Dr Ian Benson, Professor Alexandre Borovik, Professor Whitfield Diffie and Professor Michael Short – Written evidence (PSU0044)

Outreach classrooms to mitigate the UK school mathematics/informatics deficit

Authors:

Ian Benson, School of Education, University of Roehampton,

Alexandre Borovik. Professor Emeritus, Manchester

Whitfield Diffie ForMemRS, Senior Advisor, Cryptomathic, Cambridge

Michael Short, Professor of Control Engineering and Systems Informatics, Teesside University, Middlesborough

Introduction

This note has been prepared by a panel of four working mathematicians and educators in response to the House of Lords Science and Technology Committee consultation on the STEM skills gap. Together we have over 180 work years of experience of mathematically grounded innovation: in education, technology and outreach infrastructure.

This experience, gained in Britain, the United States and Russia, leads us to advocate a radical extension of university outreach: from a few specialist mathematics schools to outreach classrooms embedded in the wider education community.

Why mathematics is becoming more important

Mathematics is the second most important discovery in human history – after language. The Greek introduction of proof revolutionized science by reducing the number of things that need to be verified by experiment (which is expensive) to a minimum. Educated people may feel ignorant about whatever history, art, music, and literature they do not know: they are not as sensitive about mathematics. Mathematics should be recognized as the great element of human culture that it is and seen as essential to every educated person. Our challenge is to make mathematics more widely understood by changing the way mathematics is taught and subsequently valued. Informatics, what many call computer science, is mathematics’ modern companion. It studies the representation, processing and communication of information in both natural and engineered systems[1]. Its impact may be as great as that of proof.

The most important issue of the 21st Century is the relationship between people and the systems that they have built and will build. Several runners up are very important and more widely known:

• Conflict among the major powers
• The impact of society’s globalization, particularly travel, on contagion leading to pandemics and potential for biowarfare
• The ongoing climate disaster whose threat to fresh water and agriculture induces migration

but it is the relationship between people and the machines they make that will dwarf everything else. It calls into question how and whether human beings will be running the world at the end of the 21st Century. Mathematics stands at the center of any hope of addressing these problems successfully. In this submission we will illustrate the consequences by considering one such challenge. How can we transition the energy system away from capital intensive, centralised generation and distribution?[2]

To address the question of who will control what, we will need wider and deeper involvement of people with serious education and skills in mathematics and informatics. This has been recognised by the UK government which has begun to invest in a handful of specialist mathematics schools and in augmenting teacher professional development in mathematics and computing.

While these investments make a welcome contribution they fall well short of their potential. They are held back by the conservative nature of a testing and examination focused school accountability regime, and by a disciplinary focus that fails to properly address and exploit the symbiosis between mathematics and informatics. There needs to be a continuing mechanism for keeping mathematics up to date. This is particularly true with school mathematics and computer science. These subjects could, and probably should, be taught as one.

Mathematics may be very useful indeed, but what makes it useful is not always the same as what makes it mathematics. In mathematics, fluency means the ability to solve at least simple problems, to make proofs, to see connections between various problems and known results, and the ability to reformulate a problem using different conceptual frameworks. An essential component of mathematical education should be the mathematical world. For example, an attack on the ‘Kid’ Diffie-Hillman key exchange, based on modular multiplication instead of exponentiation, can be very instructive as a sixth form exercise.[3]

Mathematical thinking may be applied to many (seemingly) non-mathematical situations, for example by technicians operating energy networks who need to be able to build in their mind mental images of big complex systems and use them when working with real systems. It entails ‘reverse thinking’: an approach to problems or to projects in which the designer needs to work back from what has to be achieved to a starting situation. It also means understanding conceptual models rather than simply following algorithms. It means cultivating the ability to ‘see the field,’ as in football, not to keep eyes only on the ball and nothing else. It means looking at both sides of a project:  how its aims could be achieved, but also with a careful analysis, where and why it could fail – and what to do in this situation. Above all abstract thinking is an ability to remove all the unnecessary details.

The conceptual deficit at the heart of school mathematics

Mathematically educated people are stem cells of a technologically advanced society; they can retrain themselves as needed, stepping into new roles as these arise. Crises force us to learn quickly; in a technical crisis, mathematically cultured people are capable of learning what must be learned, making their training and continued professional development matters of national significance.

In the example we have chosen of energy transition the unifying element in building a new energy technology capable of mitigating global temperature rise and fossil fuel depletion is a decentralized network of sub-systems capable of integrating a far larger number of less predictable power sources (the wind may fail and clouds may cover the sun). Just as automotive technology swept society in the 20th century, creating and requiring new skills in everything from design to maintenance to accident response, so a new energy system will require skilled personnel to design it, maintain it, and cope with its failures. Mathematics will be indispensable.

Today’s students – tomorrow’s designers, operators, and trouble shooters – will need to understand everything that goes into network-function decisions. Many decisions will be made by machine intelligences; those who work with and use those intelligences must understand them at a cultural level. They must be comfortable with a new degree and ubiquity of abstract thinking.

Present day mathematics education in Britain is effectively losing the ability for such abstract thinking. It is not found in contemporary approaches such as ‘numeracy’, or ‘mathematical literacy’. Nor is it found in testing with national mathematics assessments that measure ready success on questions that require only one-step routines. In the mid 20th century these abilities were developed in school through the study of very long and complex system of proofs in Euclidian geometry, and also sophisticated and tricky proofs of trigonometric identities. There is nothing now of comparable complexity. Elementary calculus, which was once taught in the context of physics, is taught as isolated procedures. Geometry has been cut from mathematical education, trigonometry has essentially disappeared, and nothing has replaced them. There is little training today to develop theorem proving skills in mathematics.

