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What Is Mathematical Thinking?

Date: 03/08/2005 at 09:57:40
From: Penny
Subject: Can all thinking be considered mathematical

Can all thinking be considered mathematical?  For example, I have
found that there is no such thing as mathematical thinking because you
are essentially just thinking about mathematics, not the entity of
mathematical thinking--as this does not exist.  That being said, can
we reverse the tables and say that all thinking is mathematical?

I'm trying to get my head around the entity of mathematical thinking
not really meaning anything but thinking about mathematics.  There is
no such thing as thinking mathematically.

As my opening title affirms, this question explores what it means to
think mathematically.  In addition, these questions will ask the
question if you are unsure of what mathematical thinking is, how will 
you ensure that your students are meeting the goals of the mathematics 
classroom (i.e. engaging in rich mathematical thinking)?   Below, I 
have attempted to explain the position that Andrea Pitt, the author of 
‘Mathematical Thinking?’ has taken while also researching that of the 
National Council of Teachers of Mathematics (NCTM) on their respective 
views of this matter.  When reading Pitt’s article it was clearly 
evident that she feels there is no such thing as mathematical 
thinking.  In fact, she believes that people use the entity too 
loosely, when what they truly mean is that they are thinking about 
mathematics and not thinking mathematically, because this does not 
exist.  (We will explore how she is able to think about mathematics 
with her students near the end of this paper as we first need to 
understand the concept of what mathematical thinking is truly about).

This had me wondering what I believe, and whether the NCTM feels the
same way as Pitt.  After all, is this really a fictitious statement?
Is there truly no such thing as mathematical thinking?  I turned to 
the NCTM documents to help justify Pitt’s allegations.  When searching 
for a definition of mathematical thinking from NCTM, I found 
inconclusive, indirect statements of what it means.  Nowhere was I 
able to find a true definition of what the NCTM believes that 
mathematical thinking means.

Based on Principles and Standards for School Mathematics, published by
the NCTM, mathematical thinking involves connections in order to
construct some sort of mathematical understanding.  Learning to think
mathematically comes about when you have a mathematical standpoint and
apply it to a given situation.  It also means developing mathematical
capabilities using the tools necessary in the mathematics classroom.
They also state that problem solving requires thinking about
mathematics as you need basic knowledge of mathematics to solve
problems.

From the NCTM findings they use words such as reasoning, problem
solving, communicating, and connections when referring to mathematical
thinking.  These words are all encompassing proving the definition to
be very omnipresent.  These words do not signify a true meaning of
mathematical thinking, only what should be incorporated in that line
of thought.  To go along with Pitt’s view, the NCTM’s outlook of
mathematical thinking is so broad that the definition could be applied
to all other subjects in school and in life in general. For instance,
Pitt states in her article that some mathematicians, such as Dick
Tahta, define mathematical teaching as "...going beyond the
information given or trying to find a pattern..." but can you not
apply this statement to physics, or chemistry, or any other subject
matter for that matter?

The NCTM derived a list of methods for encouraging mathematical
thinking through such techniques as asking non-leading questions,
paraphrasing student’s answers to aid him/her at thinking about what
was just said, using wait time to allow sufficient time for the
students to think through and explain their reasoning.  The NCTM
believes that these procedures can help students to think
mathematically.  In comparison, Pitt alludes that these procedures are
methods of thinking about mathematics, but that does not necessarily
mean that you are thinking mathematically.

With that being said, Pitt has come to the conclusion that there is no
such thing as thinking mathematically and encourages her students to
think about mathematics in her classroom through such techniques as
those the NCTM has listed in the above paragraph.  In particular, the
NCTM believes in the use of reflective questions as these questions
can help pin point areas of the lesson that are important.  Using
these questions forces the students to explain math, not to just
answer the problems at hand without thinking about them.  In this way,
it is encouraging all students, at any level, to think about the
mathematics, which is the aim of Pitt’s classroom.  As a result,
effective discourse is the basis for promoting students to think about
mathematics.

