# Let ‘um Try!

Last week, I received an email from a mother whose concern really ticked me off.

The mother of a student entering eleventh grade this fall wrote because her son wanted to take AP Calculus next fall. He will not be allowed to take the course because he missed the cut-off grade in his precalculus course by less than two percentage points.

That’s wrong!

The College Board, under prerequisites in the current calculus Course and Exam Description, states, “Before studying calculus, all students should complete the equivalent of four years of secondary mathematics designed for college-bound students.” Her son has done that. There is nothing about achieving a certain arbitrary average.

A percentage grade does not tell you anything useful. (I’ve discussed this before. See here and the opening paragraphs here.)

How can you pass a student in precalculus and then turn around and tell them they aren’t ready for calculus? If they are not ready, fail them in precalculus.

It is certainly reasonable to council a student with an average or below average grade. You can, and probably should, sit down with them and their parents and explain that they may find the AP course difficult, and to do well they will have to commit spending more time and effort than they may be used to. Offer them extra help – it’s you job! If their grade was a D or D– you can be a little more insistent that they think it over carefully. But to flat out deny them the opportunity is just wrong.

Passing Rate.

Often teachers and administrators are concerned about their passing rate in AP courses. I once attended a session at an NCTM meeting where a teacher explained how he achieved a great passing rate each year in AP Calculus. It is easy to do. He explained that he carefully weeded out those students who were not likely to do really well. Once in the course, if they were struggling, he counseled them not to take the exam. (Why waste your money?) So, if passing rate is your concern, that’s how to do it.

He ended with, “I don’t want those students in my denominator.”

Let’s say you carefully select the students who take the course and later the exam. You have 10 students who take the exam, and they all get a qualifying score (3, 4, or 5). Well, great you have a 100% pass rate.

Now let’s say you let another 10 students into the course. Twenty kids take the exam and 15 get qualifying scores. Even better. An additional five students have earned qualifying scores. That’s what counts! You’ve done a better job! And the five students who did not qualify will benefit from having taken a college-level course and will be better prepared for math in college.

Your pass rate has dropped to 75%, but you’ve helped more students – and that’s what it’s all about.

When I was working for the National Math and Science Initiative (NMSI), part of my job was to vet schools who wanted to join our program. NMSI insisted that there be no artificial barriers or cut-off points for admittance into all AP courses. Students who had passed the prerequisite courses and wanted to take an AP course had to be admitted. And it worked: the schools had more students qualify every year than the year before.

If all your students are earning 3, 4, or 5, you are being too selective.

I gave the woman what advice I could. I hope it helps. I wish I could have done more for her and her son.

# AP Calculus Prerequisites

College Board Prerequisites

Whenever I led a calculus workshop or APSI, I always spent a little time discussing the prerequisites for AP Calculus. Unfortunately, in some schools AP Calculus is a course for only the talented and little time is spent aligning the mathematics program and courses from 7th grade on so that more students will be able to take AP Calculus. But a program that includes the prerequisite for calculus will be a good program because of this. Such a program will also benefit students who do not take AP Calculus, but still need a good mathematics program for when they attend college.

Teachers in the earlier courses are usually appreciative of guidance from the AP Calculus teacher as to what should be included to prepare students for calculus. This is part of the rationale of the AP’s math Vertical Team program.

Below in blue is the entire prerequisite paragraph from the 2019 AP Calculus Course and Exam Description p. 7. I have separated the parts and commented on each.

Before studying calculus, all students should complete the equivalent of four years of secondary mathematics designed for college-bound students:

The four years is needed. Students should not be rushed.

In some respects, this is a political statement: four years means starting in 8th grade or earlier. While some of the most talented students can probably catch up by doing two years in one or three years in two, this is not the usual case. Learning math thoroughly takes four years.

Once in my district, our junior high decided to raise the standards for their “advanced” course that taught Algebra I in 8th grade. No one told us, so the next year we found only one class, instead of two, that could be ready for AP Calculus by the time they were seniors. We tried a three-years-in-two approach. It met with only limited success. Algebra I in 8th grade is required and really should be for everyone otherwise you are denying students the chance to even consider AP Calculus when they are seniors.

courses that should prepare them with a strong foundation in reasoning with algebraic symbols and working with algebraic structures.

Using and understanding the use of mathematical notation is a must. Throughout the four years, algebra and its structure should be emphasized.  So, it’s not just 4 years of math, but four years of a good algebra-based math program. But algebra is not the only thing:

Prospective calculus students should take courses in which they study algebra, geometry, trigonometry, analytic geometry, and elementary functions.

All these courses are related and lead to a fuller understanding of high school math topics.

