Most college students have not spent much time thinking about the learning process, which is too bad, since, after all, learning is their primary goal. If pressed for an opinion, most probably would say that learning is a process of aggregation: one has a certain amount of knowledge already, and more is added to it as you learn. This is much like a ball of clay with small bits being added piece by piece; over time, the ball gets larger. That's learning.
Even if they haven't thought about it, this is the model followed by many students; but it is not a very effective model. In particular, it suggests that when a mistake is made, it is because of a lack of knowledge. The student then asks "What am I missing?" or "What is the thing I don't know that will make it possible for me to do this right?" and looks for some new information that he can add to what he knows to give him a complete picture.
The problem with this approach is that the error frequently is more basic than that: something that the student thinks he knows to be true, which he has used as the basis for the rest of his work, simply isn't true. No matter what new information is brought to bare, it will never make it right until that initial idea is changed.
To proceed at this point, the student must not just look for new ideas, but must go back an re-evaluate his old ideas to find which one is faulty, and must remove it and re-learn a correct idea before progress can be made.
"It ain't so much the things we don't know that get us into trouble. It's the things we do know that just ain't so."
— Artemus Ward
For example, if a student thinks that
(a + b)2 = a2 + b2, then no matter what new material is learned, the student will continue to get wrong answers until he goes back and identifies this wrong idea as the source of the problem, and replaces it with a correct one. Artemus Ward once wrote: "It ain't so much the things we don't know that get us into trouble. It's the things we do know that just ain't so."
In this way, learning is a destructive process as well as a constructive one. Most students don't seem to recognize that they should be constantly measuring their old ideas against new ones that they are learning, and should be discarding perhaps-comfortable old ideas in favor of more-powerful new ones. Learning is a process of change, not just addition.
All too often, students seem to discard new ideas if they don't fit snuggly into the framework they already have developed. Rather than ask whether the framework needs adjusting, they simply disregard the new idea as irrelevant with no further thought.
If it seems contradictory with what they already know, then the new idea must be at fault. This is a very limiting way of going about life; it dooms you to repeat the same errors over and over. Introspection and re-evaluation of your ideas is crucial to learning, and it is sad to see that it is so infrequently practiced today.
Learning is a destructive process as well as a constructive one. ... It is a process of change, not just addition.
These ideas are particularly pertinent to introductory math courses at college, since they challenge your preconceptions of what mathematics is about and how it is performed. You need to rebuild those concepts if you want to succeed. Furthermore, you will be introduced to ideas using a formality and precision that you may not be used to. Chances are your initial impressions of their meanings will not be correct; after all, bright mathematicians have been working out the details for hundreds of years, so it would be the height of arrogance to believe that your first impressions would be correct or complete. It will be crucial for you constantly to reconsider the things that you think you have already learned, because some of them will be wrong, and you will need to tear them down and build them up again with new insight.
When I mark something wrong on your homework or quizzes, this is intended to help you locate these faulty ideas. Don't make the mistake of asking what you didn't know that would have made it right; ask what you thought you knew that just wasn't true. Think of these as opportunities for change rather than as failures.
I know that not everyone is doing this, as some of you make the same mistake over and over and over again. This is a sure sign of the aggregate learning model. Remember, the ball of clay is rather ungainly until you begin to kneed it and work it. It becomes more uniform and coherent that way. In doing so, however, you sometimes find lumps or hard spots that need to be removed in order to obtain a finer texture. Treat your learning in the same manner.
In my own college career, I recall a professor in a literature course who asked on the evaluations at the end of the term "What book did you least like in this course?" Having committed ourselves to one, his next question was "What character flaw does this point out in you?" While this may seem harsh, it is exactly the kind of self-evaluation that makes real and lasting learning possible.