What I’ve learned about learning how to learn

I’ve nearly finished with Coursera’s Learning How to Learn course. Here are the big ideas I’ve picked up along the way:

We remember what we think about.

I first heard this from reading Daniel Willingham’s work. It keeps coming up. This is important because it seems straightforward and intuitive, but it’s actually not so easy to align our learning and teaching to this principle.

Getting good at something requires thinking about it over many weeks/months/years. At the same time, there’s only so much you can do in a single day.

Again, this isn’t a new idea, and research on spaced repetition, interleaving, etc. has been telling us this for a while. But if I take a critical look at how I learn, my learning isn’t always set up to value this kind of extended thinking and practice. And as far as I can tell, there’s not much we can do to rush the process.

No matter how well-designed the learning task, it is virtually always possible to do it in a way that doesn’t require good thinking/learning habits.

This is another tough one to deal with, because finishing a task feels so good, and making new connections and thinking hard is so uncomfortable. It’s difficult to be accountable to yourself and other people to make sure that you’re practicing the right things in the right way. And it’s ever so easy to cheat.

It’s difficult to monitor/control how learners are thinking and learning.

I need to keep sitting with this for a while, because it has pretty big implications for teaching and assessment. This idea ties in to loads of the things we teachers talk about — metacognition, motivation, assessment, revisions, the content vs. skills debate, and so on.

A big part of being good at something is the ability to have aspects of the current task remind you of things you’ve already done.

People who are good at a discipline have a vast knowledge base and are great at making connections. One thing I’ve been thinking about is: how do people develop that expertise?

Making new mental connections is uncomfortable, even painful.

For me, there’s a certain joy that comes with learning new things at a very superficial level, but practicing real proficiency is so difficult. That’s a big part of what leads to procrastination.

Even thinking ahead to focused practice can be uncomfortable.

I’ve only recently realized this about myself. This is one of the main things that leads me to procrastinate. I know how hard it’s going to be to start a task that I’ve planned for later in the day, so I spend the whole day torturing myself thinking about how uncomfortable it’s going to be to start. It’s not just studying — it’s similar to the feeling I get before jumping into cold water, or sending an email I’ve been putting off. That also might be one of the reasons that taking a class can help people learn — you know the class is happening at the appointed time, so there’s no point fretting about it, and you can’t put it off, because it’s definitely going to happen. I think it’s might be better to just set the time for doing it, and put off thinking about it until then. When it’s time to start, just start.


Obstacles to enjoying maths

I’m almost done with the Leaning How to Learn course on Coursera. It’s a great course, and I highly recommend it. The course talks about learning strategies in general, but is particularly applicable for maths. I spent a long time thinking I didn’t enjoy maths, or that I wasn’t good at it. Certainly, mathematical thinking doesn’t come naturally to me. But that’s different from not liking maths. So that got me to thinking: what is it about maths that I don’t like?

Not knowing the answer or how to approach the problem: When I look at a language or linguistics problem, I usually have a good idea how to approach it, even if I don’t know the answer right away. When it comes to maths, I often don’t know where to start, and for some reason, I feel a lot of resistance to going back to the chapter and my notes to find concepts that I can apply. It’s particularly frustrating when I know there’s a solution to the problem within easy reach — it’s very tempting to just check the answer. Why bother wasting time thinking about the problem when there’s already a solution? But that’s obviously not a very productive way to think about studying. The Coursera course talked about the importance of (limited) periods of focus, and also the importance of applying both focused and diffuse thinking to problem-solving. So now, when I encounter a problem I can’t solve right away, I make myself sit and focus on it for 25 minutes. If after 25 minutes I still don’t have any idea, I leave the problem alone. I relax, let my mind wander, take a walk, listen to music, whatever. Sometimes an answer comes to me that way. But even if not, it’s good to have the problem in the back of my mind, with no pressure to solve it. I’m not taking a maths class at the moment, so there’s no rush.

