Boston Public Schools has chosen to select the Galls-Peter projection as its projection of choice for its school as opposed to Mercator because it is an equal area projection which shows Africa as a massively huge continent relative to others. They are doing this as a way to emphasize the size of Africa, but there are many better ways to accurately display the size of Africa (over 3 times the size of the US) without stretching the shapes of landmasses to the point of being useless. It is true that the size of Africa is massive, and while Gall-Peters accurately displays the relative sizes it is hard to actually understand the meaning of this the way the map is drawn. While Mercator emphasizes relative shapes while making relative sizes absolutely distorted.
Gall-Peters:
Gall-Peters and Mercator inversely ruin the polar regions and equatorial regions so the only areas which are somewhat accurate are the temperate zones. I do not recommend either projection for this major reason. Mercator is good for drawing distances when you cross the sea, which is why it is so popular, but for someone learning geography neither projection is accurate enough. I would not put either one on the wall of a child's bedroom.
When choosing a world map which is accurate in size and useful, it is important to understand there is no one perfect way to represent the world except as a globe. This however means you can only really see about one third of the world at a time, making it useless for many purposes, so an accurate flat map is necessary.
This is the biggest challenge of cartography. It is physically impossible to accurately represent the world as a flat map. There are a few good examples, and none of them are perfect squares because it is literally impossible to make a decent map of a globe as a rectangle. Any map showing the world as a rectangle will be significantly distorted in one way or another to the point of being a poor representation of the world.
Some projections get size as close as possible, the best way to do this is a sinusodial projection the disadvantage is that the shapes of continents are almost worthless:
I am fond of the Robinson projection which the National Geographic Society uses because it has a nice balance between relative sizes and maintaining the shapes of continents. There is distortion near the poles, but this is easily supplemented by using the corners of the map to show polar regions, which National Geographic has done in many of their maps I have.
I really like a recent projection from Japan called Authagraph because it accurately shows relative shapes of continents in a way which is particularly useful for mapping the movement of people around the world. The advantage to this map is that it keeps both shapes and sizes very accurate. The areas which are distorted are now areas of the ocean which for most mapping purposes are not useful (unless you are an oceanographer, in which case it would be a terrible choice). I particularly like how every continent is shown in full, helping people understand why Antarctica is a continent and Greenland is an island (another good way to illustrate this point is a Robinson projection which focuses on elevation and ocean depths, which makes it clear where plate boundaries are).
For most purposes the AuthaGraph or Robinson projections are two excellent ways to accurately describe the world preserving both size and shape as accurately as possible. Authagraph has the advantage that it can literally be folded into a globe it is so accurate, which you cannot even do with Robinson. If I were designing the walls of a 3 or 4 year old I would likely put an Authagraph projection on the wall, and have a Robinson projection somewhere else in the house.
This is VERY cool! And you weren't just in the top 100 kid geographers, you were, I think, in the final 10! Not bad for a 10-year old.
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