There are a lot of complaints on this forum along the lines of, "I want to play a pure destruction mage but I can't because destruction is too underpowered". I take it that a pure destruction mage is one that only uses destruction in combat, without using any of the other magic schools and without using weapons or shields, but I could be wrong. I'd like to ask the people who want to be pure destruction mages, what one is exactly?

I've fought many mages in Skyrim and there isn't a single one that I would call a 'pure destruction mage' There are lots of magic wielders and some I'd call 'pure mages', but every single one of them uses magic from other schools, or weapons.

Bandit Wizards - Destruction, Restoration.
Alik'r Mages - Destruction, Restoration
Thalmor Wizards - Destruction, Restoration, Alteration, Conjuration (Atromancy & Nectromancy, including Turn & Expel)
Draugr Warlocks - Destruction, Conjuration (Atromancy)
Penitus Occulatus Battlemages - Destruction, Restoration
Falmer Shaman - Destruction, Alteration, Conjuration (Bound Weapons) Restoration
Hagravens - Destruction, Restoration.
Witches - Destruction, Alteration, Restoration.
Forsworn Shaman - Destruction, Conjuration (Atromancy including Banish) Alteration, Restoration.
Vampires - Destruction, Illusion, Conjuration (Necromancy)
Necromancers - Destruction, Conjuration (Necromancy) Alteration, Restoration (including Turn).
Conjurors - Destruction, Conjuration (Atromancy including Expel & Command) Restoration (including Turn)

...and the one's you might think are 'pure destruction mages'...

Pyro, Cryo, and Electromancers - Destruction, Restoration, Alteration.

Ironically, the common school between all these mages is Destruction - it is a 'must have' school for mages in Skyrim. But it is never the only one.

My opinion is that it is an invalid playstyle - the game has not been made for pure destruction. It also seems a bit unintelligent. I'm actually glad that those that try this on higher difficulties get creamed, . I also think that the 'pure robed mages' out there can very happy when they come across an Arch Electromancer on master, use the right spells, apparel and tactics, and defeat them.

TL;DR - You have to mix up your magic or turn the difficulty down.

What's the fun in playing with only one school of magic?

Totally agree. Same with zero cost destruction gear. It's not fun or satisfying.

It limits you, offering a tougher challenge but your right, it's an invalid play style. Not impossible, just not the kind that makes sense. Magic in TES supports itself, each school supports another. Alteration supports magic users as they don't wear armour and illusion helps to hide or distract fights. Playing one play style isn't the way the game is meant to be played, in my opinion. But you can if you want, I just don't see the appeal.

Uhmmm...roleplaying?

You're roleplaying a character in Skyrim, There aren't any other members of the population in Skyrim that use Destruction only. Why would the player character? There are even NPCs who shout, which validates that roleplay.

AFAIK, only NPC's that shout in the vanilla game are the Greybeards, and Ulfric, and both make sense.

Anyways, I never said anything against using shouting in any kind of build, as long as you're roleplaying the Dragonborn (AKA, you're doing the MQ, which not everybody does). But being a Destruction-only mage isn't odd by far.