If we want to be serious about improving education, one of the first things we need to think about is the real and perceived relationship between effort and results. Take a look at the following three cases. To keep the discussion simple, I am using unrealistically idealized curves, but they should illustrate the main principles.
Let's say that the X axis in the graph above represents days in the semester and the Y axis represents some measure of mastery.
Which of these learning would we like to see?
It is easy enough to eliminate C. Obviously we don't want kids getting frustrated by going most of the semester without any progress .
Of the remaining two, most people would probably choose A, but I am going to strongly argue for B. I want to see B because I want the students to expect B in the future. I want them to go into every new topic believing that they probably won't see much if any progress at first but eventually they will see a more-or-less linear relationship between the work they put in and what they get out of it.
The great problem with A is that it reinforces students' most dangerous misconception: that math is something you get or you don't; that understanding either comes quickly or it doesn't come at all. You simply can't motivate people to work hard when they believe that work is futile.
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