ROBERT G STEINKE <steinkedb at q.com> wrote:
> 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.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.bicycleuc.wordpress.com <http://www.bicycleuc.wordpress.com/>
>
>
>
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