The Element of Surprise

I think another hallmark of an authentic problem is that it opens the possibility of surprise.  When I'm writing a problem, either for my class or for a math contest, I usually have one or two particular approaches in mind--discussing those approaches is "the" point of doing the problem.  But a characteristic of a really terrific problem is that solving it is not just a matter of finding your way through one specific set of doors in a maze: rather, it's more like navigating over rough terrain, where what seems like a detour might reveal itself to be a quicker, or more scenic, route.  Students navigating such terrain for the first time surprise us with the new paths they discover.

One example of many:  in the geometry text used at my school, CME Geometry, students learn early on about concurrence of perpendicular bisectors.  Around that time, the text asks the following question:  suppose you need to replace a broken blade from a circular saw.  You have only the fragment shown below.  How can you determine the blade's diameter?

In general, students find this problem quite difficult; in fact, in the first edition, it was a homework problem that almost nobody could solve. 

The "ordinary" approach, suggested by the problem's placement in the text, is pretty elegant. First, construct the perpendicular bisectors of two chords.  Their intersection point is the center of the circle, from which you can now measure the blade's radius.

Over five years and about seventeen sections of geometry students, this was the only solution ever proposed.  But last year, I was surprised by the following numerical approach, discovered by two students independently:

First, construct a single chord and its perpendicular bisector as before; label the endpoints of the chord A and B, the midpoint P, and the place where the perpendicular bisector intersects the rim of the blade point C.  (Exercise: what conditions on A and B guarantee that point C is easy to locate?) If ray CP intersects the other side of the circle at D, by the Power of the Point (which my students also see early), (AP)(PB) = (CP)(PD).  So measure the three known segments, solve for PD, and compute CD = CP + PD.

The best problems are like that:  students can solve them in more than one way, with each solution giving different perspectives on the problem.  In this case, one feature of the second solution is a different perspective on what Polyá calls the unknown: the first solution gives a way to draw a particular segment (the radius), while the second solves for a numerical value (the length of the diameter).

Other examples abound, and not just in geometry.  My friend and colleague Doug reported that a student "broke" an early application of calculus--finding the maximum value of a quartic polynomial--by completing the square in x² instead of taking the derivative. A problem I use specifically in order to invite multiple solutions is this one:  How many 1x1 square tiles does it take to surround an n x n pool?  In going over solutions to this past week's Chicago-Area All-Star Math Team Tryouts, I found at least three cases where the solution writer and I had found completely different approaches to the problems.  

Even a "standard" problem can invite surprise, if the teacher creates space for that possibility.  In a talk I heard as a high school student, the mathematician and philosopher Raymond Smullyan described a surprising solution to the traditional "Each dog gets 4 biscuits; each cat gets 3" word problem.  Instead of using algebra, this solver first gave each animal three biscuits, observing "now the cats have got their share".  

The first line of my department's academic honesty policy says "Mathematics is, at its heart, a creative endeavor."  But how often do we pose problems that really give students an opportunity to make a new path, not just follow one of a few carefully laid-out trails?  How often do we give our students the chance to surprise us?  And when we exclude the element of surprise from our math classes, what do we teach students about doing mathematics?  

== pjk