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http://www.math-atlas.org/welcome.html

an introduction to the areas of modern mathematics- Last modified 2004/01/02 by Dave Rusin

http://mathworld.wolfram.com/
MathWorld is the web's most extensive mathematical resource

http://demonstrations.wolfram.com/
The Wolfram Demonstrations Project

http://demonstrations.wolfram.com
/topics.html
School mathematics is one of the topics here.

http://demonstrations.wolfram.com/
TheVibratingString/

Generally too difficult for me but I often check it to see how much I don't know about a subject.

The Demonstrations Project is fairly new and growing...I have no room to download and verify the free material myself. The Flash web samples seem good, if at times complex, and are useful themselves.

http://planetmath.org/

"PlanetMath is a virtual community which aims to help make mathematical knowledge more accessible."

http://web01.shu.edu
/projects/reals/reals.html

Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus in one real variable - Bert G. Wachsmuth

http://www.research.att.com/
~njas/sequences/

The On-Line Encyclopedia of Integer Sequences

Enter a sequence, word, or sequence number:

http://www.intmath.com
/Help/site_info.php

I developed this site when I was a lecturer in mathematics at Ngee Ann Polytechnic,
Murray Bourne, Singapore, 2008.

comments
MERLOT: This is a well-designed, mathematically accurate, technologically dependable demonstration/tutorial site that should be useful as a complement to traditional courses in college algebra, pre-calculus, and first-year calculus

Excellent! -This will go under the calculus section as well.

For my immediate purposes

http://www.intmath.com
/Fourier-series/Fourier-intro.php

For the main index- quick overview of content

http://www.intmath.com
/Help/sitemap.php

http://www.themathpage.com
/alg/algebra.htm

The Math Page - Lawrence Spector

http://argyll.epsb.ca/jreed/

Jim Reed
... I have been developing Flash resources for several years... I have taught in Alberta for 32 years. In the past, I was a grade 9 math and science teacher. I am currently teaching high school math.

http://www.purplemath.com
/modules/index.htm

Purple Math - Elizabeth Stapel

1.Preliminary Material
2.Beginning
3.Intermediate
4.Advanced
5.Word Problems
6.Appendix

Excellent, easy navigation - except that the topics are not alphebetized, (August 6, 2008), static text and illustrations, quite thorough.

I have had some difficulty in the past because of the length of some of the modules - I'm generally looking for "Quick Answers".

However, "When the pupil is ready, the teacher will appear." At almost every stage I find a need for a better understanding of some basics such as is provided here. The site will no doubt improve study habits and concentration.

http://www.algebralab.org/

2003-2008
Mainland High School

http://mathforum.org/

http://mathforum.org
/algebra/alg.lessons.htm

suggest homepage>search box

static text answers to specific questions e-mailed in over at least a 10 year period - sometimes a bit of work to find the exact question I'm looking for... keyboarding of math statements (apparently quite necessary in today's world) has been helped by reading this material...

There are navigation categories of difficulty and education level.

http://www.mathsnet.net
/algebra/index.html

"... truly interactive resources that are both wide and deep"

comment - Javascript interactive - Problems and Solutions only... quickly shows steps and answers

click on Answer button for steps...till it says REPEAT - here, this means QED (the final answer - "quad erat demonstranum", Latin - "which was to be demonstrated")

http://www.jamesbrennan.org
/algebra/

... introductory algebra - or refresher

http://www.algebasics.com
/index.html

reviewersays
...click on an equation to get a visual and auditory explanation ... well designed... uses Flash technology...easily navigated.

http://id.mind.net
/~zona/ezGraph/ezGraph.html

Graphics calculator you should be able to graph almost any polynomial, rational, exponential, logarithmic, or trigonometric function. It will allow you to enter variables into your function definition so that you can see the effect of changing coefficients easily

http://www.intmath.com
/Help/site_info.php

I developed this site when I was a lecturer in mathematics at Ngee Ann Polytechnic,
Murray Bourne, Singapore, 2008.

comments
MERLOT: This is a well-designed, mathematically accurate, technologically dependable demonstration/tutorial site that should be useful as a complement to traditional courses in college algebra, pre-calculus, and first-year calculus

Excellent! -This will go under the calculus section as well.

