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There seem to be two approaches to calculus education:

  • Early transcendentals: introduce polynomials, rational functions, exponentials, logarithms, and trigonometric functions at the beginning of the course and use them as examples when developing differential calculus
  • Late transcendentals: develop differential calculus using only polynomials and rational functions as examples and introduce the rest afterwards.

Most American universities seem to use the early transcendentals approach, but I have never seen any empirical evidence that it is superior (e.g. higher pass rates, success rates in courses which depend on calculus). So my question is:

Is there any empirical research comparing the outcomes of the early transcendentals and late transcendentals approaches to teaching calculus?

I am hoping someone has done actual controlled experiments, but I would also be interested to hear the opinions of people who have considerable experience with both approaches.

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My guess to why U.S. universities use the early approach is that most of the students have seen the transcendental functions in pre-calc. Why so? Because there are quite a few programs in various sciences that don't require calculus, but do require knowledge of log, exp, sin, cos, etc. –  Aeryk May 12 at 17:11
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If you look closely, the problems already lie in polynomials. Most students are only comfortable with power functions and quadratics. –  Toscho May 12 at 18:13
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Students being uncomfortable with exponentials and logs is precisely why early transcendentals is a good thing. They need to work on the exponentials and logs much more than a token week or three at the end. –  James S. Cook May 12 at 21:55
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A possible disadvantage of waiting a long time to do transcendentals is that there is little motivation to understand the chain rule or product rule. If you memorize a recipe for differentiating rational functions, then you're pretty much all set. However, this can be addressed somewhat by asking students, e.g., to differentiate $(x^2+1)^{1000}$, which is impractical to multiply out before differentiating. –  Ben Crowell May 13 at 0:01
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How early is early, and how late is late? To me, early would be if transcendentals were in the same chapter as the product rule and chain rule (as in the text by Larson), and late would be if transcendentals came 2 weeks after the product rule and chain rule. But AndrewC's answer says that students in the UK don't differentiate transcendentals until the second semester. That seems very late to me. –  Ben Crowell May 13 at 15:57

4 Answers 4

Disclaimer: I don't have any empirical studies, but do have substantial experience of the UK's hybrid between early and late transcendental functions in the 16-18 A-level Mathematics qualification.

How calculus is introduced in the UK

(There are a number of different examining boards with different specifications, but the split between first year "AS" mathematics and second year "A2" mathematics is specified by the UK government, and recently (2014) the January exam session has been abolished, so that all students sit all of the year's papers in the summer at the end of the academic year.)

Here is just the part of the content I feel is most relevant to your question:

In the first year, students learn how to

  • differentiate and integrate polynomials
  • deal with negative and fractional powers,
  • use derivatives to find and classify stationary points, find equations of tangents and normals
  • solve polynomial differential equations of the form dy/dx=p(x) with a simple boundary condition
  • use calculus in the context of simple rate-of-change and maximisation problems
  • use logs to base 10, 2, a etc to solve equations and manipulate expressions (no calculus)
  • solving trigonometrical problems involving sin, cos, tan (no calculus)

In the second year, students learn how to

  • use $e^x$ and ln x algebraically, as for logs to other bases in the previous year
  • differentiate and integrate sin, cos, tan, $e^x$, ln x
  • use the chain rule, product rule, quotient rule
  • integrate by parts, by substitution and by inspection (typically fractions resulting in ln (v(x)))
  • use cot, sec, cosec and associated trigonometrical identities including to facilitate calculus
  • solve seperable first order differential equations
  • use calculus with transendental functions to model growth and decay and rates of change

Note that in the first year, treatment of the derivative as a limit is encouraged by the specification but not assessed. An additional A-level qualification called Further Mathematics is available that includes the study of limits, but it is taken by a self-selecting minority of students.

Individual schools and examining boards are free to stage this as they please within the year, some covering all of first year differentiation in the first term followed by all of first year integration in the second term, others splitting the terms as calculus with positive integer powers then calculus with fractional and negative powers.

Advantages and Disadvantages I have come across for this two year arrangement

Restriction to polynomials in the first year

The restriction to polynomials in the first year helps students become familiar with them, and is very helpful at first.

However there are significant problems in the second year with over-generalisation of the method, particularly when x appears as a power.

[Admittedly, over-generalisation is common elsewhere with linearity/the distributive law being a student favourite, seemingly unbounded in the contexts in which it can be misapplied (logs, squares, square roots, 3(xy) etc etc).]

If we were to introduce how to differentiate sin, cos and $e^x$ in the first year, students might be less inclined to over-generalise, but there's no guarantee that that's the case.

