Why Study Mathematics?

Vicky Neale

• Publisher: London Publishing Partnership
• Year: 2020
• ISBN:
  9781913019136
• Edition: 1
• Topic: Mathématiques
• Subtopic: Mathématiques générales

Considering studying mathematics at university? Wondering whether a mathematics degree will get you a good job, and what you might earn? Want to know what it's actually like to study mathematics at degree level? This book tells you what you need to know. Studying any subject at degree level is an investment in the future that involves significant cost. Now more than ever, students and their parents need to weigh up the potential benefits of university courses. That's where the Why Study series comes in. This series of books, aimed at students, parents and teachers, explains in practical terms the range and scope of an academic subject at university level and where it can lead in terms of careers or further study. Each book sets out to enthuse the reader about its subject and answer the crucial questions that a college prospectus does not.

PART I

THE INS AND OUTS OF A MATHS DEGREE

CHAPTER 1

What’s in a maths degree?

it’s a real strength of the higher education sector that there are so many different maths degrees available. This means there’s a good chance you’ll find one that appeals to you, and it’s worth doing some research to explore the options that might suit you best. In our first two chapters, we’ll look at the structure of a maths degree. What topics might you study at university? What kinds of programmes are available, and what might fit your interests? We’ll also give you some idea of what it’s like to study maths at university, from admissions requirements to teaching and assessment and support services.

What do we mean by ‘maths’?

Throughout this book, ‘maths’ is short for ‘mathematical sciences’. The mathematical sciences include mathematics of all sorts (including pure mathematics and applied mathematics) as well as statistics and operational research (which is often shortened to OR). Some universities offer degree programmes that focus on particular aspects of the mathematical sciences, while others offer broader degree programmes that nonetheless give scope for specialization at a later stage. For the purposes of this book, these all fall under the umbrella of ‘mathematical sciences’, or just ‘maths’, degrees.

The names of degree programmes vary significantly. On the one hand, two programmes that are simply called Mathematics can have very different structures and content. On the other hand, programmes with different names, such as Financial Mathematics and Mathematics with Statistics for Finance, can have similar content. Some programmes have a lot of flexibility, so a student on a mathematics programme might be able to study a lot of the same content as a student taking financial mathematics, for example.

You’ll find that degree programmes with names like Business Analytics might overlap with programmes such as Mathematics, Operational Research, Statistics and Economics. So it’s important to look beyond the programme title (and the UCAS course code) to find out more about what’s involved. This book will help you to make sense of some of the topic names you might see mentioned in prospectuses and on the UCAS website (https://www.ucas.com/). This website features a useful glossary of terms as well as loads of information for those contemplating university.

Something old, something new

Some topics in a maths degree will be familiar to you from school or college. They’ll build on ideas you’ve already studied but take them further through the development of new techniques and the discovery of new applications. Other topics will be completely unknown to you. Exactly which topics fall into each of these categories will depend on what you’ve covered at school; even just within the UK, students take a variety of qualifications (A levels, Scottish Highers, International Baccalaureate, …).

Maths is a cumulative subject, where being able to study more complex material is dependent on your having grasped earlier topics. There are some mathematical ideas that everybody must study and understand so that they can progress to further ideas and techniques. You’ve already experienced this concept: for example, you needed to be comfortable using algebra to solve linear and quadratic equations in order to tackle other aspects of mathematics. The cumulative nature of this subject means that maths degrees usually include some compulsory modules on core topics. This enables later modules to assume that everyone is familiar with a certain amount of the material already.

As some subjects included in a maths degree are new and unfamiliar based on school experience, it can be hard to make informed choices about what to study at the start of a degree. These compulsory modules will help you not only to develop a secure foundation on which to build later modules, but also to explore your own mathematical interests, so you can make decisions about what to study later on when you have a choice of modules.

In some degree programmes, you’ll choose a large proportion of your modules, especially in later years, while in other programmes most of your modules will be core, with a smaller optional component.

What’s your flavour?

At school, there are standard requirements about what students have to learn: there’s a national curriculum. This isn’t the case at university. Universities have significant flexibility when it comes to how they organize degree programmes. There’s a huge variety of courses – offering differences in mathematical content and emphasis, in teaching style and in assessment methods – that can lead you to a maths degree. This means that it’s really important to research the available options before you apply, to find courses that’ll suit you.

The Quality Assurance Agency (QAA), which is responsible for overseeing standards in higher education in the UK, publishes a benchmark statement for each subject area, setting out expectations for degrees in that discipline. For mathematics, statistics and operational research, it says:1

Some courses are concerned more with the underlying theory of the subject and the way in which this establishes general propositions leading to methods and techniques which can then be applied to other areas of the subject. Other courses are more concerned with understanding and applying mathematical results, methods and techniques to many parts of the overall subject area.

