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Articles
Mathematics can be seen everywhere. In this section, I will highlight some examples in real-life where advanced maths can be seen. The articles are aimed at students in high-school and above. Any prerequisite knowledge from school maths will be identified. Click on the picture to access the respective page.
You might also be interested in this explanation of my mathematical research interests aimed at students.
If you have any questions or if you spot any typos/mistakes, please contact me at [email protected]
If you have any questions or if you spot any typos/mistakes, please contact me at [email protected]
Slides from previous talks
Don't Underestimate Exponential Growth! (2020)
Slides
Given at the Hong Kong Academy of Gifted Education.
A famous legend goes like this:
A wise man presented a king with a beautiful handmade chessboard as a gift. The king was delighted with the gift and asked the man what he would like in return. The man answered that he would like one grain of rice on the first square of the chessboard, two grains of rice on the second square of the chessboard, four grains of rice on the third square, eight grains on the fourth square and so on. The king was surprised at such a seemingly meagre request and readily agreed.
What do you think happened next?
In this talk, we will discuss how the "meagre request" was anything but meagre! A chessboard has 64 squares, if the man's request was fulfilled, the last square would contain 264 grains of rice. This number has 20 digits - it's more than the number of grains of rice on earth! This is an example of exponential growth and the legend demonstrates how people tend to underestimate the speed of exponential growth.]
We will discuss what is exponential growth and give some real-life examples where it can be seen. We will then see how underestimating exponential growth can have serious consequences - particularly during a pandemic.
This talk was inspired by this excellent article on the BBC:
https://www.bbc.com/future/article/20200812-exponential-growth-bias-the-numerical-error-behind-covid-19
It was aimed at lower secondary students. Knowledge of exponents (powers) is the only prerequisite.
Given at the Hong Kong Academy of Gifted Education.
A famous legend goes like this:
A wise man presented a king with a beautiful handmade chessboard as a gift. The king was delighted with the gift and asked the man what he would like in return. The man answered that he would like one grain of rice on the first square of the chessboard, two grains of rice on the second square of the chessboard, four grains of rice on the third square, eight grains on the fourth square and so on. The king was surprised at such a seemingly meagre request and readily agreed.
What do you think happened next?
In this talk, we will discuss how the "meagre request" was anything but meagre! A chessboard has 64 squares, if the man's request was fulfilled, the last square would contain 264 grains of rice. This number has 20 digits - it's more than the number of grains of rice on earth! This is an example of exponential growth and the legend demonstrates how people tend to underestimate the speed of exponential growth.]
We will discuss what is exponential growth and give some real-life examples where it can be seen. We will then see how underestimating exponential growth can have serious consequences - particularly during a pandemic.
This talk was inspired by this excellent article on the BBC:
https://www.bbc.com/future/article/20200812-exponential-growth-bias-the-numerical-error-behind-covid-19
It was aimed at lower secondary students. Knowledge of exponents (powers) is the only prerequisite.
Solving maths problems WITHOUT the techniques you learned at school (2019)
Slides
Given at the Hong Kong Academy of Gifted Education.
I've got some good news for you*. In real-life practice, pretty much all of the techniques you learn for calculating things such as solving equations, finding areas (and more) are NOT used. Instead, what happens is that maths problems are solved numerically. This means that the solution is found with a computer. In this talk, we give examples of how this is done for:
- solving (nasty) equation;
- calculating the area/volume of a "weird" shapes/solids
- finding the probability of a complicated event.
* Unfortunately, this doesn't mean that you don't have to learn these techniques at school!
This talk was aimed at upper secondary students. Calculus knowledge (differentiation and integration) would allow you to better appreciate the content of some of the slides (these are marked *calculus*).
Given at the Hong Kong Academy of Gifted Education.
I've got some good news for you*. In real-life practice, pretty much all of the techniques you learn for calculating things such as solving equations, finding areas (and more) are NOT used. Instead, what happens is that maths problems are solved numerically. This means that the solution is found with a computer. In this talk, we give examples of how this is done for:
- solving (nasty) equation;
- calculating the area/volume of a "weird" shapes/solids
- finding the probability of a complicated event.
* Unfortunately, this doesn't mean that you don't have to learn these techniques at school!
This talk was aimed at upper secondary students. Calculus knowledge (differentiation and integration) would allow you to better appreciate the content of some of the slides (these are marked *calculus*).