Making mistakes: I think it’s pretty common for people to think they’re not good at maths because they make a lot of arithmetic errors. In classes that I’ve taken, arithmetic errors end up being a big deal — they cause you to lose marks, and the teacher gets frustrated by them. It’s been especially instructive taking more advanced computer science courses, because it seems that people are more open about the fact that everyone makes arithmetic errors. So now I’m changing my focus. Of course, I’m trying harder to check my work, and making notes of errors that I make consistently. But I don’t beat myself up too much about forgetting a negative sign somewhere, if my approach to the problem was correct.

Having to read slowly: When I read something about language or linguistics, I can get away with having careless reading. I’m not sure whether that’s because those subjects come naturally to me, or because I’ve had more experience, or because those topics lend themselves well to careless reading. In any case, I can skip over individual words and sentences, and still have a pretty good idea of what the text is about. That gives me the illusion that I understand the material, when maybe I don’t understand it very well. Not so in maths and computer science. Sometimes it takes me an hour or more to read a single page — I have to restate each sentence in my own words, read sentences multiple times, start paragraphs over to understand how each sentence is connected, etc.. It’s slow and frustrating. One thing that’s helped with that comes, again, from the Coursera “Learning How to Learn Course”: focusing on the process rather than the product. Instead of trying to finish a chapter in a single sitting, I frame success as reading and comprehending for a particular period of time.

Time pressure: I think this a big part of why I don’t like the way most maths classes are taught. Lectures, quizzes, and tests make me feel pressured to have the right answer in a limited time. And it doesn’t help that it often feels like my classmates are able to get the answer quickly, when I haven’t even really understood the question. I’ve been in some lectures where the teacher/professor isn’t introducing information, so much as they’re confirming that students already know and understand particular concepts. I think that approach is OK when the teacher has assigned that material beforehand and wants to make sure that students understand. But a lot of times, I’ve seen professors spend entire lectures asking ‘So who’s heard about X?’ questions about material that we weren’t expected to know beforehand. That rewards students with a previous background in the material.  Anyway, enough complaining about maths classes. I’m not actually taking a class, so now I’m getting into the habit of just testing myself, and not worrying about what other people know and don’t know.


New reading strategy

I’ve been trying a different strategy for reading.

Old approach:

  • Read as much as I can
  • As I read, add words and phrases to my vocabulary spreadsheet
  • Organize the spreadsheet
  • At some unspecified point in the future, practice words and phrases on the spreadsheet

The problem with that approach was that the vocabulary spreadsheet would get very large, and more often than not, I’d never get around to actually practicing the vocabulary. I love making vocabulary lists (really!), but they don’t do much good unless I actually use them.

New approach:

  • Read until I’ve read for 15 minutes or gathered 10 new words/phrases, whichever comes first
  • Say/write something using those words/phrases, for at least 10/15 minutes (I don’t have to talk/write about what I’ve read, but I do have to use the vocabulary)
  • Continue, if I have time

Of course, this means I end up reading more slowly, but hopefully with better retention. Part of my problem was that I used to read too fast anyway, so I’d confuse vague familiarity with actual understanding.

Vectors and sorting

Today I worked on the first two exercises from Chapter 1 of Linear Algebra Done Right.

It usually takes me around 25-30 minutes for a single problem, which is discouraging, but I know that I need to put in the time, especially since I’m trying to break a 20-year habit of not doing the exercises.  In particular, I see that I need to work on:

  • Creative manipulation of equations, especially breaking up and combining fractions to get one side of an equation into a workable form
  • Taking advantage of definitions
  • Using equality creatively — if things are equal to the same thing, then they’re equal to each other.

I also worked on three sorting problems from Elements of Programming Interviews. Again, it was slow going, but I figure it’s going to hurt now, or it’s going to hurt later. There’s a big difference between thinking I can do a problem, and actually sitting down and doing it. I practiced:

  • Using max() and min() to determine intervals (this comes up a lot, and I always seem to forget).
  • Using two maps to keep track of counts and array offsets (e.g. in counting sort)

On the bright side, I’m beginning to see echoes of what I study in other fields. There was an interesting talk at work about reinforcement learning today, and I recognized the formula and could actually reason about it. That hasn’t happened before, so it’s pretty exciting.

