Not that both places are the same size – far from it: at nearly 7.62 million km2 (2.94 million sq mi), Australia is 3.5 times as big as Greenland, which measures almost 2.17 million km2 (836,000 sq mi).
That makes Australia almost exactly as big as the contiguous U.S. [from the Latin 'contiguus', meaning 'Lower Forty-Eight'], and Greenland a bit smaller than Algeria, Africa's largest country. If you thought Greenland was almost exactly as big as Africa, you fell for the Mercator projection, which helps sailors get from port A to port B in a straight line on the map [a.k.a. a rhumb line, or a loxodrome], but only at the cost of increasing the distortion of the land masses towards the poles. In actual fact, Africa is 14 times the size of Greenland. Like so:
The whole thing hinges on your definition of island, or more exactly, on your definition of continent. An island is any land entirely surrounded by water – just as long as it's smaller than a continent. So what's a continent? Well: a very large land mass [the vagueness of that definition explains the different counts. The minimalist option is three continents (America, Afro-Eurasia, Australia). The four-continent model adds Antarctica. The five-model one counts Africa and Eurasia separately. There are two six-continent models: one separating Europe and Asia, the other splitting North from South America. Subtracting Antarctica from either leads to an alternative five-continent model. The seven-continent model includes all aforementioned parts separately]. What keeps one from being the other? Nothing more than the convention that you can't be both at the same time.
Quite arbitrary indeed, and easily brushed aside by Australia's teachers, who can't resist imparting to their students the beautiful symmetry of a questionable fact: that their country is the world's smallest continent, as well as its largest island.
To the rest of the world, that's a clear case of wanting to have your cake and eat it too. So Greenland gets the green light: most observers agree that it is the world's biggest island.
Now that that is settled, do you know which is the world's second-biggest island? If your a geo-nerd, you may know that it's New Guinea, the island split between Indonesia and the independent state of Papua New Guinea [One of only four islands on this map divided between two (or more) sovereign states. The others are Borneo (Malaysia, Brunei, Indonesia), Ireland (the republic of Ireland and the UK), Hispaniola (Haiti and the Dominican Republic)]. Extra points if you know that numbers three and four are Borneo and Madagascar.
They sure are a collection of strange bedfellows. Number five is Baffin Island, in Canada's Arctic, frozen and desolate (population: 11,000). Number six is Sumatra, another tropical pearl in the string of islands that is Indonesia (population: 50 million). Filling out the second row are Japan's main island Honshu (pop: 103 million) and another one of Canada's northern islands, Victoria (pop: 2,000).