A Superior Alternative to Rote Learning

When I was taught soccer as a kid, there was one big mantra:

repetition, repetition, repetition. 

We learned to pass by standing in front of each other and passing the ball between us for 20 minutes. We did this almost every training session. The same way we learned headers. We learned shooting by shooting onto the goal for half an hour at the end of every training session.

It wasn’t fun, but it worked. After several years of weekly practice, I’m quite good at soccer.

When I was a bit older I learned to play trumpet and the mantra was again:

repetition, repetition, repetition. 

I had to repeat certain songs until I was able to play them perfectly. I’m sure this method would have worked again if I hadn’t given up after 2 years or so.

The same teaching method was used to teach me mathematics, Latin etc. in school. I learned to solve equations by solving hundreds of them. I learned to integrate by integrating hundreds of integrals. I learned Latin vocabularies by repeating them over and over again.

The story continued when I learned physics at university. To pass exams I had to know the exercise sheets by heart. Thus I calculated them over and over again until I had memorized every step.

Again it wasn’t fun but worked. Rote learning is certainly a valid approach, but is it really the best we can do?

It turns out, there is another teaching method that is not only much more fun but also far more effective. It’s called differential learning. Currently, this approach is only somewhat widespread in sports, but I’m convinced that it’s applicable almost everywhere.

Introducing: Differential Learning

The basic idea is this:

Instead of letting someone repeat the correct way to do something over and over again, you actively him/her them to do it wrong. 

For instance, if I want to teach soccer to kids, I don’t let them repeat the correct passing technique over and over again. Instead, I tell them to pass the ball in every correct and incorrect way possible.

A good way to pass a ball is to use inside of the foot. I let them do this, but also tell them to do it in every other way possible.  They have to use the outer part of their foot. They have to use the back of their foot. They have to use the bottom of their foot. They even have to pass the ball with their shin.

This way they learn to control the ball and pass it cleanly much quicker. They are immediately exposed to the differences between correct and inferior techniques. That’s why it’s called differential learning. The kids learn to adapt and find their own style. Most importantly, the brain doesn’t get bored and keeps learning and learning.

This method is surprisingly new. It was first put forward in 1999 by the German sports scientist Wolfgang Schöllhorn. However, it became popular quickly, at least in the soccer world. For example, the former coach of Borussia Dortmund, Thomas Tuchel, used it with great success.  In addition to such anecdotal evidence there is serious research going on and so far, the data looks convincing.

So ist differential learning limited to sports?

Absolutely not. It’s easy to imagine how the same basic idea could be applied in other fields. However, I don’t know any examples where differential learning is currently used outside of the soccer world. This means we need to get creative.

My field is physics, so I will use it as an example. Let’s say we want to teach quantum mechanics.

The thing is if you pick up any textbook on quantum mechanics, all you find is the standard story, repeated over and over again. I recently helped a friend who was preparing for her final exam and was shocked when I saw again how similar all the textbooks are. What you’ll never find in these textbooks is disagreement or discussions of alternatives. However, this would be exactly what we need to make differential learning of quantum mechanics possible.

So how could differential learning of quantum mechanics look like in practice?

First, let’s remind ourselves how differential learning of soccer works. Afterward, we can try to map the essential steps to quantum mechanics. To teach kids soccer, we need to identify the fundamentals: passing, shooting, headers, tackles, stopping, etc. Then we let them execute these fundamentals, but make sure that they do it in every wrong and right way possible. The goal is that the kids learn to control the ball in all kinds of situations and are able to move the ball wherever they want it to be on the pitch.

So what are the fundamentals of quantum mechanics and what do we want our students to be able to do? Our goal is that students are able to describe the behavior of elementary particles in all kinds of situations:

  • when they are alone and moving freely,
  • when they are confined in a box,
  • when they are bound to another particle,
  • when they scatter off a wall,
  • when they are shot onto a wall with slits in it,
  • when they move in a magnetic field etc.

The differential way to teach this would be to give the students the task to describe particles in these situations, together with the experimental data that tells them what actually happens. We don’t force the correct way to do it onto them. Instead, we encourage them to try it in every wrong way possible.

This way we can avoid that the students simply memorize the usual quantum algorithm* without understanding anything.

