> This conservative non-holonomic system has a seven-dimensional accessible configuration space and three velocity degrees of freedom parametrized by rates of frame lean, steer angle and rear wheel rotation.
I always adore the split between how my brain does things instinctually, but making it arbitrary completely demolishes the 'natural' flow of it. Same with complex ball throwing / bouncing trajectory calculations.
It also immediately makes me angry about how we teach math. When you learn about powers (squares, cubes, roots, etc), these things are just written out as arbitrary concepts instead of displaying them geometrically.
Hell, when I was first taught the Pythagorean theorem, it was just explained by drawing a triangle with A² + B² = C², without also drawing out the related squares of each side. Immediately doing that would instill so much more intuition into the math. In general, mathematical concepts gain so much clarity by doing them geometrically.
Sounds like a problem with your early math tutors: especially with geometry, all the examples you bring up have been taught with "what it means".
I mean, squares and cubes are just multiplication by the same factor: I distinctly remember even trapezoid surfaces, pyramid volumes being demonstrated by chopping and piecing parts together.
In the US especially, too many programs have an insistence that doing things with symbols and doing things with shapes and solids are completely different things and don't relate the two.
Primary school programs? Secondary school programs?
I could understand that in high school and uni when you need to move past "intuitive" maths into abstractions, but I'd be perplexed if this is really the "program", and not just an individual teacher (and surely, a big chunk of them too)!
We've started to swing back towards visual and geometrical models with common core/new math.
But I have a big collection of math textbooks from all over the world, and math 6, pre-algebra and algebra texts in the US have far less geometrical content and pictures than the non-US equivalents that I have. I also think that the increased emphasis on getting students through standardized tests ended up dedicating a lot of class time towards rote with short term benefits.
I always adore the split between how my brain does things instinctually, but making it arbitrary completely demolishes the 'natural' flow of it. Same with complex ball throwing / bouncing trajectory calculations.
It also immediately makes me angry about how we teach math. When you learn about powers (squares, cubes, roots, etc), these things are just written out as arbitrary concepts instead of displaying them geometrically.
Hell, when I was first taught the Pythagorean theorem, it was just explained by drawing a triangle with A² + B² = C², without also drawing out the related squares of each side. Immediately doing that would instill so much more intuition into the math. In general, mathematical concepts gain so much clarity by doing them geometrically.