The benchmark statement refers to these as ‘theory-based courses’ and ‘practice-based courses’, respectively. It continues:

While there are a few courses that are entirely theory or practice based, most have elements of both approaches and there is a complete spectrum of courses covering the range between the two extremes. It is possible for courses with the same title to have very different emphases; it is the curriculum of a course (rather than its title) that makes clear its position within the spectrum. It is important to note that all of these different emphases are valuable, and one should not be viewed as of higher status than another.

There’s a lot of flexibility about which topics are covered in a maths degree. The only topics specifically mentioned in the QAA benchmark statement are calculus and linear algebra (see Chapter 5).2 Beyond that, it’s up to individual universities to design appropriate degree courses.

Different people have different mathematical tastes. Some like nothing more than to get their hands on a large data set, to interrogate it in order to see what conclusions they can draw, and to consider the robustness of those conclusions. Others are motivated by a particular application and spend time exploring which mathematical tools can help to answer the questions they find exciting in that area. Others still are fascinated by the beautiful, fundamental questions in this subject (it’s surprising just how many fundamental questions there are for which we still don’t have complete answers) and devote themselves to curiosity-driven maths.

Within a maths degree, you’re likely to have opportunities to experience all of these facets. However, degree programmes do differ in the emphasis they place on each aspect of maths, as well as in the number of options they offer to students, so it’s really worth thinking about what style of course will suit you. Are you motivated by the use of maths in industry and other applications, and therefore interested in building a mathematical toolkit for that purpose, or would you relish delving into the background theory of how and why the tools work? You’re not restricted to one or the other: many courses combine elements of both. But when you’re researching courses, it can be helpful to consider the extent to which each might be described as ‘theory-based’ or ‘practice-based’, because viewing courses through that lens might help you to focus on the ones that’ll suit you.

The main areas of mathematics

So what kinds of topic might you study at university as part of a maths degree? These might roughly be grouped under the following headings:

• pure mathematics;
• applied mathematics;
• statistics;
• operational research (OR).

These divisions are necessarily slightly contrived, as there are lots of areas of overlap. For example, you might have already studied some calculus (differentiation and integration) at school or college: this is a good example of a topic that can be approached from both a pure mathematics perspective and an applied mathematics one. Nevertheless, these four headings can still be a helpful starting point for thinking about university-level maths topics. We’ll meet a selection of subjects under these headings in Part II. To give you some idea of what they involve, we’ll briefly describe each of them in turn below. Many of these topics aren’t introduced before university level, and in some cases you won’t meet them until the later years of a maths degree; so if the names are unfamiliar to you, or you don’t completely understand the descriptions, or you don’t know how you’d choose between topics, there’s no need to worry! The descriptions are here just to give you a sense of the kinds of topic you might find in a maths degree.

Pure mathematics

Pure mathematics tends to refer to maths that’s done for its own sake, rather than with a specific application in mind. It’s sometimes called fundamental mathematics and has a focus on rigorous proof: developing careful arguments that show with absolute certainty that theorems are true. It involves precise definitions of concepts so that mathematicians can reason precisely about these ideas. The flavour of this branch of maths is often quite abstract (or even very abstract), because it explores mathematical questions for their intrinsic interest. However, the ideas and tools developed in this way have turned out to be crucial for many applications and have had their own profound consequences for the study of mathematics. The habits of mind developed in the course of studying pure mathematics are valuable in a range of careers.

Topics in a maths degree that might be described as pure mathematics include the following.

Abstract algebra (group theory, linear algebra, etc.). Abstraction is one of the concepts that makes maths so powerful. This starts very early on: for example, when we recognize that there’s a notion of ‘five-ness’ (when we talk about five apples, or five cats, or five books, or five colours, the ‘five’ is the same in all cases). We can reason abstractly about numbers – say, 5 + 3 = 8 – without needing to know whether we’re considering apples or cats or books or colours. Abstract algebra takes this idea further. When we notice the same underlying algebraic structure, we can distil out this structure and then reason about all objects with the same structure. One important example is that of a vector space, which underpins linear algebra (see Chapter 5).

Another key algebraic object is the group, which consists of a set with an operation that satisfy certain properties. Examples of groups include the set of integers (whole numbers) under addition, the set of non-zero complex numbers under multiplication, the set of translations of the plane under composition, and the possible moves of a Rubik’s Cube. Group theory is important in many areas of maths and has applications in crystallography, computer science and particle physics, to name just a few examples. There are other important algebraic structures beyond vector spaces and groups, which you might study in a more advanced abstract algebra course.

Analysis (real and complex). Imagine drawing a continuous function on the usual x–y axes that’s negative for one x value and positive for another. The function must take the value 0 somewhere between these two x values: it must cross the horizontal x-axis. This seems intuitively clear, but how might we prove it? (The result is called the intermediate valu...