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!

what is scientific notation

ReplyDeleteScientific Notation include in the mathematics course. In the world of science some time we deal with numbers which are very small and those which are very large. In some branches of science large numbers while in others very small numbers are used.

Dear Noiln,

DeleteStudents in this course understand Scientific Notation already. They have just finished a year and a half of Calculus and probably an additional semester of Linear Algebra.

Best regards,

Florence