Every Pre-Calculus I have examined starts with functions in general, then polynomial and rational functions, followed by exponential and logarithmic functions and Trigonometry, and ending with sequences, summations, probability, and limits. Some have vectors and binomial theorem thrown in towards the end. I would rather begin with number theory, sequences, summations and binomial theorem rather than with polynomials. Is there a pedagogical or theoretical reason why this order is so common?
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4 Answers
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Mitchell,
Great question. I think that precalculus is usually a wasteland for some random classical techniques that used to be required to study calculus.
I think that you can take a completely different approach if you're interested and incorporate data and matrices in place of a bunch of algebraic manipulation.
I just taught a precalculus class and rather than the standard topics, we focus on using functions to model data and some introductory linear algebra. I believe this is much more relevant in the computer age.
If you're interested I'm happy to share more(here's a link to a talk I just gave on the class), but we used the MAA textbook by Sheldon on "Functions, Data, and Models" as a foundational text to solve problems with Python in Jupyter notebooks. The material was broken into a few sections:
Introductory work with functions
- Key Features of linear, quadratic, and exponential functions as recursive sequences through population models
- Relate recursive forms to closed forms
- Intro to harmonic motion and key features
- Use trig to model some basic periodic phenomena
Intro with data
- measures of center
- linear and polynomial regression
- non-linear regression and logarithms
- machine learning and regression
Matrices
- images as matrices through dimension and definition of matrix
- add, subtract, scalar multiply matrices and application to image manipulation
- systems of equations and matrices
- Transpose and regression again
- Trig and rotation by matrix multiplication
Either way, I think that using functions(linear, quadratic, higher polynomials, exponentials, trigonometric) to model real world situations, a little on data analysis, and some introductory linear algebra and applications is an appropriate precalculus course. Take it further depending on how advanced your students are.
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2$\begingroup$ @kcrisman just added a link to my talk on precalc with jupyter notebooks at JupyterDay Philly last week $\endgroup$ Commented May 31, 2017 at 20:02
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1$\begingroup$ I'm really not sure how any of this would count as "pre-calculus", i.e., preparatory material for work in calculus. $\endgroup$ Commented Jun 1, 2017 at 21:51
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2$\begingroup$ What Daniel said, plus I'm very curious where you're teaching that you're allowed to just do that in a precalc course. Everywhere I've taught (4 universities and 2 community colleges) had a very rigid curriculum in all classes before calculus. Some places I've been do have separate tracks. One is the more typical "precalc -> calc" approach and the other starts off with a course similar to the one you described and ends up going through courses that don't emphasize theory. $\endgroup$– user6648Commented Jun 2, 2017 at 17:47
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1$\begingroup$ @DanielR.Collins What do you have students do in calculus? I tend to have them understand how differentiation and integration help us to solve some problems we find out there in the real world. Modeling with discrete and continuous domains together lays the groundwork for the idea of a derivative and the connection to the infinitely small that is so important to understanding why we use differential equations to model all kinds of things. Further, regression is a direct application of 'theory' and the use of optimization. Statistics is largely motivated by physics and physical reasoning... $\endgroup$ Commented Jun 2, 2017 at 18:14
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Precalculus is not just a course that can be taken before calculus. Almost half a century ago, Precalculus, an American invention, started to replace College Algebra as a preparation for Calculus and in fact, if College Algebra seems to have made a comeback, Precalculus remains almost invariably a requirement for calculus.
In that case then, given that, for instance, "Functions of various kinds" are presented by Wikipedia as ""the central objects of investigation" in most fields of modern mathematics", given that the calculus is the calculus of functions, and given that the calculus is the first tool in the "hard" sciences, it seems to me that the best way to prepare students for the calculus is indeed with a study of power functions followed by "polynomial and rational functions, followed by exponential and logarithmic functions" and followed by trigonometric functions.
The problem I have with Precalculus is that it is a bag of tricks. However, this need not be the case if one is willing to use Laurent Polynomial Approximations (i.e. asymptotic expansions using power functions as gauges) to investigate said functions.
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1$\begingroup$ The question isn't about whether or not Pre-Calculus should be a study of functions, rather it is about the order in which those functions are studied. I wanted to start with number theory, sequences and summations, and binomial theorem introducing functions in this context. One reason is that students seems to see functions as just another notation for equations of two variables and the rules of functions don't seem to have a purpose. Using a function to generate a sequence of numbers doesn't led itself to this sort of confusion. $\endgroup$– MitchellCommented Jun 2, 2017 at 12:25
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2$\begingroup$ Regarding "Almost half a century ago, Precalculus, an American invention, started to replace College Algebra as a preparation for Calculus", before the mid 1950s or so (and even more so before WW 2), there were typically three courses one took before calculus (in U.S. colleges) -- college algebra, trigonometry, analytic geometry. Around the mid 1950s or so, analytic geometry courses were being phased out with the essential material shifted to calculus. Also, students began to have better algebra and trig preparation from high school, so a single precalculus course filled the gaps. $\endgroup$ Commented Jun 2, 2017 at 13:59
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1$\begingroup$ @Mitchel You are correct inasmuch as the OP mixed up two issues, the contents and their timing. What I should have said was that if one concentrates on functions, one should then start with them with the possibility of covering some less crucial topics thereafter. Hence my making a case for concentrating on functions. $\endgroup$ Commented Jun 2, 2017 at 15:16
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2$\begingroup$ @Mitchell I am guilty of THE crime: while I did reread the question, I forgot to check who the OP was. To respond to your comment: yes, students "seem to see functions as just another notation for equations of two variables". But then I would say that it is all the more important to spend as much time as is necessary to hammer in what functions are. By the way, to that effect, I use "x –––f––––>f(x) = code specifying the output f(x) in terms of the input x" and I never use the letter y. $\endgroup$ Commented Jun 2, 2017 at 15:32
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I think the main pedagogical reason is this: Calculus students need to be familiar with the elementary functions, and with the idea of functions and their graphs in general. Not every precal class gets through the desired curriculum, due to various reasons. Thus, if something is going to get cut off at the end, because of snow days or time lost for whatever reason, most teachers would rather miss out on sequences or probability than on trigonometry. Students can succeed in Cal I just fine without sequences, but if they miss out on trig, they're up a creek.
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I think the topics you want to start with are not that important to having the algebraic chops to handle calculus. I suspect you like them for theoretical reasons. But that is different from what your students need to learn.
