Post date: 27-Jun-2020 07:44:58

I just found some older writings on my computer prepared for other purposes and I thought these can serve as a biographical introduction to me. Please excuse the repetition.

Autobiographical sketch last updated around 2007/8 for Friends Reunited

My first school in London (after attending infant schools in Gothenburg; Paris - the American School in Paris - classmate Gregory Bellow - son of Saul Bellow; and one in Antibes) was St Marys Town and Country School in Swiss Cottage. I met Pete Sayers and John Maizels there. Still see them. I was there in 1951, and spent only one term there, but I have almost as many memories of that time as of four years at George Eliot 1952-1956.

I had relatively happy days at George Eliot primary school. I still see Henry Jacobsen who was there (and his twin brother Quentin Jacobsen sometimes). Steve Cousins and saw him 5/10 years ago. Saw Dave Wallis-Jones about 3/4 years ago. Sadly Gabi Weissmann died mid-late 90s. Since posting this have heard from Ruth Hilton, Robert Fleschmann, Jaqueline Sinfield and Roger Silverman.

Recently (Feb. 2003 - the day of the big anti-war demo) there was a reunion of the 1952-56 class from GE hosted wonderfully by Margaret Engler (ne Millington). Steve Cousins, Dave Wallis-Jones, Ruth Hilton, Robert Fleischmann, Jaqueline Sinfield, Roger Silverman, Alfred (& Lorraine) Levy, Angela Lewi, Rosalind and others were there. It was great!

I then went on to William Ellis 1956-1963. A mixed experience! During my A level zoology practical in 1963 I was in Tangier! (With Tony Barnett). Not surprisingly I didn't take many qualifications with me when I left William Ellis in 1963. Just 5 'O' Levels (maths, biology, physics, chemistry, English language - the sciences came naturally to me).

Just had a William Ellis reunion (2003) and saw too many mates and old acquaintences to mention! Have some dodgy photos of them too!

After WE, Autumn '63 I worked for a year as a dustman in Hampstead (picked up some quality rubbish there!) before going to an FE college to redo my A levels. That was City of Westminster College, in an Annex of the Army & Navy stores, Victoria, 1964-66. Not in touch with many from then. I remet Danny de Souza later in Gotheburg, 1968 (just picked up a book at a charity shop he wrote called "Under the Crescent Moon". Haven't read it yet). Had a couple of girlfriends from there: Myra, Lynn.

Actually I met Lynn Ellis at the Witches' Cauldron in Belsize village where we hung out in the early/mid 1960s! Sadly she died early 1970s.

In 1965 I travelled overland to Afghanistan with lots of old mates: Pete Sayers, Tony Barnett, Sue Barnett, John and Maggie Maizels, Steve Moss. In 1966 Pete and I repeated the trip alone! What a different world it was already!!

Then I went to Sussex University, 1966-1968 (thrown out) and 1971-1973 finally graduating at age 28! The missing years are a whole other story - but note it includes the Summer of Love and the Hippie revolution!

In 1974 I graduated from Bedford College, London University with MSc in mathematical logic with a distinction and then in 1985 from King's College, London University with a PhD in philosophy of mathematics (having meanwhile become a maths teacher 1976-79 at Hampstead School, West Hampstead), and then lecturer at Homerton College, Cambridge; Bedford College of Higher Education; and the University of West Indies, Kingston, Jamaica, 1979-84. Tony Barnett, my old friend from WE was also a teacher at Hampstead School while I was there 1976-79, as were Jonathon Osborne and Dec O'Reilly!

After Jamaica I was appointed as a lecturer at the University of Exeter. I am currently Emeritus Professor of Education at the University of Exeter (and I have been here 1984 to the present).

I was encouraged to start publishing my work while still a PhD student by my excellent but hard-taskmaster of a supervisor Professor Moshe' Machover, and my first paper came out in 1975, before I shifted into education.

Studies, research and teaching really got me excited about concepts, theories and practices in both research mathematics, philosophy and in education. I have developed some of these ideas in a bunch of publications and papers ranging from accessible ones in teachers' and open access journals to the most theoretical ones in top academic journals. I've written a few books on the philosophy of mathematics and education. I founded and edit a web journal from 1990 to the present located at http://socialsciences.exeter.ac.uk/education/research/centres/stem/publications/pmej/. In the past few years (2010 onwards) I've been exploring the ethics of mathematics. [Added 2020]

I live in Exeter with Jill, my lifelong partner and wife. Our grown-up daughters Jane and Nuala have flown the nest. Jane's in Leamington Spa and Nuala in West Hampstead! We love visiting London -- but are happy to have settled in Devon, where we hear the birds cheeping, see squirrels frolicking on and under the 500 year old oak in the front garden (badgers dug up a bumble bee nest under it last spring), and get the odd deer (very odd!) in the back garden eating up windfall Bramleys (and roses!) We laze out back in the pool -- when the weather allows!

I took up scuba diving 5 years ago - it's my new passion! I went off to the Red Sea for the 5th time in Dec. 2001 -- (I went with old school mate Tony Barnett -- we had a great time!). Again in Dec. 2002 with Harry, and again in July 2003 with Tony B, on a Red Sea liveaboard! A few more trips: last was Oct 2007 Tony B on a Red Sea liveaboard again! Saw 2 new (to me) species of sharks - very exciting!

I'm in touch with several of the old boys from William Ellis: Tony Barnett, Pete Sayers, Phil Howe, Steve Moss, Dave Stephens (RIP), Alan Green, Tony Jackson (RIP) (rather a dopey crew then).

