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At our university we now require one semester of Pre-calculus instead of one semester of Algebra and one semester of Trigonometry before you take Calculus I (for those who do not test into Cal I). Since we have implemented this, it seems that students who take Pre-calculus, are more likely to fail or withdraw from Calculus I than those students who have tested into Calculus I. This must mean that Pre-calculus is not preparing students for Calculus I. What are some of the underlying causes of this disconnect?

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    $\begingroup$ The cause of the disconnect is simple: students who don't test into Calculus I have poor mathematical preparation coming out of high school, and one semester of pre-calculus isn't enough to make up for this difference. $\endgroup$ – Jim Belk Mar 17 '14 at 17:28
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    $\begingroup$ You should compare students that take precalculus against others who don't, but keeping other variables constant (or do a statistical study, analyzing which variables have impact of calculus success). Perhaps the students taking precalculus just have worse math background to start with, and even while the course helps, it isn't always enough? $\endgroup$ – vonbrand Mar 17 '14 at 17:31
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    $\begingroup$ One small specific example is that at my large American university, we ban calculators in most math courses. In talking with students, they initially have trouble with this computational transition including computation of trigonometry, simplifying fractions, and logarithms and exponents. $\endgroup$ – Chris C Mar 17 '14 at 17:38
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    $\begingroup$ It seems that it would be a good idea to sample those who have dropped out or failed and ask them what was too hard for them. That way, you can identify specific, potentially remediable weaknesses in their academic preparation. Anyone else who offers an opinion is just guessing. $\endgroup$ – Confutus Mar 17 '14 at 18:03
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    $\begingroup$ The strong students test in to Cal I. The weak students don't. Even after a semester of help, they are still the weak students. $\endgroup$ – Gerald Edgar May 4 '15 at 14:56
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A blanket problem I've observed over-and-over is that the deficits that scuttle calculus students are even more fundamental than what is discussed in (typical) pre-calculus courses. Specifically, kids, as well as adults returning to school, often cannot do middle-school (pre-?) algebra. Either they get stuck and confused, or immediately and frequently commit ghastly computational errors... leaving very tiny probability that anything will turn out right.

For example, inability to correctly manipulate numerical fractions, or inexperience with it, leaves many of that demographic completely unable to correctly manipulate ratios of polynomials or any such thing.

Correct manipulation of exponents similarly...

A common further complication is that many in this population are convinced that they "know algebra" because they recall the "A" they got in 8th grade. Thus, they are disinclined to listen to "review of algebra" at the beginning of calculus. (I've found that invariably including far more "algebra" steps than minimal, all along the way, is fairly effective at (re)training people to do that low-level stuff. Leaving the low-level steps to the students is often a serious error...)

Thus, if "pre-calculus" is still predicated on the optimistic notion that the population of kids who don't test into calculus surely at least remember middle-school algebra, it won't address their fundamental deficits.

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    $\begingroup$ In addition to this, understanding what you can and cannot do with an equation that should be learned very early on is usually not revisited at all during later classes. The number of students that have issues in College Algebra because they forget that they have to multiply and divide each term or add and subtract from each side is astounding. This becomes a problem again when working with Trigonometric Identities in Precalc. Understanding that you can multiply one side by 1 in the form of cosx/cosx is lost on most students. $\endgroup$ – David G Mar 17 '14 at 18:23
  • $\begingroup$ I upvoted and agree with most of this. But if the idea in the last paragraph is to review algebra as part of precalculus, here's a counterargument: At my community college, we have a sequence of 4 classes which are mostly all basic algebra review (elementary, intermediate, and college algebra, and precalculus). Some instructors teach the first 3 of those courses with identical lecture notes. I think it would be more efficient to have well-defined courses, good placement policies, and if a student can't retain from one to the next, then they fail. $\endgroup$ – Daniel R. Collins Nov 29 '15 at 19:44
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    $\begingroup$ @DanielR.Collins, well, in my experience, many students certainly do not retain things, because it surpasses their imagination that, for example, getting the minimum acceptable grade this term will be grossly insufficient for coping later, etc. Most of their other coursework does not have this feature, after all. $\endgroup$ – paul garrett Nov 29 '15 at 20:28
  • $\begingroup$ @paulgarrett: I agree with everything you just said. I'm just holding out our experience as a case study that even nigh-endless review does not suffice for some students, and then erroneously signals that mathematics is just all-algebra review all the time (and then have students bewildered in a later class when they're asked to study new material on a routine basis). $\endgroup$ – Daniel R. Collins Nov 29 '15 at 20:37
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    $\begingroup$ @DanielR.Collins, ah, ok, I partly misunderstood. Indeed, the "problem" population seems invulnerable to "review" on its own. For such reasons, I'm not a fan of "pre-calculus" as a curricular concept... but of highly targeted "covert review" in the midst of "calculus". $\endgroup$ – paul garrett Nov 29 '15 at 20:50
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What those students need is a course that will help them see math differently. They need to learn to think mathematically. An excellent pre-calculus course can begin to do that. But at my college (for example), pre-calc has way too many topics. If I taught the content I'm supposed to, we could never dig in properly. I skip some topics, so that I can work more with students on how to think through a problem.

Also, the calculus teacher can help by going over why we take each step we do. Any review should be done not at the beginning of the course, but when it is needed. For example, when multiplying by the conjugate to simplify the expression we get for derivative of a square root (using the definition of derivative), we may need to review the idea of the conjugate (and note its usefulness in a wide variety of contexts).

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    $\begingroup$ I think one of the problems we have is that college algebra and Pre-Calculus are funded by the state and they tell us the curriculum we must teach. This curriculum does not allow us to go deep enough into the subject matter like we could with Trigonometry which was not state funded. Sounds like a good research project. $\endgroup$ – Todd Thomas Mar 18 '14 at 13:01
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My unofficial list of things not explicitly covered in precalculus courses that cause people trouble in calculus:

  • $\frac{6x + 3}{3}$ is not $2x+3$, and $\frac{2x + 2}{x}$ is not 4.
  • $\sqrt{x^2 + 9}$ is not $x + 3$.
  • $(x - 4)^2$ is not $x^2 - 16$.
  • $x^{-1/2}$ is not $-\sqrt{x}$.
  • If $f(x) = x^2$, then $f(x+h)$ is not $x^2 + h$.
  • $x^2 + x^3$ is not $x^5$, or more commonly, $e^{x+2} - e^x$ is not $e^2$.
  • $x\sqrt{x}$ is not mysterious; it is $x^{3/2}$.
  • $\ln x^2$ is not $(\ln x)^2$
  • $\sin^2 x$ is not $\sin \cdot \sin \cdot x$ because that doesn't mean anything.
  • If $\log x = 3$ then $x$ is not $\frac{3}{\log}$ because that doesn't mean anything.
  • $\frac{x}{2/3}$ is not $\frac23 x$.
  • $\frac{x}2$ is not mysterious; it is $\frac12 x$.
  • If $\sin 2x = 1$, then dividing by 2 does not yield $\sin x = \frac12$.
  • If $\sqrt{3x} = 9$, then dividing by 3 does not yield $\sqrt{x} = 3$.
  • If either of us uses the word "is" or the equals sign, like "$x^2$ is $2x$," we literally mean they are equal, not "After applying [some operation] to $x^2$, it becomes $2x$." See also the excellent "equals signs" threads on this site.
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