An explanation for the use of a false map, and why we should switch to teaching our kids a map based on true relative size, was featured in The West Wing, see the clip below.
As mentioned in the video, the map can be used to make certain areas like the US mainland and Europe much larger, giving an illusion of power and importance attributed to its size. Areas like Africa are shown to be much smaller than they truly are. As the image below shows, the size of Africa is comparable to the total size of China, the United States, India and multiple other European countries.
A sphere cannot be represented on a flat plane without distortion, which means all map projections are distorted in one way or another.
Below is an example of the Gall-Peters projection map, as mentioned in the video above.