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Mathematical Intentions

A Complete Ethnomathematical History of Our Current Secondary Mathematics Curriculum.

This website contains fifty half-hour lectures by David Dennis, in mp3 format, many supporting documents in PDF format, and interactive applets in dynamic geometry and spreadsheets.

The webiste takes apart every major idea in our current mandatory seconday mathematics curriculum (Algebra I--AP Calculus) and asks three historical questions.

  1. What were the scientific, social, religious, political and economic intentions of the people who created this mathematics?
  2. When and why were these mathematical ideas made into mandatory curriculum? and what parts of the original mathematics have we dropped from our curriculum?
  3. Why do we still teach this curriculum? and how will it be transformed by new technology?

Thus for every mathematical idea taught in High School, these three different intentions will be described. Seemingly epic in scope, this task is easier than one might, at first, suspect since much of our mandatory mathematics curriculum comes from a rather narrow piece of social history. For example, the English Civil Wars of the 17th Century and their aftermath yield a huge ammount of our current notions of school mathematics, and so the social history of the Purtitan movement must be examined in detail, along with the Jesuit response. Once we understand their intentions we can ask much deeper questions about what pieces of their curriculum we still want to require of our students. Do we still believe that the natural world is “Gods other book?” Do we continue to accept the radical pragmatisim and religious pluralism that led to Oliver Cromwell’s “new model army?”

Intentions matter! especially when mandatory curriculum is tied to high stakes testing. This site is intended to get all of the relevant historical cards on the table as quickly and as effciently as possible, specifically to aid in the reform of mathematics curriculum. Surprising answers are given to nagging questions that are glossed over in secondary curriculum, like “Where did fractional exponents come from? and what are they good for?” What was the role of tables, and how does that change with modern spreadsheets? Many historical examples are shown embedded in new transtormational computer environments, e.g. animations of Descartes’ curve drawing machines.

This website is a large scale attempt to carry out the research program described in:

Dennis, D. (2000). The Role of Historical Studies in Mathematics and Science Educational Research. In Dick Lesh & Anthony Kelly (Eds.) Handbook for Research Design in Mathematics and Science Education. Mahwah, NJ: Lawrence Erlbaum.

In that chapter I described three increasing levels of impact that history can have on curriculum development: Context, Content, and Critique. This website engages all three levels for every secondary mathematical topic, in the hope that the fruits of historical understanding, in a convienient form, can nourish the technological transformation of school mathematics.

How to use these materials

The lectures give a survey of the mathematics, as welll as its social-historical background. They are the main framework for the whole project.

Most lecture notes have associated applets: small interactive programs to illustrate a mathematical concept. When the lecture notes include a picture, the applet gives infinitely many variations on the picture, or sometimes a picture in motion.

Most lectures that describe mathematics have corresponding lecture notes. These supplement the lectures with details of the mathematics, and include exercises, some mandatory, and some for further exploration. Sometimes several lectures discuss the mathematics in one lecture notes document, and vice versa. The lectures discuss the topics in approximate historical order, with occasional departures for mathematical historical background.

Note: This website was put up during 2010, but more applets and lecture notes appear periodically. Please check back for additional material.

We welcome your comments. Send them to david.dennis@earthlink.net.

Podcasts

Lecture

Lecture notes Applets

Click to open a window to play the podcast.

The audio player should open in a separate tab or window.

PDF files to read with the lecture. These include links to the applets. There are exercises and problems for the serious reader to do.

Right-click or control-click to download instead of of opening.

Interactive demonstrations, mostly on GeoGebraTube.
Some are spreadsheets, mostly on Google Docs.
There are also links to applets made by others.

[We're moving the applets to GeoGebraTube in spring 2016.
In the meantime, some don't work. Please check back later.
Ones that do are marked with a *. ]

Right-click or control-click to download or open in a new window.

