Numb or Numbered?

It just doesn’t add up: why do so many people, including scientists, get stuck on the maths problem?

The subject is on my mind because it was raised at a departmental meeting last week where I tried to argue that A level mathematics (the qualification obtained at age 18 in the UK) should be an entry requirement for our degree programmes in biochemistry and biology. Given the increasingly quantitative nature of these subjects, I ventured, we need to be taking on students who are comfortable with maths.

The GCSE maths qualification obtained at age 16 is adequate for general arithmetic skills and a grasp of elementary algebra but does not cover many concepts that are useful in the life sciences, such as exponentials, logarithms, simple calculus (first order differentiation and integration), trigonometric functions or — a particular favourite of this X-ray crystallographer — the use of complex exponentials to represent waves. I know this because my daughter is preparing for her GCSE Maths exam next month and I have seen the practice papers.

Maths problem

My view is that we cannot rely on GCSE qualifications in maths to equip our students for a 21st Century university curriculum in the life sciences (or indeed any science). Biology and biochemistry may once have been largely descriptive disciplines but those days are gone. For more and more areas in the life sciences — enzyme kinetics, molecular interactions, bioinformatics, genetic inheritance, population dynamics to name just a few — a proper understanding requires maths skills significantly beyond GCSE level. (To say nothing of providing aesthetic appreciation of the mathematical foundation of so many living curves and curls.)

Unfortunately I did not prevail. In part I think there is reluctance to impose A level maths as an entry requirement because it would restrict the pool of applicants. Many students who do biology or chemistry at A level don’t take maths. In fact, of those students who study for A levels in the UK, only about 25% do maths. Worryingly for our science and engineering base, this is one of the lowest figures among developed nations.

That figure is probably not surprising to many people: the aversion to maths is commonplace and the refrain “I’m no good at maths” is heard up and down the land. But what has surprised me recently is the realisation that the aversion is so often also shared by scientists, a discovery that underscores the extent of the problem while at the same time pointing to a solution.

I have been interviewing life scientists for an upcoming video project and, though the sample is too small to be statistically significant, almost all have confessed to finding maths difficult or even distasteful, at least at the outset in their careers. Even TV’s favourite physicist Brian Cox was no friend of mathematics at school — by his own admission he only scraped a D at A level.

Nevertheless, each of these people faced down their mathematical demons on the way to becoming scientists; some of them now do daily battle with serious numbers and formulae. Why is that? Are these scientists the exceptions, the ones who persevered despite a lack of affinity for maths? How many others fell by the wayside?

I don’t know the answer to these questions though the testimony of the scientists reveals that motivation is the key. You have to be able to see the relevance of maths before you can rouse yourself to get stuck in. I experienced this in my physics degree during a maths lecture on Fourier Theory, which shows that any mathematical function can be represented as a sum of sines and cosines. Neat, I thought, but so what? Perversely, the lecturer omitted to make any connection with physical reality, even though Fourier himself only developed this mathematical tool to solve the problem of understanding heat conduction along an iron bar. Only later when I learned the relevance of Fourier Theory to my beloved X-ray diffraction did I really get to grips with the maths.

The abstractive power of maths is its greatest strength but can also be its undoing: the subject is too easily seen as alienating and therefore provides a ready excuse for students to give up. Is the apparent unreality of the subject also the source of the feeling that it is difficult? Perhaps so, but Cox suggested a useful remedy for that mindset, comparing the ability to do maths with the ability to play the piano. No-one expects to be able to sit down at the piano and just play. You have to practise.

And so it is with maths. But for some reason many people don’t realise the bit about needing to practise. Cox only cottoned on to it at university (and went on to get a first class degree in physics) so perhaps we shouldn’t be too hard on school students. But I do think we need to find a way to learn the lesson earlier, so that a greater number of our students might embrace the subject, and be more fearless in the face of science and engineering.

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137 Responses to Numb or Numbered?

  1. Alison says:

    I studied psychology at degree level, and I notice that the people who go on to have really stellar careers in, say, neuroscience, often have excellent maths capabilities.

    Personally, I absolutely struggled with maths at GCSE, but I fell in love with statistics and experimental design when I did psychology, and I got to grips with psychophysics quite cheerfully. I still cannot deal with the abstraction or indeed the pace of thinking in an ordinary maths lesson, but if I’m motivated, and someone can explain the connections, I’ll happily learn.

    My own 10-year daughter has severe dyscalculia, where she struggles hugely with what seem like simple concepts. I suspect I look like that to someone who finds maths easy.

  2. alice says:

    This is my personal story, for what it’s worth:

    I didn’t go further with maths simply because it was taught so badly. I loved maths when I was a kid. I used to take home extra maths homework every weekend. The abstract stuff was hard, especially as I have a very bad short term memory (dyslexic), but if I really thought it through, I’d get it and I adored it.

    I got really ahead, so when I was 13/14 my teacher told me to do an investigation where I had to sit and roll two dice again and again and again (as in, for a good couple of weeks’ worth of lessons). I found this pointless and told my teacher this, but she refused to let me do any more work until I’d sat and rolled those dice. I decided to stop, and simply didn’t do any more maths after that (a bit of revision before my GCSE, for which I scraped a B). The fact that she not only inspired my (admittedly teenage-ly obstinate) reaction and then didn’t seem to notice I’d simply stopped working is something I don’t think I’ll forgive her for.

    As an extra point, I think your use of ‘fearless’ is key. I wonder how many people don’t like maths simply because they are *told* it’s hard. Again, this is a personal thing I’d be interested if there is any research – but I find being told I won’t understand something really disabling (my brain won’t settle on the ‘ah, I get it’, I’m constantly checking myself).

    • Stephen says:

      Yes, the expectations projected by schooling can clearly have a very powerful effect. As mentioned by a commenter below, Paul Lockhart’s heartfelt analysis of the paucity of the maths curriculum (at least in the US, but it sounds from his description that the same applies in the UK), helps to explain why so many people get put off the subject.

  3. Jackie says:

    This is probably not relevant, but reading this took me back to my own struggles with the subject. I always thought I was rubbish at maths until a new teacher, Mr.Stacey, took over the class and took my frustration at not “getting it” and requests for an explanation seriously. Within a year I achieved a B grade at GCSE (this from a starting point of a projected E). A good teacher can make all the difference. I just needed to have things explained differently and he had the patience & knowledge to try several different ways of teaching the same concept. I will always be grateful to him for being willing to give up some of his “free” time to help me.

    Sadly I did not retain my knowledge of maths past leaving school, as none of the jobs I have done have required it, but I am no longer scared of it as I know from experience that with help, a bit of imagination and a lot of determination I can do it if I really want to.

  4. Caroline Kovacs says:

    Thank you for a very interesting & informative piece. I agree whole heartedly with you. I am a trainee maths teacher who went back to uni (to study maths) in her 40’s having discovered a love of maths. My own daughter took GCSEs last year. She hopes to read Biochemistry or chemistry at uni (her obsession is poisons and toxicologyd). She really struggled with GCSE (got a B) an maintains that she dislikes it, but is taking A level maths as she knows that she needs more than GCSE understanding to enable her to succeed at the degree she wants. I am very surprised that your institution does not require maths at A level for all courses. Many of the Mathematics is the grammar of science.
    On another note: why do all children feel that to do the Further Maths A level you need to be so much cleverer than for A level maths? The units are all modularised at present and hence no great leaps in understanding are required. In my opinion nothing beats Further Pure 1 which is the first taste that the average student gets of the true beauty and wonder of mathematics – you meet imaginary numbers and matrices for the first time.

    • Stephen says:

      The lack of a requirement for maths in the life sciences is a patly pragmatic decision (though A level maths is certainly required for more numerate disciplines such as physics or computer science).

      I’m chewing over my position in the light of all the interesting comments that have appeared on this thread. Clearly, the idea of always linking the maths techniques to the relevant bit of biology or biochemistry is a powerful motivator for many. However, purists would like students just to revel in the creativity of the subject. I can see both sides…

  5. Eva says:

    “Many students who do biology or chemistry at A level don’t take maths.”


    I don’t want to sound completely negative about the British education system, because I’ve met so many smart people that it’s obvious that *something* is being done right, but the fact that A levels are only in SO FEW subjects is ABSURD.

    I don’t know how it’s now in Holland, because it changed recently (but I can ask my sister, who went through the new system), but when I graduated high school, the requirements for university entry were at least SEVEN courses at the equivalent of A-level. I did eight. I only recently learned (through watching Skins, actually) that British students spend the last few years of school doing only a few subjects, and it’s completely baffling to me.

    Of those seven, Dutch was absolutely required, as well as one foreign language (usually English). Maths was not required, but in my entire graduating class only one person didn’t do maths at all. There were two types of maths to choose from: one with lots of statistics, for kids who went into economics or social sciences, and one with lots of logarithms and exponentials and graphs, for the science geeks. Because you had to take 7 courses, everyone had quite a broad background, and universities could make some assumptions: They were only allowed to ask for two required subjects for new students, and my chemistry department required maths and physics. They knew that anyone with an interest in chemistry would have already done that in high school anyway. So they knew that all students in the program had taken maths, physics, chemistry, Dutch, English, and two other courses up to the highest level in high school.

    In the UK, because you only do three, a department can’t set maths/physics/chemistry as requirements, because you’re stuck with a much smaller pool of applicants. You’d lose the biology/chemistry/physics students.

    Ideally, anyone with an interest in sciences at university would have at least *wanted* to take maths/physic/chemistry/biology (all four!) at A levels (A levels level? “the highest level”), but with the current system that isn’t even encouraged!