Sadly we've lost a few old mates from William Ellis - Alan Shoobridge died early 1970s, Pete Rasini died 1980s, and Tony Jackson and Dave Stephens died since I set up this profile! (Which must have been around 2002 - initially)

Also in touch with mates from Sussex University: Dave Fry and Jan Fry, Dave Buirski, Patrick Lane, and also Robert Powell, George Kerek, Marilyn Wheatcroft, Ann Beech -- now and then.

Also in touch with mates from those other educational establishements of the early 60s: Witches Cauldren, (eg the venerable and great Dave Young) Belsize Park, and Roaring 20s, Carnaby Street! But that's another story!!

MY THEMES

1. The search for love

2. The search for oblivion

3. The search for truth

4. The search for self-fulfilment

INTERVIEW (2010, with some additions)

Name: Paul Ernest

About you:

My first degree from Sussex University, in mathematics, logic and philosophy, and then my masters degree from London University in mathematical logic, were followed by my doctorate from King’s College London in the philosophy of mathematics. But when I became a maths teacher in a London comprehensive school the focus of my interest shifted towards mathematics education - in which I have since made my career. However, my background in logic, philosophy and mathematics, has always shaped my research agenda.

My best-known book is The Philosophy of Mathematics Education (http://www.waterstones.com/waterstonesweb/products/paul+ernest/the+philosophy+of+mathematics+education/4948203/). This first came out in 1991, and has been cited over 2,500 times in research publications.

I'm also known for "Social Constructivism as a Philosophy of Mathematics", first published 1998, which has been cited about 1000 times in research publications.

I founded, and continue to edit, The Philosophy of Mathematics Education Journal, (http://socialsciences.exeter.ac.uk/education/research/centres/stem/publications/pmej/)

This features philosophically or theoretically interesting papers on mathematics education and the philosophy of mathematics.

Having previously lectured in a number of teacher education institutions and universities, since 1984 I have been at Exeter University where I am currently Emeritus Professor of Mathematics Education. At Exeter I developed and ran specialist doctoral and masters programmes in mathematics education - which now have graduates among teachers and lecturers in virtually every continent of the globe. In the past few years I have taken on visiting professorships at various places including the University of Olso, Trondheim Teachers’ College and Liverpool Hope University - working with both staff and advanced students.

My research interests are wide, but I continue to focus on theorising mathematics and mathematics education, using philosophical, social and semiotic theories; also on the question Why teach and learn maths?, and on issues of social justice and critical mathematics education. My research has mostly been theoretical and reflective - which means I usually sit at my desk and think and write. The miracle is that others seem to find what I write interesting and helpful!

The most recent use of mathematics in my job was …

Last year (2010) I wrote an article, John Ernest, A Mathematical Artist (http://people.exeter.ac.uk/PErnest/pome24/ernest_john_ernest_a_mathematical_artist.doc) - on the mathematics in my father’s paintings, some of which he based on group theory. Typically they have a central 8X8 grid, such as in his Iconic Group Table shown here.

John Ernest, Iconic Group Table, 1970

What I wanted to do was to verify that it does indeed represent a group multiplication table and, if so, to identify which of the order 8 groups it is, out of the five different possibilities. No surprise that it turns out that the table is a group table – the piece represents the group in which all elements have order 2 - they are self-inverse - (technically the group is C₂xC₂xC₂). In addition, I identified the group operation as the set operation of symmetric difference operating on the regions viewed as Venn diagrams. To analyze this, and some other pictures, I had to retrieve my knowledge of group theory, studied long ago. I also consulted Frank Budden’s excellent book The Fascination of Groups (http://www.biblio.com/books/91666362.html) a bestiary of small finite groups, which has sat in my personal library for many years.

In fact, the quest to identify what group might be given in a table was started earlier last year when I asked myself (and others on the MathsEdList) the question Do completed Sudoku tables form groups? The answer is – no, most do not, even though the grids are permutations of 1 to 9 on each row!

Some mathematics that amazed you is …

Lots of different bits of mathematics have amazed me, from first meeting Euclid’s proof of the infinity of primes (http://www.mathsisgoodforyou.com/conjecturestheorems/euclidsprimes.htm) to reading Gödel’s original paper in which he proved his famous Incompleteness Theorems.

As a child I remember once being amazed that the size of the number 1 and the size of the gap between 1 and 2 were the same. I now know why. (1 can be understood as the step between 0 and 1, and the step between 1 and 2 is the same).

However a nicer example concerns an accidental realization. Around 1980 I picked up a book called The Creatively Gifted at a jumble sale. In it is a diagram of one person’s mental image of the number line. When this person thought of numbers between 1 and 1000 he brought to mind a picture - shown below on the left - and identified numbers with points on the line. As I read this, I realised with a shock that I too have a mental number-line image. I ran my mind over it from 1 to 1,000,000, and could see it as something like a wire on telegraph poles attached at significant points, looping up a hill as it recedes into the distance with something like an exponentially growing scale – shown below on the right – this is from Mental images of number lines, an article that I contributed to an issue of Mathematics Teaching.

I realised that whenever I think of a number, say 100 000, my understanding of that number – my concept image if you like - includes the image of its approximate location on this number line. I had never explicitly reflected on this accompanying image until that moment at the jumble sale!

In uncovering this image I also noted that two non-significant (non-rounded) number points stood out on the line, marked at 239 000 and 186 000. I recognised these numbers: they represent the distance to the moon in miles and the speed of light in miles per second, and must derive from my boyhood fascination with space travel. Thirty years later, the marks were still indelibly there in my mind, showing that I had constructed my image during my primary school years at the latest!

Something else that amazed me

is the Kerala (http://www.cs.cmu.edu/~vipin/kerala.html) contribution to mathematics. Two of my colleagues, Dennis Almeida and George G Joseph, researched the Kerala contribution to analysis – which you can read about in the Philosophy of Mathematics Education Journal No. 20 (http://people.exeter.ac.uk/PErnest/).

