Hey, I almost forgot I have a blog! (Since I’m paying for the site hosting, why not make more use of it).
This post will simply be an update, to let the readers of this blog (the number of which is non-zero) know what I am currently doing. It will be a random assortment of thoughts and comments. Right now, I am busy preparing for the semester 1 exams in the Pure Mathematics Honours program at the University of Sydney, but it is nice to take a break from study and write something here. Needless to say, it has been a very challenging semester, but also quite a rewarding one. Mathematics honours students are required to take a total of 6 courses throughout the honours year, as well as prepare a thesis. Many (?) people opt to take 4 of the 6 courses in the first semester, with the intention that more time can be devoted to the preparation of the thesis in second semester. But naturally, this means that one undertakes a lot of coursework in first semester (4 honours level courses at the same time is no joking matter), and as I am prone to procrastination, the time management has been especially challenging. Fortunately, I get along well with my thesis supervisor — who is conveniently also the honours coordinator this year — and he has been understanding and supportive during the periods when I had many assessments to submit and had not worked on the honours project!
Some thoughts on teaching
In addition to coursework, I have also been tutoring first-year linear algebra this semester. In all the maths courses at Sydney, students attend (or are supposed to attend) lectures where the essential content is presented, then they attend (or are supposed to attend) tutorials where they work through problems based on the theory. As it turns out, I enjoy teaching both mathematics and music. What both these disciplines have in common is that one can only learn by doing. For a music student, consistent practice is essential, and likewise, anyone who wishes to develop the mathematical skills necessary for a STEM career needs to practise doing mathematics through solving problems. I believe that the key idea is as follows: as a maths tutor, or as a violin teacher, one is faced with the task of helping the student develop good practise skills. This is different from delivering a lecture. Taking a tutorial is more personal, and one has to be sensitive to the various difficulties the students face. Although there will be some concepts that are challenging to most or all students, in general the nature of these difficulties will often vary according to the student. Thus it is often necessary to explain the same thing in several different ways, adapting to the responses of each particular student. In this way, one really deepens one’s understanding of the subject.
I think there are two main aspects that must be addressed when one teaches mathematics or music. I describe them roughly as “technical details” and “big picture”, although the union of these two concepts is by no means disjoint (I will explain what this means shortly) . Admittedly this attitude is derived from my music teaching, but I have found it useful for mathematics too. The first is quite obvious: in music, technical details include what is often called “basic technique”, the mechanical aspect of instrumental playing. For any instrument — including the voice! — there is an ample supply of scales, arpeggios, and études (studies) to develop basic technique. For violinists and especially so for pianists, there seems to be an endless supply of such études! In mathematics, the “technical details” are of course the theory and the methods, and the études correspond to the tutorial exercises and assignment problems. In music practice, notice that the mechanical aspects are tied together with the studies and exercises — that is, the studies and exercises are precisely the means by which one develops instrumental technique. In particular, I believe there is nothing in the music lesson that really corresponds to the “lecture”. Of course, the instrumental teacher does impart “theory” to the music student, but this is never done abstractly. It would be an absurd lesson if for example a violin teacher merely describes how to play legato and détaché while the student dutifully takes notes. Rather, it is all done through the technical studies, and most importantly with the instrument. I think there is something valuable for mathematics students and teachers in this perspective: it must be emphasised that doing exercises and problems is the primary means of develop mathematical technique. The material delivered in the lectures is of course essential, and without a clear explanation of the basic definitions and theorems, the students will not know how to begin working on the problems. But perhaps we should think of the lectures as a preparation, while the real, active learning begins when the student tackles exercises and problems.
The “big picture” aspect is slightly harder to describe, as it is less rigorous and more about developing intuition. In the music lesson, it would be terribly uninspiring if a teacher only focused on the mechanics, without giving the student any guidance on how to develop a sense of what is musically beautiful. After all, the whole purpose of developing good technique is to have the necessary skills to express one’s intentions through sound! (This is what I meant earlier, regarding the intersection of the two main aspects). A great musician not only has a clear idea of what they wish to express through music, but also knows exactly how to achieve it physically (and to do so reliably every time in concert!). In mathematics, I find that one can often get very lost in the details, and it is necessary to step back and simply ask “what is the subject about?” This is certainly not a rigorous, mathematically precise question, and hence we do not expect to have a rigorous answer. Similarly, a music student should ask themselves “what is this piece of music about?” If the piece of music is a song, a tone poem, or an opera, or anything where the musical process is guided by something extramusical, then you can answer in a definite way what the piece of music is about. But even in abstract instrumental music, this is an important question to ask. Some people will in fact be able to construct some sort of narrative to help their understanding of the abstract work. I happen to have a more abstract imagination, and therefore I prefer to consider the actual musical processes. Some questions I ask include: what are some of the important harmonies? how to the harmonies develop? how do the main motifs develop? what structures arise from these developments? and so on. (Not surprisingly, these are the same questions I consider when I compose music). In any case, the key idea is to break free of the mechanical aspect of instrumental playing for a moment and engage one’s intuition.
