The Boston Public Schools have embraced a bad map projection, specifically Gall-Peters. The so-called Peters projection is promoted as "more accurate," but this is at best misleading. Even The Boston Globe, in most respects a pretty good newspaper, neglects to mention the serious problems with Gall-Peters in its article on the Boston Schools decision here. (And that, mind you, is from a newspaper that's actually named after the globe!)
All rectangular maps (technically cylindrical projections) in actual use -- including Gall-Peters and Mercator -- necessarily distort east-west distances: They represent all parallels of latitude as equal in length, which of course they're not. For example, the equator is 21,600 nautical miles in circumference, while the 60th parallel north or south (roughly the latitude of Anchorage, Alaska) is only half as far around. All world map projections of this sort necessarily stretch out horizontal distances the farther you get from the equator, and they do so to the same degree. They differ only in how they scale distances north and south.
The Mercator projection makes the vertical scaling the same as the horizontal at any point. This does a pretty good job of representing shapes of all but the largest countries and regions, as you can see by comparing a Mercator map with a globe. It also preserves compass directions, and within small areas it comes close to representing the relative distances between points.
Of course, it also famously distorts relative areas on a global scale, for example making Greenland look as big as Africa though Africa is more than an order of magnitude larger. There's no question that students need to understand that, and that for many purposes the Mercator projection isn't the best choice for world maps used in classrooms.
But Gall-Peters is a bad choice as well. It does get one thing right. By stretching things vertically toward the equator and squashing them near the poles it correctly shows area relationships. The basic idea is to select a reference latitude north and south (typically 45 degrees) where the vertical and horizontal scaling is the same, but to exaggerate north-south distances closer to the equator and compress them toward the poles in a way mathematically calculated to create a constant scaling for areas.
Unfortunately, correctly representing relative areas on a rectangular map ends up pretty severely distorting literally everything else. Fortunately there are other equal-area projections that are less distorted.
Even better for classroom use, there are projections that achieve a reasonable compromise between representing shapes and representing distances and areas. The National Geographic Society, for example, uses the Winkel-Tripel projection, and Rand McNally uses the Robinson. These projections don't try to stretch the round Earth into a rectangle. Those projections do result in curved lines of longitude, but since we're used to looking at curved surfaces, our eyes more easily compensate for these distortions than for the weird stretchings and squashings of cylindrical projections.
To belabor the obvious, globes are better than any flat map and belong in every relevant classroom, or at least in every school. Not only do they get shapes, distances, and areas right, they also make other things clear that it's all but impossible to figure out from a map. For example, with a globe it's clear that the shortest distance between two points is rarely a line of constant compass direction. It's hard to see from a flat world map why a flight from the United State to Europe crosses over the arctic, but it's obvious from a globe.
Wikipedia has a pretty good list of map projections. The XKCD comic strip below shows some well-known (and not so well-known) ones together with amusing commentary. (For an explanation, see Explain XKCD.)
(Updated 2017 March 28 to correct at least some of my usual typos and to try to improve wording.)