# My Journey In Mathematics

How did I get to where I am today? Here’s a trip down memory lane that I hope is a fun read! Maybe there are some lessons in reflection to be learned here, both for myself and my dear reader.

## Primary Schooling

I’m not going to start with my early childhood days as I don’t have any memories from then (I think this phenomenon is called Infantile Amnesia) – so I’ll start with my earliest memories from primary school days.

### Times Tables

I remember my dad, who worked as a structural engineer before he immigrated from Hong Kong to Australia, took me aside one day to “do something fun and enjoyable”. The activity? Learning how to count pictures arranged in rectangles on multiple pages. I remember telling him, “I already know how to do this!” and proceeded to count one by one for each question – instead of using multiplication to do it in one go. I did not think it was a fun nor enjoyable activity.

From that day, he taught me the times tables song starting from 2 and eventually to 9 in Chinese, and every night I had to recite it to him before bedtime. I also quickly learned not to go into his room because he would make me recite it again before I was allowed to leave. I also had to recite it in the shower, and at random times during the day.

This initial rote learning of the times tables wasn’t too bad when I noticed the commutative property of multiplication (i.e. $$a \times b = b \times a$$) – which helps cut about half the working memory needed. By the time my peers in primary school were just learning their 5 or 6 times tables, I already knew mine up to 9 – learning the times tables at home instead of school definitely had its advantages here.

Recalling the times tables in Chinese in my head was something I did even up to Year 12 in high school.

Eventually, I noticed more patterns in the tables, like some numbers were common in the two’s with the three’s, etc. Over my primary school years, my dad also taught me about lowest common multiples, highest common factors, prime numbers, algorithms to find common factors using prime numbers, working with fractions, long division algorithms, and long multiplication algorithms. Most of primary school class in numeracy was pretty much revision.

### Spatial Awareness & Problem Solving

Something I enjoyed a lot when I was in primary school was solving cast metal link puzzles, where you had to take it apart without the use of excessive force. I think this may have had something to do with developing my spatial thinking in three-dimensions and problem solving. My favourite puzzle is one that used an unlocking strategy that behaved like counting in binary as a solution:

I also remember fondly my older brother teaching me how to play first person shooters (while I was under the age of ten) with him, and I think being able to walk and jump and visualise objects in 3D on the 2D screen helped with my understanding of geometry. (Yes, I’m well aware that the ethics of an under-aged child playing Counter Strike 1.1 or Duke Nukem games is a bit dodgy). Thankfully, there’s probably a safer environment in Minecraft to build spatial awareness through computers these days.

There’s also something about computer games that I think inspire more than a dopamine hit in the gamer. While most of my older colleagues have expressed their concerns and negative views about gaming (namely about addiction, time-wasting, social under-development, or when their husband spends too much money or time with online friends in EVE Online) – I think there’s value in playing video games (within healthy moderation) that build problem solving, motor skills, resource management, financial skills, etc.

Examples include defeating the water temple in Ocarina of Time without a guide, timing jumps in Mario, managing minerals and vespene gas and army units in StarCraft, fishing for enough lobsters in Runescape while preparing for a fight against the green dragon, or flipping weapons on the Diablo III auction house… anyway, I digress.

### Computer Programming

I think you can see my interactions with computers wasn’t exactly limited in the home. So on a more academic side of computing, I remember my brother was also learning programming and webpage design on his own when he was in high school. Hence I, the nosy little brother, also became interested in what he was doing.

He gave me his HTML4 textbook to read and from that I eventually learned how to design my own websites from scratch. In early high school, I also learned how to program interactive webpages in PHP3 – there was a time in high school when I created a puzzle website and everyone at school was playing it on the library computers. It felt good.

## High Schooling

In year 6, I sat the selective schools test and got into Baulkham Hills High School for Year 7. In that year, I remember learning algebra, equations, simultaneous equations, ratios and rates, and geometry. I don’t remember there being much emphasis on fractions, decimals and percentages as that was all covered in primary school. At the end of Year 7, I applied for a transfer to James Ruse Agricultural High School, and I think the fact my brother was a very successful alumnus there influenced their choice in selecting me to replace a student in my cohort who left to Newington College.

### Hey Look! The New Kid!

I was a new student in 2005 at the school – and the only new student. Even my student number 058001 was a bit special: year of enrolment 2005 in year 8, student number 1. I don’t think student numbers would have hidden my identity very well in school exam papers now that I think about it.

Year 8 was a very enjoyable year in learning mathematics. I remember learning Set Theory: we were assessed on set notation, unions, intersections, cardinality, etc. I’m still very grateful for this moment in my maths education, as sets are at the foundational core of all of maths – something that I believe needs to be more emphasised in today’s treatment of mathematics in schools.

