Paragraphs referring to mathematics:

On top of all that are the effects of the pay policy which froze teachers salaries for three years from 2010, and recently capped rises for most teachers at 1% until 2020. As any

mathsteacher could tell you, that means pay cuts in real terms – and more disaffection, as wages in the private sector start ticking up. […]

He echoes concerns about subjects such as

mathsand the sciences feeling the pinch. “I think that’s a combination of the economy picking up, and the fact there’s just not that straightforward route into teaching,” he says. Howes runs a course for people who haven’t got degrees in physics – an A-level is the basic requirement – but want to become physics teachers. “The number of schools I’m working with who haven’t got properly trained physics teachers is massive,” he says. “You just keep coming across the fact that trainees are working with teachers who are not themselves trained in physics.” It takes me a while to process what this means, in some cases: people without physics degrees being helped to teach physics by people who don’t have physics degrees either. […]

Barber hands me a sheet of paper recording all the vacancies he has advertised over the last few years, and how many applications he got back. A job teaching art attracted 18 – because, he says, thanks to the changes pushed through by the government, “schools are now actively reducing numbers of art teachers”, and many are going spare. By contrast, the numbers of people applying for jobs teaching English, science and computing never got any higher than four – and appointing a new head of

maths, he says, was “an absolute nightmare”.

Finding heads of departments, he says, is a particular problem, what with Ofsted ready to pounce, and results in key subjects so crucial for a school’s reputation: “If you’re the head of

mathsor the head of English, you aresoaccountable: if themathsdepartment goes down, the whole school goes down.” For this job, the school got four applications, three of which were “no good”, leaving one that he says was outstanding. The applicant came from a large academy chain, which for some reason, had given her a poor reference.

“We couldn’t understand it – what we could see in front of us, and what the reference said were completely at odds with each other,” he says. He decided to employ her, and she accepted – but a few weeks before the end of term, he received a call saying she wouldn’t take the job after all. “The next thing I know, this chain was announcing its new head of

mathsin a brand-new academy,” he says. “Presumably, they offered her a better deal.” He suddenly looks pained. “I feel like the corner shop up against Tesco. These academy chains have huge resources, and lots of lots of schools. I won’t appoint anyone I don’t think is capable of doing the job. That would still be my official line. But the truth is, you do find yourself thinking: if I don’t appoint this person and I advertise again, that’s going to cost me another £3,500 when money is really tight.”

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From the people behind the Alan Turing Cryptography Competition

MathsBombe – the new maths-based competition aimed at A-level students (but open to all UK students in Year 13 (or equivalent) or below) – started this afternoon. This is the sister competition to the now well-established `Alan Turing Cryptography Competition’ but aimed at an older group of students and featuring mathematical puzzles. If you know anybody who would be interested in this then please pass this on (or if you know of any way of promoting the competition that we haven’t thought of then please let us know!). The url is: www.maths.manchester.ac.uk/mathsbombe

Students work together in teams (of between 1-4). Every 2 weeks starting from Wed 13th Jan at 4pm, a new set of puzzles will be released. Each puzzle set contains two mathematical puzzles – one (relatively!) straightforward, and one that’s perhaps a bit sneaky and perhaps not quite so obvious! The puzzles will span the whole spectrum of mathematics from fiendish logic puzzles in pure mathematics to applications of mathematics in real-world settings.

Entrants will need to keep their wits about them and `think-outside-the-box': complicated techniques from A-level or computer coding probably won’t help! There are four puzzle sets in total (so eight puzzles across the competition). Points can be earned by solving each puzzle and there is a leaderboard that keeps track of how well each team is doing.

MathsBombe is organised by the School of Mathematics at the University of Manchester. It is the sister competition to the Alan Turing Cryptography Competition.

]]>modelling camp has 3 main aims

- To train students and early career mathematical science researchers to

engage in study groups and similar activities - To offer broader skills training – team-working, coping outside of

one’s comfort zone, introduction to modelling methodology, report

writing, and enhancing communication/presentation skills - To learn how different branches of mathematics can be applied in

various industrial settings. - The meeting will be structured to maximise time for networking and

informal discussions. - This modelling camp will be held in advance of the 116th Study Group

with Industry (ESGI), University of Durham, April 2016.

Further details, including funding options, are available on the website

http://icms.org.uk/workshops/modcamp2016

Funding has been secured for a limited number of delegates so early

registration is recommended.

Association for Psychological Science

amikulak@psychologicalscience.org

Understanding fractions is a critical mathematical ability, and yet it’s one that continues to confound a lot of people well into adulthood. New research finds evidence for an innate ratio processing ability that may play a role in determining our aptitude for understanding fractions and other formal mathematical concepts.

The research is published in *Psychological Science*, a journal of the Association for Psychological Science“Our findings suggest that human beings come wired with a sort of naïve nonsymbolic ratio processing ability and that differences in these abilities have meaningful effects on the development of mathematical thinking,” explains psychological scientist Percival Matthews of the University of Wisconsin-Madison, lead researcher on the study. “This basic perceptual ability seems to have consequences even for complex mathematical thinking like algebraic reasoning, which is quite counterintuitive.”

