Alastair Cook would baulk at being described as a giant among pygmies, but as he appeared through the classroom door at Hague Primary School in Bethnal Green, and bent his lanky frame into one of the room's undersized chairs, you got a sense, perhaps, of the stature that England's Test captain now possesses.

It is impossible to imagine Joe Root and Ben Stokes, or any other member of a young and energised England dressing room, shielding their eyes at the aura of their returning leader when the Test team reconvenes at Headingley next week. Cook, for all his legendary stubbornness, is too down-to-earth to elicit such a reaction.

Nevertheless, on May 19, after one of the longest winter breaks he has ever experienced in his ten years as an England cricketer, Cook will lead his side out in the first Test against Sri Lanka, knowing that he is on the brink of an achievement that will cement his status among the mightiest run-harvesters of all time.

Ten thousand Test runs is on the agenda, a figure no Englishman has ever before approached, let alone surpassed. Cook's own mentor, Graham Gooch, came closer than any England player before him, but ran out of gas at the age of 41, with 8900 in the bank, a mark that remained unchallenged for two decades until Cook himself ticked it off, once more at Headingley, against New Zealand last May.

And now, at the age of 31, Cook embarks on the 2016 season with 9964 to his name - just 36 runs shy. Though he has become conditioned to taking milestones in his stride in the course of a prolific career, even Cook recognises there will be an extra layer of symbolism when he completes this inevitable feat.

"Milestones are a weird thing," he says, "because when you're on 4000 runs you're desperate to get 5000 and then you want 6000. But then you realise when you're there it's not quite as important as you thought before.

"But 10,000 is a massive milestone for any batsman because you can't really argue with a bloke who's scored 10,000 runs. No matter which way you've gone about it, that's a lot of runs, in a lot of different conditions and being tested for a long period of time. So it'll be a nice milestone when it comes, but I've got to get the 36 first."

As Cook assesses his merits, it's impossible to ignore a new undertone of self-awareness in his voice. It's not so much that he has ever felt a sense of entitlement about representing England - although, when you consider that the first and only Test match that he ever missed, in Mumbai in March 2006, was more than a decade ago, he's got more reason than most to think that way.

Rather, you suspect, it's a legacy of those bruising 12 months in 2014-15 when he became so embattled on so many fronts that his form was often the least of his concerns. Between the final throes of the Kevin Pietersen saga and his sorry demise as England's one-day leader, all Cook could do was make a virtue of his obduracy and ride out the stormiest year of his career as best he could.

And now, through the satisfactions of last summer's Ashes triumph and a hard-earned series win in South Africa, he has emerged on the other side of the gauntlet - redeemed, certainly; hardened, no doubt, for no challenge has ever broken his spirit, but also ever so slightly more mortal.

The simple fact is, he's the last of a titanic breed - an endurance athlete in a sport now dominated by the sprinters. Where once Cook raged against his limitations, insisting against all empirical evidence that his formidable willpower could compensate for the gaps in his one-day technique, now he is at peace with his game, and his place within it.

"When the decision was made, it hurt a lot," says Cook. "I still would have loved to captain in that World Cup, but I understand the decision, I can see now that we were a year behind. Credit to the way Eoin [Morgan] has taken the team forward, alongside Trevor [Bayliss] and Farby [Paul Farbrace], they're playing the right way."

Cook's angular, adhesive style has never seemed more at odds with the spirit of the age, with the public conversation turning ever more enthusiastically towards the sort of fearless strokeplay that carried Cook's younger cohorts to the World T20 final - and even lit up the latter stages of Wednesday's remarkable clash between Nottinghamshire and Yorkshire at Trent Bridge.

The spirit of the age even infuses the event that has brought Cook to this corner of East London. He ventures up onto the school roof with a group of Year 6 pupils to play cricket alongside his former captain, Michael Vaughan, on a specially repurposed playground with high fences all around - though not high enough, as it turns out, to prevent several T20-style slogs disappearing into the streets below.

The pair have come as ambassadors of the charity Chance to Shine and Yorkshire Tea, the England team sponsors who are launching a young commentators competition ahead of the international summer. As part of the day, both men undergo a grilling from two of the school's budding radio broadcasters, and Vaughan is particular is left speechless by the intensity of his interrogation, which ranges from his dodgy musical tastes to the insinuation that his entire career was down to luck.

However, two moments stand out in the seven-minute set-piece - when asked who was the best player he ever played alongside, Vaughan's answer "Andrew Flintoff" draws a complete blank. Moments later, the kids are back for their follow-up - who was the best player you played against? "Shane Warne ... legspinner ... you should have seen him ... never mind." It'll make for great radio, but rarely has the work of Chance to Shine, whose mission is to reintroduce cricket to Britain's state schools, seemed more urgent.

But if there is a sense, in that exchange, that Cook is playing to an emptying gallery, then that is also precisely why he can expect to be such a towering presence during England's seven-match summer campaigns against Sri Lanka and Pakistan, then onwards into a winter tour of Bangladesh and India - the scene, in 2012-13, of his most outstanding achievement as an England batsman and captain.

Far from rendering him obsolete, Cook's insistence on swimming against the sport's prevailing tides makes him a point-of-difference player that any Test team in the world would long to build their batting order upon. What, for instance, would Australia have given for a man of his durability during that crazy morning session at Trent Bridge last summer?