I also remember learning about Circle Geometry theorems in Year 8, and this really developed my understanding of geometry proofs even into Year 10-12 mathematics.

In Year 9, I somehow caught a negative attitude towards mathematics. I don’t know why, but I remember saying that I hated it. Maybe it was because I wrote my responses to separate questions on the same sheet of paper in the Year 8 yearly exam instead of on new sheets as instructed – having thought I had done a wrong thing, I cried. That was the end of Year 8 for me.

So Year 9 didn’t go exactly that well as I didn’t really care about what I learned. I still learned some things, like completing the square of quadratics, remainder theorem of polynomials, factor theorem and graphing polynomials. However, I didn’t try very hard and didn’t do much homework so I ended up in the 3rd (out of 5) streamed classes in Year 10.

### Parabolic Turning Point

Having my teacher in Year 10 was the best thing to happen to me.

She was very inspirational, passionate and knowledgeable about mathematics. She was making classes fun every day, provided a lot of feedback on work in class quizzes, and I looked forward to being in class. I remember a funny moment when I was being naughty and was playing Tetris on a Gameboy when she was asking the class how to do a problem. No one answered her question, so she exclaimed, “… Let’s ask our top student, Ringo!” – my response: “Whaaaaat…. I’m playing Tetris.” She confiscated the Gameboy from me and commented on my score “What a low score! Let me show you how it’s done!” – I think I still had to answer the question.

I wasn’t humiliated for doing the wrong thing in class but she turned it into a fun moment.

From home, my dad was also sitting me down next to him for a couple hours per day to do my maths homework as well as to teach me more things in addition to what I learned in school.

In Year 10 at James Ruse, (not sure if it’s still done today) I had to sit a half-yearly examination, a yearly examination and then topic tests on Quadratic Theory, Radians and Logarithms – my year’s cohort didn’t have time to cover Absolute Values that was historically assessed.

The learning environment from the school and the home front really propelled me to excel that year. I did well in the examinations, and my performance in the topic tests placed me in the first streamed maths class in Year 11 and 12.

In that year I was somehow invited to participate in one of those four hour mathematics competitions run by the Australian Mathematics Trust. I was sent a letter of invitation to a summer school camp, but I regrettably did not attend as I wanted a summer holiday at the time.

It was also in Year 10 when we had to do Work Experience placements. Initially, I thought I would enter the medical field after high school as most James Ruse graduates do, but when I did my placement at a dental clinic, I knew from that experience that the medical profession was not for me. Law was also out of the equation as I did not enjoy reading long texts and writing essays. At the end of Year 10, I didn’t know what I wanted to do after high school yet.

### Senior Years

My year 11 teacher was also very approachable and knew her content very well. It made me appreciate her ability to teach when her explanation of differential calculus just suddenly made sense in my head one day when I was walking home. She taught using proper mathematical language by defining variables through set memberships like $$x \in \mathbb{J}$$, she used (and explained) logical language such as “if and only if”, and her explanations were clear.

As a side note, it is interesting that some of the staff at James Ruse used $$\mathbb{J}$$ to denote the set of integers rather than $$\mathbb{Z}$$. Older textbooks such as Rudin in the 1970’s used $$\mathbb{J}$$ – so its use kind of shows someone’s age. I’m personally a $$\mathbb{Z}$$ for Zahlen (‘number’ in German) person.

Anyway, in that year, I knew I wanted to be like her and firstly become a mathematician, and then secondly a teacher. I wanted to know my mathematics more deeply so decided that I would study it at university beyond the high school syllabus level, before taking a teaching degree to teach in schools.

My teacher unfortunately left the school to be a head of department elsewhere at the end of Year 11. Thankfully, her replacement for our class was just as brilliant. While his style was different, our class benefitted greatly from him too.

In addition to covering the HSC syllabus, I remember discussions with my teacher in extra topics such as vertical circular motion, double and triple integrals, matrices, using integrals to find arc lengths, and maybe other things I have since forgotten – these experiences inspired me to look forward to further study past high school. I also remember staying back after school to work on an integration problem (see below) with a friend – who is now a Doctor of Philosophy in Physics!

$\int_0^\frac{\pi}{2} \ln(\sin x) \; dx$

## Bachelor of Science

After graduating from high school, I learned a lot in the next three years at university while studying for a Bachelor of Science (Advanced Mathematics) at Sydney University. Firstly in academia of course, but also in humility.

See, doing well in high school mathematics brought a false sense of security coming into the university advanced mathematics degree – I thought I knew more than I actually knew. If you’re familiar with the Dunning-Kruger Effect, I was at the “Peak of ‘Mount Stupid'” coming into first year.

