Whales have very large brains but that does not necessarily indicate high intelligence. What is much more important is the complexity and flexibility of behaviour and how quickly and well an animal can learn.
Killer Whales have very complicated social lives, and different pods can be said to have developed their own unique cultures. Some feed on fish, others on marine mammals, and yet others go after seabirds. They have each developed different hunting strategies which young members of the pod have to learn as they grow up. Whales undoubtedly communicate with one-another, and Humpback Whales seem to be able to fish together co-operatively when they come across large numbers of fish. It looks very much as if they are able to plan and give one-another instructions. Bottlenose Dolphins have recently been discovered to kill porpoises and young of their own species on occasions, and this puts them right up their with Humans and Chimpanzees as the only mammals known to carry out organised murder!
Many whales use a sort of sonar to find their prey underwater, and to find their way past obstacles or to locate holes in the ice. In many ways this is similar to the way many Bats find insects at night and navigate through woodland, and while having a very complicated sensory mechanism does not indicate high intelligence, the other uses whales make of their ‘voices’ probably does.
Songs are used by whales to communicate in some way. In itself this is not indicative of intelligence (many insects also use noise in this way), but when different individuals develop their own unique songs (as Humpbacks do), or when different calls seem to refer to individuals ‘by name’ (as has been suggested for dolphins), then self-awareness and intelligence are definitely implied.
An animal that lives in water, and needs to surface regularly to breathe, cannot afford to sleep underwater! Most whales can ‘shut down’ one half of their brain to give it a rest while using the other half to keep swimming and surfacing regularly and they don’t even need brain pills. When one half of the brain has had a good sleep they simply switch halves and rest the other part. That in itself might go some way to explaining large brain size, but as was mentioned above – brain size on its own does not correlate with intelligence.
There are over eighty species of whale, and they are certainly not all the same! Talking about ‘whale’ intelligence is the same as talking about ‘ape’ intelligence – it depends which species you mean. Humans are the most intelligent apes, but which of the whales is the cleverest? Once we have worked that out we can maybe begin to decipher their ‘language’ (if they have one) and using our ‘technological edge’ perhaps even communicate in some way.