Earlier this semester, my first as a credentialed teacher, I ran headfirst into a buzz saw that nearly ripped me into pieces. I naïvely thought that my AP Calculus AB students were ready, willing, and able to take on the challenges of an advanced mathematics course. What I found tested my ability in ways I never expected. Fortunately, the guiding hand of a supremely talented principal and assistant principal provided a process, and the space, my students and I needed to work through our mutual challenges. It truly was a baptism by fire into the teaching profession; fortunately, well-trained firefighters at the ready kept the occasional flare-up from reigniting the inferno, although I tended to prefer asbestos treated clothes for a while.

**A Perfect Storm**

Many of my AP Calculus AB students had well-connected and vocal parents, who were not too pleased with what they heard from my students, and who made their concerns known to my school’s administration, and the district office. As such, less than a month into the semester, I found myself target of significant parental ire, caught in a near ‘perfect storm’ situation where the high-expectations of the College Board, and their emphasis on conceptual understanding embodied in my pedagogy collided with a student body convinced that the methods of instruction from earlier mathematics courses, which nearly exclusively emphasized procedural fluency, via a “learn by imitation” model, must solely be followed in order for them to have any chance of learning calculus. [1]

Adding to this, a strong undercurrent of insufficient preparation, predominantly by the students, impeded students’ ability to grasp the calculus concepts the first month of the course, even though I spent the first two weeks diagnosing, assessing, and reviewing prerequisite knowledge. Less than 10% of students were truly ready for the rigors of any higher level mathematics course, much less calculus. It seems they approached AP Calculus as they would any class, with minimal self-directed study, or any investigation into how best to prepare for the class. While many completed a summer take-home packet, most could not recall the concepts or apply the procedures involved. It was as if they used references and notes to do the work, just to get it done, without much understanding of what they did in the problems, or why. [2], [3]

Notwithstanding the above, and as a parent myself, I likely would have joined with these parents had my child been involved given the discontent rippling through the community at the time. Unbeknownst to these parents, as I outline above, there was a massive disconnect between student preparation for AP Calculus and the rigor required for the course, not all of which rested with the students given the “mile-wide and inch-deep” approach to mathematics curriculum as driven by state standards. Nonetheless, while earlier mathematics courses and teachers exposed students to most, if not all, of the prerequisite knowledge and skills, students’ mathematical proficiency and, more importantly, expectations were not aligned with the College Board’s expectations, as the following passages highlight.

…the College Board describes their “AP” designation on a student’s transcript as a “credible, indicator that their AP Program has authorized a course that has met or exceeded the curricular requirements and classroom resources that demonstrate* the academic rigor of a comparable college course*.”

“Success in AP Calculus is closely tied to the preparation students have had in courses leading up to their AP courses. *Students should have demonstrated mastery of material from courses that are the equivalent of four full years of high school mathematics before attempting calculus**.* These courses should include the study of algebra, geometry, coordinate geometry and trigonometry, with the fourth year of study including advanced topics in algebra, trigonometry, analytic geometry and elementary functions.”

**PREREQUISITES: Before studying calculus**, all students should complete four years of secondary mathematics designed for college-bound students: courses in which they study algebra, geometry, trigonometry, analytic geometry and elementary functions. These functions include linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric and piecewise-defined functions. In particular,

**students must be familiar with the properties of functions, the algebra of functions and the graphs of functions. Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts and so on) and know the values of the trigonometric functions…**

*before studying calculus,**– Taken from College Board AP Calculus Course Description, Fall 2010 version. [emphasis added]*

**A Much Needed Safety Valve**

The series of fourteen tweets below, sent just as the first marking period for the semester ended, offer a glimpse into my challenges, one of which consisted of the mismatch between students’ expectations for AP Calculus AB and the College Board’s expectations, represented by me, which I detailed above. Sending the tweets literally helped me regain my strength and passion. They served as a much-needed safety valve, releasing pressures built-up from the first six weeks of teaching upper level mathematics students in my school.

The tweets also offer a modicum of insight into the challenges new teachers face; even those with deep content expertise, a lifetime of experience from which to give greater context for content, tremendous passion for helping ALL students, recent AP training from the College Board, and state-of-the-art pedagogical knowledge from one of the nation’s highest ranked ed schools. OK, that did sound a bit much, but you get my drift now, I hope. More importantly, this experience was one of the best examples of the adage “That which does not kill us makes us stronger,” as derived from the Friedrich Nietzsche quote. Fortunately, I have lived to see another day, and look forward to starting my second semester this coming Monday.