Introduction of transendental functions

Introducing these is in my view necessary to motivate the product rule and integration by parts, otherwise in the presence of polynomials alone it becomes a rather bizzare way of obtaining the correct result. It also provides a much richer and diverse set of examples to use for all the calculus techniques of the second year.

I certainly use $x^2$ and polynomial products as very early examples of the product rule, because it's then clear to students that multiplying the derivative of two factors gives the wrong answer.

However I believe it's important to use these functions as much as possible and in as many different combinations from as early as possible (we do so at our college as soon as we start second year work), precisely because they behave very differently to each other and to polynomials, so that the students have a better understanding of calculus which they would otherwise perceive as primarily about powers and coefficients.

Summary

Were I at liberty to alter the split, I would introduce sin, cos and $e^x$ alongside polynomials and fractional/negative powers in the first year, including in simple differential equations.

I believe early diversity is helpful in correctly generalising concepts from examples to principles, and the longer they are exposed to these beautiful and interesting functions the better they will be able to deal with them.

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Summary: 1) Showing differentiation and integration restricted to polynomials leads students to over-generalize those rules, particularly when $x$ appears as a power. 2) The product rule and integration by parts are bizarre when restricted to polynomials, and better motivated in the context of transcendental functions. 3) Showing students the diversity of transcendental functions early helps them generalize correctly from examples to principles. The longer that students see these beautiful and interesting functions, the better they will deal with them. –  Matt F. May 13 at 3:46

Literature:

Unfortunately, I have never seen a study that compares these two approaches to teaching Calculus (nor did a cursory search through ProQuest and Google Scholar produce anything of import). I expect a reason for this is that your terminology is nonstandard. The late transcendentals approach is somewhat akin to what might be done in a (U.S.) course on pre-Calculus.

I did see at least one study that might be of interest to you in this regard; essentially, it introduces ideas around finding "turning points" for polynomials and rational functions, and later on connects these ideas to finding derivatives for the polynomials.

Cherkas, B. (2003). Finding polynomial and rational function turning points in precalculus. International journal of computers for mathematical learning, 8(2), 215-234.

Excerpt:

In today’s typical precalculus course, students are informed that a polynomial of degree $n$ has at most $n−1$ turning points. Representative graphs of polynomial functions are also given to help students recognize turning points. Yet it is only for quadratic functions that students are asked to find turning points (vertices), using the technique of completing the square. The general topic of finding turning points is left to calculus, where students learn to compute derivatives, set them equal to zero, and solve the resulting equations. In the case of polynomial and rational functions, however, it is unnecessary to wait for calculus. The purpose of this snapshot is to describe procedures that rely on the Fundamental Theorem of Algebra, spatial reasoning, and solving equations to find turning points for any polynomial or rational function with real coefficients.

The article continues a bit later on:

To gain insight into what happens behind the scenes when using this method, we first examine general cubic functions, obtaining a complete classification for the number and location of turning points in terms of the cubic’s coefficients. With the understanding gained from this investigation, we link this algebraic approach for finding turning points to derivatives of polynomials. We then illustrate these techniques with the assistance of Mathematica to find all turning points for specific polynomials of degrees $4$ and $5$.

Major drawbacks of the study are, first, that it incorporates technology as a major part (which is not something you were asking about), and, second, the Mathematica approach that is discussed for a laboratory portion of the course was not explicitly connected to the classroom-based learning.

Still, the author might agree with your skepticism about the different ways Calculus is initially broached, though perhaps he would suggest directing some of the late approach's material to what would be learned prior to a first course in Calculus. As he writes in the discussion section:

In any precalculus study of turning points, it is natural to include the related concepts of local and global maxima and minima. These visually rich ideas add to the overloading of calculus with concepts that can be introduced earlier in a revised precalculus curriculum. Presenting these concepts in advance of differential calculus introduces a spiral effect to the learning curve: students first see the concepts in the limited perspective of precalculus and subsequently revisit them in the more general context of calculus. Introducing optimization problems in precalculus that are modeled by polynomial or rational functions is part of this spiral approach. It also provides students with realistic precursors of the specific cognitive demands they will encounter in calculus.


My own thoughts:

One difficulty in comparing the approaches you have described is that it is not clear (to me) what success would look like. Your example suggestions are "higher pass rates" and "success rates in courses which depend on calculus." For the former suggestion: I think it is clear (perhaps with some thought) that this will not be a good metric. For the latter suggestion: Calculus students are remarkably diverse with regard to their goals. Are they intending to follow differential Calculus with integral Calculus? Are they intending to take Physics or Engineering courses? Are they satisfying pre-med requirements? Are they satisfying general education requirements? Etc.

