*The importance of getting a radical Narrative Mathematics Course established in schools is underlined by the recent financial verdict that interest rates are likely to go down again… back towards zero. This, it is thought, will happen sooner rather than later. These rates reflect the intrinsic demand for business investment funds, but because there are currently rather low expectations about new economic projects, there are going to be few firms willing to pay high borrowing rates. The market, in a word, will force low borrowing rates to return.*

The underlying problem here is that the economy is not rebounding as much as expected from the Pandemic… thus repeating its disappointingly sluggish performance after the crash of 2008. Adrenolin seems to be missing. Most schools are not turning out cohorts contaning lively subgroups of able, energised, motivated, well-educated would-be entrepreneurs. In the past Western civilisation relied on youth to provide an endless stream of capable, potential young leaders with promising new ideas for future projects. But the post-modern pandemonium and general disillusion seem to have all-but snuffed-out this former yearly injection of initiative.

Narrative Mathematics in schools would probably transform the subject into one of the most lively subjects on the school curriculum. Maths would be experienced by the students again and again as the best way to explore promising possibilities. But in the downbeat, disillusioned, apathetic mood which persists today, very few observers seem to have picked this up.

Here is a reform agenda which tackles the heart of the problem of teaching maths —the co-central subject with language in today’s world— which is, in turn, the weak link in education. But too often the reform agenda gets met with a yawn.

It would be nice to be able to report that the campaign for Narrative Maths in Schools which we launched last Autumn was growing fast, and that numbers of enthusiastic supporters were contributing ideas for juicy scenarios of the kind urgently needed. Getting a huge archive together of rich, authentic, realistic, tempting, curious, *workable* problems is an immense task. But there is a correspondingly immense prize at the end of the work… a thriving, popular, lively culture in schools. It is a task which must be completed before Narrative Mathematics in Schools can become a reality. And maths education is likely to be stuck in a rut until the pre-conditions for teaching from a credible Narrative Mathematics viewpoint have been met.

But the underlying malaise which today’s mindset sadly exhibits is evidently having the same effect on this Campaign… as on the problem it is needed to solve. Intellectual verve is at a low, very low, ebb. Signs of bold, progressive, fresh thinking and initiative seem to be almost non-existent. We have had relatively good support for the general idea of Narrative Maths, but also a complete absence of actual scenario-building effort.

In this instalment of this renewal blog we focus onto some new ways to liven-up classical geometry (or ‘Euclidean’ geometry as it used to be called) for use in schools. We know from the testimony of some of the greatest mathematicians that Euclidean geometry can be especially inspirational for maths-inclined young people in their formative years.

There is, however, an unfortunate stumbling block: it is an extreme archaism in the way Euclidean geometry is stillo presented in learning materials and textbooks. *They still have broadly the kind of “look” they had in Euclid’s day!* The way it is written out looks hopelessly old fashioned and out of place in today’s streamlined world. Reflecting on this difficulty, some obvious presentational changes which would make sense may be listed:

1 Points are much smaller geometric objects than lines, so points should be labelled with lowercase letters and lines with upper case.

2 Lines, arcs, curves, etc. are essentially sets of points. To say that point p is on the line L is to say that p is one element of the set of elements which make up L.

3 To say that p is on the line L we may write L ε p, shorthand for the statement that p is an element of L.

4 The angle between a line L and a line M is essentially a relationship which the two sets of points have to each other. A simple new convention might be to denote the smaller of the two angles φ and 180- φ between the two lines L and M as LM.

5 Using this convention, LM = 0 signals that the lines L and M are parallel.

GENERAL POINTS

Some of Catriona Agg’s ingenious discoveries are sufficiently striking to be incorporated into the classic elementary curriculum, for example the two pentagon result which says that the angle between two specific interconnecting lines between two different pentagons with a point in common —of different sizes and at any angle relative to each other— is always 108^{o}

The immediate visual meaning of this kind of simple planar geometry is a great stamping-ground for future mathematicians. We know this because it has retained its strong fascination as a source of puzzling but do-able problems in school mathematics for more than two millennia. It captured Plato’s imagination and provided him with a motif for explaining knowledge in general —one which lasted amazingly as the ‘Official Story’ from the time of the golden age of Greece to the 20^{th} century.

There is also a strong case for introducing more three dimensional Euclidean geometry into the maths curriculum. Confining geometry to a single conventional plane looks rather meagre and unambitious by modern standards.

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CHRISTOPHER ORMELL 1^{st} May 2023

If you would like an online copy of the P E R Narrative Maths Manifesto, send an email to per4group@gmail.com asking for this.