The class began with some introduction and administrative information as expected. The topic then moved on to the subject of communication. Essentially it was about how statements made by humans taken out of context tend to be pretty ambiguous, while computers need language to be very strained and unambiguous (programming language). I felt that this material was pretty lighthearted, which I think is a good start for a mathematics course that may seem daunting.
The topic then moved on to problem solving, with some examples relevant to python. We were also taught George Polya's approach to solving problems. What was most interesting to me about these principles was the fourth one, to review and extend. As students we've been conditioned to move along to the next problem once we get things right, never looking back. Even if we've been suggested to do otherwise, the nature of rigorous examinations doesn't really permit one to revel in the moment of a solved problem (or at least, I don't feel like you get the time to). That being said, there ought to be some value in reviewing what you did (the rights and wrongs) at that very moment when you still understand it (and, you know, because you're also really happy). What Polya suggest is that if you go over what you solved, it can help you to extend those strategies to problems in the future. My goal is to use this strategy whenever I can to solve future problems.
As for the specific examples, the ones went over in class for now were pretty reasonable. I was able to figure out the q0 through q3 problems on my own, however, in class we went over them using venn diagrams, which I thought was a very helpful way to process the true and false statements. There was also the streetcar drama problem. When I first saw this problem I actually thought that there wasn't a complete solution to it. My initial reaction to all math problems has always been to look for the numbers and just work with that. I think this problem really pushed me to find information from the text, and especially things that don't seem like givens at all (eg. the sum didn't reveal the answer).
I actually went ahead and looked at the course notes and some of the logic puzzles in there. I tried out the first one on 3 boxes using the Polya strategy:
1) Understanding the Problem
There are 3 boxes and 1 has the prize. You get to pick one of the boxes, and the host will a different box that doesn't have the prize. Now you get the choice between the box you already picked or the other unopened one. Which one would give you a better chance of winning?
So, what I know at this point is that upon first selecting from the boxes, I have a 1/3 chance of making the correct choice.
2) Devising a Plan
I started out by considering each option, whether the first pick is better or to switch to the other box. I considered sticking with the first one picked, but my logic was that it wouldn't make a difference between the two (50/50), not that the first was actually better, but that doesn't answer the question. So there had to be more to this problem concerning probabilities.
3) Carrying out the plan
I thought about how probabilities may make the two choices different. Eventually I realized that while on your first turn you have a 1/3 chance of choosing the right box, on the second turn that chance is increase to 2/3 now that one was revealed, so that must mean that the switch increases the likelihood of winning the prize. This was more of a guess at this point, but it seemed possible, yet counter-intuitive.
4) Review and Extend
I spent some time trying to unravel if the 2/3 idea made any sense. Eventually, I'll admit, I just looked it up and found that this problem is identical to the Monty Hall problem. So, I guess the switch would be correct. In the future I'd probably try to solidify my logic before finding a source for it, but it did help me make more sense of it. At this point, I'm still trying to extend this knowledge to the next set of puzzles, so perhaps next week I'll go over how they went.
The topic then moved on to problem solving, with some examples relevant to python. We were also taught George Polya's approach to solving problems. What was most interesting to me about these principles was the fourth one, to review and extend. As students we've been conditioned to move along to the next problem once we get things right, never looking back. Even if we've been suggested to do otherwise, the nature of rigorous examinations doesn't really permit one to revel in the moment of a solved problem (or at least, I don't feel like you get the time to). That being said, there ought to be some value in reviewing what you did (the rights and wrongs) at that very moment when you still understand it (and, you know, because you're also really happy). What Polya suggest is that if you go over what you solved, it can help you to extend those strategies to problems in the future. My goal is to use this strategy whenever I can to solve future problems.
As for the specific examples, the ones went over in class for now were pretty reasonable. I was able to figure out the q0 through q3 problems on my own, however, in class we went over them using venn diagrams, which I thought was a very helpful way to process the true and false statements. There was also the streetcar drama problem. When I first saw this problem I actually thought that there wasn't a complete solution to it. My initial reaction to all math problems has always been to look for the numbers and just work with that. I think this problem really pushed me to find information from the text, and especially things that don't seem like givens at all (eg. the sum didn't reveal the answer).
I actually went ahead and looked at the course notes and some of the logic puzzles in there. I tried out the first one on 3 boxes using the Polya strategy:
1) Understanding the Problem
There are 3 boxes and 1 has the prize. You get to pick one of the boxes, and the host will a different box that doesn't have the prize. Now you get the choice between the box you already picked or the other unopened one. Which one would give you a better chance of winning?
So, what I know at this point is that upon first selecting from the boxes, I have a 1/3 chance of making the correct choice.
2) Devising a Plan
I started out by considering each option, whether the first pick is better or to switch to the other box. I considered sticking with the first one picked, but my logic was that it wouldn't make a difference between the two (50/50), not that the first was actually better, but that doesn't answer the question. So there had to be more to this problem concerning probabilities.
3) Carrying out the plan
I thought about how probabilities may make the two choices different. Eventually I realized that while on your first turn you have a 1/3 chance of choosing the right box, on the second turn that chance is increase to 2/3 now that one was revealed, so that must mean that the switch increases the likelihood of winning the prize. This was more of a guess at this point, but it seemed possible, yet counter-intuitive.
4) Review and Extend
I spent some time trying to unravel if the 2/3 idea made any sense. Eventually, I'll admit, I just looked it up and found that this problem is identical to the Monty Hall problem. So, I guess the switch would be correct. In the future I'd probably try to solidify my logic before finding a source for it, but it did help me make more sense of it. At this point, I'm still trying to extend this knowledge to the next set of puzzles, so perhaps next week I'll go over how they went.
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