Schlagwort-Archive: Discussions

The Ethics of AI

I recently have to read a lot about the consequences of AI with respect to ethics and moral. Let me quickly sketch my personal view on the subject.

First, let us talk about science fiction and AI robots that act like humans. In my view, it is rather simple:

Machines are not human beings,
even when they are equipped with AI or look like humans.
Robots are not part of our society.

In fact, it is probably better for us if the robots do not look like humans at all. That helps us keep the distinction. Machines, with AI or not, should be marked as machines.

The reason I am so sure and definitive about that is not a religious one. We humans are programmed genetically to be part of a human community. Our very self and the purpose of our life depends on other humans. Being lonely, an outlaw, or even just being pushed around make us sick. Children, friends, and admiration make us happy. We are proud to take responsibility for others. It is a misconception to think that people are basically selfish.

Now you can argue that the social system we live in could be enhanced and improved by AI robots which are on our level or even above, both in moral acting and in intellectual capabilities. These robots could live among us just like our friends. Wouldn’t that possibly be a better society? Aren’t humans inherently unreliable and the machines are not?

But any AI that acts in predictable and reliable ways is not an AI at all. It would act mechanical and can easily be determined to be a machine. The very characteristic of intelligence is that it explores new paths occasionally. This makes it unreliable and not predictable by definition. Soon, most AI will even act in super-human ways that we are incapable of understanding due to our limited capacities.

Thus, my only conclusion is that we need to distinguish ourselves from the AI we have. We need to treat AI as machines that we use. Robots are not a product of nature that can live with us uncontrolled with its own rights. We need to protect mankind, our society, and ourselves.

After all that science fiction nightmares, let us talk about the current way we use AI to improve technical systems. Even that limited form of AI already poses questions of ethics and moral.

Some see a lot of problems arising with AI or fuzzy logic that are built into cars, airplanes or surveillance systems. It is indeed true that these systems have the potential of a collision between a human decision and an AI decision. But that is already true without AI technology! We all experienced a system that seems to act on its own, a computer or even a mechanical device.

Think of an airplane with an AI as co-pilot. As a first example, assume the AI commands a go-around in an unstable approach. The chances are quite high that the AI is right and the pilot has made an error. I also tend to think that the AI makes much fewer mistakes throughout the complete flight as a co-pilot would. I would definitely prefer an AI to a rookie on the right seat. The same applies to nearly all applications of AI in technology. Usually, the AI is better. It can even be used to train the pilot in a simulator, much better and versatile than a human instructor.

It must be clear, however, that the responsibility for the proper working of the AI is at the developer of the plane, car or technical system that uses it. The AI is not a human that we can make responsible or even punish for mistakes. The developer has to test the system thoroughly to make sure it works as intended. But if you think this is too difficult note that it is much more difficult to test a human, and the verdict is much less reliable.

It must also be clear at all times that the AI that makes these decisions is a machine. When we let cars drive on their own the car does not become a being on our level, however intellectually superior it may be. If it does not work it will be trashed or repaired.

The problem with AI and robots should not be that they will be superior to us in many ways. The problem is that we need to treat them as our enslaved machines, and not as part of our society.

By the way, I am not afraid of super-human AI. Indeed, humanity deserves a lesson in humiliation. But let us use AI to our benefit!

Avoiding CO2 Emissions

Sometimes I wished people (including myself) would think a bit more before they write. Looking around for CO2 sins, it is fairly easy to point the finger to flying. Indeed, aviation has tax privileges that hide the true costs of this means of transportation. This should be stopped. But in terms of CO2 output, modern airplanes are not that bad. Only the railway and especially travel buses are better. A car must be packed with at least 3 persons to beat the most modern airplanes. And airplanes should be used on distances only where cars or travel buses cannot compete at all.

