A more rigorous approach to Precalculus

I am a pure mathematics PhD student and graduate teaching assistant at a major state university. During the summers here, teaching assistants are typically appointed to teach an entire course, rather than simply leading a separate discussion section. I have been appointed to teach a Precalculus course and am of course a bit surprised.

Though the fall and spring semesters are quite standardized, we have much more flexibility over the summer. As I could modify explanations to my liking, I was considering perhaps taking a slightly more rigorous approach to teaching Precalculus. In particular, I am hoping to give more detailed explanations of objects that they will be working with. As an example, many students in my discussion sections will express belief that a line is nothing but an equation of the form $$y = mx+b$$. In reality, for $$m,b \in \mathbb{R}$$, a line with slope $$m$$ and $$y$$-intercept $$b$$ is the set of points $$L =\{(x,y):y = mx+b\}$$. I look through many different books and even the official lecture notes of my university and they don't mention this explicitly. The closest to this is one that said a line is determined by such an equation (this of course isn't wrong at all, just not the most complete). This issue I think this causes is when they try to understand why, to solve where two lines intersect, we must set their equations equal to eachother. Even with pictures, some students wonder how that geometric picture translates to the equations being set equal to one another. I personally feel that it could be alleviated through a more set-theoretical explanation since it could be explained via the sets. So to find points of intersection of two lines $$L_1$$ and $$L_2$$, one could simply look at elements of their intersection $$L_1 \cap L_2$$ and see that such a point must satisfy both of their defining features, which means that the second coordinate, $$y$$, must be equal to $$m_1x + b_1$$ and $$m_2x + b_2$$ (this would of course hypothetically be explained slowly and more carefully than I presented here).

Another issue some students in my discussion sections have had is understanding the difference of when two functions are different or the same. For example, if $$f(x) = \frac{(x-1)(x-2)}{(x-2)}$$ and $$g(x) = x-1$$, the students are likely to be convinced that they are the exact same, despite having different domains (the first not defined at $$x = 2$$).

The book and lecture notes will simply say that $$x \neq 2$$ on the left hand margin as some kind of warning against the equality of $$f$$ and $$g$$, yet it doesn't seem sufficient for students and often seems ad hoc to them. It would seem that it wouldn't be much harder for students to try to view the functions differently via the notations $$f:\mathbb{R} - \{2\} \rightarrow \mathbb{R}$$ versus $$g:\mathbb{R} \rightarrow \mathbb{R}$$, so they have different domains, but also understanding that $$f(x) = g(x)$$ for $$x \in \mathbb{R} - \{2\}$$ (or $$(-\infty,2) \cup (2,\infty)$$ however is nicer to notate it).

To me it seems that main education programs are (understandably) afraid to use any kind of rigor, or even set-theoretic notation, whatsoever, with some kind of all-or-nothing approach.

Has anyone here had any experience with teaching precalculus in this manner? Surely one would not want to go extreme and push rigor too hard on students, especially when they have likely not encountered such ways of thinking before hand, but rather a nice balance between the two. Are there any textbooks or references that follow this kind of style? Thanks in advance for any advice.

• Do not do extra stuff in precalculus. Follow the textbook closely. Many of your students may be unable to read the textbook without your help, and adding things not there will only make it worse. – Gerald Edgar Feb 25 '19 at 22:29
• I'm surprised that your text doesn't talk about graphs as sets of points, as I believe this was pretty much standard (even with set notation) for all the books I've taught from. However, this was mostly during the 1980s and 1990s, so perhaps textbooks have changed. (If anything, the books I taught from were less formal than the new math approaches used when I was in school.) Also, I would be careful in how you handle summer school. On average, you'll find the students less prepared/capable (many put the course off until summer, others are repeating the course) and the pace is 2-3 times faster. – Dave L Renfro Feb 26 '19 at 7:12
• Regarding my comments about "I believe this was pretty much standard", I often found myself translating the book's formalism into a format the students understood, rather than the other way around, which is why I said I was surprised. – Dave L Renfro Feb 26 '19 at 7:16
• Perhaps you could talk with someone at your institution who is known as a good teacher. They can tell you more about what the particular students there will need better than we can. I am concerned that your interest in rigor has you facing exactly the wrong direction. I've been teaching for 30 years, and just this semester realized how hard it can be for some students to recognize which side is opposite an angle. – Sue VanHattum Feb 26 '19 at 19:20
• @DaveLRenfro What you said is exactly what I was getting at. Having a reference in that manner and, as the instructor, making it digestible in a slow and gentle manner. I feel that the standard lecture notes and homeworks at my university do the opposite. They'll never so much as mention those types of things, yet on exams they'll ask things like "suppose $(a,b)$ is on the graph of the function" and so on, and no one will understand those questions. It may be more to weed out certain students, but I felt that it was unfair to leave out details, only to asked on them later. – Sprinkle Feb 27 '19 at 0:07