The idea is similar in mathematics: the students can churn through problems and memorise definitions and theorems, but that does not constitute an appreciation of mathematics. Quite the opposite is often true: many people develop an aversion or even fear of mathematics for precisely the reason that they experienced mathematics in high school as something tedious and repetitive, and formidably difficult but without reward. To develop a student’s intuition is a real challenge, and I do not claim to have succeeded. However it won’t be for lack of trying. For an example of how it might be done in mathematics, I highly recommend Grant Sanderson’s YouTube channel 3Blue1Brown. You will find that his videos present complex mathematical concepts with visually appealing but carefully constructed animations, and the theory is presented in a way that emphasises intuition over rigorous definition. Grant makes it clear that his channel is not a place to be overwhelmed in technical details, but rather to develop an appreciation for the mathematically beautiful.
Comments on Joseph Haydn’s chamber music
I have said to various people on different occasions that as a self-taught composer, I have probably learnt more about composition from Haydn than any other composer. I think about the string quartets very often, in particular the marvellous Op. 76 set (probably the most famous set out of all his quartets). However, may I also suggest the much lesser-known Op. 71 set, and also the unfinished quartet in D minor Op. 103 (I believe it was the last thing he wrote). Another recent source of inspiration is the wonderful set of piano trios which are called the “Bartolozzi” trios (the catalogue numbers are Hob.XV:27, 28 and 29, the keys are C, E and E-flat major respectively). In so much of Haydn’s music, and especially in the works mentioned here, one finds a masterful synthesis of contrasts: simple themes with complex developments; well-behaved, classically elegant phrases set against surprising twists of harmony and uneven phrasese; rigorous, “learned” techniques such as fugue rubbing shoulders with deliberately silly or awkward passages. The last point is important — it is essential to appreciate that Haydn’s music is full of charm, wit, and humour, and not even necessarily a refined sense of humour, but sometimes plainly outrageous. Let me share two of my favourite examples of Haydnesque shenanigans.
- The finale of the string quartet Op. 50 No. 6. This work is nicknamed the “Frog” quartet solely due to this movement. The main theme is a silly “croaking” figure in announced by first violin that is achieved using a technique called bariolage. It is up to you to decide whether this actually sounds like a frog, but my suggestion is that it is even sillier if it does not represent a frog. Here is a lovely recording by the Tokyo String Quartet.
2. The finale of the piano trio Hob.XV:29 in E-flat major (the 3rd in the set of Bartolozzi trios). This movement bears the substitle “In the German Style”, although my feeling is that it is a parody of rustic German folk music, because it is hilariously way too fast to be any proper German dance (compare with this Ländler or any of Franz Schubert’s German Dances for piano to see what I mean). Suggested image while you are listening: imagine someone recorded a video of a bunch of drunk Bavarians dancing, and now press fast-forward. Here is a delightful recording from the Trio Wanderer.
Of course, Haydn could also produce serious, profound movements. Indeed, in his last (unfinished) quartet Op. 103, the key of D minor seems to refer to a similarly dark and serious quartet, Mozart’s quartet No. 15 in D minor (KV421). I believe he was too frail and sick to complete the quartet. The quartet’s unusually sombre mood perhaps indicates that he felt the end was near. Nevertheless, it is remarkable that Haydn’s characteristic wit still shines through in this work, although more subtly than is expected.
For those who already know a lot about classical music, I encourage you to rediscover Haydn, and for those who are not interested in classical music, then I suggest that Haydn’s chamber music may change your mind.
That’s all for now, and I promise it won’t be another 6 months until the next update! But firstly I must focus on the upcoming exams. Then I intend to continue writing the “Diversions in Mathematics” series, and I also look forward to spending time to compose music and practise violin. Finally, let me express a note of gratitude to all my friends and colleagues in the maths department at Sydney. I feel welcome in this mathematical community, which is not something I imagined when I first made the difficult switch from my music degree. Your support encourages me to maintain focus when things get challenging, and our conversations, whether about maths or otherwise, are invariably delightful.