The arguments usually continue with a claim for more railway connections. This, of course, does only help if the railways are (1) driven by regenerative energies and (2) replace other means of transportation. If those conditions are not met more and cheaper railway connections only add to the CO2 footprint of mankind. The most efficient way would be to stop driving our own cars and using travel buses. We should not think that individual traffic can go on like now in the future world of eleven billion people. An option is to reduce traveling before we are forced to do so. The means to achieve this are local economies and the net.

Moreover, traveling in total adds only 15% to the global CO2 emissions if my numbers are right. And flying is only a small part of traveling. By far the major emissions come from the industry, power plants and housing. And this will be rising due to the eleven billion people that will need basic resources to live. We should have stopped that growth earlier by education, fair chances, equal rights for women and shared welfare. But now my kids have to find a way to get the planet through this sickness called „mankind“. And there are ways to do it. We need new thinking and much more science, and a new awareness of the challenges we are facing. We don’t need a brutal business that ignores all warnings.

And we do not need simple solutions which ignore all statistical facts.

Durchschnitt und Median

Es ist unglaublich! In der offiziellen Antwort der CDU auf den Youtuber Rezo findet sich die altbekannte Stammtisch-Parole, dass der Zuzug von Jeff Brezos viele Menschen unter die Armutsquote treiben würde, weil sich ja das Durchschnittseinkommen dadurch erhöhen würde und daher die 60% des Durchschnittseinkommens, die Armut definieren, ebenfalls. Muss man sich noch wundern, dass die CDU den Unterschied zwischen Durchschnitt und Median nicht kennt? Dieses einfache Faktum sollte doch irgendjemand bekannt sein, der die Stellungnahme vor der Publikation gelesen hat. Die CDU bestätigt damit auf geradezu eindrückliche Weise die Behauptung Rezos über die Inkompetenz der in dieser Partei agierenden Politiker.

Übrigens wird auch noch das ebenso dämliche Argument von der Nachkriegszeit gebracht. Dort hätte es keine Armut gegeben, weil ja alle gleich arm gewesen sind. Ich bezweifle, dass das von den Zahlen her über die 10 Jahre nach Kriegsende grundsätzlich stimmt. Auch damals gab es schon gut Verdienende und sogar Reiche. Das gleiche gilt auch für den von der CDU angenommen Sozialismus, in dem angeblich jeder gleich viel hatte. Spätestens in der Endphase der Sovjetunion habe ich da Zweifel, ob nicht doch zahlreiche Arme zu finden waren, und zwar im Sinne der Armutsdefinition am Median. Aber das ist irrelevant. Denn Armut bemisst sich auch relativ zu den Lebenskosten. Es sind steigende Mieten, steigende Fahrtkosten, Schulkosten und andere Dinge, an denen Arme scheitern.

Noch blöder ist das Argument, man könne sich heute doch viel eher einen Fernseher leisten als früher. Das wurde schon in den Nachdenkseiten auseinander genommen, und ich muss hier nicht mehr dazu sagen. Bitte nachlesen!

Mathe-Abitur – too difficult?

Maybe I should write this in German. But why not in English? After all, we need more European standards in education. And the discussion may be of interest to students in other nations. Be aware, however, that the levels of mathematics in schools are quite different between countries like India, China, Germany or the USA. But all have something in common. And that will be the point of this posting.

Currently, we have an uproar about the 2019 mathematical test for the German Abitur. In case you are not familiar with the German system, the Abitur is a final examination of the highest level of school education in Germany. It is usually written at the age of 17/18 after 12/13 years of school life and qualifies for an entrance to the Universities. The test contains centralized elements now for all of Germany to guarantee the same level for all states in Germany (the Bundesländer). By law, each state is responsible for education in its schools. The teachers have to select sets of problems from this pool to make the results more comparable.

The students claim that the test was much more difficult than last year, and contained unexpected problems. The final word about this will be out soon when the grades of this year and last year can be compared. However, petitions were signed by thousands of students to lower the requirements this year. Obviously, not only participants in the test signed. This shows the public interest in the matter, and also the problems with mathematical education in the general public.