How to find your way with algorithms (2019)
Slides
Given at the Hong Kong Academy of Gifted Education.
These days, if you are on the street and you are lost, you would use the “Map” application in your smart phone to guide you to your destination. Programmed into this app are algorithms - a “recipe” of some sort - that tell the phone how to find the shortest path to where you want to go. In this talk, I discuss:
- the concept of an algorithm and how this concept impacts modern day technology.
- the main considerations one must make when creating an algorithm.
- these considerations are illustrated through sorting algorithms and shortest path algorithms (such as the algorithm in the “Map” application).
This talk was aimed at an early secondary school audience. As such, it has no huge prerequisite knowledge.
The animations for the various algorithms can be found in the respective wikipedia pages:
- https://en.wikipedia.org/wiki/Bubble_sort
- https://en.wikipedia.org/wiki/Insertion_sort
- https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm
- https://en.wikipedia.org/wiki/A*_search_algorithm
Given at the Hong Kong Academy of Gifted Education.
These days, if you are on the street and you are lost, you would use the “Map” application in your smart phone to guide you to your destination. Programmed into this app are algorithms - a “recipe” of some sort - that tell the phone how to find the shortest path to where you want to go. In this talk, I discuss:
- the concept of an algorithm and how this concept impacts modern day technology.
- the main considerations one must make when creating an algorithm.
- these considerations are illustrated through sorting algorithms and shortest path algorithms (such as the algorithm in the “Map” application).
This talk was aimed at an early secondary school audience. As such, it has no huge prerequisite knowledge.
The animations for the various algorithms can be found in the respective wikipedia pages:
- https://en.wikipedia.org/wiki/Bubble_sort
- https://en.wikipedia.org/wiki/Insertion_sort
- https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm
- https://en.wikipedia.org/wiki/A*_search_algorithm
One Should Always Generalize (2018)
Slides
Talk given at the Hong Kong Academy of Gifted Education.
The starting point of this talk is a quote by Carl Jacobi: "One [Mathematicians] should always generalize". We talk about one of the differences between high-school maths and maths in university - the results are more general; the situations considered are more abstract. The concepts of generalization and abstraction (which is a key concept in pure mathematics) is discussed. These concepts are illustrated in more detail with two examples. The content assumes a math knowledge of around a final year high-school student specifically: exponential functions, vectors and calculus.
Talk given at the Hong Kong Academy of Gifted Education.
The starting point of this talk is a quote by Carl Jacobi: "One [Mathematicians] should always generalize". We talk about one of the differences between high-school maths and maths in university - the results are more general; the situations considered are more abstract. The concepts of generalization and abstraction (which is a key concept in pure mathematics) is discussed. These concepts are illustrated in more detail with two examples. The content assumes a math knowledge of around a final year high-school student specifically: exponential functions, vectors and calculus.
Group Theory, Card Shuffling and **Magic** (2009)
Slides
Talk given in the Enrichment Programme for Young Mathematics Talents, Chinese University of Hong Kong.
This talk was aimed at participants in the above programme. One of the aims was to use group theory to unify the content of the different courses that the students were taking. The content has no real pre-requisites but is a little more advanced compared to most high-school curricula - a keen high-school student in his/her final two years of study should be able to appreciate it.
Between slide 12 and slide 13, I performed a magic trick on a visualiser in the lecture room: you can watch Paul Gertner, the creator of this trick, perform it in the youtube video below. Part of how the trick is done is revealed in subsequent slides: apologies to the magically-inclined for breaking the code.
Much of this talk is based on this book:
S. Brent Morris, Magic Tricks, Card Shuffling, and Dynamic Computer Memories, Mathematical Association of America, 1998
Talk given in the Enrichment Programme for Young Mathematics Talents, Chinese University of Hong Kong.
This talk was aimed at participants in the above programme. One of the aims was to use group theory to unify the content of the different courses that the students were taking. The content has no real pre-requisites but is a little more advanced compared to most high-school curricula - a keen high-school student in his/her final two years of study should be able to appreciate it.
Between slide 12 and slide 13, I performed a magic trick on a visualiser in the lecture room: you can watch Paul Gertner, the creator of this trick, perform it in the youtube video below. Part of how the trick is done is revealed in subsequent slides: apologies to the magically-inclined for breaking the code.
Much of this talk is based on this book:
S. Brent Morris, Magic Tricks, Card Shuffling, and Dynamic Computer Memories, Mathematical Association of America, 1998
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