Tony Gardiner writes:

The mathematical “house” is fortunate in having “many mansions”, inhabited by a remarkable variety of workers. By any account Geoffrey Howson – who died on 1 November 2022, aged 91 – was a significant player throughout much of the period 1950-2000. However, like so many other workers, he operated effectively, but quietly, so may not have been noticed. Nevertheless his life offers interesting insights into how UK society has changed since 1931 (when he was born as the seventh in a family of seven children), and into how UK mathematics and mathematics education worldwide have evolved since the 1950s.

Geoffrey belonged to the generation, who emerged in significant numbers (perhaps for the first time) in the 1940s. Their families had never been to secondary school – let alone university. Yet – thanks to structural changes and committed teachers – they somehow emerged in small numbers at age 18, ready to take on whatever challenges the post-war world might present.

Geoffrey always remained faithful to his roots (a solid Yorkshireman, from a deprived, but proud, mining community). Yet he came to excel in mathematics, in university politics, and in international mathematics education – as well as in the world of opera, Bauhaus design and embroidery, and medieval church architecture.

Geoffrey went to Castleford Grammar School (founded 1906), and was probably the first from that school to study mathematics. He went on to Max Newman’s department in Manchester, where his teachers included: Max, Walter Ledermann, J.W.S. Cassels, Bernhard Neumann, Graham Higman, Kurt Mahler, Arthur Stone, James Lighthill (MA President 1970), M.B. Glauert, Charles Illingworth, and Bernard Lovell. He was Graham Higman’s second PhD student (proving that the intersection of two finitely generated subgroups in a free group is finitely generated). He also attended Turing’s lectures on morphology – interrupted only by Turing’s death.

Invitations from Reinhold Baer (Illinois) and Saunders Maclane (Chicago) were put aside in order to complete National Service (when he taught RAF trainees about guided missiles). He then moved to the Royal Naval College in Greenwich in 1957 (where he taught the new generation of future naval commanders about similar things).

In 1962 he went to Southampton to manage the School Mathematics Project (SMP). This was the UK equivalent of “new math”, but much more humane and less abstract. At its height SMP materials were “used” (in some sense) in 60% or so of UK secondary schools. But SMP remained a Teachers’ Cooperative, with no government support. Geoffrey’s job was officially to edit and to manage the program of new textbooks. In practice, he had to coordinate the writing (planned and completed by a remarkable group of full-time teachers); the production of draft materials; the revision process; and to deal with the publishers and the exam boards (since no project could survive if there was not a corresponding tailored public examination at age 16 and 18).

Geoffrey became a representative spokesperson for “modern maths” developments in the UK, and so came to interact with those similarly placed in other countries – in both East and West – producing many reports, and editing collections published in the 1960s, 70s, and 80s. He published and edited a huge variety of books and papers – all written in a thoughtful style. His goal was to inform and enlighten, rather than to engage in “theoretical research”. He became a leader in Mathematics Education internationally, but was never really appreciated by the new breed of “Maths Education” researchers. His contributions were mostly pragmatic comparisons, surveys, and analyses, designed to inform and to allow improved judgements to be made. He was also very active in supporting teachers’ colleges and those working in polytechnics.

He helped to salvage ICMI/ICME after it came unstuck around 1980. And it is a mark of the man that he managed this (with Jean-Pierre Kahane) while remaining great friends with those who had been part of the previous regime. He was recognised in other countries but not much within the UK.

He was Head of Department and Dean 1990-92 and may have helped in building up parts of what is now a very strong mathematics department. He also Chaired the LMS/IMA/RSS committee that produced the report “Tackling the mathematics problem”: this was a rare instance of the three scholarly societies acting together on a matter of mutual concern, and then having a significant impact on subsequent policy-making.

They don’t make them like that any more.

My answer to a question on Quora:  Which are the procedures you are following to memorize or learn so fast?

One my favorite procedure is called “understanding”. Prior to memorising, try to understand. Very frequently, the need to memorise disappears – you simply understand.

Another procedure that I try to popularise is called “interiorisation”: making something external a part of yourself.

I remember my conversation with my grandson, at that time 5 years old. I asked him: “What’s new at your new school”. He: “I do not like the school. Too many rules. Hard to remember.” I: “Why should you remember the rules? You simply have to follow them. I’ve seen in the lunch room at your school a poster:”Do not mess with food”. What is so difficult about it?”

My grandson fell in deep thought. We walked along a street on our way to his home, in silence. I didn’t distract him. It should be a law of the land: never distract young children when they think.

My answer on Quora: Do the children of math teachers always pass algebra?

My passed.

I do not wish to generalise, and I do not know statistics on this specific issue, but I see some reasons for children of mathematically educated people be a bit more confident in their mathematical studies at school. For example, they have never seen emotions of fear, or dislike of, mathematics in their parents. It is not about genetics, it is about inheriting certain social / cultural capital: values, habits, motivation. I intentionally avoid the word “intellect”, this is not about intellect either.

My answer on Quora: How does category theory relate to other branches of mathematics?

