I'm teach Proofs 101 in Fall

Farmer Ben only has ducks and cows. He can't remember how many of each he has, but he doesn't need to remember because he knows he has 22 animals and 22 is also his age. He also knows that the animals have a total of 56 legs, because 56 is also the age of his father. Assuming all the animals have the usual numbers of limbs, how many ducks and how many cows does Farmer Ben have?

I adopted a liberal arts math textbook (Crossing the River with Dogs, by Johnson, Herr and Kysh) for my Introduction to Higher Mathematics course for math majors. I love the book; it has loads of figure it out questions and great rhetoric about doing them efficiently that apply broadly to life as well as to higher mathematics. But, I think that, with the exception of those who study at Harvey Mudd College, most liberal arts math students think questions like this are contrived and silly (like, what's the guy going to do next year? Get a one legged duck?), even though the author clearly thought he gave a believable context. I also think that I'd have to work pretty hard to get the liberal arts math students to be willing to sit there and figure out the answers to these funny questions. (Yes, you should go ahead and figure it out. You'll get that satisfied "I did it" feeling that's all the rage these days.)

Math majors on the other hand revel in this sort of thing. They will not be fazed by such a scheme for Farmer Ben to remember how much livestock he has. They probably have an equally contrived method for remembering their own phone numbers.  They also desperately need to recognize math as a creative endeavor, and they need to practice exercising that creativity. My past incarnations of this course were all about how to write mathematical proof, and somehow in trying to get the writing straightened out (which must be done, don’t get me wrong) I focused the students’ attention away from figuring out the tricks and ideas that actually allow us to figure things out.

While the problem of Farmer Ben’s livestock can easily be solved using algebra, that is not the assignment in the text with this exercise. The assignment says draw a diagram to solve the problem. (Yes, you should go back to the problem and work out a good picture to draw that captures the would lead, for example, my brilliant 6 year old to figure it out.) The content in this section includes student solutions to problems in which they had to draw diagrams, giving the reader examples of multiple correct solutions (that actually have little in common), and examples of language they should use to explain their work. The problems in the section are not all algebra problems, but instead are all problems whose answer can be diagrammed effectively. Another example is about the number of high-fives a certain soccer team did at the end of a game if everyone high-fived everyone else.

Probably all (OK maybe this is optimistic) of the students in the Introduction to Higher Mathematics course would solve the problem with algebra and with no difficulty, but choosing a useful diagram would require some thought. Up to this point we have largely asked these students to answer questions that they could solve by copying a process from a similar problem, and they are reluctant to explore what they understand without following such a model.  I have plenty of mathematical ideas that include the high level vocabulary and concepts that my students are ready to study for which “solve this problem by drawing a diagram” is an excellent exercise. 

Other sections in the text include other strategies that are valid at all levels of mathematics, like “Making a systematic list,” “Eliminating  possibilities,” “Solve an easier related problem,” “Look for a pattern,” and so on. My job is going to be to develop assignments that enable them to practice these problem solving techniques while also learning sufficient high-level vocabulary. I will be building that bridge (over the river, so that we can walk across with the dogs without ado) over the summer. We’ll see how it goes!