Tag Archives: maths

Mathematics, Jewishness, and Direction

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When I was nearly 18 I was part of the British team to the International Mathematical Olympiad (IMO) in Bucharest (see my account of the experience).  The US team were Jewish1, all eight of them.  While this was noteworthy, it was not surprising. There does seem to be a remarkable number of high achieving Jewish mathematicians, including nearly a quarter of Fields Medal recipients (the Maths equivalent of the Nobel Prize) and half of the mathematics members of the US National Academy of Sciences2.

Is this culture or genes, nature or nurture?

As with most things, I’d guess the answer is a mix.  But, if of culture, what? There is a tradition of Biblical numerology, but hardly widespread enough to make the substantial effects. Is it to do with the discipline of learning Hebrew, maybe just discipline, or perhaps is it that mathematics is one of the fields where there has been less prejudice in academic appointments3.

I have just read a paper, “Disembodying Cognition” by Anjan Chatterjee, that may shed a little light on this.  The paper is an excellent overview of current neuroscience research  on embodiment and also its limits (hence ‘disembodying’).  One positive embodiment result related to representations of actions, such as someone kicking a ball, which are often depicted with the agent on the left and the acted upon object on the right.  However, when these experiments are repeated for Arab participants, the direction effects are reversed (p.102).  Chaterjee surmises that this is due to the right-to-left reading direction in Arabic.

In mathematics an equation is strictly symmetrical, simply stating that two thinsg are equal.  However, we typically see equations such as:

y = 3x + 7

where the declarative reading may well be:

y is the same as “3x + 7”

but the more procedural ‘arithmatic’ reading is:

take x, multiple by three, add seven and this gives y

In programming languages this is of course the normal semantics … and can give rise to confusion in statements such as:

x = x + 1

This is both confusing if read as an equation (why some programming languages have := read as “becomes equal to”), but also conflicts with the left-to-right reading of English and European languages.

COBOL which was designed for business use, used English-like syntax, which did read left to right:

ADD Tiree-Total TO Coll-Total GIVING Overall-Total.

Returning to Jewish mathematicians, does the right-to-left reading of Hebrew help in early understanding of algebra?  But if so then surely there should be many more contemporary Arab mathematicians also.  This is clearly not the full story, but maybe it is one contributory factor.

And, at the risk of confusing all of us brought up with the ‘conventional’ way of writing equations, would it be easier for English-speaking children if they were introduced to the mathematically equivalent, but linguistically more comprehensible:

3x + 7 = y

  1. Although they did have to ‘forget’ while they were there otherwise they would have starved on the all-pork cuisine[back]
  2. Source jews.org “Jews in Mathematics“.[back]
  3. The Russians did not send a team to the IMO in 1978.  There were three explanations of this (i) because it was in Romania, (ii) because the Romanians had invited a Chinese team and (iii), because the Russian national mathematical Olympiad had also produced an all Jewish team and the major Moscow university that always admitted the team did not want that many Jewish students.  Whether the last explanation is true or not, it certainly is consonant with the levels of explicit discrimination in the USSR at the time. [back]

The Cult of Ignorance

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Throughout society, media, and academia, it seems that ignorance is no longer a void to be filled, but a virtue to be lauded.  Ignorance is certainly not a ‘problem’, not something to be ashamed of, but is either an opportunity to learn or a signal that you need to seek external expertise.  However, when ignorance is seen as something not just good in itself, but almost a sign of superiority over those who do have knowledge or expertise, then surely this is a sign of a world in decadence.

Although it is something of which I’ve been aware for a long time, two things prompt to think again about this: a mailing list discussion about science in schools and a recent paper review.

The CPHC mailing list discussion was prompted by a report by the BBC on a recent EU survey on attitudes to science amongst 15-25 year olds.  The survey found that around 1/2 of Irish and British respondents felt they “lacked the skills to pursue a career in science” compared with only 10% in several eastern European countries.  The discussion was prompted not so much by the result itself but by the official government response that the UK science community needed to do more “to understand what excites and enthuses young people and will switch them on to a science future.”  While no-one disagrees with the sentiment, regarding it as ‘the problem’ disregards the fact that those countries where scientific and mathematical education is not a problem are precisely those where the educational systems are more traditional, less focused on motivation and fun!

