Saturday, November 06, 2010

Learning: school versus university

Two days ago, a young man about to finish high school told me he couldn’t decide whether to study mathematics, informatics or physics. If I would have any advice? Though physics is interesting at school, he doesn’t really like it because there is so much to learn, he said. Math on the other hand he likes better because once understood, one just has to do it.

My first reaction was if you dislike learning, a university isn’t the right place for you no matter what field you chose. Then I thought he might be disliking not learning in general, but a particular sort of learning. It might be useful to distinguish the following four types of learning:

  1. Physical learning
    Is the training of motion sequences through practice and exercise. Plays a major role for sports, playing an instrument, driving, and so on. It’s aiming for the goal, doing your scales practice, filling cuvettes till you manage without spilling, etc.

  2. Learning by doing
    Is learning from cause and effect, trial and error. Omnipresent theme of children’s toys and school education. Many science museums too work with push here - look there. In contrast to physical learning though, the emphasis is not on learning a particular motion but understanding a relation.

  3. Knowledge gathering
    Is the classical learning of facts and data. Avogadro’s number is about 6 x 1023. The capital of turkey is Ankara. The milky-way is about 100,000 light-years side to side. The can-opener was invented 48 years after the can. Etc.

  4. Conceptual understanding
    Is the learning of explanations and relations, theories and concepts. What is chaos? How does the stock market work? What makes airplanes fly? Why doesn’t the moon fall down on Earth?

Learning at school as well as at the university is typically a combination of these 4 types of learning. But the composition depends on the field, and it may substantially change from school to university. Languages for example are generally heavy in knowledge gathering. You just have to memorize that vocabulary, no way around it. And you can’t lead any argument in history without the names and dates. Biology, chemistry, physics and mathematics necessitate type 3 learning in declining order. Lab work is the contribution from 2.

At school, you will generally do well just by learning the facts and it is, at least in my experience, also often where the emphasis of the educational system is (except for classes like sports and music which rely on type 1 learning). Especially in mathematics however, it is possible already in school to replace type 3 learning with type 4 learning: You can either memorize a table with functions and their derivative and integrals, or you understand what a derivative and an integral is. You learn the steps you have to do to calculate the intersection of two planes in a 3-dimensional space. Or you understand what the equations mean. Pupils who fly through math are typically those who understand the concepts, rather than learning a scheme for computation.

When you finish school and start studying math or physics, the relevance of memorizing facts drops dramatically. Who cares if you know Avogadro’s number - you can go look it up if you need it. Sure, it’s handy to know the distance from Earth to Sun, but it’s not going to impress your prof. In mathematics, the break with school practice is particularly dramatic. What you’ve done at school doesn’t prepare you for studying mathematics at all, except that some symbols might look vaguely familiar. To quote my younger self:

[W]hat's called mathematics in school has little to do with mathematics. It should more aptly be called calculation. Don't get me wrong, it is essential knowledge to be able to multiply fractions and calculate percentage rates, but it has about as much to do with mathematics as spreading your arms has with being a pilot. Problem is, that's about all most people ever get to know of mathematics. The actual heart of math however is not number crunching or solving quadratic equations, it's the abstraction, the development of an entirely self-referential, logically consistent language, detached from the burden of reality.

Both Stefan and I can recall from our first semesters those students who tried to continue type 3 learning that had worked so well at school. You can indeed just learn by heart what your textbook says what the variational principle is, and reproduce the relevant sentences when asked. You can memorize every example discussed in class, and learn technical terms by writing down definitions on a stack of index cards. This might get you through the first few semesters, but it’s not going to work in the long run. Both Stefan and I have seen dropping out the fellow students who proceeded this way, one after the other.

To come back to the young man’s question. If what you dislike in physics at school is the emphasis on type 3 learning, chances are you’ll do just fine at the university. There’s still the lab exercises where you have to stare at glowing wires for several hours or find anything else on the oscilloscope besides the 50 Hz curve, but if I managed that you can do it too.