This is exactly what goes wrong in the standard approach. Like the kids learning soccer by repeating the “correct way” to do something over and over again, students of quantum mechanics usually only learn to apply the standard quantum algorithm again and again.

Instead, through differential learning, they would not only be able to describe what the particles do in all these situations but actually, understand why the description works.

That’s just one example, but it’s easy to apply the principles of “differential learning” to any other topic. I would love to see people implement it in all kinds of fields. So, if you know any existing course that makes use of “differential learning” or has any ideas of how and where it could be used, please let me know.

*The algorithm is so simple that it is easily possible to apply it without any deeper understanding: Write down the Hamiltonian for the system in question, put it into the Schrödinger equation, solve it and while doing so take care of the boundary conditions. The solution is a function of space $x$ and the square of the absolute value of the solution gives you the correct probability to find the particle at any place you want to know about. You can simply memorize it, together with the Schrödinger equation and you’ll be able to solve almost any problem your professor throws at you in an exam.

PS: There are, of course, still lots of details missing in the alternative quantum mechanics course outlined above.  However, it’s on my to-do list for next year to fill in the gaps and develop a fully-fledged quantum mechanics mini-course that applies the principles of “differential learning”.


Everyone can Contribute

There is a problem in the tech community called “Devsplaining“. This notion is used to describe when “experts” condescendingly explain to others the “proper way to code”.

The thing is that there is really is no proper way to code. Usually, the “experts” unnecessarily complicate stuff. As a result, most beginners think they need to study coding for years and know everything about the latest technology before they are capable of contributing anything valuable. With this kind of mindset, most never share anything they create. In tech, this means that people never launch their project or never even start because they think they aren’t ready.

Exactly the same problem exists in physics, mathematics and probably most other scientific fields. The problem is so widespread that there isn’t even a name for it. It’s just so normal.

As a result, many people feel they are not good enough to contribute. This is a problem because people who overcomplicate things exclude many brilliant minds who really could make a difference.

It’s nonsense that you need to spend your best years doing complicated calculations to prove they are good enough. It’s nonsense that you must be capable of doing the most complicated calculations before you can add something. It’s nonsense that you need to master every mathematical aspect before you can contribute anything significant to the field.

Novel deep insights can originate from everywhere.

Possibly from a rigorous proof.

Possibly from a long and complicated calculation.

Possibly from a simple thought experiment.

It’s not too hard to find examples for each of them:

Moreover, making huge discoveries is not the only possibility to help physics move forward. To quote Sir William Lawrence Bragg

“The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them”

There is a famous cartoon (based on a quote, often attributed to Einstein) that summarizes the problem nicely:

If you now roll your eyes because you have seen this cartoon already 100 times, please let me explain.

The thing is that while in exams the task usually is comparable to the task “climb the tree” here, the overall “task”, for example, in physics is much broader. Maybe more like “understand the tree”. Climbing the tree is a viable possibility to understand some aspects of the tree, but certainly not exhaustive.

The thing is that “Devsplainers” try to convince everyone that the only thing worth doing is to climb the tree and that you have to do it in a very specific way.

However, with a broader goal like “understand the tree” in mind it’s easy to imagine how each of the animals in the cartoon could contribute.

The fish and bird could together figure out where the water comes from that are crucial for the tree. The elephant could use his strength to open up a “window” to look inside… You get the idea.

As an aside: From this perspective, it’s clear how problematic it is that students usually are only asked to “climb the tree” in exams. Many students who could contribute in other ways are filtered out. However, this is a quite different story I don’t want to dive into here.

So how could the situation be improved?

I have two concrete ideas.

  1. I’m currently building a “Travel Guide to Physics“.  The goal here is to show that nothing is really complicated. There are lots of different ways to tackle a subject and not every explanation is suited for everyone. Instead, everyone needs to find explanations that speak a language he/she understands and that’s what the Physics Travel Guide helps with.
  2. I’m trying to get more people to share what they learn. The thing is, as I like to emphasize, beginners need explanations from beginners to understand something and not polished stuff from “Devsplainers” who overcomplicate everything.  If more people would share what they learn, everyone could learn more easily. Currently, we “are an army of wheel-reinventors” and I would like to help to change that. Moreover, sharing notes while learning is an example of how students can start contributing early on. Now, why aren’t more students doing this? There are two major obstacles:
            1.  Many don’t know they are “allowed” to do this or that others would care about their notes and
            2. many don’t know how to do publish their notes online.

