Early vs. late transcendentals

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

I know that this is an old question, but it should be pointed out that in the United States at least, "early transcendentals" vs. "late transcendentals" refers to the placement of exponential, logarithmic, and inverse trigonometric functions. I've never seen a textbook which didn't introduce the derivatives of trigonometric functions in the first semester, but I have seen a number of texts which delay exponential and logarithmic functions until the second semester. As a simple example -- compare the table of contents of James Stewart's calculus texts. The one with "Early Transcendentals" in its title covers logarithms and exponential functions before integration and the one without that in the title covers them just after integration is introduced. Both cover trigonometric functions just after polynomials.

Where I teach we use the early transcendentals approach. I was resistant at first since I had both originally learned and previously taught using the other approach, but I now see the advantages of the earlier approach. The chief advantage is that it is better for the science and engineering students who (at least where I teach) often make up the majority of a first semester calculus class. In particular, our biology majors are only required to take a single semester of calculus and it would be a shame if they ended their calculus without seeing the link to exponential growth and decay.

I see two disadvantages to early transcendentals. One is that it pushes back integration to the very end of the first semester. I am chronically pressed for time towards the end of the first semester. I am able to get through the Fundamental Theorem of Calculus, but often at the cost of being very hand-wavy about things like Riemann sums. The early transcendentals tends to force me to cover integration more superficially than I would like.

A second disadvantage is that it is almost impossible to give a rigorous approach to exponential and logarithms without knowing about the Fundamental Theorem of Calculus. I don't know of any adequate definition of a^x for x irrational that can be shown to make sense using only the tools of early calculus. At best you can say something like if a^x exists and if certain limits exist, then it is a differentiable function. The entire discussion is an almost hopeless hand-wave. In contrast, with the late transcendentals approach you can define ln(x) as the definite integral of 1/t from 1 to x and work out the properties of both logarithms and exponential functions from there. This is clearly a better approach for pure math majors. In addition to being more rigorous, it leaves them with a better understanding of what the Fundamental Theorem actually means.

Answer 6

After teaching calculus for a long time, I have found that actually doing math during class is absolutely essential. By this I mean that you should justify as much as you possibly can, and tell a story about how these ideas arise. The other major way I have seen math taught is through a slide show where concepts are listed and then applied, as if they are magically true and the only interesting thing to do in math is applying these concepts to specific problems. When math is presented as a sequence of concepts that are applied to solve problems, students do not experience math as a coherent language that itself leads to new concepts derived from familiar ones. Under this approach, I cannot define ln(x) until one can integrate functions, knows the mean value theorem, and of course can use limits. If f(x)=ln(x) is introduced before integrals, how do you define it? As the inverse to g(x)=e^x? OK, what is the definition of e^x, and how do derivatives and inverses relate? By going down the path of early transcendentals, you can find more derivatives earlier in your college career (by 2 months or so), but you can't integrate until later in your college career, and also don't know the definitions of many of the functions that you are using, nor how they are related.

Early transcendentals leads the professor to justify the concepts like all of the facts about ln(x) by saying "because I said so" which I think is unacceptable at the college level. This is OK in grade school when math is taught by people whom are not experts in math. However, the fact that this is OK in grade school contributes to the US scoring behind many countries in math proficiency, since math is taught as a sequence of disconnected and arbitrary concepts to be memorized for the test and then discarded.