[1] Ironically, it is the first foray into calculus, learning the concepts of limits, continuity, and differentiability, that is the most conceptually challenging for students. It is also challenging since they need to bring to bear much of their prerequisite knowledge and procedural skills simplifying and solving expressions in various algebraic forms (e.g. rational, exponential, and trigonometric) just to finish problems in learning these new concepts. However, once the concepts of limits and continuity and the introduction to derivatives using the definition of a derivative is completed, and more handy rules such as the power rule, product rule, quotient rule, or chain rule appear on the horizon, procedural fluency regains a larger place in the pedagogical mix. Unfortunately, circumstances prevented us from working through this natural evolution. Had that been possible, I believe it would have enhanced students appreciation for calculus, strengthened their ability to persevere in the face of adversity, and prevented significant angst among all parties.

[2] While most, if not all, students had taken precalculus the prior year, retention of the material they learned, and the skills needed to apply them were nearly non-existent. Students struggled recalling the simplest of concepts or procedures in algebra, trigonometry, functions, or limits. Worse yet, prior exposure to many of the derivative rules at the end of their precalculus year without a sufficient understanding of their origin interfered with their ability to focus on, understand, or apply the definition of the derivative. All of these conditions poured fuel on the fire raging in and out of the classroom at the time.

[3] I also believe that many students today are overloaded with AP courses, some taking three to four at a time, too much homework, multiple extracurricular activities, and other obligations leaving them little time for themselves, or an adequate amount of time to rest and sleep. As a parent, I am concerned about the overly competitive nature of school today, and its impact on adolescent development. See “The Race to Nowhere” for one perspective on this dilemma today. Challenge Success is another informative site on the issue.

Hey, Dave, forgive me for doing this, but I’m going to critique this post on its merits as a write up of an event.

If I hadn’t known what happened, I wouldn’t know what had happened after reading your post.

You were tasked with teaching AP Calc to kids who weren’t prepared. That much anyone can get. But how bad was it? Could they graph a line? Could they factor? Where, specifically, was their deficit? Why did the parents react by demanding your head? What specifically happened? Should readers infer that you flunked all their kids, or what? And then what happened?

It’s really unclear. You give the readers 3 paragraphs from the College Board–for no reason I can see–but nothing about what, specifically, happened to make your semester so stressful. As someone who knows what happened, I’m stunned at how you managed to write all these paragraphs yet never once clearly state the general outline of your problem.

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Hi Cal. I did not want to get into the small details…my goal was to describe the larger issue of the students believing they needed to be spoon fed (even wrt prerequisites) versus my pressing for them to let go of their preconceived view of the world and strive for conceptual understanding…also, keep in mind, problems can be framed in many ways…there were layers upon layers of problems here…I may not have scratched your itch for a particular perspective…I might consider various examples for follow on posts.

Thanks for reading and commenting! I value your thoughts and suggestions.

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I had to repost this comment from another blog post here…it is so right on and directly relates to my post here. It is from: http://blog.mrmeyer.com/?p=12428#comments.

This is used IAW CC-BY per dy/dan’s disclaimer: Everything I post here is licensed CC-BY unless otherwise specified. Same goes for everything you post here too.

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9 Jim Doherty

on 29 Dec 2011 at 5:39 am

@Jon

I absolutely agree that this is the ‘right’ way to approach teaching but I cannot count how many times a student has told me, with complete honesty, that s/he wished I would just tell them the way to do a problem. This year one of my seniors in Honors calculus told me that she always thought that math was about knowing which formula to use and remembering that formula. I wanted to weep but this was not her fault, it was the fault of how she has been assessed and rewarded along the way. So, I don’t know where I am going with this, but wanted to join in here…

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Hello,

Curious future math teacher,

can you tell me what the pass rate for the class? AP test? was. Thank you.

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Before I arrived: 30%…after year 3: 82%. Waiting for this year’s results…

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Sorry to keep pressing this question, but I just graduated from Univ. here in Illinois, and I may be getting my first job. It is one AP Calc., a non-AP Calc, and 2 sections of precalculus. So, what was your pass rate for the AP class? and then for the AP test ? I am interested in the data only during the first year (year 1) you taught AP calc. Thank you.

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Year 1 – Course: 100%, AP Exam: 58%

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