Absolute case numbers and incidence, reproduction numbers and exponential growth - a basic understanding of maths is needed to grasp the coronavirus infection situation. Specialist didactician Astrid Fischer on a discipline between practical benefits and abstract concepts.
The current infection situation in the coronavirus pandemic is expressed in new figures and mathematical terms every day. What tools do we all need - or would we need - to be able to deal with this?
Some of these concepts require a lot of basic knowledge. For example, to understand exponential growth, i.e. the increase of the increase, you need to know what multiplication means. This is not as trivial as it seems. You also need to understand what numbers actually are. It's not enough to be able to recite the number sequence in the correct order, it's about understanding what the numbers mean: How do the basic operations work and how are they related? In their school days, many adults simply had to memorise the multiplication tables - today, this is less of a focus. Unfortunately, the fact that someone knows the multiplication tables by heart does not mean that he or she fully understands what multiplication means. More advanced tools for dealing with the numerical material of the pandemic are functions whose equations are not mentioned in the media, but are displayed graphically. You have to be able to interpret them. The relationships between these different representations and the information they contain are quite complex.
This means that some people lack a basic understanding of maths. Why is that?
There are many reasons. In maths, the learning content builds on each other. If you only know the number words but have no concept of numbers and quantities and don't know what the basic operations mean, then you can't understand what multiplication means, for example, and consequently can't understand a linear function. Things are repeated from school year to school year, everything builds on each other. This is a strong feature of maths. The content becomes more and more abstract. Adding is easier than multiplying, which is only dealt with in the following school year. If adding is not understood, it becomes problematic.
Does this lead to deeper and deeper gaps in understanding over the school years?
Yes, and this can lead to them disliking the subject. These children and young people are helpless in the face of new material: They can't understand it because they can't link it to what they know and build up expanded ideas about it. And that's why they can't deal with it creatively. This is different in humanities subjects. In maths, geometry is an example that is not based on an understanding of numbers and algebra and sometimes offers pupils the chance to get back into it. To a certain extent, this also applies to stochastics, the theory of probability, but again, a lot of maths is involved. So withdrawing from the subject is an auxiliary strategy: if it's "cool" not to be able to do maths, you don't have to face up to it. Because basic things are not understood and you feel insecure, you withdraw and shut down.
Can this be remedied?
It's very difficult to catch up again - and it's a dilemma in maths lessons in the upper years of school. It's difficult when tenth-graders lack the basics from the fifth grade. I, for one, have not yet found a solution. Because it's not simply a matter of taking note of information, it takes years of development to get used to the increasingly abstract concepts and internalise them. The lessons are deliberately designed in such a way that the children and young people usually only have to take the next abstract step in the following school year.
How can schoolchildren be made enthusiastic about maths so that as few as possible lose touch along the way?
It is important that maths lessons focus on ensuring that pupils really understand the concepts - and not just give superficially correct answers. To pick up on the example from before: So it's less about memorising the multiplication tables and more about understanding number spaces and arithmetic operations. Teaching that repeatedly encourages students to make and use connections mentally promotes a deeper understanding than if they are simply taught by rote. Semi-written arithmetic methods, for example, which repeatedly emphasise one's own thinking with arithmetic steps in the head, have an advantage over purely routine tasks such as written addition. It's about learning with understanding, and this also creates a positive relationship with the subject: when you understand, you have some control and don't feel at the mercy of it.
So understanding makes the subject fun all by itself?
Pupils - basically all of us - need interesting tasks where we have to puzzle or solve problems. Where we can contribute our own ideas and pursue and discuss them - where it's not just about "right" and "wrong". Of course, this is also important, but it's not the only thing. There are often different solutions, it doesn't always have to be the standard solution. Allowing children to think for themselves leads to a sense of competence. If we only set simple tasks so that the pupils experience "I can do this", it quickly becomes boring. It's much more about discovering something new. For some children and young people, enthusiasm for the subject is indeed a "no-brainer", but it is and remains exhausting to deal with maths and learn how to deal with abstract concepts. Some people don't enjoy sports lessons because they shy away from the effort involved - this is probably also the case with maths.
How can simple mathematical relationships be made accessible to a wider population?
For some contexts, this is even easier. When it comes to coronavirus case numbers, even someone without a deep understanding of multiplication can understand what it means when the curve is steeper or flatter. The fact that the diagrams are somewhat simplified and the function graphs are not labelled as such is certainly a good thing, because otherwise some people might not follow. It is more difficult to visualise how quickly the numbers can explode. A doubling is only seemingly small. You realise this at the latest when you follow the well-known example of grains of rice on a chessboard. If you start with one grain of rice on the first square and double the number on the following square, you end up with a twenty-digit number! If you only think additively, you can't estimate how dangerous and potentially explosive the growth is in the early stages. This realisation with regard to the coronavirus infection may have come as a delay and somewhat of a surprise to some people.
Does the pandemic also offer an opportunity for maths lessons, namely in the teaching of mathematical content with a high degree of topicality and undeniably high relevance? Whether you have the class calculate at what point the healthcare system is overloaded or how likely a superspreading event is...
The pandemic certainly provides an opportunity for this - and it shows very clearly how important maths is for our modern society. However, I personally have difficulties with the claim that maths is supposed to show exactly that. In the subject of German, I don't hear this regular demand that the practical benefits must be recognisable. Interpreting poetry, for example, is not in question there either. Nevertheless, there is this demand on maths, which is shared by many teachers. Students also sometimes have the idea that they have to embed everything they teach in a real-life context. But maths is also about abstract concepts, it is a science of structures. While it is important that real contexts have their place in the classroom, I don't think they belong at the centre of mathematics lessons.
Interview: Deike Stolz
