Aims

The Information Revolution has resulted in innovative digital products which have changed and continue to change human life and culture: fake news are spreading with the aid of bots, job applicants are assessed by AI systems which have been proven to be biased in their judgement and contact tracing apps harvest our most private data. The aim of this course is to explore these issues through reading, building small computational systems, essay writing and debate.  

Content

Each week will provide a 45min lecture on one ethical issue, such as bias in machine learning systems. The 1.5h workshop session will be used to explore how the ethical issue can be addressed in the building of systems. One example will be the design of a chatbot which won’t develop sexist language. In addition you will be set a weekly task which will either require you to write an essay or to develop the coding project started in the workshop.

Who’s it for?

We will be coding small artefacts, such as chat- and twitter-bots to help us investigate what an ethical code of conduct for the production of digital products may need to include. These coding challenges will be accessible to non-computer scientists, yet scalable and open ended to ensure that more experienced programmers are equally stretched. There is a high expectation to work effectively in small mixed ability teams, where respect for each other’s skill set is paramount.

What our students say

Iris

Such a fantastic course!  As a specialist school, lessons follow a particular style, and content specification.  This has allowed seminar-like discussions, detailed writing and deep, considered thought that, although I hadn’t realised, I’d really missed!  AMAZING COURSE!

Ryan

Over such a short period, I’ve definitely improved my ability to see where other people are coming from. Unlike at debate club, none of us aim to win or persuade anyone that we are right but simply understand where the others are coming from. I have begun grasping the history to explain clearly why certain actions can be justified from different viewpoints such as the ancient Greeks or modern utilitarianism. This course will prepare me for wider life after EMS when I need to justify my decisions to someone with a different value system which is quite a common occurrence.

Molly

I now know the wall does not exist…but it does (and doesn’t).

Aims

The programming language C++ is an incredibly popular language and despite being much older than Python, Java or JavaScript. The aim of the course is to offer all students at EMS an introduction to the powerful features of C++, to gain an insight into how to write fast and scalable code bases. The course will challenge you to write built-in features of Python yourself and gain the full understanding of object-orientation preparing you for any coding module offered by a degree within STEM.  

Content

Each week will provide a 45min lecture on a concept, such as input and output stream in C++. The 1.5h workshop session will be used to practice the skill in implementing the concept. You will also be set a task which requires you to apply the learnt concept independently.

Who’s it for?

Non computer scientists will benefit from learning general coding techniques, such as object orientation, which will help prepare for any STEM degree where coding is taught a supplementary skill. Computer Scientists will benefit from getting practical experience in coding with pointers and references. This will not only benefit all those considering a degree in Computer Science, but also widen their career prospects.

Aims

The aim of this course is to introduce students to the rigour and precise nature of pure mathematics. It is a stepping stone between A level mathematics and university mathematics and is designed to make the first lectures in a mathematical university course feel a little less strange and daunting.   

Content

We will look at what it means to make a formal definition in mathematics and how we use it to prove general statements. We start the course by looking at various methods of proof which you have already come across, such as proof by contradiction and proof by induction. For the majority of the course we focus on sequences of real numbers and the notion of convergence, including the formal definition.

Each week there will be a 45 minute lecture where you will be expected to take your own notes. There will also be a 90 minute examples class where you can work through exercises and try to understand and apply the ideas from the lecture. 

Who’s it for?

This course is for anyone who would like to learn what “real” pure mathematics is about at university level. It will teach you how to make precise statements and how to write down formal proofs. It is an introduction to the ‘analysis’ courses which are commonly the first rigorous parts of any mathematics degree. It is particularly useful for students who are thinking of studying mathematics at university or any related university course that has mathematical analysis as part of the degree.  

We strongly recommend that you complete this module if you are interested in exploring continued fractions or sums and integrals.

What our students say

Chloe

The Limits sessions were great, having a taste of what university is like is what the courses are about, and I feel I know more of what to expect now in university. The topics had different complexity which made some weeks easier then others, but this it was made the course interesting and engaging.

Tom

This course gave me a good indication of the university learning style. I also enjoyed the more in-depth look at proof.

Fraser

The extension material was challenging and really interesting. The problem solving classes allowed me to start to explore these problems and then I would complete them at home.

Aims

This course is excellent preparation for mathematical programming at university which is very likely to be needed in any mathematics degree and is often an area students are under-confident with. Some of the projects are based on genuine undergraduate programming projects. In doing the course we will also develop a number of new mathematical ideas which will be important in undergraduate study.  

Content

The course will consist of a 45 minute lesson on the mathematical and programming theory required for each project, followed by a 90 minute supported session working on the project. The projects come with a “hints” sheet.  Some content would have been taught in the ‘further pure with technology’ course. 

Who’s it for?

