by Steve Meiring
We are all a little weird and– Dr. Seuss
Life's a little weird,
And when we find someone whose
Weirdness is compatible with ours,
We join up with them and fall in
Mutual weirdness and call it Love (of Mathematics).
What better way to introduce you to the world of Stella’s Stunners than to let you know at the outset you have entered a realm where the usual rules of learning seem not always to apply. Somehow, you found your way to this website, presumably because you have a strong interest in problem solving. Or perhaps your arrival here is motivated by the desire to stimulate deeper and more thoughtful mathematical interactions for your students through good problems. If so, you have come to the right location.
This introductory essay is an electronic orientation for what you can expect to find here and to provoke your desire to explore all that is to be discovered.
To start: A Stunner is a non-routine problem like the following:
Ignacio runs the Zaragoza Deli in New York City. He has a 2-pan balance and a set of 4 brass weights. He can weigh out any whole number of ounces up to 40 using just these weights. What is the amount of each brass weight? (Weights are used in any combination on either or both pans of the scale.)
Problems provide the medium for exchanging ideas in mathematics. The better the problems, the more intriguing are discussions and discoveries that prompt the thinking that leads to the solution ‒ as well as implications for how that thinking is directed toward other problems. Even more compelling are discoveries we make about our own thinking processes, and how we are changed by the experience of confronting good problems.
Being human, we react to challenges in which we invest time. Our reactions might be relief at being finished, elation at success, self-doubt for admitting failure, enlightenment of discovery, agitation because the result was not worth the effort, or myriad other emotions unique to the individual. The most important reaction occurs when a problem takes on a life of its own. The solution search no longer driven by someone else’s direction, it becomes a personal drive for understanding and reflection.
The point is that interesting, fanciful, challenging problems command our attention over an appreciable time span, and in the process become memorable. Good problems fascinate us; they provoke intriguing conversations; and they tag along as unbidden companions as we go about other activities. Once the challenge of a good problem has gotten hold of you, you can’t let it go. Stunners make mathematics a more fun activity in which to become personally invested to learn.
In that context, this site is a repository of more than 700 such problems, assembled by now retired teacher Rudd Crawford, over the course of a long teaching career, which started in Concord, Massachusetts and which ended in the small Ohio college town of Oberlin. Used with students for much of his teaching career, these non-routine problems brought a new sense of purpose to his students’ learning and, in the process, helped to enrich the traditional program.
Of consequence to you, this problem collection is offered as a professional service. Problems are free for the taking. The collection is organized by course, by title, by topic, and into problem sets used by Rudd and his Oberlin HS colleagues to nurture student problem solving on a recurring basis. One of many features of this site is a selection basket for which the web visitor can choose particular sets of 6-8 problems from the collection itself or self-select individual problems, assemble, and print out a personalized problem set. More about these organizational features in the user guide section, "The Problem Library."
An Odyssey of Teaching Discovery
Accompanying the Stella collection is a series of short essays written by Rudd in his authentic and understated teaching style. These essays are gound in the User Guide, within the section titled "Using the Library with Students." The essays tell a story of the discoveries made by a classroom teacher, eventually leading to how to use Stellas (student name for Stunners) to complement the normal fare of textbook learning.
The story begins in the late 1960s with Rudd making the discovery that his personal affection for brain-teaser problems, and small collection on note cards, could be used with a student when nothing else seemed to work. In "How I Got Started", we learn how this notion stayed with him into the 1970s, whereupon he confronted junior high school students who were bright, but bored, with the customary textbook work. Once again, he found that learners flourished in an environment liberally sprinkled with non-routine problems.
When they (and he) eventually reached the high school together, those same students clamored, “We haven’t done anything hard in math since middle school – give us some more of those problems.” This impetus and a student named Solomon caused Rudd to begin pondering. That pondering led to how to offer a new, problem-solving dimension within the framework of traditional learning – practically, instructionally, and programmatically. It also led Rudd to think deeply about the nature of mathematics, how students learn it, and how they are shaped by that learning.
