A Few everyday objects demand as many compromises as the umbrella. It must be light enough to carry for hours yet rigid enough to survive a sudden gust; it must open wide enough to shelter its user and yet fold small enough to be forgotten in a bag. Its canopy has to shed water without tearing, and the whole device has to be cheap enough that losing one is no disaster. That these demands pull in opposite directions explains why the umbrella, for all its apparent simplicity, has been reinvented so many times. Its earliest forms solved none of these problems, because they were never intended to. Carved reliefs from ancient Assyria and Egypt show attendants holding broad canopies over the heads of kings, and in China and India the number of tiers on a parasol could signal the rank of the person beneath it. Shade and ceremony, not shelter from storms, were its original purpose.

B The first significant step towards protection from rain is generally credited to Chinese craftsmen, who stretched paper, often made from mulberry bark, over a framework of bamboo and then brushed it with oil or lacquer until it repelled water. Bamboo was an inspired choice: it is light, springy and, unlike most woods, inclined to bend rather than snap. Oiled-paper umbrellas of this kind spread to Korea and Japan, where they remain a traditional craft, although the hours of handwork each requires now confine them largely to festivals and the tourist trade. In Europe, by contrast, the parasol long remained a fashion accessory for the wealthy. When travellers in the seventeenth century brought back descriptions of its use in wet climates, the waterproof versions that resulted were made from oiled silk or cotton stretched over ribs of cane or whalebone, a material taken from the mouths of baleen whales.

C Technical progress mattered little, however, while social attitudes stood in the way. In eighteenth-century England, the umbrella was widely regarded as an effeminate or even a foreign affectation, and a gentleman caught in the rain was expected to hire a sedan chair or a carriage. Jonas Hanway, a merchant and philanthropist who had travelled in Persia, is usually credited with breaking this taboo by carrying one through the streets of London for some thirty years from the 1750s. He was jeered at, and the drivers of hackney coaches, who rightly saw the device as a threat to their earnings on wet days, are said to have tried to run him down. By the time of his death in 1786, however, the habit had spread so widely that 'a Hanway' had become a common nickname for the object itself.

D Even once it had been accepted, the umbrella remained heavy and unreliable. Whalebone was expensive, absorbed moisture and eventually split, and a large example was considerably heavier than its modern equivalent. The decisive improvement came in 1852 from Samuel Fox, a Yorkshire manufacturer whose firm drew steel wire for products that included the stays of women's corsets. Reputedly looking for a fresh outlet for his surplus stock, Fox developed what he called the 'Paragon' frame, in which each rib was a strip of steel rolled into a U-shaped cross-section. The curved profile gave the rib great rigidity for very little weight, much as a sheet of paper folded along its length resists bending far better than a flat one. Within a few decades, steel frames had driven whalebone from the market, and the principle behind Fox's design survives in many of the umbrellas sold today.

E The next problem to be tackled was size. A full-length umbrella is awkward on a crowded train, and in the early twentieth century a number of inventors experimented with ways of making the shaft and ribs collapse. The most influential solution came from Hans Haupt, who had been injured in the First World War and who, walking with a stick, found a conventional umbrella difficult to manage. In 1928 he applied for a patent on a design with a telescopic shaft and hinged ribs that folded down to a fraction of the usual length. It was later sold under the brand name Knirps, a word that German speakers still use for any compact umbrella. The gain in portability came at a price, however: every additional joint is a point of weakness, and compact models remain more prone to failure than their rigid predecessors.

F The greatest remaining enemy is wind. A conventional canopy behaves rather like a sail; when a gust catches it from below, the ribs are forced backwards and the umbrella turns inside out. Designers have attacked this problem in two broad ways. One is to let the air escape, through a vented double canopy whose overlapping layers lift to release pressure. The other is to reshape the canopy itself: in 2006, three engineering students at Delft University of Technology in the Netherlands launched an asymmetric umbrella which, like a weathervane, swings round to face the oncoming gust and is claimed to withstand winds of around 100 kilometres an hour. Such designs cost considerably more than the mass-produced models that dominate the market, and this points to a wider difficulty. Cheap umbrellas, assembled from steel, plastic and fabric that cannot easily be separated, are rarely mended and almost never recycled, so that vast numbers are thought to be thrown away after a single season. For an object that has been refined over three thousand years, durability has proved the hardest quality of all to achieve.