I have blogged before about my concerns regarding basic numeracy, but that was about ‘honest ignorance’, people who should know not knowing.  However, there is a common attitude to technical subjects that makes it a matter of pride for otherwise educated people to say “I could never do maths” or “I was never good at science”, in a way that would be incongruous if it were said about reading or writing (although as we shall see below technologists do do precisely that), and often with the implication that to have been otherwise would have been somehow ‘nerdy’ and made them less well-balanced people.

Sadly this cult of ignorance extends also to academia.

A colleague of mine recently had reviews back on a paper.  One reviewer criticised the use of the term ‘capitalisation’ (which was in context referring to ‘social capital’) as to the reviewer word meant making letters upper case.  The reviewer suggested that this might be a word in the author’s native language.

At a time when the recapitalisation of banks is a major global issue, this surely feels like culpable ignorance.  Obviously the word was being used in a technical sense, but the reviewer was suggesting it was not standard English.  Of course, ‘capital’ in the financial sense dates back certainly 300 years, the verb ‘capitalise’ is part of everyday speech “let’s capitalise on our success”, and my 30 year old Oxford English Dictionary includes the following:

Capitalize 1850. …. 2. The project of capitalizing incomes 1856. Hence Capitalization.

Now I should emphasise it is not the ignorance of the reviewer I object to; I know I am ignorant of many things and ready to admit it.  The problem is that the reviewer feels confident enough in that ignorance to criticise the author for the use of the word … apparently without either (a) consulting a dictionary, or (b) while filling out the online review form bothering to Google it!

This reminded me of a review of a paper I once received that criticised my statistical language, suggesting I should use the proper statistical term ‘significance’ rather than the informal language ‘confidence’.  Now many people do not really understand the difference between significance testing (evidence of whether things are different) and confidence intervals (evidence of how different or how similar they are) – and so rarely use the latter, even though confidence intervals are a more powerful statistical tool.  However the problem here is not so much the ignorance of the reviewer (albeit that a basic awareness of statistical vocabulary would seem reasonable in a discipline with a substantial experimental side), but the fact that the reviewer felt confident enough in his/her ignorance to criticise without either consulting an elementary statistical text book or Googling “statistics confidence”.

So, let’s be proud of our skills and our knowledge, humble in accepting the limits of what we know, and confident enough in ourselves, so that we do not need to denegrate others for doing what we cannot.  Then ignorance becomes a spring board to learn more and a launching point for collaboration

PPIG2008 and the twenty first century coder

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Last week I was giving a keynote at the annual workshop PPIG2008 of the Psychology of Programming Interest Group.   Before I went I was politely pronouncing this pee-pee-eye-gee … however, when I got there I found the accepted pronunciation was pee-pig … hence the logo!

My own keynote at PPIG2008 was “as we may code: the art (and craft) of computer programming in the 21st century” and was an exploration of the changes in coding from 1968 when Knuth published the first of his books on “the art of computer programming“.  On the web site for the talk I’ve made a relatively unstructured list of some of the distinctions I’ve noticed between 20th and 21st Century coding (C20 vs. C21); and in my slides I have started to add some more structure.  In general we have a move from more mathematical, analytic, problem solving approach, to something more akin to a search task, finding the right bits to fit together with a greater need for information management and social skills. Both this characterisation and the list are, of course, a gross simplification, but seem to capture some of the change of spirit.  These changes suggest different cognitive issues to be explored and maybe different personality types involved – as one of the attendees, David Greathead, pointed out, rather like the judging vs. perceiving personality distinction in Myers-Briggs1.

One interesting comment on this was from Marian Petre, who has studied many professional programmers.  Her impression, and echoed by others, was that the heavy-hitters were the more experienced programmers who had adapted to newer styles of programming, whereas  the younger programmers found it harder to adapt the other way when they hit difficult problems.  Another attendee suggested that perhaps I was focused more on application coding and that system coding and system programmers were still operating in the C20 mode.

The social nature of modern coding came out in several papers about agile methods and pair programming.  As well as being an important phenomena in its own right, pair programming gives a level of think-aloud  ‘for free’, so maybe this will also cast light on individual coding.