In the essay, "Stella’s Biography" we make the acquaintance of Stella, an older and eccentric Dutch baroness – friend to Rudd and, up to the time of her death, a consultant to the Oberlin HS math department. Among her many eccentricities was a penchant for carrying around vexing problems in her leather purse until they were solved, and even after, in a sort of celebration. Stella became the source of the stunners visited upon students. She also served as a model for students of a mathematician who relishes the challenge of good problems and who will persist with them until they are vanquished.
"Stella in My Classroom" recounts how Stellas came to be offered in sets of 6-8 problems considered independently by students over the course of the calendar week – besides regular coursework. By week’s end, solutions and student thoughts about the stunners were recorded in their notebooks and turned in for comments and grading. This essay further describes how students came to understand that heuristics arise naturally as what one does when you don’t even know how to start on a problem.
The section on grading is a concession to the reality that a ‘good’ performance on any problem set is about half the problems solved. It also presents an alternative scheme (and philosophy) for how to evaluate problem solving in a way that is constructive and not self-defeating.
The Argument for Stellas
Perhaps the most important essays about teaching discovery are "What Students Gain from Problem Solving" and "Feedback from Former Students." “What Students Gain...” is an insightful look at the purpose for learning mathematics as measured against how we typically go about it. Implied without it being stated is this generalization: we usually learn mathematics, lesson after lesson, in a manner that depends upon the teacher to predigest the material for us, to work out sample problems, and then to assign exercises that coincidentally happen to fit the lesson just taught. When difficulties are encountered, students are encouraged to quickly seek help. The pared down problems and exercises are mostly bereft of the contexts that often make problems more interesting.
This kind of a lesson is efficient for learning lots of mathematics and to automate the skills, ideas, and processes to be acquired. Doing so frees up the mind to learn new things the next day. The limitation is that the knowledge acquired is customarily at an introductory level. It is not immediately integrated into a deeper and well-connected hierarchy of understanding that makes that knowledge quickly and easily accessible to address new problems presented in a different context. Consider, for example, this problem:
A chicken and a half lays an egg and a half in a day and a half. How many eggs will 12 chickens lay in 12 days?
It is safe to say that most secondary students are not very good at solving such a whimsical problem. The question is Why not? And what does that tell us about the relative accessibility of their learning?
One answer is that dispositions the student brings to the task influence her chances for success. Students are accustomed by daily lessons to think of math as a task or situation you identify as something recognized, for which you recall from memory an appropriate technique for solution. This preconception applied to the above problem presents a conundrum. It is not recognizable. Habituated to dispatching such a task quickly and routinely, you take a superficial ‘swipe’ at solution, multiplying the information in the first sentence through by 2/3 . Eventually, you arrive at the incorrect answer of 12, or a more reasonable, yet still wrong result of 144.
Missing in this pedagogical sequence is the opportunity for students to learn HOW to address situations for which they have not been explicitly taught. This goal entails learning the type of heuristic thinking and learning skills necessary to apply what you understand to something new. Stellas offer that opportunity in the equivalent of a ‘virtual lab,’ where the emphasis is on thinking your way through to a solution, rather than the mechanics or elegance of the solution steps. Stellas provide a new and complementary dimension to textbook-style learning.
"Extending Problems" raises two other considerations for Stellas. First, good problems often beget other good problems to solve. Take the example of the balance problem above. Why are the required weights all powers of 3? This is a suitable problem for students and teacher alike.
Extending problems provides an opportunity for the Stella strand and regular coursework to intersect. A Stella problem can foreshadow a textbook lesson to come, or it can revisit a previous topic learned, but in a different way. For example, the chicken-and-egg problem could be identified as an extension of direct variation. Suppose the student begins by exploring the problem and mulls, “If I double the number of chickens, what should happen to the number of eggs produced?” This quickly leads to a method of solution for building up the ratios until the required condition is met. This is an illustration of using Stella problems to revisit previously learned ideas and to extend students’ understanding.
After the chicken problem has been considered, perhaps now recognized as a direct variation of two independent variables, it sets the stage for a subsequent textbook lesson on compound variation. In the midst of that discussion, the teacher can innocently pose this question, “Where have you seen an example of a compound variation that depends on two independent variables and a constant rate?” One answer ($i = P \times r \times t$) presents another avenue for solution to the chicken-and-egg problem as well as a connection to a taught schema for solving a class of problems.
