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I don't know-- usually the answer to such questions about academic priorities is "because it was cheaper", but they just seemed to emphasize geometry much more
by lambdaphage 13y ago
I don't know-- usually the answer to such questions about academic priorities is "because it was cheaper", but they just seemed to emphasize geometry much more at the K-12 level. It's a generalization, of course, but a pretty robust one. One of my professors came from Kazakhstan, and once casually remarked that a certain problem on a homework set was "impossible unless you were Russian", since the proof was easy if you knew a certain proposition from Euclid, but extremely tedious without it.
EDIT: this interview with Izaac Wirzsup comparing the Soviet and US systems confirms my prejudice:
Another extremely harmful feature
of [the US] school mathematics programs
is that only about half of our students
take geometry, and for only one
year, generally in a concentrated high
school course. Students cannot be
expected to master the material taught
in this way. Moreover, they are not
being taught solid geometry, and they
rarely have a workable perception of
three-dimensional space, which is so
essential for studying science,
technical drawing, or engineering.
Soviet children study geometry
extensively for ten years, including
two years of solid geometry.
- nawitus 13y agoI've also noticed that the Chinese have a pretty high focus on geometry.
- netcan 13y agoJut from my own common sense it seems like geometry is important for understanding the relationship between abstract things and concrete things. It's easily understandable that shapes are described by geometry and it seems obviously useful. The square footage of a house. The volume of a bath. If you try to describe what Calculus is or does, it's abstractions of abstractions. Rates of change or 'angle of a curve for a certain values. I think it's hard for students to see this as something useful or even see how it's a description of the world that opens up ways of understanding it.
- jjoonathan 13y agoI don't see calculus as an abstraction of abstractions. The fundamental idea is completely geometric: "break the domain of a problem into a bunch of pieces that can be easily described and related (e.g. by physics) then put the pieces back together." Time is a first-class dimension. Abstractions only enter the picture when you want to separate the problem of picking a mesh from the problem of representing mesh elements. Differential operators perform the task of "breaking into pieces" in a mesh-invariant way. Differential forms are mesh-invariant pieces. Integration is the mesh-invariant description of putting the pieces back together. It's convenient that differential forms can be interpreted physically (by normalizing, associating with geometric elements, etc) but I'd hesitate to associate them with any single physical interpretation (e.g. rates of change) because doing so de-emphasizes the generality of the approach; you can have a rate with respect to distance, area, or volume just as easily as a rate with respect to time. Leibniz notation makes the hop from the geometric approach to the "operator that maps a function to a function" approach seamless, and since the latter description isn't nearly so intuitive, I've always suspected that the geometric approach could profitably be taught first.
- Tloewald 13y agoMy mother learned math in the French system in the 50s and it also stressed geometry.
- atmosx 13y agoThanks for the extensive resp.
- S4M 13y ago> One of my professors came from Kazakhstan, and once casually remarked that a certain problem on a homework set was "impossible unless you were Russian", since the proof was easy if you knew a certain proposition from Euclid, but extremely tedious without it. Do you remember what was the problem, by any chance?
- lambdaphage 13y agoI don't remember-- it must have been either differential geometry or topology, but I think the theorem in question was the inscribed angle theorem: http://www.proofwiki.org/wiki/Inscribed_Angle_Theorem http://www.proofwiki.org/wiki/Inscribed_Angle_Theorem. Not a difficult theorem, but you had to know it well to be able to see the application immediately.