    • Stephen says:

      Well, the business about the specialisation imposed on 16 year olds by the structure of the A level curriculum should probably be the topic for another post. I agree with you that it’s a shame students in the UK drop so many subjects in their final 2 years.

      Reform has been discussed many times but the system seems entrenched. I think the universities are reluctant to see a broadening which would have to be at the expense of the depth of students’ understanding of the subjects they study.

  6. cromercrox says:

    ‘any mathematical function can be represented as a sum of sines and cosines’

    Wow!!! Never knew that.

    I agree with you, Stephen, from my own experience. I went to study a degree in the biological sciences (went in to do Zoology with Biochemistry at Leeds, eventually came out with Zoology and Genetics instead). I did Biology, Chemistry and Maths at A-level. I chose Maths over Physics because the Physics teacher, although inspirational as a teacher, never turned up to his classes on time. Once at University I found my Maths training excellent. Because I’d done some calculus I just got Michaelis-Menten kinetics in a way that my Physics-A-level compadres just didn’t.

    On the whole, though, maths was a struggle. I scraped a E at A-level, so re-took the second year and managed the C I needed to matriculate. The problem was that I loved maths, but it didn’t love me. Crox Minor (13) experiences the same frustration, though Crox Minima appears to have more of a flair for it.

    As to the abstraction issue – I always liked that part the best (I loathed Applied Maths) – though I can see others find it offputting. My sister was a hopeless case at Maths at school but has recently got an adult-education diploma in something called something like ‘Everyday Maths’, which teaches maths through concrete everyday issues, such as working out your change in shops, paying pills and so on – things to which most people can relate more easily than the sum of sines and cosines.

  7. Eva says:

    (P.S. My school encouraged eight subjects, even though the requirement was seven. I realize “I did eight” sounded like showing off, but it was the default for some schools. We had to make an effort to drop the eighth course, and I could never decide if Economics of German was my “extra”, so I kept them both. It was just laziness.)

  8. Jenny Koenig says:

    I think it’s a complex mix of attitudes and culture. The same Nuffield Foundation report that you mention points out that the UK is one of few countries not to have compulsory study of maths at age 16-18 and I think this sends out the message that it’s not so important. I think we need to also look at the maths content of A level biology and chemistry and the maths qualifications of A level biology and chemistry teachers. You could argue that it’s more important for students to see the maths embedded within the other science subjects. (In fact in the USA they’ve looked into this in a lot of detail:

    I’m currently running a survey to ask academics in bioscience the question: “Is there a mathematical skills shortage amongst bioscience graduates? Closing date is 30th April 2011

    The link for the survey is here:

    Please take part! (Further info at

    But going back to your post – does the fact that you don’t require A level maths for biochemistry and biology send out the message that maths isn’t important? And does it mean that it limits the maths content of the bioscience degree if a large proportion of students who are entering the degree haven’t done calculus, logs etc?

    • Stephen says:

      Thanks for those links Jenny – I’ll check them out properly once I’ve got through my backlog of comments. I promise I’ll do the survey (which I see should only take 10 mins so others might be able to help out too – hint, hint).

      To address your question, I suspect the lack of a maths requirement does send out a message and, yes, it does limit the scope of the material that you can teach on some courses. Although most of our students in fact do have maths, a significant fraction do not and we have to make sure to cater for them. So, for example, it’s difficult for me to give a thorough-going mathematical treatment of crystallography to our final year students. Some would argue that we can make up the deficit by incorporating more maths into our own curriculum and perhaps this is the way to go. But we would have to devote a serious amount of time to it to make up for the lack of two years of A level training.

  9. rpg says:

    I echo Alice’s experience. I loved maths at school, and was naturally good at it. Was offered the chance to take the O Level a year early, but turned it down because I wanted to enjoy myself a bit more.

    Then had two completely awful maths teachers who put me right off it. Finally, a term before the O Level we got an amazing teacher, but I was pretty much fed up by then. Stormed the exam, of course.

    When it came to A levels, I did Phys, Chem, Biology. I insisted on the Biology (it was a different school) because my O Level school had refused to let me take 3 sciences, saying I didn’t need O biology and could do it straight at A level. So I did (after an argument with my new headmaster!)

    In retrospect, I would have loved to have taken maths as well, because I enjoy it. Strangely enough, when I got to Oxford (Biochemistry) I was bored in the cell biology courses in the first year, because I knew the biology already, but my best Prelim paper was Physical Chemistry (and it was one of the highest scores in the year), which naturally is math-heavy.

    I’d quite like to take some maths courses, actually. Would have helped with the NMR and crystallography I did, sure, but I managed anyway.

  10. ergates says:

    It frightened me to death when I did it at school in the mid 70’s-just managed to get a B at O level. But I was fascinated by physics and chemistry and went on to do a degree in biophysics (where not having A levels was a real problem to start with). But once I could see what maths could actually do it was a lot easier, and I’ve read a lot of maths books since just for fun. So my experience has been the reason why I (and lots of other people) found it so hard was the abstract and frankly dull way it was taught to me.

    The way maths is being taught now to my 7 year old son is so much more intuitive, to the point where he finds it easy to learn and maths is just another way of looking at the world. We spend a lot of time fixing my motorcycle and an old land rover (which he loves doing) so that he gets to see maths in action- such as why the different sizes of different sockets and allen keys matter, how 3 dimensional structures like bolts work, measuring stuff with micrometers etc.

    Does “TV’s favourite physicist Brain Cox” count as what my wife would call a Freudian penis?

  11. Jacky says:

    The problem with maths GCSE is that it is a one size fits all. It has to range from those who just don’t get it in their time at school to gifted 16 yr olds who can walk the exam without breaking a sweat. It is taught as a mathematical tool at GCSE with almost none of the beauty of numbers. My son is taking his GCSEs this year and he really likes maths so much so that he intends to do further maths A level and possibly take it higher. However, I can’t say I have ever seen a ‘love’ of numbers and a concept of some ‘magic’ about it that is the inspiration of true maths. If he has to wait until he does A level or even further maths A level to see that, then I am not surprised that 99% of the population cannot get past the idea that maths is a tool, hard or easy, to be struggled through and learned or not at GCSE. Maybe we should have two types of maths GCSE the ‘maths tool’ to teach basic arithmetic and skills for everyday life, and the ‘maths beauty’ GCSE for those who might just see slightly more to this maths idea. We could then set this GCSE as the minimum requirement for university level science degrees.

    • Stephen says:

      It was my impression that the GSCE exams are varied according to ability — I thought there was more than one size. But, having scanned my daughter’s exam paper I see that there’s little in the curriculum to get your teeth into. She also happens to be doing Additional maths and has loved getting to grips with differentiation and integration on that course – something to do with puzzling her way through the intellectual challenge, I suspect.

  12. Matt Pook says:

    I loved maths for years until I came up to GCSE level. While still managing an A*, I never saw any real relevance and whenever I asked my teacher ‘Why are we learning this?’ her only response was ‘So you can be an architect.’ Knowing that I didn’t want to be an architect, I gradually lost motivation. It wasn’t until I did Meteorology in my first year of uni that I really began to see the use of maths and physics in my degree. If there was an applied teaching style towards maths, then I’m sure it would be more relevant to scientists but seeing as the scope of GCSE style questions tended to be about men mowing their lawns or the chance of picking out red balls and blue balls, it seems pretty useless.

    I also know a lot of people who did statistics as part of the A-level maths course, and I never really found much scientific relevance in this. Having done basic stats (t-tests, Mann-Whitney, Spearman’s rank, chi-squared) in A-level biology and environmental science, I really dislike mentioning such vital scientific stats tests to friends with A-level maths with stats and them having no idea what on earth I’m talking about! I also now find stats quite logical having come across it at an earlier age.

    I think mostly it’s a lack of emphasising the relevance of maths to a wider range of disciplines.

    • Stephen says:

      The influence of teachers again – that’s becoming a theme. What an important role they have to play.

  13. Stephen says:

    Wow – thanks to all above for your contributions. A lot to digest there. I’ve been away from my desk all morning and am about to leave again but will endeavour to respond/summarise later today.

  14. Beki says:

    Really interesting post, and as another maths-hating scientist it struck a chord with me. I actually disagree that maths should be a requirement at A level, I’d hate to think that I couldn’t have done an amazing undergrad degree (biochemistry & genetics) that has made me love so many aspects of science. When I was at school I hated maths – I felt that I struggled (although I was actually doing well), the concepts and explanations just didn’t seem to make sense and I do think that was my teacher’s lack of ability to explain something in more than one way. She’d just keep saying the same things, I’d get immensely frustrated, go home and get my friend’s maths teacher dad to explain it properly (which is probably the only reason I did well!).
    At A level I wanted to do biology, chemistry and physics but was told that without A level maths (or lunchtime classes) I couldn’t and so I gave up a physics, a subject that I’d loved before that.
    So when I turned up at uni, by now with no enthusiasm for maths, a belief I was useless at it and a mindset that I would never understand it, the stats/number-crunching classes scared the living daylights out of me. But I got on with it, worked hard on the things I didn’t understand (with the help of friends and demonstrators) and ended up doing a structural biology PhD (EM/crystallography). Throughout that I did find the maths hard (particularly with mental calculations and confusing principles) but with a piece of paper, a calculator and a bit more time I was fine.
    I was so dead set against maths after my GCSEs I don’t know what I’d be doing now if I’d had to have maths at A level, and it makes me sad to think of all those people like me – who would fine on the course with a bit of extra graft – who would no longer have the chance to be scientists.

