Mercator Projection v.s. Peters Projection

In the show the West Wing cartographers make very valid points about the Mercator projection and the Peters Projection. They state that all public schools should change the map they are using to teach with. They said that the Mercator projection is teaching kids the wrong way. On the Mercator projection Greenland is the same size as Africa, which is actually 14 times the size of Greenland. So basically the proportions are off. As the Peters projection it has everything proportioned to what they actually look like. The mercator projection also shows that Europe is larger than South America but is actually smaller. Europe is showed on the Peters Projection as smaller and only 0.9 million square miles where South America is 3.8 million square miles. As well as Alaska appears to be three times as large as Mexico but actually Mexico is larger by 0.1 million square miles.