Update post (June 2018)

calvinhobbes-almost-started

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!

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Diversions in Mathematics #2: Hilbert’s Hotel

In this instalment, I introduce the concept of infinity in a simple and (hopefully) entertaining way, which puts into practice the counting concepts introduced in the previous Diversion. In fact, Hilbert‘s infinite hotel was one of the ‘stories’ that got me seriously interested in mathematics in the first place, and so it is a pleasure to share it here. This is a very well-known piece of story-driven mathematics. I hope that experienced mathematicians who happen to come across this blog do not tire of hearing (reading) it again, and that they see the value in telling the story to the general public.

Just before we start: I assume knowledge of the definitions and notations introduced in the previous instalment, namely, the very basics of set theory.

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Sight-reading, AMEB exams, and studying maths

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 one AMEB 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.

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A Brief Ramble on Mathematics and Creativity

I recall that I was interested in mathematics from a relatively early age, probably from primary school. I was never a particularly talented mathematician — that is, I was certainly not a Wunderkind who won Olympiads, nor did I accelerate through school — but I always did well in maths, and most importantly, I have always had great curiosity and interest in the subject. More than likely my good marks were the result of my curiosity and passion, not the other way round. Oh alright, stereotypical Asian parenting also plays a role, I’ll happily admit. You may have read news articles or psychological studies about the phenomenon of math anxiety. I was rather skeptical about this when I first heard of it, but it seems to be a real psychological condition! You can read this rather dramatic article about this so-called phobia of numbers. Needless to say, this was not a problem for me. I enjoy maths, and I generally do well in maths, which only serves to increase my interest and confidence. When I don’t do well, there is still the enjoyment of a good challenge, of having grappled with a difficult problem or concept, even as I fall miserably short of the desired solution. Incidentally, if you ask a musician why they devote so much of their daily routine to practice and rehearsal, usually the response is along these lines: I enjoy it; music inspires me and I hope to inspire others; it is rewarding and fulfilling; this is my passion, and so on. In both cases, music or mathematics holds intrinsic interest for certain individuals, and they are motivated to master the skills or understand the concepts in their area of interest, and perhaps go as far as to extend current knowledge by exploring and developing new techniques and ideas. In short, I would like to propose that creativity is essential in mathematics. Obviously this is not the same kind of creativity necessary for artistic development, but nevertheless some form of it is needed. Now this is not a theorem which can be proved rigorously, but I hope you will allow me to conjecture a little!

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