Geometry And Discrete Mathematics 12 Class
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Decoding Patterns of Success Case Study: How I Got the Highest Grade in my Discrete Math Class November 25th, 2008 A Hallway Encounter During my sophomore year at Dartmouth I took a course in discrete mathematics. The tests were not calibrated to any standard scale, so it was difficult to judge how well you were doing. On the midterm, for example, scores around 50 to 60 out of 100 were at the top of the class, whereas for the final those would be failing. Rewind, then, to the end of the winter quarter, and imagine my surprise in the following scenario. It’s the day after the final. I’m walking through a hallway when I encounter the TA: “Yougot the highest grade,” he said. “On the final?” I asked, somewhat surprised.
“No, for the entire course.” This was hard to believe. The course had 70 students. Three of them were from Eastern Europe where, educated in the old Soviet-style talent-tracking system, they had already studied this subject in high school! I didn’t think of myself as a math person. Before this class, I had shown no particular talent for the subject. I was trying to just hang in there with a decent grade.
My victory, as we like to say here on Study Hacks, was tactical. In this post I will explain how I achieved this feat, and how following similar strategies can help you dominate even the most thorny technical courses No Tolerance For Lack of Insight At the high-level, my strategy was exactly what I spelled out in my post of two weeks ago: learn the insights. But I want to dive into the details of how I accomplished this goal for this specific class. Think of this as a case study of the insight method in action. Here was my specific strategy:.
Proof Obsession: Discrete math is about proofs. In lecture, the professor would write a proposition on the board — e.g., if n is a perfect square then it’s also odd — then walk through a proof. Proposition after proposition, proof after proof. As the class advanced, we learned increasingly advanced techniques for building these proofs. I soon developed a singular obsession: I wanted to be able to recreate, with pencil and paper, and no helper notes, every single proof presented in class. No exceptions.
Discrete Appl Math
Lack of understanding of even one proof wouldn’t be tolerated. My Obsession in Practice Here’s how I learned every proof. I bought a package of white printer paper.
As the term progressed, I copied each proposition presented in class onto its own sheet of paper. I would write the problem as the top of the sheet and recreate the proof, from my notes, below.
I tried to do this every week — copying the most recent material onto its own sheets — though I often got behind. While doing this work I would sometimes — okay, many times — realize I didn’t quite understand the proof I had copied in my notes.
In these cases, I would break out the textbook, or do some web searching for the problem, to see if I could make sense of what I was writing down. This usually worked. In the worst case scenario, I would ask the professor or the TA for help. Not understanding the proof was not an option. I wasn’t practicing transcription; I knew I had to learn these. About two weeks before each exam I started scheduling sessions to aggressively review my “proof guides.” I always worked on the second floor of the Dana Biomedical Library on the outskirts of campus. (Think: dark, concrete-floored stacks, with desks tucked away at then end of long rows, each illuminated by a single, bright incandescent bulbstudy heaven.) I did standard Quiz and Recall: splitting the proofs between those I could replicate from scratch and those that gave me trouble, and then, in the next round, focusing only on those that gave me trouble, and so on, until every sheet had been conquered.
By the day of the exam, you could give me any problem from the course and I could rattle off the proof, without mistake and without hesitation. Lots of Work, but Not Hard Work In retrospect, it’s not surprising I did well in this class. Most of the other students — even the Eastern European students — started studying for the exam 48 hours in advance, trying, frantically, to review as many of the high-level techniques as possible. Not surprisingly: a lot of details were missed. They knew the basics. But they lacked mastery.
Consider, by contrast, my approach. If you add up the time I spent copying out the proofs on the white paper, add in the time required to track down help for the proofs I didn’t understand, and then throw into the mix the time spent reviewing, the total is somewhat staggering. To try to do the same a few days before the exam would have been literally impossible. This doesn’t mean, however, that my life was hell. If anything, this was a relaxing term. The secret was that I inlined my work throughout the term. I never spent more than 2 hours at a time working on these proofs.
I never stayed up late. I never ground through material. I kept attacking it fresh, time and time again.
There are two lessons I hope you take from this case study:. Conquering a technical class requires a massive amount of work. There is no short-cut. If you’re pulling high school bullshit and trying to wait until a few days before to learn everything you slept through in class, then you’re screwed. You need to grow up and leave that behavior in the past. Conquering a technical class doesn’t have to be painful.
The key is to define your challenge — learn every insight — come up with a plan for winning the challenge — e.g., in my case, using proof guides to learn every single proof — and then putting the plan into motion with time to spare. No cramming necessary. Know thy enemy and it becomes a lot less fearsome Related Articles About Technical Classes. ( Photo by ).