A typical example of this attitude I heard in a radio interview with a mathematician who organizes the Mathematical Olympiad in Germany. The question was: „If the math test in the Abitur was already that difficult why do we need mathematical Olympiades?“ It simply reflects the general tenor of „Who needs mathematics?“. The interviewed mathematician replied very politely, where I would have been much more direct and would have asked: „Why do we need a high jump?“ The interview goes on with asking if the math problems of the Abitur could be used for the Olympiade. Again, the answer was polite, but a bit off in my opinion: The Abitur is more on reproducing known techniques, and the Olympiade is on creating ingenious solutions. This answer made me write this posting. I think it is wrong and misleads the public view on mathematics.

The truth is that the successful participants of the math Olympiad all practice quite a lot. They have meetings, they read books, and they study old problems. I once hosted one of their sessions and saw how serious they take mathematics. They have a lot of tricks up their sleeves that the innocent student has never heard of. Practicing is the secret to success. It is the same as in all other competitive activity, including the Olympic Games. You do not expect a high jump professional to be excellent in table tennis. It is not what he learned and improved with thousands of repetitions. And being an expert in Algebra does not make you a brilliant applied mathematician, as well as the other way around. Human intelligence relies on learning and recognizing, and not on brilliant extraordinary ideas. Those happen only to the ones that have studied the most.

Getting back to our „Mathe-Abitur“, we have to face the truth that we have to teach the necessary skills before we can ask students to apply them. There is no principle difference between reproductive tasks and so-called transfer tasks. It is just so that the easy problems make it obvious which skills have to be applied, and the difficult problems hide this and may require more than one skill. The student needs to learn to sort out the mathematical content from the given text and see how he can apply the learned material to the problem. This is a skill of its own.

In summary, we cannot really judge the difficulty of an exam without knowing the details of the teaching that prepared for the exam. The only other way is to study previous exams. Clever students do, of course, look at these on their own. For most students, this must be part of the teaching. At this time, the only way to convince me that the problems were too hard would be a petition by the math teachers. They are the experts. A serious drop in the grades would also need an investigation, with input by those that really face the students, the math teachers.

Wie funktioniert Mathematik?

Ich kam kürzlich auf einen Blogeintrag von „Halbtagsblog“ über das Mathematik. Dort wird ein Bild gezeigt, dessen Inhalt man erst nach geraumer Zeit erkennt, weil er durch eine Schwarz/Weiß-Filter verfremdet ist. Die meisten sehen das Dargestellte erst, nachdem man Ihnen gesagt hat, wonach sie suchen sollen. Der Blog ist auch ansonsten ganz interessant, insbesondere für einen, der die Früchte des Schulunterrichts vor sich sitzen sieht.

Ist Mathematik wie das Erkennen von Mustern in scheinbarer Unordnung? Der Gedanke wirkt verlockend. Es ist nicht von der Hand zu weisen, dass beides Beharrlichkeit und Geduld (engl. persistence) erfordert. Er scheint auch zu erklären, warum manche in Mathematik talentierter sind als andere. Sie haben einfach das bessere Auge. Was mir daran gefällt ist, dass es somit möglich erscheint, durch Wiederholung und mit der Zeit Erfolge in Mathematik zu erzielen. Das ist ohne Zweifel ein guter und positiver Gedanke.

Dennoch möchte ich widersprechen. Beharrlichkeit und Geduld sind nämlich für alles Lernen notwendig. Das gilt für das Tennisspielen oder den Spracherwerb ebenso wie für das Trompete Blasen. Damit wird es als Leitfaden speziell für die Mathematik untauglich. Eigentlich ist es insgesamt zu sehr eine Binsenweisheit als dass es nützlich wäre. Die Weisheit von Hollywood-Filmen „You can reach everything you really want!“ ist schon zweifelhaft.  Als didaktische Richtschnur für Mathematik hilft sie nicht viel.