For my immediate purposes

http://www.intmath.com
/Fourier-series/Fourier-intro.php

For the main index- quick overview of content

http://www.intmath.com
/Help/sitemap.php

http://www.mathtutor.ac.uk
/integration/Main.html

http://www.mathtutor.ac.uk
/viewdisksinfo.php?disk=6

http://www.mathtutor.ac.uk/
Video tutorials, with diagnostics, summary text and exercises, take you through more than eighty topics in the way you choose... you can view and work with them online.

http://www.youtube.com/watch?v=EX_is9LzFSY

20 minute Calculus I video
YouTube

Great overview or review. Not a deep study.

CAVEAT, this video is fine and shows something relatively new in e-learning presentation, a good advertisement for its producers - I hope it stays online, but related videos on YouTube show what is wrong with some classes and what students actually are thinking about instead of calculus - vids apparently by students themselves, no doubt offensive to some but reality to others. I have not activated the link here since some do not want to see this kind of material. You can copy and paste the URL if still interested. I would not have put the URL here if I did not think it worthwhile in itself.

http://www.khanacademy.org
/videolibrary.jsp

khanacademy

Salman Khan (Sal) and Jonathan Goldman
We're a not-for-profit organization with the mission of giving access to knowledge to anyone, anywhere.

http://www.youtube.com/
watch?v=Nk2q-_jkJVs&feature
=SeriesPlayList&p=
AD5B880806EBE0A4
Introduction to Harmonic Motion

http://www.youtube.com/
watch?v=xoUppFlif04
Harmonic Motion Part 2 (calculus)

August 6, 2008 - All things considered, having reviewed two or three of these vids on Limits etc., and finding them very informative, I would think my immediate course of action would include watching most of the calculus vids here.

July 7, 2009 eleven months later and I'm finally getting back here - but much better prepared.

http://www.themathpage.com
/aCalc/calculus.htm

The Math Page - Lawrence Spector

http://www.youtube.com/
watch?v=jbIQW0gkgxo&feature
=SeriesPlayList&p
=590CCC2BC5AF3BC1

Lec 1 | MIT 18.01 Single Variable Calculus, Fall 2007
Derivatives, slope, velocity, rate of change
51:32

http://mathforum.org/math.topics.html

http://mathforum.org/
dynamic/sketchpad.links.html

http://www.math.uri.edu
/~bkaskosz/flashmo
/Barbara Kaskosz

Flash

http://www.math.uri.edu
/~bkaskosz/flashmo/derplot/
Derivative plotter

http://www.math.uri.edu
/~bkaskosz/flashmo/sequences/
Sequences and Series Plotter

http://www.math.uri.edu
/~bkaskosz/flashmo
/developers.html
Flash Mathlets -- Materials for Developers
On this page we provide links to the materials authored by Barbara Kaskosz of
the University of Rhode Island and Doug Ensley of Shippensburg University.

epsilon delta applets - Limit Concept

http://www.slu.edu
/classes/maymk/Applets
/EpsilonDelta.html

http://math.hws.edu
/javamath/config_applets
/EpsilonDelta.html

Graphics for the Calculus Classroom
Douglas N. Arnold

http://www.ima.umn.edu/
~arnold/graphics.html
for first year calculus.

http://www.ima.umn.edu/
~arnold/calculus/differential
/differential-j.html
The Java version:

http://www.ima.umn.edu/
~arnold/complex-j.html
for complex analysis.

http://www.ima.umn.edu/
~arnold/complex-maps
flash animations for visualizing complex functions.

http://www.themathpage.com/
aCalc/calculus.htm

TheMathPage: An Approach to Calculus - Lawrence Spector; Borough of Manhattan Community College, CUNY

http://www.calculus-help.com/
go to Tutorials and Fun Stuff
Michael Kelly

http://cow.math.temple.edu/
Calculus on the Web
... internet utility for learning and practicing calculus. Gerardo Mendoza and Dan Reich

http://www.mc.maricopa.edu
/~dschultz/LimitStudent.html
#MapleAutoBookmark1

http://www.mc.maricopa.edu/
~dschultz/

David Schultz

http://www.univie.ac
.at/future.media/moe/
galerie.html

maths online

interactive multimedia learning units...

http://hyperphysics.
phy-astr.gsu.edu/
hbase/hmat.html

HyperMath - Carl R. (Rod) Nave; Georgia State University

http://www.zweigmedia.com
/RealWorld/

Online interactive tutorials - Algebra Review, Finite Mathematics, and Calculus

may be on a different server as well:

http://people.hofstra.edu
/Stefan_Waner/RealWorld
/tutindex.html

http://www.ies.co.jp/
Excellent Java applets (International Education Software)

http://www.ies.co.jp
/math/products/calc/menu.html

http://www.ies.co.jp
/math/products/trig/applets
/graphFourier/graphFourier.html

http://www.ies.co.jp
/math/products/trig/applets
/graphSinX/graphSinX.html

http://www.math.gatech.edu/
~carlen/applets/index.html

Graphing Applets for Calculus - Eric A. Carlen

http://www.math.gatech.edu
/~bourbaki/JavaProjects.html

...a collection of applets designed for use in conjuction with the calculus courses at Georgia Tech.