I started studying mathematics and only changed to physics after my bachelor’s degree. That was possible because I had taken all the necessary classes and the department of mathematics had a respective agreement with the department of physics. It didn’t cause me any problems, and pretty much all of the additional math came in handy at some later point. I don’t know much about the requirements for informatics, but what I know from friends is that the first semesters are very heavy in math too. So in case of doubt, I’d recommend to start with math because the change to either physics or informatics will be easier than if you do it in any other order. However, since the time I was in my first semesters many regulations have changed to accommodate the European master’s program. I don’t know if, or under which conditions, it is still possible to change field after the first semesters.

In summary: Don’t expect that physics or math at the university is a continuation of what you’ve done at school, neither for what success or boredom is concerned. Best is to primarily follow your interests because you will need perseverance and motivation.


  1. Surprisingly many people are learning with communication, in discussions in particular. They usually don't read original articles, only popular info about them. Such people have broad but shallow knowledge. They're not motivated to learn something new without actually talking about it, or at least writing about it on the blog (not actually feasible way for mathematicians, as the typing of symbols is tedious).

  2. i just wanted to add that informatics might not be that mathy at all. we only had 2 courses (both within the first year) for if i remember correctly 12 credits (out of 180....).
    if he is doing math there is no way he could easily switch back to CS (at my university at least)

    how come you switched your master?

    @zephir: terence tao seems to have no problem writing his mathematical blog posts ;)

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  4. Yes "the German rote system" as we call it here in the US is an excellent way to do well on homework and tests, but horrible in retention, meaning a week later it's not uncommon to have forgotten much of what you learned. University, if done right, involves more experimentation and critical thinking, the whole annoying Socratean "answer a question by asking a question" thing. Regarding the 3 things he's considering:

    1) PHYSICS - He just got a strong does of Classical Newtonian stuff, and Electromagnetism. It SEEMS like a lot, but I think the easiest way to convince him to go forward or not in this direction is to explain Heisenberg's Uncertainty Principle to him. It's so counter-intuitive yet 100% important. It will either excite him or piss him off. It's always fun to watch students new to the game say "that CAN'T be right" regarding uncertainty.

    2) MATHEMATICS - It's all about "proof", not in understanding it but in coming up with the same. Applied Math is down the road. Learn the Pure stuff first, including the frontiers of research which are many. To get him started have him read The History of Mathematics section in the Macropedia volumes of The Encyclopedia Brittanica, and The Foundations of Mathematics in the same after that. Wikipedia is so dry it will discourage him. Math in Wiki has no soul, although it's accurate.

    3) INFORMATICS - Is this what they're calling Computer Science these days? Interesting. (I just noticed it includes Information Science as well, um, OK).

    Well whatever, this is the hot field right now, as it serves all the other fields. The 1800's was the best century for Math, the 1900's for Physics, and the 2000's, so far at least, for this. So I would advise him to pursue this field, but only after considering the other 2.

    Or just tell him to look into the biological sciences and astronomy. They're exploding too.

  5. Plato vs Meno: How the learning is possible

  6. Hi Zephir,

    Yes, that's true. Communication serves well as reflection and helps to shape what one has learned already, find out inconsistencies, add facts, hear new explanations etc. I'd say it's generally a combination of type 3 and type 4 learning. I believe Phil said in an earlier post that there's many people who learn "along the way" rather than to seek out a particular topic. Picking up most information and knowledge from communication, popular articles, etc is exactly such learning "along the way." Best,


  7. Hi Tom,

    Where did you study? I know a handful of people who did informatics (all of them in Germany) and the first few semesters was loads of elementary logic, stochastic and statistics etc. I don't know though how much of that was by choice rather than mandatory.

    How come I switched major. That's a long story. The short version is a) I didn't really know what the pure math was good for. I wanted to understand something about nature. b) The dept. of math was broke and the dept. of physics offered me a job and a desk.