Margaret-Anne Storey gave a fascinating keynote about the use of comments and annotations in code and again this picks up the social nature of code as she was studying open-source coding where comments are often for other people in the community, maybe explaining actions, or suggesting improvements.  She reviewed a lot of material in the area and I was especially interested in one result that showed that novice programmers with small pieces of code found method comments more useful than class comments.  Given my own frequent complaint that code is inadequately documented at the class or higher level, this appeared to disagree with my own impressions.  However, in discussion it seemed that this was probably accounted for by differences in context: novice vs. expert programmers, small vs large code, internal comments vs. external documentation.  One of the big problems I find is that the way different classes work together to produce effects is particularly poorly documented.  Margaret-Anne described one system her group had worked on2 that allowed you to write a tour of your code opening windows, highlighting sections, etc.

I sadly missed some of the presentations as I had to go to other meetings (the danger of a conference at your home site!), but I did get to some and  was particularly fascinated by the more theoretical/philosophical session including one paper addressing the psychological origins of the notions of objects and another focused on (the dangers of) abstraction.

The latter, presented by Luke Church, critiqued  Jeanette Wing‘s 2006 CACM paper on Computational Thinking.  This is evidently a ‘big thing’ with loads of funding and hype … but one that I had entirely missed :-/ Basically the idea is to translate the ways that one thinks about computation to problems other than computers – nerds rule OK. The tenet’s of computational thinking seem to overlap a lot with management thinking and also reminded me of the way my own HCI community and also parts of the Design (with capital D) community in different ways are trying to say they we/they are the universal discipline  … well if we don’t say it about our own discipline who will …the physicists have been getting away with it for years 😉

Luke (and his co-authors) argument is that abstraction can be dangerous (although of course it is also powerful).  It would be interesting perhaps rather than Wing’s paper to look at this argument alongside  Jeff Kramer’s 2007 CACM article “Is abstraction the key to computing?“, which I recall liking because it says computer scientists ought to know more mathematics 🙂 🙂

I also sadly missed some of Adrian Mackenzie‘s closing keynote … although this time not due to competing meetings but because I had been up since 4:30am reading a PhD thesis and after lunch on a Friday had begin to flag!  However, this was no reflection an Adrian’s talk and the bits I heard were fascinating looking at the way bio-tech is using the language of software engineering.  This sparked a debate relating back to the overuse of abstraction, especially in the case of the genome where interactions between parts are strong and so the software component analogy weak.  It also reminded me of yet another relatively recent paper3 on the way computation can be seen in many phenomena and should not be construed solely as a science of computers.

As well as the academic content it was great to be with the PPIG crowd they are a small but very welcoming and accepting community – I don’t recall anything but constructive and friendly debate … and next year they have PPIG09 in Limerick – PPIG and Guiness what could be better!

  1. David has done some really interesting work on the relationship between personality types and different kinds of programming tasks.  I’ve seen him present before about debugging and unfortunately had to miss his talk at PPIG on comprehension.  Given his work has has shown clearly that there are strong correlations between certain personality attributes and coding, it would be good to see more qualitative work investigating the nature of the differences.   I’d like to know whether strategies change between personality types: for example, between systematic debugging and more insight-based scan and see it bug finding. [back]
  2. but I can’t find on their website :-([back]
  3. Perhaps 2006/2007 in either CACM or Computer Journal, if anyone knows the one I mean please remind me![back]

mathematics goes reality TV!

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In 1978 I was on the British team for the 20th International Mathematical Olympiad (recollections of the trip). It was in Romania and the event was prime time news … and I was one of a group interviewed for the news of the event. the following year the 21st IMO was held in London and there was no press coverage that I found whatsoever. OK mathematics is hardly a spectator sport, but the complete British lack of interest in anything remotely intellectual was disturbing.

But now … nearly 30 years later … perhaps things are changing. On Sunday BBC2 are showing a 90min documentary about the olympiad team. Maybe maths will get sexy!

Beautiful Young Minds1
BBC2 Sun 14 Oct, 9:00 pm – 10:30 pm 90mins
Beautiful Young Minds tells the story of some of the brightest mathematical brains of a generation. Each year, exceptionally gifted teenagers from over 90 countries compete for medals at the International Mathematical Olympiad. The film follows a group of brilliant teenagers as they battle it out to become the chosen six selected to represent the UK.

  1. unfortunately the BBC’s own page on this disappeared at the end of the week – why do they do this! – but there are many descriptions and reviews of it on the web including one at plus.maths.org [back]