Category Archives: Teaching

kids, exponential growth and 42

Last week, I was lucky enough to attend the W3C workshop on ebooks in NYC. This allowed me to visit some old and very dear friends. In a conversation with one of their kids, I pulled out a classic that I like very much.

Today, I did some fact checking and — lo and behold — the answer was not 52 but 42! That is, of course, fantastic.

Anyway, the question I asked was: how thick is an piece of regular office paper if you fold it 52 42 times?

The answer is: it would reach all the way to the moon!

That usually surprises kids (and non-kids) and is a nice example for the surprises of exponential growth. In fact, it also surprises me and I’m always somewhat nervous when a kid takes me up on the offer of checking that the number is actually correct.

For this you first have to decide what paper you’re looking at. A piece of A4 paper (I’m German after all) is on average 0.1 mm. That’s actually hard to estimate but it’s what I eventually found on the interwebs; if you have the time, I invite you to delve into the art of density and calipers.

When you fold it 42 times, it’s as if you stacked test $2^{42}$ pieces of paper on top of each other. So the thickness is $2^{42}$ x 0.1mm, which is ~439,804 km (and a kilometer is 1,000,000 milimeter).

The moon is on average 384,400km from earth, and 405,410km at its farthest — so we’ll get there no matter what day. If, that is, we could fold a piece of paper 42 times.

For what it’s worth, the world record for folding paper is 13 times — achieved by high schoolers on MIT’s campus in 2011.

Hanging out with Sam

Sam already wrote about the g+hangout that he hosted last Saturday and I wanted to jot down some of my own thoughts on the experience. I thoroughly enjoyed the whole thing and I’ve been looking for good imagery to describe it. I only came up with two, but maybe you can help me with that in the comments.

  • Reinhold Messner going on an small hike with a group of amateur friends.
  • A good go player playing a high-handicap game against a weaker opponent.

It is always a pleasure to join people to think about a problem, but I was surprised how much fun it was when I already knew I could solve the problem. This probably had something to do with the fact that I hadn’t thought about this particular topic since essentially my Vordiplom (after 2 years at university). This brings me to my first image. Just like a professional mountaineer on an amateur-level hike, I knew I wouldn’t have to worry about being dangerously out of my depth (though I stumbled myself one or two times). However, this didn’t make me enjoy doing mathematics with the others any less than Messner would enjoy any hike in good company, being challenged every once in a while and taking in the majestic nature surrounding us.

Equally intriguing was the interaction with the other participants. A Go player will play a high handicap game against a weaker opponent not because she wants to show off her strength, but because it levels the playing field and makes it enjoyable for both sides. Sam did a wonderful job at facilitating the exploration as well as the condensation of ideas, much like placing handicap stones at just the right level. That way, all parties involved could enjoy the artistic process that is doing mathematics.

Coda

I would suggest this idea to anyone — join such an interaction, at least once in a while! It is also yet another great example of how the internet allows us to connect to people and activities that might not be as readily available locally. Finally, it’s another example of the great potential the net offers for researchers to engage with a community of people that have dabbled in mathematics at some point in their lives but now have few chances to interact with math, let alone professional mathematicians. We can help people to keep their interest in mathematics alive and kicking while having fun ourselves — if that’s not worth it, I don’t know what is.

The candy stores of academia

Over lunch I had one of those really excellent discussions you can enjoy with grad students at the University of Michigan. Among a thousand interesting topics we talked about the question of career options for grad students after graduation. Of course, we started debating the usual sad story: being academically brought up at a research university your supervisors will hardly ever discuss alternatives to a career in academia because, well, they obviously did not pursue alternative career paths and hence have little to share on the topic. Of course, most faculty will argue that they are in the education business (not the certification business) so they need to focus on educating you about (their) research topics. At this point it’s not difficult to start a rant for a day or two…

Thankfully, the grad student gave it an interesting twist by mentioning the experience of a friend higher up in the academic food chain at a university in a city far far away. Said friend did, in fact, try hard to do just what we both complained about and invested a large amount of time in pointing out the risks of an academic career. Unfortunately, it turned out that the students could not be bothered to consider alternatives, would be annoyed and generally opposed to information on the subject.

This is, unfortunately, not really surprising. Universities are, in essence, the candy stores of academia. The students are like small children in awe of people who seemingly live in the candy store. What child would believe them when they say that you shouldn’t eat candy all day long?

The single most important subject

Recently, I had a few conversations about teaching mathematics varying from elementary school to undergraduate level. These are not refined, in depth analyses, just some quick thoughts that I think worthwhile to ponder every once in a while.

