Mathematical Intentions
A Complete Ethnomathematical History of Our Current Secondary Mathematics Curriculum.
This website contains fifty halfhour lectures by David Dennis, in mp3 format, many supporting documents in PDF format, and interactive applets by Susan Addington in dynamic geometry and spreadsheets.
The webiste takes apart every major idea in our current mandatory secondary mathematics curriculum (Algebra IAP Calculus) and asks three historical questions.
 What were the scientific, social, religious, political and economic intentions of the people who created this mathematics?
 When and why were these mathematical ideas made into mandatory curriculum? And what parts of the original mathematics have we dropped from our curriculum?
 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 Puritan 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 “God's 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 well as its socialhistorical 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.
Rightclick or controlclick 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 the GeoGebra website in 2018.
In the meantime, some don't work. Please check back later.
Ones that do are marked with a *.
]
Rightclick or controlclick 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 
Similar
Triangles *
Similar
Quadrilaterals *
Similar
Pentagons *
Triangles
Inscribed in a Circle *
Three Similar
Right Triangles *
The Geometric
Mean *
Square 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 
Babylon
Square Root Algorithm *
Hindu
Square Root Algorithm *
European
Square Root Algorithm *
Making a
Logarithm Table *
Virtual Slide Rule,
by Derek Ross *
Descartes's
Logarithm Machine
Descartes's
Two Mean Proportionals
Machine
Descartes's
Logarithm
Graph
Slope 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.
Cone from a
Rotating Line
*
Symmetry of
an Oblique Cone
Tangent Plane
to a Cone
Circular Sections
of a Cone
Conic Sections
Conic Sections:
Parabola 1
Conic Sections:
Parabola 2
Conjugate Diameters
of an Ellipse
Conjugate 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) 
Parabola constructed
by a
Rhombus
Parabola by
Focus and Directix
Quadrivium Logo
Ellipse Construction
Van 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) 
Adjustable Pascal's Triangle * 
19. Summation and Difference 


20. The Summations of IbnalHaitham 
Al Hazen's Summation Formulas 
AlHazen1
The Sum of Integers: 1+2+3+...
*
AlHazen2
The Sum of Squares: 1^{2}+2^{2}+3^{2}+...
AlHazen3
The Sum of Cubes:
1^{3}+2^{3}+3^{3}+...
ParabDome 
21. Characteristic Ratios 
Newton and Characteristic Ratios 
ParabArea
CharRatioXtoN 
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

WallisTable *
WallisNewtonTable * 
31. Purity and Abstraction 


32. Culture, Motion, and Language 


33. The Challenge of the Cycloid 
The Cycloid: Tangents, Velocity Vector, Area, and Arclength 
CycloidArea
CycloidTangent 
34. Trigonometry 

TrigSegments 
35. Leibniz and Transmutation 
Transmutation 
[In progress. Please read the
Lecture Notes for explanations.]
Transmutation_Lin
PascalIntSin
TransmuteCircle
TransmuteConics 
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 

Complex Addition *
Complex Multiplication *
Complex Reciprocal *
Complex Squaring Function *
Complex Square Root Function *
Complex Exponential Function *
Complex Logarithm Function *
Complex 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 100120 MB, so they may take a while to download.
A bibliography for those who want to check on the details. Includes both history of mathematics and math education books and papers.
