I guess what you need is a H-bridge.
However, we should sort out what happened here first. When no button is pressed, you have a closed circuit that goes through the ground (the outer black wire on your diagram), through the motor and then through the outer red wire, thus making the motor spin. When a button is pressed, all you should get is a short circuit (and eventually sparkles).
Now that i think of it, it may be to complicated with a H-bridge for this, though you could try because you'll probably learn a lot from it. Here is a link to it: http://www.societyofrobots.com/schematics_h-bridgedes.shtml
The easiest approach here would be using two batteries, connected parallel to the motor but with opposite directions and with a button wired into each of them. This way you'll create two separated circuits. You could still create a short circuit through pressing both buttons, because it will make the batteries connect serially.
Good luck, and prove your friend they are really wrong