I’m also doing discrete math and things like recurrence, Big O notation, and inductive math are hard. That’s why you have to learn this, proof by proof. As each proof gives you trouble, get help until you understand it. Then — and this is important — review the proofs using the quiz and recall method.
This means you try to recreate the entire proof from.scratch. without peeking. If you can do this, then you’ll remember. What about for such seemingly abstract concepts from Green’s and Stoke’s Theorem?
That’s slightly different. Presumably you’re not being asked to re-derive Stoke’s Theorem. You need only understand what it says and how to use it. The relevant practice there is probably describing the theorem from scratch (i.e., writing it out without looking at your notes) and being to do walk through one or two sample uses, explaining every step as if lecturing a class Are you considering writing a book for Grad School?
I’m actually moving in the opposite direction: my next book is about college admissions. I think grad school is an interesting subject, but (a) the experience widely varies depending on the program; (b) the market is small; and (c) I haven’t yet graduated so I don’t feel qualified yet to say that I’ve succeeded. I have a math final on Monday, under these circumstances, how do you suggest that I apply your technique? To the best of your ability.
Rosen Discrete Mathematics
Hit the main proofs and concepts first and see how much time you have left. Start right away.
Work early in the day. Work in isolation! Hi Cal, this post is exciting cuz discrete maths is what i’ll be doing soon!
I have been looking closely at how I can develop insights am stuck at a few places where things seem to be so basic that the textbook does not go into explaining whyfor example, how do we.understand. innately what the truth table for ‘if p then q’ means? In two instances where the hypothesis (p) is false, the conditional proposition is true no matter what q isand this baffles me because I don’t understand why.
The above is just an example of confusion we would face when trying to develop clear, good insights, and we can deal with this in two ways: 1) accept the results for the proposition above ie memorise or 2) clarify with someone. I’ll take it to my tutor soon, but I’d appreciate your thoughts on how you clarified/understood seemingly basic propositions (or for the benefit everyone: concepts). Thanks for the great post! This is like a dejavu to me. My parents are mathematicians. When I was growing up (in a post-Soviet and not Eastern European country btw)my dad used to make me to the exact same.Although I never understood what was the fuss about going over a calculus, a geometry, a physics and all other science books was all about (since I thought I was a diligent student anyway), each week and from elementary till high school (except for grade 11)he would have those sessions where I had to solve a randomly picked problem or prove a theorem or something else.
Needless to say, I would get penalized if I couldnt do them (talk about zero tolerance). The end was exceptional-I was the highest ranking student both in high school and in medical school. But up until now I was thinking that I was simply a diligent person myself and he just made it hard on meMakes sense Now if I could only become the same way againKeep harping on it Cal, you are doing everyone here an unbelievable favor! This is absurd. Learning proofs is about learning proof techniques and being able to recognize cases that are analogous to, but not the same as, previous proofs covered by a professor. Yes, a lot of mathematics is about memorization and repetition; however, you’re espousing only that. The focus should be on developing techniques, not the ability to copy what the professor has done previously.
This method will not carry you through higher-level mathematics courses where the emphasis will be on ingenuity, not regurgitation. Gail, you have misunderstood. The article states, in bold type no less, the following strategy: “I wanted to be able to recreate, with pencil and paper, and no helper notes, every single proof presented in class.” If you are rewriting a proof you have already seen, you are not, in any significant way, focusing on technique. Instead, a student of discrete math–or any proof-based mathematics course–should be attempting proofs that use similar techniques but are solving different propositions. For example, if the professor proved the statement “x is even if and only if x^2 is even,” then the student should not attempt this same proof; instead, the student should try to prove (or disprove) a different bi-conditional using a similar technique. I have to say that I was a bit disappointed by this post.
It was wonderful, but the whole problem lies in how to understand your mathematics and physics. Once you understand a lesson, it is easy to remember it. Can anyone tell you how to achieve full understanding? Unfortunately, probably not. I wish you wrote a piece on how to prepare a written maths or physics exam consisting of problems. If you have to work and rework around, let’s say, 300 problems to pass your quantum mechanics exam, what is the best way to do it, having in mind again that 70% of it is probably the understanding on how to solve it. P-Q can be read as if “event P” occurs then so must “event Q”.
“True” means that the proposition is valid (without flaw). Assume for example that event P is the event that Peter sneezes and Q is the event that Quincy gets sick. Let the Bold Proposition be P-Q: “If Peter sneezes then Quincy will get sick.” Then in this case if P is false and Q is true we have “Peter did not sneeze and Quincy got sick”. Observe that this doesn’t poke any holes in the Bold Proposition since the claim was that Peter sneezes forces Quincy to get sick (so the BP is valid/true). The only way you could prove that the Bold Proposition to be flawed is if Peter sneezes and Quincy doesn’t get sick.