      That’s why I created Physicsnotes.org. My goals with this project are

            1. to motivate people to share their notes and show them that they don’t need permission.
            2. To give them the tools and knowledge to do so.

      (I really regret that I didn’t publish my notes while I was a student. While some of them are now published as a book, the majority of them no longer exist.)

    So these are just my ideas. I’m certain there are more and I would love to hear them. Moreover, I think a crucial first step would be to give the problem a name like the tech people did with “Devsplaining”. With a fitting name, people could start to call people out for overcomplicating things and frighten beginners. So if you have an idea, please let me know.


Unfortunately, repetition is a convincing argument.

I recently wrote about the question “When do you understand?“. In this post, I outlined a pattern that I observed how I end up with a deep understanding of a given topic. However, there is also a second path that I totally missed in this post.

The path to understanding that I outlined requires massive efforts to get to the bottom of things. I argued that you only understand something when you are able to explain it in simple terms.

The second path that I missed in my post doesn’t really lead to understanding. Yet, the end result is quite similar. Oftentimes, it’s not easy to tell if someone got to his understanding via path 1 or path 2. Even worse, oftentimes you can’t tell if you got to your own understanding via path 1 or path 2.

So what is this second path?

It consists of reading something so often that you start to accept it as a fact. The second path makes use of repetition as a strong argument.

Once you know this, it is shocking to observe how easy oneself gets convinced by mere repetition.

However, this isn’t as bad as it may sound. If dozens of experts that you respect, repeat something, it is a relatively safe bet to believe them. This isn’t a bad strategy. At least, not always. Especially when you are starting, you need orientation. If you want to move forward quickly, you can’t get to the bottom of every argument.
Still, there are instances where this second path is especially harmful. Fundamental physics is definitely one of them. If we want to expand our understanding of nature at the most fundamental level, we need to constantly ask ourselves:

Do we really understand this? Or have we simply accepted it, because it got repeated often enough?

The thing is that physics is not based on axioms. Even if you could manage to condense our current state of knowledge into a set of axioms, it would be a safe bet that at least one of them will be dropped in the next century.

Here’s an example.

Hawking and the Expanding Universe

In 1983 Stephen Hawking gave a lecture about cosmology, in which he explained

The De Sitter example was useful because it showed how one could solve the Wheeler-DeWitt equation and apply the boundary conditions in a simple case. […] However, if there are two facts about our universe which we are reasonably certain, one is that it is not exponentially expanding and the other is that it contains matter.”

Only 15 years later, physicists were no longer “reasonably certain” that the universe isn’t exponentially expanding. On the contrary, we are now reasonably certain of the exact opposite. By observing the most distant supernovae two experimental groups established the accelerating expansion as an experimental fact. This was a big surprise for everyone and rightfully led to a Nobel prize for its discoverers.

The moral of this example isn’t, of course, that Hawking is stupid. He only summarized what everyone at this time believed to know. This example shows how quickly our most basic assumptions can change. Although most experts were certain that the expansion of the universe isn’t accelerating, they were all wrong.

Theorems in Physics and the Assumptions Behind Them

If you want further examples, just have a look at almost any theorem that is commonly cited in physics.

Usually, the short final message of the theorem is repeated over and over. However, you almost never hear about the assumptions that are absolutely crucial for the proof.
This is especially harmful, because, as the example above demonstrated, our understanding of nature constantly changes.

Physics is never as definitive as mathematics. Even theorems aren’t bulletproof in physics because the assumptions can turn out to be wrong with new experimental findings. What we currently think to be true about physics will be completely obsolete in 100 years. That’s what history teaches us.

An example, closely related to the accelerating universe example from above, is the Coleman-Mandula theorem. There is probably no theorem that is cited more often. Most talks related to supersymmetry mention it at some point. It is no exaggeration when I say that I have heard at least 100 talks that mentioned the final message of the proof: “space-time and internal symmetries cannot be combined in any but a trivial way”.

Yet, so far I’ve found no one who was able to discuss the assumptions of the theorem. The theorem got repeated so often in the last decades that it is almost universally accepted to be true. And yes, the proof is, of course, correct.
However, what if one of the assumptions that go into the proof isn’t valid?

Let’s have a look.