1. How Did We Get Here?

   

2. Puritan Printheads

   

3. The Firing Line

   
4. A Car is not a Horseless Carriage    
5. Observable Units    
6. Corporations and Kinship    
7. Puritans and Race in America    
  Navigating GeoGebra (how to run the applets)  
  Geometric Constructions  
  Similarity, Geometric Arithmetic, and the Geometric Mean

Thumbnail of Similar Triangles appletSimilar
Triangles
*

Thumbnail of Similar Quadrilaterals appletSimilar
Quadrilaterals
*

Thumbnail of Similar Pentagons appletSimilar
Pentagons
*

Thumbnail for Triangles Inscribe in a CircleTriangles
Inscribed in a Circle
*

Thumbnail of Inscribed Right Triangles appletThree Similar
Right Triangles
*

Thumbnail of Square Root Construction appletThe Geometric Mean *

Thumbnail of Square Root Construction appletSquare Root
Construction
*

8. Slide Rules and Logarithms

9. A Discourse on Method

10. Square Roots and Logarithms

11. Linkages and Logarithms

Square Roots

Graph paper in inches and tenths

Napier Rods

Slide Rules and Logarithm Tables

Napier's Mirifici Logarithmorum Canonis Descriptio (1614)

PaperSlideRule

 

 

 

 

Descartes's Logarithm Construction

Thumbnail of Babylonian Square Root Algorithm appletBabylon
Square Root Algorithm
*

Thumbnail of Hindu Square Root Algorithm appletHindu
Square Root Algorithm *

Thumbnail of Medieval European Square Root Algorithm appletEuropean
Square Root Algorithm
*

Thumbnail for Making a Logarithm Table spreadsheetMaking a
Logarithm Table
*

Thumbnail of virtual slide rule by Derek RossVirtual Slide Rule,
by Derek Ross *

Thumbnail for Descartes Log appletDescartes's
Logarithm Machine

Thumbnail of Descartes 2 Mean Proportionals appletDescartes's
Two Mean Proportionals
Machine

Thumbnail of Descartes Logarithm Graph appletDescartes's
Logarithm Graph

Thumbnail of the Logarithm Graph Slope appletSlope of the
Logarithm Graph

12. Geometry Fades Away    
13. Flattening Apollonius

Apollonius and Conic Sections (information with problems)

Parabolas and Coordinates (problems/activities)

Some of these 3D applets need the Cabri3D plugin
(free; Mac and Windows only). We're updating to GeoGebra.

Thumbnail for the applet A Cone from a Rotating LineCone from a
Rotating Line
*

Thumbnail of the Cone Symmetry appletSymmetry of
an Oblique Cone

Thumbnail of the Tangent to a Cone appletTangent Plane
to a Cone

Thumbnail of Circle Sections of a Cone appletCircular Sections
of a Cone

Thumbnail of Conic Sections appletConic Sections

Thumbnail of Parabola Conic Section appletConic Sections:
Parabola 1

Thumbnail of 2nd Parabola Conic Sections appletConic Sections:
Parabola
2

Thumbnail of Conjugate Diameters of an Ellipse appletConjugate Diameters
of an Ellipse

Thumbnail of Conjugate Diameters of a Parabola appletConjugate Diameters
of a Parabola

14. Parabolas and Tangents

Functions of a Curve: Leibniz's Original Notion of Functions
and its Meaning for the Parabola (paper by Dennis and Confrey)

 

 

 

 

Ellipses (problems/activities)

Thumbnail of the Parabola Rhombus appletParabola constructed
by a Rhombus

Thumbnail for the Parabolas by Focus and Directrix appletParabola by
Focus and Directix

Thumbnail of Quadrivium Logo Ellipse appletQuadrivium Logo
Ellipse Construction

Thumbnail of the Elliptic Orbit appletVan Schooten's
Elliptic Orbit Machine

15. Hyperbolas and Ratios Descartes's Hyperbola Construction  
16. Refraction and Efficiency    
17. Pascal and Nature    
18. Pascal's Triangle and Induction

"A Treatise on the Arithmetical Triangle"

(Translation of Pascal's paper, with an investigation/project
written by David Pengelley)

Thumbnail of Pascal's TriangleAdjustable Pascal's Triangle
19. Summation and Difference    
20. The Summations of Ibn-al-Haitham Al Hazen's Summation Formulas

Sum of 1st powersAlHazen1
The Sum of Integers: 1+2+3+... *

Sum of 2nd powersAlHazen2
The Sum of Squares: 12+22+32+...