  8. Hi Bee,

    I support your point of view, since I've had a similar problem in school. Physics and maths seem to be boring. The only interest I've had has been chemistry. I thought always, that my school in particular has been the reason for this. So I started after school studying metallurgy, but find the physics lessons so interesting to change to physics, and later to theoretical physics.

    Best, Kay

  9. Vienna University of Technology.

    Just had a look at the curriculum and it is 15 credits math, 6 credits statistics. for the entire 5 years.
    In addition to math/stats we have 10.5 credits theoretical CS and logics over 5 years. so ye, it isn't much.
    We also have no physics or electrical engineering.

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  11. Hi Bee,

    It’s interesting how you categorize the methods of learning and yet I would agree with Steven that it all comes down to considering the differences between route and conceptual learning, with me finding the former not allowing one come to have things understood well at all. In fact as a consequence of my own life experience, I’ve found that most people can be categorized as being primarily either route or conceptual learners.

    One such example is respective to the professional engineers I interact and deal with on an everyday basis (this is not to have it taken I referring to Steven), as being one’s employed by the company I work for or those that form to be consultants and clients. What I have discovered is that the majority being route learners, having at best shallow knowledge and little interest respective of the underlying fundamental physical principles on which their profession is both founded upon and depends.

    In the beginning I was annoyed to discover this, as I thought all should be conceptual learners and then later realized each has a function and bring things both separately and in combination having utility. The only problem I’ve found with route learners, being is if one is to depend upon them you need to attempt to ascertain the source(s) of the knowledge, as the temptation of many, being rather than have it known that simply they don’t know something, is to express an opinion, regardless if it be based on reliable source or more importantly careful consideration. Fortunately, most if not all understand the difference between having a structure or mechanism fail, as opposed to their thoughts in terms of the consequence regarding other concerns.

    This has me mindful of what Richard Feynman said, in an autobiography of his I read respective of his work in physics, with him confessing rather than relying on remember things, he had to learn as to almost reinvent most of it for himself. This might appear for many as a long and inefficient method, yet it’s evident with his success it can work quite well; that is even in respect to nature, for which he was to discover and then latter share with all. The thing I still struggle with, is not finding as it needing to transpire this way, or so much the process evolved (route), yet rather as to where in terms of space(s) it is enabled to do so:-) Confucius long ago once tried to have it all made simple in saying, ”I hear, I know. I see, I remember. I do, I understand.”; yet I think “to do” only has one to be confident that things work as do, while only in asking why can one hope to better understand.

    In respect to the young man in question, I would steer him towards things such as engineering if looking to university as a path leading to a career; although I wouldn’t have him believe he stands to be a great one, that is if preferring to continue in restricting himself to route, respective of the limits it imposed on discovery, creativity and ultimately understanding more generally.



  12. the development of an entirely self-referential, logically consistent language, detached from the burden of reality.

    Contemporary physical theory is lost: Quantum gravitations, SUSY, creating mass within the massless Standard Model. Euclidean geometry contains no mistakes as an abstraction. Try deep sea navigating with it. Science is empirical.

    find anything else on the oscilloscope besides the 50 Hz curve

    60 Hz hum in the New World! Theory is universal, knowledge is local, data is only data. (400 Hz would be more efficient point of use, and no more 50/60 Hz fluorescent light headaches).

    Education trains a nacent adept to separate what he is told from what he sees from what he imagines. All discovery is insubordinate. Question authority with empirical reality. Higher education also identifies people who should be doing other things (inverse diversity). Fiat putting them in charge is not one of those other things

  13. Hi and belated congratulations about the twins. Mine are in school now. You can clearly see both their profiles in the gaps in my publication list. Hope you're more efficient.

    I think the choice for this young student is not so important, since so many people I know have switched among these subjects: e.g., half my (maths) department is composed of former physics majors and a sizable minority comes from engineering.

    As far as job prospects goes, I would normally assume any hard science is a good choice. I would tell the student to relax and concentrate on finding a good university.

