from Guide to Undergraduate on Nov 20, 2022
To maximize lecture, listen like you're about to reteach it.
I usually fell asleep in lecture, and when I didn't, I took notes on slides I didn't understand, in a notebook I'd never open again. To understand and maximize lectures, listen like you're about to reteach that lecture.
Classes are stressful for a variety of reasons: Sure, the material is difficult, and sure, there are also many weekly deadlines — across homeworks, projects, labs, etc. However, I realize now in retrospect that the stress was aggravated by another serious problem: I was sitting through hours of lectures and labs, furiously taking notes^{1} that I didn't understand or would never revisit. This created a dual problem: I (1) not only lost the many hours of my day, which I could've spent on assignments but (2) I also didn't learn anything during those many hours.
Lectures and talks don't always need to be this way. In industry now, I still see maybe 34 hours of "lecture"like environments several days of the week — either I'm learning from a speaker or sitting in a large group meeting. Despite my previously lackluster interest in lectures, I do find these new "lecture"like hours very worthwhile.
There are several reasons for a renewed interest in lecturelike hours. This post covers the key ideas that have made lectures more worthwhile for me.
Find out why this topic is important.
There's an important distinction between school lectures and all the talks I do attend now: I know in advance what I'm attending the talk for. This explains why some of the most popular lectures are review sessions. We all know in advance why this lecture is important: It covers exam material.
There are certainly other instructorenforced techniques: mandate lecture attendance for a grade, iClickers to track inlecture poll participation, Google Forms for attendance, and other tactics. This certainly made lecture important for our grades. However, to really learn from lecture, we know to know why this topic is important. To do this, we need to understand the motivation for a topic and how it relates to the rest of the content.
 Skim the introductory paragraph in advance. The goal is specifically to identify the motivation for that lecture. Take just a few minutes to read the introductory paragraph of the lecture notes. If notes for your semester aren't available, find notes from previous semesters. If notes for the course aren't available, search for the topic online.
 Skim the note for references to other notes. The last resort is to skim the entire note and see which other topics it references. Understanding how the different topics interact can help you establish the topic's importance on your own.
 Ask course staff for the topic's motivation either on a course form or in office hours. I'm sure instructors would enjoy answering a higherlevel question, "What is the motivation for topic X, and how is it related to the course topic Y?". At the bare minimum, your instructor should be enthusiastic about and understand the importance. For example, the first note of Discrete Mathematics^{2} introduces "Propositional Logic" by saying this lecture introduces notation you'll need later on. However, after discussing with the instructor, you may find a broader motivation: This particular course trains you to form rigorous arguments and poke holes in incorrectlybuilt arguments. As a first step, you need a way to translate arguments into math. This way, you can easily identify argument flaws by recognizing a mathematical pattern.
For an example of a motivating example in an introductory paragraph: the "Stable Matching Algorithm" note opens with the following, which states that this lecture simply introduces a new problem to apply what you've learned, to:
Since we don't have the luxury of picking lectures we're interested in, the least we can do is to determine why the lecture is needed at all. Do this as often as you can, so you understand the relationship between what you're learning today, with everything you learned previously.
It's worth noting that a common piece of advice is to read the lecture notes in advance. That way, you go to lecture with question in hand. This is certainly ideal, but it's a little too idealistic. I often didn't have time to do this, so instead of recommending a 2 or 3hour prelecture commitment, I'm instead recommending a short 510 minute prelecture skim.
Ask "How would I teach it?"
Here's a key idea that changes your approach to lectures: Listen as though you're going to reteach lecture. In particular, throughout lecture, actively ask yourself: "How would I teach this?" This forces you to realize when you don't truly understand the lecture. An even easier litmus test is: Prepare a response for your friend, who asks "What did they cover in lecture today?" Instead of preparing a 30second response, ask yourself how you would spend 10minutes summarizing lecture. This forces you to add detail and explain ideas rather than mention topics. In preparing to reteach lecture, use the following tactics to maximize the cohesion of your summary:
 Understand the takeaway first and foremost. The most important time to do this is when you're lost in a proof. It's okay to be lost, and don't get hung up on line 2. Try to determine how this proof is related to the rest of the lecture. Knowing the takeaway is far more important. Once you've established the takeaway, understand how this fits into your outline.
 Anticipate the next step or concern. Again, especially during a proof or a multistep solution, try to anticipate the next step. This is precisely what you'll be doing on an exam or for a problem set, so might as well practice here in lecture. Granted, you're learning the topic for the first time, but even if your guess is wildly off, providing any guess at all will improve the stickiness of the actual next step your instructor takes. After the fact, summarize how we go from each line to the next in a proof. Justifications can be as broad as "algebra", more specific like "expanding squared term", or specific applications of what you learned, like "applying Baye's Rule". Don't make the mistake of getting hung up on the first confusing step. Treat it as a crossword puzzle: Figure out as many steps as you can, and for any remaining steps, ask for help.
