Camera phones at the National Gallery stoke fears that technology is leaving us incapable of deep engagement with anything

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How long do you need to look at a painting to really appreciate it? There are many answers to this question. As long as you like, is one. Longer than you think, is another. The art historian James Elkins wrote that it took him about 100 hours, over three years, to learn to really see a Mondrian painting. He recounts meeting a woman who had spent an hour looking at the same Rembrandt work four times a week for at least two decades – or about 3,000 hours.

Bendor Grosvenor, who runs the influential arthistorynews blog, takes his estimate from the late Kenneth Clark, the museum director who is now best known for presenting the landmark BBC documentary series Civilisation. "Clark had a good line," he says, "that the time it takes to look at a picture properly is roughly the time it takes to peel and eat an orange. I think that's about right."

There's no right answer; but there is some evidence of what most people do in practice, much of it quoted by Elkins in a 2010 essay on the Huffington Post. In summary, if museum-goers are eating oranges, they're eating them bloody fast. The Louvre says that the average visitor looks at the Mona Lisa for 15 seconds. New York's Metropolitan Museum of Art found people looking at each work for 32.5 seconds. Another study, conducted by Rutgers University, found that the median time spent on each work was 17 seconds. On the universal fruit-attention scale, this is more like the time taken to eat a grape.

There's another point, too: it's not just a question of how long we attend to a painting, but of the quality of that attention. This is, of course, much harder to quantify. But there's a common hunch afoot that over the past decade or so, this has changed radically, and for the worse.

Technology gets the blame, and you know the argument: the rise of social networks and the smartphone have made us fatally incapable of concentration, and nowadays we are more interested in telling people what we are doing than doing it.

So, we share our dinner with the internet, instead of with our date. Our novels go unfinished as we flip through finely crafted 140-character miniatures. And when we go to a gallery, we don't look at the art: we take a selfie, with it. Even if you are left cold by art, your view of this contention matters: if it is true, then so is the broader claim that the 21st century is leaving us incapable of deep engagement with anything.

Those who fear that this is true would have been distressed, last week, to hear of a couple of radical developments at one of our greatest museums. The National Gallery was introducing WiFi throughout its public spaces, it explained, so that visitors might be better able to access information about the pictures that they see, and to "interact with us more via social media". Because that development would necessitate the greater use of smartphones and tablets, it would become all but impossible for gallery attendants to keep track of who was looking up details of a painting and who was taking a picture of it. And so it would give up its status as a valiant defender of the gallery as a screen-free zone, and join many other museums in letting visitors snap away.

Not everyone is delighted about this. "On the one hand, museums should do everything they can to make themselves accepting and accessible," art historian and broadcaster Dr James Fox observes via email. "On the other hand, few things irritate me more than seeing some doofus staring at a painting through his iPad in order to take a bad photo of a picture that's been photographed much better many times before." More worrying still is the fact that "photographing artworks more often than not comes at the expense of actually looking at them." Fox quotes Robert Hughes's observation that people nowadays don't go to museums to look at artworks, but to have seen them.

The artist, writer and publisher Jasper Joffe takes a similarly trenchant view, except that to his mind the situation is so awful that a bit of photography could hardly worsen it. "I am in the final stage of grief: acceptance," he says, also via email. "Already people go around galleries in a state of anaesthetised indifference to the objects on display. Headphones on, piping in the canned thoughts of audio guides, eyes only on the wall panels telling them what to think, so what difference will the jostling of selfie-seeking Facebookers make?"

Those involved in the National Gallery's decision are not so pessimistic. Instead, they say, they are simply exercising their responsibility to make the collection as widely accessible as possible. "I spend a lot of time watching how people look at the art, and I don't think I've seen any great change in approach," says Dr Susan Foister, the gallery's director of public engagement.

"Yes, you always want people to be drawn in by a single work – but we have six million visitors a year, and probably there are six million ways of looking at the art. We think it's important to offer lots of ways in. The National Gallery has always been a public space. You have to consider that other people may not enjoy it the way you do."

