Dunno if I agree with this, just sounding it out seeing how it feels.
>>Can literacy be taught with hitting students with a stick when they err?
>And then the answer is yes, as this is how it is done.
(19th century).
>Now, why would be want to do otherwise? Why would we ever want people to learn less?
Maybe you don't learn less by not being hit? Even if my grandparents all had vastly better handwriting than myself. Perhaps I learned some other things not being met with such hostility in my education? I don't know for sure.
At some level of complexity we learn "too hard for me to integrate by hand" and also stop learning how to perform more and more complex integrals. We stop. Where is the most productive place, to maximise useful learning, to stop?
Maybe not being hit with a stick cost me beautiful copperplate handwriting. I feel it was more productive for me to not be hit. I can see someone taking a different view on that with regards themselves. "Never did me any harm" is still a pervasive way of thinking about what others call child abuse. Not many wish they were hit more.
What do you get out of integrating by hand? At each step ratcheting up complexity from y=x, through each increasingly complex function type to where you stopped? Are there diminishing returns? Or do you get past a threshold opening up whole new areas of understanding otherwise not experienced? Something else?
I note anyone can integrate y=x and get 1/2x^2 + c so I don't think avoiding this integration by hand is the point.
> At some level of complexity we learn "too hard for me to integrate by hand" and also stop learning how to perform more and more complex integrals. We stop. Where is the most productive place, to maximise useful learning, to stop?
I preface this by saying I find math very difficult. However, it is the struggle to understand in mathematics that leads me to understand the concepts. It often seems too hard, but I found that if I stuck at it I actually started to understand what was going on.
Not popular advise, I realise, but those who can stick with it seem to find maths almost fun, despite the pain.
>Can Calculus be taught with differentiating or integrating by hand?
And then the answer is yes, as this is how it is done.
Now, why would be want to do otherwise? Why would we ever want people to learn less?