This course is primarily aimed at students who wish to study a mathematics degree and want to prepare for programming content. However, it is open to anyone who wants to improve their programming whilst looking at some new areas of mathematics. 

What our students say

Fraser

The problems were interesting and having a new topic each week made the course fast moving and different each time. This worked really well especially as it meant that if in one topic I did well and found another harder, both only lasted for one week. The ability to progress through different challenges made the difficulty level work no matter the topic.

Aims

To gain an insight into the topic of topology (the study of properties of geometric objects preserved under continuous stretching). This will give you a taste of how Mathematics at a higher level can use techniques from familiar topics to study new areas.  

Content

After a general introduction you will be introduced to knot polynomials and other invariant properties. Knots are twisted loops in 3 dimensional space. We will not be tying knots that you may have met in scouts! 

I have taken some content from a 30 hour third-year undergraduate course in knot theory, to produce a short course accessible to A-level students.  Some of the knot invariants will require substantial use of polynomials and matrices. Some spatial awareness is useful but not essential. 

Who’s it for?

I hope it is accessible to anyone, all are welcome at the very least. Ideal for anybody who is applying for a Mathematics degree to gain an insight into an area of topology. 

What our students say

Yuki

It’s knots, but knot as you know them.  You learn about the various invariant properties, and the notation used in knot theory.  However, I’m frayed I can’t rank the knots, because they’d all be tied.

Chloe

The course was very interesting, looking at properties of different knots and links including Alexander and Conway Polynomials.  Would highly recommend.

Aims

Any degree in a STEM subject is likely to involve at least one module applying a programming language to explore data – this course will give you a taste of what to expect, introducing some of the tools and techniques used to analyse the enormous data sets which now exist, and will give you an insight into the fun that can be had exploring data with R. 

It will also give you a flavour of how statistics can be used in the real world and help you to decide whether you would like to move towards a career in this direction.

Content

The discipline of Data Science is younger than you are and is still evolving. It is all about using data to solve problems. The core job of a data scientist is to understand the data, extract useful information out of it and apply this in solving the problems. These are highly valued skills in the employment market.  We will extend and add to some of the theory already encountered in Statistics, but with a focus on using R to explore and visualise data using real datasets. 

 Cleaning data  

Data analysis using R  

Data presentation and visualisation  

The central limit theorem and confidence intervals  

Statistical Modelling and simulations  

Bootstrapping techniques 

Who’s it for?

Anyone interested in taking a STEM degree at university or wishing to develop employability skills with data science.  If you are interested in what a career in Data Science may involve, this will be a good course for you. 

What our students say

Fraser

Amazing for developing new skills on R for different types of projects exploring different uses of data science.  Learning new skills required for web scraping of data was a particular highlight.

Ryan

The best bit of Data Science was the time we spent learning by playing around with it ourselves.  We were encouraged to dive right into some difficult areas like simulating things beyond that would be possible to solve with analytical techniques from degree level mathematics but we were always supported in the seminars by both our brilliant peers and knowledgeable teacher.

Aims

To prepare you for working in a university-style. 

To develop a greater depth of understanding of calculus. 

To build on the skills learn in limits and apply the ideas in a new context.

Content

How does integration work?  And what is it used for aside from the slightly arbitrary calculation of areas under curves?  This course will show you a bit more of the formal set up behind integration, and will then explore a whole host of contexts with curved surface areas. 

Who’s it for?

Anyone interested in taking a mathematical based degree at university or anyone who is just curious to better understand the techniques which are met at A-level but not fully explored. 

What our students say

Isobel

Really interesting course, I learnt a lot and have a new way of thinking about integrals, CoM etc. It was really well explained. I liked the casual assignments, and I was actually motivated to do the work because it was interesting.

Yuki

The course was well spaced out and Ben was great at teaching the new concepts.  I rate it g out of pi-squared, would recommend.

Aims

This course will be delivered as pairs of lectures and problems classes, to give you a taster of university-style learning. There will be an emphasis on rigorous proof in the lectures, where this is possible at a pre-undergraduate level, with a limited number of examples. Time for exploring the concepts will mainly be through the weekly problems sheets. This is also one of my favourite areas of Mathematics so I hope you will enjoy studying this beautiful topic!

Content

Continued fractions are an exceptionally elegant way of representing real numbers. We will look at both finite and infinite continued fractions, their convergents, using these to approximate real numbers and selected applications to solving Diophantine equations. The final lecture will extend methods of solving Diophantine equations to look at infinite descent.   

As part of studying infinite continued fractions means proving that these are well-defined, it is strongly recommended that you have completed the Limits course as a pre-requisite

Who’s it for?

Anyone who enjoys playing with numbers and finds mathematical problem solving satisfying! Especially recommended if you are thinking of a Maths degree (particularly if you already know that you like pure maths or number theory) but also appropriate for anyone who enjoys recreational Maths. 

