With the unique seasons and long periods of darkness in the Winter and long periods of daylight in the Summer how did the Inuit and other Native Americans of the Arctic keep track of time? A solar calendar might not be effective in Summer given that the Sun never sets and a lunar calendar wouldn't be effective either given that during the Summer the moon might not be visible. How did they know when to sleep in the Summer and when to wake up in the Winter?
"Traditional" Inuit conceptions of time are very different from modern linear time and there's quite a bit of literature going over it that someone who's actually specialized in Inuit anthropology would be better placed to cover. Macdonald's Arctic Sky^1 is the classical text that goes into a fair bit of detail on how some Inuit groups conceived of time and how they related it to place, but it's very difficult to find outside an academic library.
Personally, I like how Stern (2003)^2 puts it:
Prior to contact with Europeans at least some Inuit accurately reckoned seasons using lunar phases and chronological time from the positions of certain stars. Names of months were usually associated with natural phenomena or seasonal subsistence related events (such as the spawning of fish or the calving of caribou), and thus varied considerably by locale and latitude. The indigenous calendrical system enabled Inuit to count days and lunar months, and to accurately predict a number of annual astronomical events such as the winter solstice and the return of the sun above the horizon.
Essentially, there were many different ecological calendars for each individual region and group. The ecological year would begin with the Winter Solstice and proceed through 6 seasons that corresponded to ice and other environmental conditions. Months would subdivide the seasons and new months began with each new moon. There was also a leap month occasionally added to keep things in line with the solar year.
However, I'd also like to point out that it's perfectly possible to keep solar/lunar time even above the Arctic Circle. While the Sun doesn't technically cross the horizon during Summer or Winter, the light in the sky is far from constant. Here's a picture of Winter Polar Noon in Russia for an example. However possible it is to maintain linear time there, the local traditional calendars better reflected expected weather and resource changes for traditional communities throughout the year.
[1] MacDonald, J., & Schledermann, P. (1998). The Arctic sky: Inuit astronomy, star lore & legend. Arctic, 51(4), 394.
[2] Stern, P. (2003). Upside-down and backwards: Time discipline in a Canadian Inuit town. Anthropologica, 147-161.
