For years the Precalculus class at Aragon has served as the hurdle/gateway to AP mathematics. The unfortunate effects of this “Harry Potteresque sorting” result in my receiving annual calls and emails from desperate parents and students. I could have just kept quiet and reveled in the boon the class gives my tutoring business. However, I felt compelled to speak out publicly and address the problems with the system, particularly when they came to a head early in the 2105-2016 school year.

My personal goal is to teach and try to inspire students to love math and science, and I have grown extremely weary of instead having to put band-aids on what I consider is a broken system.

If you are or will be soon a parent of an Aragon student(s), I urge you to read the following comments carefully and take them into consideration when your student chooses his/her math courses. I also encourage you to contact the school and lobby for other math options such as I suggest in the article below.

A disastrous CPM precalculus textbook experiment two years ago was the reason that I began writing public commentary on local education, first on my local Nextdoor.com site and later on this blog. Earlier articles include the following:

Good news regarding precalculus from Aragon

Good News re Aragon Precalculus

By the end of the 2015-2016 school year, good sense prevailed, and the textbook was dropped in favor of a more conventional text. **Unfortunately that year of students suffered the consequences; this past school year was the very first time that none of my students decided to move on to Calculus BC after taking AB. I believe the CPM ****experiment played a role in their decision due to their weaker knowledge foundation. ** Instead they opted to take the supposedly “easier” AP Statistics class (more on that topic in an upcoming article.)

However, first let me note the good news about recent changes in the Aragon precalculus curriculum.

A new teacher, Kris Reiss, joined the department during the CPM year, and he made a concerted effort to assist students who were struggling in the program. He provided a free after school tutorial, and, contrary to CPM orthodoxy, made an effort in class to explain concepts rather than leave students to discover them through CPM’s guided questions method. Mr. Reiss continues to do his after school help sessions, and I always hear favorable comments from his students. I would like to personally thank him for his efforts to help our local students.

Second, this year Mr. Behrooz Shahrvini taught an accelerated algebra 2 / precalculus class. About 3 years back, the middle schools, in response to Common Core demands, made it more difficult for students to get the required prerequisite math to take calculus in high school. Because Mr. Shahrvini teaches the multivariable calculus class at Aragon, this had a direct impact on him.

He responded by teaching an early morning “zero period” geometry class at Borel, in addition to the classes that he teaches at Aragon *and* at CSM. Despite these efforts, a student recently told me that there were not enough students to offer multivariable calculus at Aragon next year.

Mr. Shahrvini is the most accomplished math teacher at Aragon, and is always well-regarded by his students with whom I have worked. Aragon should not lose his talents. The school also attracts a cadre of highly accomplished students whose parents have provided them with extracurricular math training at places like the Russian School of Math. It makes sense to provide a multivariable class for those students who may be ready for it.

However, as I have mentioned previously, this cadre of overachieving math students can create an unfortunate consequence for their friends who are also smart but have not had extra after-school math training. This second group of students feels compelled to try to keep up and consequently take steps like skipping from precalculus directly to AP Calculus BC because “their smart friends are doing it” *and the school allows it*. I have always vigorously counseled against it except in exceptional circumstances for students who are objectively prepared. Skipping from precalculus to BC leads to many students rushing through the calculus curriculum and coming away with a very cursory knowledge of the subject which is not long retained.

Personally I think that more students should be encouraged to take both calculus AB and then BC, and that Mr. Shahrvini should play a role in teaching some of the resulting extra BC classes where his efforts would also be appreciated. More Aragon math students would graduate with a better foundation in calculus. I remain concerned that the AP math classes are viewed as a required hurdle for college admission, and are taken primarily for that reason. However, because the pace of AP calculus is too fast, I almost always advise my students to retake calculus when they attend college if they need to really understand the subject for their STEM majors. Berkeley seems to agree with me.

**Returning to precalculus issues**, the chapter/section tests in the class have become more reasonable than several years ago. Back then it was fairly common to have so many challenging questions on some tests that students who had diligently completed the homework and understood the core concepts, still ran a risk of getting a D or an F in the class due to their lack of mathematical insight, i.e., they could carry out the computational methods but were not clever enough to solve problems which might require a novel combination of recently learned ideas. This lack might be grounds for denying them an A and possibly even a B, but it was far too often keeping them from even passing the class. The class tests are not quite so strict now (challenge problems are now usually for extra credit), but many students still struggle because of quantity of material reasons described further below.

If students do not pass precalculus, Aragon has only one option for them called Finite Math. The gap between Finite and the two other classes, AP Calculus and AP Statistics, is wide, and I advocated that Aragon offer a somewhat easier non-AP Calculus class to fill it, but to date this has not been done.

**So why does precalculus remain so hard?** Of course, part of it is due to the intrinsic nature of the subject itself, but the problem is compounded by the amount of material covered in the class. The precalculus teachers might seriously consider dropping some of the topics from the course, e.g., linear programming as just one minor instance.