These functions include linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions.

This is a list of the types of functions that should be included. They are the basic functions studied in the calculus. Linear and simple polynomial functions start in Algebra I and the others are added later. Piecewise-define functions also start early – the absolute value function is a piecewise-defined function.

In particular, before studying calculus, students must be familiar with the properties of functions, the composition of functions, the algebra of functions, and the graphs of functions.

The algebra of functions means learning how to add, subtract, multiply, divide, and compose functions and how doing so affects the properties and graphs of the resulting functions. The graphs of these functions and how doing algebra, composition, and transformations affects the graph is important.

Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts, and descriptors such as increasing and decreasing).

The list of the language functions is too short. Some terms such as increasing, decreasing, maximum and minimum values, concavity and others often considered the province of calculus all come up in the study of functions and can and should be discussed when they arise using the correct terminology and notation. There is no need to wait for calculus to use them to describe functions, graphs and transformations. An informal use and understanding of continuity and limits should be included. Asymptotes should not be overlooked (they are the graphical manifestation of limits and continuity or the lack of same). The more students learn before calculus, the less you’ll have to do in calculus.

Students should also know how the sine and cosine functions are defined from the unit circle and know the values of the trigonometric functions at the numbers $0,\tfrac{\pi }{6},\tfrac{\pi }{4},\tfrac{\pi }{3},\tfrac{\pi }{2}$ and their multiples.

Yes, with all the technology available these basic trig facts should be learned (learned, not just memorized); they are always tested on the AP Exams.

Students who take AP Calculus BC should have basic familiarity with sequences and series, as well as some exposure to parametric and polar equations.

Here I disagree. Parametric equations, vector equations and polar equations should be a part of the curriculum for all students. Students who do not take BC calculus, may well take more math courses in college and should understand these ways of working with the plane and with functions defined in different ways.

This list does not define the entire high school math program. There are other topics that can and probably should be included – statistics, systems of equations, linear algebra and matrices, proofs, probability to name a few. What it does define is what should be included so that students will be ready for calculus.

What I think is missing here is the use of technology. In the world today mathematics is done with technology. The proper use of technology should be an integral part of the program from before Algebra I.

AP Statistics is a great course. Students who have completed Algebra II should consider this course. However, AP Statistics it is not an algebra-based course. About three-quarters of the course and its exam is writing; there is very little algebra involved. Therefore, students should not be taking AP Statistics instead of AP Calculus, or if they are not taking calculus, instead of a third year of Algebra. The AP Statistic prerequisites state:

Students who wish to leave open the option of taking calculus in college should include precalculus [i.e. a third year of algebra] in their high school program and perhaps take AP Statistics concurrently with precalculus.

Students with the appropriate mathematical background are encouraged to take both AP Statistics and AP Calculus in high school.

AP Statistics 2019 Course and Exam Description p. 7, emphasis added.

The point is that students should not have a year in high school without an algebra course. A year in which to forget their algebra before going to college where they may need it again is not a good idea.

I like to think of all the mathematics courses before calculus as “precalculus.” In many schools, “precalculus” is the name of the last course before calculus. That’s okay, I guess. What I disagree with is that often the precalculus teacher, with the good intention of preparing their students for calculus, teaches them “derivatives.” By which they mean the rules for computing derivatives. This really does not help the students or the calculus teacher.

Derivatives are limits and derivatives are slopes; computing derivatives is the least of your worries. If students have learned all the other precalculus topics (including parametric, vector, and polar equations) well and there is time left, consider delving further into limits and continuity. Limits seem to be more difficult to understand and some repeating of the topic when students arrive in calculus will do no harm. Leave the calculus for the calculus class. (The exception is when the precalculus class is intentionally meant to get an early start on the calculus; when it is taught by the calculus teacher or a teacher who is aware of the Essential Knowledge and Learning objective of the AP Calculus course.)  – Just my opinion.

High School Prerequisites

Some high schools add their own prerequisites to enter AP Calculus courses. This usually means students have to earn a significantly higher score than just a passing grade in the precalculus course(s). I do not agree with such a policy.  It excludes students who may benefit. If your student passed the precalculus course, even with a low grade, how can you say they are not ready for calculus? What will make them more ready? True, they may have to struggle, but that won’t hurt them. You may want to council them (and their parents) and explain, without discouraging them, the amount of work and time required in a college level course like AP. Explain the amount of time and work they will have to spend once they get to college in a course that meets far fewer times then AP Calculus to cover the same material. Even if they end up without earning a qualifying score on the AP Exam, they will still benefit by putting in the time and effort. If they want to try, encourage them.