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Topic: Mathematics education on the arXiv?
Replies: 2   Last Post: Jul 29, 2013 10:57 AM

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Clyde Greeno @ MALEI

Posts: 220
Registered: 9/13/10
Re: Mathematics education on the arXiv?
Posted: Jul 29, 2013 1:13 AM
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One crucial arena of math-ed that sorely needs *mathematical* attention by
competent mathematicians is perhaps best called, "Mathematical Knowledge for
Teachers' Education."
Perhaps it could be a sub-category. I am speaking about knowledge within
mathematical theories ... rather than mathematicians thoughts about
what/how mathematics should be taught [e.g. Wu's opinions about teaching

For a *mathematical* example:
the commonly taught non-sense about the real-domain, real range, quadratic
3(x-5)^2+7, having non-real solutions is self-contradictory (and
thoughtful students are troubled by it). The mistakenly asserted "complex
roots" are 5 [+/-]sqrt[-7/3], which makes no sense for real valued
functions of real variables.
BUT, when that parabola is vertically flipped over its vertex, its "image"
- - 3(x-5)^2+7, whose real-number solutions are 5 [+/-]sqrt[7/3] ... as
"image roots" for the original
Its not a new mathematical discovery, but it is important for teachers of
algebra (and of algebra-teachers) to know.

There probably are dozens of such MKTE insights that should be made
accessible to all teachers of core-curricular mathematics, ASAP. Indeed,
there is a nationally crucial need for mathematicians to mathematically
re-examine the mathematical foundations of core-curricular mathematics ...
and to collectively assemble a mathematically solid body of MKTE [e.g. math
majors who teach school mathematics should know the logic of converting
between decimals and percents].

I am personally ready to publish several such *mathematical* papers about
little known MKTE essentials, and to nurture collaborative development of a
library of such items. Some exemplary papers are available if they are

- --------------------------------------------------
From: "Alain Schremmer" <>
Sent: Sunday, July 28, 2013 7:21 PM
To: <>
Subject: Re: Mathematics education on the arXiv?

> On Jul 27, 2013, at 5:21 PM, Dana Ernst wrote:

>> Greetings! My name is Dana Ernst and I am an assistant professor at
>> Northern Arizona University. I am a mathematician that dabbles in math
>> ed.
>> The virtues of the arXiv are well known. Yet, there is currently no
>> dedicated category on the arXiv for mathematics education research. The
>> math.HO ? History and Overview category lists mathematics education as
>> one of the possible topics, but it doesn?t appear to be commonly used
>> for this purpose. In contrast, there is an active physics education
>> category (physics.ed-ph). Unfortunately, at this time, there is not a
>> culture among math ed folks to utilize pre- print servers like the arXiv.
>> However, if there is going to be a cultural shift, there needs to be a
>> dedicated repository for math ed papers. Authors need to know where to
>> submit papers and readers need to know where to look. A category called
>> History and Overview doesn?t cut it. A precedent has been set by the
>> physics education crew and we should follow in their footsteps. It is
>> also worth mentioning that Mathematics Education is listed as one of the
>> American Mathematical Society?s subject classification codes (n!
>> umber 97).
>> I've contacted the arXiv and they are open-minded to adding math.ED -
>> Mathematics Education as a category. However, they will seriously
>> consider it, they want to know that there is support from the community
>> and that it will get used. As a result, I have created a petition on
>> If you are in favor of the arXiv including math.ED ?
>> Mathematics Education as a category, please sign the petition. If you
>> would also utilize this category by uploading articles related to
>> mathematic education, please leave a comment (on the petition)
>> indicating that this is the case. You can find the petition here:
>> The arXiv mentioned support by at least 50 people, but I'm shooting for
>> 100, so if you are in favor, please take a minute to sign the petition.
>> If you are curious or want to know more, check out the short blog post
>> that I recently wrote:
>> I'd love to hear what y'all think about this. Feel free to comment on
>> the blog or reply to this email. I'm especially interested in hearing
>> from people that are willing to help out.
>> People that have responded to me via email, Twitter, Google+, and my
>> blog post have been pretty supportive, so I'm hoping to see this come to
>> fruition. It is worth noting that 3 people so far have expressed their
>> support but also their skepticism that math ed folks will go along with
>> the idea. In short, all three people have said something like, "math ed
>> researches have been shunned too many times by mathematicians, and as a
>> result are protective of their territory." Maybe this is true, but I'm
>> not okay with it. Let's close the divide. As a mathematician that
>> dabbles in math ed, I feel pretty passionate about this.

> (1) arXiv is indeed a very nice "container".
> (2) Unfortunately, there is just about nothing worth placing therein:
> "math ed" is no more a science than, say, "economy". There are just
> "Educologists" who want you to believe that they have just discovered the
> ultimate "sugarcoating" to make "math" palatable to those, let us tacitly
> agree, mostly mindless students.
> (3) The only work in mathematics education I respect is that of Z. P.
> Dienes but it pertains only to children.
> (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
> <>
> on the subject.
> (5) "Mathematics is not necessarily simple" (Gödel Incompleteness
> Theorem, algebraic statement by Halmos) but the only known way for adults
> to learn mathematics is by gaining "mathematical maturity" by
> experiencing the "compressibility of mathematics" by "reading pencil in
> hand". Hence the importance of the text. However, textbooks have very
> rarely been good at presenting even those parts that are simple. (One,
> rare, exception being Fraleigh's A First Course in Abstract Algebra.) Of
> course, books used to be written for the colleagues whether because they
> may review the book and/or because they may let their students buy it.
> (Now the books are written for---or increasingly by---the editor and are
> at an all time low.) Yet
> (6) The difficulty in learning mathematics resides only in:
> -- the degree of abstraction of what I am considering (= how far removed
> is it from the real world, e.g. when I am counting marbles, I ignore
> their colors while when I am operating in a group of moves, I am ignoring
> a lot more than the nature of the universe in which I am making these
> moves.)
> -- the degree to which the information is concentrated in the language or
> even left as "going without saying".
> Both can be dealt with in an honest text via a Model Theoretic setting.
> The idea of level and the concomitant idea of prerequisite are artificial
> constructs that can usually be easily dispensed with.
> For a specific example of what I mean, consider my
> <>
> While not particularly well written, it is essentially without
> prerequisite and, In fact, it is really a text on differential calculus:
> Given
> f(x_0+h) = A(x_0) +B(x_0)h +C(x_0)h^2 +D(x_0)h^3 + [...] (an
> extrapolation of decimal approximation)
> we need only give a name to the functions
> x ------> B(x),
> x ------> C(x),
> ...
> to have the derivatives---up to the factorial needed to make things
> recursive.
> Thus, all that is needed to learn, say, the differential calculus beyond
> a good text, is only a willingness to:
> -- stop and consider,
> -- insist on things making sense.
> So, an efficient scenario is to let the students unable to jump to this
> or that level go through a sequence such as described in
> <>
> (7) Last, but not least, while we are always ready to brag about the
> latest little "educational" gimmick we are using, we are quite unwilling
> to depart significantly from the "true and tried"---rather understandably
> if you are not tenured or need a promotion. (How many people do you know
> who are not lecturing?)
> Regards
> --schremmer
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