Information is not Knowledge

In the search for an answer to the question, I have come to the realization that a certain message from various sources (among them is Zuming) is extremely important for people to understand. This is a place where conventional math classes have a huge failing, and before people have realized it, they've assimilated the same failing into all of their studies, both inside and outside of school.

I realized that this is a problem when Dan sent out his latest email. Here is an excerpt from it:

"For a small subset of you, it was your mathematical knowledge that prevented you from making the calculation. To this subset, I assure you that we will attempt to teach you the mathematics. However, at the meeting last Tuesday, I believe that the majority of the people who did not take the relevant derivative, even after repeated requests from me, had the capacity to do so but lacked the motivation."

What is this saying? It's basically saying what I said in my last blog post: that people don't want to work when they know that they can. But I have to question something that Dan said. Do people actually have the capacity to do what they have the information to do? There are some people who "know" the product rule but would almost certainly be unable to do the problem that Dan posed. That is, they have the information but not the knowledge.

I have observed this failing in many places, not just at TJ. Perhaps the most surprising place for some of you would be red MOP. Yes, not everyone at MOP is good at math, as shocking as that may seem. They may be better than you. That doesn't make them good.

I was walking through the blue room (the main lounge) one day and I overheard some red moppers talking about abstract algebra. Some of the things I heard were along the lines of "Yeah, rings are really cool...except I don't really know what they are." This was when I realized the truth of Zuming's statement, but the importance had not yet hit me. What I did know at that point was that informing them of what a ring is would be completely useless. They would surely forget it by the next day, and most of the effect would be lost on them.

Now I think I understand what's going wrong. People love information. When they feel they understand one piece of information, they immediately grab for the "next" piece. Then why they think they're done with that piece, they grab for the next, and the next, and the next. What is the result of this process? It's a ton of information, but no knowledge. When you follow a trail of crumbs, you eat one, then the next, then the next. But did you ever wonder why the second crumb comes after the first, and not before? Sometimes it's obvious - the second explicitly cites the first. But what about when the first cites the second? Why do we learn about real numbers before we learn about Cauchy sequences or Dedekind cuts? Why is calculus the "next" step after algebra and geometry? The people who don't consider these questions are doomed for failure. They will never understand where they are going, and when the line of breadcrumbs dies out, they won't know where they are, unable to find any familiar landscape, hopeless until they find a new trail of bread crumbs that won't help, but rather just lead them further into the wilderness.

So what is the answer? I am not saying that you should be contrary and head off into a completely different section of the woods from where the crumbs lead you. This will probably lead you to just as much failure. After all, there are crumbs there for a reason: this path has worked for other people before, and it will work for you too - if you walk it the right way. You should walk this path, but don't swallow bread crumbs as fast as possible. Savor them. Enjoy the scenery. Maybe even deviate from the path in order to learn the surrounding area. Then come back to the next crumb and do the same thing. Each time, you'll feel more and more comfortable with the area, allowing you to move further and further out into the wild, while still always knowing where you are.

All around me, I see people who have been adversely affected by the bread crumb trail. There was a trail of crumbs. They ate the crumbs; they did their homework. Every once in a while, there was a signpost; they took a test. They got to the end of the trail, marked by a final signpost: the final exam. Where were they? They had no idea. What they did know is that there was another trail of crumbs waiting for them: the next class.

People tend to like hammers, like the Fundamental Theorem of Calculus, or Muirhead's Theorem, or Combinatorial Nullstellensatz. But think: If you have no idea how to use them, how useful are they? Not at all. Information is not knowledge.