A Manifest for better University Teaching in Math
The Current State
Teaching at the university level is completely different from teaching in schools. We assume that adults are capable of autonomous, self-guided learning. Thus we teach by „reading“ to them. In math, that is writing on a blackboard. Students are expected to make notes to conserve our wisdom for their studies at home. In math, this means copying every word, formula, and figure from the blackboard, carefully avoiding transmission errors and possibly correcting the errors we make on the blackboard.
Simple copying is a boring, time-consuming, and inefficient way to learn.
The old-timers often disagree. They argue that it has a learning effect, especially if it is followed by preparing a careful summary at home, something that only very few students do and only in their favorite subjects. Nowadays, students have to be present about 20 hours a week and cannot be expected to spend the same time to prepare summaries, and then the same amount of time again for solving exercises.
The truth is that most university teachers are too lazy to prepare each lesson separately. So they reproduce the same script year after year. A minority does indeed prepare the lesson mentally with the result that they talk and write at the same time, but with different content. This is a most common mistake with slides. Note:
When using slides you must read aloud everything that is written on the slide!
Otherwise, your audience will be reading and not listen to you.
Blackboards do not foster this mistake in the same way as slides, because you have to turn your back to the audience while writing, and it feels strange to talk without eye contact. Some do it nevertheless. But yet you are easily tempted to talk and explain while the students still copy your content from the blackboard. You should always remember that you have just uncovered your writing by stepping aside from the blackboard. The following is very obvious.
When using the blackboard be silent while the students copy from the blackboard!
Since we are talking about the current state and the mistakes that are done in teaching we should become aware that the times are changing. Be aware that after the Bologna process the universities changed drastically. When I studied math I took two subjects, such as topology and probability theory. In these subjects, I had to pass a final examination with no grading. The other subjects in my secondary subject I did not take too seriously. They had to wait until I prepared for the final examination, sometimes all at once in one written test.
Nowadays, it is not uncommon to have six subjects at the same time, like complex functions, algebra, numerical math, an introduction to statistics, and two classes in the side subject. All these modules have tests at the end of the term, and all must be passed and are graded. The grades usually go into the final grade of your degree.
Students have less time for your class than you think.
Most modules in our university have 5CP with 4 hours of presence. The semester has 30CP, so that adds to 24 hours of presence. You need a bit of time to learn or do exercises. I would say you need at least another 18 hours just to keep up with the subject. So you should not be surprised if students do not spend as much time on your subject as you expect.
We also have classes with 10CP in the first semesters and 6-7 hours of presence. This is much better. But even then this is only 1/3 of the total amount of studying. So students can only spend two full days on your subject, almost one day completely taken by copying your content from the blackboard.
Towards better Teaching
The first change would be to have fewer subjects at the same time. There is simply no way a student can learn a lot of different math subjects at the same time in any sustainable way. The student will concentrate on a few of them, maybe one or two. It might be yours that is neglected!
The number of graded exams should be limited to two or three in each semester.
There might be one or two more subjects but they should not be graded and the exams should be very easy. And in no way should they count for the final grade.
This concentration of fewer exams can actually be achieved by collecting modules from several semesters into one bigger module in one semester. The problem is often to organize this, i.e., to make room for larger modules. The staff that is responsible for organization finds it easier to squeeze small modules into the available space. We should not let them do this.
Math takes time and concentration on one subject without too much distraction.
The next step is not to waste the time of your students. If you write your script onto the blackboard and they have to copy it from there in the class this process is simply a waste of time. Students do not think while they write. If you want to get the feeling for this process go into a research talk and write down everything that appears on the blackboard, or even worse, on the slides. Do that while the speaker is explaining what he just did and try to split your attention to reading, writing and listening. Research talks are not meant to be written down verbally. Some take notes, but nobody writes down all presented content. So why are you expecting your students to do just that?
Have a script available for the students and explain it in the class!
