I’ve been teaching mathematics for over 30 years at the community college level. Most people at the time of taking a course don’t have a sophisticated enough understanding of math to really appreciate “intuitive explanations” because they don’t have intuition.
Take parametric curves. I explain that they generalize the concept of a function. Every function can be parametrized in a trivial way. They don’t really understand this concept. They have a hard time parametrizing a function and do so only becuase of a formula.
The fact is most people need to go through the mechanical process of doin g before they can get to a point of understanding. It takes almost the entire semester for me to convince beginning algebra students that the reason that 2x + 3x is 5x is because of the distributive property. And when they do understand it they don’t understand why that is important.
Later on when things click for someone they will often say things like, “Why didn’t they just tell this when we took the course?” Usually we did. You just didn’t have a sophisticated enough understanding of things to grok it at the time you took the course.
I am sure you are right, and a little convo here isn't going to do the topic justice. So many aspects to how people understand and learn things.
And I know picking on your example isn't in the league of a general solution.
But if 2x (which is x + x) is two apples in a box, and 3x (which is x+x+x) is three apples in another box, then you put those two boxes in a bigger box (another +), people already intuitively can see the distributed property of scalar multiplication vs. addition of some unit, they just didn't have a name for it.
Likewise, a 3x4 square of paper next to a 7x4 piece of paper can be easily seen to be a 10x4 piece of paper. Multiplication of numbers over added numbers.
So one way to introduce distribution is to start by showing examples of several places where people already understand the concept of multiplication distributing, and use it every day, but just didn't know it was one concept with a name.
Once people can recognize distribution as an already familiar relationship in everyday life, then the symbols can be visited as the way we write down the already known and useful concept so we can be very clear and general about it.
Anyway, that's just a reaction to one example, which may not mean much.
Yes, almost everyone gets that. Then you need to explain that x is really 1 x. Then you need to explain that -x is really (-1)x. Everything is great. We all understand. Now simplify (1/2) x + 3x. You’ll lose most people at this step. Then explain (1/2) x - (2/3)x. More confusion. Now explain that ax+x is (a+1)x. You lost a lot of people at this step. Now explain that xy+ x^2y is (x+x^2)y and that this is just the distributive property “in reverse”.
Sometime later a person will really grok all this and then say, “why didn’t they just tell us this is all just the distributive property?”
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.
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.
Take parametric curves. I explain that they generalize the concept of a function. Every function can be parametrized in a trivial way. They don’t really understand this concept. They have a hard time parametrizing a function and do so only becuase of a formula.
The fact is most people need to go through the mechanical process of doin g before they can get to a point of understanding. It takes almost the entire semester for me to convince beginning algebra students that the reason that 2x + 3x is 5x is because of the distributive property. And when they do understand it they don’t understand why that is important.
Later on when things click for someone they will often say things like, “Why didn’t they just tell this when we took the course?” Usually we did. You just didn’t have a sophisticated enough understanding of things to grok it at the time you took the course.