It's pretty clear you've never instructed a remedial math class.
My university would conditionally admit students who had a math score below a certain threshold on their ACT, I think it was 19 or something. Anyway, as part of their admittance criteria, they had to attend an after class lab for an additional hour an a half for a total of three hours per week dedicated to learning pre-algrebra. There were four modules where they would do some reading, work through some example problems through interactive software, and then have some homework to work through that was almost identical to the examples. Students would try and try and try and try to learn the material, and they would take days to work through the problems on their own (often with my guidance, giving pointers on how to think about the problems) to finish the module so they could take the quiz and pass the module. Near the end of the semester many students made appreciable progress, but for others the inability to retain and apply what they've spent so much time on results in tears, especially because they don't know if this requirement will keep them from being able to graduate.
Given infinite time, could these guys all have figured out pre-algrebra enough to pass? Maybe. But the amount of time it takes them to learn math concepts that are very easy for us means that it's entirely impractical to expect them to ever achieve proficiency in advanced mathematics.
> There were four modules where they would do some reading, work through some example problems through interactive software, and then have some homework to work through that was almost identical to the examples.
This just does not look like a universally effective way of teaching to me, irrespective of the topic. It's hardly any wonder that some people fell through the cracks if they were unfamiliar with the subject in the first place. What about leveraging stuff that's actually been tried and tested, like the Khan Academy videos and their automated interactive, school-like environment?
My university would conditionally admit students who had a math score below a certain threshold on their ACT, I think it was 19 or something. Anyway, as part of their admittance criteria, they had to attend an after class lab for an additional hour an a half for a total of three hours per week dedicated to learning pre-algrebra. There were four modules where they would do some reading, work through some example problems through interactive software, and then have some homework to work through that was almost identical to the examples. Students would try and try and try and try to learn the material, and they would take days to work through the problems on their own (often with my guidance, giving pointers on how to think about the problems) to finish the module so they could take the quiz and pass the module. Near the end of the semester many students made appreciable progress, but for others the inability to retain and apply what they've spent so much time on results in tears, especially because they don't know if this requirement will keep them from being able to graduate.
Given infinite time, could these guys all have figured out pre-algrebra enough to pass? Maybe. But the amount of time it takes them to learn math concepts that are very easy for us means that it's entirely impractical to expect them to ever achieve proficiency in advanced mathematics.