They found that quite a few of the series that were fundamental to the development of analysis, such as the Maclaurin Series (http://mathworld.wolfram.com/MaclaurinSeries.html) and the Taylor Series (http://mathworld.wolfram.com/TaylorSeries.html), were discovered in Southern India a couple of centuries before they were known in Europe. What they were investigating is whether this knowledge was transmitted to Europe and so sparked the well-known revolution in calculus and analysis. In my view the evidence for the transmission is weak, but the undisputed fact that another culture developed these complex results so far in advance of the later discoveries in Europe is amazing - and reminds us to remain humble about the origins of western science and mathematics.

Why mathematics?

I had a certain facility in mathematics at grammar school, but it was only when I did A level pure mathematics at the City of Westminster FE College that I became switched on to it. I had a brilliant teacher called Derek Yandell who inspired me to take it further. So I applied to the recently opened Sussex University to study mathematics alongside philosophy. I don’t think I really knew what I was letting myself in for!

I'd read Russell's An Introduction to Mathematical Philosophy (http://people.umass.edu/klement/russell-imp.html)

but didn't understand it! However, by a happy accident I fell in love with mathematical logic, which was in no little part due to an inspiring lecturer Dr Yoshindo Suzuki. The external examiner for the undergraduate degree was Imre Lakatos (http://www.lse.ac.uk/collections/lakatos//), and he suggested that I study for a PhD with him. Instead I opted for an MSc in mathematics at London University specialising in mathematical logic. I passed that with distinction, which was rather reassuring after flunking so many courses in the 1960s!

Sadly Lakatos died that year (1974) so I never did get to meet him. However, my MSc lecturers John Bell and Moshe Machover continued to inspire my fascination with mathematical logic. I went on to take a PhD supervised by Moshe Machover, applying mathematical logic to the problem of meaning in the philosophy of mathematics.

By now I was married to Jill and with a daughter, Jane, and needed some means of support. So I became a school mathematics teacher, thus opening the door into the world that would dominate my interests for the rest of my professional life – the world of mathematics education!

A significant mathematics related incident in your life was …

When I was in my first teacher education position at Homerton College, Cambridge during 1979-81 I attended a course by Keith Hirst on divergent series. At one sticking point in a proof I got lost and Keith asked everybody who didn’t understand to raise their hand. I put my hand up and suddenly realised that I was no longer ashamed of, and so needing to be secretive about, my difficulties - I could openly admit them! Previously the mathematics I had studied had what I might best describe as a ‘macho’ ethos, where you never showed your weaknesses. I realised that after some years in school-teaching and teacher education I was free from this ethos. Keith explained the sticking point, and we all understood easily - but it meant more to me than just that.

And that was a deep lesson I carried with me throughout my teaching career. No matter how good you might be, many carry this burden of shame that they have an inner flaw they cannot admit. It is now called the ‘imposter syndrome’, you feel a fake! I never had it full-blown but I understand some do, and that sensitive teaching and handling can help overcome it.

Another lesson was doing my PADI open water scuba diving course in Teignmouth, Devon in 1998. There were four of us, three boys aged 15, 16 and 17 and me a 54 year old professor of mathematics education. When it came to calculating the residual nitrogen pressure in your body tissues based on repetitive dives I was stuck. There was this chart and rule of thumb and I just couldn’t get it. I was ashamed that these boys seemed to be sailing through, and me with my position, just couldn’t get it. I really felt a pressure as we were having a test on it. Well the next day I did get it and passed all the theory and practice elements and got my PADI certificate. But I learned something more about pressure (emotional, not nitrogen), negative attitude, test anxiety which stayed with me. For example, I have always opened viva exams for doctoral degrees with words designed to put candidates at their ease and to give them confidence (except in two cases when we knew they could not pass).

Once you yourself have passed beyond some of these attitudinal barriers you forget how intense and debilitating they can be. I try to never forget! Indeed one of my set talks is about the importance of errors, that we all need to make them (knowledge cannot advance without them), and we need a learning atmosphere that accepts, encourages their admission, and does not castigate or demonise them.

A mathematics joke that makes you laugh is …

Bertrand Russell defined the order of a joke as follows.

“Why did the chicken cross the road?” is a first order joke. Whereas “A chicken met a duck at the side of the road and said 'don't cross that road, you'll never hear the end of it!” is a second order joke because it refers to a first order joke, namely the first example. Russell goes on to define the hierarchy of jokes inductively in this way. At the end he claims that transfinite order jokes do exist, but they are incomprehensible to us and evoke only the inaudible laughter of the gods! Of course this itself is a transfinite order joke! So this is Russell’s paradox in a jokey way!

The best book you have ever read is …

Excluding fiction, I got a lot out of Kuhn's The Structure of Scientific Revolutions (http://plato.stanford.edu/entries/thomas-kuhn/), Karl Popper’s (http://www.tkpw.net/) and Imre Lakatos' works, especially Proofs and Refutations, and Frege's (http://www.iep.utm.edu/frege/) writings. I especially value Philip Davis and Reuben Hersh's The Mathematical Experience (http://www.cut-the-knot.org/books/hersh/index.shtml). The book I read and got the most out of during my studies is A Course in Mathematical Logic (http://www.elsevier.com/wps/find/bookdescription.cws_home/501742/description#description), by J. L. Bell and M. Machover.

Who inspired you?

In addition to my maths and logic teachers named above (John Bell and Moshe Machover) I was inspired by philosophical lecturers Peter Nidditch and Jerzy Giedymin who I met as an undergraduate. When I was doing my 'A' Levels in 1966 it was Derek Yandell, a great lecturer at City of Westminster College.

