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Topic: Mathematics education on the arXiv?
Replies: 2   Last Post: Jul 31, 2013 9:06 AM

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Clyde Greeno @ MALEI

Posts: 220
Registered: 9/13/10
Re: Mathematics education on the arXiv?
Posted: Jul 31, 2013 2:40 AM
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"The problem "math educators" face is entirely different: mathematical
thinking is about establishing the truth or falsehood of a statement."
# To me, that's a bit myopic. I'll take "mathematical thinking" to be about
the internal development of (what amounts to) a personal mathematical
"theory" ... perhaps as guided by teachings from others. The logic of
statements surely is of concern, especially at the higher levels of such
theorizing. But without the processes of theorizing, the statements have
little substance. Agreed: "... the goal of a math class is to get the
students to learn establishing whether or not things make sense and to
insist on things having to make sense. " And that means learning how to
theorize, well.

>>E,g, "identify at least 3 kinds of triangle and at least 3 *pertinent*
kinds of non-triangles"?
"This is exactly the kind of things that is killing mathematics in the mind
of people."

Once again, we are talking in two very different contexts.
True, as an instructional tool, a particular exercise of that kind has a
very limited purpose and, for students, may seem to be only a perfunctory
challenge.
But regarding tools for mathematical learning, the student (or teacher) who
cannot surface examples and contra-examples of a mathematical concept or
conclusion has little or no conceptual meaning for it ... and has not
internally re-abstracted in order to own it ... and is left with only
superficial formal knowledge, thereof.

For sure, the Mathematical Knowledge for Teachers' Education must include
knowing the mathematical roles of examples and counter-examples. [Just
think about the discussions that go on in graduate mathematics courses!]
Moreover, I am certain that good research can readily disclose that students
or teachers who master the art of examples/counter-examples benefit,
thereby, even among young children.

Cordially,
Clyde

- --------------------------------------------------
From: "Alain Schremmer" <schremmer.alain@gmail.com>
Sent: Tuesday, July 30, 2013 8:00 AM
To: <mathedcc@mathforum.org>
Subject: Re: Mathematics education on the arXiv?

>
> On Jul 30, 2013, at 2:22 AM, Clyde Greeno @ MALEI wrote:
>

>> #Children internally develop their personal "theories" by thinking about
>> their experiments with whatever things they are trying to manage ...
>> even if adults call it "playing." Adults do the same, whether or not
>> someone calls it "playing."

>
> play: engage in activity for enjoyment and recreation rather than a
> serious or practical [ODE]
>

>> "No. What is necessary is that the students be helped reading the
>> text---assuming of course that the book is honest."
>> # That depends on what you are trying to accomplish. If the goal is only
>> to equip non-readers with the mathematics of carpentry, reliance on a
>> textbook probably would get in the way.

>
> Would-be carpenters learn how to use, say, a framing square, nothing
> else. And they do not need to be cajoled into learning to do so. The
> problem "math educators" face is entirely different: mathematical
> thinking is about establishing the truth or falsehood of a statement. It
> is about not being defenseless when some so-called expert (usually
> self-appointed) says something directly relevant to your life which may
> be false. E.g. "the need for austerity". To quote Colin McGinn, "One of
> the central aims of education, as a preparation for political democracy,
> should be to enable people to get on better terms with reason?to learn to
> live with the truth." And so, at least as I see it, The actual contents
> do not matter all that much since it is only with differential equations
> that mathematics starts being applicable.
>

>> But if a concurrent goal is to prepare them for higher levels of
>> education, collateral instruction in reading is a worthy undertaking."

>
> No. "to prepare them for higher levels of education" is, in a way,
> teaching to the test.
>

>> "Yes, but arXiv is for research papers, that is assertions that are
>> either provable from previously established knowledge or reproducible
>> experimentally." [About teaching students how to learn mathematics.]
>> #It seems to me that there is much room for high quality research about
>> at least some "how to learn" tools ... such as "concept analysis:
>> identify at least 3 kinds of examples of that concept, and at least 3
>> *pertinent* kinds of counter examples.

>
> E,g, "identify at least 3 kinds of triangle and at least 3 *pertinent*
> kinds of non-triangles"? This is exactly the kind of things that is
> killing mathematics in the mind of people.
>

>> I suspect that "pencil in hand" includes some such particular tools.
>
> Maybe but what the standard meaning is that, as you are reading it, you
> write it yourself. I don't know what the brain mechanism is.
>

>> " No. Mathematical processes are *developed as needed*. " [About MKTE in
>> arXiv dealing with "mathematical processes"]
>> # Whoa. It does make sense for students to become consciously aware of
>> what mathematical processes are called for *at the time*. But within the
>> body of research in Mathematical Knowledge for Teachers' Education, all
>> of the important processes should be aired and explored ... albeit the
>> papers might be read only "as needed."

>
> Ok. I didn't express myself correctly. I meant that as one is facing a
> "problem", mathematical or not, one develops a way to deal with it. And
> thus, *coping* is what is important to learn. But it is a mindset.
>
> And, I almost forgot: the only point you did not respond to was:
>

>> (4) The only work I respect in adult education is Atherton, J. S. (1999)
>> Resistance to Learning [...] in Journal of Vocational Education and
>> Training Vol 51, No. 1, 1999 That's hard to find but he wrote
>> <http://www.doceo.co.uk/original/learnloss_2.htm>
>> on the subject.

>
>
> Now *there* is something worth putting in arXiv. As I wrote to Atherton,
> "believe it or not, it is only after forty years of teaching that,
> reading you, I realized the destructiveness of what I had been doing all
> these years. Now that, at least, I am aware of the danger, I try to dance
> around it."
>
> Best regards
> --schremmer
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