One concluding statement that Pitt has disagreed with is that we could
say all thinking is mathematical; instead of the reverse, having no
such thing as mathematical thinking.  She believes that mathematical
thinking is the same sort of thinking for all subject areas and that
it is too broad to be solely included in the mathematics domain, but
she does not believe that all thinking is mathematical.  I am not sure
if I believe this or not...



Date: 03/08/2005 at 14:00:06
From: Doctor Edwin
Subject: Re: Can all thinking be considered mathematical

Hi, Penny.

Nice question. You've obviously been thinking about it (mathematically
or otherwise) for some time.

I will throw in something to suggest that there is in fact something
you might call mathematical thinking--a well-known example from
cognitive science.

If I say to you, "All Kpelle men are rice farmers.  Steve is not a
rice farmer.  Is Steve Kpelle?" You will instantly answer, "No,
because otherwise he would be a rice farmer."

But when the question was asked of Kpelle rice farmers, a variety of
answers came out, mostly along the lines of "How should I know?  I've
never met him."

Some answers were more interesting, like, "If you have never even
heard of someone, you can't answer questions about what they do for a
living," or, "With a name like Steve, probably not."

This phenomenon is not confined to the Kpelle.  The experiment has
been repeated in many places where schooling is rare.  Check out this
answer:

"Everyone in the village who owns a house pays a house tax.  Alfonse
doesn't pay a house tax.  Does Alfonse own a house?"

"I would say that if he can't afford to pay the house tax, he probably
can't afford to own a house."

Some people were able to answer these questions the way we expect, and
some were not.  As I alluded to above, the difference was schooling.
As little as three months (!) of schooling would grant the ability to
answer these kinds of questions correctly.

I think that the essence of mathematical thinking (okay, maybe just AN
essence) is the ability to lift the abstract structure of a situation
away from the specifics and answer it based on that alone.  I don't
have to figure out two people times three meals, because I already
know what two times three is.  There is a formal reasoning (that is,
reasoning about the form) that is apparently inculcated in schools,
among other places.

What were they being taught during those three months?  Certainly not
logic puzzles, syllogisms, or what have you.  Probably some
mathematics, probably some writing.  Whatever it was, the teachers
were certainly teaching something else in addition to whatever they
thought they were teaching.  I would suggest that it was a type of
mathematical reasoning.

It may be hard for us to see it as separate from other types of
reasoning.  I am reminded of the saying, "I don't know who discovered
water, but I'm sure it wasn't a fish."  If mathematical reasoning is
everywhere we look, and especially if it's part of what we look WITH,
it may be hard to notice.

I'm going to leave this question in the "incoming questions" bin, in
the hope that other doctors will have more to add.

- Doctor Edwin, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 03/08/2005 at 14:23:32
From: Doctor Ian
Subject: Re: Can all thinking be considered mathematical

Hi Penny,

>For example, I have found that there is no such thing as
>mathematical thinking because you are essentially just thinking
>about mathematics, not the entity of mathematical thinking--as this
>does not exist.

I would disagree.  I think there is very much a kind of thinking--
which can be used to think _about_ anything at all--that is
"mathematical" in nature.  And it differs radically from other kinds
of thinking.

>That being said, can we reverse the tables and say
>that all thinking is mathematical?

The world would be a very different place if that were true.

>Trying to get my head around the entity of mathematical thinking not
>really meaning anything but thinking about mathematics.  There is no
>such thing as thinking mathematically.

See

    http://mathforum.org/library/drmath/view/52370.html 

>When reading Pitt’s article it was clearly evident that she feels
>there is no such thing as mathematical thinking.

Maybe she's just never experienced it.  I might make the same kind of
claim about religious ecstasy, because I've never experienced that.
But a lot of others, who _have_ experienced it, would disagree.

>When searching for a definition of mathematical thinking from NCTM,
>I found inconclusive, indirect statements of what ‘it’ means.  No
>where was I able to find a true definition of what the NCTM believes
>that mathematical thinking means.

Perhaps their purpose for using the term is something other than
providing a clear definition.  In fact, perhaps providing a clear
definition would actually run _counter_ to their purpose.