All code is copyrighted by the authors. It is however placed here for public non-comercial use -- you are free to use it as long as you do not include it in or bundle it with (in any way) any product being sold at any price. So academic use and freeware is fine.

Apparently good applets - includes epsilon delta.

November 5, 2008 - Dates from 1996 - Good but probably too complex for my purposes, insufficient introductory material.

http://math.hws.edu
/javamath/config_applets
/index.html

Java Components for Mathematics Version 1.0 Hobart and William Smith Colleges August 2001

A set of thirteen configurable JCM applets is available. Note that there are no restrictions on using these applets

http://www.martindalecenter.com/
GradMath.html

math dictionaries and encylopedias; mathematics courses, tutorials and databases; courses by subject; related mathematics information and databases

http://archives.math.utk.edu/visual.calculus/

Visual Calculus (Mathematics Archives) - Larry Husch; University of Tennessee, Knoxville (UTK)

http://archives.math.utk.edu/
articles/atuyl/confrac/index.html

http://archives.math.utk.edu
/topics/calculus.html

November 5, 2008 - Finding a lot of Drills and Problems possibly better suited to classroom than references from Musemath. Insufficient introductory material for my current purposes.

The Flash animation I see here consists mostly of animated words and equations (slideshow), few (any?) graphics. The Content may be fine but I am having difficulty staying with this presentation.

Revisit 1x later.

http://www.karlscalculus.org/
Karl's Calculus Tutor
- problems worked out for you step-by-step

http://www.karlscalculus.org/
notation.html

http://homepage.mac.com/
nshoffner/nsh/CalcBookAll/
Calculus.htm

Calculus I: an Electronic Book Using LiveMath - N. Scott Hoffner

Free calculus book with interactive LiveMath examples.

the interactives require an activeX download - not yet tried

November 5, 2008 - A lot of "word problems" possibly good for classroom use but likely to scare away my audience. - keep for now

http://www.sosmath.com/
calculus/calculus.html

http://www.sosmath.com
/fourier/fourier2/fourier2.html
Fourier Sine and Cosine Series

Not highly interactive
having some difficulty with structure of lessons and comprehension.

November 5, 2008 - A popular site, maybe just not my style, leave for now, good at least for Fourier.

http://www.maths.abdn.ac.uk
/~igc/tch/ma1002/index/index.html

Notes - Math 1002 calculus at University Of Aberdeen: from Ian Craw

I downloaded pdf version but you can use html for ease of navigation)
Very Good

..."any teacher who can be replaced by a teaching machine, deserves to be".

http://www.macalester.edu
/aratra/index.html

A Radical Approach to Real Analysis 2nd edition. copyright 2006 David M. Bressoud In the second edition of this MAA classic
Published 2007 MAA

Functions of real
variables

http://tutorial.math.lamar.edu/

Paul Dawkins
a complete set of free online (and downloadable) notes and/or tutorials

http://math.dartmouth.edu/
~klbooksite/toc.htm

http://math.dartmouth.edu
/~klbooksite/

Principles of Calculus Modeling
An Interactive Approach
by Donald Kreider, Dwight Lahr, and Susan Diesel
Department of Mathematics, Dartmouth College
Department of Mathematics, Norwich University

http://ugrad.math.ubc.ca
/coursedoc/math101/notes
/integration/area.html

http://www.ugrad.math.ubc.ca
/coursedoc/math100/index.html

java applets

First impression - some are well thought out

Go to Course Notes

November 5, 2008 - Integration fits well with CourantRobbins_406 animation

http://www.calvin.edu
/~rpruim/courses/m162
/S02/mathlets/

good applets
http://www.calvin.edu
/~rpruim/courses/m161
/F01/java/EpsilonDelta.shtml

http://www.calvin.edu
/~rpruim/courses/m161
/F01/java/SimpleGraph.shtml

Randall Pruim
Department of Mathematics and Statistics
Calvin College
2002

http://ocw.mit.edu
/ans7870/resources
/Strang/strangtext.htm

MIT OpenCourseWare

Calculus

Gilbert Strang

"The point is that the world doesn’t end with polynomials, sines, cosines, and exponentials. There are an infinite number of other functions out there waiting for you and some of them are useful. These functions can’t be expressed in terms of the elementary functions that you’ve grown to know and love. They’re diffrent and have their distinctive behaviors."

