Last fall I took the "Introduction to statistics" course at the University Berne (see for example my post about Boxplots). It was my last course after 7 years of studying, but it turned out to be the most insightful one I had so far. Working part-time, I had lots of time to spend on a single course, and also to reflect a bit on learning as a habit and gaining some deeper knowledge about my own learning patterns and difficulties I had, and I'd like to share some of those thoughts.
1. Write your own summaries
I remember using a summary from a friend ("Neat, I don't have to do it myself!") for an oral exam in a history subject. I tried to get a grip on the things he wrote, in the way he wrote it - but it was just not possible. The information was already quite dense andtotally okay for a quick overview, but when I sat down with my docent, I didn't remember anything from it. Crafting your own summary can be fun too, and help you to prepare the learning material in a way you're getting the most out of it (see Get your gear ready).
Write down your Eureka!-moments. When you learn, sometimes it just feels like a knot unravels and you really get that warm, exciting feeling of a deeper understanding in a certain topic. Most of the times these moments are about small things – a procedure which finally makes sense, or you get what has been used to solve a certain exercise. I noticed that writing those down has two primary benefits: You remember better what exactly you understood, and also it’s easy to understand that particular thing again really quick, which is crucial in exam preparation, when looking through the important stuff again.
Sometimes it's even a good idea to just focus on these little moments exclusively, to just be aware where your very own difficulties are when solving exercises, and take corresponding measures. For example, I noticed that I sometimes had trouble with integrals, and although it wasn’t even part of the lecture but a prerequisite, it came up quite a few times before I decided to do a little refresh on the topic.
Keep digging. When summarizing an exercise or a concept (or ideally explaining it to someone) you typically understand a whole lot but also come across some unknowns. While it is important to focus on important things, digging deeper has a similar property to asking questions, pondering on it helps you get an even deeper understanding. When producing a summary, those things you don’t understand emerge much clearer than when you just consume and learn without actively recreating anything, so it is a good practice to at least write them down, and maybe look at them another day or discuss them with other people.
2. Ask questions
One of the most important things I learnt was to ask questions. I used to be (and still am sometimes) a bit of a loner when it comes to learning, I'm just much more focused when I'm alone and also the planning is much easier. If I'm tired, I skip a learning session, if I feel like it and I'm making good progress I can learn all into the night.
Use the available human resources such as course assistants, other students, ask questions, talk to people, send emails. Understanding isn't just about reading text books or solving predefined exercises, but also about an active communication with other human beings. Sometimes I could ponder literally for hours on a single problem, and taking it with me into class and discussing it with other people solved it within minutes. Communication is key.
Write down your questions. Writing down clarifies the exact problem of understanding and sometimes even solves the problem, or at least breaks it down to what exactly you don't understand. Think it over another time, ask other people for help - sometimes what's difficult for you is really easy to explain for others.
Follow the exercise schedule. If a question hour is provided, be sure to use it - It's like a free private coaching! Make use of that, write down your problems, ask questions. Thinking "Oh, I didn't manage to finish the exercises this week, I'll do them next week" is a vicious cycle and really hard to break, even if you have plenty of time available. This is why: If you try to solve a problem, fail, think about it for some time, boil it down to what exactly you don't understand and then solve it with your tutor - this is experience and understanding on the subject you'll never be able to get just from looking up the solutions some weeks before the exam.
3. Be aware of your different focus levels and use them
Different parts of learning require a different level of concentration - for example, memorizing words or copying notes from a friend don't take as much concentration as getting a new concept or trying to solve a difficult question. Often learning combines at least some different activities, which you can easily split up into high- or low-concentration tasks.
Doing so is crucial, especially if you're working and can't (or don't want to) spend all of your time learning, you have to cleverly manage your time.
Create a high-concentration environment without any distractions and visit it regularly. I have some space at home, but also used to stay at the library in the university or in my company after work, in a quiet, distraction-free room to really be able to do some serious studying. This is where the real learning happens and where I gain most of my insights. Use it only if you feel like it and if you're ready, it doesn't make much sense for trying tired or exhausted after a long day of work. Sometimes it's also hard to really focus and don't procrastinate by doing things you could do later, like organizing stuff or browsing through forums for a specific problem.
Save other activities like organizing your material or learning stuff by heart for low concentration environments, such as a train ride, a break at work, watching TV, etc. Doing so can save you a lot of stress, as you know exactly which tasks to do when. It takes away the feeling which puts you under pressure when you are doing other things, like "I should be doing some studying right now" - no! You're aware that all you can do now, casually, is some repetition or light reading and that you're ready to deal with complex problems in the sessions where you are highly focused, well rested, and undistracted.
4. Get your gear ready
In humanities, I used to learn out of the textbook and take some notes, learn from these and maybe read through all of it again before the exam. Then I started studying computer science and suddenly became aware that this just wasn't enough anymore. I had to find efficient ways to not only learn new stuff, but also being able to remember it for a long time.