A A colony of ants routinely makes choices that would tax a well-organised human committee: where to forage, how many workers to commit to each task and, when disaster strikes, where to live. Remarkably, nobody is in charge. Despite her title, the queen issues no orders; her role is purely reproductive. Each worker responds only to what it can detect in its immediate surroundings, chiefly chemical signals and brief contacts with nestmates. Biologists have spent four decades explaining how such local information can add up to decisions that are, for the most part, remarkably sound.

B Some of the earliest answers came from laboratory experiments on the Argentine ant. In the late 1980s the Belgian biologist Jean-Louis Deneubourg and his colleagues connected a nest to a food source by a bridge that split into two branches of unequal length. Within minutes, most of the traffic was using the shorter branch. The explanation lies in pheromone, a chemical that foragers deposit as they walk. Ants taking the shorter route complete the round trip sooner, so pheromone accumulates more quickly along it, attracting further ants, which deposit still more — a process known as positive feedback. The same mechanism, however, has an obvious weakness. When the researchers offered the shorter branch only after the colony had already settled on the longer one, most ants carried on using the established route: a well-marked trail, once laid, is difficult to abandon.

C Yet colonies must also adjust their effort as conditions change. For more than thirty years, Deborah Gordon of Stanford University has studied red harvester ants in the deserts of the south-western United States. She found that the rate at which foragers leave the nest depends on the rate at which successful foragers return: when seeds are plentiful, ants come back quickly, and their arrival stimulates others to set out. Working with the computer scientist Balaji Prabhakar, Gordon noted that this resembles the method by which the internet adjusts the speed of data transmission to the capacity available. Her long-term records have also shown that, in dry years, the more cautious colonies — those that keep more foragers at home and so lose less water — go on to found more daughter colonies.

D The most detailed studies of collective choice, though, concern house-hunting in Temnothorax, a genus of tiny ants that nest in rock crevices or hollow acorns. When their home is damaged, a minority of scouts set out to inspect possible replacements, judging cavities by such qualities as their darkness, internal area and the width of the entrance. A scout that finds a promising site returns and leads a single nestmate there in what is known as a tandem run, the follower keeping in contact by tapping the leader with its antennae. Nigel Franks and Tom Richardson of the University of Bristol argued in 2006 that this qualifies as teaching, because the leader slows its pace whenever the follower falls behind. Stephen Pratt, now at Arizona State University, showed what happens next. Once the number of ants at a new site exceeds a threshold, or quorum, scouts abandon tandem runs and instead pick up nestmates and carry them, a method roughly three times as fast.

E Crucially, the quorum is not fixed. In experiments led by Franks, colonies forced to emigrate under harsh conditions lowered their threshold, reaching a decision faster but more often settling on a poorer site, whereas colonies in more favourable circumstances took longer and chose more reliably. Pratt's group has also compared colonies with individuals. In one study, adding a plainly inferior third option altered the preferences of individual ants tested alone, but did not mislead whole colonies. In another, led by Takao Sasaki, colonies outperformed solitary ants when two nests differed only slightly in darkness, although the advantage disappeared when the difference was large. The reason seems to be that relatively few ants ever compare the alternatives directly; most visit a single site, and the comparison is effectively carried out by the colony as a whole.

F Army ants provide a more dramatic illustration. Species of Eciton in Central and South America hunt in vast raiding columns and, where the forest floor is uneven, workers link their own bodies to form living bridges across gaps. In a 2015 study in Panama, Chris Reid, Simon Garnier and their colleagues inserted an adjustable V-shaped obstacle into raiding trails. The bridges that formed began near the point of the V and then gradually migrated away from it, so that the route became shorter. Yet they consistently halted well before reaching the position that would have provided the most direct path. The researchers proposed an economic explanation: every ant held in a bridge is unavailable for carrying prey, and beyond a certain point the distance saved no longer justifies the labour withdrawn from the raid. No individual calculates this balance; it emerges from each ant's tendency to join a busy bridge and to leave a quiet one.

G The findings have reached well beyond biology. In 1992 the Italian computer scientist Marco Dorigo described an algorithm in which virtual ants lay virtual pheromone on the routes of a network, and he first tested it on the travelling salesman problem. Variants of this 'ant colony optimisation' have since been applied to scheduling, vehicle routing and telecommunications traffic. Biologists caution, however, against treating colonies as flawless problem-solvers. Ants that cling to an outdated trail or settle too hastily on a poor nest are reminders that collective intelligence is tuned to the conditions in which a species evolved.