In Wahrheit ist Mathematik eine Sammlung von Techniken, mit denen wir versuchen, die Welt zu erklären. Dieses Grundmuster beginnt beim 1,2,3-Zählen und zieht sich bis zur Quantenmechanik. Wir bauen dabei immer auf dem auf, was wir von den Alten übernommen haben. Beharrlichkeit und Geduld, gewürzt mit ein wenig Kreativität, helfen uns, neue Möglichkeiten zu entdecken. Dadurch entwickelt sich die Mathematik fort. Die besten Mathematiker sind die, die das umfassenste Wissen auf ihrem Spezialgebiet mitbringen, gepaart mit dem Drang, dieses Wissen auf Neues anzuwenden und anzupassen. Bloses Warten auf einen Einfall hilft meist überhaupt nichts.

Für den Lehrer bedeutet das, das er Mathematik als Technik darstellen soll, mit der man etwas anfangen kann. Wenn das nicht gelingt, verlieren die Schüler das Interesse. Oder, besser gesagt, das Interesse beschränkt sich auf das in den Tests Benötigte.

Was aber kann man mit der Mathematik anfangen? Hier sind wir bei der entscheidenden Frage für einen guten Unterricht. In einem Mathe-Wettbewerb zu brillieren oder später selber Lehrer zu werden, kann nur eine Teilantwort auf diese Frage sein. Ich habe an anderer Stelle so viel über Mathematik versus Welt geschrieben, dass ich hier nicht darauf eingehe. Aber ein Lehrer, der darauf keine überzeugte Antwort zu geben weiß, wird niemals ein guter Lehrer.

Smartphones, Schulen und Digitale Initiative

Die Bundesregierung plant angeblich eine digitale Initiative, um die im Vergleich zu anderen Ländern an deutschen Schulen zurückliegende digitale Infrastuktur zu verbessern. Zumindest taucht eine solche Initiative immer wieder in Sonntagsreden auf, mit dem Hinweis darauf, dass Deutschland für die „digitale Zukunft fit gemacht werden soll“. Es ist auch nicht überraschend, dass die Freunde der deutschen „Wettbewerbsfähigkeit“ sich kräftig einmischen, die von neuen Schulen träumen, an denen die Schüler für die „Arbeitswelt von Morgen“ fit gemacht werden.

Auf der anderen Seite stehen die altbekannten Bremser. Viele Lehrer und Eltern scheinen sich Sorgen zu machen, dass „Kulturtechniken“ verloren gehen. Als Beispiele werden oft das Rechnen und die Rechtschreibung genannt, gerne auch das handschriftliche Schreiben und die schöne Schrift. In den Chor mischen sich die Anhänger der sprachlichen Ausbildung in Latein und Griechisch, den Grundlagen des christlich-römischen Abendlandes und des logisch, analytischen Denkens. Das Internet mit der vorherrschenden Sprache Englisch und dem Video als Hauptmedium ist das grasse Gegenteil dieses Erziehungsentwurfs.

Auf dieser Basis ist es sehr schwer, über eine Einbindung des „Digitalen“ in die Schule vernünftig zu diskutieren. Ich würde raten, erst einmal verbal und emotional abzurüsten. Weder ist ein Tablett im Unterricht der Untergang des Abendlandes, noch kann ein Unterricht rein über vernetzte Medien gelingen. Wie immer liegt die Wahrheit doch in der Mitte.

Wer Jugendliche im Netz beobachtet, sieht doch, dass sie die Möglichkeiten der neuen Geräte im Wesentlichen zur sozialen Interaktion nutzen. Das Surfen ist eben nicht gleichzusetzen mit dem Fernsehkonsum. Smartphones sind deswegen so beliebt, weil man Nachrichten senden und empfangen kann. Die überwiegende Zeit verbringen Jugendliche in sozialen Netzen wie WhatsApp oder Facebook.

