[Nevermark]

Ah, I get it.

A different approach I have thought about, which would really tear the textbooks apart, is introducing every concept in its simplest form as early as possible.

Then when it is eventually expanded on, its familiarity will aid in taking further steps more quickly and intuitively.

For instance, something as simple as adding up the area of a fence of varying heights, or the area of multi-height wall to be painted, being referred to as integrating the area, in early grade arithmetic, creates a conceptual link for down the road.

Systematically going over K-12 materials, just making similar small adjustments to terminology and concepts to be highlighted, would be interesting.

[skhunted]

As I see it the issue with your integrating example is that area is the correct word for finding “area”. Integrating is not finding the area. The indefinite integral is not about area. The definite integral in dimension 1 has to do with signed area. I don’t think having people ingrained to think finding area is “integration” would be a good thing. Especially since most people don’t take calculus.

To your point, people do constantly try to tweak things to make subjects easier to understand and more intuitive.

[lupire]

That's how Riemann integrals are defined in class. Vertical slices.

Tossing the word "integral" at younger children won't make that easier or harder.