Classical Mathematics

Page with geometrical diagrams cropped smallThis page is about our live online course in Classical Mathematics that we have since retired. If you’re looking for a course in Classical Mathematics, please see our new curriculum

Classical Mathematics is an online four-year high school course which roughly follows standardized high school math curricula. In Classical Mathematics, students study and learn essentially the same mathematical concepts and processes as they would in standard high school textbooks. The difference is that Classical Mathematics is driven by primary sources (think a Great Books approach to math) and takes a chronological approach to mathematical concepts.

High school students typically study math (and science, for that matter) by memorizing a series of processes and concepts without knowing where these processes and concepts came from.

While there is much to be said for this textbook approach (it is well organized), it simply isn’t the way mathematics developed. What students lose by studying math by this artificial approach are the philosophical and human elements. Mathematics is not the cold and unchanging discipline that high school texts present. Mathematics is a complex inquiry into the structure of rationality and the universe. The formulas and methods that textbooks casually present have rich histories and are often surprising solutions to difficult problems. When we study math in context and learn not only the solutions which we moderns have been handed, but also the questions which lead to them, we experience math in a different way. Math ceases to be merely an art of number tricks which allows us to balance our budgets and solve problems we’ll never encounter in ‘real life.’ Math itself comes to life, in much the same way that literature does.

Thus, my goal is to simultaneously solve several problems that plague students under contemporary math instruction:

  • memorizing formulas to pass tests rather than grasping the underlying concepts.
  • thinking that mathematics has nothing to do with other liberal arts (like philosophy or literature, for instance).
  • thinking that mathematics is a purely practical discipline which we must learn to solve problems––problems which we’ll probably never encounter.
  • not understanding that mathematics is a historical development in which perplexing problems lead to surprising discoveries, many of which explain the rules and formulas we use today.

Just as the modern high school math is designed to lead students to calculus, the goal of this course is likewise to prepare students for calculus over four years––or to put it another way, the goal is to get to Sir Isaac Newton’s Principia.

The first year is primarily geometry, beginning with Euclid’s Elements and proceeding to Greek developments in algebra and arithmetic.

In the second and third years students will study the sources of modern algebra, further developments in geometry, and trigonometry.

During the fourth year, as I mentioned, students will study the foundations and beginnings of calculus.

Please note the following:

This is first and foremost a math course and only secondarily an “history of math” course. Students will learn real mathematical concepts & processes and work math problems. Students will learn much math history, but the primary goal of the course is for students to study math within a Classical and Socratic approach.

 

Click below to read or download the Classical Math & AGA Comparison Chart (PDF format) which outlines how Classical Mathematics I-III compare to standard Algebra 1/ Geometry/ Algebra 2 programs.

Classical Math & AGA Comparison Chart 


Classical Mathematics I

This course is no longer being offered as a live class. We are translating Classical Math I into a year-long curriculum for home and private schools. The curriculum will be available for purchase this fall (2017). Please contact Daniel Maycock at help@polymathclassical.com if you have further questions about our curriculum or wish to be notified when it’s available.

Please see this link for a description and preview of our Classical Math I curriculum.

The majority of this first year will be spent in Euclid’s Elements.

Written over two-thousand years ago, Euclid’s Elements still provides the basis for all our high school geometry textbooks. Modern textbooks make an effort to distill the most famous and most useful theorems from the Elements and present them within a more relevant and practical context. Unfortunately, more is lost than gained by this approach.Detail from the beginning of a medieval treatise on geometry

What modern geometry textbooks miss is the logical precision and the systematic nature of Euclidean geometry. Because they only hit the highlights, most textbooks present a fragmented view of geometry. As a result, students conclude that geometry is confusing and that the proofs are unnecessarily difficult. After all, why must we prove that two triangles are congruent which have two sides equal and the angle between them equal? Isn’t it obvious from the diagram? But Euclid reminds us that to measure angles and lines would be too imprecise: the world contains no perfect circles nor perfectly straight lines. Instead, we are forced to rely upon our reason where we would be tempted to trust our figures and our eyes.

By presenting a coherent and systematic whole, Euclidean geometry does much with comparatively little. First, it teaches deductive logic. Second, it emphasizes precision and certainty. Third, it teaches complex abstract thought. The ancient philosopher Plato thought so highly of its benefits that he wrote, “Geometry will draw the soul toward truth and create the spirit of philosophy.” Thus, the Greeks considered geometry a prerequisite to philosophical thought.

Finally, perhaps the most fascinating aspect of geometry is lost by most textbooks. What one discovers after spending some time with the Elements is that by talking about lines and figures in such a systematic and logical way, one is really exploring the limits of human rationality. Geometry is much more than a study in lines and circles. It’s an inquiry into our own rational natures, and that makes it beautiful, human, and sometimes even frightening.

Details

  • This 32 week online course meets once per week for 2 hours.  There is also an optional weekly discussion seminar midway through the week to allow students to get extra help solving problems and grasping concepts.
  • We will cover Books I-III, V-VII of Euclid’s Elements, selections from Nicomachus’ Arithmetic, and selections from Mathematics for the Nonmathematician by Kline.  See the Classical Math & AGA Comparison Chart (PDF) which outlines the concepts we will cover over the course of the year.
  • Class time will be spent discussing and working out propositions. Students will be expected to work through propositions during the week and will take turns presenting the proofs to the class. Students should expect to spend up to 2 hours per school day studying & working problems.
  • During the week, students will be responsible for keeping up with weekly assignments & practice problems posted on the course Study page, which will become available to them after registering.
  • Three essays will also be assigned.

Both students who excel at math and students who think math was invented for the punishment of mankind are welcome. Students enrolling in this class should be strong readers.

Required Books & Materials*

*Reading selections not listed here will be provided in electronic format.

 


Classical Mathematics II

This class is no longer being offered live.

Prerequisite: Classical Mathematics IWesterner_and_Arab_practicing_geometry_15th_century_manuscript

Students wishing to take this course must first complete Classical Mathematics I.

Required Books & Materials

(Book list may be revised without notice)

Details

  • This 32 week online course will meet once per week for 2 hours.  There will also be an optional weekly discussion seminar midway through the week to allow students to get extra help solving problems and grasping concepts.
  • We will plan to cover further selections from Euclid’s Elements, further selections from Diophantus’ Arithmetica, and broad sections of Apollonius’ Conics and Viete’s Analytic Art. We will also plan to begin Descartes’ Geometry and learn to graph on the Cartesian coordinate plane. See the Classical Math & AGA Comparison Chart (PDF) which outlines the concepts we will cover over the course of the year.
  • Class time will be spent discussing propositions & theorems, and carefully examining algebraic & geometric concepts. Students will be expected to work through propositions during the week and will take turns presenting the proofs to the class. Students should expect to spend up to 2 hours per school day studying & working problems.
  • Students will be responsible for keeping up with weekly assignments & practice problems posted on the course Study page, which will available to them after registering.
  • Three essays will also be assigned.

 


Classical Mathematics III-IV

These courses are still in development.

 

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