My experience in teaching Precalculus is that you will be surprised about what you actually need to teach compared to what you thought you needed to teach.

For example, if you ask students to compute and simplify the difference quotient for the function $$f(x) = x^2$$, even after several lessons and group work, you may be surprised that by far the most difficult part of this is the idea that $$f(x+h)$$ is an expression and that it is equal to another expression. You will get student work that is confusing to you at first. For example, the following was real work from a student:

$$f(x) = \frac{f(x+h) - f(x)}{h}$$

$$f(x^2 + h) - f(x^2)$$

$$\frac{h}{h}$$

$$1$$

Then you will either dismiss the student as hopeless or you will identify that the student has two critical skill deficiencies (the ones discussed in two of the all-time great MESE questions here and here) that can be overcome.

As a result I would offer the advice to not try anything extraordinary on your first time through. Instead, take all that extra time and effort and use it to think about why students make the errors they make and resolve to restructure the course to teach them the things they actually lack (for me, I would restructure the course to be basically a class on mathematical maturity, the equals sign, and the idea that students can tell whether things are true without asking the teacher).

After you teach the class once, you may be inspired to try something new and radical and that would be great. However, postpone any unconventional solution to a problem until after you have seen the problem.

• Thanks for the advice. I still wholly planned the main lectures to be based off of lecture shells the university recommends. I left out an important context that this is a course that is double the regular time, so instructors are left about an extra hour after the main lecture to do whatever they please, including extra lecture, group work, or Q and A. I’ll just stick to a traditional route. – Sprinkle Feb 25 '19 at 22:36
• @Sprinkle If you have the extra time, you might consider the addition of some collaborative work sessions. These can be quite effective, particularly for students who are entering your class with some disadvantages (such as a weak background in mathematics from high school). The paper Studying Students Studying Calculus by Triesman may prove useful. (It might be more advantageous to adjust the teaching and learning style, rather than to try to rewrite the curriculum.) – Xander Henderson Feb 26 '19 at 2:54
• @Sprinkle Don't take my advice to mean "do everything the traditional way" -- instead, plan time to do some crazy stuff but wait to decide what is needed until after you see it. An analogy might be a calculus student who says "My professor is about to give me a really hard integral. I am considering using integration by parts." Our reply would be "Wait no, I mean maybe, but look at the problem first!" – Chris Cunningham Feb 26 '19 at 13:48
• @ChrisCunningham I think you make a very good point. Most students will run into many mistakes and sometimes It's just best to help them correct the mistakes rather than trying to teach in a way that "corrects them" before it even happens. – Sprinkle Feb 26 '19 at 23:43
• @Sprinkle: It's just best to help them correct the mistakes rather than trying to teach in a way that "corrects them" before it even happens. --- This is something I did way too much of early in my teaching, and even later I often had to make a conscious effort to avoid it. Fortunately, with the rise of the internet in the last 25 years, I've found a less problematic outlet for these tendencies! – Dave L Renfro Feb 27 '19 at 7:55

The notion that more formalism and precision will help beginning students learn is a common one among mathematicians, especially ones who are near the beginning of their teaching experience. It is rooted, I believe, in a conflation between two closely-related notions:

1. Some ideas come logically after others;
2. Some ideas are taught sequentially after others.

Since you are a mathematician, you can think of the curriculum as a partially-ordered set, in which some topics come before others, but with two different order relations on the same underlying set. (This paragraph is, in fact, a self-conscious an example of using unnecessarily formal language to explain a simple idea.)

