Re: [Phys-L] [Norton AntiSpam]Re: The great barrrier (was Proportional reasoning)

["John Clement" <clement@hal-pc.org>]

I really doubt that graphing packages have an effect here because students
often do not use them.  They do use graphing on the calculator because that
is used in math.  But students have a big problem with graphing.  They don't
often don't understand that making an uneven scale is something that you can
not do if are analyzing data.  I have seen all kinds of funny scales.
Indeed some students line up the points at a 45 degree angle on the grid
lines, and then put the scale numbers according to the points.

So I never allow them to use graphing packages until they have all
demonstrated prowess with manual graphing.  Of course they are given
graphing instructions.  This is a sign of innumeracy which is actually much
more widespread than you think.

A large part of the problem with algebra is that many students have not
gained the thinking skills needed before they had algebra.  It is assumed by
mathemeticians that students can easily learn algebra in 9th grade, but that
is not true for the 75% who do not have the necessary advanced thinking.

The big problem with education is that the standards for what should be
learned are often chosen by people who have no clue as to what students are
actually ready to learn.  When you acquire higher level thinking, you do not
remember what it was like to think at a lower level.  This happens all the
way along.  When you acquire language many of the prelanguage memories are
lost.  When you acquire the ability to think symbolically you don't remember
what it was like to not think symbolically.  As you acquire understanding
you put things into a logical order.  So you think that the order you have
created is the correct order to teach things.  But this is FALSE.  The
correct order may be very different.  The higher order thinker doesn't
understand that they learned many things which were not noted on the way to
becoming a higher order thinker.  They also do not really know how they
acquired the higher order thinking.

An example of this sort of thing is the acquiring of the understanding of a
zig-zag path being longer than the straignt one between 2 points.  Children
learn this automatically, but nobody remembers how they learned it.
Experiments show that there is an age below which this can not be learned.
And experiments have shown that this can not be taught didactically.  But
there is an age range in which it can be taught kinesthetically.  If a child
hops both paths or 2 children simulataneously walk both paths they do learn
it.  But in this age range telling does not teach, and walking both paths is
insufficient.

By the time most students get to our classes we observe symptoms of
problems, but the underlying causes are often not evident.  Math teachers
always tell me they can't go back and treat the lack or proportional
reasoning because it was already taught, and they have other curriculum to
push.  In other words they will noodle the students with useless things they
are incapable of using or understanding.  We now know how to treat dyslexia,
but for good treatement it needs to be done before age 7 because the higher
plasticity allows activation of the portions of the braing that are inactive
in dyslexia.  The logical course is to screen all children and give the ones
with dyslexia Orton-Gillingham treatement at the necessary early age.  That
costs money, which the better off parents spend, but the poor parents can
not afford.  But having people who are functionally illiterate is a big drag
on our economy, so the schools should be given public money to do this.
Vaccination has wiped out most deadly childhood diseases, so testing and
treating could wipe out dyslexia as a reading problem in the US.  It will
not necessarily treat other problems with dyslexia.

It is easy to say the students have an aversion to using letters, but I
propose that it is a lack of understanding of the use of symbols in math and
science.  If you gave the Lawson test you might see that those students
would score low.  Treating the symptoms would then be useless.  Instead you
have to get their thinking up, which will not be done by simple didactic
instruction.  Or maybe they have low working memory.  It turns out that
there is now a computer program which improves working memory and the
improvement seems to be permanent.  I personally think that introducing
algebra a X marks the spot is a BIG mistake.  It ought to be introduced
first by bringing in the idea of variables without using symbols.  Then
allow students to use whole names instead of single letters.  Finally use
single letters as shortcuts in writing.  Then always use a variety of
letters.  But along the way elementary teachers need to have their
curriculum materials changed a bit.  So when the problem is adding 5 apples
to 6 apples the box for the answer should be number=__________ apples.  The
current method of apples=_____________  makes students think of apples as a
variable when in reality it is a unit and number is the actualy variable.

So when you see these symptoms, perhaps a lot of digging is needed to figure
out why these symptoms are there.  My example of the 45 degree graph is
partially conditioned by the texts where all graphs are at 45 degrees with
evenly spaced points on the grid lines.  Such graphs always go through zero.
So students try to make all graphs go through zero.  But those texts are
just reinforcing other misconceptions and are not necessarily the main root
cause.  The Minds on Physics authors are acutely aware of many of these
texbook problems and they have students do things to make them more
flexible.  For example they have them read a map which is printed tilted at
20 deg on the page.  They are carful to put points in between grid lines and
to never use unitary spacing for the grid blocks.  It is amazing how many
students count the blocks to get an area on a graph and never look at the
scales.  You can learn a lot about basic hidden student problems by using a
few of the MOP activities.

John M. Clement
Houston, TX

> I'm not sure how this fits in, but is clear to me that for at 
> least some students who rebel at using letters (and it may be 
> just an aversion to letters for these students) that they are 
> treating number symbolically; e.g. ultimately canceling it 
> out when appropriate and never performing actulal arithmetic 
> with the numbers (like adding 2 + 3 to get 5, they will 
> cancel in the sense of (2/2 = 1) or 3+(-3)=0, operations 
> you'd be able to do with symbols; but for some unknown reason 
> to me, putting in a symbol that looks like an arabic numeral 
> gets them past a barrier that using a latin letter fails to do.
>
> I'd comment that using many graphing packages do not help as 
> you often have to explicitly put in a number for a parameter 
> in the formula you are plotting in order to get the 
> computer/device to actually make the plot.  I wonder if this 
> is affecting the situation being discussed here to the 
> detriment of flexibility in thinking for our students.