## Background of Subject Advisor David Martin

I am a past Chair of Ilkeston U3A where I enjoy leading a 'Maths for Fun' group. I am a qualified radio amateur and enjoy all things scientific, films and strolling with friends.

I am both a Chartered Mathematician and a Chartered Mathematics Teacher. After a degree and doctorate in mathematics (specialising in computer based mathematical models of molecules) I engaged in post-doctoral research in theoretical chemistry. Following research I trained to teach, teaching initially Computer Science at a Sixth Form College and then Mathematics Learning Support and as a Teacher Trainer at a College of General Further Education (FE).

I have enjoyed working to network organisations and at international level chaired meetings of heads and staff looking at how pupils at their schools and colleges in France, Germany, Poland, Sweden and the UK may participate together in shared activities including that of project work and multilateral visits. I have had over the years the privilege to have been engaged in mathematics education debate as Chair of the National Association for Numeracy and Mathematics in Colleges, as a member of the Schools and FE Committee of the Institute of Mathematics and its Applications (IMA), as a member of the inner circle of the National Association of Mathematics Advisors, as Honorary Secretary of the Joint Mathematical Council of the UK (JMC), and as Chair of the British Congress of Mathematics Education Committee of the JMC (with its role of drawing together members of twenty one National Mathematics Organisations in the UK for a combined conference). I continue involvement in Education as a member of the Derby Diocesan Board of Education and as a member of the membership committee of the IMA.

I value U3A as an opportunity to be kept engaged intellectually, physically and socially and in my role as national subject advisor for mathematics and statistics to be of help in the resourcing and networking in this area.

## Starting a new group

**Preparation:** In setting up a new group it is useful to gather together four or more interested members that will form a stable core from which the group can expand. Where possible, consider shared leadership of the group and by persons who between them have both mathematics knowledge and experience of teaching. The group is likely to have a wide range of mathematical abilities, knowledge and commitment and if so then it will benefit from both a gentle start and help from the more advanced members while the less secure settle in.

**Programme:** Those with limited mathematics knowledge can be easily discouraged and it is essential to design the first few meetings to maintain their engagement. They must not feel lost after 15 minutes and so including material accessible to a range of levels is useful. A starter activity such as a puzzle to solve in the first few minutes can be a useful warm up exercise. It is useful to involve members in various ways in suggesting puzzles, offering different ways in which they have solved problems, and in providing ideas for topics. Material covered may be, for example, to develop a mathematical skill, to examine an application of mathematics, to look at the history of mathematics through the life of a mathematician or the development of a mathematical topic. This material can be usefully revisited from different angles to strengthen understanding.

**Level:** U3A Maths Groups operate at various levels based on the mathematics assumed, including i) Pre GCSE/O level of Maths for Fun and Mathematical Puzzles ii) GCSE/O level of Exploring mathematics through equations, calculus, matrices, groups, basic statistical methods and applications through to iii) Applying advanced level mathematics in topics such as projective geometry, fractals, number theory, logic, infinity and medical statistics. The level of the group can emerge by discussion with the group and may start at a fairly basic level and become more advanced over the years.

**Style:** The way a group operates will vary from group to group. In some a member of the group may give a lecture or lead a seminar while in others a workshop approach may be more appropriate. Whatever the style, getting members to ‘do mathematics’ is useful both to develop understanding and to keep members engaged .

**Time:** The group may be interested in meeting weekly, fortnightly or monthly. Weekly meetings need significant commitment while monthly meetings can offer slower progress. Contact can be maintained between meetings by email, including providing a useful reminder a few days before each meeting.

**Location:** Groups may meet in member’s homes which is suitable for small groups and enhances social interaction or in a rented venue suitable for larger groups which can be expensive and needs arrangements for carrying or storing equipment. Wherever the Group meets it is very useful to have some form of whiteboard, while a laptop, large screen and DVD player are also useful.

## Ways of working with groups during lockdown

Circumstances vary from group to group and while some groups are on hold, others are continuing with established methods of keeping in touch through for example email and WhatsApp groups, while others are trying something new such as moving to virtual meetings using for example Zoom, replicating physical meetings. Methods of working can include communication via Closed Facebook pages and use of YouTube videos in preparation for a virtual meeting. Some may wish to try a MOOC, and where a group are all studying the same MOOC, meeting together virtually before, in the middle and at the end of the course. Many are working on the weekly Maths Challenge that is uploaded to www.u3a.org.uk each Thursday with sample solutions for the previous week’s problems.

## Some sources of mathematical resources

Videos and podcasts

Gresham College - Professor of Geometry lectures over the years cover many topics starting with a roughly GCSE / O level base.

In our time Radio 4 podcasts

Khan Academy - A virtual school offering free mathematics presentations at a range of levels.

LMS - Since 1982, the London Mathematical Society has hosted its annual free Popular Lectures to inspire interest in mathematics.

Ted Talks – Some interesting talks by key people. Search on Math

YouTube - Search on Mathematics for a range of interesting talks.

Other sources of mathematical resources

BSHM - The British Society for History of Mathematics resources page points to useful webpages on various aspects of the history of mathematics.

JMC – The Joint Mathematical Council of the United Kingdom details on its website over 20 UK national mathematics organisations and provides links to each of them.

Mathematics Matters - The Institute for Mathematics and its Applications (IMA) has created some very useful short articles called Mathematics Matters on the uses of mathematics.

National STEM Centre - The National STEM Centre is based at the University of York and is collecting together for access from its website, and for those visiting, copies of resources in Science, Technology, Engineering and Mathematics.