You are certainly not the first person to ask a question on this site that is based on the premise that teaching "the basics" would be more effective if we were really careful about the "foundations". But Hilbert's Grundlagen is not a book for beginners! Regarding a related question (Why would you teach Calculus before teaching Real Analysis?), I wrote the following answer:

You may as well ask: Why teach elementary school children how to perform whole-number arithmetic without teaching them the Peano axioms first? Why teach high-school Algebra without starting with the basics of groups, rings and fields? Why, for that matter, teach children to read first, instead of starting with the fundamentals of grammar and linguistics?

Everything we know about how learning takes place tells us that complicated ideas are formed and absorbed by building them on a foundation of simpler ones. It is important to realize that "simpler" does not mean here "more fundamental", i.e. logically more primary; rather "simpler" is here used in the sense of "easier".

A human being is not a formal system, and our minds do not develop according to deductive principles. Almost nobody understands anything complicated the first time they encounter it; we master things by revisiting them, over and over again, gaining deeper insight into it each time. We teach people Calculus first because if we taught them Analysis first they would have no experiential or conceptual substrate on which to build.

This applies even more so to Precalculus students, who are even closer to the beginning of their understandings of functions, and therefore have even less of a foundation on which to build more sophisticated understandings.

• Why teach elementary school children how to perform whole-number arithmetic without teaching them the Peano axioms first? --- I recently made an attempt to push this way further (after all, why stop at Peano axioms) in this comment, which will be clear if you know a bit of logic and follow the links I give. – Dave L Renfro Mar 15 '19 at 19:27

Pre-calc that includes a lot of work on function domains is not so strange. I took a class like that in 80s. It was even called "functions". Covered a lot of domain, range stuff, and inequalities and a little derivatives and antiderivatives. Really, if you just look at a few different courses, texts, you will see this--try a GT track at a high school--won't need to ask the web if this is something new to the world.

I think that you need to think about the qualities of your students, not just the coverage of the course. It's not something new to the world to have a course with the content you mention. Has really been the trend since the 60s or so. However, the content that stronger kids take may not be as suitable for weaker kids. You may need to do more patching up of bad algebra 2 and trig and a little prep for calc. And less domain stuff (like really, how often is x = 2 exactly anyways!)

Actually, that you didn't discuss the student quality shows a big whole in your thinking (worse than missing x = 2). Teaching involves content, recipients, and methods. You are ignoring a lot of "z variables" and only thinking of x versus y, if you say "I'm a pure math guy, let's cover more content, more strictly". Now kids at a major state uni aren't going to be bottom of the barrel. At the same time, if they are taking pre-calc in college, that is remedial in this day and age. Calculus is the expected freshman college starting course. And better kids validate that, even. But not even being in calc for frosh year? Those are weaker kids.

You need to consider how strong the kids are versus if you can add more load on their backs. An easy thing to look at would be how they do on the other content (e.g. in standard content courses). If they are crushing it, sure add more plates on the bar. If they are struggling, then absolutely don't add more load--are you crazy? If they are kind of doing OK (not too much of a gut, not getting killed), than I would still lean to not toughing the course up, especially given your own low experience and status.

This doesn't even address what Dave said about summer section capabilities. I'm not sure, might find they are stronger or weaker (need to check). Plus I actually find students can often concentrate easier on a single course in the summer than during the year.

Based on the likely weaknesses of your students (and yourself, as being a new relatively new teacher), I recommend not to change the content when given more ability in a summer section. Concentrate on working on your pedagogy instead. Teach the rock bottom standard course and see if you can horse the kids up on that, better. You need to think about your own progression as a teacher and not change too much at once. It sounds like you will have more responsibility in the summer. Learn to deal with that first, versus changing the content also.

• What is "GT track"? – Tommi Mar 1 '19 at 8:38
• @TommiBrander Possibly GT = Gifted/Talented... – Benjamin Dickman Mar 15 '19 at 19:45

One of the important first steps in course design is to know your audience.

In this case, do your students need that level of understanding? It sounds like you're wanting to approach this like a math class but you need to keep in mind that many undergraduate math classes (including pre-calculus) are so-called "courtesy classes" that are taught by the math department for other departments like physics and engineering. Your average engineer doesn't care that "a line with slope m and y-intercept b is the set of points L={(x,y):y=mx+b}." They just need to know how to find the equation. That's the kind of distinction that would be important to a math major but you probably won't find a lot of them in a college pre-calc class.