If you weren’t doing this job you would be …

… doing I don’t know what. When I was at school, chemistry was my first love - linked to my childhood love of, and experiments with, fireworks and bombs. A careers adviser suggested I should be a biochemist when I was 18 years old, because I was also studying Zoology and Pure Maths. But I have learned that I need people in my work – something I didn’t know when I was younger.

I worked for (almost) a year as a dustman in the Borough of Hampstead - from Kilburn almost to Gospel Oak, from Hampstead Village to Swiss Cottage. That was Summer 1963 to Easter 1964. My official title was Refuse collection officer. But as he was dismissing me for gaps in my attendance the foreman said "Look, you're not really a career dustman. After all you have 'O' levels!" And of course he was right!

I worked for two years as a computer programmer from January 1970, during my study break. I suppose I could have stayed in that industry and watched it really take off from the inside! But I did find writing simple computer applications for businesses unsatisfying. Of course had I continued, the job would have grown and developed out of recognition. But through a series of lucky chances and choices I ended up doing what fascinates me the most!

AUTOBIOGRAPHICAL NOTES

Paul Ernest was born in New York in September 1944, son of a Jewish-American artist John Ernest and Swedish psychologist Elna Adlerbert. She was visiting her brother Bo in USA and had travelled there on a Norwegian ship when Norway was invaded by Germany leaving her stranded there for the duration of the war. She decided to attend university and graduated from Cornell University before getting married in 1943 to John Ernest. Paul’s father was always known as Ernie, although his name was Jachiel Cohen, which he changed for a number of reasons including the avoidance of anti-Semitism.

John Ernest (1922-1994) was later to become a founder member of the British constructivist art movement in the 1950s and1960s. His work is highly prized and is represented in museums and private collections throughout the UK, Europe and USA, including a number of works in the Tate Britain Gallery, V&A, and Arts Council collection, London. He was also a gifted amateur mathematician and discovered a mathematical result in graph theory. Elna Ernest (1920-2013) worked as a clinical psychologist until her retirement in 1980 and then lived and painted full time in Spain. Paul’s only sibling Susan Ernest, born in Paris 1950, is also a painter and lives in Spain.

In 1946 Paul and his parents moved first to Gothenburg, Sweden and then to France in 1949, where they lived in Paris and Antibes. The family moved to London to settle permanently at the time of the Festival of Britain in 1951. By this time 6 year old Paul was trilingual.

The family lived in West Hampstead and Paul attended both primary and secondary schools in the Hampstead area, including William Ellis Grammar School in Parliament Hill. The 1960s was an exciting time for growing up around Hampstead, and Paul aspired to the Beat and protest Generation, participating in four annual anti-nuclear Aldermaston marches as well as many peace, anti-Vietnam war and anti-apartheid demonstrations during the early 1960s. Much of his youth was spent at the infamous Belsize Park coffee house, the Witches Cauldron. He later became a fellow traveller of the Mod and later the Hippie movements in the mid 1960s. He attended live concerts by Tamla Motown artistes, the Beatles, Rolling Stones, Miles Davis, John Coltrane, Thelonious Monk, Jimmie Hendrix (5X), Pink Floyd (4X), Captain Beefheart and others now forgotten.

Although ostensibly a student of science and mathematics, Paul spent much of the 1960s reading modern and classical literature and poetry, philosophy, mythology and mysticism, and in the appreciation of modern painting, classic film and early music (medieval to baroque), and learning meditation and yoga, as well immersion in the popular culture of the time.

Like many of his generation, Paul Ernest dropped out from his studies in the 1960s, and then travelled extensively in Europe, Morocco, Turkey, the Levant, Iran and Afghanistan. In between bouts of study he held a number of jobs including refuse collector, security guard and computer programmer, designing and installing the first computerized accounting system for H.M. Treasury, London in 1971. He met Jill, his wife to be in 1970, and they were married in 1972. Jill undoubtedly contributed to his new found seriousness and resuming study in 1971 he graduated in mathematics, philosophy and logic from Sussex University in 1973, was awarded an MSc with distinction in mathematical logic by London University in 1974, and finished his doctorate on the philosophy of mathematics and logic, in 1984.

Paul was accepted to study for an undergraduate degree entitled Philosophy and the theory of science (with mathematics) at Sussex University, in the Logic Division of the School of mathematical and physical sciences. After an interruption in his studies, by the time he graduated it was called Logic with mathematics. The degree fundamentally remained the same with 50% pure mathematics and 50% philosophy of mathematics, philosophy of science, the history and philosophy of logic, mathematical logic, epistemology, and some other scientific aspects of modern philosophy.

For his masters degree Paul was registered at Bedford College, University of London, situated in Regent’s Park. His tutor was the well known mathematical logician and historian of mathematical logic G. T. Kneebone, although he took his main courses at Chelsea College (mathematical logic, proof theory, recursion theory with Moshé Machover) and London School of Economics (Boolean algebra, model theory, axiomatic set theory with John L. Bell)

For his doctoral studies Paul was a student in the Department of the History and Philosophy of science at Chelsea College of Science, London University, with supervisor Moshé Machover. By the time the PhD was awarded Chelsea College had become part of King’s College London, and the Department was absorbed into the philosophy Department.

For his doctoral studies Paul made an in-depth study of mathematical logic, the foundations of mathematics and the philosophy of mathematics and developments in these fields from the 1870s to the 1970s. His doctoral thesis is entitled ‘Meaning and Intension in Mathematics’, and is based on using proof theory (semantic tableaux) and model theory to provide a formal definition of the meaning of mathematical expressions with both intensional (proof based) and extensional (truth based) components. The thesis defined a measure of the distance between the meanings of any two mathematical expressions, and proved the equivalence of a number of formal relations involving meaning, such as synonymy, and meaning distance 0. It also drew on Wittgenstein's meaning as use doctrine.