Sum of 3rd powersAlHazen3
The Sum of Cubes: 13+23+33+...

Parabolic DomeParabDome

21. Characteristic Ratios Newton and Characteristic Ratios

Characteristic ratio for the parabolaParabArea

Characteristic ratio of x^nCharRatioXtoN

22. Wallis and Cromwell    
23. Reading, Writing, and Arithmetic    

24. Wallis and Fractional Exponents

25. Wallis and Pi

26. The Impact of Wallis

27. Newton Reads Wallis

28. Newton's Binomial Series

29. Interpolation and Continuity

30. Experiments with Technology

Wallis and Fractional Exponents

Wallis and Negative Exponents

 

 

 

 

 

 

Newton's Binomial Series

Euler and the Exponential Base e

NewtonInterpolation

 

 

 

 

Characteristic ratio of (1-x^2/3)^1/2WallisTable

 

 

 

 

Thumbnail for WallisNewtonTableWallisNewtonTable

31. Purity and Abstraction    
32. Culture, Motion, and Language    
33. The Challenge of the Cycloid The Cycloid: Tangents, Velocity Vector, Area, and Arclength

Thumbnail for cycloid area appletCycloidArea

Thumbnail for cycloid tangent appletCycloidTangent

34. Trigonometry   Thumbnail for Trig Segments appletTrigSegments
35. Leibniz and Transmutation Transmutation

[In progress. Please read the
Lecture Notes for explanations.]

Thumbnail of Transmutation of a Line appletTransmutation_Lin

Thumbnail of Pascal's Sine Integration appletPascalIntSin

Thumbnail of Transmutation of a circle appletTransmuteCircle

Thumbnail of Transmutation of a Conic appletTransmuteConics

36. Testing the Leibniz Rules    
37. The Textbooks of Euler   See the complex number applets with Lecture 45
38. The Children of Euler    
39. Weierstrass and Godel    
40. The Quest for Certainty    
41. The Shape of Averages    
42. Personality and Structure    
43. God's Other Book    
44. Mathematics and Art    
45. The Future of Curriculum  

Thumbnail for complex addition appletComplex Addition *

Thumbnail for Complex Multiplication appletComplex Multiplication *

Thumbnail for Complex Reciprocal appletComplex Reciprocal *

Thumbnail for Complex Squaring appletComplex Squaring Function *

Thumbnail for Complex Square Root appletComplex Square Root Function *

Thumbnail for complex exponential appletComplex Exponential Function *

Thumbnail for complex logarithm appletComplex Logarithm Function *

Thumbnail for complex sine appletComplex Sine Function *

46. Does Algebra Teach You to Think?   Wolfram Alpha: online Computer Algebra System *
47. John Henry and the End of Algebra Lyrics to the folk song John Henry  
48. Culture Jamming    
49. The Future of the Obsolete Marc Antony's speech from Shakespeare's Julius Caesar  

Lectures, in bulk

Bundled in .zip format, 5 at a time. Download for your mp3 player. Warning! These are all 100-120 MB, so they may take a while to download.

Lectures bundled in sets of 5
01-05 06-10 11-15 16-20 21-25
26-30 31-35 36-40 41-45 46-49

References (alphabetical; by subject) and further reading

A bibliography for those who want to check on the details. Includes both history of mathematics and math education books and papers.

Quadrivium

Mathematical Intentions
Measuring the World

Applets

Contact us

Last updated November 29, 2010

Copyright 2009-11 David Dennis and Susan Addington. All rights reserved.