Also ist es offensichtlich, dass ein Smartphone im Unterricht stört. Es ist einfach gleichzusetzen mit dem Schwätzen, einer unvermeidlichen Unsitte, mit der Lehrer seit Urzeiten kämpfen. Die Schüler nutzen ihre Smartphones auf genau diese störende Weise im Unterricht, wenn man es nicht ausdrücklich untersagt. Sie nutzen sie auch als Ablenkung von den Hausaufgaben. Diese Trivialität des guten Lehrens und Lernens wird gerne hergenommen, um Smartphones in Bausch und Bogen zu verdammen. Wir sollten statt dessen dahingehend wirken, dass die dauerende soziale Interaktion über ein Netz da unterbleibt, wo der Kontakt mit dem Gegenüber oder die Konzentration auf eine Sache wichtiger sind.

Die Anhänger des händischen Rechnens und der Beherrschung eines korrekten Deutsch sollten über diese Lernziele durchaus einmal nachdenken. Rechner und Schreiber waren in den ganz alten Zeiten zwar angesehene, aber dennoch rangniedrige Spezialisten. Heute ist der allgemeine Erwerb dieser Fähigkeiten allerdings eine Grundlage für unser modernes Arbeits- und Gemeinwesen, also die Voraussetzung für ein funktionierendes Bürgertum. Ohne Lesen und Schreiben zu beherrschen und mit Zahlen umgehen zu können, ist eine höhrere Ausbildung nicht möglich. Die Ausbildung aller Bürger in diesen Techniken ist also absolut notwendig für die Gesellschaft, in der wir leben.

Nun schreitet aber die Technik voran und auch die Welt als Ganzes. Ich habe selbst als Berufsmathematiker seit 40 Jahren keine schriftliche Division oder Multiplikation mehr durchgeführt. Mein Latein, das ich in neun langen Jahren erworben und verfeinert habe, brauche ich sehr selten und nur für Hobbies, und mein fünfjähriges Altgriechisch habe ich komplett vergessen – bis auf die Buchstaben, weil sie in der Mathematik verwendet werden. Meine Rechtschreibung habe ich im Wesentlichen nach der Schule verfeinert und ich schaue auch gerne mal im Online-Duden nach. Ein wesentlicher Teil der Kommunikation findet in Englisch statt, einer Sprache, die nolens volens (die Eliteerziehung schimmert immer durch) die allgemeine Sprache einer zusammenwachsenden Welt geworden ist. Meine altsprachliche Ausbildung behindert mich im Englischen nur. Übrigens ging sie auch auf Kosten einer naturwissenschaftlichen Bildung. Auch schreibe ich fast nie mit der Hand, und wenn, dann unleserlich. Tafelvorlesungen versuche ich mit Druchbuchstaben zu schreiben. Das geht genauso schnell und ist leserlicher.

Man schüttet natürlich nicht das Kind mit dem Bade aus. Selbstverständlich ist die Grundschule dazu da, die Grundlagen dieser Kulturtechniken zu erwerben. Es geht nicht ohne. Es ist lediglich so, dass wir uns überlegen müssen, was uns wichtiger ist. Wollen wir weiterhin stupide Techniken einüben, die in der Welt „draußen“ später nie benötigt werden? Oder wäre es da nicht besser, die nutzbringende Anwendung der zur Verfügung stehenden Techniken zu lehren? Damit ich nicht falsch verstanden werde, füge ich hinzu, dass für Kunst, Musik und zum Beispiel so etwas Exotisches wie Kaligraphie in der Schule Platz sein muss. Wir müssen auch das wertschätzen lernen. Aber gerade deswegen sollten wir Inhalte aus dem Unterricht entfernen, die nur Zeit verschwenden.

Und sobald man diesen Schritt vom Arbeits-, Schreib- und Rechenknecht hin zu einem selbständig denkenden, suveränen Menschen gemacht hat, wird klar, dass die digitale Umwelt Teil der Schule werden muss.

Is Math the Language of the Universe?

To call math the language of the universe sounds great for mathematicians and propagates it to being sort of a divine subject. If God made the university with math we mathematicians become untouchable priests. That is nice and feels good. But is it really the right way to perceive our subject?

