The greatest obstacle to discovery is not ignorance, but the illusion of knowledge.
— Daniel Boorstin
One of the challenges facing anybody who is learning something new is that you always start from somewhere already. What’s already there may either help or hinder or both. In many cases, learning something requires a rearrangement of what’s already there. Another way of saying this is that learning anything sufficiently interesting requires unlearning as well. Near as I can tell this is unavoidable, and so we can save ourselves the trouble of deciding whether it is a good thing or a bad thing.
What I’ve just said will either land for you as something so obvious it is perhaps not worth pointing out explicitly, or it will seem like a strange statement for which I should be prepared to offer ample evidence in support. I intend to flesh out in greater detail how learning and unlearning shows up around math and what impact it could have on the way we relate to math education.
A simple example to give flavor to what I’m talking about is the way children interact with “+” or “×” or “-.” It is very typical to see children interact with these symbols as telling them to do a particular thing. “+” is a command to add numbers together, “×” a command to multiply two numbers. So, when Jana sees “3 × 4,” the action Jana knows she’s supposed to take is to figure out how much 3 times 4 is and report on the answer. Jana is supposed to produce the number 12. Years later in school, Jana is introduced to formulas such as t + 1. Most teachers are well aware that children have a difficulty with this thing called “t” which is kind of like a number but isn’t quite a number. What is less clear that children also have a difficulty with the “+” in “t + 1.” For in “t + 1” the addition is no longer a “do it now” command that the student is supposed to execute and produce a number. In “t + 1” the addition is there to be thought about, picked up and held up against the light and looked at, and treated as something that has an existence other than “stop everything else you’re doing until you’ve produced” the third number that is the product of these two other numbers.
My point in giving this example is not that we should somehow introduce “+” to kids from the very beginning as something other than “produce this now.” In fact, my point is quite the opposite: that it is natural and unavoidable that later learning demands a reorganization of earlier learning, and sometimes to the point of completely unlearning something that seemed so clear and obvious at the time.