http://www.cut-the-knot.org
/index.shtml

Interactive Mathematics Miscellany and Puzzles

http://home.scarlet.be
/~ping1339/

MATH-abundance

http://plato.acadiau.ca/courses
/educ/reid/resources.html

Mathematics Education Resources at Acadia's School of Education - David A. Reid, Acadia University

http://aleph0.clarku.edu/
~djoyce/
general index

http://www.clarku.edu/
~djoyce/complex/
Dave's Short Course on Complex Numbers

http://www.mathgv.com/

A powerful graphing program for the PC, - free.
Greg VanMullem;

http://curvebank.calstatela.edu
/home/home.htm

National Curve Bank - Gray, Venit; California State University, Los Angeles

http://ourworld.compuserve.com
/homepages/davidbowers
/Getty98/main.htm

"...It is an increasing anomaly that learning material which aspires to encourage the use of technology to assist the conceptual understanding of dynamic processes in mathematics does so by using static printed screendumps."

10 year old site - using animated gifs

http://www.understanding
calculus.com/index.php

Understanding Calculus is a complete online introductory book that focuses on concepts.

http://www.calculus.org/

http://eqworld.ipmnet.ru/

... to sail more easily through the ocean of mathematical equations (differential, integral, functional, etc.) and their solutions.

links page looks good:

http://eqworld.ipmnet.ru/en
/education/edu-mathsites.htm

Shodor Foundation
http://www.shodor.org/

A non-profit research and education organization dedicated to the advancement of science and math education, specifically through the use of modeling and simulation technologies.

http://cs.jsu.edu/mcis
/faculty/leathrum/Mathlets/

Mathlets: JavaTM Applets for Math Explorations - Tom Leathrum

http://www.themathpage.com
/aBookI/plane-geometry.htm

The Math Page - Lawrence Spector

http://www.math.tamu.edu
/~dallen/history/euclid/euclid.html

The Elements an excellent illustrated overview which may contain all or most of what I need for my purposes

http://www.themathpage.com/
ARITH/arithmetic.htm
Arithmetic

The Math Page - Lawrence Spector

http://www.themathpage.com
/areal/real-numbers.htm
Evolution of Real Numbers

http://www.aaastudy.com/

AAA Math features a comprehensive set of interactive arithmetic lessons...Immediate feedback prevents practicing and learning incorrect methods

http://www.harcourtschool.com
/glossary/math_advantage
/glossary8.html

http://www.arcytech.org
/java/b10blocks/b10blocks.html

http://www.newbanner.mcom
/CEL/Math/algebra2.htm
Montessori school use of Dienes blocks

http://nlvm.usu.edu
/en/nav/frames_asid_189
_g_1_t_2.html?open=activities
&from=topic_t_2.html
National Library of Virtual Manipulatives - NLVM

http://argyll.epsb.ca
/jreed/math9/strand1
/multiply_base10.swf

http://argyll.epsb.ca
/jreed/math9/strand1
/divide_base10.swf
Jim Reed

Some websites with Dienes or "Base 10" (or some other base) Blocks which should be useful to the planned animation. I have already used the principle (or something similar which arose from the context of developing the visual demonstrations) in two animations (equal_temperament_ratios_2.html, pp 13-25 and prosthaphaeresis_1.html p 12) - now that I understand them as an established teaching tool, I can plan accordingly.

July 7, 2009 - I am not trying to create a manipulative, This is to be an interactive animation which refers to the familiar Dienes Manipulative and introduces some calculus concepts. Some preliminary work: Dienes Blocks

The NLVM one has an adjustable rectangle which may be useful for the continued fraction analogy in equal_temperament_ratios_2.html

July 7, 2009 - the following linked February 2009 report is generally favorable towards virtual manipulatives.
http://www.nmsa.org/Research/
ResearchSummaries/Mathematics/
tabid/1832/Default.aspx

'...Web-based manipulatives can "enhance the knowledge and understanding of learners, while creating a conceptual understanding of mathematical theories beyond the mere formulaic models of traditional mathematical coursework" (Crawford & Brown, 2003, p.176).'

http://seeingmath.concord.org
/sms_interactives.html
Free for Public Use - requires latest Java, not seen yet

http://www.concord.org
"Our innovative professional development programs integrate face-to-face and online interaction with web-based learning, video case studies, and dynamic interactive software.

