With a push outside off stump, Alastair Cook's series with the bat was done. His tally of 23 runs in four innings was the leanest series of his career.

Before this series he spoke about coming through dark times in Australia with the double-century in Melbourne. It was his second double in 12 innings which is impressive. The problem has been what has come either side of those huge innings: outside of those doubles, he has made 216 runs in 16 innings. Shortly before his 243 against West Indies, his 88 against South Africa at The Oval was an innings of absolute class in demanding conditions. But is one major innings per series enough?

"It is an easy story to write when a slightly older player isn't scoring runs," Cook said a couple of weeks ago. "Is he going to give up? Is he thinking about it? Phil Mickelson is to me a prime example… I'm not saying I'm anywhere near Phil Mickelson's league, but he hadn't won for five years [when he won the Mexico Championship in March]. But people weren't doubting his hunger or desire to play, just because he wasn't quite getting the results.

"So it's amazing how a little period in a cricketer's life can get transformed into a lot bigger than it is. If you hang about in Test cricket long enough, there are going to be little periods when you play really well and little periods or slightly longer periods when you don't play well."

This is not a time to write him off. He has proved doubters wrong at numerous times in his stellar career. But neither should the issue be ignored. The start of the English season, and a spell with defending County Champions Essex, is shaping as important with runs needed to stop a debate igniting.

In this series he has been outdone by Trent Boult bowling at his best, so there is no disgrace in that, but he has looked out of nick in New Zealand. Perhaps that was caused by the far-from-ideal lead-in period which consisted of a muddled four days in Hamilton and nets. The early signs at Seddon Park were that he was actually hitting the ball nicely, but Boult has proved a demanding prospect.

The second-innings flick down the leg side in Auckland can be put a little down to bad luck, but in the other three innings he has been unpicked by classy swing: edging to the keeper/cordon twice and losing off stump to a beauty in Christchurch.

There were hints when he began his innings on Sunday that there was positive intent. There was a good stride into a delivery from Tim Southee as he drove through the covers and after lunch he pulled the same bowler strongly through square leg, then in the same over flashed over the slips.

A curious aside in Cook's career against New Zealand has been the difference in facing Southee and Boult. Southee has never dismissed him 11 matches; when Boult found the outside edge it was the ninth time he had removed the left hander in nine Tests.

The challenge of a left-armer will follow Cook in the first part of the English season with Mohammad Amir set to lead Pakistan's attack in the two-Test series. Amir has spoken of managing his Test workload, but surely the England series won't be one to make way. He has removed Cook five times in eight matches. India will also bring an impressive seam attack with them - arguably their best ever - and both touring sides also likely to be among the better-prepared visitors.

At the start of the English summer it will be 18 months since Cook gave up the captaincy. He has spoken about how he is now finding his place in Joe Root's team, knowing when to offer advice without compromising the current captain, but for a player who invested so much in leading his country it will have been a significant change. There is the carrot of another crack at Australia on home soil in 2019, but after 153 Tests only Cook will really know how much more he has to give.