    • NB – the first bit of this is also relevant to Eva’s comment about kids in the UK not taking a big enough range of subjects post-16.

      Back in the distant 1970s, when almost all A level students in the UK did only THREE A levels (rather than the 4 or even 5 that now seem to be common), the private schools at least often used to implement a system in which you also did “subsidiary subjects” in the 6th form.

      If you had approx 30 lesson slots/week, and the A level subjects took 6 each, then you could do three A levels, have 6 slots / wk “private study” (i.e. homework) and also do three “subsidiary” subjects at 2 lessons/wk. In a system like that it would have been easy to do subsidiary “maths for science” lessons. Indeed, this sort of system was not THAT far off the multi-subject Baccalaureate set-ups prevalent in European countries. So for instance, doing A levels in Chemistry, Maths and German (sic) in the 6th form, I also had 2 lessons/wk each in French, Russian and Computer Programming.

      On the subject of compulsory A level maths, UK Univs long ago conceded that they need to teach science students basic maths relevant to their sciences, so I can see why Stephen’s suggestions didn’t fly. Many Univs run 1st yr units in “Data handling”, or simply in maths, for 1st yr science students. Again, this has been going on a long time; as a Chemistry BSc student at the start of the 80s (and even with A level Maths) I had to do a compulsory maths course consisting of (a term each of) calculus, statistics, and vector algebra.

    • Stephen says:

      Thanks for that counter view Beki – I think you’ve identified a weakness in my position. I certainly wouldn’t want to put people off the life sciences by insisting on maths. But I would want them to come to university with a feel for the subject and a belief that it would be a way to help them access the science more powerfully.

      • Beki says:

        And looking back I agree that I did feel a bit like a fish out of water! But overall, I think the idea of choosing what I thought was horrible, boring, difficult maths was overridden by the chance to choose subjects I really enjoyed. Something I’m getting from this whole thread is that perhaps we A) don’t have the chance to do a big enough range of subjects in the UK, and B) we need something like an AS/short course in maths for life scientists which you could even pick up in your final year.
        Thanks for a really interesting discussion, it’s something I thought about a lot through my academic career so I’m glad people are considering it carefully!

  15. As people have already alluded to, I think the reasons Maths has been on a long-term decline are various. And it IS a long-term decline – people in biosciences were bemoaning the declining standards of numeracy of science undergrads back in the 1970s. Goodness knows what those same people would say now.

    I would venture that there is a small group of people who just naturally “get” maths, and are fascinated by its abstractions. This is the group that I suspect provides university maths students and (in turn) professional mathematicians. However, there are never going to be that many of these folks, hence the reported scarcity in the UK of school maths teachers educated to degree level in maths.

    For most other people recognising the USEFULNESS of mathematics has, I think, to be a large part of motivating kids to do it, especially beyond age 15. This means real-world examples, and practise with numbers and data. You can see this very clearly in bioscience degrees, where students are taught stats repeatedly, but only really “engage” with it when they have data of their own that needs statistical analysis, notably in final year project work.

    Speaking personally, I never had any great enthusiasm for maths (or physics!) because they were the only subjects at school that I actually had to work at to do well in. I was looking forward to ditching Maths after O level but was “persuaded” to do Maths A level by my dad – who told me maths was useful for other things, like sciences and economics. I roundly cursed him a good number of times during my A level years for talking me into it, as I disliked A level maths lessons, though I passed OK by dint of a lot of practise on past exam papers. However, even my rather feeble (and nowadays greatly faded) grasp of maths and mathematical ideas has proved useful over my years in science, both in teaching and research.

    • Stephen says:

      I’m very wary of accepting the view that there are some people who are naturally able to do maths. That’s just the flip side of the “I can’t do maths” attitude. I’m unsure of my ground here but don’t people ‘get’ the things that they are really interested in? Isn’t it a matter of sparking that interest?

      That said, I’m probably guilty myself of thinking that certain people have a ‘natural’ ability for music. I’m a mass of contradictions…

      • I don’t think it’s completely (or even mostly) innate. My husband tells me that he really struggled with maths at school, because he didn’t see the point – it was all abstract concepts with no explanation of how it might fit into “the real world”. However, when he started to apprentice as a carpenter in his early 20s, he had a “WOW – this stuff is actually useful and important!” moment, and taught himself trigonometry from scratch. Once he’d mastered that, he kept going with algebra, just for interest. He still uses what he taught himself every single day, and he’s a much bigger maths geek than I’ve ever been – he doodles trig problems on napkins in restaurants, has a stack of maths text books, and was super excited to buy a state of the art scientific calculator app for his android phone! So it wasn’t innate ability that was lacking, just a failure by his teachers to put things into a context that he actually cared about.

        • Stephen says:

          Great example Cath. BTW, we’ll now want to test him when we finally get to meet in the UK… 😉

          I’d still be interested to hear of a similar case with someone who learned a musical instrument after childhood. That would really convince me to completely abandon the ‘natural talent’ trope (if it is such).

          • beckyfh says:

            Musical ability is absolutely about motivation and practice. My mum took up playing the flute when I was young and now plays in several (amateur) orchestras and groups. She loved listening to music, admired those who played and enjoys the social life that goes along with group playing. I suspect many adults could do the same, and that many children get put off by bad teaching.

  16. Jim says:

    My experience of mathematics has haunted me my entire life and was always one of my dirty little secrets through high-school. The times-tables were learnt by rote at primary school, and I managed to avoid ever learning them at that stage; I guess I was pre-occupied. This didn’t bode well for high school where the principle maths education was working your way through progressively more challenging work cards, with only a small amount of clarification by the teacher. It was the only subject I got middle-setted for, doing well in all others. I did manage to get an A at GCSE, but it was a painful experience and I didn’t do A-level maths, but did A-level chemistry and four other A-levels.

    Studying biochemistry at university was a real shocker, but I found ways to make the maths work. I discovered that I was damned good at statistics, which I’d never done before, more because they were logic agreements and data-handling than ‘maths’. I never had a problem calculating amounts, quotients and volumes in chemistry, but likely I used more cumbersome methods than a maths whizz might use, I still get the correct answer.

    These days I do calculations swiftly and easily. After a decade of actual laboratory research, I can do some fiendishly complicated calculations for my needs, but would probably still struggle with pure maths.

    I realised during my A-levels and subsequent degrees, that there is nothing I can’t turn my mind to, and do well; like foreign languages, which were also prescriptive and un-imaginatively taught at school, I later discovered an aptitude for picking them up by total immersion. it’s probably about time I stepped up and exposed myself to pure maths again, to see if the years I spent during PhD study, in which I deconstructed all the conditioned learning of my prior education and really learned to think, has availed me with new insights into the subject.

    • Jim – snap – approaching my 50th year, I still calculate dilutions/amounts/volumes exactly the way I was taught to do it in O-level chemistry classes.

    • Stephen says:

      Jim – I find it difficult to believe that anyone who can cope with chemistry (my own personal demon) could not deal with maths. 😉

      I think that the confidence you obviously accrued in your A levels and degrees is the key – it’s gives the momentum needed to take up new challenges. A lack of confidence is probably what stalls most kids at school.

  17. I don’t really subscribe to the view that maths is getting worse, at least among biologists. It has always been bad. I recall in my first year (16 or so students!) most had not done calculus, and most didn’t bother to learn the small amount needed for the first year physical chemistry class.

    Thoughout my life I’ve been involved in endless discussions about how to improve students’ abilities at maths. Time and time again it has been proposed that a good way would be to incorporate quantitative work in most of the courses that they do. Time and time agai, it doesn’t happen. The reason for that, I fear, is that most academic staff are, to vaying degrees, unable to cope with the maths themselves. It took a while to realise this, because it is very rare to hear them admit the deficiency openly. In fact it is quite dangerous to even mention the subject because you are apt to lose freinds (those ones who are afraid that you might show them up).

    This is a sad state of affairs, and I don’t know what to do about it. For some time now we have preferred people with A level maths, and that seems a very reasonable requirement to me.

    • Well, I’m more than happy to admit that I’m crap at maths, David. And I got an A in maths A level in 1979, though to this day I’m not sure how… beyond working through every question on every past paper for the previous six years in the three months before the exam.

      Not sure what one can conclude from this, other than that: (i) practise helps you do maths, even if it doesn’t come naturally to you; and (ii) crap-ness at maths is a relative thing, as are many abilities in science and elsewhere.

      • yes I think it is very much a matter of practice. I learned some because I needed it, and because I found its certainty a nice change from biology. My biggest mathematical achievement was probably finding the inverse Laplace transform for the length of a burst of single ion channel openings (1982) and much later (1998), the related distribution of the burst of openings that follow a concentration jump to zero concentration, the relationship between that defines the relationship between single molecule activity and the macroscopic time course of synaptic currents.

        Much of what I needed for this was taught to me by a real mathematician, Alan Hawkes, and lunchtime tutorials in linear algebra from Hyman Kesteleman. It is interesting that none of this would have happened if I had not met these people in the Academic Staff Common Room. That is one reason that I think the current attempts of UCL to deprive us of the common room is very misconceived See

  18. chall says:

    I love maths, something with that “solving the problem and getting an answer in the end” I think that has always made me happy. Structured and organised thinking?! (ok, maybe a bit OCD)

    That said, it was a slightly odd experience taking math at uni since I then struggled with the questions “what does a MSc in Maths do as a job after education?” and since no-one could tell me (apart from being a Professor and working with equations and making sense of the world) I ended up shifting towards the microbiology/biology field. One thing do stick in my mind though, my Professor positivly glowing when explaining that “everything in life can be summed up in ‘ln e’ the most beautiful thing in the world” 🙂 That passion for the maths still makes me happy.