Paul began his studies in the 1970s with a traditional Platonist/absolutist philosophy of mathematics. His studies of Intuitionism, Lakatos and Wittgenstein led him to begin to question this philosophy. However, it was the immersion in mathematics education, first as a mathematics teacher 1976-1979 and then more decisively as a mathematics teacher educator 1979 on, that led to the full humanisation of his philosophy of mathematics, ultimately leading to his development of his version of social constructivism. It also led to his highly critical stance towards absolutism in philosophy of mathematics, arguing forcibly against his own beliefs of a decade earlier, that he had subsequently repudiated. A first version of his social constructivism drawing on radical constructivism was published in 1991, and the fully developed social constructivism as a philosophy of mathematics using Vygotsky’s work and semioticsappeared in print in 1998. A paper of 1994 details the significant differences between these two positions (Ernest, P. 1994 ‘Social constructivism and the psychology of mathematics education’, in Ernest, P. Ed. Constructing Mathematical Knowledge: Epistemology and Mathematics Education, London, Falmer Press, 1994, pp. 62-72.)

Since 1976 he has devoted himself full time to the teaching and learning of mathematics and to mathematics education, beginning with a 3 year stint as a mathematics teacher in a London comprehensive. This was followed by a series of positions in mathematics teacher education at college / University level. Becoming a mathematics teacher educator started changing Paul’s interests towards to mathematics education, as he reflected on pedagogical issues and mathematics curriculum in order to teach these subjects to trainee teachers. It also accelerated the humanization of his philosophical views. Then after 4/5 years as a teacher educator with published reflections on problem solving, investigational work and other pedagogies he realized how important he believed people's personal philosophies were for their practice, and that his earlier studies in philosophy of mathematics were a valuable resource to apply in reflecting on and theorizing teacher belief systems. Paul attended his first international conference in 1986 – PME10 in London. For the next ten years he attended every annual PME conference religiously. This served as a Summer school, introducing Paul both to the leading edge theorizing in mathematics education, and also to many of the leading authors and researchers in the field personally (E. von Glaserfeld, H. Freudenthal, U. D'Ambrosio, N. Herscovics, J. Kilpatrick, L. Steffe, D. Robitailles, D. Wheeler, R. Hersh, P. Davis, T. Tymoczko, D. Bloor, R. Thomas, C. Hoyles, N. Balacheff, A. Bishop, R. Lesh, F. Lester, C. Keitel and rising stars who now have their own places in the Pantheon including D. Grouws, P. Cobb, J. Confrey, M. Niss, O. Skovsmose, A. Sfard, S. Lerman, J. Evans, D. Pimm, J, Mason, R. Noss, and many, many more.)

His published papers focused increasingly on teacher belief systems. After a few papers on relations between the psychology of learning mathematics and mathematics education, teachers attitudes and beliefs, and the mathematics curriculum making connections , he wrote his widely cited 1991 book (The Philosophy of Mathematics Education) in 12 months, from around Easter 1989 to Easter 1990. He has never - before or since - been so driven!

Paul worked as a mathematics teacher educator at Homerton College, Cambridge 1979-81, and Bedford College of higher education, Bedford, 1981-82. He enjoyed an exciting two years as lecturer in education at the University of the West Indies, Jamaica, 1982-84, where he designed a distance education course for mathematics teachers which until recently was still transmitted around a dozen Caribbean territories by satellite. He took up his present post at the University of Exeter first as a lecturer in 1984. He was promoted to reader in 1994 and full professor in 1998. Paul Ernest is currently Emeritus Professor of the Philosophy of Mathematics Education at Exeter. He has also been a visiting professor Oslo University (2006-2013), University College, Trondheim (2007-2010) and Brunel University, London (2011-13).

He has taught and developed student materials at all levels, from undergraduate teacher education, Postgraduate Certificate of Education (primary and secondary), through to Masters Degree and doctoral degrees in Mathematics Education (both EdD and PhD). He also chaired the development of the overall research methodology programme attended all doctoral students in education at Exeter University. He developed and directed the specialist doctoral and masters programmes in mathematics education taught in a unique distance learning format that attracted students from almost every continent. For these programmes he authored doctoral course handbooks on

1. The Psychology of Mathematics Education

2. The Mathematics Curriculum

3. Mathematics and Gender: The Nature of Mathematics and Equal Opportunities

4. Mathematics and Special Educational Needs

5. Research Methodology in Mathematics Education

Paul Ernest’s research addresses fundamental questions about the nature of mathematics and how it relates to teaching, learning and society. His academic interests include

· The relationship between the philosophy of mathematics and mathematics education, and more generally the philosophy of mathematics education, ethics and values in mathematics education, and the philosophy of research methodology

· The mathematics curriculum and its aims, curriculum ideologies, etc

· critical mathematics education and all aspects social justice in mathematics education including issues of gender, special educational needs, class, etc

· psychology of mathematics education and learning theory in mathematics education

· the semiotics of mathematics education and language in mathematics education

· his earlier publications also explored mathematics teaching pedagogy and the use of information and communication technologies and computers in mathematics education

· His most recent publications challenge and critique traditional and received conceptions of:

A. The certainty of mathematical knowledge and truth

B. The nature of mathematical objects

C. Aesthetics and mathematics

D. Mathematics and values

E. Ehics and social responsibility of mathematics.

He has published over 300 books, chapters and papers in learned journals, and given over one hundred presentations at national and international conferences in countries including France, Spain, Denmark, Italy, Hungary, Sweden, Norway, Finland, Belgium, Greece, Germany, Cyprus, Abu Dhabi, USA, Mexico, Canada, Brazil, Japan, India, Malaysia, Brunei, Thailand, Australia, Barbados, Bermuda, Nepal, Poland, China, South Africa, Ethiopia, Ireland, England, Scotland and Wales.

