By Colin Capenito, Jack Eames, Boyd Hall, Lennon Isaac, and Kerrie Verbeek, VI Form
Brownies, Icing, M&M’s, and Calculus
Step by step, movement by movement, the task became a disaster. The lack of clarity in unison with a rudimentary understanding of an instructional lexicon, allowed only a few to complete the task at hand. It is not often that you are asked to create a dessert while following instructions verbatim, especially in calculus class. In efforts to shine a light on the importance of communication in calculus, Ms.McBride’s Advanced Calculus class was tasked with the job of creating an instruction sheet to make a brownie that was cut in half and covered in icing, then to have a member of the class follow the instructions in the most literal way possible. After almost every group failed, the message became clear: clarity is paramount in calculus. Like making an intricate dessert by hand, effective communication is paramount in the realm of calculus. One must be able to inscribe their thought process on paper as they surmount difficult problems not only to prove the legitimacy of their work but to show their reader a fully translatable math problem.
During a class period, we were split into small groups and tasked with creating the aforementioned set of instructions on how to construct a brownie sandwich. The catch? One groupmate would have to step outside while these instructions were made, and would then need to follow the instructions word for word. No one in the group could give the person building the sandwich any help outside of the instructions, which emphasized the importance of creating a clear set of rules. In math, writing clearly is just as important. If a problem is given and the work shown is a garbled mess, the reader will not be able to understand how the solution was reached.
Calculus, like baking, is a recipe. When solving a problem, it is essential to carefully and individually write out each step. The omission of steps done to reach the end goal, whether it be a solution to a problem or a beautiful brownie sandwich, leaves the reader of the “recipe” confused and unable to replicate the work. Steps must be written down in clear order, so any reader, even one who has never completed a calculus problem before, is able to follow the work. Without an understandable “flow,” the work done on the problem and the answer are useless because no one can decipher how to learn from or replicate the problem. Even the student who completed the problem might not remember how to duplicate their work after a few days. Knowledge retention must never be undervalued, especially in a math class, so a cumulative understanding of the subject matter is key. Going back to look at your work proves to be far better than initial learning, so why cheat yourself out of those study benefits?
In some classes, getting the right answer is the absolute concern. This is not the case in our Calculus class, for we are told that how we get to the answers is what truly matters. Just like making a brownie sandwich, if a person somehow manages to do the incorrect steps but succeeds in the task, it is not as meaningful because it may be a one-time success. In calculus, the same is true. If a student is unable to apply the appropriate method to reach the answer, then getting the answer may be invaluable and not repeatable. Mrs. McBride does not focus on the answer, only about understanding the process. Calculus students know that they will not receive full credit for a problem when they do not show their work, even if the answer is correct. Different problems will have different answers, but understanding the work (which means effectively communicating the process and logic) behind that problem allows students to solve many problems.
The importance of communication is critical to convey to others what we want to accomplish knowing that no one is a mind reader. It required the use of proper notation and sequencing to say what you need to say and help others understand why you did what you did. An example of using proper notation to explain the behavior of a function might be written:This is different from saying f( 5) = 7. This notation tells the reader that the limit as x approaches 5 for the function f(x)is 7. This function has a hole at x = 5 and there is no y value at x = 5. If the writer of this solution forgot to include any piece of this notation, a reader would be misinformed. Forgetting to write “lim” would prohibit a reader from knowing that an approximation of the value (not the actual value) was found and it would be incorrect.
Communication is essential to validate one’s answer. If asked to describe how a function behaves (and possibly use that understanding to predict the future) we might use our understanding of derivatives to express the way a function is changing. To do that, we need to use correct notation. For example, the first derivative represents the rate of change at a specific point of a function, which gives insight into whether a function is increasing or decreasing. The first derivative can be described by using the notation f ‘. The second derivative, expressed by f’’, represents the change in concavity or the rate of change of the rate of change. The third derivative, f’’’, is the rate of change of the rate of change of the rate of change. Each derivative expresses different information about the function. If one does not clearly communicate with proper notation, then the reader could misunderstand how the function behaves.
In addition to concise communication, for a calculus student to be successful in expressing his or her workflow, they must be able to show their workflow in an organized, linear fashion. Being able to see each individual step the student took is not only important for the teacher to take a step inside the mind of their student but for the students themselves in the event that they choose to study or learn from their mistakes.
In calculus, as well as other areas of life, communication is instrumental to the success of a task. Although the answers to the problems we do in our class are not important within the context of our lives now, learning proper sequencing and precise notation now could make a huge difference when using math in our careers when the answer could impact the lives of many. Without being able to properly communicate the calculus done, one is limited to being a solo worker and not being able to share his or her work with the broader community because he or she will not be understood. Being a great calculus student and being able to effectively communicate one’s thoughts articulately are dependent on one another. Being an effective communicator is to the benefit of all.