# Why teach the algebraic Calculus?

In the context of a standard undergraduate Calculus sequence, I've noticed there is a big emphasis on teaching the algebra part of Calculus. What I mean by this is that a student may feel more comfortable finding, say, a limit by computing a bunch of values approaching some number, or by going from the graph. But yet, it seems we spend an inordinate amount of time on finding certain limits by factoring, or rationalizing, etc.

Same goes for being able to find a derivative from the definition, or with finding indefinite integrals by hand. Integration is an algorithmic task which we can only perform for hand picked cases at great pains, and there is a huge class of functions for which there is no antiderivative in elementary functions. Given that CAS can do it much more easily, why exactly do we require students to learn this?

While I understand some of these things are interesting in their own right to a mathematician (for example, partial fractions brings to a discussion of what polynomials are irreducible over the reals), I'm not sure who in the real world actually goes around trying to find an indefinite integral by hand, or calculating derivatives from their definition. Again, I do understand these are important skills, but I frequently hear the argument that these are the most important skills, as opposed to having an intuitive "feel" for limits, or for when derivatives come into models, for example.

So my question is:

Do engineers/scientists,etc. actually need to know the algebraic part of Calculus? How?

Concrete examples would be greatly appreciated. Thanks!

• Have you heard the argument that students learn algebra in calculus and learn calculus in differential equations? I heard this quite often when I was a student (1970s for up to early graduate level) and afterwards. There's also the scaffolding argument. That said, in the U.S. there has been a significant decrease overall since the early 1990s in the algebraic manipulation aspects covered in calculus (e.g. partial fractions less emphasized, trig. substitutions less emphasized, curve sketching without calculators, etc.). Commented Mar 3, 2021 at 17:38
• calculating derivatives from their definition --- Actually, this seems to me more relevant, since the focus is on the conceptual definition/meaning of the derivative and not on memorization of short-cut rules. For example, finding the derivative of something as simple as $f(x) = x|x|$ at $x=0$ probably requires one to go back to the definition of the derivative (although it would be nice if the student understood that appropriately gluing together two graphs having a horizontal tangent at $x=0$ gives a graph having a horizontal tangent at $x=0).$ Commented Mar 3, 2021 at 17:40
• @DaveLRenfro: I agree with you about calculating derivatives from their definition. If you weren't familiar with doing this, then how would you know that the derivative of sine is cosine? Would it just be another 'rule' taken on faith?
– Joe
Commented Mar 3, 2021 at 17:47
• I have many students who come from a poor K12 school system. I often spend significant time explaining how to simplify complex fractions (just did that today) and other basic algebra techniques, in my calculus course. I focus on the conceptual, so we do a lot with the definition of the derivative. You can't model without algebra, can you? I think optimizing problems are one of the most important parts of the course. And I promise you, my students have more trouble with the basic reasoning steps in modeling than they do with the algebra. Commented Mar 3, 2021 at 20:43
• Many universities offer an alternative calculus sequence, usually titled something like "Survey of Calculus", which include almost no "algebraic calculus".
– user13395
Commented Mar 3, 2021 at 23:48

It's certainly not possible to omit instruction in these topics for all students. After all, someone has to be able to write the code for the computer algebra system. Someone who wants to be a mathematician or physicist needs these skills, and using a CAS is not a substitute. For example, they will encounter multivariable versions of some of the single-variable techniques and theorems, and they can't be prepared to deal with those unless they have gotten their hands dirty with them in single-variable calculus.

On the other hand, it is ridiculous that people who want to be pharmacists, dentists, and physical therapists are being forced to learn how to integrate using hyperbolic trig substitutions. Here in California, calculus requirements have been gradually becoming universal for students on these tracks. It started with the UC system, and now it's gotten to the point where AFAICT all such students are being told by their counselors to take calculus and calculus-based physics.

So this is not an issue that math departments created, and it's not one that they can solve. Math departments are acting as service departments here.

The reason that these requirements have been inflated in life sciences and allied health fields is simply supply and demand. There are too many students who want to do these majors and go into these professional programs. The market has responded in two ways. First, there is the phenomenon of Caribbean med schools and vet schools, and also the less selective DPT programs. I've had some very good students go to those schools and have successful careers, but lower-performing students from Caribbean med schools end up not getting internships, which is a horrible outcome. The second way that the market has responded is by simply making students jump through hoops in order to get into the impacted majors at better schools, and the better professional programs. There, forcing students to learn hyperbolic trig substitutions is simply a way of filtering them by diligence and intellectual ability. They could be required to learn Homeric Greek instead, and it would serve the same purpose.

