The 5 Commandments of Calculus
Karen Sleno is an expert teacher at Flushing High School where she has taught everything from Algebra 1 through AP Calculus and serves as the department chair. Additionally, she is an adjunct instructor at Mott Community College and at the Center for Talented Youth at Johns Hopkins University. In 2022, she won the Michigan Department of Education Regional Teacher of the Year award in Region 5 for her teaching expertise gained over 30 years in the profession. She is a College Board consultant for AP Calculus and has held various roles (including question/exam leader) at the annual AP reading. Her efforts in education and in the AP program earned her recognition as Educator of the Year for her district in 2015.
It’s not surprising that the thought of grading AP Calculus exams for 8 hours a day for 7 consecutive days may sound like the worst punishment one could suffer. I might have thought that myself once. However, once I experienced the AP reading, that opinion took a sharp 180. Not only is my time there rewarding due to the people that surround me, but I also learn so much about how the current AP exam is scored that the week really doesn’t feel like work at all. In fact, those 7 days have become the highlight of my year! But this isn’t a post about the AP reading; it’s about something that happened near its conclusion one year.
So often, student responses fail to earn points not due to a lack of calculus knowledge, but rather a lack of communication skills. Over and over, we see unequal expressions that are set equal or a missing parenthesis that changes the meaning of an expression completely. Those types of errors are painful to see because they could have been so easily avoided. One year, I said to a colleague, “I wish I could share these ‘fixes’ with every math teacher out there!” And now, here is my opportunity to do just that! May I present…my 5 Commandments of Calculus (or any math, really)!
I. THOU SHALT NOT USE VAGUE PHRASES SUCH AS “THE GRAPH”, “THE SLOPE”, OR “IT”.
We’ve all seen it. “Decreasing because it’s negative”. What exactly is negative? “The parabola opens up because it’s positive.” What is? Allowing such explanations in our classroom means that students will use those explanations on the AP exam, too, and if there is any uncertainty about what “it” is, full points will likely not be earned. Instead, insist that explanations use names of functions consistently and correctly, even if it sounds silly to student ears. “The function f is decreasing because f’(x) is negative”...yes!
II. THOU SHALT NOT CLAIM THAT TWO THINGS ARE EQUAL WHEN THEY ARE NOT.
“Stream of consciousness math” is what I’m referring to here…the tendency of students to use the equal sign as an indicator of their next step. The problem with this is that if the two expressions they are linking are not truly equal, a penalty will occur. For example, suppose a student is finding the average value of a function f(x). Here is what we might see:
The result is correct mathematically but not in presentation. The student realized that average value requires division by the length of the interval, but in this case, it came as an afterthought. This is not just a calculus difficulty, however; any process (quadratic formula, area/volume to name two) opens the door for the equal sign issue to rear its ugly head.
III. THOU SHALT NOT SAY TOO MUCH.
This may be the saddest of them all. The student writes a beautiful response that would earn full points, and then says something incorrect because the response didn’t end when it should have! How to remedy this? Remind students that short, concise answers that meet the requirements are all that are necessary to earn full points. Save the essay writing for AP English!
IV. THOU SHALT ALWAYS LABEL THY ANSWER.
Forgetting to use correct units is one of the most universal errors in any math class; even students in my consumer math class sometimes forget to include the $ with their results. Fortunately, it’s also one of the easiest to remedy with consistent reinforcement. While the College Board is often kind enough to remind students with the words “using correct units” in the question prompt, we should expect this as an automatic signature to any contextual problem, whether in finance, algebra, geometry, or yes, calculus.
V. THOU SHALT CONSISTENTLY USE CORRECT MATH PUNCTUATION/GRAMMAR.
What? Grammar in math? Oh yes, indeed! Consider this treatment of a logarithm:
What is the argument of this logarithm? It should be x based on what was written but sometimes students use this notation to mean ln (x+2).
Here is another great example:
See it? That first quantity in the numerator should have parentheses around it, but they are missing, so what we read is only 2(3), not (4x + 2)(3). Expecting good grammar/punctuation in class means that students will “perform” what they “practice”.
Someday, math classes everywhere will enforce the 5 Commandments every day with every student, but until then, I will preach the word to all teachers who will listen. And maybe, just maybe, in a not so distant future, those AP Calculus students whose work we read will remember what they “shalt” or “shalt not”.