    @Kay: in school I used to think mathematics was exciting, physics would have been if only I could make sense of it, biology was kind of ok, and chemistry was desperately dull. Maybe some people (i.e., me) are really meant to be mathematicians. We're weird like that :-).

    Speaking of which, @Steven: mathematics is going fireworks all over the place! Fermat! Poincaré! Plus all the wonderful stuff laymen usually never hear about because the maths you learn at school does indeed stop in 1850.

  14. I am studying in a Physics Bachelor program in Heidelberg University and sadly I have found out that the most successful among my fellow students are those who simply excersize and memorise a couple of problem examples from every part of the semester`s curriculum. Most of the people I talk to are really bored from what we learn and don`t have any intention of gaining any deeper conceptual understanding as long as they pass their exams expetionally well.

  15. Hi Martin,

    I would think this observation being more of a reflection of the students, rather than of the curriculum or its presentation. I suggest this because you mention they attempt to simply cherry pick through it, rather than looking for connections or relevancy. Sadly this only serves to reinforce my opinion (being one I hope to being proven as incorrect) as this resultant of something innate respective to each individual, which has one to become more of a route learner than a conceptual one and vice versa.

    It would be interesting if a study could be devised to be conducted in an attempt to reveal if some sort of ratio exists or better a quantitative/qualitative distribution between the two and finding it even more significant if this can be extended to discovering if this remains the same or different respective of the cultural/sociological spectrum or ones place in time. It’s sort of like asking if Plato’s contention being true or rather it being simply a case of wishful thinking.

    ” Whereas our argument shows that the power and capacity of learning exists in the soul already; and that just as the eye was unable to turn from darkness to light without the whole body, so too the instrument of knowledge can only by the movement of the whole soul be turned from the world of becoming into that of being, and learn by degrees to endure the sight of being and of the brightest and best of being, or in other words, of the good.”

    -Plato [The Republic, Book VII, “Allegory of the Cave”]



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  17. Hi Bee,
    I am a keen follower of your blog - a software tester by profession having deep interests in history of science, philosophy and learning.
    I would say type 4 (conceptual learning) is fundamental to every type of learning hence there are only 3 types of learning (phyisical, learning by doing and knowlede gathering/memorising)with the conceptual learning at the bottom as base learning.
    What do you say?

    Shrini Kulkarni

  18. Sigh!

    The perpetual question of mothers to be.

    Good luck with your twins.

    I have twins.

    My circle of life has just been extended ... I have just become a gran.

    As a result, I'm trying to learn about the subject "learning".

    I want to be able to stimulate, through his senses, the "motivator" for that young mind.

    If the basic skills of using the senses are not developed there will not be progress.

    If there is no interest there will be no progress.


    ps. I know that my granson is the prettiest baby in the world. ;-)

  19. Hi Shrini,

    First, I was suggesting that any sort of learning can be decomposed into the 4 basic types of learning I listed. I don't know what you're trying to say with one being fundamental, thus there's only 3. The contribution of type 4 learning differs greatly among tasks, with some having essentially no contribution from 4. Take the example of learning vocabulary that I mentioned. Or learn a poem by heart. You can learn the whole thing without understanding a word about its content. Likewise, you don't need to understand why a bicycle doesn't topple over to learn to drive one.

    Second, if you want to continue commenting here, please avoid the self-advertisement, ie drop adding your blog's url to your comments. You're new here, so you might not know, but I don't like that.



  20. Hi Phil,

    I think your distinction between route learners and conceptual learners is more about what people chose to learn, not how they learn it, which is what I was concerned with. Of course both isn't entirely unrelated in that people will chose to learn things they are good at learning with the sort of learning they like. But you can be a route learner with any of the learning types that I've mentioned. Best,


  21. Hi Bee,

    I suppose I could have it wrong, yet the experiment I was talking about you could say I have been running for years, only not in a formalized manner. That is I haven’t kept a record of the data set, that is other than in my own mind. This is resultant of the nature of my job, having it required that I explain things to people, which includes fellow employees, clients, consultants and suppliers. That is when it’s appropriate I first attempt to have things explained conceptually, which if grasped has much of what people needing to know having then able to have answered or realized for themselves.