Approaches to texts

As I’ve written in previous articles, maintaining literacy in a language requires us to explore and create a huge variety of texts on a regular basis. Another thing I’ve been thinking about is varying approaches to texts.

For example, I might read, watch, or listen to something in order to:

  • enjoy the content
  • learn the content
  • learn to imitate the language and/or style
  • maintain communication with someone

I might say or write something in order to:

  • reinforce content knowledge I’ve been working on
  • reinforce language knowledge I’ve been working on
  • practice a particular text type or style
  • maintain communication with someone
  • relax and/or have fun

There are probably other goals that I’ve missed. The thing is, depending on which of those goals I have in mind, my approach to the texts will be different. The goal will also define what I do when I’ve finished the text — how much time I spend reviewing, fixing mistakes, looking up things I didn’t know, starting over from scratch, and so on. So what I’m wondering now is, in addition to planning to explore and create a variety of texts in different genres, how can learners also vary their approaches to those texts? Should learners approach the same text in different ways? Or focus on one approach per text?

More vocabulary adventures: looking words up when writing

It’s been a while since I came up with (and failed to follow through on) the German language challenge more than a year ago, but I haven’t completely given up on German during that time. I’ve been reading a book of short stories, and keeping up with Deutsch im Fokus.

There’s something missing though. I mentioned in a previous article how it’s important to explore a variety of texts in different genres. That’s necessary, but not sufficient — it’s not enough to just read or listen to things. It’s also important to create a variety of spoken and written texts (including spoken conversations), ideally for an audience who will engage with you, such as friends and colleagues. There are loads of obstacles to accomplishing that. For me, one of the biggest obstacles is the resistance to look words up.

For example, if I wanted to write this blog post in German (or even in Mandarin, which I know much better), I’d need to look up quite a few words and phrases. Some of those are domain-specific, like ‘internalize’ or ‘shared literacy’. Others are common turns of phrase that I might not have encountered in my reading and listening, such as ‘now that I think about it’. In cases like those, there might not be a direct translation into the other language, so I’d need to consult a translation dictionary to find the closest equivalent.

When I first started studying languages, I looked up tons of words, all the time. I probably didn’t come even close to remembering (or even practicing) all of them, and that never used to bother me. But now I’ve started to feel like I’ve failed if I have to look up a word when I speak or write. I’m not sure why that’s changed. It’s probably a combination of factors: I’m more interested in efficiency than I used to be, I’m more aware of how language learners rely too much on the dictionary to avoid actually learning and practicing, my idea of ‘success’ in language learning has changed, and so on.

I also think back to my time at the Princeton in Beijing program, which for me was the most ideal language learning experience. It was definitely an immersion experience, but not just in the sense of using Mandarin all the time. We had a community of shared literacy — we were always talking with each other about what we read and heard, using the phrases and structures we learned from those texts. There was such a huge overlap between the texts we explored and the topics we talked and wrote about.

Using what you know, and only what you know, is an important skill. It only gets better with practice, it’s not very satisfying, and it’s tempting to avoid.

But looking up words is inevitable, at least for me. Especially since I’m currently studying computational linguistics, but I’m not reading about that in other languages, there’s not always going to be much overlap between the vocabulary I encounter when I read and the vocabulary that I need when I write. There are great resources available to fill that gap. One of my favorites is Reverso Context.

I still stand by the idea that if students in a class are turning to the dictionary for every word in an assignment, then something is going wrong, either with the teacher’s design of the assignment, or with the student’s approach to it.

But instead of feeling like I’ve failed when I look up words, I think a better approach will be to take steps to avoid the pitfalls of looking up too many words. The biggest problem in the classroom is that students look up a word or phrase to complete a particular task, and then they never use it again. Or the next time they want to use it, they have to look it up, as thought they’re learning it for the first time. So if I look up a word or phrase, I will have a specific plan for using it in a variety of contexts from then on. That includes a plan for keeping track of what I’ve learned (basically, spaced repetition, but with the goal of using it in some context, not just recalling the meaning).

We’ll see how it goes!