NIST – The NIST Digital Library of Mathematical Functions was developed over a number of years to provide a reference tool for researchers and other users in applied mathematics.

NCETM – The National Centre for Excellence in the Teaching of Mathematics is a Government funded organisation that focuses on providers of training for teachers of mathematics and has useful resources to help in the development of mathematics group leaders.

NRICH – The NRICH site is based at Cambridge University and contains stimulating activities to try out in mathematics.

TSM Resources - Douglas Butler, former head of mathematics at Oundle School, maintains an excellent website of very useful links to mathematics related websites.

Wikipedia – A developing free encyclopaedia

Wolfram Alpha – This is part of Wolfram who have developed the powerful mathematical package ‘Mathematica’. This site allows you to enter a range of questions to get some useful information. Try for instance entering ‘integrate (1+x)^(1/3)’ or ‘1,1,2,3,5’.

## Using puzzles to develop mathematical thinking

Puzzles can be a useful way to engage in mathematics in a non-routine way. They offer many advantages and some potential challenges such as:

- Puzzles can provide a different way of looking at mathematics.
- Some card tricks and games have underlying mathematics worthy of further investigation.
- Puzzles can be set in a realistic context and can be engaging for those who would otherwise lack patience working through textbook problems.
- They provide a way in to develop investigative approaches to learning mathematics and a stimulation to investigate mathematics topics that arise during puzzle solving.
- Puzzles can often be solved in different ways and so they offer an opportunity for groups to discuss their different ways of solving them.

And now for some of the potential challenges:

- If the focus of a session is the development of a particular skill, then puzzles with their different methods of solution may not provide the desired focus.
- In any group there are likely to be both those who like puzzles and those who are not particularly interested in them.
- Puzzles while engaging can also be more stressful than straightforward problem solving.
- Computing provides a powerful tool to solve some puzzles, but there is a danger of a focus on the computing rather than on the mathematics.

## Some tips for developing mathematical thinking

- Encourage fluency in key mathematical skills to build a toolkit for thinking
- Find a counterexample e.g. to n^2+ n + 41 is always prime
- Generalise e.g. from 1, 1+3, 1+3+5,… to the general formulae
- Conjecture e.g. all quadrilaterals tessellate
- Give an example of… another…and another e.g. a shape with area 9
- Define e.g. a square is a shape with…is this necessary and sufficient?
- Compare and contrast e.g. x^2+y^2 = 4 and x^2+y^2 =4x
- Consider impossible things e.g. construct a 4, 5, 10 triangle
- Odd one out e.g. 15, 16, 17 – the only triangle number, square, prime.
- Is it always, sometimes or never true e.g. ‘division makes things smaller’
- Ordering e.g. 60% of £ 18, 20% of £ 58 and 30% of £38 increasing
- Use puzzles to engage in non-routine mathematics
- Justify or prove e.g. that the product of two odd numbers is odd
- Create your own question
- Explain your solution to another and compare solutions
- Draw a diagram that helps to illustrate the problem or solution
- How is this linked to another area of mathematics?
- Devise your own notation e.g. for describing a polygon
- How might your method of solution be used elsewhere?
- Can you revisit your solution and make it more efficient?

Further details on many of these can be found in ‘Thinkers’ by Chris Bills et al, ATM (2004)

## Ways of working in mixed ability groups: Benefitting both the novice and the expert

- Enable experts to act as tutors to novices and as presenters
- Utilise members areas of interest e.g. in music, astronomy, topology
- Use recognisable contexts e.g. taxes for percentages
- Partition sessions into shorter varied slots
- Include history of mathematics to add a human dimension
- Look at various uses of mathematics e.g. IMA Mathematics Matters
- Ask questions with several possible answers e.g. describe a Platonic solid
- Encourage all to share their own solutions
- Think low threshold high ceiling problems e.g. NRICH
- Include talks by experienced communicators e.g. YouTube, LMS, Gresham
- Use puzzles to stimulate interest in developing new mathematical topics
- For part of some sessions split the group

## Some tips for teaching mathematics online

- Participants engage with pre-set problems and resources before meeting online
- Presentations are mixed with some joint activity to enable social interaction.
- For large groups (e.g. over a dozen) consider muting all participants. Questions and comments to be submitted via Chat.
- For large groups utilise breakout groups to enable discussion in smaller groups
- Consider utilising polling to keep participants engaged
- Build in a break away from the screen for long sessions (over an hour)
- Utilise share screen to share PowerPoint, or audio resource, or video resource, or whiteboard
- Consider setting up a second camera, which may be a smartphone or web camera to view written work.
- Record the on-line session, if appropriate, to enable participants to review the session or to catch up if missed.
- Consider using an online session in conjunction with offline e.g. emails, WhatsApp, or a closed Facebook page

## Some tips for problem solving

Many refer to George Pólya’s four principles.

1. Understand the problem

- What are you asked to find or show?
- Do you understand all the words used in stating the problem?
- Can you restate the problem in your own words?
- Can you think of a helpful diagram?
- Is there enough information to enable you to find a solution?

2. Devise a plan

- Guess and check
- Make an orderly list
- Consider special cases
- Use direct reasoning
- Solve an equation
- Find a counterexample
- Estimate expected results
- Look for a pattern
- Draw a picture
- Solve a simpler problem
- Work backwards
- Use a formula

3. Carry out the plan

- Persist with the plan
- If it continues not to work discard it and choose another.

4. Look back

- Could you have solved it in a different way?
- Could you use the method for other problems?

(see https://en.wikipedia.org/wiki/How_to_Solve_It for further details)

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