By explaining I mean any communication that is not written down by the students. It is about looking, thinking, imagining and if possible discussing. It may include small sketches or mind maps. The important thing is that it is activating the process of reflection on the subject. One of the best ways of doing this is by connecting it to previous knowledge or experience. In any case, these explanations are not meant to be written down and read later. For this, we use the script.
I have done classes without a script and actually am doing one right now. I am completely aware and ashamed of doing so. As a lame excuse, this is not one of the classes I do on a regular basis. Moreover, I try to cope with this defect by carefully waiting until the blackboard content has been copied. Only then I start to explain what I did and encourage the students to think and reflect on what they just saw. Moreover, I give them time to recover while I carefully clear the blackboard. The good side effect is that I do not cover an excessive amount of details. It is still enough since this class is one of the rare modules with 10CP.
You and the students should know what you expect your students to accomplish.
Do you expect your students to study every detail of every proof you give? That is exactly what many teachers do. Of course, it is not possible to achieve this completely. Consequently, the process of learning is a frustrating thrive for perfection. The argument behind this ambitious goal is to keep the standards as high as possible. Asking for less sounds like giving in to the „mediocre students of today“. But this attitude does not work and hinders the students from becoming true mathematicians.
Use the exercises to make your goals as clear as possible!
You should set clear goals for your students. These should be goals they can reach. So, device exercises that actually are possible to do. Do not underestimate the positive motivation of a goal that has successfully be mastered and the negative impact of a thrive for perfection which is impossible to reach. In fact, it is not possible to completely master a subject in math. Luckily, there will always be some open question in our subject, even on an „elementary“ level.
Get the students interested in the interesting math that is not covered in your class.
The part of the subject that you have covered in your class should be a good basis, but it is never complete. Make that clear! And make clear that you expect the basics to be mastered, and that those basics are covered in the exercises that you gave. You can add exercises that you do not expect to be solved but by one or two students in your class. You expect the students to understand and try these problems to practice their persistence. These exercises should be clearly marked, however.
In the same way, you might extend your class to subjects which are not covered in the exam. Make clear that you do this because these subjects are interesting. Make sure that they really are. Be aware, that the students will only accept this if they get the feeling that they can actually accomplish the subjects that are part of the exam. Otherwise, they will claim that you are wasting their time with unnecessary stuff which just confuses them.
The future has already started. We now have the internet. It has its good and bad sides. Usually, we tend to completely ignore the net. We argue that its content is not reliable enough. In view of our own background, it is not surprising that we want the students to use books and research papers. But these limits are no longer appropriate. Restricting students to printed material is wishful thinking.
Be aware of the Internet!
In the net, you will find academies with thousands of videos about all topics in math. Many of them are actually very well done and enjoyable ways to learn a topic. The idea of the academies or the virtual universities is „self-guided“ studying. You can stop a video or a presentation and repeat it at any time. No doubt do we meet problems. Some videos, even by high-class universities, are actually very bad. But so are the classes, to begin with, and they do not get better by recording them. The purpose of a video in the net can only be to design teaching on a very high level and make that public. It takes effort and time to do this.
There is also material like scripts and exercises publicly available in abundance. Clearly, this material might not always meet the requirements or follow the presentation of the attended class. But students tell me that with a bit of search they always find something that helps them better understand my teaching. I have no quarrels with that.
Use the net for social interaction!
This is what young people today are used to. Social media is by far the biggest part of the net, and the most attractive. You can view that as an extension of learning groups or student meetings. Sadly, is it often also a replacement for the actual being-together of people. But for us teachers, it is a chance that we just are beginning to explore. Currently, I use our platform only to present material like exercises, scripts and code snippets, and occasionally links. It is a one-way communication. The students have their own meeting points.
This ends my „manifest“ towards a better teaching of math. This blog does not get many comments. But it may be worth to discuss a little bit about the teaching of math in the comments. You are welcome to do so!