(October 8, 2008) I have no diskspace to download the latest Java software so I cannot try this out yet.

Looks worth trying.

Dienes

"...in fact the depth of understanding needed to model dynamic mathematical systems using concrete materials often far exceeds that needed to treat mathematics simply as a set of computational skills (Bell, Fuson, and Lesh 1978)

http://thinkzone.wlonk.com
/Numbers/NumberSets.htm

http://thinkzone.wlonk.com
/index.htm

Keith Enevoldsen's Think Zone

Real Number Sets

I particularly like the number line and general layout here...

http://www.coolschool.ca/
content/showcase.php?type=math

some Flash based interactives....

http://vels.vcaa.vic.edu.au/
essential/discipline/mathematics
/glossary.html#connective

Mathematics Glossary

http://www.mathsisfun.com/

The idea behind the site is to offer mathematics as well as some fun bits, and to combine the two wherever possible.

developed and maintained by Rod Pierce
General

http://onlinestatbook.com/rvls.html

Rice Virtual Lab in Statistics

This is the most I've ever been interested in statistics

The applets can be downloaded and used without restriction.
Last updated: 6/28/2008

http://www.statsci.org
/teaching.html

statistics Teaching Resources

http://www.quantitativeskills.com
/sisa/index.htm

SISA (Simple Interactive Statistical Analysis) allows you to do statistical analysis directly on the Internet.

http://www.c3.lanl.gov
/mega-math/

MEGA Mathematics - Los Alamos National Laboratory

The MegaMath project is intended to bring unusual and important mathematical ideas to elementary school classrooms so that young people and their teachers can think about them together

October 2007: ...for roughly the last ten years, the project has been unfunded and the site unchanged.

http://www.thiel.edu
/mathproject/atps/default.htm

This book has been placed on the Web by the Science Education Team at Los Alamos National Laboratory to help students better prepare for careers in mathematics and science...The topics are essential for success in college level mathematics but tend to be taught inadequately...1998-2008

http://www.thiel.edu
/mathproject/CNAT/Default.htm

complex numbers

http://www.bymath.com/

Dr. Yury Berengard is a mathematician, an engineer, a scientist and a teacher.

http://www.bjpinchbeck.com
/math.html

"We also thought that it would be cool to make a homework helper site that others could use so they would not have to surf all over the WWW to find real good educational links."

http://www.bjpinchbeck.com/
general research topics

http://homepages.gac.edu
/~mcs121/2006F/javaApplets.html

calculus applets

http://www.sciencegems.com
/math.html#5

Frank Potter's Science Gems - Mathematics
Latest update: Sunday, April 21, 2008.
http://www.sciencegems.com

links list
from k1 to GroupTheory/Topology etc

http://illuminations.nctm.org
/LessonDetail.aspx?id=U163

To Fret or Not To Fret
Exploring the Math Behind the Placement of Frets on Stringed Instruments

The National Council of Teachers of Mathematics

http://wims.unice.fr/

WWW Interactive Multipurpose Server

http://www-history.mcs
.st-andrews.ac.uk/history
/index.html

School_of_Mathematics_and_Statistics
University_of_St_Andrews,_Scotland

http://www-groups.dcs.
st-and.ac.uk/~history
/HistTopics/The_rise_
of_calculus.html

A history of the calculus

http://www.maths.uwa.edu.au/
~schultz/3M3/Table_of_Contents.html

http://www.maths.uwa.edu.au/
~schultz/3M3/L28Weierstr,
Dede,Cantor.html

Last update: 19 August 2000
Author: Phill Schultz,

September 3, 2008

Apparently a lecture on the origins and development of calculus. Well done, I had seen it some time ago but needed some study to appreciate it.

http://www.math.rutgers.edu
/~cherlin/History/Papers1999
/schmarge.html

Conic Sections in Ancient Greece
Ken Schmarge

...a systematic introduction to the work of the Greek geometers involved in the development of conic section theory...