Here are several of the topics covered during just the second semester:

- review of exponential and logarithmic functions and their domains, ranges, and graphs
- trig functions and their graphs (with a lot of tricky problems involving transformations of the graphs of the parent cos, sin, tan, sec, csc and cot functions)
- trig identities
- solving equations containing trig functions
- vectors and vector products
- vector application problems
- parametric and polar equations (a brief description hiding a lot of details)
- systems of equations and matrices including linear programming problems
- conic sections: their equations in both Cartesian and polar coordinates. Sketching the graphs given the equations and determining the equations given information about the graphs
- arithmetic series
- finite and infinite geometric series
- limits of functions, continuity
- definition of the derivative including derivation of basic derivative rules
- Computation of derivatives using more advanced rules such as the product, quotient (and sometimes even the chain) rule.

Each of the short bullet points above conceals a wealth of rather intricate detail. Looking at the school calendar, there are about 16 weeks in spring semester to teach after one removes holidays, standardized testing periods, and “dead” week for final review. That leaves only slightly more than a week (4 or 5 class periods) for each of the topics above plus testing.

The pace is frenetic, and I am rarely surprised when I begin my final review sessions with students (including the ones who end up with A’s in the class) that they have forgotten almost everything farther back than the previous couple of weeks.

The only way that a student can master this amount of material is through a considerable amount of practice (this is where those taking extracurricular math have an advantage), and too many students do not seem to have an idea of what kind of time commitments are in store for them when they take this class. I frequently get calls from parents that begin, “My child always got an A in math prior to taking this class.”

I personally think that high school is still a place where students should be educated instead of merely washed out of the system (save that task for lower division college…), though this is admittedly a topic for debate. The way the Aragon math department operates the precalculus program, it definitely disagrees with me. I have no sway over how they run their curriculum and can merely suggest that it might be slightly more humane to curtail some of these topics or (and this offends current teaching orthodoxy) do a better advance job of streaming students into different classes depending upon their math ability. In other words, put the “Russian School” kids and their like into a faster paced class, but this would probably be too complicated to schedule.

Finally, if the precalculus teachers can not bring themselves to reduce the number of topics taught, **at the very least consider giving the final exam on just a subset of the most important topics.** Please put together final exam review sheets that list just those topics and do not test students on topics that are not on the review sheet! Given the lack of precalculus retention that I observe repeatedly, it is a major time sink for students to review the entire set of current topics in the limited time allotted for this purpose before final exams. They also have to prepare for their other finals!

If changes along the above lines are not or can not be made, students will simply have to be better informed, prepared to work very hard, and commit the time to the class at the trade off of less time spent on other classes and after-school activities such as sports (which also seems anathema to too many people possessed of the “we can have it all” attitude). *In light of all these demands, it is rather ironic that the SMUHSD Board of Trustees is currently puzzled by a student survey showing that a significant fraction of local students are “sad”…*

Finally, if the Aragon math department wants to continue to run the program in the current manner, they should at least seriously consider giving those who can’t get strong grades in the class better options than just Finite Math.

FYI: This year at San Mateo High, the precalculus classes didn’t cover the following topics:

– polar equations. (parametric equations were seen). (Students missed the opportunity to see it before Calc BC)

– systems of equations and matrices including linear programming problems – Too bad, matrices are very useful for understanding data structures in computer science.

– conic sections: (Cartesian and polar coordinates)

– Computation of derivatives using more advanced rules such as the product, quotient (and sometimes even the chain) rule.

But they had a chapter on probability because with the changes the last few years with common core, some students have not seen any probability and statistics in previous classes!

A.

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Thanks for your comments, Anne.

If I go through the Aragon curriculum point-by-point (which would have made an already long article even longer) I can give a reason why almost everything on it should be included, but for several items the reason is similar to what you indicated above in one place: the students will see the topic again in Calculus BC. However, most students will have forgotten these brief and hurried introductions not long after the ink is dry on their finals, so including topics that won’t be seen again until two years later is only justified if your intent is to skip calculus AB.

At some point one has to draw the line and admit that it just might be possible that too much is being attempted. Topics that will not come up again until BC would be potential candidates for elimination. Of course there will always be the happy genius or the child who has received copious extracurricular training who will be trotted out as the justification that the workloads can be handled by some students, but the real question is what is the effect on the entire class?

Not being a faculty member, I unfortunately never get to see the overall class grades for precalculus. When I ask my students what the class average was on any particular exam, this information rarely seems to be announced. However, if the current system is really working well, why does this parental question keep popping up? “My child always got A’s in math until they took this class!”

To phrase a possible answer in a way that a mathematician might understand, perhaps the double-sided limit does not exist and the function is not continuous 😉.

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