Programming and maths resources

Studying computational linguistics and web design is forcing me to grapple with maths and programming. I’ve known for a while that I want to be literate in those fields. The trouble is that I haven’t been taking the time to practice and let my mind wander around maths and programming topics. It’s still All Language All the Time in my head. I’ve been trying to turn my inner monologue on maths and programming. Here are some texts I’ve been using.

Programming and web design

I’ve written about 100 pages of a programming workbook template similar to my language learning workbook template. I’ll post it once it’s presentable. I’m getting ideas for the workbook from resources that f friends and colleagues have recommended, such as the following:

The Odin Project

This website is comprehensive and well-structured. They give us learners everything we need to find information, check our understanding, and test ourselves. At the same time, the materials are organized so that we’re completely responsible for making the most of all the available resources. And after finishing all of the coursework, we can point to specific useful things we’ve made.

I particularly appreciate the way that the practice problems are designed to make learners feel lost, yet there are resources available to guide us when we get stuck. I’m not used to not knowing how to solve a problem. These days I rarely encounter problems that completely stump me. I don’t always get the right answer, but I rarely have the feeling of being totally lost. I’ll try one technique, take it is far as I can, and if I hit a dead end, start over and try again with something else. On the other hand, if I see a maths or programming problem, I don’t even know where to start. I know that’s natural, but it’s been harder for me to overcome the temptation to not even try.


For some reason, I’m intimidated by GUIs, even though I remember making them with Java in high school and thinking they weren’t that bad. I like this tutorial because shows how to do the same thing in a few different languages.

Programming course websites:

There are so many!

Natural language processing

Natural Language Processing with Python by Steven Bird, Ewan Klein, and Edward Loper

This was the first text I found on the subject. I’m about halfway through it now. It’s really clear.

Foundations of Statistical Natural Language Processing by Chris Manning and Hinrich Schütze

This was the second text I started reading after Natural Language Processing with Python, and it’s *way* more challenging. When I started reading, I was totally lost by the first exercise. I looked at the problems and I didn’t even understand what they were asking. Fortunately, It’s starting to make more sense now that I’m reading the Jurafsky and Martin text, and now that I’ve found a more accessible introduction to set theory and probability.

Speech and Language Processing by Daniel Jurafsky and James Martin

This is the text that’s really got me interested in natural language processing. It’s much more technical than Natural Language Processing with Python, but the writing is superb. I’m hoping that after I finish this text, I’ll be able to understand Foundations of Statistical Natural Language Processing without feeling completely hopeless.


A Probability Course for the Actuaries: A Preparation for Exam P/1 from Arkansas Tech University

SticiGui from UC Berkeley.

I did just fine in my middle and high school maths classes, but I never really had a decent conceptual understanding of maths. It’s caught up with me,  as Daniel Willingham has warned it will. So I really need to start over. After a few months of banging my head against a wall trying to understand the statistics and probability sections in Foundations of Statistical Natural Language Processing, feeling like a hopeless idiot, I found the two resources above. The formulas mostly make sense now. It’s a huge relief.

A First Course in Abstract Algebra by John B. Fraleigh

My mathematician friend lent me this book. I enjoy it, and it’s very clear. I wish I had known earlier what university-level maths is all about. I would’ve stuck with it for longer.

My favorite part about this text is that I’m no longer intimidated by jargon. I’ve always been more comfortable with linguistic jargon than I am with maths or programming jargon. I know it’s irrational: ‘morphotactics’ and ‘semantic field’ don’t faze me at all, so why should ‘commutative ring’ and ‘finite state tranducer’ make me nervous?

All those terms describe things or ideas that are worth thinking and talking about as a group, so it helps to give those collections a name. Fraleigh’s text is very accessible. The definitions are clear, the steps are explicit, and for the most part he avoids phrases such as “obviously” and “by an elementary algebra technique” which make me want to throw the book across the room.


I haven’t given up on learning languages, though. I’m still going to be using and updating my language learning workbook template. I’m working towards using the ideas in that template to make an online resource for learning and practicing languages. I’ve been meaning to be more active on lang-8, and now I’ve got lots of things to potentially write about. I want to improve my written Chinese, so I’m working through a few books from the 别怕作文 series. As for Spanish, French, and German, Reverso is a fantastic resource.