An important condition, already mentioned in the abstract of the original paper is Poincare symmetry. This original paper was published in 1967 and then it was reasonably certain we are living in a universe with Poincare symmetry.
However, as already mentioned above, we know since 1998 that this isn’t correct. The expansion of the universe is accelerating. This means the cosmological constant is nonzero. The correct symmetry group that preserves the constant speed of light and the value of a nonzero cosmological constant is the De Sitter group and not the Poincare group. In the limit of a vanishing cosmological constant, the De Sitter group contracts to the Poincare group. The cosmological constant is indeed tiny and therefore we aren’t too wrong if we use the Poincare group instead of the De Sitter group.
Yet, for a mathematical proof like the one proposed by Coleman and Mandula, whether we use De Sitter symmetry or Poincare symmetry makes all the difference in the world.

The Poincare group is a quite ugly group and consist of Lorentz transformations and translations: $ \mathbb{R}(3,1) \rtimes SL(2,\mathbb{C}) .$ The Coleman-Mandula proof makes crucial use of the inhomogeneous translation part of this group $\mathbb{R}(3,1)$. In contrast, the De Sitter group is a simple group. There is no inhomogeneous part. As far as I know, there is no Coleman-Mandula theorem if we replace the assumption: “Poincare symmetry” with “De Sitter symmetry”.

This is an example where repetition is the strongest argument. The final message of the Coleman-Mandula theorem is universally accepted as a fact. Yet, almost no one had a look at the original paper and its assumptions. The strongest argument for the Coleman-Mandula theorem seems to be that is was repeated so often in the last decades.

Maybe you think: what’s the big deal?

Well, if the Coleman-Mandula no-go theorem is no longer valid, because we live in a universe with De Sitter symmetry, a whole new world would open up in theoretical physics. We could start thinking about how spacetime symmetries and internal symmetries fit together.

The QCD Vacuum

Here is another example of something people take for given only because it was repeated often enough: The structure of the CP vacuum. I’ve written about this at great length here.

I talked to several Ph.D. students who work on problems related to the strong CP problem and the vacuum in quantum field theory. Few knew the assumptions that are necessary to arrive at the standard interpretation of the QCD vacuum. No one knew where the assumptions actually come from and if they are really justified. The thing is that when you dig deep enough you’ll notice that the restriction to gauge transformations that satisfy $U \to 1$ at infinity is not based on something bulletproof, but simply an assumption. This is a crucial difference and if you want to think about the QCD vacuum and the strong CP problem you should know this. However, most people take this restriction for granted, because it has been repeated often enough.

Progress in Theoretical Physics without Experimental Guidance

The longer I study physics the more I become convinced that people should be more careful about what they think is definitely correct. Actually, there are very few things we know for certain and it never hurts to ask: what if this assumption everyone uses is actually wrong?

For a long time, physics was strongly guided by experimental findings. From what I’ve read these must have been amazing exciting times. There was tremendous progress after each experimental finding. However, in the last decades, there were no experimental results that have helped to understand nature better at a fundamental level. (I’ve written about the status of particle physics here).

So currently a lot of people are asking: How can there be progress without experimental results that excite us?

I think a good idea would be to take a step back and talk openly, clearly and precisely about what we know and understand and what we don’t.

Already in 1996, Nobel Prize winner Sheldon Lee Glashow noted:

[E]verybody would agree that we have right now the standard theory, and most physicists feel that we are stuck with it for the time being. We’re really at a plateau, and in a sense, it really is a time for people like you, philosophers, to contemplate not where we’re going, because we don’t really know and you hear all kinds of strange views, but where we are. And maybe the time has come for you to tell us where we are. ‘Cause it hasn’t changed in the last 15 years, you can sit back and, you know, think about where we are.”

The first step in this direction would be that more people were aware that while repetition is a strong argument, it is not a good one when we try to make progress. The examples above hopefully made clear that only because many people state that something is correct, does not mean that it is actually correct. The message of a theorem can be invalid, although the proof is correct, simply because the assumptions are no longer up to date.

This is what science is all about. We should always question what we take for given. As for many things, Feynman said it best:

Science alone of all the subjects contains within itself the lesson of the danger of belief in the infallibility of the greatest teachers in the preceding generation. . . Learn from science that you must doubt the experts. As a matter of fact, I can also define science another way: Science is the belief in the ignorance of experts.