Find suitable forms of representation. Once you get the idea of how much you'll be learning and what stuff to remember, you'll have to process and represent it in a way which lets you get a quick hold on it, so once you'll look at it you'll go "oh wait, I know this one and how it works". Obviously, this will not be possible with everything, so you have to write down more complex things too. I generally distinguished between:
- simple facts or formulas, which I would write down on these little memory cards
- algorithms or complex procedures, which I would first try to understand, then summarize them and put them down with some good examples in a dedicated notebook. Adding some colors, graphics and extra facts about certain problems really turned it into a neat summary of nearly the whole lecture and I enjoyed working with it very much. I also recognized that I'm the graphical learning type and benefit from using colors and fancy drawing stuff - I think this is quite individual, others try to remember things with little stories or even by singing them :-) As long as you find out what gets you going.
Gather every bit of relevant information you can get. I used to have one big Dropbox folder with literally everything about the lecture in it - descriptions, links to websites (Wikipedia articles, specialized pages like Matroids math site), exercises, solutions, all the stuff from last year’s lecture, script, my own notes from the lecture, other notes, etc. etc. I really liked the freedom to just pull out my tablet anywhere - for example, I was in the Netherlands (twice) in the time before the exam, visiting family, and did a good part of the learning there - just because I had everything ready. I even scanned in my whole 70 pages of notes on the lecture the day before I left to have everything available.
5. Get used to regular learning
One thing that really helped me as a quite spontaneous person were the fixed learning sessions I planned on a regular schedule, spread across the week. I had mine on Monday morning, Wednesday afternoon and Saturday morning, also because work let me. They changed over time (see Adapt your learning behaviour over time), becoming more frequent as the exam came closer. I arranged with myself, that no matter what other things were around, I would always use these sessions to do some studying, and after some trouble in the beginning I managed to integrate them into my daily routine.
These learning sessions were also the place where I tried to get "in the zone", to create a high-focus environment as already described, so I could rely on those sessions to work through the stuff that popped up during my work week, which I used to write down on a list especially for that purpose. It also took me a while to realize that I wouldn’t be able to use my time given efficiently when learning at home, too much distractions. It’s really hard to learn while you could do some housekeeping or cleaning, too :-). Going to a quiet room at the university or the library helped a lot.
Identify important topics. Depending on the style of lecture, you'll have some kind of exercises or papers to turn in, which gives you a rough estimate of what to expect later on. As we had weekly task assignments to hand in, there were lots of topics covered and the idea on important topics was quite vague.
Throughout my studies, I noticed some indicators for important topics:
- An exercise (or part of it) keeps coming up through multiple task assignments.
- A certain concept is the foundation for other exercises, or there's a common thread through a part of the exercises. Those kind of exercises also make nice exam questions, where the different parts of a questions are intertwined.
For example, I once drew a dependency model (of topics) in linear algebra:
As you can clearly see, the orange marked balloons are really the foundation for pretty much everything else, so it would be a clever thing to invest some extra time into them. (Also, there were lots of exercises which combined some of the marked bubbles.)
- Carefully listening to what the prof said gave me some good hints. Usually the exam won't consist of topics the prof finds unimportant or boring.
- Asking the course assistants also helped me a lot, usually because of their experience they have a good sense for more important topics and will tell you so.
Divide and conquer. When faced with a big task, for example, learning to integrate, it's mostly too big to cope in a single session. Also beginning is really hard if you see the dimensions of the things you want to cope with and you stand before a mountain of work. Dividing the task is at utmost importance, before you despair. Most of the time it is a good idea to begin by looking at practical examples like a specific task: How can you integrate 2x^2? Building on some basic tasks and expanding the scope by conquering certain parts ("Oh, powers of x are mostly treated the same way. Let's learn some other common rules of integration."), you'll be able to gradually build up knowledge.
Don't get lost. This one's sometimes a hard one to spot – finding the compromise between breadth and depth on a certain topic. It's tempting to just stick to something you're understanding instead of moving on and facing the stuff that gives you a headache. I know some people who were really confident before the exam, which I knew studied a lot, and would then say afterwards, "Wow, what was that – I just totally blew it because I learnt the wrong things!". As trivial as it sounds, learning the right things is important and success can be misleading and a poor guide in this regard.
Ask yourself how much time you've got left – realistically – and adjust your learning depth accordingly. You don't want to end with some important topics uncovered! Identifying the most important topics as mentioned before is extremely helpful not only for learning the right things, but also for planning and time allotment.
7. Track your progress
Keep a diary of what you do and learn - just a few key-words or topics you've been working on, one line per learning session. It pushes your self-confidence to see what you already accomplished, and it's also a neat summary of the topics you already covered or where you should invest some more time and effort.
Get a grasp of the important things already described in section Prioritize, and make a list of exercises, or of topics (or both!), to see how you’re going on a big scale. Draw a mind map and see what needs you attention and what you already covered sufficiently.
8. Apply your knowledge in real life
Another thing that really helped me was the application of what we learnt (quite easy in statistics, but also useful on other subjects) - not just in the weekly assignments, but testing it on some real life data. And to maintain your interest to invest some additional time - since it is not directly part of the syllabus, strictly speaking, and an additional workload - it is important to search for some interesting examples.