>Based on Principles and Standards for School Mathematics, published
>by the NCTM, mathematical thinking involves connections in order to
>construct some sort of mathematical understanding.  Learning to
>think mathematically comes about when you have a mathematical
>standpoint and apply it to a given situation.  It also means
>developing mathematical capabilities using the tools necessary in
>the mathematics classroom.  They also state that problem solving
>requires thinking about mathematics as you need basic knowledge of
>mathematics to solve problems.

That sounds pretty muddled, doesn't it?  It also sounds untrue.
People solve problems all the time without using mathematics.  ("The
Smiths are coming over for dinner tomorrow.  What should we serve?
Mrs. Smith is allergic to seafood, and last time they were here,
Johnny wouldn't touch the chicken we served...")

Having said that, given any particular problem, there is definitely a
mathematical way to approach it, which doesn't necessarily mean
reducing it to sets of equations, or even using what we normally think
of as "mathematics" to solve it.  The mathematical approach, as
illustrated in the joke I cited earlier, is to reduce the problem
before you to one or more simpler problems, and to keep doing that,
over and over, until you end up with a set of trivial problems, i.e.,
problem whose answers you already know because you've already solved
them in the past.

Contrast this with the many other ways that you might go about trying
to solve a problem, e.g., looking up the answer, asking someone else
to solve it for you, trial and error, guessing, prayer, solving a
different problem instead, and so on.  These are _not_ mathematical
ways of approaching the problem.

We joke about this all the time in my family, because I'm usually
looking for the "mathematician's solution" in any given situation.
For example, say we need to go pick up the dry cleaning, and we also
need to go drop off a movie that we rented.  My wife would get out a
map and try to find a route _from_ the dry cleaner's _to_ the movie
place; but I would drive to the first one, then drive back home,
because now the second problem (getting to the movie place) is one
that I already know how to solve.  It's more wasteful in terms of time
and fuel; but it's more economical in terms of allowing me to use my
limited cognitive resources for things I find more interesting than
figuring out how to get from point B to point C.  And it is, I
believe, a distinct way of looking at the world.

There's another very nice story about Enrico Fermi, which I think
illustrates the difference between mathematical thinking and other
kinds of thinking.  One evening Fermi was watching his wife knit, and
suddenly he got up and went to his study.  An hour later, he came back
and announced that, topologically, there were only two ways to tie the
kinds of knots she was doing, and showed her the other would work.  "I
know, dear," she said.  "The other one is called perling."

Now, what makes his approach mathematical isn't that he _used_
"mathematics", i.e., topology, to investigate the problem.  What made
his approach mathematical is that it occurred to him in the first
place that there might be some way to reduce what his wife was doing
to some axiomatic description, and then use that description to
consider _all_ the ways that it might be done, all at once, instead of
one at a time.

_That_ is mathematical thinking, and it's not really very much like
any other kind of thinking at all:

    http://mathforum.org/library/drmath/view/52350.html 

So to sum up, the way it looks to me is that "mathematical thinking"
is primarily a very sophisticated kind of laziness, which can be
learned.  When confronted with a problem, the mathematician tries to
avoid having to do any more work than absolutely necessary by going in
one of two directions: down (breaking the problem into smaller
problems that have already been solved), or up (abstracting the
problem so that he can solve not just the one in front of him, but a
whole bunch of others that might arise in the future).

Note that to learn this kind of thinking, you don't necessarily have
to work with what we commonly think of as "mathematics".  But it's a
lot easier, because mathematics provides an imaginary world in which
it's possible to control, with great precision, the amount of
complexity that a particular problem involves.  By abstracting out the
millions of details that arise in any _real_ situation, we can give
students the luxury of focusing on the task at hand, i.e., learning to
transform problems into other problems, and to look for ways to reduce
sets of individual rules or formulas to more general ones, e.g.,

    http://mathforum.org/library/drmath/view/54685.html 

Now, whether this kind of thinking is being taught in classrooms is
another story.  I think it's not, largely because nearly every math
teacher I've ever explained this view to has responded in the same
way: first, by saying "That makes perfect sense"; and second, by
asking "Why didn't anyone ever explain that to me before?"

I hope this helps.  Write back if you'd like to talk more about this,
or anything else.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
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