Engineering majors are a middle ground. I teach physics to engineering majors, and my main complaint is that they are force-fed the random set of topics of second-semester calculus without gaining any clear picture of the structure of the subject or being able to separate what's important from what's not. A typical symptom of this is that they can do an integral in 5 minutes that would take me 30 minutes (because the techniques are freshly practiced for them), but when I ask them to do a Taylor series, their reaction is something like, "I don't remember that." To me, this is the wrong way around. Taylor series are massively important and practical, whereas forgetting one's integration tricks is not a problem in an era of computer algebra systems.

Since you asked about engineers and scientists, the answer is easy; yes they need to know about algebraic aspects of calculus. There is no way to fully understand physics without having a complete and functioning toolset of algebra. Every trick we do in calculus and more are needed to derive theorems and solve problems in physics. Do engineers and scientists need to understand Physics ? That depends, do you want them to understand things from basic principles or are you content that they adopt a blackbox mentality that anything too physical they have to leave to experts. I would hope we want them to gain a mastery of physics that allows them to reasonably create their own models from applying appropriate physical law etc. to the problem at hand. Algebraic skill is a base level skill towards that goal. Furthermore, mastery of power series and comprehension of special functions far beyond what we even dream of teaching in the usual DEqns course would be helpful. For practicing physicists, if they actually care about understanding the mathematics used in their discipline, so much more than this is needed.

Like everything else nowadays, the question is do we want to put in the effort to do it correctly, or do we just want to do the bare minimum. Maybe I'm just getting old :)

I find that the algebraic manipulations are similar to those in physics and engineering (and to a lesser extent chemistry). Much more so than looking at a graph on a calculator or the like. Both the specific calculus tricks as well as the work similarity (multi-step rearrangements) needed in HW problems in physics/engineering.

Note that engineers often don't do calculus after they leave school. Maybe someone working for NASA, but that's a rarity. Lots of people doing HVAC and the like. And that is really just algebra and arithmetic and drafting and keeping track of details.

As a business consultant, there is a lot of manipulation of numbers and formulas and the like in Excel (or even estimates in meetings in your head/discussion). Basically word problems (which is simple applied algebra...making sure the ratios are right side up). But you'd be amazed how many people struggle with this.

So, if I hire an engineer or a consultant, I would prefer him to have had a classical algebra calculus background. It's kind of overkill for what he does...but it makes sure he has what he needs. Would be leery of kids that did a bunch of concept and graph looking stuff, but couldn't do multi-step calculations. I would not underestimate the value (and mental load) of being able to keep track of multi-step problems for math majors either.

For a nursing student it's definitely overkill that they can handle partial fractions integration. Arguably, it is overkill for doctors as well. Although, I might just appreciate the "weeder" aspect of filtering out the less bright pre-meds. At a weaker school, I can also buy the value for business/econ/social science students of stripping out a lot of the intricate tricks. But some basic algebra calculus still of benefit. But just less volume of it. (At better, say top of state school and up, I would expect all the kids to handle a typical Thomas Finney calculus course.)

To provide a different angle from the great replies already offered, I think other factors include:

1. the technology required for a computational approach has only (relatively) recently become readily accessible to a wide audience (& still isn't in many places);
2. depending on the approach, the use of software can hijack the course, making it more about learning to use a specific software tool, as opposed to learning the underlying ideas;$$^{\color{red}1}$$
3. numerical approaches have non-trivial limitations -- not just issues related to floating point (e.g. a naive choice of $$\delta$$ values can easily lead to erroneous conclusions);

Of course, there are a bunch of other reasons for the current emphasis on an algebraic approach -- probably enough to fill several volumes! Personally, I've find that students benefit from a mix of approaches.

As to your explicit question, I do think physicists and engineers benefit from being comfortable with algebraic manipulations, but certainly not at the expense of a broader understanding of the core concepts of calculs (which, unfortunately, does seem to happen, a lot, although I'm not sure it's fair to place all the fault on algebra...).

$$^{\color{red}1}$$: In a sense, you could argue this is exactly what happens with the algebraic approach, except the tool that hijacks the course is algebra; this points to Dave Renfro's comment about "students learn algebra in calculus and learn calculus in differential equations."