Let me tell you about a mistake I made recently. Some months ago, I made a few calculations which seemed promising, and I thought they could lead to some new results. When I showed the calculations to my supervisor, he was skeptical, and rightly so. Our project involved some fairly abstract concepts in the theory of operator semigroups, so his immediate advice was to check the calculations against some simple, concrete examples . So of course I did that, and quickly realised that the calculations were totally wrong! Moreover, they were wrong for a rather silly, elementary reason — a pitfall that I thought I should be capable of detecting with my current mathematical experience.
To clarify what I mean by “elementary”: no, it wasn’t a silly error in the sense of a missing minus sign or multiplying two numbers incorrectly. Mathematicians say that something in their field of study is “elementary” if it is a basic fact in that particular field. However, to paraphrase my supervisor, “basic does not mean easy”, so if I were to describe the mistake to a non-mathematical audience, it would still be extremely difficult. Another interesting point is the following. When you do research in pure mathematics, quite often you deal with abstractions upon abstractions, and it is easy to forget the humble origins of these abstract concepts. For example, the concept of a linear operator is fundamental to many areas of maths, and as cool as it is to be working with linear operators on infinite dimensional Banach spaces or whatever, it is also worth checking a statement about abstract linear operators on, let’s say, a 2-by-2 matrix as encountered in 1st year university maths (or even high school, depending on where you were educated). This comic sums up the sentiment quite well (excuse the pun).
Now this brings us to the main point of this blogpost. The wrong calculation I described above is not the mistake I want to write about; rather, I want to focus on what happened next. I assume that most people do not enjoy looking at their failures. I am not sure what I did with those few pages of calculations: either I threw them out, or I shelved them away somewhere I had forgotten. In any case, recently I wanted to return to the same problem with a different perspective, but now I have lost my point of reference. Because I had misplaced or thrown out the incorrect calculations, I cannot really remember what I had got wrong the first time! To be fair to myself, I have a vague idea of the kind of error it was, and it is very unlikely I will make exactly the same error again, but nevertheless it would have been more efficient to keep the original calculations, wrong as they are, so that I can remember exactly what the lapse of judgment was. As I have learned by now, mathematical work is not only about putting one true statement after another. It is equally important and even illuminating to know what kind of result cannot be true. In hindsight, throwing away those calculations was a bigger mistake than the calculations themselves.
There are certain similarities with the performing arts, although I have to admit that the comparison gets fuzzy on closer inspection. For a start, mistakes in mathematics are objective errors: you are objectively wrong if you claim that 1 + 1 = 3, for instance. On the other hand, speaking from a background in Western classical music, even though there are certain objective errors (e.g. playing a wrong note), usually we are unhappy with a performance because we feel that we could have done certain things better. One could say this is a “subjective” error. Despite this fundamental difference though, both mathematicians and performing artists undeniably improve by learning from their mistakes, objective or subjective. Indeed, I would say that learning from mistakes is an essential part of development in the two disciplines.
As a violin student, I used to dislike making recordings — what a way to highlight one’s faults and weaknesses! However, in hindsight I should have taken more opportunities to record my playing, and to embrace whatever mistakes (objective or subjective) occurred during a performance. One should not take this viewpoint too pessimistically. The idea is not to focus only on failures; rather it is about using those failures strategically to improve one’s understanding or problem solving technique in the case of mathematics, or one’s expression capabilities in the case of the performing arts.
I end this post with a nice little comic, which comes with some wise words from the great mathematician Paul Halmos:
On 19 and 20 January, I had the pleasure of teaching at the Zhang Violin Summer School for young violinists (AMEB grade 3 to A.Mus equivalent). This was a 4-day intensive course, where students received training in preparing solo repertoire and chamber music, as well as Dalcroze eurhythmics. I was invited to take chamber music lessons, a task that I gladly accepted. There was quite a wide range of ages: the youngest students were in years 1 and 2, while the eldest ones were senior school students. As a result, there was a similar range of experience in chamber music. Teaching at the summer school has reinforced my belief of the high importance of chamber music in music education, and in this blogpost I would like to share some of my thoughts on the subject.
