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.