The mysterious effectiveness of trivial pursuits

This is probably not worth posting about, but I’m curious to see if anyone will respond.

I usually take notes while reading a math book or article. There may be obvious utilitarian reasons to do so, for instance I may want to reorganize a proof in order to understand it better, fill in some gaps, work out the details of the omitted calculations, and generally rewrite selected parts more to my liking. No, I didn’t really expect that you’d be surprised to hear any of that.

What’s more interesting is that I also seem to benefit from the physical process of writing out my notes in longhand, the old-fashioned way. This summer I spent some time reading a paper that, in addition to proving a significant result, is a perfect example of how I’d like all math papers to be written. Aside from a couple of minor calculations that I wanted to work out in detail, there was no need to add, clarify or reorganize anything whatsoever. Even so, I still found that copying bits and pieces of the paper by hand helped me understand it better, as if the motor activity fired up some part of my brain that would otherwise remain disengaged.

It was not the first time I noticed this, and I don’t think that this is just me, either. I even have a vague recollection of reading a popular science article that said something to that effect. (That was a long time ago and I can’t find it now.) It works in other settings, too. You might expect that taking notes during a lecture would divide my attention and leave me less able to focus on the subject matter, but actually it’s quite the opposite. This is of course assuming that I can follow the lecture, at least in principle; some topics are too far out of my reach, and some lectures can’t be saved. But I digress.

Now, here’s where I have a question, especially to the readers under the age of 30 or so. I’m told that handwriting is out of fashion these days, all but replaced by typing and texting. This article reports that the younger generation in China loses the ability to write Chinese characters by hand because they don’t have much use for handwriting except to sign the back of their credit cards. Over here, I suppose it helps that we have a somewhat less complicated alphabet, but it’s still true that there are fewer and fewer reasons to write anything in longhand. Even signatures might become obsolete eventually, replaced by PINs, passwords and biometric technologies.

This isn’t going to be a “kids these days” rant. I’m not particularly eager to go back to the days when “cutting” and “pasting” meant using scissors and glue, respectively. I don’t really miss the logarithmic tables, either, in case you were interested. Nor would I want to type all of my papers the way I typed my Master’s thesis, on a borrowed mechanical typewriter with no word-processing capabilities and no math symbols. Those had to be filled in by hand, in the spaces you’d have to leave between the typed letters and numbers. For corrections, I used the white-out fluid that a colleague had brought from a trip abroad. Also? The wired kids these days might not even know the literal meaning of the expression “carbon copy”, or they might look at old issues of scientific journals and wonder about the instruction for authors to submit the “original copy” of the manuscript. I could explain it all to them in more detail than they’d ever want.

But I said I had a question. It’s this: if you didn’t grow up doing a lot of handwriting, if it’s typing and texting but not longhand that feels natural, do you still take handwritten notes when you read a math paper, for reasons similar to those I described? If not, is there something else that you do instead? Taking notes in TeX, for example, does not do it for me. Obviously I use TeX to write papers and exchange notes with collaborators, but typing in TeX distracts me from thinking about the mathematics involved, whereas writing in longhand helps me focus on it. Does anyone here see it differently?


Filed under mathematics: general

17 responses to “The mysterious effectiveness of trivial pursuits

  1. I’m not quite in the generation that grew up viewing electronic input methods as preferable to paper; being instead in the immediately preceding generation in which both methods were available but the preference was reversed. So I can’t directly answer the question from personal experience. But I can make two points:

    * Doing _anything_ actively related to reading a paper is going to help with understanding it, when compared against passively reading the paper. In passive mode, one can be fooled into thinking that one understands something when in fact one only understands a sufficiently large fragment of it that the illusion of complete understanding can be obtained. This is similar to how the brain can take a partially heard sentence and complete it into what it thinks to be a coherent and complete sentence, which can sometimes lead to misheard lyrics and other auditory illusions. In contrast, if one is actively trying to transcribe a partially heard sentence, one will be much more aware of any gaps in comprehension.

    * I think electronic input methods, while undeniably convenient for large tasks (such as writing a paper or lengthy notes) still have a “transaction cost” that, while small, is still non-negligible enough (e.g. a minute of effort) to deter using such methods for micro-tasks such as scribbling out a few lines of computation. It’s still easier for me to grab a pen and paper (or a piece of chalk and walk to a blackboard) if I want to work out a paragraph of a mathematical argument, than to open up a LaTeX file or to scribble on a tablet PC or something. (I don’t own an iPad, but perhaps the transaction cost is closer to zero for that device, though.) So there is still a niche for pen and paper here, though perhaps that niche will shrink in the coming decades (especially as “electronic paper” type technologies become more ubiquitous).