Allow me to go out on a limb and suggest that the fundamental (the word choice is no accident...) goal of learning Calculus is to make sense of the Fundamental Theorem of Calculus. In particular, I think the goal must be for students to see the relation between differentiation and integration.

For polynomials (say, with coefficients in $\mathbb{R}$ or $\mathbb{Q}$), they have the nice property that differentiating and integrating is not so tough: The derivative of such a polynomial is a polynomial; the antiderivative of such a polynomial gives a family of polynomials.

But you are suggesting rational functions, too, and now we will have a problem. In fairness, you are asking about an approach to teaching "differential Calculus"; but I think it is nice to have an earlier familiarity with the more commonly appearing functions so that, when the time for finding antiderivatives comes around, some of the cognitive load can be reduced.

The question of which functions should be covered is a difficult one, particularly with regard to finding antiderivatives. Even after going through the full Calculus sequence, a student might rightfully ask: What is the antiderivative of $\sin(x)/x$? What is the antiderivative of $x\tan x$? By a theorem of Liouville, these antiderivatives cannot be expressed in elementary functions; cf. former and latter.

And so there will be picking and choosing when it comes to which functions are to be introduced, and to which derivatives or antiderivatives are to be taken. Given the nature of the AP examinations and the subsequent courses in Calculus in most mathematics departments, I think it wise to introduce at least the functions mentioned in what you call the early approach.

For students who do not intend to take the AP examination (or an equivalent one) and departments that can organize themselves around this nonstandard approach - perhaps they have a different idea of the fundamental goal of learning Calculus? - it would be interesting to see whether lightening the cognitive load early on will pay dividends later, or whether introducing bits and pieces earlier on will make their re-appearance later more palatable.

Conclusion:

In most situations, I would advocate sticking to the early approach. With regard to theory: It would be interesting to compare the two (perhaps such studies exist and I am just unaware of the terminology!). With regard to practice: For a mathematics educator who is interested in the late approach, perhaps its inclusion in a pre-Calculus course would make for a better fit.

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Thank you for the reference, and for your thoughts. The idea of introducing "turning points" in precalculus appeals to me for the same reason as the late transcendentals approach: it helps students appreciate the tools of calculus in the context of examples which are not too far out of their comfort zone. –  Paul Siegel Jul 8 at 14:34
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Regarding your comments about measuring success: let me first specify a specific curriculum that I have in mind. The first part of the course would be on derivatives (definition, computational tools, curve sketching, applications). All examples in the first part would be polynomials and rational functions. The second part of the course would extend the library of functions to include trigonometric, exponential, and logarithmic functions. The third part of the course would be integrals and antiderivatives. –  Paul Siegel Jul 8 at 14:40
    
I see from AndrewC's answer that this is not necessarily what is meant by "late transcendentals"; maybe what I describe should be called "intermediate transcendentals". But with this curriculum it would be reasonable to use higher pass rates as a metric of success: the material covered overall is the same as usual, just in a different order. It would even be possible to give the same final exam as one would in a traditional early transcendentals course, permitting controlled experiments. –  Paul Siegel Jul 8 at 14:45

I'm not sure, if my experience is considerable, but I'd like to point it out:

The late "transcendentals approach" has the huge drawback, that students often miss out the mere algebraic properties (like linearity) of these functions. And in all but the simplest calculus examples involving transcendental functions, these algebraic properties are important as well.

The early "transcendentals approach" has the smaller drawback, that all steps of calculus need more time as they need to be applied to all function classes.

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Can you elaborate on the algebraic properties that you're referring to? Are you referring to the observation that there is no polynomial counterpart to identities like $e^{x+y} = e^x e^y$? –  Paul Siegel May 12 at 19:32
    
@PaulSiegel Yes, I'm referring to algebraic properties like $\mathrm{e}^{x+y}=\mathrm{e}^x+\mathrm{y}^y$ –  Toscho May 13 at 14:16
    
Since when $e^{x + y} = e^x + e^y$?! –  vonbrand Jul 8 at 18:06
    
@vonbrand Sorry, typo: $\mathrm{e}^{x+y}=\mathrm{e}^x\cdot\mathrm{e}^y$. –  Toscho Jul 16 at 11:20

Many places have Calculus I as a co-requisite for University Physics I. The physics instructors like students in thier class to be familiar with derivatives of exponential functions before the end of the semester, hence a need for early transcendentals.

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