I think Archimedes had some great non-infinitesimal methods for discovering the area and volume of shapes. Some very visual methods involving his method of exhaustion for the volume of a sphere for example. These ideas were transcribed from some Palimpsest discovered with various drawings on them too but almost hidden. These methods are known now so why aren't they taught in high school (a way to introduce High School student to Calculus)?
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I think I answered this question here:
Are there any theorems that become "lost" and discarded over time?
Mathematics is very large. There are many beautiful things we could teach. But the time available is limited. Therefore we teach what is considered most important. Geometry (parts of Euclid) and algebra in high school. Calculus and linear algebra in the College. This is a very small part of mathematics.
answered Dec 28, 2014 at 21:14
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$\begingroup$ This doesn't answer my question about Archimedes methods and how they should be taught in high school ( with modern terminology). $\endgroup$– 201044Commented Dec 30, 2014 at 4:20
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$\begingroup$ Thew short answer to your question is that Archimedes methods should NOT be taught at high school. However reading Archimedes himself is highly recommended to a young student with a strong interested in mathematics. $\endgroup$ Commented Dec 30, 2014 at 20:09
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$\begingroup$ Why shouldn't Archimedes methods be taught in High school to encourage a fascination with mathematics? $\endgroup$– 201044Commented Jan 1, 2015 at 6:21
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$\begingroup$ Archimedes methods where visual and involved physics style thought experiments like finding the center of mass in an object the shape of a triangle ( actually a thin triangular prism) and balancing it , or something like that ( if I got that right). So relating visual images and physics thought experiments to find out a geometric idea is a fascinating mix of math techniques that could be taught in high school. $\endgroup$– 201044Commented Oct 22, 2015 at 4:07
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$\begingroup$ Didn't Archimedes using his method of exhaustion and other arguments determine the ratio between the volume of a cylinder with a circular base and a sphere of the same diameter fitting snugly in it. So indirectly discovering the formula for the volume of a sphere way before 'pi' and modern terminology developed? These arguments the forerunner of calculus might be very useful to students.. $\endgroup$– 201044Commented Feb 27, 2016 at 0:13