[Edu-sig] Python and pre-algebra
mokurai at earthtreasury.org
mokurai at earthtreasury.org
Sat Jul 2 20:38:04 CEST 2011
On Fri, July 1, 2011 2:06 pm, kirby urner wrote:
> A weakness in contemporary geometry is this belief that it's best to start
> in the abstract / ethereal realm of the infinite this and the
> dimensionless that.
The real origin of geometry was the necessity of resurveying all
agricultural land in Egypt after the annual Nile floods.
> Anti-experiential. Wow 'em with less than self evident axioms and
> definitions and pretend that's elite and superior (a kind of snobbism).
See Thorstein Veblen, The Theory of the Leisure Class.
> Regular old objects of everyday space are more polyhedrons than
> polygons. The Earth is an oblate sphere.
Oblate spheroid, approximately. The Earth-Moon orbital equations currently
have several hundred terms, to do with their figures and their mass
distributions. Many of those terms have been worked out using the
retroreflector placed on the moon by astronauts, allowing us to measure
its distance with an accuracy of a few inches.
Kirby likes to focus on certain aspects of polyhedra, like his Martian
Math. But polyhedra are endlessly fascinating in a multitude of ways, in
crystallography, in topology (the Euler formula V-F+E), in
error-correcting codes, in the proof of Fermat's Last Theorem (the
relationship between elliptic functions and modular forms), and many other
areas. The Bit in the original Tron movie had two forms, both stellated
polyhedra (and both rendered for the movie in APL by Judson Rosebush).
Then there are the much wider range of non-Euclidean polyhedra and the
higher-dimensional polyhedragons, starting with the well-known tesseract.
> Keeping geodesy in the picture, ala Google Earth, Google Mars etc.,
> ... so easy to pull ahead of the rank and file. One needs to be free
> of district strictures though, straitjacketing "standards".
and the tyranny of the Right Answer, which prevented the recognition of
negative numbers, non-Euclidean geometry, and complex numbers for
centuries in each case, and is now holding back non-standard analysis,
which allows us to use infinitesimals correctly in calculus.
H. Jerome Keisler, Elementary Calculus: An Infinitesimal Approach
> At Saturday
> Academy, we are not bound and gagged the way they do in some
> other schools I won't name in this post.
>> Ratios and Proportions
>> Standard Math (3 sections) Supported by McDougal-Littel Course 1
>> Problem Solving Strategies
>> Fraction operations
>> Decimal operations
> Maybe get some polyhedrons in here.
> I was just visiting with Father Magnus Wenninger in Minnesota.
> He's one of the premier polyhedronists.
> I met a young guy recently (he was waiting our table) who
> saw the polyhedrons we had (lunch meeting) and correctly
> named them. I was amazed and asked him which school
> system he'd attended. Minneapolis Public Schools.
>> Number Sense (Prime factorization, GCF, LCM, Divisibility Rules)
>> Our district is generally supportive to adding new software to the
>> computers, however requests are only honored during school breaks
>> spring, summer) as they want to keep the computers available for student
>> and MAPS testing. (http://www.nwea.org/) :-(
> Typically, they'll teach GCF using prime factorizations and bleep
> over Euclid's Algorithm. That's a fork in the road. What I call
> "digital math" includes Euclid's. Here's Guido's version:
> def gcf(a, b):
> while b:
> a, b = b, a % b # modulo arithmetic
> return a
This is a variation of the general pattern for recurrence relationships
def fn(a, b):
a, b = b, f(a, b)
which actually works better if written as a Generator.
If f is + and a, b = 0, 1 we get Fibonacci numbers.
> Milo on mathfuture thinks number theory was expunged from
> Lower48 curricula during the anti-German backlash of
> Woodrow Wilson and WW1. Planar Euclidean geometry
> became the new pavement. More like the Russian curriculum
> in some ways. We could use a lot more ethnography of math
> education. Many full time anthropologists should be tasked
> to this important work, observing and reporting.
> One thing you can exhibit using the Python window (one of many)
> is this idea of types. We all know that objects (like dogs and shoes)
> come in types. "What type of thing is that?" So then in Python we
> have this "type" function that spits back the type of a thing.
> <class 'int'>
> <class 'float'>
>>>> import decimal
> <class 'decimal.Decimal'>
> <class 'set'>
Understanding types makes it possible to form a correct notion of adding
apples and oranges. Ordinary addition doesn't work on most combinations of
types, but one can define addition and the other basic arithmetic
functions for anything. As I said the other day, this is called a graded
ring of Laurent polynomials, which is a fancy way of saying "algebraic
expressions" like 2/x + 3y.
It is important to understand that there is not just one kind of addition,
but a vast multitude of kinds, operating under different algebraic rules
and forming a multitude of categories of algebraic structures with all
sorts of mappings between them. Shopping cart arithmetic is but one of
them, and in shopping cart arithmetic, you *can* add apples and oranges.
> This is rather generic language, almost like basic English (which
> it is, translates to other languages pretty easily). Weaving together
> an "object oriented" patter with everyday ordinary speaking is a
> goal of my math classes. It's a language class. Nomenclature
> matters. Dot notation: noun.verb( ). results = thing.action( inputs ).
> noun.adjective. More grammar.
> Lights go on when students realize how much computers deal with
> alphanumeric data, not just numbers. There's this stereotype from
> the outside that it's all "number crunching" meaning glorified
> arithmetic. It's as much about text, about parsing, about markup.
> I like to dive in with some ideas about tcp/ip and shared infrastructure.
> To this end, I project 'Warriors of the Net', admittedly pretty basic:
>> My plan is to begin with my advanced math students.
>> On a side note, I have enjoyed reading the personal stories you have
>> sharing. Mine is that my first job out of college was working for the
>> defunct Teletype Corporation, a part of the now defunct Western
>> Electric, a
>> part of the now defunct Bell System, a part of the perhaps soon to be
>> defunct AT&T?! I spent ten years in public relations, took time off to
>> children, then returned to the workforce to teach middle school.
>> Again, I appreciate your support, and I look forward to collaborating
> The Baby Bells are striving to get back together they say.
Yes, one of them bought AT&T and assumed its name, and I get TV and
Internet from them.
> like one of those summer science fiction movies where the alien
> Globs are seeking to rejoin and form the Mother Glob.
> Anyway, I should get back to the day job (teaching Python as it happens).
> Great chatting,
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