In January I participated in a panel discussion at the Teaching Excellence Symposium here at Rice. This led to a number of fascinating conversations with various members in the Rice community, including an interview I did with the Office of Information Technology. The OIT runs a news blog where they feature innovative teaching happening on campus. Here's an article they wrote up on my most recent class: Keeping it Real in Calculus Class. There are some other great articles on their blog that I encourage you to explore. I appreciate Carlyn, Liz, and the rest of the OIT staff for providing this valuable resource.
In A classification of the 'whys' and 'hows' of using history in mathematics education, U.T. Jankvist surveys a large body of research affirming the value of including the narrative of the historical development of mathematics in a math course. From this research, he concludes that there are two primary answers to "why" history should be included. One answer is that there is value in teaching the history itself, as it allows us to "show students that mathematics exists and evolves in time and space", dependent upon persons in a number of cultures. The other answer sees history as an important tool in aiding students' learning of mathematics. The arguments he collected in favor of this latter position include:
I find a number of these arguments compelling. In particular, I've noticed that providing historical connections provides context and gets students engaged in material.
For instance, the disputes between Newton and Leibniz regarding the credit for developing the Calculus are well-known; however, there is also a fantastic story of a feud between Descartes and Fermat that can be found in George Simmon's Calculus Gems: Brief Lives and Memorable Mathematics:
Before the development of the Calculus, Fermat has developed a method of finding tangents. "When the famous philosopher [Descartes] was informed of Fermat's method by Merseene, he attacked its generality, challenged Fermat to find the tangent to the curve x^3+y^3=3axy, and foolishly predicted that he would fail. Descartes was unable to cope with this problem himself, and was intensely irritated when Fermat solved it easily."
I incorporate this story into the lecture on implicit differentiation in a single-variable Calculus course, posing the challenge of Descartes: find the tangent line to the curve (called the 'folium of Descares') at an arbitrary point. It presents the need to develop a method of differentiation to solve such problems, which motivates us to develop implicit differentiation in order to save our honor and make Descartes "intensely irritated".
There are a number of other excellent historical accounts that I've found to have great value, such as the historical development of finding the volume of the sphere in a multivariable calculus course (noting Archimedes' insights and the very clever Cavalieri's principle). Many excellent books (such as Calculus Gems) have conveniently packaged these stories.
In another post I shared what I've found to be principles of effective application problems in a calculus course. Here, I'd like to continue that discussion by sharing an additional principle: use real data.
Below is an example taken from a vector calculus book's section on level sets:
While I appreciate the simplicity of the model, I'm left unsatisfied with it merely hinting at topographical maps.
Fortunately, more than enough real data is readily available online. For instance, TopoQuest has topographical data for most of the United States. Here's a map I shared with my class of Texas' highest point, Guadalupe Peak.
But why limit ourselves to mountain climbing?
Here's a write-up on a method for combining data with Google Maps. In particular, data from the Houston Police Department was used to create a countour map of violent crime in Houston:
I've found incorporating real data into the classroom to be an effective way of engaging students in the material. Especially data that is local. Our campus is located in downtown Houston.
Thus this is more than just a contrived example. Students become invested when they see how the material they're studying will connect with the lived experiences of those in their immediate community.
Real data examples help to make this connection. Even if not fully flushed out, they indicates that the things we're learning have a purpose. They let students know that calculus can fight crime.
It's one thing to tell students that the gradient of a function lies perpendicular to its level set at a point.
It's quite another to show that a crime-fighting hero can calculate the gradient of a crime density function to find the hottest crime spots.
That's right, real heros use vector calculus.
And perhaps that's the extra motivation a student occasionally needs.
Below is a typical 'application' problem from a standard Calculus book.
I believe that there are a number of compelling reasons why applications should be included in a Calculus course: they capture student interest, they encourage problem solving skills, and they demonstrate the power of mathematics. But in all three areas, this example fails. Very few college freshmen are interested in corrals, the pre-labeling of the example with x and y robs the student from any opportunity to develop and exercise problem solving skills, and honestly the solution is unlikely to leave the student with the impression, "Wow, math matters."
Simply put, the application appears artificial and uninteresting. There's no obvious reason why such a problem is preferable to simply giving the student an equivalent problem in strictly geometric terms with no mention of ranchers or corrals.
So what do we do? One approach is to teach 'only the math'; however, I'm concerned this robs the student of the above mentioned benefits, Thus I've settled on a different option.
In Dan Meyer's excellent TED talk, "Math class needs a makeover", he gives some suggestions for presenting application questions. Here is a slightly modified list I've created to guide me in choosing applications:
Below is one such application I've developed and used in teaching.
Soda Can Problem
Holding a soda can, one can naturally pose the question, "How did Coca Cola decide on the dimensions of this can?" After all, they could have made it taller and skinner or shorter and fatter, but for some reason the standard can size was settled on.
Notice that a soda can holds 12 ounces (355 ml). Thus we're really asking what are the best dimensions so that our volume comes out to be 355 cubic cm. Of course, 'best' means cheapest and cheapest means using the least amount of aluminum possible in the construction of the can.
Now we can formulate our question more precisely: what dimensions of a can (that is, a cylinder) minimizes the material of the can (that is, the surface area) but keep the volume constant at 355 cubic cm.
Diameter and radius arise as natural choices to determine the dimensions of the can, and from these one can derive expressions for surface area and volume.
This is how I've begun my lectures on related rates in single variable calculus and Lagrangian multipliers in vector calculus. We then go on to develop the necessary calculus to solve the problem.
Then comes the best part: at the end of the lecture when we calculate the solution, it is actually interesting.
We calculate what the dimensions should be to minimize surface area, then measure with a ruler the actual can to see if they agree.
Coca Cola would be saving a significant amount of material (and hence money) if they made their cans significantly shorter and fatter.
So why don't they? Maybe none of their product engineers know Calculus. More likely, a shorter fatter can size, although cheaper to make, would be awkward to hold. Perhaps they intentionally paid more for greater consumer satisfaction.
How did they model that trade-off? What other factors did they consider?
The student is left with an interesting result but also a host of related questions. Questions that calculus can continue to help them answer.
A handful of summers ago, I spent a couple months as a volunteer teacher in Papua New Guinea (PNG). While there, our team discovered an incredible hesitancy for students to speak-up during class time, and when they did, it was barely louder than a whisper. I suspect there are some good reasons for this: English--the language of instruction--is the second (or third) language students learn; moreover, a number of cultural norms seem to encourage students to remain quiet. To combat this, I did an activity where I had my class simultaneous shout the answers to various questions.
In the American University, there also appears to be influences that discourage students from asking questions. Perhaps the most common one being the fear of asking a bad (i.e. 'stupid') question. This post is my attempt to combat it.
The irony is that instructors crave for students to ask questions--even the 'bad' ones--because they reveal what aspects of the lecture the students are understanding and what ideas haven't been communicated effectively yet. But this is precisely what the student is afraid of: that by asking a question they will reveal how much they don't know (i.e. how lost they are).
Of course, being a student is all about not knowing (that's why one is taking the course in the first place). When one signs up and attends a calculus course, it automatically signals the instructor that s/he doesn't know the material.
So the fear of asking questions can't be grounded in not knowing, for that's in a student's job description. Rather it seems the fear boils down to the other aspect of being a student: learning. The student may fear that asking questions reveals he/she is an ineffective learner. "If I ask about that, everybody will know I didn't understand it the first time."
A few points to consider: