In years to come, when people look at the scorecards from this series, they will presume that England's bowlers had no answer to New Zealand's resolute batters.

And there's more than a bit of truth in that. Certainly there were long passages of play, not least when they were bowling at BJ Watling, when England's bowlers were made to look pretty toothless. Like sparrows pecking at a tank; like ducklings attacking a tiger.

Statistician Andy Zaltzman worked out that this series represented England's worst collective strike rate in a series in their Test history: they averaged 115.7 balls per wicket. Their collective bowling average was, at 55.8 runs per wicket, their second-worst in Test history following the drubbing in the 1989 Ashes when it was 57.7. They're ugly figures.

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So there's no denying that England need to improve their potency with the Kookaburra ball. And they need to improve their potency and their plans on flat wickets. They need to be honest about the level of assistance their bowlers receive in England and decide if they are content to be home-track bullies or if they want to give themselves a chance to compete abroad consistently. They have currently won just one - and lost five - of their most recent seven series away from home. The next two English winters include tours of India (2020-21) and Australia (2021-22).

But alongside the improvements England need to make to their batting and bowling, they also have to acknowledged they have a problem with their fielding. For their bowlers in this series have been let down by a series of dropped chances which have damaged their figures and undermined the side's hopes of success.

The final day of the series provided a perfect example. Kane Williamson, of all people, was dropped twice. The first chance, when he had 39, came off a glove down the leg side and the second, when he had 62, came when he spooned the ball gently to midwicket. If the first, offered to keeper Ollie Pope, should have been taken 95 times out of 100, the second, offered to Joe Denly, should have been taken every time. Truly, it was as easy a chance as you will ever see put down in international cricket. Jofra Archer, the bowler, shrieked in anguish.

The Denly drop might, to some extent, be dismissed as an aberration. He has proved an outstanding fielder in his England career to date - the catch he took at Lord's to dismiss Tim Paine off Archer during the Ashes will linger long in the memory - and he looked distraught at this error. Moments later, thrown the ball to shine it between deliveries, he dropped it again.

The Pope drop was, perhaps, more avoidable. Selecting such an inexperienced player - Pope had kept in just five first-class games ahead of this Test, the least by any England keeper in more than a century - as reserve keeper on a Test tour was always going to be a risk. While he had, to that point, performed pretty well, the purpose of selecting an experienced, polished performer is to achieve consistency. Even if we accept the logic that Jonny Bairstow required a break to refresh himself and work on his game, Ben Foakes was available and should have been selected.

Ollie Pope put down a straightforward chance off Kane Williamson AFP / Getty Images

England's continued failure to take advantage of such a good player as Foakes - and a player with a good record in the limited international opportunities he has enjoyed - is puzzling. Equally, it might be worth reflecting if their use of Pope - asked to bat out of position during his first spell in the team and asked to keep in his second - is really giving him the best chance to flourish. It might be best not to take too many risk with such a talent.

It was poor reward for Ben Stokes' efforts, too. Stokes, who has been suffering from knee pain during this game, delivered a sustained spell of something approaching leg theory with six men on the leg side and a third man on the boundary almost behind the keeper. To have expended such effort and generated such lift from a docile surface was an admirable achievement. It deserved better reward.

The same might be said for the Denly drop. On that occasion, Archer had produced an outrageous piece of skill to deceive Williamson. Archer changed his grip even as he gathered to deliver and, instead of bowling a conventional seamer, delivered a knuckleball. He also didn't use his front arm in delivery.

To have the skill to do that, to produce an error from Williamson on such an unresponsive surface, and to see such a chance go down must have been galling. And it does have to be taken into account when reflecting on his figures for the series - he finished with two wickets at a cost of 104.50 apiece and a strike rate of 246 - as a whole. Yes, the England bowlers struggled. But their fielders hardly gave them the best opportunity to succeed.

It would be wrong to blame a last-day lack of concentration, too. While that might have been relevant, it has been a problem that has afflicted England throughout the tour. Tom Latham, the first-innings centurion, was dropped by Stokes on 66 - again off Archer - while in Mount Maunganui, Watling was dropped, again by Stokes, at slip when he had 31. He went on to score 205.

Let's be clear: the best side won this series and, even if both those Williamson chances had been taken on the final day in Hamilton, it was too late to force a win. It rained from around 2pm. But the point remains: for an attack that struggles as much as this to create chances in such conditions, it becomes even more important that the standard of catching improves.

"It's frustrating for the bowlers," Root admitted afterwards. "They worked very hard to take chances on a very flat wicket. It's very frustrating.

"Catching is something you want to pride yourself on as a team and we want to get better at. We work very hard in practice and know that in other circumstances those drops could potentially cost us a chance of a winning a game or even losing it. I'm sure Joe won't hear the end of that one for a while.

"It just proves that when you do get your chances, you've got to be right on it and you've got to take them because good batting sides will make you pay if you don't."

Joe Denly reacts after dropping Kane Williamson Gareth Copley/Getty Images

The good news, from an England perspective, was the encouraging update provided by Root over Stokes' knee. But it did seem odd that Stokes, with the injury cloud hanging over him, bowled more overs than anyone bar Sam Curran in New Zealand's second innings and long after any hope of winning the game had disappeared.

"Hopefully it's not a long-term thing and it's something that calms down very quickly," Root said. "It can depend on if the strapping is not quite right.

"With Ben, you're always trying to make sure he's being honest with you. He'll continue to keep bowling unless you pull him off. He'll do absolutely everything for his team-mates so that sets a great example for the rest of the guys. At the same time, you don't want him to hurt himself."

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Maybe there was good news on Archer, too. He has, very clearly, not enjoyed a successful tour. But the figures do not tell the whole story. As well as arriving in New Zealand with very little experience of a red Kookaburra ball or these conditions, he also arrived with some unrealistic expectations as baggage. There were times he really struggled.

Increasingly, though, he bowled with skill, intelligence and stamina. To see him on the final day - going wide on the crease, changing his action and bowling knuckleballs, cutters and, of course, that wicked bouncer - was to see a young man who had soaked up these learning experiences and put them to good use. He didn't win much reward, it is true, but if these trials come to have value in India or Australia, well, perhaps it was time well spent.

Either way, if England's bowlers are to enjoy success in either nation, they will require far more consistent support from their fielders.