...the culmination of the subject came at the hands of Apollonius...(his) Book V introduces the idea of "maximum" and "minimum" lines to refer to tangents and normals, respectively. The propositions and relationships it proves, which today are more easily shown using differential calculus, are rigorously explored in the classic Greek geometric fashion (Heath, 1961, pp. lxxv-lxxvi)...

http://web01.shu.edu/
projects/reals/history/

Historical Tidbits - Paul Golba

http://www.math.rutgers.edu
/~cherlin/History/Papers1999
/kijewski.html

Isaac Newton

Kerry Kijewski

History of Mathematics Term Paper, Rutgers, Spring 1999

... Newton claimed that his method of fluxions was conceived in 1665. Its fullest treatment is in De methodis fluxionum et serierum infinitorum, dated 1671... which was actually a letter that was never meant to be published.

This gives some dates and titles, some helpful explanations of Newton's conception of calculus, and suggestions for further reading

Kijewski, ibid

The starting point was that geometrical magnitudes arise from continuous motion... In other words, a line or curve is generated by the continuous motion of a point, a surface by the motion of a line and a solid by the motion of a surface...Newton called the line a "fluent" because he thought of it as a flowing quantity ... The fluxion, or rate at which it flowed, was the point's velocity .[5, p.213] Dotted letters were used to represent the fluxions. These fluxions could, in turn, be taken as fluents and this led to higher order fluxions, which were represented by two dots. There is an obvious relation: the velocity of a point determines the nature of the curve, and a curve of a given nature can only be generated by a point with a certain velocity...two problems arose: given a relationship between fluents, how one can determine the corresponding relationship between fluxions, and the inverse, how one can determine fluents on the basis of fluxions...These processes are known today as differentiation and integration.

Some of the Newtonian explanations here might be animated to give some idea of his approach.

In general his work may be better suited to animated demonstrations but Leibniz' formal symbolic calculus is the accepted form today.

Where possible they might be compared in animation.

Newton's `Principia':
Book Two, Lemma II

http://scanserver.ulib.org
/is/scanserver/newton/xml
/doc.scn?rs=0&br=0.0&rt
=0&rp=http%3A%2F%2F
scanserver.ulib.org%2Fis
%2Fscanserver%2FSearch
.asp%3FgroupId%3Dnewton
%26bookId%3Dnewton%26
query%3Dbook&pg=260&
auth=c2b421b91241a34ea8

http://www.maths.tcd.ie/pub
/HistMath/People/Newton
/Principia/Bk2Lem2/PrBk2Lm2.html

Scholium

"The foundation of the general method is contained in the preceding Lemma."

These are I believe Newton's own final words on the subject, from his own revised edition of the Principa written with half a century of reflection.

The book is generally considered the most important work in science, from within its framework. an image of our place in the material world has been drawn.

http://www.maths.uwa.edu.au
/~schultz/3M3/L19Leibniz.html

By 1676, Leibniz had invented the notation that made calculus easy to learn and apply: dy/dx , INT y and INT y dx . The first account of this appears in Nova methodus... published in his own scientific journal Acta Eruditorum in 1684.

By 1690, Leibniz had discovered most of the ideas of elementary calculus, including differential equations, but he did not write up a complete treatment of this material, which was first done by L'Hospital (1651--1704) and Jean Bernoulli (1667--1748).

There is some good historical treatment of the subject on this site,

This apparently translates some of Leibniz's original discussions. Here is an animation (currently little more than a placeholder - eventually it may explore some ways to make the explanations a bit more appropriate for the NetAge.

Leibniz

http://www.princeton.edu
/~mike/articles/canons
/canons.htm

INFINITESIMALS AND TRANSCENDENT
RELATIONS:
THE MATHEMATICS OF
MOTION IN THE LATE
SEVENTEENTH CENTURY

Michael S. Mahoney
Princeton University

Mathematical for Descartes was what could be grasped in a single intuition or could be reduced in clearly understood steps to such immediate apprehension.

04/17/08

Descarte's ideas of visual mechanical linkages as logical steps is discussed a bit in Origins 12TET 2, page 29 and Origins 12TET 3, page 14 .

Apparently, in 1637 it was most important for Descarte to demonstrate the validity of the new algebraic operations by picturing them in terms of well known, existing geometrical methods.

"A quantitative relation," wrote Leibniz, (11) "is a way of finding one quantity by means of another", and, as will become clear below, an equation of whatever degree constituted a modus inveniendi. (modus inveniendi - a way of discovery)

...Taking algebraic equations as themselves basically intelligible, he moved to widen the concept of the equation by shifting focus from the notion of a compound relation to that of a modus inveniendi, a way of finding (one quantity by means of others), thereby encompassing the new, transcendent relations that were so interesting both mathematically and physically.