Matthews and colleagues Mark Rose Lewis and Edward M. Hubbard, all of the University of Wisconsin-Madison, hypothesized that we must have some basic ability that allows us to reason, in an informal way, about proportions.

“The human brain existed way before we had things like math and reading, which means that however we do these things, we have to sort of recycle abilities that the brain already has,”

Matthews explains.

To investigate this hypothesis, the researchers recruited 181 college undergraduates to participate in their study. The students completed four tasks in which they had to compare ratio quantities represented by pairs of dot arrays or pairs of line segments.

Students saw a pair of dot groupings or a pair of line segments on a computer screen for 1.5 seconds, and were asked to indicate which option in the pair represented a larger quantity. In making this assessment, the students essentially had to intuitively “feel out” and compare ratios composed by dot (or line) pairs on one side versus dot (or line) pairs on the other side.

Importantly, the ratios represented by the dots and the lines varied from trial to trial.

The students completed several additional tasks that gauged their formal, or symbolic, math ability.

In one task, they were presented a paper-and-pencil assessment of general fractions knowledge. In another, they were given a fraction – such as 1/7 – and were asked to represent the fraction by clicking on the analogous position on a number line that ranged from 0 to 1. The researchers also examined students’ scores on the algebra placement test that they took when they entered university.

The results revealed an association between the students’ ability to gauge nonsymbolic ratios and their competence in symbolic math: Students who were adept at processing pictorial ratios by comparing dots and lines also tended to be good at comparing fractions and estimating fractions on a number line.

But the association also extended to more general math skills: Students who scored higher on the pictorial ratios task also tended to be good at solving algebraic equations.

“Far and away the most surprising finding was that this ratio sensitivity was correlated with performance on university entrance exams,”

says Matthews.

“It is easy to see how sensitivity for the magnitudes of what amount to nonsymbolic versions of fractions might be correlated with fractions processing performance. However, it’s another thing for it to be correlated with something as complex as algebra.”

Matthews and colleagues are hoping to find evidence-based strategies that will make teaching and learning fractions more manageable, given the difficulty that many people have in mastering fractions. These new findings, they argue, serve as an important for step in that endeavor:

“At the end of the day, we want to encourage psychologists, math education researchers, and educational practitioners to consider how actively trying to leverage these abilities might help improve fractions learning,”

says Matthews.

]]>I was always deeply uncertain about my own intellectual capacity; I thought I was unintelligent. And it is true that I was, and still am, rather slow. I need time to seize things because I always need to understand them fully. Even when I was the first to answer the teacher’s questions, I knew it was because they happened to be questions to which I already knew the answer. But if a new question arose, usually students who weren’t as good as I was answered before me. Towards the end of eleventh grade, I secretly thought of myself as stupid. I worried about this for a long time. Not only did I believe I was stupid, but I couldn’t understand the contradiction between this stupidity and my good grades. I never talked about this to anyone, but I always felt convinced that my imposture would someday be revealed: the whole world and myself would finally see that what looked like intelligence was really just an illusion. If this ever happened, apparently no one noticed it, and I’m still just as slow. When a teacher dictated something to us, I had real trouble taking notes; it’s still difficult for me to follow a seminar.

At the end of eleventh grade, I took the measure of the situation, and came to the conclusion that rapidity doesn’t have a precise relation to intelligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn’t really relevant. Naturally, it’s help to be quick, like it is to have a good memory. But it’s neither necessary or sufficient for intellectual success.The laurels I won in the Concourse General liberated me definitely from my anguish. I won first prize in Latin theme and first access it in Latin version; I was no longer merely a brilliant high school student, I acquired national fame. The Concourse General counted a lot in in my life, by helping me to get rid of a terrible complex. Of course, I was not instantly metamorphosed, and I’ve always had to confront the same problems; it’s just that since that day I know that these obstacles are not unsurmountable and that in spite of delicate and even painful moments,they will not block my way to accomplishment, which research represent for me. Fortunately, I had an excellent memory. For instance, in twelfth grade, in math, I believe that at the end of the year I remembered every single thing I had learned, without ever have written anything down. At that point, I knew my limits but I had a solid feeing of confidence in my possibility of success.

This type of competition is an excellent thing. Many young people feel self-doubt, for one reason or another. The refusal of any kind of comparison which reigns in our classrooms as a concession to egalitarianism, is all too often quite destructive; it prevents the young people who doubt their own capacities, and particularly those from modest backgrounds, from acquiring real confidence in themselves. But self-confidence is a condition of success. Of course, one must be modest, and every intellectual needs to recall this. I’m perfectly conscious of the immensity of my ignorance compared with what I know. It’s enough to meet other intellectuals to see that my knowledge is just a drop of water in an ocean. Every intellectual needs to be capable of considering himself relatively, and measuring the immensity of his ignorance. But he must also have confidence in himself and in his possibilities of succeeding, through the constant and tenacious search for truth.