    On the other hand, if I have already paid the transaction cost, for instance when writing latex/wordpress lecture notes for my classes, then I do indeed find it very convenient, useful, and instructive to transcribe arguments from a textbook to my lecture notes file, and then I seem to get the best of both worlds; the active learning experience usually provided by pen and paper, together with the digital record provided by electronic input.

  2. That’s certainly true about active vs. passive reading. My question was more about which specific activities people do or do not find helpful in shifting from passive mode to active engagement, and how that depends on the environment they grew up in. In my case, taking notes in TeX would surely qualify as doing something active related to the paper, but it would not enhance my understanding of it, in fact it would likely be counterproductive. Even when I have to produce a set of typed notes in the end, I do most of the mental work while scribbling stuff by hand. It’s not necessarily a question of the initial investment, either. The paper I mentioned in the post is about 40 pages long and the initial effort of, say, starting a TeX file would have been totally negligible compared to the remaining work of actually reading the article.

  3. I’m also from the “before electronics” generation, and I find “active” learning for me to be correlated only with writing with a pen or pencil; in fact, I have never tried to take notes in TeX while reading a paper. Thinking about it now, one reason is probably that it is much too difficult to achieve non-linearity (writing bits of computations all over the page) as I go along. Also, I like my TeX files to — at least — compile, and requiring correct syntax will just be too much work.

    On the other hand, I have a very good friend (same generation) who is a blind mathematician; for him, actually working in some form of LaTeX has been the standard practice since the late 1980’s/early 90’s (and an immense improvement in doing mathematics only in his head — although he could carry along in his head much more information than I can — and with Braille notes).

    Also, when I was taking graduate classes in the US, I had the habit of writing them in French; this was mostly a straight translation, but it also involved simple adaptations and minor reformulations, and I always had the impression that this forced me to think more clearly about what was being said instead of copying the blackboard without thinking.

  4. The main reasons I typcally tex things up are that a) I have something nice to look at the end and b) I’m horribly disorganized so would lose track of everything if I used paper. I do like to work out short calculations on paper while I’m still trying to figure out what’s going on, but I don’t think I’ve ever tried to take handwritten notes. Since I can type faster than I write, I can make a much larger quantity of notes this way.

    One of the main reasons I like TeX *especially* for long calculations is that, with long formulas, it’s easy to copy and paste and make small modifications bit by bit rather than rewriting the formula over and over, which makes preparing a detailed, checkable-by-brute-force proof much easier.

  5. To add to Akhil’s comment, a LaTeX writeup does do wonders for organizing mathematical thoughts.

    For an individual paper, however, the “transaction cost” for inputting electronic notes remains too high for me; I relegate that to longhand notes in a notebook that I constantly carry with me. I also find diagrams easier to draw, by hand.

    When I need to organize the collective knowledge from _several_ papers, then LaTeX is quite handy (including the end bibliography format). It’s easier then for me to see any gaps in knowledge (e.g. “Wait, has anyone proven that case yet? Is it hard ..?”)

  6. Richard Séguin

    I’m approaching 60 in age and definitely come from the pre-electronic era. If you’ve ever had to write a mathematics paper with a typewriter, LaTeX is definitely a miracle of modern technology. I now write in LaTeX from skeletal hand written notes with little distraction from the minutiae of the LaTeX syntax and generally produce better text. Still, it seems that I’ve lost something over the years. I used to have neat and readable handwriting (well, at least printing) and the process of patiently committing my proofs to letters on the page by hand so that the letters formed complete sentences did seem to etch the ideas into my brain, while at the same time slowing my impatient brain a little bit so that I’m double checking what I was thinking and writing. Now after years of writing mathematical and non-mathematical text on the computer, writing a lot of computer code, together with some aging, my hand writing is quite impatient and erratic and no longer neat at all. There is nothing like doodle time with a pen and a blank piece of paper for ideas to flow, but if the hand writing aspect of it no longer feels good and natural, then I’m probably not benefiting from this as well as I could.