An excellent question. Category theory is important on its own, and has important applications in a number of other mathematical theories; however, the crucial and the most fundamental impact of category theory is invisible and under-reported, it is of cultural nature. It can be compared with the influence of set theory: 99% of mathematicians use only a modicum of naive set theory, ignoring deeply penetrating and frequently very hard results of set theory as the live research discipline which continues to develop and flourish.

There is a telling example: the theory of games of chance was created in the 17th century and gave birth to probability theory; the latter was already quite developed by the time when, in the 20th century, the concept of a deterministic game had finally crystallized – in a paper, of all people, by Zermelo, who proved that chess was a deterministic game: for one of the players, there is a strategy, that is, a function from the set of permissible position to the set of moves, which achieves at least a draw. His paper was published in 1913 (see Zermelo’s theorem (game theory) – Wikipedia). Why did this happen so late? The word “set” in the definition of a strategy came to use only in the second half of the 19th century.

The same is happening with category theory: the vast majority of mathematicians use its ideas and terminology in a very rudimentary and naive form, frequently even without realisation that they are doing so. In the work that I am doing, it had happened to be very important to remember that an algebraic group was a functor from the category of unital commutative rings to the category of groups. Some my colleagues who work in the same theory continue to insist that a group is a fixed set with some operations on it. When my co-author and I recently solved a certain problem which was open since 1999, we were able to do that only because for us a group in question was a functor – not much deeper than that. Why it was not solved by someone else earlier? Because for them a group was just a set.

My answer to a question on Quora:  Is it natural for a mathematician to forget specifics of some parts of math?

I think it is natural. I love to surprise my students by saying that I hardly remember any trigonometric formula beyond \(\sin^2x+\cos^2x=1\)– but I can deduce most standard trigonometric formulae on the spot. On a number of occasions I offered my students a game: give me a wrong formula, say, for \(\sin(x+y)\) — and I will instantly explain you why it is wrong. I have always won. I am not special; I believe that the vast majority of my mathematician colleagues can recover statements of L’Hopital theorems even if they do not remember them exactly.

And there is one more aspect of mathematical memory. The great mathematician Andrew Wiles said in a recent interview Andrew Wiles: what does it feel like to do maths?:

I really think it’s bad to have too good a memory if you want to be a mathematician. You need a slightly bad memory because you need to forget the way you approached [a problem] the previous time because it’s a bit like evolution, DNA. You need to make a little mistake in the way you did it before so that you do something slightly different and then that’s what actually enables you to get round [the problem].

So if you remembered all the failed attempts before, you wouldn’t try them again. But because I have a slightly bad memory I’ll probably try essentially the same thing again and then I realise I was just missing this one little thing I needed to do.

My answer to a question on Quora: Is it normal that one doesn’t understand a math procedure unless demonstrated by a teacher? Like for example, I cannot grasp what the textbook is trying to say but if I watch a video tutorial on how a procedure is done, I can understand right away.

Today I attended a seminar of mathematics education experts. They are seeing inability of many students to adequately understand written/printed texts as a serious and growing problem facing school mathematics education. Mathematical texts are difficult. What are other (that is, non-mathematical) difficult texts that you have read in your life? For example, have you ever tried to read fine print in an insurance policy? By complexity, this could easily beat maths textbooks. Ability to read and understand a difficult text is a useful skill — a skill for life.

My answer to a question on Quora:  Why do I need to learn conics in math?

This is the second best kept secret of mathematics education: it does not matter what students are taught, what matters is how deep their learning, and how efficient is the network of connections between mathematical facts that grows in their minds in their minds: in how many steps they can get form Fact A to Fact B? Conics could be included in high school courses of mathematics, and could be omitted, it does not really matter. What matters is whether students develop specific mental skills of mathematical thinking.

In one of the best mathematics high schools in the world, conics are used as a training ground of mathematics problem solving, that is, solving problems not seen by students ever before — simply because the theory of conics, if taken seriously, is rich, and because it is really hard to find in the literature good books on advanced level but elementary theory of conics. Students, I was told, really have to work from scratch.

I can offer you one problem; I think it was used at one of the International Mathematical Olympiads of yesteryear.

Assume you are given a sheet of paper with a parabola printed on it. Using only straightedge and compasses, construct the Cartesian coordinate system in which this parabola has equation y=x^2.

People who can solve this problem have reasonably deep understanding of conics.

My answer to a question on Quora: How do I start studying mathematics from the beginning until I get the gold medal in the Mathematics Olympiad?

I know a number of my fellow mathematicians who won gold medals at the International Mathematical Olympiad ( I think you mean that Olympiad). I doubt that any of them got this medal after starting studying mathematics specifically with this aim. The only fruitful way to learn mathematics is because you are interested in mathematics, because you wish to understand mathematics, because you wish to learn to recognise that specific feeling of joy which comes from understanding mathematics.

I once met a young girl who told me that she loved mathematics because she felt that mathematics biought her closer to  the God. This is a good reason for studying mathematics, but it is essentially the same as the one just described by me, but expressed from a different viewpoint. But let us think a bit: does winning a gold medal brings you closer to the God? I doubt that. Medal is from people, not the God.

An answer to a question on Quora: Is pure math hard? What type of people would study pure math?