First of all, I remark that it is a happy coincidence that the summer school was held at Barker College, where I was a former student. It was at Barker that I first experienced playing chamber music — specifically, the string quartet. As it turned out, I played string quartets throughout my entire education at Barker College, and it was one of the most important factors in my development as a musician — not only as a performer, but also as a composer! The main point that I will emphasise is this: playing chamber music makes great demands on the musicians involved, in technical aspects (i.e. the physical part of instrumental playing) as well as conceptual (i.e. intellectual understanding of the music). These demands are quite different from those encountered in solo playing, and furthermore, I will try to argue that chamber music should play a significant role in a student’s musical development.
At the beginning of the year, I promised that I would try to write more regularly. This has clearly not been achieved! In my defence, studying mathematics full-time requires much dedication, patience, and practice — not unlike learning a musical instrument. But now I have time to write since I have completed my semester 1 exams.
This would seem like an awkward mix of topics, but hopefully I can convince you of the similarities. The connection occurred to me recently, as I have been tutoring a student for theory and sight-reading in preparation for a grade 4 AMEB violin exam. (The AMEB is the Australian Music Examinations Board, a bit like the local Aussie version of the ABRSM, which is a world-wide music education organisation). In addition to preparing a set of pieces for performance, the student sitting the exam must also answer questions relating to music theory and history. I have only ever taken oneAMEB exam in my life, so I don’t know exactly what kinds of questions are asked during a typical exam. Based on this student’s learning materials, I can deduce that, at this early stage in the progression of grades, they are likely to be questions regarding the fundamentals of music history and analysis, such as: “What is a concerto?”; “What is the form of this movement?” (binary, ternary, ritornello, and “through-composed” are the expected possible answers at this grade — sonata form comes later!); “The music of Mozart is representative of which period of music?”; “You just performed a piece by Handel, can you name some other pieces by Handel?”; “What does allegro moderato mean?; and so on. This is not particularly challenging. A student who is curious and motivated will probably know the answers already, via searching on Wikipedia and other sources on the internet. These basic concepts of Western classical music may also be covered in high school music classes, if the school is fortunate enough to provide them. For the average student, these facts can be imparted easily by the teacher during a lesson, with the additional advantage that explicit examples from the music being practised may be used. With a little more effort, the fundamental concepts of musical analysis and theory can be similarly acquired, or else taught in the lesson too. Remember, at this early stage (grade 5 or below), the student only needs to recall the basic facts.
I have decided to record some of my thoughts on composition on this blog. In the good ol’ days, these things were usually penned down into a notebook or included in letters to friends and colleagues, but now we have the Internet!
Finals of the Kendall National Violin Competition
For this first notebook entry, I would like to say a few words about my recent trip to Kendall, NSW, where I attended the finals of the Kendall National Violin Competition (hereafter KNVC). Despite its setting in a quiet rural town, the KNVC is nevertheless one of the most prestigious in Australia, and attracts the best young violinists from around the country.
(*My apologies for not writing sooner, it has been a very busy time at Uni with many assessments due recently!)
There are some words which (bad) writers of popular science like to throw around in order to sound impressive. Quantum is perhaps among the most frequently abused terms, for instance. Another would be statistically speaking. (Did you really analyse the sample distributions, and compute the variance and correlation coefficients? Didn’t think so). Another buzzword, and the one pertinent to our discussion here, is neuroplasticity. Fortunately, unlike quantum field theory and statistics, we don’t have to pretend to understand neuroplasticity, since there are many cases where it is clearly visible, and I believe its core concepts are readily grasped by the general public. In addition, popular science titles like The Brain That Changes Itself by the neuroscientist Norman Doidge certainly increase public awareness and understanding of this issue. We can observe directly that the brain is capable of remarkable change and adaptation. One of the most dramatic and convincing examples of this is the phenomenon of “phantom limbs” — the ability to feel pain or itching or other sensations in missing limbs. If you are interested, I will save you from googling for unreliable sources, and link this journal article by Ramachandran & Ramachandran (2000). The important and (at the time) revolutionary idea is that the brain is not a static organ. The brain map can be reorganised — new neural networks can be created, and the physical topography of the brain is susceptible to change. This is reassuring for those with dystonia. It means that there is a possibility of “unlearning” the dystonic movements, and creating new neural pathways to replace the old, misbehaving ones. Many dystonia researchers now recognise re-training as a viable treatment option, and from a musician’s point of view, I believe it is also the most useful and least harmful. Medical treatments can work very well for some individuals — and depending on the severity of the symptoms, may even be essential — but the prospect of taking botox injections several times a year, for example, is not exactly pretty. However, re-training requires considerable time investment and dedication… then again, isn’t this exactly like learning a musical instrument!