  7. Wow, thanks! I didn’t even realize that I had such a diverse group of readers.

    I agree with janus about using LaTeX for organizing thoughts – it’s quite satisfying to have a typed up and printed set of notes. But that usually happens at a later stage, once I’ve come up with something to organize in the first place. Also, now that I think about it, I use diagrams and “nonlinear” notational shortcuts quite extensively. That would be hard to reproduce in a TeX file.

    I’ve never had a “neat” handwriting in the old-fashioned sense. We had handwriting drills back in grade school – dipping pens, desks with holes where the ink bottles were placed, the whole circus – but after that, I mostly needed to be able to take notes quickly while paying full attention to the lecture or whatever, so my handwriting evolved accordingly.

  8. From an even older generation, I remember reading an interview with Serre (I don’t remember where; it’s not the one in Vol. 4 of his Collected Works) where he said that once, at the beginning of his career (late 40’s/early 50’s), he didn’t dare write to someone for a reprint of a paper (I think that might have been Eilenberg), and hence he just copied the whole paper by hand for himself (this was before copying machines, at least in France)…

  9. I am from your “target generation”, but even so I’m more in a bridging age than you intended. I also have the problem where Chinese words don’t come to mind when I now pick up a pen, but that’s more due to disuse (it’s been 12 years since I seriously written in that language) than electronics. I’m sure if I were forced to take notes in Chinese I’d prefer the pen to the computer. That said, in English I reckon I’ve written more on my computer than by hand, and unlike Terry, as soon as I pass the preliminary investigation stage (as soon as I have some semblence of an idea of the right way to proceed), I tend to work directly in LaTeX. Half of the computations in my dissertation were done in a text editor on the computer. So I’d say I use computers a lot, and satisfies one of the hypotheses of your question.

    On the other hand, when something is more nebulous to me (just trying out ideas, seeing what works and what not), or when I am trying to learn something (from reading a paper or verifying a computation), I prefer to write it down. Actually, I prefer the office black board over paper and pen(cil) (but I’d prefer paper and pencil over white board if that’s what’s available). I am not exactly sure why, but I have a guess.

    Part of the issue is active versus passive learning. Writing things down does help when trying to focus my efforts. It helps to make things take shape, and helps me imagine what’s going on. Part of the issue is speed and changing gears. I can just as well focus when I use the computer, but for whatever reason my use of computers is more single-tracked, whereas writing by hand can be more free. I think when I am tackling an unfamiliar idea, I want to let my brain wander a bit and explore the landscape. And in those situations the computer feels too restrictive. It’s hard to describe, but a lot of times when composing an essay or when reading a paper or when just thinking about some mathematics, I will simultaneously explore several directions at once, the directions often cross feeding each other. I can keep a lot of “other ideas” in the short-term memory in the background while working on a “current idea”. And when I jot those ideas down on paper, I can do them orderly without affecting the stored “other ideas”. But when I work at a computer, I feel that something else competes for the short term memories and the “other ideas” gets dumped a lot quicker.

    I guess one way to say it is that if I want to be really, really focused on only one thing, then I can benefit from using the computer. But if I want some creativity or leeway for free thought and exploration, writing by hand gives me that flexibility. I think this is also why I prefer to take notes by hand in lectures (when I do so at all). Letting my brain wander around comparing what I just heard to what I know is a more effective way for me to learn than to just single mindedly soaking up word-for-word what is being said by the lecturer. I think I feel more comfortable when something is “intuitively obvious”, and developing that intuition requires comparing what I just heard or read with everything else that I know to draw some sort of analogy.

  10. Oh, I just had another inkling why using a computer (be it typing or writing on a tablet computer) uses up more “peripheral brain power”: with black boards or pen/paper, I don’t have to think about erasing a portion of the board or flipping to the next page in the note book. Going back to a previous step in the computations is either looking on a different spot on the board or at a different sheet of paper. The physical aspects of writing on a computer (in the typing case, interaction with the keyboard; in the tablet case, clicking the mouse/stylus to flip pages or scroll up and down; I’ve never heard of anyone expending brain power on how one’s hand interact with the pen or chalk one is using to write) always intrude a little bit and perhaps end up a bit distracting. And I think it is because the brain is dealing with this peripheral distraction that I can feel like I am more focused/engrossed when I am using a computer.