"It is all about appreciating what you have done, having been around for a while," he says. "If you do have a little bad run, you know you will turn the corner and you will score runs again because you've done it so many times before."

His status is borne out by a glance down the list of Test cricket's leading run scorers. A slew of retirements in recent seasons, including Sri Lanka's legends, Kumar Sangakkara (12,400) and Mahela Jayawardene (11,814) and West Indies' final link to their golden age, Shivnarine Chanderpaul (11,867) means that the top of the table is preserved in aspic, potentially never to be challenged - except by the man himself.

Younis Khan endures for now, on 9116, but at the age of 38 and with run-ins with the PCB becoming a daily occurrence, his ambitions may not extend too far beyond this summer's four Tests in England.

And further down the list, with Michael Clarke (8643) and Pietersen (8181) gone, and AB de Villiers (8074) fluttering like a pennant to the IPL's winds of change, the only other active Test cricketer with even half as many runs to his name is a man who would dearly love another crack, but may already have played his last.

Cook claims not to have spoken about England's impending selection meeting when he met up with Ian Bell for dinner in Birmingham last week. Instead, he says, the pair met as brothers in arms, friends reunited after a rare winter apart, following Bell's surprise omission from the South Africa tour.

"I hadn't seen Belly since the UAE, so I thought it made sense to catch up," Cook says. "I played a hundred and however many Test matches [106] with the guy, and Alice [Cook's wife] knows Chantal [Bell's wife] really well, and the kids. It's an amazing thing, when you're in the team together for so long and then you don't go on one tour and you don't see each other for five months. I thought it was really important to catch up with him as a friend and see how life is. It was a really nice meal."

Cook speaks of his England career with a wistfulness that reflects both the changing dynamics of his sport, and the long absences from the action that his axing from the one-day team has created. He also acknowledges - perhaps for the first time in an eternally youthful career - that the end is now nearer than the beginning.

"I wouldn't want to go back and start on nought and have to prove again I can do it," he says. "Every time now you turn up for a Test tour or series, this could be closer to one of my last ones as history suggests not many people play many more games.

"I think that's why playing one form of international cricket and having that bit more time off really makes you look forward to the occasions more and more, and the times we are spending with people like Jimmy Anderson and Stuart Broad - they won't be playing for England forever."

The end is not yet nigh - not by a long chalk. On Thursday, it was confirmed that Cook had agreed a new two-year deal with Essex, the county for whom he has once again heralded the arrival of spring in his time-honoured fashion, with a trio of hundreds against Gloucestershire, Sussex and Worcestershire.

"It's gone all right," he shrugs, as he shares the credit for his success with the new ECB edict, that the visiting team can opt to bowl first, thereby encouraging the home teams to lay on more batting-friendly tracks.

A separate ECB edict, on the other hand, briefly threatened to undermine his preparations, when - in the wake of a spate of facial injuries, including a career-ending skull fracture for Somerset's Craig Kieswetter - Cook was ordered to stop wearing his trusty old helmet with its adjustable grille, and don instead a new fixed version.

The saga looked like boiling over into a national incident when Cook fell for 1 in his first outing with his new lid, against Sussex in Hove. However, a second-innings hundred restored his equilibrium, even though it remains clear that the England captain isn't entirely enamoured with the decision.

"I understand the rules and regulations, but I find that I don't see the ball as well in the new helmets, which makes it quite ironic in one sense," he says. "Are you safer with a helmet that is meant to be safer, but you can't see the ball quite so well? It's not ideal, but we are trying to find a solution. It shouldn't be a headline anymore."

Away from such irritations, Cook's heralding of spring has followed another familiar pattern in the months since the end of the South Africa tour - lambing season on his wife's family farm in Bedfordshire, an annual ritual that, in taking his mind completely off cricket, plays as much of a part in his mental preparations for the season as any number of net sessions.

"I came home on February 7, and straight away the first lamb was born and I didn't pick up a bat again until March 5," he says. "Then I threw myself into Essex's pre-season. You train your ass off, doing all the hard stuff and putting yourself through the wringer to give yourself the best chance to score runs."

As far as his career is concerned, it may already be autumn, but Cook's England ambitions appear to be as undiminished as his appetite for runs. He claims he has no plans to relinquish the captaincy before the 2017-18 Ashes in Australia - and seeing as he will turn 34 on the eve of that year's Boxing Day Test, he ought still to be near the top of his game as he bids to avenge the whitewash tour two years ago.

But having lost the argument when it came to his one-day leadership, Cook has learnt the hard way how quickly the team dynamics can shift. In particular, he knows that the time may come, sooner rather than later, when Root - England's coming man in all formats - needs to be handed the keys to the kingdom.

"I think I'll go for the right thing for me personally," Cook says. "When the next generation have played a few more games, we might see the side moving in a different direction. It happens to a lot of captains who've been around a while. But if Rooty or whoever - but I think we know it probably will be Joe - gets the call, he's ready to do it."

Regardless of how the future maps out for Cook, it's the present that concerns him for now - seven Tests against Sri Lanka and Pakistan, the only two teams that England have failed to beat in their most recent Test series encounters.

"Nobody expected Leicester City to win anything," he says, a caveat that sports journalists are going to have to get used to hearing in the coming months.

"We're playing against two very good sides, so we're not just going to rock up and win. We've got to go out and produce the goods and people have got to stand up. We did win in South Africa and we played well, but I still don't think we played to our potential and this side can still get better and better."