I’ve been bad about updating lately. This is mostly because the semester has really set in and I’ve found lots to do offline. I wanted to share an idea that I’ve been thinking about lately.
There are a number of things that we try to do in teaching: introduce new concepts, convey interesting information and provide students with practice using new tools and concepts. However, one practice that can make teaching very challenging (and might be neglected) is developing students’ habits of mind.
The tacit knowledge which we use daily to solve problems and design experiments is rarely communicated directly to students. In fact, as a student, I thought that many of these habitual practices were “tricks”. By the way, we don’t do them any justice by calling them “tricks”.
One such nugget of tacit knowledge is how we as physicists use Taylor series. Actually, we are often using MacLaurin series; I’ll come back to this in a moment.
We hoped to communicate some of these ideas early in the semester. We had developed a few clicker questions and a tutorial.
On the tutorial, most students demonstrated a good working knowledge of Taylor (MacLaurin) series (although they faltered a bit with interpreting their results). This might be because they had just answered several clicker questions and an example problem. Or it is likely because the tutorial had been altered (by the instructor) to include far more calculations, so they just got better at performing calculations. In this tutorial and on their accompanying homework, students received a lot of practice cranking through Taylor (MacLaurin) series. Some problems had a physical context, some didn’t.
However, many students were unable to perform four key functions:
- Identify the small parameter in the equation to be expanded
- Execute the expansion efficiently
- Compute a Taylor (not MacLaurin) series
- Interpret the results
Now, only the first three of these are related to Taylor series. The last one is part of a larger problem which students at this (and other) level exhibit, the disconnection between math and physics (saved for a future post). I’ll discuss the first three, each with an example.
Identify the small parameter
Consider the following equation which compares the acceleration due to gravity between the surface of the Earth,
, and some distance above it,
.

Students were given an equation similar to this and asked to determine the value of
in that
was near the surface of the Earth. A student (we’ll call him “Lionel”) came to my office to ask about this problem.
Lionel works with a large (~10) group of students and mentioned that his study group was unable to satisfactorily answer this question. I asked what he and his group tried to do. He immediately said they had no idea which variable was the “right” one and they tried
,
and
, but the group wasn’t sure what to do. The group was unable to identify the small parameter and thus tried all the possible variables. In addition, they were at a loss as to what value to expand about, they decided that zero was the logical choice.
Each of these variables could be used to determine a Taylor expansion of the function
, but which one is the right choice?
Most physicists would form a small parameter given the context of the problem. In this case, by dividing out
in the first term, you’ll get a small parameter
. You can then compute the MacLaurin series in
, because this value is close to zero. It’s also possible to use
, if you compute the Taylor series about
.
Executing the expansion efficiently
One of the major benefits that I see to being a physicist is the ability to build a quick model and get an approximate answer quickly. So, I’ve committed a few expansions to memory (e^x, sin x, cos x, ln(1+x), etc.). Our students are still learning the benefits of this tacit knowledge.
When our students compute a Taylor series (after they’ve gotten through the mess of identifying the small parameter), they tend to use the full form of the Taylor series.

Now, this method will always work. But it can be very inefficient.
Suppose a student has been asked Taylor expand a function that is predominantly polynomial, maybe with a ln(1+x) added to the end of it. Such an equation appears if you consider the vertical position of a particle that experiences linear drag as a function of its range.
I interviewed a few students who were attempting to Taylor expand this function. All but one (who was a very strong student) tended to take derivatives of the entire function and evaluate them. Most physicists have adopted the technique of searching for the non-polynomial terms and replacing them with the first few terms of the recalled expansion. The binomial expansion was my savior for graduate E&M.
While students might eventually produce the correct answer (after a significant amount of algebra; none did in my interview), such a habit is good for physics students to acquire. It will make their life easier and enculturate them into the spherical cow club.
Perform a Taylor Expansion
Many of the examples of Taylor expansions in undergraduate studies tend to actually MacLaurin series (a Taylor expansion about 0). So when students are asked to compute a Taylor series then tend to fail.
Now, you might think, “well, if all the examples are around x=0, why should we bother with others.” Changing the point about which we compute the Taylor series exposes the fragility of students knowledge of Taylor series. In addition, many problems in science and engineering do not produce nice functions where the expansion variable or the point about which the expansion should be performed are immediately clear as our cooked-up examples. So, a thorough working knowledge of Taylor (not just MacLaurin) series is important.
On a recent exam, we asked students to compute the Taylor expansion (up to the first non-vanishing, non-constant term) of a position function about some non-zero value for the time,
.

Roughly 50% of the students computed the expansion about time equals zero; many of whom plugged in the non-zero time value to obtain a function with no time dependence. Only 10% of students solved this problem correctly.
What does this all mean?
For me, this is a lesson in the failure to fully explicate the tacit knowledge which physicists use that we want our students to employ.
These three problems might be mitigated by additional activities: Students might explore MacLaurin expansions by choosing different parameters and evaluating they expansions utility. They might evaluate the efficiency of computing a Taylor series using the formal method and the “search and expand” method. Finally, some experience with Taylor series around non-zero points would not be the worst thing in the world.
I’m planning to investigate each of these issues in detail with a few more interviews and some new questions. These investigations will hopefully codify students’ challenges with Taylor series and help to improve the quality and focus of activities that we are developing to meet these challenges.
But, ultimately this work begs the larger question, “what tacit knowledge do you assume your students are using?”