    It’s really a useful tool, and sometimes I cringe when I realise how much it’s used in the “behind computer programs and algorithms” without people seemingly think about it….

  19. phayes says:

    “You have to be able to see the relevance of maths before you can rouse yourself to get stuck in.”

    Nooooooo….!!! 😉

    • Stephen says:

      Thanks for the reminder – a pdf of Lockhart’s Lament has been sitting on my computer for months. I guess I have no excuse now not to read it.

    • Stephen says:

      OK – I’ve read Lockhart’s Lament now and have immediately regretted not doing so before. What a passionate, articulate defence of mathematics for its own sake. I’ve no idea how this creative approach to maths — treating it as an art — could be introduced in the classroom but would love to see someone try.

      • Caroline says:

        As a trainee secondary school maths teacher I am not at all sure how this would work with my pupils. Student teachers tend to get yrs 7, 8 and 9 to teach and it is hard to get children all engaged to the same extent. Increasingly the GCSE course starts for many children in year 9.
        Am trying investigative tasks and more “functional maths” where the children have to decide for themselves how they are going to tackle a project. Tricky as they want more direct instructions which is precisely what you are trying to avoid.
        Referring back to previous threads, I am not “good” at maths (took first degree in languages in early ’80s) but just love the subject now. There is nothing to beat the feeling of success when you have really sweated over a problem and you finally arrive at understanding and “right answer”. If I ever get the chance I would dearly love to do MSc just for the love of it, however having just completed a maths degree and being part-way through PGCE need to earn some money & finance kids through uni for now.

        • Stephen says:

          You’re absolutely right Caroline. While there is so much to commend Lockhart’s approach, it is difficult to see how it could be incorporated into the the UK maths curriculum with a class of mixed-ability children. Even Lockhart, who I believe has gone back to teaching maths in schools, acknowledges the difficulty. I wonder how he tackles the problem.

          But I guess not everything has to be taught in a way that gives the children all the time they need to explore (can’t see this working with disruptive elements in the class anyway) — but at least some chance to explore/discover could be a powerful motivator kids willing to give it a go.

          • phayes says:

            And if we must have a fairly prescriptive curriculum it should be held to high standards. Vector calculus may be clumsy and antiquated but at least it’s not pathological…

            “It is a major scandal that orthodox methods continue to be taught at all to young statisticians, economists, biologists, and medical researchers; this has done irreparable damage in these fields for decades.” –E.T. Jaynes.

  20. Jenny Koenig says:

    The following quote comes from a fantastic article “Mathematics Is Biology’s Next Microscope, Only Better; Biology Is Mathematics’ Next Physics, Only Better.” Joel E. Cohen, PLoS Biol. 2004 December; 2(12): e439

    “Those who understand the calculus, ordinary and partial differential equations, and probability theory have a way of seeing and understanding the world, including the biological world, that is unavailable to those who do not.”

    It makes me sad to think that bioscience courses don’t get across this quite fundamental idea.

    • Stephen says:

      Thanks for pointing out that paper Jenny – will take a look now that I’ve finished Lockhart’s Lament. Here is a direct link for anyone who wants to do likewise.

  21. Tom Crick says:

    Amazingly, I encounter a similar problem for entry onto my Computing and Information Systems degree programmes — A-level Maths is not a requirement (which is the case for a number of other CS departments across the country), so you have numerous issues with students who have not encountered serious maths for at least two years. And then get frustrated when we have to cover a lot of content in a first year discrete maths module.

    Maths and computing underpins most of the science and engineering degree disciplines; when I did my undergraduate degree, I had numerous friends studying physics and engineering who were horrified at the amount of maths and computing they had to do in the first year. But it shows that mathematical abstraction, problem-solving, deeper analytical skills and especially computational thinking are key parts of all of the sciences.

    I just hope that the National Curriculum Review in England reflects this…

    • Stephen says:

      My goodness, Tom, I am astonished at that. I know that at Imperial the COmputer dept demands A level maths of entrants and recommends that they also have Further Maths — they find that students equipped with Further maths cope best with the course. Interestingly, they’re pretty relaxed about whether the students have Computer Science at A level…

  22. I was the only person in my school who took both biology and maths, and I was one of only two who took biology and chemistry. (Although given that only seven of us took biology and five took chemistry, perhaps that’s not so surprising. I think there were around 20 who took maths). My peers on my biochemistry and genetics course at undergrad had a mix of background, and I have to say that the (very few) people who hadn’t taken chemistry struggled much more than those who hadn’t taken maths – but that was way back in the descriptive days of biology 🙂

    I used to love maths, especially algebra – it was so logical compared to other “messier” subjects. I still enjoy fiddling around with budgets and other numbers as a nice break from the other messy stuff. And of course as soon as I saw the GCSE problem you included in your post, I had to do it (and I did indeed prove that 9×2 – 17x -85 = 0. QED!)

  23. Svetlana Pertsovich says:

    Very interesting subject.

    And what is that picture in your post? Is it the example of mathematical task, which your children at 16 are solving in the secondary school? Indeed?! Oh, it is disaster, if it is so… 🙁 Your Ministry of Education must be beaten and dissipated.

    I remember that we were solving much more complicated tasks in our secondary school in USSR. We had trigonometry, exponentials, logarithms, limits and simple differential/integral equations. Though I can’t say that our school program was ideal. For example, in those times some idiot in our Ministry of Education decided that we had not to learn combinatorics. I don’t know – why… However, probably differential equations and trigonometry were included in the school program “to replace” it 😉

    In any case you have frightened me by this example, because today our parents terribly complain that the education in secondary school became worse than in the past. So I tried to learn what our children are solving today in our secondary school. Here is the examples of the math tasks from different sources:

    Our children of 15-16 y.o (i.e. 10th year of education in 11-year secondary school) have these tasks. I think that these tasks are like our old ones… Though our Ministry yet deserves a harsh criticism, because it has made a lot of other silliness to ruin our secondary education (e.g. religious education are being planted in the schools!) and our bureaucrats in Ministry are terrible too.

  24. Stephen says:

    Wow – I have been bowled over by the huge response from so many people to this post. I have tried to respond to as many as possible. If I haven’t replied individually, it’s only that I couldn’t think of anything beyond the platitudinous or banal to say – but I am really grateful for your contributions.

    A couple of themes have emerged – the huge importance of teachers and the desire to link maths to science, so that its relevance is more immediately apparent.

    Although phayes railed against this utilitarian view — and I can see where he’s coming from — I think our views are not so very far apart. Like Lockhart, he would rather see maths taught as an art and a joy in itself. I agree but don’t think that is inconsistent with wanting students to experience the joy when they discover how maths is connected to some previously mysterious aspect of biology or biochemistry (e.g. my own epiphany with Fourier theory and diffraction).

    • phayes says:

      Oh you’re absolutely right there Stephen: one of the things I most adore about maths is its relevance and utility, and only a very foolish mathematician would declare that his or her ostensibly obscure intellectual ‘inventions’ could have no practical application:

      “There seems to be no part of (so-called) pure mathematics that is not in immediate danger of being applied.” —M. Hazewinkel.

      And I didn’t and wouldn’t actually rail against the utilitarian view in the same way Lockhart does: I think I’d probably have to write a long essay myself to explain my position clearly, but some of what I’d advocate in maths education would likely best be described as more utilitarian than what is done now.

      • Stephen says:

        Thanks for expanding on your original comment and apologies for my mis-calculated extrapolation of your views. 😉

        What has really struck me about the response to this post is the number and breadth of the people weighing in with opinions. both here and on Twitter. That says to me that there is a widespread appreciation of maths, despite the frustration experienced by many in their experiences at school. It seems that there is a well-spring of interest sitting there to be tapped.

  25. Stephen says:

    I’ve no idea what a GCSE looks like. I had 2 semesters of calculus before college. I really needed at least 3 more semesters, through Elementary Differential Equations to make it through engineering. I actually had 6 more semesters – Partial Differential Equations, statistics, and something with matracies. I’ve made my living as a computer programmer since. I can’t say i needed even calculus for that even once. But i digress.

    Partial Differential Equations is Rocket Science. The rocket starts at rest, becomes less massive as it burns fuel, drag goes up with speed but down with altitude, where does the rocket end up? This was, by far, the most fun i’ve ever had with math. By contrast, i can’t think of the slightest relevance of Green’s Theorem. In fact, most of the math i’ve learned has been taught with little relevance.

    When i was in high school, my uncle gave us a Japanese abacus and two books. With it, i learned to do 20 digit divides as mental arithmetic (takes about 4 minutes). In 35 minutes, i computed the sin(37.2 degrees) using Taylor series expansion to 11 significant digits as mental arithmetic. It took about 3 months to learn the technique. At the time, i thought anyone could do it. I’m less sure of that now, but certain that the techniques could help anyone. The technique leads to reliability of calculation. Calculation becomes mechanical, and with practice, can become essentially error free. This leads to a reduction of the fear of math. This fear stems from the fear of failure. Starting a problem is starting an investment. What if it’s a waste of effort? Even with calculators, one has to know what you’re doing – and that means knowing how to do it. Once the arithmetic is totally mastered, much of the rest of math comes easier.

