# So what was precalculus, again?

As the first math teacher in a brand new school (9 years ago), I’ve had a bunch of time to consider that question. It’s sort of a mess.

There is no branch of mathematics called “Precalculus” (there is Algebra, there is Geometry, there is Trigonometry. Hmm, there’s not really Calculus, except for The Calculus, but there is Analysis. Maybe we should rename Calculus as “Beginning Analysis (with actual functions and numbers)”)

So what belongs here? And why?

Doing it just because we’ve done it before is a fairly lousy reason. I want to steer clear of that.

In the next days and weeks I will describe different approaches and variations we’ve taken. This year we made some changes, I’ll go into those as well, and the results seem positive. Emphasis on “seem.” I still want to ask, “what was precalculus again?” to have some sort of yardstick to measure our progress against.

JD,

We call our “pre-calc” class Math Analysis…but it really doesn’t make it easier to decide what it “should” contain. I’ve been trying to figure this out for a lot longer than your nine years, so good luck. I look forward with interest to reading what you’ve done, what you think works. I’d like to get it right (or at least right-er) before I hang up my chalk.

“math analysis” is indeed more accurate.

i’ll tell anybody who’ll listen that when

we refer to “calculus” in school mathematics

we’re “really” talking about *differential*

calculus and *integral* calculus; that

*a* “calculus” is a set-of-calculational-

-techniques (or a rock-like buildup on

some organic structure). thus, for example,

“propositional calculus” (a subject that

deserves a much greater place in the

curriculum… the differential and integral

c’s get far too *much* attention by contrast).

forgive me if i hope you’ll give a good bashing

to stuff that usually *is* included but ought *not*

to be (calculator-based approaches, e.g.);

my usual contrarian streak showing through

i suppose. anyhow, looking forward for sure.

I’m interested to see what you conclude. It’s always confused me. There are so many regional/etc difference and haziness around what belongs in Algebra 2, Trigonometry, and Precalc.

Or you could just take the NYSED approach and throw it all in one year and call it done.

I’m also very interested in this. I probably have the least experience here, so I’d like to hear about what paths others have taken.

To add to the problem, in California, the standardized test students take when they take Pre-Calc is called “Summative High School Math” which is a combination of Algebra 1, Geometry, Algebra 2, and sprinkles of Stats. Topics that I cover or that students practice don’t align with a test that I’m held accountable to. So I end up juggling demands from state accountability measures, district expectations, requirements from our course submission to UC (a-g) system, expectations from calculus teachers, and student needs.

I always thought of it as “Algebra III”. For us, it included things like trigonometry, logs done right, linear programming, 3rd order+ functions and relations, prob/stats, etc.

Since calculus requires a solid algebra foundation, that was a major reason for the focus on algebra..

That’s a nice set of ideas. And there are more out there. Before jumping to the next segment, let me share what I went through, honors track, suburban northeastern high school (ok, Amity Regional, outside of New Haven, Connecticut) back about 30 years ago (78-82).

8th grade algebra, traditional american, non-descript text (later they told me that we used 2, one was Dolciani, the other was similar)

9th grade was geometry – we sort of taught ourselves, the 3 of us, and played a lot on the computer. (we punched our programs into paper tape)

10th grade was algebra II – traditional.

Up to that point I am fairly certain we had no, as in zero, trig. No circles, no functions, no triangles, nada.

11th grade. I thought I took trig. Looking back, it was called Math Analysis. Spent almost half a year on trig. A full quarter on counting and probability. The rest was all over the place. I recall extracting square roots by hand.

12th grade. Calc. I thought AB. Then I recently disagreed with a professor at a local college about the content of AB, and I looked back, and no, I did not take AB. I took two terms of University of Connecticut Calc (1 and 2)

So what role did the math analysis course play? It filled in 100% of my trig. It let me flex my math muscles without a test/goal. It hit assorted odd-ball topics. And it taught me better probability and counting than any of my NY State educated colleagues ever got (their high schools didn’t teach the stuff, then taught it badly in Sequential 1, 2, and 3. Or they took it in college, and never had a chance to play for weeks on end with conditional probability and interesting counting questions).

Was it worth it?

Hmmm. I like that I took it. I needed the trig before calc. I could have done just that? (shaking head vigorously NO. The chance to play, that’s what I gained, what I wouldn’t have traded…)

If I were designing pre-calc for students who would take my calc (college calc 1), I’d emphasize algebraic trigonometry (algebra with trig functions–solving trig equations, knowing the formulas and identities, that sort of stuff). I’d also make sure I got in the binomial theorem. and a fair amount of graphing rational functions by hand (knowing without plugging in numbers what the function should look like: where the roots and asymptotes are, what the roots and asymptotes look like–odd and even multiplicities that sort of thing). Fractions are good too–complex rational functions, and manipulations of rational functions. These are the sorts of things that I think would make my students more successful in calculus. It’s not that they haven’t done these things at all before, but they aren’t fast/confident enough with them and so they get distracted by the algebra and can’t focus on the calculus.