 Organize a hierarchy of topics for that lecture. As lecture progresses, construct a written or mental model of the lecture's outline. Fortunately, if your lecture is accompanied by lecture notes, the notes already provide a basic outline. This gives you a better idea of your misunderstanding: Is it just an example proof? Or is it a key idea for understanding all of lecture? Just knowing where you are in lecture can make a big difference in your confidence. Feel free to modify the outline to your liking; the more you modify the outline to make it your own, the more understanding you demonstrate and the more it sticks. For example, I would use the following outline, slightly distinct from the original Stable Matching Algorithm note.
 proposeandreject algorithm
 Setup: Hiring managers and candidates. All managers rank candidates...
 Steps: Every manager sends offer to top candidate. Candidates...
 (omitted background on SMA)
 properties of algorithm
 stability: matching is stable if there is a pair...
 (all the various lemmas for stability)
 optimality: (define optimal and pessimal pairings)
 (all the various lemmas for optimality)
There may also be other techniques for extracting value from a lecture — the best way to come up with these techniques organically is to continue walking into lecture with an eye for how you'll reteach the lecture afterwards. Then, after lecture, there are several steps you can take to solidify the material, outside of working with your notes.
"Use" lecture.
Homeworks and projects are of course designed for you to leverage what you learned in lecture. However, I personally found several other tactics more useful, during my own undergraduate years.
 Compile lecture notes for the class. In some of my classes, the instructors didn't have typed notes to release. They did come to lecture prepared — but with written notes that weren't ready to publish. In light of this, I latex'ed notes in class. However, the most important for me was sitting through my latex'ed notes after class. At that point, I would go through proofs and painstakingly justify each step or rework algebra that didn't make sense to me. I did this for several class, including our machine learning (CS189), designing information systems (EE16B), and graduate convex optimization (EE227C) courses. I treated my original set of notes only as a reference, which I used to guide my rewritten proofs. This certainly counts as "reviewing" in the 5 Rs of Cornell note taking; however, specifically for math courses, this is a bit more than just rereading. This involves reworking solutions and proofs from scratch.
 Teach your friends. Your natural next step after lecture is then to actually sit down with your friends and to teach them the topic. Study the notes—both the official lecture notes and the ones you took—to prepare for your own minilecture. Believe it or not, your version of the lecture will always be far more useful for your friends. This is because you better understand what your friends already know and what they may be confused by. As you teach, you may stumble on parts you find confusing; this is a great opportunity to identify what you truly understand.
 Write practice questions and quizzes. Funnily enough, I did a lot of this as a TA, so you could say that's when I began to truly and deeply understand the material. As you write questions for the material, you'll notice a mental switch into a different mode: You'll start to look for ways to extend existing questions, patterns across questions, and content in the notes that are testable. These are all great practices for "using" lecture.
Homeworks and projects technically serve this purpose, where you're asked to "use" lecture. However, the hectic nature of the school year make these assignments less effective.
Don't overdo lecture.
You don't need to follow the steps above religiously for each lecture. In fact, here are several caveats about lecture that seem counterproductive.
 Don't try all techniques all the time. It's important to try all of these techniques at some point. However, it's counterproductive to do them all the time. You should see which ones are most efficient for you and which ones you're most motivated to actually do. For me, I found extrinsic motivation: I was interested in compiling lecture notes, because my peers loved them when I posted on Piazza. I likewise loved writing extra quizzes, because my students loved them when I uploaded them to my website. Finally, my friends loved teaching each other. I don't know why, but I was ever so grateful.
 Leave lecture when it isn't working for you. If you're ever nodding off in lecture or you're no longer paying attention, the solution is simple. Get up, and head out. You've lost interest, or you've already missed so much that listening won't do much. However, don't squander that time. Go straight to a library and simply pull up the slides or notes. Speed read if you need to, but simply ingest the material on your own. Feel free to head out just the same when lecture time is over. Keep that lecture time dedicated to lecture material, even if it's not spent in lecture halls.
 Take a break when you're burned out. Recognize when you're burnt out. Forcing yourself to drink from a firehose will do nothing when you're completely out of it. Intentionally take a day to completely disconnect and defuse the pressure. Don't think about work on that special day. The school year is a marathon and not a sprint. Use your best judgment for when disconnecting is a possibility, but you're far more productive when you're energized.
These tips above are geared towards your sanity. Take care of yourself, and maximize your learning efficiency rather than your time spent on learning. With those caveats in mind, here are again my tips for maximizing lecture:
 Understand why this lecture topic is important. Ask your TAs or instructor if need be.
 Ask "How would I teach it?" throughout lecture and afterwards. Listen like you're about to teach.
 "Use" lecture in some way. Usually, assignments cover this, but redigesting lecture for your friends and classmates is equally effective.
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There are a multitude of resources that teach how to take notes, so I won't cover that here. Instead, this post covers tactics to maximize learning the content during lecture itself — your mental model and ability to relate different concepts can make or break your digestion of lecture during lecture itself. ↩

The examples in this post use notes from the course Discrete Mathematics and Probability Theory (CS70) at UC Berkeley. You can find a guide for this material in Guide to Discrete Mathematics and Probability Theory. ↩
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