Assume you are living as a small animal on a flat surface moving just a few meters in each direction. You will, e.g., discover that walking 10 steps into one direction, turning right at a rectangular angle (that is the angle that splits the straight angle into two equal parts), then walking another 10 steps and continuing like this brings you back to were you have been. If you are a really clever animal you will discover the Pythagoras and all of the beauty of plane geometry. Why would you want to do this? The simple answer is that a mathematical model of your world is the only way to describe what you see and predict what you expect to happen. It will be enormously useful. It does not „explain“ your world, nor „is it“ your world. It is just your model of the world, accurate enough to do science that you cannot do in any other way.

As soon as we as men look up and a bit further to the horizon, we discover how ships vanish behind the horizon. So the surface we live on does not stretch straight in all directions when we define „straight“ as the way a ray of light propagates. The surface bends somehow. Doing more accurate observations will yield a more accurate model where we live on a ball instead of a flat surface. With just a little effort we can even come up with a rather precise measurement of the diameter of this ball. And we can now predict that walking along a square as above will not exactly bring us back to our starting point. To test this will take some effort, however, as well as to test the prediction that the angles in a triangle no longer add to 180 degrees if the triangle sides follow the surface. We might even predict that we are not capable of reaching India from Spain with the resources of our sailing ships unless we are lucky to find land in between. It is an aesthetically pleasing and useful „explanation“ of our world, but as we now know it is not accurate. Moreover, we cannot „see“ the ball that we live on and have only indirect proof of that fact, such as the shadow of the earth on the moon among a lot of other phenomena. So it is just another model which helps us describe and predict.

The same happens to the planets, the moon and the sun. The first model with the earth in the center had to be made too complicated to explain what we really see in the sky. It turns out that there is a far more simple way to describe and predict the skies. We only have to put the sun into the center and the planets in circles around them. But wait! More accurate measurements show that this is again not a precise way to describe what we see. Indeed, the planets seem to be moving on ellipses with the sun in one focal point. How can this be? The genius that found a simple „explanation“ valid for centuries was Newton. He gave us a model with an interaction called gravity between bodies that decreases with the inverse square of the distance. Using mathematical tools, he showed that the result was just what we see.

Again, this model had flaws. E.g., the movements of Mercury are not exactly explained. This fact was known for a long time. But it took another update of the model by another genius, Einstein, to explain this effect. We always encounter new phenomena that do not fit with our model. Currently, we have a problem with invisible mass and energy in the large universe, and also with awkward behavior in the tiny things.

The history of this goes on and on. E.g., the special theory of relativity is just a model to explain why we do not measure a different speed of light when we move relative to the light source. The combination of time and space into a four-dimensional mathematical model yields very pleasing formulas to describe and predict this. Without these formulas, modern technology would not work. But I would strongly argue against stating that our world „is“ a four-dimensional space-time. This is just a model that helps us to describe the phenomena we see and predict the outcome of new experiments and observations. One of the most striking predictions was the curvature of light around stars by Einstein. He just found a more accurate way to model the straight lines that light rays follow, and to model our measurement of time.

In conclusion, our mathematical models are useful, but they are not identical to the world. Why then are they so elegant? An explanation for this may be that they smoothen statistical facts that we are incapable of seeing. E.g., we describe air by pressure and flow, or even chaotic turbulence, with simple formulas while in fact air is formed by movements of zillions of individual particles. On a more elementary level, we „neglect“ the resistance of air. But even if we find the one world formula like in the „standard“ model I bet it will only yield an approximation of the world, and we will soon discover that it does not represent the complete truth.

However, I could be wrong with this bet. We might be able to model the world as far as is forever possible to us humans. As useful as such a model might be, it will fail to „explain“ the world on a grand scale. We will have to be contempt with our approximate math models which served us so well over the centuries.

About Infinity

I recently came across one of the many videos on Youtube dealing with the problem of infinity. At the end of this posting, I am going to link some interesting ones for you. But the one I want to talk about deals with a very bizarre generalization from the finite to the infinite.