Victoria Siddall, director of the Frieze Masters art fair, is similarly sanguine about the impact of photography. "People definitely use cameras more, but if it means they spend less time looking at the work I've not noticed," she says. "I feel like not to allow it is a losing battle. It's part of the culture of the way people look at things today." In any case, she adds, social networks have made people more interested in art, not less so. "Seeing all these works online just makes people want to see them in person more," she says.

That Frieze Masters now sells out annually, is she says, "in part fuelled by having 80,000 Instagram followers".

It's hard to adjudicate between these claims. All the same, I went to the National Gallery last week to see how the new rules were affecting things. I arrived just before it opened for the morning. Outside, tourists were paying human statues to get their pictures taken, and as I filed in with the queue that had been snaking down the steps it felt a little like a continuation of the same thing: free, but still somehow the product of a transactional mindset. In the first gallery, there were smartphones and tablets everywhere, many more taking pictures than using the WiFi to find more information. Instead of a gallery, the space felt like a catalogue, the visitors reconfigured as shoppers, picking something off the rack to try at home later. Today, the real permanent collection is the one we all store on the cloud.

As I sat down in front of Lorenzo Costa's luminous high-altarpiece (1505), I saw one woman, five-foot-nothing in an enormous visor and a bright yellow sweatshirt, approach with her camera about six inches in front of her nose. Her view of the work came through the screen on the camera's back. She paused, waited for someone to get out of the way, took a quick snap, and then took another of the wall text. Within an instant, she went to a Girolamo Romanino nativity scene and did exactly the same thing.

She moved into the next room, her camera so firmly in place that you might have thought it was hanging off the visor. I got up and followed her for a bit. She stopped at Moretto da Brescia's Portrait of a Young Man, Quinten Massys's Ugly Duchess, and Hans Holbein's The Ambassadors. At each one the same routine: aim, snap, wall text, snap, and on.

I found the whole thing a bit depressing: an indication of Joffe's "anaesthetised indifference". But when I approached her and asked how she was finding her visit, she broke into an enormous smile. Her name was Li Shu. "I love it here!" she said. "I've been many times. I always find something new." Was she pleased to be able to take photos? "Yes," she said. "I'll send them to my daughter. She loves art."

Her obvious pleasure left me feeling chastened, and like a complete snob; however she was looking at it, Li Shu was obviously far more serious about art than I am. As Bendor Grosvenor told me later: "If people were really only happy to look at a snap, they wouldn't be in a museum in the first place."

Outside of that first room, the visitors quickly spread out, and as they did so the urge to take pictures seemed to diminish: that acquisitive impulse from the outside world soothed, perhaps, by the gallery's silence. Nor did I see anyone take a selfie. I took some myself as I went round, feeling utterly self-conscious.Whether media confection or permanent new practice, the term and the act seemed utterly out of place here: museums are still good at quieting one's internal monologue, and hence the need to be at the centre of everything. If it is true that the modern age is one of self-obsession, they are therefore more precious than ever.

I still can't say I feel totally cheerful about the invasion of the cameraphone. And yet I don't blame the National Gallery for changing the rules. We have, as the Harvard academic and internet philosopher Lawrence Lessig puts it, shifted from a "read-only" culture to a "read/write" culture. To blame all of these changes on photography, to worry that the presence of a few cameras is the source of our inattention and self-obsession, is to confuse cause and effect.

There are lots of better ways that museums can try to encourage people to engage with art than banning the inescapable instruments of modernity. Susan Foister mentions an event at the National Gallery called Looking Without Talking, at which visitors are invited to sit quietly in front of a masterpiece for five minutes and see what happens. At Tate Liverpool, community groups are rewriting wall texts, removing the dead hand of academic language and making forbidding works of art vivid again. There are as many such ideas as there are galleries, and as our institutions adjust to an era of constant connection, there will surely be many more of them.

In the meantime, if you don't like cameras in museums, the solution is simple: don't take one. A punter with an iPhone is no more obtrusive than one with a sketchbook unless you have a chip on your shoulder.

As Li Shu said, heading off in the direction of the Impressionists, "Paintings are for everybody!"

Who, I wondered, was her favourite artist? "Van Gogh," she replied. "The Sunflowers. I can look at them for hours."