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Preparing for Further Study in Mathematics

Take a deep dive into the rich subject of mathematics and discover theories, concepts, and problems at an undergraduate level.

1,414 enrolled on this course

  • Duration

    6 weeks
  • Weekly study

    3 hours

Examine key mathematical concepts and theory in preparation for a maths degree

When considering a mathematics degree, people often have little idea how the subject develops from school to university and its use in professional contexts.

On this six-week course from Manchester Grammar School, you’ll be introduced to mathematical ideas you may encounter as an undergraduate, gaining an overview of higher level maths.

Explore concepts of advanced mathematics, from number theory to non Euclidean geometry

Advanced maths can open doors to many careers, including finance, engineering, and computer science.

On this course, you’ll examine advanced mathematical theory, including computability and metric spaces, and expand your understanding of mathematical analysis.

Using a case study to demonstrate your understanding of advanced maths, you’ll put your learning into practice and prove you have the necessary skills to take the next steps in your mathematical studies.

Learn how to explain complex theories, using proof of contradiction, induction, and contrapositive

In maths, proof is a series of logical steps used to verify, or disprove, a mathematical argument.

Through group discussion and a range of examples, you’ll examine the surprisingly complex nature of providing absolute proof in a mathematical context, learning how proof can be subjective.

Investigate the history of mathematics

You’ll gain a critical view of how maths has developed from ancient civilisations to modern mathematics, being able to explain key moments in mathematical development.

Using this knowledge, you’ll be able to explain the evolution of mathematical thinking and discuss the potential future of maths and its applications.

By the end of this course, you’ll understand some complex theories and concepts of advanced maths, as well as their applications in a variety of contexts.


  • Week 1


    • What is Mathematics?

      We start by looking at what we mean by Mathematics, and how this changes depending on our age and experience.

    • Age-appropriate Mathematics

      How does Mathematics change as we go through the education system?

    • Numbers and Notation

      An introduction to some of the notation used to describe Mathematics.

  • Week 2

    Historical Context - Modern Maths starts in 1687?

    • Classical Mathematics

      Mathematics could have come from any one of a number of civilizations around the world. Eventually, it was the Mathematics of Ancient Greece which came to have most influence on the future course of the subject.

    • Newton, Leibnitz and Calculus

      The Dawn of Modern Mathematics

    • Modern Mathematics

      How has Mathematics developed since 1900?

  • Week 3


    • The History of proof

      Following on from last week, we will be having a look at how the idea of 'proof' has evolved over time. This is (surprisingly for Maths), sometimes a rather subjective idea. Let's see if we can come to a common agreement...

    • Some methods of proof

      In this week, we will be looking at the idea of rigorous proof in Mathematics, and why it so vitally important to Mathematicians. We will also look at some of the common techniques used by Mathematicians.

    • What makes a proof good, or bad?

      How do we know that a submitted Mathematical proof is actually rigorously correct? Here we look into some known fallacies and common misconceptions which arise all too often in Mathematics, and see how we spot them.

  • Week 4

    Topics in Maths 1

    • Analysis

      We briefly met Analysis when we talked about Calculus in Week 2. This section will give you a flavour of how precise work needs to be in professional Mathematics.

    • Group Theory

      A brief introduction to Group Theory - explaining what a Group is and why it is important in Mathematics.

    • Number Theory

      In this hour, we have a look at some of the advanced Mathematics that can be done, using the simple ideas of natural numbers.

  • Week 5

    Topics in Maths 2

    • Computability

      The idea of Computability predates computing machines, though not by much. It is a mixture of the practical and very abstract, and is a relatively modern area of Mathematics.

    • Metric Spaces

      This is an example of the kind of Mathematics which is produced by generalising the rules followed by a familiar concept - in this case, distance.

    • Non-Euclidean Geometry

      A look at what happens if we play with the rules of Euclidean Geometry

  • Week 6

    Some Real Mathematics

    • Some Real Problems

      Some real problems from post-school Mathematics. Some proofs are given, some are indicated and some unsolved problems are posed!

    • Further Resources

      If you have enjoyed some (or all) of the topics covered in this course you might like to use some of the following resources to continue your studies...

    • Thank You

      A thank you from your course creators!

Learning on this course

On every step of the course you can meet other learners, share your ideas and join in with active discussions in the comments.

What will you achieve?

By the end of the course, you‘ll be able to...

  • Explore the history of Mathematics.
  • Appreciate the need for precision and clarity in mathematics at a higher level.
  • Develop an understanding of a selection of topics in higher mathematics.
  • Produce a piece of work exploring an aspect of modern maths and understand its importance.

Who is the course for?

This course is designed for those considering an undergraduate degree in mathematics. It’s also suitable for anyone interested in situating mathematics into a historical and cultural framework.

Who will you learn with?

I completed a BSc At Leeds University and spent three years working on research in Functional Analysis at Sheffield University. I have been involved in teaching Mathematics for the last 30 years.

I studied pure Mathematics at the University of Manchester, getting a PhD in Ergodic Theory and Dynamical Systems in 2014. I have taught Mathematics for 8 years.

Who developed the course?

Manchester Grammar School

Our history dates back to the time of Henry VIII, when The Manchester Grammar School was founded in 1515 by Hugh Oldham, Bishop of Exeter, to provide ‘godliness and good learning’ to the poor boys of Manchester.

The School proceeded to build a reputation as one of the country’s leading educational establishments, a position it still holds today as an independent day school.

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  • Complete 90% of course steps and all of the assessments to earn your certificate

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