[eggy]

Algebraic Geometry is all you say it is, however, working with young engineers who have not had more geometry in their curriculum, I've noticed they don't have that intuitive spatial connection to reality other than through CAD or their lived experience. I am reading Tristran Needham's book, "Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts", and it is amazing. I love Sir Michael Atiyah's famous quote that Neeham uses at the start of the book:

"Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine."

And it continues: "...the danger to our soul is there, because when you pass over into algebraic calculation, essentially you stop thinking: you stop thinking geometrically, you stop thinking about meaning."

Being someone who worked with their hands in the real world building things before the current Maker movement took off, and then went on to physics, engineering, and more abstract matters, I see this deficit in the young engineers with a Master's in Mech. Eng., but no real-world experience, or intuition of the world around them and geometry. Algebraic Geometry has its place and it has achieved much, but I strongly opine that children should play in the real world a lot, learn mathematical concepts with a strong geometrical underpinning, and then move into things like Algebraic Geometry. I have watched my children's curriculum and it is no where near as heavy in geometry as it once was for me.