[Numeracy 642] Re: That old thinking style thing

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Chip Burkitt chip.burkitt at orderingchaos.com
Fri Dec 24 14:22:27 EST 2010


Learning math is really a lot like learning a language. Mathematicians
become fluent in it, and it involves using symbols to produce abstract
expressions which can then be manipulated to produce general results. As
interesting as it is to solve 4x^x + 4x + 1 = 0, it is more interesting
(to a mathematician) to solve Ax^2 + Bx + C = 0. Mathematicians become
adept at viewing concrete problems as instances of a more general
problem. This, I think, is precisely what many students have difficulty
with. They want a solution to /this/ problem, and they don't see what
value there is in thinking in more abstract terms. In fact, the value is
not immediate. It lies in providing a means of solving a whole class of
problems. But mathematicians don't memorize abstract formulas and then
apply them to concrete problems. They find ways to derive the formulas
in the first place. For example, I never forget the quadratic formula in
part because I always remember how to derive it using simple algebra. It
is in finding ways to solve whole classes of problems that the true
power of mathematics lies. The challenge for us as teachers is to
facilitate the student's ability to recognize patterns and do the work
of abstracting. For testing, the problem often is to interpret the test
question as an instance of a general type of question whose solution is
known. Many students struggle both with the process of abstracting a
problem to form a more general solution and with the process of
interpreting a problem as an instance of a solved general problem. For
example:

A flag pole casts a 40' shadow at the same time that a 9' road sign
casts a 12' shadow. How tall is the flag pole?

Solving this kind of problem requires that the student recognize this as
an instance of similar triangles. If the student does not make that
interpretive leap, nothing else about the problem will make sense. If I
show the student how to solve /this/ problem, it will have no effect on
his or her ability to solve other similar triangle problems that are
worded differently. For example,

Bob pushes a load up a 15' ramp. After pushing it 3', the load is 1' off
the ground. How far off the ground is the load at the end of the ramp?

This problem looks utterly different from the previous problem, but to a
mathematician they appear identical. How can we encourage our students
to think like mathematicians?

Chip Burkitt

On 12/23/2010 9:15 AM, Michael Wolf wrote:

> One of the things that always strikes me about my students is that in real life, if they had a problem that was equivalent to a simple one or two step equation, they would find the answer without too much difficulty. It seems to me that I spend too little time grounding the math in the real world and actual experience of my students. I like the going to the movies example, for instance, because I think that would feel somewhat "real" to the students.

> On the other hand, it is also difficult for them to abstract or generalize the movie example or many others into a general strategy. In fact, while they could solve the movie ticket price problem as a practical matter, they may need some serious convincing that "writing an equation" is helpful or neccesary in order to think about the problem. I am always torn ebtween letting them find their own ways of organizing and expressing the information versus showing them the conventional or textbook approach. Complicating things even more is limited time. If we had unlimited time, it would be much easier to let them evolve their own approaches and algorithms for solving problems. When I have to get them through a relatively fixed amount of material in a semester or quarter or whatever, the temptation to "show them how to do it" becomes muich greater. I imagine we all live in this awful place of never knowing how long we should allow them to explore the problem versus telling th

> em what to do.

>

> Michael Wolf

> Harcum College

>

>

> -----Original Message-----

>> From: Susan Jones<sujones at parkland.edu>

>> Sent: Dec 22, 2010 2:02 PM

>> To: The Math and Numeracy Discussion List<numeracy at lincs.ed.gov>

>> Subject: [Numeracy 638] Re: That old thinking style thing

>>

>> Lots of the students I work with would rather enjoy it -- but wouldn't really think about the "X" aspect. They'd be confounded by my explanation (because of the whole variable thing), and then at some point realize "Oh! You want me to write equations that start with {whatever number I'd said x equaled}!" and they'd write, perhaps 13 + 3 = 16, 13 x 2 = 26. However, (thinking as I type), I'm thinking that even this would be a challenge to them, and perhaps a worthy one. Perhaps ... pehraps I *could* introduce it that simply -- as in, "write an equation that *starts* with 12, has something done to it, and gets an answer." So, once I had a little collection, then I could rewrite a set of them as "X - 2= 10" and "X x 2 = 24." Hmmm... that just might convey that sometiems elusive idea that X stands for a number.

>>

>> THat would be drifting well away from "real" stuff, but I think might be useful for developing fluency with the abstractions. I often try to explain the "standing for a number" with something like "the price of a movie ticket." So '3x + 4 = 13" might mean you bought three movie tickets, and also had to pay four bucks for parking if you went in a car instead of by bicycle. However, if they're focused on "how do I figure out the code ?!?!?" then that gets filtered out. And... I could try to make up a story for each of the ones they came up with and challenge them to do the same.

>>

>> 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

>>

>>

>>

>>>>> <tjdclaire at cox.net> 12/21/2010 10:14 PM>>>

>> 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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