What our students say

Fraser

I particularly enjoyed the new ideas that this course introduced. For example, proof in the form of x is a solution, f(x) generates solutions, these solutions are the only solutions. In addition, introducing methods for solving equations such as x²−17y²= -1 was new. New ideas about maths are really interesting and I would recommend the course to anyone who is interested in new maths in a similar way to me.

Chloe

The course had great structure, and everything was laid out well, so it made it easier to follow and understand. It was more like a university-style course, but the content was not too difficult, and could be followed well. I feel it was a great experience of more university-style learning whilst still being accessible. I found the course very interesting and would highly recommend.

Alex

Equally as difficult as it is interesting. On multiple occasions I was listening to large chunks of information before catching myself and realising that I understood absolutely nothing. The problem sheets helped a lot though to provide some context to what felt like very abstract maths. I know this sounds quite negative, so just worth adding that is was quite a comfortable cluelessness- picture a dog at a beach, watching the tide roll in and out, not understanding why but enjoying it nevertheless- and felt very rewarding when you understood something (however rare), much like watching University Challenge. I felt that, particularly towards the end of the course, the content was really interesting and engaging, whilst also acting as a little crystal ball, illuminating the maths that one day, hopefully, I will actually be able to understand. Maybe. We’ll see. P.S. Fermat’s Last Theorem was a nice addition.

How will this prepare you for post-EMS study?

This course will be delivered as pairs of lectures and problems classes, to give you a taster of university-style learning. There will be an emphasis on rigorous proof in the lectures, with a limited number of examples. Time for exploring the concepts will mainly be through the weekly problems sheets.

Content:

Linear algebra is a branch of pure mathematics that studies linear equations and linear maps using vectors and matrices. Since linear algebra is algorithmic and can be used to model natural phenomena it has numerous applications in physics, engineering and computer science. In this module we will explore the concept of vector spaces and linear maps, as well as learning how to use matrix algebra to solve systems of linear equations.

Who’s it for?

This course is primarily aimed at anyone who is interested in mathematics and hopes to go on to study the subject further at university. However, due to the applications of linear algebra in physics, computer science and engineering, the techniques taught in this lecture course with be useful for anybody who is interested in these subjects.

It is recommended (but not essential) that you have completed the Limits course as a pre-requisite. It is also recommended that you have studied the Extra Pure module.

What’s the workload like?

The problem sheets are designed to take you longer than the time we’ll have in the weekly problems class. It is possible to complete enough questions in the problems class alone to get the hang of the material, though a more solid grasp will come with additional independent work (approximately 1-2 hours per week).

There will be a choice of two questions (one computational, one more abstract/proof-based) for you to submit each week, though you can also ask for feedback on any attempts at the most challenging questions.

Previous students say…

“The course had great structure, and everything was laid out well, so it made it easier to follow and understand. It was more like a university-style course, but the content was not too difficult, and could be followed well.”

“I learned a lot of interesting stuff! You get to solve the equation in Brahmagupta using an especially elegant method. Each lecture builds on the ones before, leading to some amazing results that you wouldn’t expect to understand if you’d looked at it before starting the course, but you can understand and work through them in the course! The lecture style really helped me prepare for university – I made my own notes for practice, but there were also notes available.”

“Demanding and intriguing.”

How will this prepare you for post-EMS study?

This course gives you the opportunity to explore one of the main topics that would otherwise be in Mechanics Major. Concepts will be explored by applying techniques from pure maths in addition to new, mechanics-specific methods. There is a lot of overlap with Physics content, but this course will explore topics from first principles and extend your thinking to harder problems.

Each week will have a lecture followed by a problems class.

It is strongly recommended that you complete this course if you are taking STEP III, as you will then be able to access mathematical questions on circular motion.

Content:

This three-week course models situations involving circular motion and uses calculus to derive the equations for circular motion with variable acceleration. We will also investigate rotational energy and the derivation of the formulae for moments of inertia. The workshops will have challenging problem-solving questions, including angular momentum and the old STEP circular motion questions.

The course will cover:

• Angular speed and acceleration of an object moving in a horizontal circle
• Angular speed and acceleration of an object moving in a vertical circle
• Use of vector notation to describe circular motion
• Solving problems where an object does not stay on the circle
• Kinetic energy of a rotating rigid object
• Moment of inertia calculations
• Angular momentum modelling

Who is it for?

Budding physicists, engineers and applied mathematicians. You do not need to be taking Physics to take this course, though it would give you more familiarity with the topic.

What’s the workload like?

This course can be completed in the timetabled sessions. The problem-solving class is run as a collaborative session with the opportunity for students to see how others tackle complicated mechanical problems.

Previous students say…

“It’s a good course to add onto knowledge learnt in Mechanics Minor.”

“The course covers all Maths to do with circles. There isn’t very much to do with circles in the normal curriculum anymore, so this course covers circle-related Maths from what is required to maintain movement in a circle to moments of inertia.”

