One thing I've learned a lot about in my Discrete Math class, somewhat inadvertently, is how my fellow students learn and the ways they struggle. There are only twelve of us in the class, and our prof typically does only a little bit of lecturing before having us work on problems and lecture each other. Quite frequently we get sent to the board with markers to write out solutions and, more importantly and unusually, explain things to each other. He will often have several of us explain the same concept for the benefit of the class.
There is, of course, a range of ability within the class, and each person's ability also changes from task to task. Sometimes I have seen other people pick things up more quickly than I do, and at other times I've been stunned as the rest of the class has seemed to turn into morons.
One of the topics my class has really struggled with has been probability. I also had some probability this semester in my Prob & Stats class (as you'd expect), and people seemed to struggle with it there too, though it's not as obvious in a large and non-interactive classroom. But I've heard from some people that, when they took that class, they had no problem with the statistics part, but did badly with probability.
I am no expert at probability and I have learned new things in both of these classes, but probability has always come somewhat easily to me. There seem to be a very small number of principles, and then you just have to figure out how to apply them to the situations in the problems. The principles themselves seem to make logical sense. Because there is a not a lot of fiddly algebra and everything makes logical sense (rather than seeming arbitrary), I do very well with it.
I have guesses about why other people struggle with it. I think one problem is that all the problems are word problems. A couple of students in my prob & stats class were talking once about not being able to figure out what the questions were asking - questions like "What is the probability that no more than 4 valves fail?" And in Discrete, even my professor has struggled to understand the questions sometimes, though he's obviously quite smart.
I think the other problem in probability is that, because it's about applying a few rules to diverse situations, it's not a field of math where you are given extremely specific question types and shown how to solve them. In calculus, for instance, you typically have a section of very similar questions and you are shown methods that always yield the correct results. In probability you really have to analyze the situation and think of equivalences between disparate types of events.
These are just guesses. There have been times when I've really wanted to tear my hair out at the struggles my classmates were having with things that seemed very straightforward to me, and I've had to remind myself that they haven't seemed dumb or underprepared on all topics, but that there is apparently something specifically difficult about probability.
What have your experiences been like?
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3 comments:
My experience is that the fuzziness (and uncertainty?) of probability bothers people. People aren't good at dealing with probabilistic situations in real life, either. Statistics is simple, the mean either is, or isn't, 5. Probability isn't so simple, and seemingly small distinctions matter. People always underestimate the importance of moderately low probability events, and overestimate the importance extreme ones, and simply misconstrue what the actual probabilities are. Correlation simply escapes people. (Hence the mortgage crisis - a moment's thought is necessary to reach the conclusion that mortgage defaults would be strongly correlated, because of the effect of economic downturns, but no one thought about it in those terms, and so they were shocked when their 'safe' - because they consisted of a huge number of mortgages with estimated probability of default, which, had they been independent, would have simply converged to a pretty stable mean value - derivative instruments based on mortgages went belly up due to a (very small and not particularly unusual) systematic rise in default risk due to economic conditions.)
The Monty Hall problem is a classic example. Everyone reaches an intuitive answer and is absolutely convinced of its correctness, but 90% of people are simply wrong, and they can't be convinced of the actual answer. You can show it to them mathematically, actually set up and run the game repeatedly to demonstrate the answer, walk them through it with props, math, logic, and they are still convinced that their answer is right. (Believe me, I spent a good week trying to convince a staff of financial analysts, economists, and engineers - not exactly the most innumerate or probability-impaired groups - and still was left with people saying, 'ok, mathematically that might work, but it isn't RIGHT'.)
That's funny about the Monty Hall problem. I think it's pretty simple to explain (as do you, I'm sure).
On the first day of my Computer Science 3 class, the prof brought up the Monty Hall problem. The class was unable to come to a consensus as to the answer, so he said, "This is a good time for experimental computer science," and sent us home to write up a program to find out the actual odds.
I love having the ability to incorporate that kind of very basic "research" into probability. Ed and I have been working problems lately and I've always felt reassured thinking that, if we couldn't come to an agreement about the answer, we could always write a program to find out.
I once told my mom she should just play the numbers 1-6 in the Lotto. She would never do so, of course, because the odds are so low that those numbers would come up. Um.
When I worked in a grocery store, on the occasions when someone's bill would come out to an even dollar amount, they'd often say, "What are the odds of that!" and I'd have to restrain myself from saying "About 1 in 100." (It's actually lower than that because not all totals are equally likely, but you get the idea.)
I remember about 10 years ago, Robert was playing a game of chance with some of my family members and my cousin (then about 14 or so) said at one point, "Huh, what's the odds of that" and Robert said what the odds were. My cousin's response was, after a pause, "So you can actually figure that out? Cool."
My mom one time mentioned, "Does Robert realize that when people ask that question, they don't actually want to know the answer?" I said, "Yeah, he just can't help himself."
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