    I started writing up a new way to teach this on my blog. My idea is to reduce the barrier to entry as low as possible. Use your fingers to get started. Use a small number of rules instead of a zillion cases. Speed isn’t as important as knowing what you’re doing.

    In general, we need to change the way we do education. Not just of math, but everything. If my idea is a good one, that doesn’t mean it should be taught to everyone. It means it should be tested in a small pilot. One classroom. If it works, test it in one school. If that works, test it in a region. These tests should including effectiveness and costs. But under no circumstances should these tests be evaluated by a political process. Otherwise, we’ll end up with the most incredibly stupid ideas on top. In the USA, that would be “No Child Left Behind”.

    • Stephen says:

      Going out on a limb here (memory a bit foggy) but I think Green’s theorem is pertinent in electromagnetism and electrostatics. But isn’t it cool that there’s a relationship between what happens around the boundary of a plane and what’s going on within it? 😉

      FWIW I agree that the approach to teaching has to change to get kids more interested in the curriculum (not just in maths). In the UK part of the problem is the rigidity of the national curriculum and the pressure on schools from league tables to ‘teach the exam’. But that is just the view of a parent (and husband of a teacher). I would be very interested to hear more views from teachers on this.

      • phayes says:

        Yes – EM and any field theory, and thermodynamics and… probably anywhere else a diff. manifold appears. I think the clumsy Gibbsian vector calculus still being taught instead of a modern geometric approach might be a good example of excessive utilitarianism.

      • Stephen says:

        I had E & M physics. No mention of Green’s. Well, 2nd semester physics usually comes before 3rd (or was it 4th) semester calculus.

        I took thermodynamics and differential equations at the same time. This was a mistake. One really needs differential equations to grok thermo. I ended up re-taking thermo. Much easier when not trying to guess the math. I should have talked to my adviser about my schedule earlier. And either the course catalog didn’t mention it, or i missed the dependency.

        As a parent, i’ve come to the conclusion that education wasn’t very good, and has gotten much worse. The “No Child Left Behind” system gets called “No Child Left Untested”. The theory goes that teacher’s degrees, teacher testing, and teacher seniority do not reflect in student’s test scores, and are therefore a poor measurement for teacher competence. But why then are tests considered a good measure of student competence? Multiple choice tests are cheap, but tend to measure mostly test taking skills (process of elimination) and regurgitation of (mostly useless) facts rather than competence. In fact, the best measure of competence anywhere is peer review. And the question that a reviewer should ask is “is this someone i would hang out with”. It’s all counter intuitive, but the reviewer should not attempt to test the peer. There’s no way to institutionalize this. My engineering school emphasized projects. They at least talked about teaching students how to learn. Mistakes were made, but it’s still the best approach i’ve seen. I’ve heard from educators who’ve thought about it advocate the project approach in secondary school. At the very least, subjects should be taught together. The write up of astronomy observations is also language arts. Improve the quality of the report and get double credit. This encourages quality, while creating time to allow it to happen.

        I have half a mind to change careers to education. Oddly, i’m not qualified.

  26. Matt Cliff says:

    I had to coach my wife, an ex-professional musician, through her Numeracy skill’s test in order to qualify for primary school teacher status. I managed to persuade her that she could do it by simply reading the questions slowly and not panicking, which turned out to be true. She now specialises in maths teaching, more or less repeating this advice. My experience of bioscience under- and post- graduates is that they often have the same problem. One-to-one tuition is required to get over the fear, but once they get it, they get it. To exclude people without A-level maths from biosciences courses may actually remove many able mathematicians from the subject area. I think teaching proper maths as part of a bioscience degree course is a good idea, as I have never learnt matrix algebra and coordinate geometry properly, which for a structural biologist/NMR spectroscopist is a definite handicap.

    • Stephen says:

      That’s a telling story and echoes what some others have also said. These examples — of people over-coming a fear of maths with the right guidance, encouragement and instruction — go some way to weaken my insistence on A level maths as a university entry requirement (though that would have to be matched by an insertion of enthusiastic maths teaching in the university curriculum). Ideally, maths at school should be a more postive experience for more people.

  27. At the bottom end of our ability range, I recall trying to lead a student step by step through the problem of solving for x the equation p = x/(x+k). After much puzzling, we reached an intermediate stage: I asked ‘can you think what we might do next?’ After some thought the answer came back ‘multiply both sides by one?’

    That’s pretty dispiriting.

    I also remember asking a postdoc applicant, who had finished a PhD measuring the exponential decay of synaptic currents, what the equation was for exponential decay. They had no idea. the computer gives you the time constants. I didn’t offer the job, but soon afterwords the candidate got a lectureship in a Russell Group university. I guess that illustrates the problems.

    For many years we had a remedial course in the first year of physiology and pharmacology degrees, to revise basic physics and algebra. It was called course B14, i think. After it had been running for some years, student feedback recorded that B14 was a bit difficult so they wanted a remedial course before taking it

    Incidentally, if anyone wants to learn matrix algebra in a week, you might try our summer school,

    • Stephen says:

      The fun never stops as UCL! 😉

      Seriously, a summer school looks like a great format for getting people over their hang-ups with the subject. Intense, small-group tuition, with the space and time to be a bit ‘exploratory’ sounds ideal.

    • Liz says:

      I’m a third year undergrad geneticist, and I was always into maths at school. I took Additional Maths in my GCSE year (basically because there were a few of us who would otherwise have been bored), and got an A in A-level maths. My university makes all first year scientists take a maths course, depending on their existing level of maths. I took the “maths for biologists” course (oh, and I didn’t do Biology at A-level…), having been persuaded down from the maths course that most physical scientists took. It turned out to be mostly stats, with a bit of modelling. On the plus side, it convinced me that stats were interesting, since stats at school had been counting the number of letters in words in different newspapers… ugh. On the down side, I have never studied matrices, and I can’t even remember most of the stats. I’m planning to teach myself more maths and refresh my stats knowledge over the summer, because I suspect my fourth year course (systems biology) will require them!

      There was also a first year course for those without A-level maths, but very few people took it. Then again, it is Cambridge, so probably a higher percentage of students have Maths A-level than at most universities. I hear they’re planning to bring in a second year course version of maths for biologists – sadly too late for me, but I do think biology is starting to require more maths, especially in the area I hope to go in to, and I’m sure there are more biologists out there who are interested and regret not having the chance to do more maths! Actually, I think the comments on this post show that there definitely are.

      As someone who has always been interested in maths, I don’t really have any suggestions for getting more students interested, other than to not make them count hundreds of words in newspapers for coursework! More world-relevant coursework would require more creative approaches from teachers, I suspect, but would probably help.

    • Liz says:

      Argh, I’m sorry, I didn’t mean to reply to this comment! I was originally going to say that the matrices course sounds brilliant, but then it kinda expanded, and I thought I’d reply to the post itself instead. Clearly failed, oh dear.

      • Stephen says:

        Thanks for your comment(s) Liz. I’m afraid I don’t think I can change their position but not to worry.

        It is certainly interesting to hear that Cambridge is now introducing a 2nd year maths course for biologists (though isn’t everyone still doing ‘Natural Sciences’ at that stage? I’m not so familiar with the Cambridge system). I think that’s the way the universities will have to go — hopefully keeping the maths relevant all the way.

        • Liz says:

          Well, I don’t know when it will be introduced. It’s more of a rumour! I’m guessing it would be billed as a continuation of the first year course and so would mostly be taken by those interested in biology, as the first year course is now. There’s quite a distinction between “phys” and “bio” from the start, actually, including a fair amount of good-natured competition!

  28. Emily says:

    ” A-level maths” is very different to “maths with a reason for doing maths”.

    A-level maths is about practice, very little more. When maths is put into context by your own research, wanting to apply stats, modelling, whatever – its a different ball game – you learn it because you need to. But more importantly you understand the reason for doing it. A-level maths may well be a vague marker for general intelligence – but it is far from indicating ability for research.

    Research is about hard work, linking concepts and creativity…. not cramming relatively short-term memory with rote learning and unrealistically confined examples.

  29. Comments from a mathematician were requested from twitter, so here are my thoughts. As you point out mathematics is abstract, that is where its power comes from. It is also what makes it hard to teach. The abstraction can make things seem irrelevant until you find the particular concrete setting that makes the answers relevant to you. Even being shown how the theory can be used in certain situations might not be helpful if those situations are not of interest.

    From my own experience I can tell a story that is very similar to many of those above. The only difference is that for me the subject was not “maths” but “algebraic number theory” a particular branch. I studied it as an undergraduate and it seemed to be asking questions that I did not find particularly relevant. During my PhD however I started to realise how deeply it was tied into the geometric questions I loved. With this new motivation I was able to dig deeper and get entranced by the way ideas were raised and problems solved. I am still not particularly well versed in the area, but now I can use it and appreciate its beauty.

    • Stephen says:

      I think if, from the very start, you can’t see the point of a particular maths technique, it is difficult to whip up the motivation to master it. However, I wonder if a little success goes a long way? I would suppose that demonstration of some utility — enough to facilitate mastery — would imbue the student with sufficient confidence and experience to take on trust any subsequent maths teaching, even if the applicability were not immediately apparent. Who knows, they might even develop a taste for the subject?

      • In teaching and engagment with maths I often use the phrase “corrupting people into mathematics” and agree that success can be essential. In part because you need to undertsand something of the mathematics before you know how it can be used. We need a variety of motivations. Puzzles for those who like puzzles, designing building and construction for other, sceince for others.

