[^^The Pond Normal School] (^to the home page)
See also: -[zix teaching methods]- (teaching folder)
-[Different Models]-
-[GEOMetry]-
-[Traditional Maths topics]-
Algebra, Geometry, Analysis (eg, Calculus, Number Theory, Topology)
-[nmaths: Approximately]- (1 = 0; approximately)
-[seol-three: web: Math [sic] wizards dot com]-
On this Page: {nmaths - Intro}
{Negative maths}
{Ways of Thinking}
{Methods}
See also: -[Different Models]-
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{Changing the Maths Cur.}
*** {Connect a Million Minds .com}
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{Links}
Nmaths: Intro
Negative Maths
Primary ref: is Martinez' book {links-down-here}
In this section: {}
{Taking for granted the way we apply the rules}
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{Refs}
{Links}
Taking for granted the way we apply the rules
THere's an old joke/riddle that runs like this:
A hunter travels one mile south, and
then one mile west and shoots a bear,
and then travels one mile north and
finds himself where he started out.
What colour is the bear?
White of course - his camp is at the North Pole; actually
there are infinitely many solutions to the problem at the South Pole
-- but that's another story. In his excellent intro to general
relativity, Malcolm Ludvigsen ([P. xi]) starts off with:
A trive living near the North Pole might well
consider the direction devined by the North Star
to be particularly sacred. It has the nice
geometrical property of being perpendicular
to the snow [on the ground], it forms the axis
of rotation for all of the other stars on the
celestial sphere, and it co-incides with the
direction in shich snowballs fall. howerer, we
know this is just because the N.P. is a very
special place. At all other points on the
surface of the Earth tyhis direction is special
-- it still forms the axis of the celestial
sphere -- but not that special. To the man
in the moon it is not special at all.
In fact it wouldn't be the axis of the rotation of the
delestial stars. Similarly as Martinez points out, how we make
so many tacit assumptions in even simple arithmetic; eg, from
Pp. 213-214, we have:
BEGIN BLOCK QUOTE - Martinez
...
But what about division? Division is not commutative:
a / b != b / a
[ in genearal; 1/1 is a special case as are 2/2 etc.
Iconospherically we would say:
MATH (in SCI) x ABS (rules of maths) --> ALT maths systems
See also {Negative Math, Note 1}
]
Note the tacit implications of the ORDER of the operations when we
"take" the numbers". 3 / 2 Means take 3 FIRST then take 2 and.
[Just taking 3 means trouble since so few numbers will divide it
evenly]
By lacking the commutative property, subtraction and diviion also
share a property that is easily illustrated in terms of
the physical considerations. Both of thewse operations pre-suppose that one
quantity is to be taken FIRST in time, and then
some alteration to be perofrmred on it. Eg, given 4 apples, divide them by 8.
It matters, for the result, what quantity we BEGIN with: Eight or Four.
We can, to be sure, also give examples in shich we formulate, say, multiplication
in termes of one quantity given first. Yet we can also formulate multiplication
as an operation concerning quantityes that are given simultaneously. For, eg:
<> <> <> <>
<> <> <> <>
<> <> <> <>
Here the three rows of unites and four columns of unites constitute the 12 units.
hence, we can write: 3x4 = 12
[but this follows the "RC" convention - "RC Cola" - rows then columns]
THe cols and rows, and actually all the quantities, appear simultaneoously.
Likewise, we can represent addition by means of a simultaneous arrangement of
units:
<> <> <> <> <>
such that: 2 + 3 = 5
Pp. 214,
END BLOCK QUOTE
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{Refs}
Refs
Ludvigsen, Malcolm (1999). General Relativity - A Geometric
Approach.
DD# 530.11 L9470
Martinez, Alberto, A. (2005). Negative Math - How Mathematical
Rules can be positively bent.
DD: 510.M385N
LCCN: QA 155.M28
ISBN 0.691.12309.8
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Links
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Notes - Negative Maths
(notes for this section only)
Negative Math, Note 1
Might we not ascertain a similar operation, w/clear phsical
meaning , that yet has the the commutative property? Possiby
some would commute, while others would not; ie,
I: There does not exist a "C" | a / b = c = a / b
for a,b elements if NC-sET1
II: Burt, there do exist some C(i) | A(i) / B(i) = C(i) = A(i) / B(i)
for A(i), B(i) elements of COMM-SET2
Thus (in therms of venn/set diagrams)
/ gives
a --> b ------> c
b --> a ------> d c != d
Or in terms of operations tables:
a / b | r s t u
--------------------------
m | z w x
n |
p | etc
q |
b / a | m n p q
------------------
r | z v y
s |
t |
u |
Note that above, only when a=m, and b=r does a/b = b/a = z
otherwise a/b != b/a
{Jump back to text, above}
Ways of Thinking
Associative thinking, and analogical thinking.
In the 2-dimensional Euclidean plane we can find
the intersection of two lines using the WAY of
"two equations in two unknowns"
Similarly, we can take a differential equation
and restrain it via a set of constraints and
get a certain solution; a diff set of constr's
gives a diff (or no) soln.
What happens when fractals collide?
That is, fractals as graphs of an equation.
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Methods
See also: -[Different Models]-
Changing the Maths Cur.
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http://www.google.com/search?hl=en&q=%22changing+the+math+curriculum%22
{
Connect a Million Minds .com
-[www: connect a millions minds.com]-
MTGK Institute
www.mtgk.com
5022 Tennyson Parkway
Plano, TX 75024-3151
(972) 473-8377
Ways of Thinking
-[Change of Context]- (c of c)
-[Hanover]-
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Methods
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-[Programmed Learning]- (nobody here in this maze, but just us mice!)
Links
In this section: {<><>}
{Teaching Maths}
{Mathematical Proofs} (and such)
Teaching Maths
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-[www: TheMathLab . com]-
Interesting site with lots of "get invovle"
activities for maths
In the "teachers only", some good guidances
Games
mini-lectures (five mins each of: Talk, Try the idea, Report!)
boardwork
groupwork
individual work
projects
videos
writing assignments
discovery lessons
computer practice
Internet research
spreadsheet explorations
humorous stories
lively historical anecdotes and facts
one on one peer tutoring
experiments
timed drills
self checking worksheets with answer banks
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the sites www.coursecompass.com/ and MyMathLab.com
seem pretty $-oriented "improve that grade!"
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Mathematical Proofs
(and such)
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-[Goldberger's paper on why proofs are necessary]-