Initially the spherical cow was just a cute thing for my classroom. I made a rubber stamp of a headless, Holstein-marked circle and used it to indicate completion of homework assignments. I later redrew the cow to add some personality, and a head, and made a corresponding new and improved rubber stamp.
Consider a Spherical Cow, A Course in Environmental Problem Solving,” by John Harte, which I didn’t know existed, and I now highly recommend. (Harte also published a follow-up, “Consider a Cylindrical Cow.”) This led me to think more about justifying the spherical cow as a mascot for the class.
In order to understand the complexities of the world, we use simplifications and approximations to develop a first-order understanding of a given phenomena. A physicist trying to understand a cow's energy consumption, heat radiation, or milk production would not initially worry about the bumpiness of a real cow – a spherical stand-in would be sufficient to get an initial sense of the situation. I made an animation with Flash to try to convey this, but it was insufficiently integrated into my course for my students to really get what I meant. (Not all computers or browsers are Flash-friendly, so the space below may be blank.)
Harte quotes Aristotle who states the idea behind the spherical cow much more eloquently:
It is the mark of an instructed mind to rest satisfied with the degree of precision which the nature of the subject permits and not to seek an exactness where only an approximation of the truth is possible.
Representing the world is at the heart of Modeling Instruction, and representations necessitate discernment of what is relevant and what is irrelevant. Of those things that are relevant, we often simplify or idealize them in order to distill out the fundamental pattern, or underlying model. From that foundation, we can begin to reincorporate more of complexity of the real world, allowing us to increase the precision of our predictions.
Over the years, I’ve become a bit smoother at integrating the spherical cow message in the flow of class activities: In introductory physics, we don’t worry about the complexities of our moving buggies, the accelerating wheel and axle, friction, and air resistance, at least initially. Instead, we simplify the subject of our analysis by using particle models – this works great for introductory kinematics, dynamics, and momentum. Sometimes we encounter problems (Atwood’s machine) or concepts (energy) that can only be analyzed when considering the internal structure of a system, and when we need to account for that complexity, the cow becomes less spherical.
I now have kids stating in class, “Oh, so we should just spherical cow this problem.” Success!
With standards based grading, I don’t use the spherical cow rubber stamp for marking homework completion anymore. Instead, I’ve used the spherical cow as a stand-in “particle” to illustrate various physics principles, such as these examples from the front of my acceleration, circular motion, and electric circuits curriculum packets:
Gary Abud's post about why and how to use branding in your classroom.) Through trial and error I've come to recognize the value of most everything in Gary's list, and it's given me ideas about how I can extract more value out of the spherical cow in my classes.
There it is – more than you ever wanted to know about the spherical cow. Our art department is considering getting a silkscreen printer. Maybe we’ll actually take the plunge and make shirts this year. I’m hankering to add the cow to one of my ties.