        Dan Meyer does a great job of finding ideas and making resources that draw students into a question that it takes maths and imagination to solve:

  30. Jon Turney says:

    As one who took a degree in biochemistry at UCL decades ago, I don’t recall drawing heavily on my maths A-level in my studies. But it does strike me as unlikely anyone would get far in biology these days without some grasp of maths.
    My main comment, though, is that it is intriguing to see how the reliance on, yet ambivalence toward maths plays out in different disciplines. One daughter, who has steered clear of science, took a degree in maths, largely because aptitude fostered enjoyment. She is now following that with a Masters in social science. She is amused/irritated by the fact that even in the quantitative methods classes the general approach is a semi-apologetic assurance that no real mathematical understanding is required. The outcome, I suspect, is students who can crank out an analysis using methods they have learnt to apply by rote, but with no appreciation of how the statistical techniques involved actually work, hence none of their strengths and weaknesses. I’d guess much the same happens in medical school…

    • Stephen says:

      That’s an interesting example, Jon. And of course, such students may go on to be teachers/lecturers and so the feel for the subject is lost.

      I suspect many university lecturers in science are apologetic about any maths content of their courses for fear of upsetting/mystifying students with naught but a GCSE in the subject. I always try to be re-assuring (but it is the tragedy of the human condition that nobody can feel anybody else’s pain…)

  31. Erika Cule says:

    I took the degree in question (Biochemistry, 2005-2008) and took my A-levels in 2003 so my experience might be a bit out of date – the syllabus of both courses has changed since I took my exams.

    Among my peers on the Biochemistry degree, most had A-level maths. Is there a difference in the distribution of grades of students with/without A-level maths when they graduate? I do not recall a difference but I nor do I remember exactly who had which A-levels. There was maths content to the degree when I took it and as it was of the applied variety most students took the “Austin” approach (practise, practise, practise) and managed fine despite using concepts that they were coming to for the first time.

    If there is no difference (in grades), then the challenge comes when you want to introduce more mathematics to the course (as is increasingly appropriate in life sciences) and have to spend some more time covering the mathematics needed with students who have never seen it before.

    Making A-level maths a requirement for admission would, I think, narrow the type of applicant, and I think this would be a shame. Because UK students only take three or four subjects at A-level, and chemistry is (rightly) a requirement, requiring maths leaves only one or two free choices. It would be a brave student who applied to study for a life sciences degree without A-level biology (although I know some do and that biology is not required), and then you are looking, really, for applicants with biology, maths, and chemistry for A-level. Not only is this quite an unusual combination (see Cath’s comment – although this is exactly what I took), surely you do not want only students with these three A-levels? It leaves no room to study for a creative subject, nor for a language (required if you want a year abroad) or anything else you might fancy.

    One option would be to make AS-level maths a requirement. I think it is in the first half of the A-level syllabus that, for example, calculus, logarithms and exponents, and basic statistics, are met. And I think A-level physics would be a realistic alternative as I think these topics are met there?

    • Stephen says:

      There’s a lot of substance to that view — this comment thread has been a great learning experience for me.

      Insistence on AS level maths (taken after 1 year of study at age 17) would be a useful intermediate. Of course, if university biol/biochem etc depts were going to go down this road, it would only be fair to give schools reasonable advance warning. But the linkage of maths with *all* the sciences in this way, might help elevate the worth of the subject in students’ minds.

      Inspirational teaching is a different matter and would also need to be addressed.

      • Stephen says:

        Or perhaps, you improve secondary school math, demonstrate somehow that the students have really improved, and then make use of it in higher ed. What if it’s impossible? Then no amount of warning helps. Only those students who were going to get it no matter how badly you taught it will continue to get it.

    • It would be a brave student who applied to study for a life sciences degree without A-level biology (although I know some do and that biology is not required), and then you are looking, really, for applicants with biology, maths, and chemistry for A-level.”

      A teensy bit off topic, but this one interested me. What Erika says (students almost all have A-level Biology, while A-level Chemistry is compulsory) is true for our life science students intake, and mostly for our medical student intake too. Conversely, also in accord with her comments, far fewer have either Physics or Maths A-level., and almost none have both.

      But…. ’twas not always thus. Back in the late 70s it was a given that to study Medicine in England you had to take A-levels in Maths, Physics and Chemistry. Biology was not regarded as necessary, or even particularly useful. This, I think, reflects in part the basis of “pre mol biol era” medicine in physiology, a subject which, in its classical form, was/is very much the application of physical and mathematical principles to understanding biological processes. (See also some of David Colquhoun’s comments on the thread)

      Nowadays, of course, a lot of our teaching to the medical students consists of trying to explain the physical analogies and quantitative aspects that are useful for understanding the body’s processes. Students that naturally think like this stand out, because they are now the exception rather than the rule. This was brought home to me a couple of years back when one medical student – who I had identified as having an unusually well-developed ‘scientific’ way of looking at things – told me he was worried because he “hadn’t done Biology A-level” and “thought all the others were well ahead of him”. He had, of course, done Maths and Physics.

      • rpg says:

        Biology certainly wasn’t required for Biochemistry in Oxford back in the day. Hence my comment about snoozing through a third of my first year lectures. It was definitely a chemistry course, although I didn’t know that (didn’t really think of Oxford) when I chose my A Levels. Had I known, I’d have fought to take 4 A-levels* (as it was I took the Bio and Chem S levels), especially seeing as Biology A is piss-easy anyhoo.

        (*But then it was a crappy state comprehensive with an intake of 6-fingered retards. I could probably have taught the courses myself)

      • Stephen says:

        Thanks – that’s a useful historical perspective Austin. I’m in the middle of reading Denis Noble’s Music of Life and I suspect he would concur.

  32. Karura says:

    Hi Stephen, I’m at Imperial too (although I’m just a lowly PhD student), and although I love maths and physics now, I have to agree that at school the teacher makes a big difference- alongside some very good, inspirational teachers, I had a couple who made both subjects seem so obscure and unnecessarily difficult that it would be hard for anyone to view them as anything more than a chore. Similarly, my sister, who is now doing her GCSEs, has been lucky enough to have a great maths teacher and loves the subject, but she doesn’t get on at all with her science teachers and has consequently taken a major dislike to physics.

    On the other side of the fence, I have tutored Materials undergrads in maths, and let’s just say they weren’t the most able and motivated of groups. They don’t seem to realise they need the maths until it’s too late.

    • Stephen says:

      I would appear that history repeats itself in many maths lessons… 😉 The problem of the students not seeing the point seems so widespread.

    • Stephen says:

      Oh, and I meant to add: there is nothing lowly about being a PhD student! It is a fine and honourable thing. You should be proud.

  33. Duncan Hull says:

    Interesting post, I’ve got Physics, Chemistry and Biology and A-level but maybe this as unusual as Maths, Biology and Chemistry? Isn’t the trouble with requiring maths for undergraduate biology is (laudible though it maybe), you’ll prevent lots of people from applying. At the University of Manchester they used to run a course called “Biomathematics” which was compulsary for undergraduates without A-level. It was thinly disguised statistics that many people hated, but this seems like a better solution?

    Anyway, didn’t Hollywood make a film about mathematically illiterate biologists? It was called Numb and Number I think? Or as Chris puts it Biologists “are the drummers of the science world”, leaving Chemists and Physicists as the rock stars, singers, song-writers and guitarists… heh!

    We are not worthy 🙂

    • I’m pretty sure the “Biomathematics” course you mention is still with us, Duncan, though it is now called “Data Handling” in order to make it more obvious why it is useful..!

    • Stephen says:

      Numb and Number – ha, ha! – and like the drummer metaphor, even if it is a bit demeaning…

  34. I took A Level Biology, Chemistry, Maths and Further Maths because I thought I wanted to study either Biology/Biomedical sciences or Maths at university. In the end I chose Biology/Genetics, and my degree course at UCL stipulated Maths and Chemistry as essential requirements. Biology was optional.

    Although I wouldn’t say I’ve ever used a tremendous amount of what I learnt in those maths A Levels, I think continuing quantitative training post 16 is of high importance if you want to undertake further scientific study, even if you don’t excel at it. A friend of mine studied Geography and Biology at a university that didn’t require post GCSE maths. Although she had struggled with it and got a D in Maths at A Level, she found that she was in a far better position than most of her classmates, and things finally ‘fell into place’, while her fellow students were learning things from scratch.

    These days I’m in epidemiology, so my interests in numbers and living things have found a pleasant harmony. I’m glad I had that foundation in maths, as it’s given me the confidence to not shy away from the quantitative challenges or be one those people (who also exist in research as well as the general population) who claim “I don’t do numbers.”

    Someone asked why students perceive that you need to be so much smarter to do A Level Further Maths. In my day (10 years ago) I think there was definitely a step up in difficulty. Half of my Further Maths class didn’t finish the course. I definitely found it a lot tougher, and I was only one of two of us to get an A grade in it. I think also you have to really love it to spend twice as much of your study time on it.

    • Stephen says:

      Thanks for the comment. In common with what some others have said, I think a little facility at maths can take you a long way, especially if you have enough confidence in the tank to persevere.

      On the question of further maths, I agree it is a step up in conceptual complexity. My own my recollection of the subject is a bit murky. But my son, who is taking maths and further maths this year, poor chap, has confirmed for me that, while he finds the maths pretty straightforward, some of the stuff in FM has been pretty challenging. Which is more or less as it should be I think.

      • On the plus side, doing Further Maths makes the maths course seem comparatively easy (although selection bias etc). But it certainly consolidates the maths syllabus.