The ride down towards the “Valley of Despair” started from the first week when I realised my understanding of a function from HSC was incomplete: I had no idea what a co-domain was, and terms such as injectivity, surjectivity and bijectivity were completely new. You can read more about my thoughts about HSC functions here: The Incomplete Treatment of Functions in HSC Mathematics

By the time epsilon-delta definition of a limit was introduced in class, I knew the journey in university was going to be a fun one – one that I am grateful for doing as it has opened my eyes to a world of mathematics beyond what’s constrained in the HSC.

Maths at university was the real deal: no more “informal treatments” of concepts.

In second year university, the most important subject that I took was “Real and Complex Analysis”, which opened my eyes to understand how and why mathematics works – not just focus on computation as is often emphasised in high school. Terence Tao opens one of his books with this quote:

It is a fair question to ask, “why bother?”, when it comes to analysis. There is a certain philosophical satisfaction in knowing why things work, but a pragmatic person may argue that one only needs to know how things work to do real-life problems. The calculus training you receive in introductory classes is certainly adequate for you to begin solving many problems in physics, chemistry, biology, economics, computer science, finance, engineering, or whatever else you end up doing – and you can certainly use things like the chain rule, L’Hˆopital’s rule, or integration by parts without knowing why these rules work, or whether there are any exceptions to these rules. However, one can get into trouble if one applies rules without knowing where they came from and what the limits of their applicability are.

Terence Tao, analysis i., 2016

I entered university thinking I wanted to major in Applied Mathematics, but this course helped me appreciate proof and rigour of understanding why things worked more. Although I got better results in the few applied units of maths that I chose at university, I had more enjoyment and personal satisfaction from studying pure mathematics, and hence chose more of those subjects.

My lesson in humility was felt deepest when I hit “The Valley of Despair” really hard in third year when I was studying Modules and Group Representation Theory – it was something that I was clearly not prepared for and I was way out of my depth in that course. It’s definitely something I want to revisit somewhere down the track when I am more mature and ready for it (probably next year).

In the end, I graduated from university with 10 third year level mathematics subjects completed:

• MATH3061: Geometry and Topology
• MATH3065: Logic and Foundations
• MATH3962: Rings, Fields and Galois Theory (Adv)
• MATH3963: Differential Equations and Biomaths (Adv)
• MATH3964: Complex Analysis with Applications (Adv)
• MATH3966: Modules and Group Representations (Adv)
• MATH3969: Measure Theory and Fourier Analysis (Adv)
• MATH3977: Lagrangian and Hamiltonian Dynamics (Adv)

I didn’t go on to do an Honours year as I was set to do the Master of Teaching degree the next year. I sometimes regret not doing my Honours as there were some cool things such as Algebraic Topology that I didn’t study, and it also means entry into academic life in a post-grad setting would require more hoops to jump through – something I thought at the time I wouldn’t consider.

## Master of Teaching

The next two years completing a teaching degree was a huge transitional shock for me as the type of academia and cohort I was with changed drastically.

For the last three years leading up to this point, I was studying a rigorous advanced mathematics degree – to then do a course with pass/fail (i.e. mark out of 1) assessments and having to read educational philosophy texts and discuss more humanitarian/social work ideas – I felt like I was in the wrong place.

There was a time when I was feeling burnt out with the course that I started to look for jobs in industry, but I always remembered my high school desire to be like my Year 10-12 maths teachers so I stuck with it and am now teaching at school. The best parts of the degree were the teaching practicums where I learned more about teaching by doing it.

## Study Hiatus and Return

In the 10 years since graduating from my undergrad degree, I had the opportunity to teach high school mathematics at school and learn a lot about working in a school from my colleagues, especially from my heads of department. I also got to learn a lot about living at school from working in a boarding house for almost 4 years.

In that time, I also kept reading mathematics for my own enjoyment – revisiting my old undergrad notes such as Group Theory when I taught that Option to a Further HL maths class, Measure Theory when the HSC introduced random variables to the course, and other random tidbits here and there.

It was in 2019 that I got the itch to learn more again and so I decided to complete a Graduate Certificate of Data Science. That course helped me really understand and appreciate Probability and Statistics (as I didn’t really do much of that in my undergraduate studies), and the application of it through the use of computer programming to data analysis.

Completing this course helped me realise that I still wanted to learn, and I am much more academically mature now compared to when I was an undergrad – so I have started a Master of Mathematics at UNSW in 2022. Read more about it here: I’ve Applied for a Master of Mathematics Degree!

So this is where I am now – currently studying a Master of Mathematics and learning more about mathematics each day! Let’s see where this goes…