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Re: Mathematics education on the arXiv?
Posted:
Jul 29, 2013 1:13 AM


One crucial arena of mathed that sorely needs *mathematical* attention by competent mathematicians is perhaps best called, "Mathematical Knowledge for Teachers' Education." Perhaps it could be a subcategory. 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 fractions].
For a *mathematical* example: the commonly taught nonsense about the realdomain, real range, quadratic function, 3(x5)^2+7, having nonreal solutions is selfcontradictory (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" is   3(x5)^2+7, whose realnumber 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 algebrateachers) to know.
There probably are dozens of such MKTE insights that should be made accessible to all teachers of corecurricular mathematics, ASAP. Indeed, there is a nationally crucial need for mathematicians to mathematically reexamine the mathematical foundations of corecurricular 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 needed.
  From: "Alain Schremmer" <schremmer.alain@gmail.com> Sent: Sunday, July 28, 2013 7:21 PM To: <mathedcc@mathforum.org> 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.edph). 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 openminded 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 >> change.org. 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: >> >> http://www.change.org/petitions/arxivorgaddmathedmathematicseducationcategorytoarxiv >> >> 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: >> >> http://danaernst.com/mathematicseducationonthearxiv/ >> >> 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 > <http://www.doceo.co.uk/original/learnloss_1.htm> > 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 foror increasingly bythe 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 > > <http://www.freemathtexts.org/Standalones/RAF/index.php> > > 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 derivativesup 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 > > <http://www.ams.org/notices/201303/rnotip340.pdf> > > (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 > > > **************************************************************************** > * To post to the list: email mathedcc@mathforum.org * > * To unsubscribe, email the message "unsubscribe mathedcc" to > majordomo@mathforum.org * > * Archives at http://mathforum.org/kb/forum.jspa?forumID=184 * > **************************************************************************** **************************************************************************** * To post to the list: email mathedcc@mathforum.org * * To unsubscribe, email the message "unsubscribe mathedcc" to majordomo@mathforum.org * * Archives at http://mathforum.org/kb/forum.jspa?forumID=184 * ****************************************************************************