He enjoys theorizing and some would say risky speculation about philosophical aspects of mathematics and mathematics education and their role in society. Currently he is working on challenging ideologies, myths and hidden assumptions about mathematics and underlying mathematics education.

Paul Ernest has a wide variety of personal interests including Computing/Internet, Academic writing, Cooking/Food and Wine, Gym/Keeping Fit, Literature and popular fiction, Cinema, Art/Museums, Music from Popular, Blues & World to Classical, Travel/Sightseeing, Scuba diving (especially diving with sharks).

MEDITATION AND MYSTICISM

One of the interests and practices that plugged the hole left when I straightened out in 1969 was that of meditation, yoga, and a fascination with mysticism, spirituality and the occult. I have been interested in Buddhism for a long time, especially the Tibetan form and Zen Buddhism. It was big in the 60s. I have also had some interest in other forms of mysticism (e.g., Christian mystics, Sufis, Taoism) and had read The Cloud of Unknowing, Revelation of Divine Love, and St John of the Cross’ poems, although still an atheist. I learned meditation in 1970 (the well known Trancendental Meditation technique) and I meditated everyday for 2 years. It was a very, very helpful practice. I use the TM mantra technique - you imagine a word sound silently e.g., "om" and try to keep it at the front of your mind without straining. You relax your body as well, e.g., sitting in a chair. Mostly I do it in bed at night to relax and try to get back to sleep if I wake up. While 'saying' my mantra I also try to relax into my visual field and experince whatever visual comes up. In other words - turn off logic and reasoning, lightly try not to go with the random flow of ideas.

For a while I did Yoga everyday, last thing at night, ending with a 20 minute headstand. It got to be a bit of a tedious duty after a while and I gave it up.

One aspect of developing the spiritual side of myself was giving up all ‘intoxicants’ including alcohol and tobacco. While I did slip back onto tobacco I managed to give it up completely at last in 1982. I gave up drinking alcohol in 1970 for 30 years. I do now enjoy a glass of good wine with my dinner, but I don’t enjoy the ‘fuzzy’ feeling if I have more than one.

BAD BOY

I was a bad boy in lots of ways: trespassing, stealing, setting off fireworks and bombs, smuggling home slow worms from Sweden. But I was rarely nasty and never violent. I also had empathy and spoke up when I say injustice, and took some unfortunates under my wing. I dont feel bad about my transgressions (except slightly - the story below), and as Sydney Silverman (MP and Roger's dad) once said to me "We all stole when we were kids!" He was a great man with a big heart and the world is a poorer place for his early demise.

GROWING UP IN THE SIXTIES - LOVE, DRUGS AND WISTFUL DREAMS

It all began when my father Ernie told me and my sister Sue about his wonderful mescaline experience. It must have been 1959 or early 1960 when he recounted his experience. This included the supernaturally glowing colours, the intricate patterns and shifting kalaedeoscopic designs he saw after eating the dried peyote cactus at Asa Benveniste’s party. This fitted so well with his abstract paintings and artworks. I was 15 and Sue was 9 and we were riveted by the description of all the wonders he saw and experienced. It sounded even better than Walt Disney’s film Fantasia, my favourite movie during my primary school years. I’d heard Ernie and Elna’s stories about smoking lots of pot in Bohemian circles of artists and writers in New York and Paris, where we lived before moving to London in 1951. But this was magic, and we each decided it was something we had to experience.

On New Year’s day 1960, aged 15, I started keeping a daily diary. It only lasted a couple of weeks but I still have the wine-red hard bound book of blank pages in which I wrote. I think Paul Vaughn gave me the book. When I looked at it later in my teens it embarrassed me, It was full of tedious tales of everyday life, breakfast etc, punctuated with lists of the girls I was sweet on. (See section Joys and Torments of Youth where much of it is reproduced). There was Anne Heller in the flat immediately below us at number 13-15 Frognal. She lived with her single mother, a photographer, who worked nearby with Anna Freud at the house in Maresfield Gardens, and they holidayed in Walberswick. I think I kissed her once, maybe more times. When she moved to the USA that year I sent her a love letter every day, an aerogramme, for over a year. She replied a few times.

I met her again 5 years later and I was a bit embarrassed by the memory of my exposed immaturity and naked neediness. When I did remeet her, near where she was staying in Arkwright Road, Anne said a mutual friend had recommended me as a source for drugs. She wanted something to boost her confidence, because she had to give a talk. I was a bit bemused by my rep, but I did keep steady supplies of drugs. I gave her some speed (amphetamines) with a warning. She had attended Camden School for Girls when I first knew her. I knew lots of girls from there – it was almost a sister school to my own William Ellis Grammar School, which I left Summer 1963.

