**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:

“You…got 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 How to Ace Calculus 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. 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 bulb…study 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, with high energy, 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**

- How to Ace Calculus
- How to Solve Problem Sets without Staying Up All Night
- The Quarantine Method
- Use Technical Explanation Questions when Studying for Technical Classes

(*Photo by foundphotoslj*)

So, basically – recopying your notes is the “insight” of this post?

I think you missed completely point of the article. He was not just re-copying his notes, he was going through every proof from scratch and making sure he got it right and understood every step. That is a far cry from just re-copying notes.

That is the

oppositeof the insight of this post. I taught myself to be able to recreate every proof taught in the course from scratch. Building the study guides was just the setup phase for the insight-driven review that followed.What does it really mean to “recreate every proof taught in class from scratch?”

It really does sound like you just re-copied notes. I mean, I assume you didn’t start from basic axioms and proved individual operations for every problem. For example, if you proof involved set operators, I’m assuming you didn’t prove them, you just used them.

He’s not talking about recopying your notes. Recopying your notes won’t get you to the deeper understanding you need for a certain problem whatever the problem is whether it’s be proofs, a physics problem or an engineering problem. You need to understand the foundation of what’s going on behind the problem and how you can use what you learned to tackle it. Also note that he says he doesn’t do it for more than 2 hrs and he comes it at with fresh energy. That’s very important to learning as research studies have shown that we learn best when we use both focused and diffused modes.

Good article!

Hmm..

I also do something like that at math. Just review ALL the questions asked in the book. And in the period, I make ALL the questions, not the one the teacher gives to us, but all of them.

It works.

Thanks Cal! This method helped me do well in Organic Chem, where the professor would write examples on the board, but take less than straight forward routes to solve them. By writing each type of problem on a paper, and practicing them until I could do them with no hesitation or problem, I was able to do well on his tests.

Awesome,

exactlythe follow up I was looking for after the Calculus Insight post, which is one of my favorite. Thanks for sharing Cal, it means a lot to us 🙂If I could go back in time and take your blog with me, I would. Just so I could re-take discrete math and gleefully be the one person in the course who actually gets it.

I found that a technique similar to this worked in my climatology course this term. We needed to be able to use the locations of semi-permanent features (meteorologically) and how they interact to create specific climates. I constantly was printing copies of maps and writing in the features to create a study guide for each season. It might not be writing proofs, but making sure you know how things work is a key part of any natural or applied science course.

I agree. Just what the doctor ordered, Cal. Makes me feel better already about returning to science courses.

I used that technique for my molecular bio class. I was able to recreate every lecture and detail in my notes, and I studied with my friend who crammed. Before each exam, I was teaching the material to him because I knew it so well. I just got one of the highest grades in the entire class and got the single highest score on the second exam.

Awesome post! A great follow up to the post on learning insights. Of course, I’m a bit biased- I’m going to be taking exactly that course next term. =) You can bet I’m going to be making some proof guides…

Did you ever watch a movie called Training Day? I watched this in my second year of college, and suddenly had a slight envy of policemen, and gained respect for their attitude and outlook. In it, Denzel Washington’s character says:

“It’s not what you know. It’s what you can prove.”

I know you already know this. But it’s nice to have it in quote form, hmm?

And what you can prove in discreet math is by what you know!

It’s great to hear stories of you guys using this method in a variety of courses — from molecular bio to climate science — with equal success.

@Hung-Su: If I ever write a math textbook, I’m putting that quote on the cover.

this is like olympiad sciences/math in a nutshell:

insight>anything.

@Amy,

I had a question about your Molec. Bio class. Did you use a similar technique to what Cal used, that is constantly writing on white sheets of paper or did you just keep writing out notes on regular paper – kept shrinking them until you had known everything?

Thanks for expanding on your use of the method.

I just wrote them out as if I were lecturing to the class. I made sure to use the logical progression of thought like the original researchers did in their experiments, and I was telling a story with my notes. This made me learn everything within a specific context, and it fit together better and forced me to fill in the holes of what I was learning.

Every scientific fact we learned, we had to know and interpret the original experiments and papers, so I asked myself, “what is the evidence for this proposition?” Then having learned that and seeing the correct evidence, I would then ask, “what would this then lead us to conclude or ask next?”

This is still hard. I go to the professor and he can’t explain things well. I read the book which is not as advanced as our assignments. I don’t know what to do. I keep going through tutor after sucky tutor. I just do not understand how to do this. The only thing I haven’t tried is doing every problem in the book. Which I shall do tomorrow and then I can really bech out if that doesn’t work.

Without insights, simply doing problems won’t help. You need to understand *why* the main concepts presented in class are true.

What subject are you taking? Can you give us some sense of the type of material you’re struggling to learn insights for?

I’m having trouble with Calc 2 and manipulating these sequences and series. I also feel like sometimes a simple problem is hard because I forgot how to do something I did in earlier problems.

I’m also doing discrete math and things like recurrence, Big O notation, and inductive math are hard.

Great post Cal. Keep em coming.