Let me first tell you the finite version. It is about a very nice trick which first seems impossible. Assume you are in a group of friends (i.e. finitely many, of course) with blue and red hats. Each of your friends can see all the other hats, but not his own. Your problem is that you want each member of the group to announce the correct color of his or her hat, one after the other. This is clearly not possible, but you can make it so that each but one announces the correct color. If you want to find a solution to this problem, go ahead. I’ll be waiting.

You certainly found the trick: The first person announces „red“ if the number of red hats he sees is even, and „blue“ if it is odd. You take this first round. At this point, all your friends will know the color of their hat. Neat, isn’t it? Moreover, you get a 50% chance to get your color right.

The video I am discussing generalizes this to an infinite number of friends, countable many. Now you can play the even-odd-trick only if you are in non-standard analysis and have a non-standard infinite natural number at your disposal. All standard friends will then get their hats correctly. But this is using a theory which is even more fantastic than the solution I show you below. If you up for a thought adventure go ahead and find a solution. Hint: You have to use the axiom of choice.

The proposed solution uses an equivalence relation on all sequences in the set {red,blue}. Two such sequences are considered equivalent if they are equal from some point on, i.e., they differ only in a finite number of places. This yields equivalence classes and you agree with your friends on a representative element of each and every equivalence class. Using this, each of your friends can come up with the same sequence of colors, and this sequence will agree with the correct sequence of hat colors from some point on, i.e., differ only at finitely many places. Note that all of you have the same sequence! So you, on number one position, only need to announce if you see an even or an odd number of discrepancies to this sequence. Then everyone will know if his hat color agrees with the color in the sequence or not.

A trained set theorist will immediately spot that the axiom of choice is needed to define a function Phi that maps any sequence to one representative. If this function exists it will compute the same representative even if we change a finite number of colors in the sequence. That is why the trick above works.

However, this is all too absurd for me to be considered of any value. Other mathematicians may disagree and find this to constitute a nice application of set theory. To me, it’s useless garbage. The problem is that there is absolutely no constructive way to compute the above wonder function Phi. To compute it you need to inspect the complete sequence at once. And this is not possible. You cannot get close to Phi(S) by inspecting the first N elements of the sequence S like you can get closer and closer to the square root of 2. After N elements of S, you know nothing at all about the Phi(S) you can choose.

In the video, there is another algorithm where you and your infinite number of friends announce their color at the same time. By announcing the color in the representative sequence they will get „at most a finite number“ of colors wrong. In the finite case, this means they might get all wrong. This is again a useless generalization to the infinite case.

There is another trick if friend N can only see friends with a higher number as if you are standing in line. Again, if the first in the row announces the even or odd number of discrepancies, every following announcement (one after the other) can be made correctly. This does also work in the finite case as you will observe easily.

Finally, here are some interesting videos about infinity.

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

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!


Jauch und Mathematik

Ich kann nicht glauben, dass der Bericht so vollständig stehen bleibt:

„Auch bei der folgenden mathematischen Frage musste Scherbaum sich helfen lassen: Ein Kreis mit einem Umfang von 3.141,6 Metern hat einen Durchmesser von ziemlich genau …? A: 100 Metern, B: einem Kilometer, C: zehn Kilometern, D: 100 Kilometern. Scherbaums Telefonjoker? Passenderweise ein Mathelehrer. Doch dann der Super-GAU: 29 Sekunden entfleuchte dem Pädagogen kein einziges Wort. Kurz vor Abbruch der Verbindung dann ein entspanntes: „Schwer zu sagen“. Danke für Nichts! Ein betagter ehemaliger Physik-Student aus dem Publikum wusste mit B dann aber doch die richtige Antwort.“

Ein zweiter „Mathelehrer“ hatte dann nach diversen Berichten die richtige Lösung parat. Peinlich ist schon, wenn der Kandidat das nicht wenigstens abschätzen kann. Aber jemand, der irgend etwas mit Mathematik zu tun hatte und seine Nerven halbwegs zusammen hat, sollte das leicht wissen. Ich kann mir das nur durch hochgradige Aufregung erklären.