  11. Olof Sisask

    I’ve grown up with both writing longhand and typing, and both feel quite natural to me. What medium I use to write something generally depends on what stage of understanding of the subject matter I am at: if I have a decent grasp of what I want to write, whether it be an essay or notes about a paper, then I’ll jump straight in to writing on a computer, whereas if my understanding is a bit too hazy, for example if I don’t at all follow an argument in a paper, then I’ll generally start writing and working things out on paper.

    In other words, I prefer typing something on a computer in order to formalise an intuitive understanding that I have of it, whereas in order to develop such an understanding in the first place I prefer to write on paper. I feel like there is generally ‘active engagement’ in both: one helps to develop intuition and the other helps solidify it. Naturally there may be jumps between the two: often after I’m done typing something there will be a whole bunch of envelopes with scribbles all over them next to me.

  12. By coincidence, just this week I tried for the first time using a PDF annotator (named, appropriately enough, “PDF Annotator”, but there are many such software packages), to write on a PDF of a paper I was reading on my tablet PC. It actually simulates the pen-and-paper experience quite well, while keeping a digital record that one can store or forward as necessary (and also having such features as the ability to add text boxes in addition to freehand drawing, or to cut and paste annotations as necessary). I plan to use this in grading homework (about half of my homework nowadays comes in electronic form – one can see the generational shift occur with each new incoming class).

  13. I do not understand anything if I do not write it down longhand on paper first. I am 42 years old, so this may be due to the fact that I grew up before computers really took over. Even taking all the modern advances into account, I do not see a good substitute for paper and pencil when I need to check a calculation. It is not that I prefer it to other methods. I just do not believe that suitable electronic methods really exist in this regard.

  14. On the subject of Terry’s latest comment: I’ve never tried using a tablet PC or many other devices that you can write on using a stylus, but the few touchscreen devices I have used have always had a noticable lag between moving the stylus and the writing appearing on screen. Does anyone know if this is still the case or if the technology is good enough for the writing to appear nearly instantly? Is the resolution good enough for reasonably small writing, or does one need to make over-sized motions?

    If e-paper starts to allow accurate and responsive (manual) writing as well as reading then I can imagine switching completely to an electronic system. It would make it so much easier to keep everything organised, and would take up a lot less physical space than pads of paper.

  15. My tablet PC is a brand-new 2010 model (a Panasonic CF-C1 laptop/tablet convertible, to be precise). There was no noticeable lag, and once it was calibrated, I had no problem with accuracy. Responsiveness was still a little touchy when writing rapidly. If one is using the stylus as a marker, it behaves and feels almost identical to a physical marker pen; if one is using it at a finer point resolution, it behaves like a cheap roller point pen (with maybe one in ten strokes not coming through well, though this could be my own penmanship at fault).

    It’s still not quite electronic paper though – it’s not terribly convenient to write on the screen in laptop mode, and it takes a few seconds to rotate into tablet mode and then I lose the keyboard. Also, to input handwriting, I need to open up special software; I can’t simply scribble into this textbox I am using to write this comment, for instance (though I can use the handwriting recognition tool to input text. which I just did for this parenthetical comment; it works quite well but my handwriting speed is slow compared to my typing speed.)

    It may be that a side device, like a touchscreen mousepad, that is somehow connected to one’s main computing device, would be closer to a genuine e-paper experience, but I don’t know if there is anything like this that is of usable quality that is available commercially.

  16. But if tablet PCs become good enough, it will simply mean that electronic technology will have caught up to paper… As for keeping things organized, it was pointed out by Umberto Eco in mid 90s that paper is still the best way to store information if you want to be sure that the information you write down will be in good condition and readable, say, 20 years down the line. I believe that the point has not lost all of its validity even today. But I do not deny that an efficient table PC would likely help to keep things well organized in a short run.

  17. I have a very hard time trying to take notes in a lecture. Maybe I didn’t learn the habit early enough, but it’s distracting for me. The amount of things that I’d have to write down to get anything I’d halfway understand if I looked at it later, especially if it’s a totally new topic, is so great that I don’t succeed so much.

    On the other hand, a lecture that’s close to what’s already written in a textbook or paper can be very useful even if it’s completely redundant. However, a good lecture for me is one that’s more about what you’re thinking roughly than precise technical details, but maybe the better you know a subject, the more warranted an assumption it is that you already know most of the guiding principles.