      • Erika Cule says:

        It works both ways – there does exist a (perhaps outdated?) perception that biologists “can’t do maths”. When we had to take a course either in humanities or a course in the business school in the second year of my degree, I went for a course in the business school. In the first lecture, the lecturer asked the class

        Does anyone here not know calculus?

        …silence from the students…

        Okay then, do we have any biologists here?


        I don’t think that the lecturer’s comment was massively helpful.

    • I realise no-one’s probably reading the comments now but just popping by to say I’ve realised I mis-remembered. Maths was not an essential requirement for my degree. However, the stipulation was ‘Chemistry plus biology or maths or physics’ which suggested that maths was considered as useful for a bio degree as biology was.

      • Stephen says:

        Thanks for the clarification. I’m hoping to do a follow-up piece to this so it’s good to have the facts straight in the comments.

  35. Adam Rutherford says:

    I did Maths A-level (along with physics and chemistry; yes, no biology, despite career in biology). Absence of biology was not a hindrance to doing genetics, and I gravitated towards population genetics, which is a mathematical subject. Statistical analysis of large data sets is also essential in experimental models of population genetics, but I was too much of a whippersnapper whe doing research to fully understand the nuances of the processes that my superiors selected. That was Pom, who is head of biology at UCL now.

    In recruiting PhD students at ICH (UCL) we insisted on them doing some simple molar calculations on how to make various solutions (on the grounds that this was much of the donkey work of a genetics PhD). Many strong candidates could not do it.

    • Stephen says:

      Heh – I remember being shown that calculation on what may well have been the first day of my biophysics PhD (I only have O levels in Biology and Chemistry). I think if I’d been asked it at interview I might have flunked!

      Of biology and chemistry the one I feel the lack of these days (now that I have to teach biochem) is definitely the chemistry. It so much more conceptual and I haven’t made the time to learn enough to get a feel for the subject. I’m pretty sure I’d be up to it intellectually – it’s just that I no longer have the time. That’s why I’d rather students were able to retain more subjects at 16-18.

  36. We were having this same discussion when I was studying for my PhD in a chemistry department around 1992!

    I think there’s scope for being more accepting of “pragmatic maths and numerical methods”. Packages like Mathematica make algebra much easier, what you need to learn is how to use them and how to be confident in the answers they give.

    Allowing your colleagues in the maths department to teach maths to your undergraduates normally doesn’t work out too well.

    • Stephen says:

      1992, eh? I suspect this topic has been rumbling even further back than that.

      Never used Mathematica, but if it gives answers without understanding, it’s probably not what I’m looking for in the education system.

      And am completely with you on the futility of getting the maths dept to teach bio/biochem undergrads. We tried that before and it’s didn’t work out well.

  37. Steve Caplan says:

    I had always liked and been good at math in high school in Canada, taking all the most advanced courses, some calculus and linear algebra, and even a class in statistics. I was confident. My problem came when in university in Israel, I suddenly realized that despite all my advanced courses, I was way behind the average Israeli student.

    To compound matters, although math, computer programming, statistics, and of course physics and inorganic, organic and physical chem were all major components of the biology program, the studies were not coordinated. So our physical chemistry professor, shocked to realize that we had not yet completed the integral calculus at the level he needed for his own course, just bulldozed ahead leaving everyone to their own devices to figure out the equations. Although I eventually survived, this was traumatic for me, and certainly not a good way to keep biologists on good terms with their mathematical abilities.

    • Stephen says:

      Thanks for the tip – won’t be moving my family to Israel anytime soon… 😉 But you raise an important point about integration/co-ordination of the syllabus across the degree program. Boring work maybe, but makes life a lot more interesting for the students.

  38. Rivka says:

    I love maths and think I’m good at it but I didn’t do A-level mostly because I went to a rubbish school with a tiny sixth form and they wouldn’t put it on unless at least 2 people wanted to to do it – I was the only taker.

    I have used maths throughout my scientific career thus far and I’m married to a mathematician. While I fully agree that a knowledge of maths is beneficial to biological scientists I’m reluctant to push for the formal requirement of an A-level because I think it runs the danger of excluding those from disadvantaged/clueless backgrounds.

    • Stephen says:

      Thanks for the comment Rivka – I’m hoping you didn’t have to sit any kind of exam before getting married…

      And I am inclining to your way of thinking, which chimes with what some others have said, that too draconian an insistence on maths as an entry requirement would exclude some promising students.

      Though encouraging them to keep maths to AS level might be a healthy compromise…

  39. Stephen says:

    I did a project in college that had to do with student scheduling. When i graduated, i noticed that only one in four of my class made it. A year or so later, i noticed that half of the school population were 1st years. There are two things that might account for this. Almost every degree program had at least one course that seemed to be geared towards weeding out weaker students. Totally unnecessary, IMO, as engineering is hard enough as it is. In Mechanical Engineering, it was “stress”, taught over two semesters. It should have been at least three, just to give time for the homework. One way to get through it is to take fewer other courses. But i wanted some Electrical Engineering background. It turned out that the EE introductory course was their wash out course. I dropped it in the first week when it was clear that the homework could only be completed if all other work was dropped. The really sad thing is that, in industry, collaboration is how things get done. But how do i talk to the EE if there’s no common language?

    But i don’t think collaboration is the ultimate key. You can’t put a math-only head, a biology head and a quantum physics head in room and expect greatness. At least, i don’t think you can. Collaboration is great, but it’s not a substitute for competence.

  40. On the topic of over-specialisation at A Level, I think the increasingly popular International Baccalaureate makes all students take some kind of maths course as one of the 5/6 subjects they study. This can be at a basic or more advanced level. While a whole A level in maths might seem like too much of a prescriptive requirement, something akin to the basic level course in the IB might be a happy compromise.

    In my view, it’s the ongoing exposure to quantitative thinking and methodology that is probably most important, rather than the specific content. To stop at GCSE and then resume in a science degree strikes me as a less than ideal way to go about things.

    • Stephen says:

      I think the IB could be a useful approach. Under the current A level system, many student take an additional AS subject in year 12 (age 16-17) which is dropped for their final year. Perhaps life science depts could encourage more students to at least get to that level.

  41. Caroline says:

    I have thoroughly enjoyed all the posts and comments.

    I can see that it would be a concern for a university to retain as large a pool of prospective students as possible by not restricting application to those with A level maths. However, as a parent of a prospective student, when A level choices were made some thought was given to possible uni courses, hence my daughter is taking maths, as she did not want to miss out on a place for want of the A level. I am also worried that with the cost of a degree escalating (tuition fees will be circa £27,000) I did not want my daughter to be at a disadvantage through lack of maths skills and risk struggling on a course, when I know full well that university life will pull her in many different directions. Some of the courses that we have looked at do indeed offer specific maths courses for students without A level maths, but many require the A level. If the A level is not a requirement then adequate maths tuition & support must be given.
    Back to the Further Maths issue… The MEI A level course is now modular. A level maths = Core 1 – 4 plus two other modules most often Statistics 1 or Decision 1or Mechanics 1 or S2/D2/M2 depending on which 1st module is taken (A/S maths = C1 + C2 + S1/D1/M1 ). Further Maths = Further Pure 1 (a personal favourite) plus 5 others from a choice of around 10 modules. I admit that Differential Equations at A level would be hard (I found them tough at degree level) and need a bit of work, but overall an A/S Further Maths would not be unachievable for a student considering a maths/science university course. I strongly suspect that the A level is easier than it used to be: I speak as a mature student who took the modular A level (and FP1) in a year 2006/7.

    • Stephen says:

      Right now, I’d settle for AS Maths — rather than further maths — as an entry requirement. That would give the university a stronger base to build on.

      But I have the sneaking suspicion there’s no one solution to this problem, given the heterogeneity of students’ abilities and interests.

  42. 13loki says:

    Having been through school in Australia, I always find it strange how little is done in 6th form here. I teach science now, and my pupils are always amazed when they ask about A levels – my equivalents are Maths,Further maths, Physics, Chemistry, English, Economics and Further Economics (oh and a little-talked-about AS equivalanet drama, well, a girl needs an outlet) I did do more than necessary, but only by one subject. If they’d let me, I might have done biology, too (but I never was great at staying in the lines in colouring in).
    English was the only compulsory subject, but the next year it changed to English and Maths. Most schools arranged it so it was impossible to not do at least the AS equivalent, although we did have 5 different levels of maths to choose from. My university wouldn’t look at you for a science degree without at least 1 science and maths, and if your maths wasn’t good enough you had to do a maths elective.

    Teaching in the UK now, I don’t know how any university can deal with the general lack of maths knowledge they must encounter – our school is talking about offering (insisting on) a maths for science short course at the end of year 11 for anyone doing a science A level and not doing maths. Most of us spend a significant amount of lesson time teaching students the maths they need – they have often even learnt it in maths, but can’t seem to apply it in another subject.

    • Stephen says:

      Thanks for that perspective 13loki. Is the ‘maths for science’ course your school is pondering likely to be a full year course – like an AS?

      • 13loki says:

        The maths for science will most probably be a week at the end of GCSEs, along with one or two lessons a week at the start of year12 – up to a term. No funding, so teachers would be doing it in their frees, but hopefully it makles our teaching easier

  43. nico says:

    Wow, this a mammoth comments thread! Stephen, you obviously tickled a sensitive spot.

    I don’t have time time to read all comments unfortunately (childcare and all that), so others will probably have made my points earlier, but that comment box is irresistible!

