The Dodecahedron is a character from the book The Phantom Tollbooth by Norton Juster. He lives in the city of Digitopolis at the base of the Mountains of Ignorance. On page 148 he has this conversation with Milo and the Humbug.

“I’m not very good at problems,” admitted Milo.

“What a shame,” sighed the Dodecahedron. “They’re so very useful. Why, did you know that if a beaver two feet long with a tail a foot and a half long can build a dam twelve feet high and six feet wide in two days, all you would need to build the Boulder Dam is a beaver sixty-eight feet long with a fifty-one foot tail?”

“Where would you find a beaver as big as that?” grumbled the Humbug as his pencil point snapped.

“I’m sure I don’t know,” he replied, “but if you did, you’d certainly know what to do with him.”

“That’s absurd,” objected Milo, whose head was spinning from all the numbers and questions.

“That may be true,” he acknowledged, “but it’s completely accurate, and as long as the answer is right, who cares if the question is wrong? If you want sense, you’ll have to make it yourself.”

My students have had so many years of maths with problems that do not make sense that they have trouble in math class when required to determine what a reasonable answer is. As an instructor of maths, I have to carefully read all of the application problems in our textbooks so I can weed out the unreasonable questions so I can work to teach my students how to determine if an answer is sound.

I ask the following question early on in most of my lower level courses: If I can mow my lawn using the John Deere lawn tractor in 6 hours and it takes 8.5 hours using the Yard King mower, how long will it take to mow the lawn using both tractors? The only thing I ask them to do is identify what would be a reasonable answer to this question.

It is sad to me that over half of my students simply average the two numbers and say that 7.25 hours is a reasonable answer. They don’t even think about the actual situation with which they all have experience, they just do a quick calculation and don’t think at all. When confronted, I can get most of them to tell me that it would have to take less than 6 hours.

I now make finding a reasonable answer an outcome in each one of my classes. I model how I determine a reasonable answer to my students and I have them practice it weekly.