Another one of the girls I listed in my diary was Sunita, a girl with an Indian father and a German mother, who lived in Hampstead Hill Gardens. For a date we went to see the Bergman film The Face (Ansiktet) at the Hampstead Playhouse cinema in Hampstead Heath village. We sat in the back row and snogged. Only at the end did we admit that, to be more attractive, we had both left our glasses at home, and couldn’t see any of the film. We were ‘going out’ for about 10 days before she dropped me. This involved dates and snogging, a little bit of 'feeling up', but nothing more. Sex was pretty chaste for me until Summer 1963. In 1959 I went to some teen parties at Steve Goldblatt’s house near Platts Lane. He had mid teens boys and girls there, maybe 8 or 10 in all, and we played games like spinning the bottle and postman’s knock. This involved, if you were lucky, kissing a member of the opposite sex, which I found very, very thrilling! We had a few drinks at these parties, got a bit tipsy, my first experience of mind altering substances. Steve introduced me to Buddy Holly and The Teddy Bears (Phil Spector’s first group). I remember him telling me that Buddy Holly had died in that fateful plane crash with Richie Valens and the Big Bopper in 1959. I had Richie Valens’ 45 single ‘Oh Donna’ which moved me much more than the B side ‘La Bamba’. I was also moved by Paul Anka’s ‘I’m just a lonely boy’ and ‘Diana’. My early record collection included LPs by Elvis, Little Richard, Lonnie Donegan but I also liked Fats Domino and the Everly Brothers.

There were about half a dozen other girls I was sweet on listed in my diary of 1960.

Apart from Ernie’s mescaline and pot stories I can’t remember what else tweaked my early interest in drugs, except literature. Early in the 60s I read Aldous Huxley’s mescaline book the Doors of Perception, Beat writings like Allen Ginsburgs Howl, Kerouac’s On the Road, and first saw Burroughs ‘Naked Lunch’ in my friend Celia Lawes' big sister’s hands. I wanted that book. It was banned and I bought my own copy in Paris summer 1961. I also read Rimbaud and liked his philosophy of the ‘disordering of the senses’, as well as trying to get into Baudelaire for his love of opium and hashish. I read books like Otto Shenks’ Book of Poisons, Robert de Ropp’s Drugs and the Mind, and Martindale’s Extra Pharmacopeia at the local library. I identified with the beats and wore a salt and pepper speckled roll neck sweater and sandals, clear markers of beat aspirations, even on a 16 year old boy.

One day in spring 1960 I spoke to my friend Peter and told him about my growing but still theoretical interest in drugs. He said “Paul, we weren’t going to tell you this because we weren’t sure you could keep your mouth shut, but we’ve been experimenting with drugs.” It turned out that these were prescription drugs stolen from their parents. Jonathon’s father took Ritalin, now known for being the drug they give ADHD kids. It also gave a marvelous euphoria. I also started taking my parents pills. My mother took Preludin for weight control. She kept them in an antique silver heart shaped pill box in her handbag. I could pinch half a dozen pills every couple of weeks or so, without her noticing. She stopped getting them after a couple of years when she realized she was getting dependent.

I think my first speed experiences were on Preludin. It gave me a great warmth, loving feeling and an urgency to share all my thoughts, feelings and hopes through talking to whoever I was with at the time. My father Ernie was an insomniac and took Seconal to help him sleep. This made you feel a bit stunned and alien in yourself, and made walking seem like wading in treacle.

Tony’s mother was on Spansules, amphetamine with barbiturates in slow release pellets. Again these gave me a softening and long lasting energy and need to chatter. I forget who contributed what, but between us we had a reasonable range of psychoactive prescription medicines. I remember my first small doses of speed, amphetamine or Preludin. I was awash with joy and love and spent hours talking to a now forgotten girl sitting out on a garden wall somewhere in Hampstead. I was almost overcome with bonhomie, joy at life, sharing my every thought, wish and aspiration with her. The conversation was so intimate, so disclosing, so intense, so sharing, so warm, so caring and loving - but only in terms of private feelings. It was a wonderful and uplifting experience. But I only felt this on a little bit of speed. Like many drugs, less is more.

In about 1961 we found that for 2 shillings you could buy a nasal inhaler called Nostroline that contained 350 mg of amphetamine oil. It was mixed with essential oils to help ease cold symptoms. We could buy Nostroline over the counter at a chemist off Highgate Hill, near to school, and one tube is 70 standard doses of amphetamine. There were about 10 awful tasting strips of impregnated paper inside each inhaler tube. Eating one of those would keep you awake all night talking and other obsessive behaviours. We would often eat several sheets and suffer from the symptoms of amphetamine overdosing. Peter ate a whole tube once and his heart stopped for a short while. He claims he restarted it with fist thumps to his chest as he fell to his knees. Peter’s tale is a bit dubious, and, what’s more, he now denies it.

Large doses of amphetamine lead to raised pulse rate, incessant talking, tall tales and lies, obsessive scratching and other OCD behaviors. As I discovered later in the 1960s very large repeated doses of amphetamine lead to sleeplessness, delusions, hallucinations, feelings of paranoia and aggression, which becomes full blown amphetamine psychosis. Jules Holley was an afficionado of speed - amphetamines in all their varieties - and he manifested it through intense all night talking sessions of such intimacy that I have loved him ever since - you feel like you have really become soul mates when you have been through that together. Unfortunately Jules also manifested his speed binges in terms of tall stories, lying, theft, paranoia and aggression. But all that came later.

From about 1960 onwards, I would meet my friends and at the Witches Cauldron, a coffee bar in Belsize Park. My circle of friends grew as I met new people there, and also at pubs, the Belsize Tavern, the White Bear, and at a few other coffee bars in Hampstead and at private parties. These bars included El Serrano, The Loft, 321 (aka Moon & Sixpence) in College Crescent, and briefly the 01 club, next to Belsize Park tube station in 1961. This, run by Bernie Osgood’s cousin, was where I first hear the song Locomotion by Little Eva, which still send chills up my spine.

