[^^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]- {} {Changing the Maths Cur.} *** {Connect a Million Minds .com} {} {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} {} {} {} {} {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 -[]- -[]- -[]- -[]- {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 -[]- -[]- -[]-## Links

-[]- -[]- -[]- -[]- -[]- -[]-## 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. -[]- -[]- -[]- -[]- {

## Methods

See also: -[Different Models]-## Changing the Maths Cur.

-[]- -[]- -[]- 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]- -[]-

## Methods

-[]- -[Programmed Learning]- (nobody here in this maze, but just us mice!)

## Links

In this section: {<><>} {Teaching Maths} {Mathematical Proofs} (and such)

## Teaching Maths

-[]- -[]- -[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 -[]- the sites www.coursecompass.com/ and MyMathLab.com seem pretty $-oriented "improve that grade!" -[]- -[]-

## Mathematical Proofs

(and such) -[]- -[Goldberger's paper on why proofs are necessary]-