[Edu-sig] a non-rhetorical question

[Laura Creighton]

Late: I have been busy with Europthon

In a message of Sun, 08 Jul 2007 16:35:17 EDT, Jay Bloodworth writes:
>On Sun, 2007-07-08 at 21:46 +0200, Laura Creighton wrote:
>> 
>> Do you have many students who are good at geometry and still rotten
>> at algebra?  Also what do they say when you ask them 'what don't
>> you understand here?'
>
>
>I wouldn't say rotten, but it's not unusual to have students who do
>significantly better in geometry than algebra.  Again, it could just be
>a year of brain maturity that makes that so.

I'm curious about this.  I don't think I have ever heard about
'brain maturity' -- except in terms of 'reading readiness'.  So
I have always assumed that if a student was having a really dreadful
time learning something, and if it wasn't as a result of having an
eidic memory, or a learning disability or some other unique way of
experiencing the world, then it was most likely due to lack of
relevant pre-requisite experience.

And in my case, I built up a model where experience in Geometry is
a prerequisite for learning Algebra, which I why I think that one
ought to teach Geometry first.  Of course, other people may have
different ways of teaching Algebra which doesn't depend on a background
in Geometry.  I'd be really interested in seeing such lesson plans.

One hard thing, of course, is to tell whether your students really
have no relevant experience in Geometry -- or whether that is the
sort of stuff that they have picked up, on their own, simply as part
of being in the world.

>"What don't you understand?"  Usually if a kid can answer that they
>don't have a problem:
>
>Ex: 2x + 3y + 5x = 7x + 3y
>
>Kid 1:
>"Why don't you understand?"
>"Where did the 7x come from?"
>"From combining like terms.  I added 2x and 5x."
>"What are like terms?"
>"Terms with the same variables to the same powers.  2x and 5x both have
>x to the first power and no other variables."

Aha.  I go after this differently.

I give you 2 oranges + 3 apples + 7 oranges.
How many oranges do you have?

If that doesn't work, try MONEY.

It is amazing how many children I know who have problems keeping track of
apples and oranges but have no problem with 20, 50, and 100 Kronor notes.

Long before we start talking about 'variables' and 'powers' and the
like -- which is all part of the 'notationally true' world, we need to
nail down the absolute truths about addition that the order in which
you add terms does not matter.  Which we can then formulate as LAWS
about addition.

And we have to show that this is different from subtraction, where
the order matters, very, very much.  

To get _more_ notational truths across, I just have them set the
variables to things that take a lot of writing.

Pretty soon they are sick of writing:
3 refrigerators full of gorrilla food + 12 refrigerators full of
penguin chow + 6 refrigerators full of gorrilla food = 9 refrigerators full
of gorilla food + 12 refrigerators full of penguin chow

and understand perfectly why mathematicians decided they would rather write

3g + 12p + 6g = 9g + 12p

Sometimes python is useful here.  For the die-hards who don't want to
believe that addition is associative:

It works for letters.

>>> 3 * 'g' + 12 * 'p' + 6 * 'g'
'gggppppppppppppgggggg'
>>>

They can go count them.  And then write a program that counts the
number of each letter in a string.

Try it with words:

>>> 3 * 'orange' + 5 * 'apple' + 2 * 'orange'
'orangeorangeorangeappleappleappleappleappleorangeorange'
>>>

Writing that program is more interesting.

>versus
>
>Kid 2:
>"Why don't you understand?"
>"I just don't get it."
>
>Not a great example, because most students can do a little better with
>like terms than Kid 2.  But the point is that the "don't get it" kids
>I'm talking about can't really tell you what they don't get.  They see a
>string of symbols on line one and another on line two and claim to see
>no connection between them.  And though I can often ask a series of
>questions to determine what they don't get and to explain it - "Do you
>see where the 3y comes from? Good.  How about the 7x?  Okay, do you see
>the 2x and 5x?  etc." - they'll still say they don't get it.

Yes.  I know lots of Kid2s.  And they are quite correct.  They don't
get it.  They cannot draw you a picture about it, either.  (But if
you can get them to draw silly pictures, it may help them get it.)
They cannot spot the problem for the notation.  It is an abstract
way of representing .. what? Nothing they can see as a problem.

Most of the Kid2s I have met have a real problem with understanding:

Let X be 'refrigerators full of badger steaks'.
They need the lesson above.

But before I want to do very much more algebra, I want to teach the
kid2s how to get out graph paper and draw

y = 2x + 17 
y = 2x + 4 + 3x and the like:

As part of a geometry course, where you construct things.

Because then, in addition to other things, they will know that
the above are abstract representations of lines.

Will your students already know this before you get to teach them?

>That was probably a longer yet less complete answer than you were
>looking for.  Nonetheless, I hope it helps.

It helps.  Thank you.  Sorry that this is so late.

Laura

>
>Jay