A Few educational practices are as universal, or as fiercely contested, as homework. Parents complain that it devours family evenings; teachers defend it as the bridge between the classroom and independent study; and politicians occasionally intervene, as when the French president François Hollande proposed in 2012 that it be abolished and replaced by study in school, on the grounds that it favoured children whose parents could help them. What is striking about this quarrel is how little of it engages with the considerable body of research that now exists. That research does not settle every question, but it is clearer than either side tends to admit, and it points towards a conclusion that ought to disappoint both those who would abolish homework and those who equate a heavy workload with high standards.

B The most frequently cited synthesis is the work of Harris Cooper of Duke University, whose team reviewed American studies conducted between 1987 and 2003. Their conclusion was that the link between homework and achievement depends heavily on age. Among students in the final years of secondary school, those who did more homework generally performed better, and the association was substantial. Among children in primary school, by contrast, it was close to zero. Cooper has popularised a rule of thumb of roughly ten minutes a night for each year of schooling, so that a pupil in the second year might do twenty minutes and one in the final year around two hours. The guideline is sensible as far as it goes, but it is a heuristic derived from correlations, not a dose established by experiment.

C That caveat matters more than is usually acknowledged. Most homework studies are correlational: they record how long students report working and compare this with their results. Such data cannot reveal the direction of influence. A conscientious pupil may both do more homework and earn higher marks for reasons that have little to do with the homework itself; conversely, a pupil who is struggling may take far longer over the same task, which would make extra time appear harmful. Research by Ulrich Trautwein and his colleagues in Germany illustrates the point. When they separated the frequency with which teachers set homework from the time individual students spent on it, they found that regular assignments were associated with better results, whereas the sheer number of minutes a student logged was not, and was sometimes linked with lower attainment. Effort and concentration, it seems, count for more than duration.

D International data reinforce this impression. An analysis published by the OECD in 2014, drawing on the PISA assessments of fifteen-year-olds, found that students who spent more time on homework tended to score higher, but that beyond about four hours a week the additional benefit became negligible. It also found that students from socio-economically advantaged homes did more homework than their disadvantaged peers, and that this gap was associated with differences in performance. Some of the highest-scoring education systems assign relatively little homework — Finland is the example most often quoted — which suggests that what happens during lessons matters far more than whatever is added afterwards.

E Opponents of homework have seized on findings of this kind, and some of their concerns are well founded. A 2013 survey of more than 4,000 pupils at high-achieving secondary schools in California, carried out by Mollie Galloway, Jerusha Conner and Denise Pope of Stanford University, found that they averaged over three hours of homework a night, and that many reported stress, lack of sleep and physical complaints such as headaches. Writers such as Alfie Kohn have gone further, arguing that the research offers no justification for homework at all in the primary years and that its chief effect is to erode children's enthusiasm for learning. Yet to move from 'too much homework causes harm' to 'homework should be scrapped' is a non sequitur. The Californian sample was drawn from unusually affluent and competitive communities, and it would be rash to generalise its findings to schools where the more common problem is that pupils do very little work outside class.

F The more productive question, then, is not whether homework should exist but what form it should take. Here the research offers useful, if tentative, guidance. Tasks that ask pupils to retrieve and practise material already taught, such as short quizzes or spaced revision, appear to be more effective than assignments that introduce new content, which tend to leave pupils dependent on whatever help happens to be available at home. Feedback matters too: work that is set but never discussed sends the message that it is a ritual rather than a means of learning. In my view, these considerations argue for less homework in the early years, and for homework of better quality throughout, rather than for any single prescribed quantity.

G The equity problem identified by the OECD deserves particular attention, because it is the strongest argument the abolitionists possess. If homework rewards children who have quiet rooms, reliable internet access and well-educated parents, then each additional assignment may widen the very gaps that schools are meant to narrow. Abolition, however, would not remove the advantage; affluent families would simply buy private tuition instead. A better remedy is for schools to offer supervised study sessions on their own premises, where every pupil has access to the same resources and the same adult support. Homework, in short, is not a moral question but a practical one: its value depends on the age of the learner and on the care with which it is designed.