Mahoney Ibid

citing J.E. Hofmann

"... Leibniz strove for a technique of representation which is simplified and formalized down to the [detail] by means of appropriate symbols, yet which cannot be immediately grasped but must be learned. Whoever can acquire this has an unimaginable advantage over the uninititated, even when he has no particularly deep insights into the connections: the formalism thinks for him.

My thinking is that the animations, and probably to a greater extent their actual construction by students in some software program provides some of the useful "insights and connections."

That's pretty much the way I think I'm learning this stuff - along with this historical approach. (and I still need to try making up for some formal education lapses by going to the basic problem oriented math courses).

Mahoney Ibid

That the straight and the curved coalesced at the level of the infinitesimal was a premiss of the calculus, indeed its raison d'être.

http://www.math.vanderbilt.edu
/~schectex/courses/whystudy.html

,,, Newton and Leibniz knew how to correctly give the derivatives of most common functions, but they did not have a precise definition of "derivative"; they could not actually prove the theorems that they were using.

.... They explained a derivative as a quotient of two infinitesimals (i.e., infinitely small but nonzero numbers).

Ultimately, the biggest difference between the infinitesimal approach and the epsilon-delta approach is in what kind of language you use to hide the quantifiers:

The numbers epsilon and delta are "ordinary-sized", in the sense that they are not infinitely small. They are moderately small, e.g., numbers like one billionth. We look at what happens when we vary these numbers and make them smaller. In effect, these numbers are changing, so there is motion or action in our description. We can make these numbers smaller than any ordinary positive number that has been chosen in advance.

The approach of Newton, Leibniz, and Robinson involves numbers that do not need to change, because the numbers are infinitesimals -- i.e., they are already smaller than any ordinary positive number.

To a large extent, mathematics -- or any kind of abstract reasoning -- works by selectively suppressing information. We choose a notation or terminology that hides the information we're not currently concerned with, and focuses our attention on the aspects that we currently want to vary and study.

http://www.maths.tcd.ie/pub
/HistMath /People/Leibniz
/RouseBall /RB_Leibnitz.html

D.R. Wilkins

http://www.maa.org/editorial
/euler/how%20euler%20did
%20it%2035%20foundations
%20of%20calculus.pdf

How Euler Did It
by Ed Sandifer
Foundations of Calculus
September 2006

...Euler published his differential calculus book, Institutiones calculi differentialis... in 1755.

...Euler is stuck with the paradox that the quantity dx is, in some sense, both zero and
not zero. ....Essentially, he introduces some rules for the use of infinite and infinitesimal quantities,
roughly equivalent to our techniques for manipulating limits...

I have not gone very deeply into this material. (at all)

Jerome Keisler's book on Non-Standard Calculus is now available on-line (free, April 18, 2008)

http://www.math.wisc.edu
/~keisler/calc.html

http://mrfronius.com
/download4.php?file=limits.pdf

Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus

Judith V. Grabiner,

http://www.maths.uwa.edu
.au/~schultz/3M3/L27Cauchy.html

Lecture 27 The Calculus of Cauchy.

Phill Schultz
11 September, 2000

The early teachers at the École Polytechnique set a precedent of writing textbooks and Cauchy followed the tradition. He wrote three texts on Calculus, in 1821, 1823 and 1829, and they set the pattern which nearly all calculus texts have followed to this day - definitions of limits and of continuous and differentiable functions, numerous graphical and analytic examples and exercises, simple applications, definite integration, d.e's. He didn't define real numbers or functions, but he did have dependent and independent variables. He defined derivatives as the limit of the difference quotient...

http://askville.amazon.com/simple-
explanation-calculus-concept-limit
-delta-epsilon/AnswerViewer
.do?requestId=2465032

"Need a simple explanation for the calculus concept of limit, especially delta and epsilon"

http://en.wikipedia.org
/wiki/Karl_Weierstrass

Karl Theodor Wilhelm Weierstrass 1815 – 1897

At the time, there were ambiguous definitions regarding the fundamentals of calculus, hence theorems could not be properly proven. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and other eminent mathematicians such as Cauchy had only vague definitions of limits and continuity of functions. ...