Studying mathematics requires both motivation and an understanding of what the subject you are learning to do consists of. There is a broad awareness that physicists work in all sorts of industrial design activities, have specialized laboratories (particularly accelerators), fly experiments in space, and win Nobel prizes. Mathematical education should cover much more culture of mathematics that is currently taught giving students a sense of not only the history of mathematics but, in particular, where mathematicians work: from university jobs, to intelligence agencies, to hedge funds and helping to discover new and better medicines. The UK needs an intellectually ambitious mathematics education framework: one that encourages a rich combination of childlike curiosity, persistence, fruitful frustration, and the solid satisfaction of structural sense-making.[4]

Beyond Specialist Mathematics Schools: An outreach classroom proposal

Mathematics schools largely inspired by the Russian institutions of the same name were announced by the Cameron–Clegg coalition in 2011. We believe that this initiative is on too small a scale, and is too conservative in ambition. It was intended that these schools undertake outreach to local schools, but very few have been reached. The aim was to establish 12 schools over a three year period. The first two mathematics schools, the King’s College London Mathematics School and Exeter Mathematics School, opened in 2014 and a total of 9 are planned or in operation. Internationally specialist mathematics schools form a continuous spectrum – ranging from ordinary schools with standard syllabus, but with good mathematics teachers, through good schools with good mathematics teachers, to schools like Louis-le-Grand in Paris and Fazekas in Budapest. Teresa May’s plan ‘to open a specialist mathematics school in every UK city’ – has been found to be flawed in two ways: ‘in every city’ was unfeasible – there are 59 cities in England and Wales and ‘school’ was unfeasible as well. Universities did not take up the government’s offer.[5]

One of us has been experimenting since 2004 with university outreach to a network of primary schools in which mathematics and informatics have been taught together from Key Stage 1.[6],[7],[8] This initiative has been managed by a social enterprise, asset-locked to Churchill College in the University of Cambridge.[9] The 2014 national curriculum reforms accelerated this program. Two Department of Education (DfE) initiatives in particular were important. For the first time all four arithmetic operations, and fractions as functions, were introduced at the same time for small numbers in Year 1. And, there was a statutory entitlement for all students to write small computer programs from Key Stage 1. Sadly the subsequent advice from the National Centre for Excellence in the Teaching of Mathematics (NCETM) has silently rowed back from these commitments. NCETM have shown little ambition to exploit the symbiosis of computer science and mathematics.

Mathematicians are re-educatable, able to change their role, metamorphose – and inevitably have to be autodidacts in the process. Indeed, who will teach them in their professional future? They have to teach themselves and learn from each other. Students need a rich diet of challenging problems which go beyond application of procedural recipes, stimulate mathematical thinking, and require the use of deeper intuition and sharing of intuition.

In that aspect, mathematics is actually not much different from the arts. Part of the skills that children get in music schools, acting schools, ballet schools, and art schools is the ability to talk about music, acting, ballet, and art with their peers and with intuitive, subconscious parts of their minds.

Funding for Outreach Infrastructure

As an alternative to setting up and running whole schools it is possible, as undertaken in Russia and Ukraine, for university mathematics and informatics departments to run outreach classes in mainstream schools. Experience of specialist mathematics schools around the world suggests that practical classes should be run in groups of at most 8–10 students. They can be drawn from neighbouring secondary schools to attend lessons during the school day. These students should be selected from the approximately top 1% of 13-14 year old learners on the basis of strong abilities and motivation for a deep and systematic study of mathematics and informatics at advanced level. Motivation is the key: it should be expected that majority of these students will eventually progress not only to undergraduate, but also postgraduate studies in mathematically and/or informatics intensive disciplines. Helping mathematically inclined children to recognise their potential in this way does not necessarily mean supporting the privileged. Classes rather than schools will also make the endeavour affordable to smaller universities.

An effective program of outreach classrooms that builds on university mathematics and informatics teaching will need an expanded pool of teaching and mentoring resources. Ultimately we will need new curricula, new textbooks and a new approach to professional development. We believe that the UKRI research and development budget is the appropriate vehicle to create these resources and to draw working mathematicians into a program of system wide renewal. It should tap students and recent graduates, working mathematicians and alumni willing to devote time to mentoring. This work should dovetail with the promotion and extension of voluntary outreach programs and resources that support UK mathematics, for example the Mathematics Olympiad and STEP (Sixth Term Examination Paper) Mathematics. The latter is a well-established mathematics examination designed to test candidates on questions that are similar in style to undergraduate mathematics. STEP is used in undergraduate admissions by the University of Cambridge, the University of Warwick and Imperial College London. Cambridge produces sufficient preparation sources, but the majority of students (and the majority of school teachers) are yet to use them. This outreach initiative will help to rectify this gap.

In conclusion we recommend a program of experiments in outreach classrooms at a regional and national level, involving the universities and the regional/national education leaderships. This work should be primed by the UKRI budget and funds should be released to support a trial focusing purely on schools outreach, for mathematics with informatics only. As international experience proves, if such initiatives are successful, they will enhance their students’

• ability to learn by absorption
• ability to engage the subconscious when doing mathematics
• ability to compress mathematical knowledge
• capacity for abstract thinking.

6 September 2022