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”.