The single, most important subject in school is language

The single, most important subject in school is the native/primary language — especially, for mathematics. This seems to strike people as odd. But I find this statement trivial. Too many people seem to think that everybody understands language anyway and that’s that. As a mathematician and science fan I often found that the lack of proficiency in languages compromises people’s abilities to deal with everyday concepts in our scientific world. The complexity of language is a challenge not to be taken lightly. Even professional journalists are frequently overwhelmed by the complexity of scientific writing — over at scienceblogging.org you will probably find some science blogger crying out against a piece of misunderstood and/or abused science everyday.

In mathematics, it is however even more vital than in the sciences and the humanities. The sciences have empirical facts and the humanities have both emotional and factual aspects that allow us to understand their concepts with information outside of the language being used. If you don’t know what I mean, just look up to the stars, look at some original historical documents or listen to a poem read aloud.

In mathematics the situation is different, language is both alpha and omega. The initial step in understanding a new concept in mathematics is alike to Galileo’s problem: his first telescope was of such primitive nature that if you didn’t not know what you were looking at, you could easily get confused and not see what’s there. Written mathematics is often the only way to make a first step, maybe blindly following the formalism, maybe trying to find understanding by meditating over prose. Once this initial step has been taken we have diagrams, we have computer animation and most of all fruitful discussions; we have all sorts of helpful tricks to add to this initial understanding. But that is not enough, one cannot stop there. In the end, when push comes to shove, we only trust a detailed proof in written or spoken language (and if Doron Zeilberger ever reads this: code is valid for me). There are no other facts, no independent empirical facts, no historical facts that can significantly support a mathematical thought. The abstract thought that gives rise to mathematics can, it seems, only be exchanged neutrally using language.

At school level, it might seem that language is only needed for word problems. But these are often the worst examples — head over to Dy/dan to understand that. Language is used uselessly; lots of words for no mathematical content. At university on the other hand you often find only our highly specialized language without any supposedly superfluous understandable language. For example, you will encounter the tradition that refuses to find a way to write proofs as they are discovered (which actually often makes them more accessible); the university variant of dy/dan’s pseudocontext are epsilon-delta arguments written the ‘flawless’ but inaccessible way: let $\epsilon > 0$; set $\delta$ to $\frac{ 2 \sqrt{ln(\epsilon) – 5.2346}}{3\pi}$.

So, teach language more! It’s the one and only subject that guarantees a) life long, self-governed education, b) citizens that can understand complicated (political) issues and c) students that have a better chance of excelling at mathematics (and in the long run produces better mathematicians, hooray)!

Supervisor Precognition

I am currently working on a nice little proof. I was initially very confident that I could find the proof — mostly because I already gave a different proof for the same observation, albeit with completely different tools.

This time around the proof will probably end up as a series of stronger and stronger lemmas. Already 3 weeks ago I thought I had shown the strongest intermediary lemma I have formulated so far. Unfortunately, Andreas shot it down, then I found a different proof, and Andreas shot it down again, then I found another proof, and Francois shot it down. As painful as this sounds (and really, really is), I am so lucky to have such colleagues.

Finally, I think I have a found a proof for the intermediary lemma that will live (well, maybe I should wait until Francois sees it later today…). The creepiest thing about this proof, however, is Andreas’s precognition.

When I showed him the failed second attempt (which failed pretty much at line 1…), we discussed the phenomena involved and Andreas made two comments about the problem itself. The first was that due to the setting, an indirect proof seemed to him to be the way to go; second, he gave a very simple example, a special case of the problem that should turn up in some general form. And as you might expect, these two predictions came true — in almost every respect.

The first time I ever heard of such precognition was from a student of Stevo Todorcevic when I was just starting out on my PhD — and it scared the hell out of me to hear that he predicted a complication that the student only found after 3 months of work on that problem. I call this phenomenon ‘supervisor precognition’. It’s not like supervisors in mathematics always have an idea for an actual proof of a problem, usually not even for a strategy. However, supervisors often have a small but brilliant insight into the situation as such and might spot some critical properties far ahead, long before the student actually gets there.

I know that this strength has as much to do with experience as it does with mathematical talent, but it is both annoying and wonderful. Annoying, since I would like to pretend that I could be as productive without these amazing insights from other people. For Hikaru no Go fans, it feels like Akira playing Sai in the first game — it’s a move from far above, so to speak. To use a metaphor that I don’t really like, if we fight through a jungle to get on a mountain top, this precognition maybe is the equivalent of a greater height, allowing to see not the exact path ahead, but a few major obstacles ahead.

I think this precognition is perhaps the most important strength of a good supervisor. On some level, this precognition needs to be present to guide students to their own, independent research. Students should look out for signs of it (and ask around who’s got it) and researchers should try to develop it (if anyone can tell me how, please tell!). Now if only my proof was finished…