    I think you might be surprised to learn that even when the simplest of concepts are explained, how few people actually are able to do anything with them, other than perhaps reiterate what I’ve explained. When this doesn’t work I switch to the route method, as to only give them the specific facts or instructions that they feel to needed at the time or for a specific situation.

    The most interesting observation I’ve made is in the few instances where I am exposed to a conceptual learner, in being able to identify them by their reaction, which first has me finding them extending the dialogue beyond the scope of their initial query, yet what I find being just as significant having them often thank me as to how things were explained. Anyway I won’t rattle on about it further, only to say I have found there to be a distinction to be made, not just between how things are learned, yet how differently people come to learn them.



  22. Hi Bee,

    That's a delicate question, but I find your answer quite reasonable.
    I found science so boring before university, and it blew my mind after... Surely some people have experienced the contrary, loving math and physics at school and hating it at university when "type 3" learning became too hard.

    From my experience, math is the cornerstone of our way to understand nature, but I admit I may be biased, because of my attraction to it.

    Anyway, I like and agree with the way you describe the "actual heart of math", except when you end with "detached from the burden of reality". I believe the attachment or detachment of math to reality is only a question of interpretation, which consists merely in conventions. So I would refer to it as "our" burden and not that of reality, but that's just biased me (again).

    Best regards,

    PS : congrats with the couple of epsilons ^^

    About your remark to shrini, I already posted some comments here (with the pseudos tiger or mad), but I didn't think it'd be so rude to give a link. I deleted my post. Sorry about that.

  23. Hi Phil,

    I wasn't saying you got it wrong, I was saying I was talking about something else. You're talking about people's individual style of learning. I'm talking about possible modes of learning, not necessarily tied to people. What personality profiles are more or less often to find, I don't know, and your experience might indeed provide a good cross-section. Best,


  24. Hi Shrini,

    We're talking past each other. It might very well be that some sort of learning is more or less good, but that's wasn't the point of my listing. With four points was just simply saying, you can distinguish between these different sorts of learning, and the general sort of learning is a composition thereof. I actually don't think that knowing why a bicycle doesn't topple helps anything with learning to drive it. In any case, as I said later on, type 4 learning is in my impression what gets you forward, so I don't even think we disagree in content. Best,


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  27. Hi Bee,

    Yes I understood you’re focus being on the methods of learning, yet I thought also as it relates to the advice the young man was seeking. That is I was simply attempting to have emphasized that one’s way of thinking provides a clue as to what they might be best suited for.

    As for instance, I find it interesting that you connect music to physical learning, as although I greatly appreciate it, however despite all my attempts at playing an instrument have not been rewarded with much success.

    Therefore similarly I would suggest that no matter how much some may come to appreciate the value of conceptualization, they still struggle to have realized what it provides in not being able to do it well. So perhaps Confucius should have added the proviso, as not being able to do is a hindrance to understanding.



  28. Interesting Post!
    I'm a physics graduate student in
    Korea, and your post made me to think about the physics education
    I got here one more time.
    Actually, physics education in my
    undergraduate courese was not much
    different from that of high school.
    Professor teach, students practice,
    and take exams.We had lab courses,
    but such courses were never treated seriously when I was an
    undergrad student.
    What you posted about "University
    education" worked at graduate school,not at undergraduate school.
    Since my work is oriented more to research rather than the classes,
    "just remember things" tactics can't work.
    Actually, this feature helped me
    a lot, because I had to prepare
    learning skills needed for research
    for long time by myself.
    Doing experiment designed by myself,debate on physics phenomenon with frirends,etc...
    Of course, this kind of things never helped to get high GPA both
    in high school and undergraduate school.


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