Six maths learning questions that research can answer

Welcome to the July 2017 Transum Newsletter. Before the news here is this month’s puzzle sometimes known as the Mystic Rose.

Mystic Rose

Joining points with chords to divide a circle into regions

If there are two distinct points on the circumference of a circle, a chord drawn between these two points will divide the circle into 2 regions.

If there are three points joined by three chords the circle will be divided into 4 regions. Similarly four points joined by six chords produces 8 regions and five points produce 16 regions.

How many regions will be produced by six points joined by fifteen chords?

The answer is not the obvious one! The answer will be at the end of this newsletter after this month’s Transum News.

A brand new online exercise called Train Timetables was written on my laptop as I flew from Malaysia to England then travelled from London to Wolverhampton at the beginning of last month. It is quite overwhelming the number of different styles train, plane and bus timetables take and I’m surprised that people can actually extract relevant information from them. Some of them are particularly hard to read. The online exercise is based on standard train timetables but I have decided to collect photographs of some of the more obscure timetables and add them as an extra level to the exercise. If you have spotted any good specimens please send them to me.

Another brand new exercise called Functions has also been written to cover the GCSE content and provide a strong base for A-level and IB courses. The ordering of the questions was carefully considered to provide progression without forfeiting consolidation. There are six levels and the higher levels include inverse and composite functions.

When I was in London I attended the excellent MathsConf10, a maths conference for the enthusiastic Maths educator. It was a wonderful day and I chose to attend some excellent presentations.

The first was titled ‘Angles in Depth’ and was presented by the prolific blog personality, Jo Morgan. As the presentation progressed I was rapidly making notes on how the Transum angles exercises can be enriched with some to the ingenious examples Jo had found. Though she limited her presentation to adjacent angles on a straight line and the angle sum of a triangle there seemed to be an endless supply of good ideas for activities, puzzles and exercises.

Another presentation I attended was about Filtered Maths Education Research. Cambridge Mathematics produces Espressos for teachers: filtered research reviews to be enjoyed over coffee, discussed at department meetings, or as a basis for digging deeper into CPD issues of interest.

The research answers the following questions:

  1. What are the issues in learning and assessing times tables?
  2. How does assessing confidence affect learning and testing in mathematics?
  3. Is there any value in applying ‘traditional’ and ‘progressive’ models to mathematics teaching?
  4. What is ‘number sense’ and how does it affect mathematics learning?
  5. What are the effects of attainment grouping on mathematics learning?
  6. How does maths anxiety affect mathematics learning?

Colleen Young presented a rich overview of the many excellent free resources for learning A-Level Mathematics. She emphasised the advantages of using the free resources provided to the Boards other than the one you are teaching for to enable your students to appreciate a diverse learning experience.

Liz Henning Investigated making connections from the word problem to bar modelling to abstract approaches with an emphasis on explicit mathematical language and understanding.

Dani Quinn and Rose Dalders shared how they have introduced an alternative approach to marking and feedback that focuses only on quizzes, not books. They have seen improvements in pupils’ outcomes, higher-quality feedback for both pupils and teachers, and – importantly – reduced workload for teachers.

The conference was organized by LaSalle education and you can read more about their forthcoming conferences here: https://completemaths.com/events

The answer to this month’s puzzle actually depends on whether the points (vertices) are evenly spaced around the circumference of the circle or whether they are spaced to produce the maximum number of regions. In the first case the answer for six points is 30 regions.

If however the points are not evenly spaced an additional region exists at the point where the three diameters intersected in the first case. The maximum number of regions is 31 and the formula is:

Regions Formula

This formula was brought to my attention by Paul Metcalf, a colleague I had when I first started teaching (at the beginning of the 1980s). It was good to meet up with him on my recent travels and learn how busy he is keeping himself not only running his own hotel but also freely giving his time to support national mathematical organisations.

The wonderful thing about this puzzle is that most of us, given the sequence 2, 4, 8, 16 …, would have been convinced that the answer was 32. Did you think the answer was 32?

That’s all for this month.

John

P.S. A Mathematician can’t remember whether he’s been going out with his girlfriend for one year or two but he knows it’s <3