An example: We had some data provided from students the years before us, a questionnaire filled out with some questions about origin, habitation, income and so on. One category was "smoking", and they could answer with "yes", "no" or "occasionally". I looked at the categories men/women and compared their smoking behavior, which where nearly identical, relatively speaking, in the "yes" and "no” categories. I then used Fisher's exact Test to show that if you pick an occasional smoker at random (from a group like the one the data came from, and with as many men as women), the odds would be nearly 1.7x higher for that person to be a woman.
Without even thinking if the group size was representative (it wasn't) or the group was skewed (it was) the data still yielded a new and unexpected result, which helped me understand the meaning of statistical odds and simultaneously showing some results on real data.
Connecting your knowledge. The even bigger benefit of practicing methods and techniques is the newly gained experience when to use which concept from the vast amount of information and skills you're learning. Asking yourself the questions, "Is this concept appropriate in this case? Will the result of this test have any meaning?” and even remembering what procedures are available at all helps you to connect and classify the new knowledge.
Another big advantage is that if you choose good examples, they are more likely to stay in your memory. Even now, months after my exam, I still recall most of my own data experiments in detail - opposed to the given tasks we had to solve.
9. Be persistent
As time goes by and you're in a continuous flow of learning, sometimes you just get thrown out of the routine, either by a lack of motivation, getting sick, going on holidays or whatever. I thought for a long time that this (immutable) process was just plain failure - that I was not able to be a really continuous learner. Over time I realized that only my reaction to these interruptions could be changed, all other things are plain luck and not in my influence.
Get going again - maybe the simplest rule, but also the hardest - it's normal to lose one's rhythm, but the difficult part is to get going again after a break and getting back on track. One good practice is to think long-term: I want to succeed, therefore I have to start learning again, if not today then tomorrow. It's not really a decision if, but when to start learning again, and every day I procrastinate is a day closer to the exam, unused.
Actively care about your motivation. One thing that really helped me stay motivated was the application of stuff we learned, so I tried to find some interesting problems to solve (see Apply your knowledge). It's really fueling to have some real life stuff to work with and the feeling to already making use of the newly acquired knowledge.
It also helped me to set some intermediate goals, which would typically split a learning session into three or four parts, and which also were a good indicator to take a break and grant myself with a cup of coffee, a walk around the block etc. Enjoying a free afternoon is so much more rewarding after you had a good learning session or some items ticked off the list.
10. Adapt your learning behavior over time
There are usually several months of studying involved for a single lecture, and it doesn't make much sense to just stubbornly go through everything in chronological order (see Prioritize), or even to stick to a certain way of learning from the beginning until the end, because we don't work that way. Our brain learns by repetition and actively using the new skills (see Apply your knowledge).
Because you make quite a big progress from understanding nothing and being thrown at with new concepts to gradually getting a grip to – hopefully – a wholesome understanding, also your way of learning should change. Besides your own progress, there are also the circumstances that change, for example the lecture finishes and you'll get an additional month or two until the exam is due, with more time on your hands to spend.
In respect to those two factors, I used to split my learning into 3 stages:
- Lecture stage
- Understanding stage
- Reviewing stage
Lecture stage. The beginning of the course with lectures to attend can be quite stressful, so the main motto here was for me: do the weekly exercises, write down everything you hear, survive :-)
Understanding stage. Like preparing for a marathon, this is the main stage after warming up to lay a good foundation for everything that follows. It is the last time it makes sense to catch up on things that were left behind earlier. Usually this is the most intense stage in the whole learning process. This is also the time to work out a neat summary (see Write your own summary). Here you should try to acquire and deepen your knowledge, apply it, and do some additional work, if time allows.
Reviewing stage. In the weeks before the exam, you should recognize a shift of what you do and learn, the focus lies more on reviewing already learnt material rather than looking at completely new problems. Here you should have seen everything at least once, and coming across something you don't understand yet you have to decide whether to start working on it or to leave it aside (see Prioritize). Focus on learning by heart, connecting the knowledge, and understanding the higher purpose of the concepts, why and when to use them.
Prepare for the modus operandi of the exam. Last of all, you should prepare yourself specifically for the exam – what kind of circumstances do apply, what do you have to know by heart? How many tasks will there be, how much time do you have for each of them? Also, most of the time writing a summary is just for the sake of learning, not for bringing it with you at the exam. So plan enough time not only to write your summaries, but also to look at them and memorize – eventually it has to be in your head, not on paper.
Learning with memory cards really did the job for me, working with a simple system where cards get laid aside if I know them, and from time to time I would repeat the whole stack. Then, in the days before the exam, things I didn’t know or understand by then I learnt by heart, to be at least able to write something down, should it come up in the exam.
Never try to understand something on the day before the exam. One of the most important rules regarding the exam - if you don't get it two days before the exam it's just too late. If you try to get something new, it will only distract you and make you feel unsure, focusing on things you can't do instead of relying on everything you've learnt so far.
Good luck ;-)