My derm used liquid nitrogen on age spots on my arm and hand 5 days ago. Are they supposed to heal this way? (photo)

What is referred to as age spots can be many different lesions and will require different treatments. Dermatologist often use liquid nitrogen to treat lesions by freezing them. Depending on the depth of tissue destruction there could be a whitish scar or a darker scar. The appearance is normal at this point, but once the areas are healed you should definitely cover them with a sun block to avoid hyper pigmentation.  

It looks like an appropriate treatment was performed. This liquid nitrogen type of treatment can take a few weeks to fade.In my practice I use an intense pulse light or fraxel treatment for age and brown spots.   There are many different brands of intense post light treatments that are effective. In particular, I use a Cutera and a viora platform, as well as a 1550 and 1927. Best of luck, and thanks for the question. 

Sometimes in response to thermal injury (hot or cold), the skin can react in this way. For any of those spots where the skin has broken down, I would recommend applying a topical ointment such as Bacitracin or Aquaphor until the skin has healed over. This may take a week or so. At that time, these areas will assume a more pink quality. To speed the resolution of the pink color, a topical steroid (hydrocortisone vs. triamcinolone) can help. I would recommend following up with your dermatologist so he or she can monitor the wound healing and address any prolonged pigmentation problems, should they occur. Good luck!

Liquid nitrogen is an option which works by damaging the top layer of skin. This causes the skin to peel off and will require a minimum of 1-2 weeks to heal and time for the color to resolve. If the color persists a laser which targets dark color such as a q switched yag or picosecond laser can remove the residual spots.

The liquid nitrogen caused blistering, which is it why it is used sometimes to treat age spots. Depending on the size and thickness of your spots, other options may have been used too.  In general, thick age spots (seborrheic keratoses) can also be electrodessicated ("burned") and scraped off, called curettage.  I find this heals well.  Thinner lesions may be amenable to laser treatment, such as with the Yag laser.  Apply Vaseline or Aquaphor to lesions and keep them out of the sun to promote healing.  Be well.