Weierstrass also formulated similar definitions of limit and derivative still taught today.

http://math.rice.edu
/~lanius/Lessons/

Mathematics lessons that are fun / Cynthia Lanius

http://www.ux1.eiu.edu
/~cfsjy/mts/math/_link.htm

1995-1999 Shu-Chen Jenny Yen
MONTESSORI MATH ALBUM
INDEX

Shu-Chen Jenny Yen's On-Line Montessori Albums contain 240 developmentally appropriate activities for 3-to-6 year old children

http://www.amblesideprimary.com
/ambleweb/numeracy.htm

Numeracy Manipulatives and Games
The vast majority and content of the pages are made by the children during lessons, at breaktimes, at home or at Computer Club...At Ambleside we involve children in all parts of the process of designing and creating content for our website. Many of these games have been designed by, or with children.

http://www.thinkingblocks.com
/TryIt.html

To begin any Thinking Blocks program, please click on the corresponding colored oval.

http://www.globalclassroom.org/
ecell00/javamath.html

Education links for kids (math and others)
(Extensive)

Patti Weeg
"The world is our classroom! All are welcomed in...

http://vels.vcaa.vic.edu.au/downloads
/vels_standards/velsrevisedmathematics.pdf
math standards

Victorian Essential Learning Standards
East Melbourne
Victoria
Australia

These preliminary standards seem well expressed - quite challenging.

Something of a learning tool in itself at times.

http://www.utdanacenter.org/
k12mathbenchmarks/downloads
/secondary_strand_may08.pdf

Apparently derived from:
http://www.achieve.org/node/337
Achieve
Created in 1996 by the nation’s governors and corporate leaders, Achieve is an independent, bipartisan, non-profit education reform organization based in Washington, D.C. that helps states raise academic standards and graduation requirements, improve assessments and strengthen accountability.

Academic standards which may be compared with those above.

The idea is: what do people need to know and when are they most likely to know it.

http://ocw.mit.edu/
OcwWeb/hs/calculus
/calculus/

We have selected relevant material from MIT's introductory courses to support students as they study and educators as they teach the AP® Calculus curriculum.

In general, the U.S. standards for "Advance Placement" in math for (I'm guessing) 14-17 year old students.

http://mathmistakes.info/index.html

Real Mistakes from Real Student Work and Math Facts Students Should Know - Russell Blyth

Have gotten some good results here. Seems helpful to me - took a bit of playing around to get the idea - explanations immediately helpful.

http://nlvm.usu.edu/

The National Library of Virtual Manipulatives

http://argyll.epsb.ca/jreed/

Jim Reed
... I have been developing Flash resources for several years... I have taught in Alberta for 32 years. In the past, I was a grade 9 math and science teacher. I am currently teaching high

http://www.webmath.com/

type the math problem you're working on- instantly analyze your problem, and when possible, provide you with a step-by-step solution, instantly!

http://www.netn.net/27111.htm#MATH

National Education Telecommunications Network

http://www.mathnerds.com/

MathNerds

Created by the Center for Advancement of Teaching, Xavier University,.. Team members who respond to questions are generally professional mathematicians holding advanced degrees in mathematics.

Aug. 13, 2008

Listed as possible resource, I have not yet used it.

http://www.umiacs.umd.edu
/~ramani/cmsc460/lecture19_fft.pdf

University of Maryland Institute for Advanced Computer Studies

Computational Methods
CMSC/AMSC/MAPL 460
Fourier transform
Ramani Duraiswami,

http://www.wmueller.com
/precalculus/index.html

http://www.wmueller.com
/precalculus/families/00.html

http://www.themathpage.com
/aPreCalc/slope-of-a-line.htm

The Math Page - Lawrence Spector

http://www.univie
.ac.at/future.media/moe
/galerie/fun2/fun2.html

gallery of graphs of functions

August 4, 2008

maths online - I find this page very good for my current level of understanding - challenging, interesting applets that make me think and result in some learning...

Some of the external links are broken and it may be that at least this English version of the site is no longer being maintained (2003??)...it asks for "at least 800x600 res" and the instructions and explanations are getting difficult for me to read.

Be sure to get as much as useful here before it fades into cyberPast.

PGC
http://www.accesscom.com
/~lillge/pgc/

The Plane Graphic Calculator is a software tool that interactively displays a graphical view of mathematical functions.

Version 1.1 June 2001

http://www.themathpage.com
/aTrig/trigonometry.htm

The Math Page - Lawrence Spector

Simple, straightforward, static illustrations, quick answers to short relevent problems, no animations.

...I animated some of the trig identities from here in the Prosthaphaeresis animation. It turned out to be somewhat irrelevant to Musemath so I have not tried to improve on it.