How will this prepare you for post-EMS study?

This course serves as an introduction to vector calculus and the fascinating world of fluid dynamics, which has applications ranging from predicting the weather and modelling climate change, to understanding how planes fly. The Mathematics that will be introduced will build on many of the pure and applied topics from Year 13 such as differential equations, Maclaurin series, planes, matrices and dimensional analysis, and show how these techniques can be used to describe real-world phenomena.

The challenging content of this course will help to prepare students for any applied Mathematics courses at university, particularly in demonstrating how calculus is both versatile and ubiquitous. Students will begin to link together areas that appear as distinct topics in A-level, but at university are often interconnected.

Content:

The lectures and problem sheets from this course will include:

• Introduction to vector calculus (multivariable functions, partial differentiation, vector fields and the gradient operator)
• Describing properties of fluids (finding the divergence and curl of a vector field and interpreting the results, finding the path traced by a particle in a fluid)
• The Navier-Stokes equation (chain rule for multivariable functions, the material derivative, and deriving the momentum equation)
• Dimensional analysis (deriving the Reynolds number and comparing turbulent and laminar flow)

Who is it for?

Applied mathematicians, physicists and engineers. Anyone who would like to explore an application of their A-level Pure Maths.

It is recommended, but not necessary, that you have studied the Extra Pure module.

What’s the workload like?

All work for the course is contained within the timetabled contact time. Students are expected to take notes during lectures, and submit solutions to question sheets completed during the problems class.

Previous students say…

“It was an introduction to fluid dynamics both in ideas and in the Maths we can use to describe fluids.”

“Basically starts with an introduction to operators in vector calculus, and ends with deriving Euler’s equation (and Navier-Stokes to some extent). It is an interesting course, especially if you’re more into applied Maths.”

“This is a course about asking questions, and trying to understand the answer. If you enjoy pure maths you will love this course. I have learnt to love vector fields, multivariable calculus and composite functions.”

Aims

Introduce different models of light, including a basic quantum mechanical model for describing photons. This will be solid foundation for university level physics and also a remarkable and interesting topic to explore.  

Content

Optics – traditional models of refraction  

History of light – the important scientific discoveries that have lead to our current understanding of light  

Understanding photons  

Quantum behaviour – photon models for propagation – why does light appear to travel in straight lines?  

 The speed of light – why is it constant and why does it appear to change in different media?

Who’s it for?

Recommended for anyone considering a degree in physics. This course will not be suitable for non-physicists.  

What our students say

Tom

Best course ever – very very interesting.

Harry

This course really helped me to look at light and photos in a new way, and explained to me the nature of light with a deeper understanding.

Jess

It’s a great course to do if you really want to think in depth about what makes up our universe. Light’s a fascinating topic that I’d definitely recommend learning more about! If you’re doing this course, make sure you go into it with an open mind.

Aims

An interdisciplinary course with the ultimate goal of explaining how information is digitised and sent. The course will provide a good example of the application of mathematics and physics in the context of computer science and also give a broad understanding of modern communication systems. 

Content

Information and signalling – What is information and what forms does it take? How can we get information from one place to another quickly?  

Analogue signalling – looking at ways of sending a continuous signal  

Digitisation –converting analogue signals to digital and the reasons for doing so  

0’s and 1’s – sending digital signals and the theoretical limits to data transfer speed  

Possible extension 5G – what makes 5G better (and can it give you corona virus)?  

Who’s it for?

Recommended for computer scientists. It will be accessible to non-physicists. Anyone who is interested in how their phone can access the internet, or wants to know why their download speed is slower when they get further from their WiFi router.   

What our students say

Ellie

A good introductory dive into signalling, looking at signalling basics and also more in depth in specific areas like noise. Not aimed towards physics or computer science, finding a middle ground making it good for all EMS students.

 

Iris

Fantastic teaching! A subject that I took on a bit of whim ended up so gripping that I remember not being able to do anything but listen intently.

 

Ryan

Last year, I had played around with the practical side of sending bits of data between two Raspberry Pis along a wire so I was glad to be given the opportunity to develop a basic understanding of the theory of signalling.

Aims

In this short course we will discuss what things affect the length of day on different time scales. We will then focus on the equation of time and attempt to calculate how this affects our length of day throughout the year. This will be done using python (or a method of your choice).  

It will help show you how computers can be used to calculate a solution to problems and prepare you for university by lectures followed up by project work.  

Content

• The equation of time. 
• Torsional oscillations. 
• Tidal locking.  

Who’s it for?

This is ideal for anyone who is interested in astronomy and Earth and planetary science. It is a great course to improve your python programming if you are still a beginner and see how programming is an essential skill if you are thinking of heading off to a geophysics/physics/applied mathematics degree.