Even talented students struggle with fundamental concepts in mathematics and physics. They cannot reason with graphs and have no feel for physical magnitudes. Their instincts are Aristotelian; in their gut they believe that force is proportional to velocity. With such handicaps in mathematical and physical reasoning, they can learn only by rote. I’ll discuss these difficulties and how the art of approximation can improve our teaching and students’ learning. Students then cannot conceal misconceptions behind mathematical agility, and can enjoy estimating the size of raindrops, the pitches of xylophone slats, or the distance that birds (and 747’s) can fly without refueling.