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In the right (or wrong) hands, bar graphs and pie charts can become powerful agents of deception, tricking you into inferring trends that don’t exist, mistaking less for more, and missing alarming facts. The best measure of a chart’s honesty is the amount of time it takes to interpret it, says Massachusetts Institute of Technology perceptual scientist Ruth Rosenholtz: “A bad chart requires more cognitive processes and more reasoning about what you’ve seen.”

Entertaining the idea of a peculiar long number such as pi puts me in mind of its cousins twice removed, prime numbers, which have so many strange properties. Prime numbers are those numbers, such as 3, 5, 7, 11, and so on, that can be cleanly divided only by one and themselves. Primes and pi suggest a benign infinity, a pleasing order—pi because of its endlessness and its relation to the circle, and primes because no matter how far you travel on the number line you will always encounter a prime, as Euclid proved in 300 B.C. Recently, Yitang Zhang solved a problem involving prime numbers called bounded gaps that had been open for more than a hundred years. Zhang proved that no matter how far you go on the number line, even to the range where the numbers would fill many books, there will, on an infinite number of occasions, be two prime numbers within seventy million places of each other. (Other mathematicians have reduced this gap to two hundred and forty six.) Before, it had not been known whether any such range applied to prime numbers, which seem to behave, especially as they get larger, as if they appear at random.

Gerd Gigerenzer's test In 2006 and 2007 Gigerenzer gave a series of statistics workshops to gynaecologists, and kicked off every session with the same question: A 50-year-old woman, no symptoms, participates in routine mammography screening. She tests positive, is alarmed, and wants to know from you whether she has breast cancer for certain or what the chances are. Apart from the screening results, you know nothing else about this woman. How many women who test positive actually have breast cancer? What is the best answer? nine in 10 eight in 10 one in 10 one in 100 Gigerenzer then supplied the doctors with data about Western women of this age. (His figures were based on US studies from the 1990s, rounded up or down for simplicity - recent stats from Britain's National Health Service are slightly different.) The probability that a woman has breast cancer is 1% ("prevalence") If a woman has breast cancer, the probability that she tests positive is 90% ("sensitivity") If a woman does not have breast cancer, the probability that she nevertheless tests positive is 9% ("false alarm rate") In one session, almost half the gynaecologists said the woman's chance of having cancer was nine in 10. Only 21% said that the figure was one in 10 - which is the correct answer.

A newly published paper suggests that personality similarity affects teachers' estimation of student achievement. That is, how much you are like your teacher contributes to his or her feelings about you — and your abilities. "Astonishingly, little is known about the formation of teacher judgments and therefore about the biases in judgments," says Tobias Rausch, an author of the study and a research scientist at the Otto-Friedrich-Universität Bamberg. "However, research tells us that teacher judgments often are not accurate."

For example, a recent study from Israel showed that teachers gave girls lower grades on math tests when they knew their gender. And lots of researchers have looked at the importance of having teachers who share the racial and socioeconomic backgrounds of their students.

Graphs can be as manipulative as words. Using tricks such as cutting axes, rescaling things, changing data from positive to negative, etc. Sometimes putting zero on the y-axis is wrong. So to be sure that you are communicating the right things, you need to evaluate the message that people are taking away. There are no absolute rules. It all depends on what you want to communicate.

The bottom line is that humans are very bad at understanding probability. Everyone finds it difficult, even I do. We just have to get better at it. We need to learn to spot when we are being manipulated.

Mathematics and art are generally viewed as very different disciplines – one devoted to abstract thought, the other to feeling. But sometimes the parallels between the two are uncanny. From Islamic tiling to the chaotic patterns of Jackson Pollock, we can see remarkable similarities between art and the Mathematical research that follows it. The two modes of thinking are not exactly the same, but, in interesting ways, often one seems to foreshadow the other. Does art sometimes spur Mathematical discovery? There’s no simple answer to this question, but in some instances it seems very likely.