[With thanks to Jonathan Crabtree]

]]>Over the last decade, many students have asked us how to get involved in research. To address this need, we are partnering with MIT PRIMES, which has trained many outstanding high school student researchers over the last several years. MIT PRIMES/AoPS CrowdMath will allow mathematically sophisticated high school students to collaborate on unsolved problems under the mentorship of outstanding mathematicians. CrowdMath begins with a series of Resources for students to discuss over the next couple of months. On March 1, we will release the official research problems, which will be based on material students learn while discussing the Resources.

Our goal is to discover new knowledge! Should we succeed, we’ll produce a research paper based on our collective work.

Visit the MIT PRIMES/AoPS CrowdMath pages for more details.

]]>I’d like to share with you that the latest English version of *Lines and Curves* is now available on the Springer website (they made thousands of their books freely available online): http://link.springer.com/book/10.1007/978-1-4757-3809-4

A later addition, 07 January 2016:

Sorry, it looks like it was a mistake on the part of Springer: only few hours our book “Lines and Curves” was free on their site. In a case that some kids want to have the book there is the link to the first English edition :

https://archive.org/details/StraightLinesAndCurves

https://archive.org/details/StraightLinesAndCurves

One can download the book for free in different formats.

]]>The announcements in English and Swedish respectively can be found at

http://www.su.se/english/about/vacancies/vacancies-new-list?rmpage=job&rmjob=832

http://www.su.se/om-oss/lediga-anst%C3%A4llningar/lediga-jobb-ny-lista?rmpage=job&rmjob=832

The closing date for applications is January 15, 2016.

]]>When I was in high school, we once had a visit of a few Chemistry students from the local university. They were visiting high schools all over the city to inspire young students to apply for university-level chemistry education. Their promo event went like this: they showed us chemicals. That was it. They mixed chemicals of various colours, and created colourful smoke, steam, and liquids in test tubes of various shapes. And then they went home.

This sort of presentation inspires non-scientists and gives the wrong idea of what science is! You should become a Chemist if you love doing Chemistry, not if you love the end product. Doing Chemistry means: these are your starting chemicals, and this is your equipment – which methods and in which order will you do them to arrive at the desired chemical end product? That is what scientists love, that is what gets you the best brains in the class; the colourful chemicals attract non-scientists.

And if you think about it, the same goes for mathematics: “there is an infinite amount of prime numbers” – may be true, may not be true, but let’s find out, let’s either prove or disprove it, we shall see for ourselves. That is exciting. The knowledge of the mathematical proof is much more exciting than the knowledge whether the statement was true or false.

And that is not what we teach, we teach answers, because answers can be graded. I sincerely believe that if the greatest mathematical minds there ever were were born today, they would be disgusted by today’s mathematical education, and would go on to pursue other fields. It is said that Gauss was a rather annoying pupil because he always finished early during the math classes, so his desperate teacher, in an attempt to keep him occupied for the rest of the class, told him to sum all numbers from 1 to 100 when he was 6 years old – he immediately came up with n*(n+1)/2. This formula is what some high school students need to make a “cheat sheet” for and hide it into their shirt sleeves during an exam, because they have not developed the skills needed to derive this formula by themselves – to them it is just a series of symbols.

So, to answer the original question, why is it so difficult to teach mathematics then? – I think the education of mathematics will not change until we find a way how to put an exam grade on mathematical creativity – which is something you can not grade. All we can grade is the “hard work” – reducing a fraction, deriving/integrating a function, all the tasks that no mathematician really enjoys, because it is calculation, not math. So, the only thing we could probably do, is to rename the high school subject from “mathematics” to “calculation”, as it used to be named just a few decades ago in the Czech Republic.

Sorry if I seem way to passionate about this subject matter. It is because I work at a photonics laboratory as a physicist, but the more I do this job, the more I regret I did not study mathematics instead. It was not my fault though – mathematics on the high school level is the most mind-numbing discipline imaginable. It was not until I went to the university when I first encountered what mathematics is about, but that was too late, I was already on Physics. As insulting to scientists as it may sound, from my experience in a modern high-tech physics lab, I notice that all sciences are nothing but subsets of mathematics. I choose the word “subset” very carefully – mathematics is without any doubt the broadest science (not a natural science though!) that defines pure reasoning, true wisdom in its purest form, that the human brain is capable of. Natural sciences, such as physics, take a part of mathematics and put it into the context of atoms. Not the other way around. It is clear that mathematics was always steps ahead than technology – it can not obviously be the other way around.

Please, feel free to disagree with me, I would be happy to be wrong here, as the modern state of math education is sadly not a happy topic. I have heard someone say that a major revolution in math education came during the Cold War, as both the Soviet Union and the US started training mathematicians as “soldiers” – it was believed it would be the mathematicians who would build the best nuclear bombs and win the war, not the infantry soldiers. That is when the strict military-like math grading was enforced. But I found no evidence to prove or disprove this explanation.

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