    1. I find it extraordinary that people can state that “they don’t do math” when faced with something as simple as calculating a percentage or sharing a bill, and not be ashamed of themselves (conditions like discalculia aside). Can you imagine being handed a menu in a restaurant and saying “sorry I don’t do reading”?? This seems to me to be more accepted in the UK than in the rest of Europe (that I know of), and I am absolutely persuaded that it is in large part because of the image of maths, and because the minimum maths necessary for GCSE are very weak, and can be dropped off altogether afterwards. Back in my land of Asterix and camembert, even “literature” A-level (Bac L) students have maths all the way to graduation. If you do “science” (Bac S) with bio or phy/chem specialty, maths are weighed as heavily as your specialty subject! I know you cannot do much about what is going on in schools, but I think this is definitely the root of the problem.

    2. Related to that is the fact that maths are, on the whole, badly taught by people who do not necessarily want to or enjoy teaching maths, or by people who enjoy it too much (applicable to both the UK and France in my opinion). The former will at best be just competent enough, and the latter will not see the need to make links to “real life” maths because it is interesting to them to talk about imaginary numbers per se. This could take decades to improve on a national level, but with a well-designed curriculum is absolutely possible.

    3. About the issue that prompted your post, namely to request that students applying to Imperial have at least 1 A-level in math seems to me eminently sensible. If even the most elite universities in the country (and arguably the world) cannot make a reasonable demand that their science students are at least familiar with some basic maths, then who can? Imperial College is popular enough that you should not have a dearth of applicants following such a decision, and I can guarantee that other universities will follow suit. A friend of mine, who teaches biomechanics at a less prestigious institution, has the same problem as you. He would like to see applicants to have at least 1 A-level in math, if only to spare himself to have to teach basic algebra at university, and because an increase in mathematical fluency in the population at large can only be beneficial. The cliche that “50% of people do not know what 50% means” comes to mind.

    • Stephen says:

      Thanks for the comment Nico. I’d be interested to know if you think that the French school system, where many take maths to 18 (is it compulsory?) has an impact on students’ perception of the subject. I imagine there are still plenty who dislike it or don’t see the point but I wonder if it has more status — something that might induce more to persevere with the subject than in the UK?

      • nico says:

        Sorry about the dealy in replying, I have had to clear my desk of all the outstanding stuff at work before my holiday, and also this:

        The French attitude to maths is somewhat the same as in the UK, but it is also seen as a high status things to learn and master. Anyone studying to Baccalaureat level (~A-level, except we have 8-10) will do maths up to 18, and it is compulsory. This results in a love/hate relationship with maths. Because it is seen as a high status skill however, no one will admit to not being able to split a bill three ways. I remember my grandmother boasting to me that she knew her multiplication tables better than me, and she stopped at eq. GCSE (I was young!).

        Further, to enter any of the “Grandes Ecoles” there is an entrance exam that is heavy on fairly advanced maths, usually people go through 2 years of “classes preparatoires” before even sitting the exam. Some argue that there is too much emphasis on maths for these exams.

        My limited experience is that at undergrad level, all of my French mates were much more fluent in all areas of maths than our Scottish classmates (again, not at an insitution quite as prestigious as Imperial!). Integration was something we wouldn’t bat an eyelid at, whereas they hadn’t heard of it. Same when I taught a Top-up chemistry class to students who didn’t quite have the grades to get into uni, I had to give them a quick primer in algebra in order to explain how equations (chemical and mathematical) are balanced!

        One major issue in France is the amount of time spent at school. In the years leading to the Bac exam, I usually had weeks of around 36 hours of lessons (including about 2 hours PE), plus homework. In classes preparatoires it would be about 40 hours, no PE, and ridiculous amounts of homework.

  44. Sandy Gray says:

    This post is extremely interesting and I would agree with those that say a poor teacher inhibited them from progressing further in maths. I started out with an excellent teacher and I used to take home extra work and get support out of school hours in maths. Unfortunately he moved to another school. My next teacher was fine however there was not the same enthusiasm. However I have always enjoyed maths and we even got a taste of those aspects you mention Stephen that GCSE does not equip you with. I found them fascinating and later in life thoroughly enjoyed learning statistics for my final year degree progect. Since then I have completed an OU foundation level course in Maths and could see it applied to many aspects of everyday life. Fascinating. Subsequently I went on and used these skills in further courses in Physics and Chemistry. So to me it is seems absolutely inconceivable that students are expecting to get somewhere in the Sciences without an understanding of maths past that obtained at GSCE level. It should be a requirement. Indeed I am often hearing (I work at a University) of students struggling in Biochemistry/Physics and Biology because they cannot cope with the maths.

    • Sandy says:

      PS I mean to add that I agree with some of the others here that AS level would be acceptable

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  46. Marie Frusher says:

    When studying a biology and Chemistry joint honours there was a compulsory maths module for all chemists, if you had done A level maths you did the faster maths course (which module you needed to be on was confirmed by a maths test in the first week. It made an easy module pass for me and gave me a quick refresher of the topics that I should know from A level and put everyone at roughly the same starting point. If A level maths isn’t required then maybe an alternative should be offered such as a basic course as I believe biochemistry needs a reasonable understanding of maths and having it definately put me at an advantage in my career years after my degree.

  47. Scott Peters says:

    I was most certainly put off of doing a more physics based degree. Speaking as one who hated maths from a very early age (I failed ‘O’ level maths 5 times) It was the only O level that I failed but I was set on a career in science. I had to go through the technical side which limited my options. I did however end up doing maths and statistics involving calculus which I found surprisingly easy passing the exam easily. From then on it was degree and masters and I even started my PhD at the IOP. So I would not support a greater emphasis on maths especially as we increasingly use computers to do the stats with our only role to feed in the data from experiments. I feel that more emphasis would result in people being put off……just as I almost was.

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  50. Aaron Brown says:

    Having read through this rather long discussion, it brings back complete nightmares for me.

    I absolutely HATED maths and, unfortunately, still do to this day. I was pretty good at it through Infant school and during the earlier years of Junior school. However it was at some point during my time in Junior school (mid/latter years) that the study of maths took a nasty and sour turn for me. My remaining time in Junior school saw me developing a greater and deeper hatred of maths. The turning point for me was purely down to the method of maths teaching and delivery by a particular teacher.
    I gradually became more and more frustrated with maths as i slipped behind the more talented pupils. I was most definitely not the only one. My deepening frustration was not purely because of the fact that i was rapidly falling behind, but that this particular teacher refused to believe there was a problem with his method of delivery. My head of year suggested that he “needed to find a way to get him (me) to relate to the material”. This fell on deaf ears as my teacher continued to teach maths in his same old blinkered “one size fits all” way. By the time i drew close to completing my Secondary education i detested maths that intensely that i point blankly refused to attend maths classes. I barely scraped a C. Even then, i put that down to 80% Luck.

    What made all of this so painful was the fact that i absolutely adored and loved Biology. Everything about it. I would read Pop Science, books and journals covering biological subjects ranging from microbiology to marine biology, from genetics to ecology, until my eyes were sore. Any publications, programmes, studies or other sources of knowledge relating to Biology (microbiology in particular) i could find would be snapped up and studied intently. I will never forget reading Darwins Origin of Species for the first time. It captured the excitement that i should imagine drives all scientists. The words on the pages jumped clean out at me and captured my imagination, i read it unti it quite literally began to fall apart.
    I was 17 and adamant that i wanted to study for the appropriate A levels at college with a view to then attending University for degree study of Biology. I enrolled at Southdowns College taking A levels in Biology, Chemistry and maths. I opted for maths purely because, despite my hatred for the subject, this was reccomended by a friend of my fathers who worked as a bacteriologist. I worked extremely hard in maths at A level. The Biology was a breeze as was, for the most part, Chemistry. Why? Because i was ultimately interested in them. I failed my maths exams. Twice. My score was, in fact, worse the second time around. I fell victim to paralysis by analysis. This was sadly the final straw for me and i completely dropped out of further education, with possibly the worst hatred for maths you can imagine. I joined the British Army, 29 Commando Royal Artillery where i spent the following 12 years ultimately enjoying myself and seeing the world.

    I eventually left the military with a new passion……….International Relations.I had also studied and achieved fluency in spoken Arabic and spoken/written Mandarin, i had studied part time for A levels in Politics, Sociology and Psychology (all of which i passed convincingly and with relative ease). I applied to Portsmouth university to Study BA (Hons) International Relations and Languages, opting for German as a language and eventually graduated 4 years later (Note: 4 years, third year being a placement abroad). I have since worked for NATO, The UN and am now employed as a European Marketing Director for a leading Global Software and Services company. I am completely grateful for the direction my life eventually took. Though, even now i live with a “what if” with regards to my wish to be a biologist all those years ago.

    My opinion on maths has still not changed to this day. I can do the fundamentals, though algebra, statistics and calculus still absolutely escape me to this day. If we wish to see less pupils devloping a hatred and/or fear of mathematics the answer is simple. We must find a way to allow “ALL” students of all ages to relate to the material. I am a firm believer in one thing above all. Being “Interested” in a subject is essential. It has nothing to do with laziness. It is a simple fact that many teachers, even now, deliver maths education to a “one size fits all” standard. This breeds failure. YES dedicated and regular maths practise will improve ones grasp of it, but for a good proportion of children a lack of interest allied to an ever deepening frustration will simply lead them to close their minds to the subject.

    Now is the time to address it. Prevention is most certainly better than cure.


    A lost maths cause.

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