What about for such seemingly abstract concepts from Green’s and Stoke’s Theorem? I realize that that’s a rather specific question though…

Great post. Technical classes always gave me problems, the more I read this blog the more I see WHY. Looking forward to reading your books over the winter break!

PS. Are you considering writing a book for Grad School? I’d love to read your insights on what to do as an undergrad to get in, and HOW to survive it once you’re there.

Keep up the great work!

I have a math final on Monday, under these circumstances, how do you suggest that I apply your technique?

Right. 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.

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…

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.

To the best of your ability. 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, I have a few questions:

1) What did you major in college?

2) How can somebody with bad memory adapt to such a system of success? For example, my math course is divided into 18 units. I may get something close to 100 on unit 3, but by the time I’m on unit 4, when someone asks me for help on unit 3, I would have a lot of trouble helping them.

3) You should get a plugin that notify me of comment updates 😉

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 why…for 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 is…and 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 exactly like the mega problem sets.

So, the entire point of your post is that, we should all understand the material and not just regurgitate it? I hadnt realized that the age-old rote memorization technique was useless when every problem was different! Who knew that comprehension was preferable to not? Thank you so much for this…

I always study two days before exam. No wonder I never did exceptionally well. :p

Studies show that regular study over time improves long term memory, whereas with cramming around half the material is lost by the end of the week. How many of those proofs do you think you can still recreate?

The questions in the comment sections are f… interesting. Thanks for taking time and answering them. Also, think about, implementing RSS feed in the comments.

now i know what’s my problem with real analysis class.

Great stuff. Thanks for the insight.

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 me…Makes sense… Now if I could only become the same way again…Keep 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

onlythat. 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.In reply to Jim: It appears you have misread the post.

He doesn’t memorize proofs, he ensures that he can quickly develop a proof, through to completion, and he used the proofs presented in class as a starting point. Those proofs are (hopefully) chosen to allow a variety of solving methods, and inability to solve any of those proofs, shows a weak spot in comprehending that method.

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.

Jom, By the way, what have you done that’s so great? Do you create anything, or just criticize others work and belittle their motivations?

nice ad hominem there Andres, sidestepping the central issue like a pro as usual. real smooth.

I’m going tonight for my first class in discrete math, I’m hoping your post is going to help me. Thank you!

Hi, I am currently taking discrete math with application in computer science. Thank you for this post and I will apply the strategy here in my class. I, however, have a question. My professor went by really fast and every time I stopped to ponder what he did he was already several steps ahead. Should I just focus on taking notes and try to understand it later or understand the material right then and risk getting behind during lecture? Thank you.

Does anyone have any tips on applying this technique to General Chemistry 2 – Acid Base equilibrium concepts?

Go to the end of your chapter on A/B chemistry and do ALL the problems which you have the solutions to. Then active recall all the steps and talk through the steps showing WHY.

This is amazing. I’m taking my first “Intro to proofs” class called “Introduction to Higher Level Mathematics” in the fall. The syllabus states that this class determines your success in higher level classes. THERE WILL BE NO LACK OF INSIGHT DURING THE FALL 2014 SEMESTER!

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.

Thank you.

Not so much a question about how to study, but a question on propositional logic that I’ve never been able to get an explanation for. In a standard conditional statement p–>q I understand why if p is true and q is true the conditional is true and that if p is true and q is false the conditional is false. What I don’t understand is why when p is false and q is true that the conditional is true.

Ryan. 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…

Are these in addition to Mega-problem-sets? It seems like you are sort of recopying your class notes here, which is good is for understanding them, but were these “proof guides” in addition to mega-problem-sets and/or class examples?

If n is a perfect square, it doesn’t follow that n is odd. I hope the professor didn’t prove that it is!

True. 4 is a perfect square. So are 16 and 36, etc.

25 is odd and so is 81… Both perfect squares.

I totally feel this. TT____TT I have hundreds of bond papers and ballpen refills. And, just a very short usual convo to share when I’m studying in the library with friends:

Non-math friend: Why do you rewrite your notes?

Me (math major): That’s just how we do things.

Cal, I read your post at the start of this semester, took your suggestions on proof obsession to heart, and just found out I got a solid A in discrete math!

Hi! I stumbled on your blog and glad I found it!

I was actually struggling with Discrete Math but thank God I found this.

I’ll look through your blogs and books and hopefully enjoy it ! 😀

from the Philippines

jk :DDDDDD

Just reading this blog post made my discrete math endeavors 50% less stressful. Bravo…

I am taking this class right now. My problem is that I get lost in the middle of the proof or I don’t know where from to start proving a statement :((

But how are we defining odd in that proposition?

Thanks for sharing the work flow. This sounds like what I did in the 80s when I took calculus I and II. Now I’m back in school taking Discrete Structures and wondering how on earth I will hack my way through it.

I’m glad I read this. Recopying the proofs and getting the understanding in place will help. I found it made a big difference in how I studied for two semesters of P-Chem, too.

First class and the professor gave us a quiz. I know I failed that thing. Not proud of it. Just a wake up call to get my butt in gear NOW.

You were surprised you aced the course after you put in the maximum possible amount of work?

I would be surprised if you had achieved any other result.