FWIW, I think it is telling that James Stewart, author of Stewart Calculus (which, judging by the fact that he has a $23 million showplace house in Toronto, I’d argue is a fair bet for being the standard textbook), was so exasperated by the state of his students’ preparation for Calculus I classes he went back and wrote a whole Advanced Algebra and Precalculus textbook sequence that puts functions (including hand-graphing and evaluating) front and center in the book. It covers all the things that LSquared advocates, and has clear, consistent, and very logical explanations, although but his problem sequences are not always the best (they tend to make a leap of faith about understanding about 75% of the way through that I do not always find to be warranted).

David Cohen’s Precalculus textbook, on the other hand, has some rather confused explanations of topics but has such beautifully sequenced practice problems I am only too happy to forgive him his clumsy explanations.

It’s not just the topics that make Precalculus a worthwhile class for the not quite-top-tier math students (like me) who need Calculus (or whatever we call it). It’s also the scaffolding to support their development as mathematical thinkers, as well as a pace that suits the levels of emotional maturity and organization they need to develop to be able to work through complicated algebraic problem-solving at an advanced level. Most (not all, but most) students who get shoved through all their algebra requirements before they hit 14 or 15 simply don’t have the retention, organizing skills, or self-awareness to tackle the algebra and algebraic trigonometry they will need to thrive in Calculus.

The truly able students will always get what they need from whatever class and/or curriculum we toss at them. But the moderately able students seem to benefit from that added, non-punitive year in which to consolidate their algebraic and trigonometric understanding.

– Elizabeth (aka @cheesemonkeysf on Twitter)

At Clatsop CC we teach in three 10 week terms and have a College Algebra/Pre-Calculus/Trigonometry sequence that covers

College Algebra – Review of rational expressions/equations, quadratic formula, polynomial/rational functions including inequalities, Rational Roots, Exponents/Logarithms

Pre-Calculus – Functions (we spend half the 10 week term on functions) notation, increasing, decreasing, max/min, composition, inverse and then word problems which are essentially calculus max/min word problems with the TI83 finding the max/min. The reason for this is that when I taught Calculus, I saw so many students struggling to set up the problems, they never even got to differentiate them! In the second half of the term we cover circles and parabolas as conics, then paraboloids of revolution and finish with sequences and series. One year, we had time for some discrete Permutations/Combinations, but I think I’ve added in more word problems in the functions section and we don’t have time anymore.

Trigonometry – Right triangle trig, graphing trig functions by hand, identities, trig. equations, law of sines, law of cosines.

This seems to work well for our students, there are always places we could trim or add topics to make them more or less challenging and so forth, but this works well. The students are challenged but not deluged…

When I taught Pre-Cal I asked AP Calc teachers what they considered to be the properly prepared student. The consensus was a well-prepared student should know how to graph any function no matter what family the function comes from and be able to graph without any self-doubt. A well-prepared student should be able to find any asymptotes and zeros, determine end behavior, sketch a function, state a function’s domain and range, and understand the reason(s) a function looks the way it does (number sense). All function families should be covered including Linear, Polynomial, Trigonometric, Logarithmic, Rational, Exponential, Step, (anything else?). Then, students should be able to model a data set in order to make predictions and/or generalizations about a data set. I started teaching before the graphing calculator came on the scene. While I use the calculator some, I emphasize paper and pencil sketching skills. I believe kids should be able to produce great graphs on the back of any envelope, or paper napkin for that matter. There’s usually time for binomial and counting etc. When I took pre-calculus in 1979 we used Brown-Robbins…that’s about all I remember. I think that course included trig. I agree with all the trig identity work…that should go into Pre-Calc. I spent three years being the pre-calc person at a small independent school. I really enjoyed the course. I like the whole graphing approach and trying to get kids to delve into function behavior. Some kids were intrigued by the modeling aspect. As a resource I consulted a book produced by the North Carolina School for the Math and Sciences…I think that’s what the school is called.

I’m doing student teaching right now at school where pre-calc (called advanced topics) is optional. Most freshman take geometry, most sophomores take algebra 2, and then they are ready for Calculus. Trig is one (of 8) units in algebra 2. I think this seems an excellent idea. It allows students who so desire to get to Calculus quite soon, but there is also the option of taking the advanced topics class (pre-calc) as further preparation before calculus.