Next, we come to the essay on "Feedback from Former Students." As modern research will attest, emotions often influence how we think as much as do our reasoning skills. What we believe to be true (whether a problem can be solved by us), what we think is an appropriate learning task (is it fair to ask me to start out not knowing what to do), and what importance we assign to a task (our passion to persist in spite of adversity) are direct measures of how much of our sense of self and our intellectual resources we will commit to the task.
Stellas present a learning environment in which the rules of engagement are identified upfront as different from textbook exercises. By definition, Stellas are hard; you are expected to persist over a period of time; and not knowing how to proceed is a given. Sometimes you get the problem; and sometimes the problem gets you. Students are encouraged (and expected) to express their feelings in their writeups about Stellas and themselves after they have emerged from the experience of tackling something that is mentally challenging.
The Gestalt of Stellas
Science education as a discipline has long had a handle on a unifying principle they define as scientific inquiry. In a nutshell, it means that science has adopted the notion that for students to learn science appropriately, they sometimes must have opportunities where they must think like scientists:
Inquiry means that students are handling science; they are manipulating it, working it into new shapes and formats, integrating it into every corner of their world, and playing with it in unknown ways. Inquiry implies that students are in control of an important part of their own learning where they can manipulate ideas to increase understanding. As students learn to think through the designs and developments of their own inquiry, they also develop a sense of self-responsibility that transcends all subject areas.
What Rudd’s Stella strand provides are opportunities for students to apply what they have learned and sometimes to discover mathematics as it was created. They have the occasional chance to act like mathematicians in the virtual laboratory world of Stellas. That experience changes them in the way in which they think. They learn habits for learning like Persistence, Resilience, Focusing over time, Flexibility, Curiosity, and Love of subject matter.
As quotes from Crawford’s former students attest: Stellas taught them more than just how to do and how to learn mathematics. Stellas taught them how to apply their minds generally.
In looking over the various essays and problems that comprise Stella’s Stunners, one is struck by several things. One, none of us gets smart all at once – students or the teacher. It took the better part of a teaching career for Rudd to refine the Stunners you will find here and the teaching methodologies that support them. At each and every turn in the essays, you find illustrations of teaching methods that are dead on.
For example, problems were given out as sets rather than individually (a Stella used as a class toss-up question primarily benefits the solver). Assigned sets were constructed to give every student a chance to participate and some measure of success. Sets worked on over a span of days enabled students to learn what persistence looks like and how their minds work when given time and sufficient focus. He employed an evaluation scheme that validated the importance of the work and that assigned credit in a constructive way that valued alternative approaches.
He created a safe environment where students could struggle and meet success or failure in applying their knowledge, knowing that they were not competing against everyone else, but only against what they themselves were capable. Requiring formal writeups of problems taught students how to do a technical report. Stella, herself, was a model for student expectations as well as the nemesis toward whom students could focus their vexations.
None of this came about easily nor quickly. In one sense, Rudd had to practice what he expected of his students. Just as they had to learn how to learn differently, he had to learn how to think differently about teaching. He started slow. He experimented with better students first. He learned how to adapt and to adjust to what worked and what didn’t. He talked with other colleagues who were interested in the same things. This is one more serious piece of advice for you, the reader, in experimenting with these materials.
But mostly, the essays are a story of a love affair among a teacher, his students, and challenging, quirky math problems. One doesn’t put that much energy and time into something requiring this degree of hard work and creativity without loving what you are doing. That same affection for his profession and for what good teaching should aspire to has led to this resource of Stellas made available to you, the website visitor.
Steve Meiring has been a long-time friend of the Stella program and himself has a life-long interest in problem solving. Now retired, he has conducted hundreds of workshops on problem solving and authored Problem Solving -- A Basic Mathematics Goal I & II as a mathematics consultant for the Ohio Department of Education. He is also the author of an unpublished manuscript, Think About It: A Resource Guide for Secondary Math Teachers and Students.