Unlike my immediate school friends, mostly grammar school boys and girls, as well as some girls from Parliament Hill School, I met a much wider set of people through the Witches Cauldron. This included working class boys and girls, bikers, and would be beats, poets, aspiring bohemians, and youth and young adults more generally. As well as chatting with interesting acquaintances and comrades, not to mention chatting up girls, this involved sharing information about parties, especially for Saturday nights. I think the maximum number ever shared was a list of 10 parties to go to one Saturday night. Typically these were around Belsize Park, Swiss Cottage, Chalk Farm, Camden Town, Hampstead village, but also in Highgate, Golders Green, Holloway, Hampstead Garden Suburb, Finchley and further afield suburbs of North London like Edgware, Tottenham, Totteridge. We also went to parties in Notting Hill, Westbourne Grove, Kilburn, Chelsea. I remember going to one in Oxford once, in the back of someone’s van. Coming back I found my legs sellotaped together and bar soap rubbed into my hair. Sometimes it was our friends having a party. Or friends of friends. Or acquaintances of acquaintances. Sometimes middle class girls, and it usually was girls, wanted to invite the famed Hampstead or Witches set to their parties, and shared details via our friends and acquaitences. I was invited by a girl I knew to a party in Bishops Avenue, already known as millionaires’ row. I told my friend Paul Vaughan, then studying art at Hornsey College of Art, who had 100 invitations printed and distributed. When I got to Bishops Avenue there were scores if not a hundred young people milling in the street near to the party house, from which they were excluded. I was let in as an actual invitee, but was caught by some adult, parent or otherwise, opening a ground floor widow to let others in. I can’t recall if I was ejected or just reprimanded!

Mostly at these parties we would chat, dance, get drunk, snog girls, smoke dope, throw up, and stagger home or to the next party or spend the night sleeping on the floor. Some companions smashed places up and stole things. I never felt like doing this, except for one thing. I developed an unerring nose for finding the liquor cabinets, which I cracked open and drank and shared the contents. I remember leaving one party in Hampstead Heath village in the wee hours with half full bottles of gin and whiskey. I dropped one and it cracked and started to leak so I quickly poured what I could save into the other bottle. A week later I went to a party with Paul Vaughn and Vaughn O’Leary and I drank the dreadful gin and whiskey mixture down on the tube going to Chelsea. It was more than a pint, and when I got to this small party of 8 or 10 of us I was drunk and became even drunker very quickly. Afterwards, I was told I put my hand down the back of one girl’s jeans and got smacked for it but all I remember was running out of the room to the toilet with my cupped hands brimming over with vomit. Later I slept in the parent’s living room and despite promising to use the fireplace if needed to, I threw up on the carpet. This was the first time I had a two day hangover. It was not infrequently that my Friday or Saturday nights ended with me being sick.

At another party, perhaps in Chiswick I was quite drunk and lay down on the pavement outside the party house. Some wit decided it would be funny to bounce empty beer bottles off my stomach, and one flew up and down hitting me in the teeth and sheering off one of my two front teeth raggedly. I remember trying so hard to turn the clock back, to reverse time, so that I could have my tooth back in place. It didn't work and ever since one of my two front teeth has been a crown, until that failed ten years ago, and now it's an implant. Not that you would notice it, they do such a good job these days.

A little later, I’m not sure when, around 1962. when going to parties I extended my foraging to the parent’s drug cabinets and bedside tables. I pilfered the full range of uppers, downers, tranqs and other drugs, leaving decent quantities behind so as to reduce the risk of being caught. I recall drugged up weekends at Tony Jackson’s parents’ flat, overlooking Regent’s Park, where I found a suitcase full of drugs bought in France, as well as UK prescription drugs. There was plenty to take but I drew the line at silver coloured sex hormone pills. I didn’t like the sound of their effects. That must have been 1963 because by then I was using heroin. We also had stoned weekends at Pete Rasini’s near Highgate, when his parents were away. We would move in for the weekend, maybe 6 or 8 of us, including girlfriends and have a very happy time, maybe playing poker, dancing to music, not to mention getting stoned, very stoned. I remember taking my then girlfriend Cheryl Molyneux (a delightful and very pretty young dancer aka "bubbles") to parties at Tony Jackson's, and sharing a bath with her. The image remains, her lovely young body but me too out of it to be any use to her. But she was very tolerant and loving. Probably there were some of Tony Barnett, Alan Shoobridge, Pete Rasini and Phil Howe there, as well as Tony Jackson, although he was not a user himself. I lost touch with Cheryl from Fordwych Road, West Hampstead, and to this day have no idea what happened to her. She probably got married and changed her surname.

So around 1960 we were experimenting with our parents’ prescription drugs. My primary school friend Roger S, always more advanced than me, went to Paris on his own in the Summer of 1960 and stayed at the Beat Hotel, No. 9 Rue Git-le-couer. He came back with some marhijuana. He, Paul Vaughn and I smoked some at Paul’s parent’s flat in Greencroft Gardens, Autumn 1960. This was the very flat where Ernie, my father, had taken mescaline a couple of years earlier. I don’t remember much of what happened that first time except we all got the giggles and the other two chased me up and down Greencroft Gardens pretending to hold a spider. As a archnophobe I was genuinely terrified, a terror probably made all the more hysterical by the pot. It all ended in laughter. Paul Vaughn was a year older than me, and more advanced too. He lived with his mother, the painter Pip Benveniste, and her second husband, the poet and printer Asa Benveniste, and a couple of brothers, Mark and Jasper. They were good people who treated us kids like equals, and lived a true bohemian and artistic life style. I remember them throwing the Yi Ching for me with yarrow stalks. I can’t recall which hexagram came up. Paul and Vaughn O’Leary used to go down to Eel Pie Island to dance the Stomp, where they heard the Rolling Stones before they became famous.

Asa Benveniste, poet and founder of Trigram Press

See Asa Benveniste - Wikipedia