Teaching Calculus using IBL

Written by: Dr. Christine von Renesse

Do you think it is possible to teach calculus without ANY lecture? (or any other passive teaching method like watching videos, reading the book etc.)

Well, I have been teaching calculus (I, II and III) for the last 7 semesters without any lecture. And I love it.
This blog will give a brief introduction about how I teach calculus I,II and III using inquiry,
providing resources, tools and some classroom videos along the way. Please email me if you have questions
that are not addressed here, as it is impossible to capture it all in one blog.

Resources:

I was fortunate to have the calculus 1 and 2 materials developed at Rockhurst University ( iblcalculus.com ) available when I started out. I used many of their activities, left some out and adapted a few, according to my audience.
Last semester I developed calculus 3 materials with Amy Ksir, and now appreciate even more how much work Mairead Greene and Paula Shorter invested in creating their materials. A first draft of our calculus 3 materials can be downloaded here .

The following video clip shows you what most of class time looks like for me. This group in calculus 3 is grappling with the idea of tangent lines and tangent planes to surfaces in 3-dimensional space.

Exploring Definitions and Theorems:

What I like about the iblcalculus.com materials is that the students develop most definitions and theorems themselves. Here is an example from calculus 2:
Instead of giving the students the definition of a series, telling them when a geometric series is convergent and then let them grapple with the proof and examples, the students first investigate the area and perimeter of Koch’s snowflake. They naturally develop an example of a series and ask just the right question: Is this infinite or not? How do we know when a (geometric) series is convergent or not? Looking at many examples on wolfram alpha can then lead to a conjecture. Now the students are ready to think about how to prove their conjecture. Notice that any language needed (series, geometric series, convergence, divergence, sum notation, limit, …) can be provided during the exploration, when the students need and want the vocabulary to express their thinking.

Similarly, we let the students in calculus 3 make conjectures about how the second partial derivatives can be used to classify critical points. The following video shows the culminating whole class discussion about $f_{xx} f_{yy} -f_{xy}^2$.

If you would like to learn more about how to run a successful whole class discussion you can read the blog .

Representations and Context

The iblcalculus.com materials show a "red thread" throughout the two semesters: exploring and making use of representations. Functions in calculus 1, for instance, are investigated numerically (tables), symbolically (equations), graphically and using narrative. The goal is for the students to be able to flexibly switch between different mathematical representations in order to solve problems or prove conjectures.
Similarly, we decided to let the students explore different representations of function in calculus 3: using parameterizations, implicit equations, and graphs. As an example you could see students comparing two representations of tangent planes in the first video clip above.
As important as flexibility with representations is the ability to see a mathematical idea in context. In the iblcalculus.com materials you can see many examples from biology and physics. In the calculus 3 examples we used temperature and profit to give context to the optimization investigations. The reason for choosing a consistent context goes beyond knowing how to apply the mathematics to a real world problem. Many conjectures and proofs in calculus make sense more deeply if you see them in a context.

Proof and Sense Making:

Using inquiry in my calculus class has let me realize that even though I have proved many of the theorems before, I had not really made sense of many of them. The fundamental theorem of calculus is a great example of that. I realized that I was not able to explain (just using words – not symbols), why the theorem made sense to me. I had to really think about it myself! I realized that while formal proofs have their place they may not be enough to really understand something and that I would sometimes choose sense-making over a formal proof in calculus. See also our Proof as Sense Making Blog .

What are the big ideas (How to decide what to cover less of)?

It is so hard to decide to leave things out and yet, even in a lecture class, we do it all the time. There is just too much math out there ☺. So maybe this is a better question: what do my students need to know so that they will be able to independently understand the topics I did not cover during class? Using this idea I, for instance, let the students explore and prove only a few of the convergence tests for series, relying on the fact that the rest of the tests can be easily found in a text book or online.

I have learned so much more in this class alone than I have learned in all of my other
classes combined. In my other classes, you only needed to memorize the material and spit
it out for the tests.

Calculus Student

How to prepare students for the next semester?

To make sure that my students also have the computational abilities they need in the following semesters, I teach some of the computational sections very differently -- using almost no conjectures and proofs. In calculus 1 for example, we don't prove all the derivative rules. Instead we focus on disproving the tempting incorrect rules. Here is my computing derivatives exploration. Notice that at this point students already have a good sense of what a derivative is and how to use it in context for optimization. They also have proved how to take the derivative of monomials. In this part of the course the students that have taken calculus before usually need very little time and are ready for harder problems, or some proofs. See below for more thoughts about differentiated instruction and IBL.
I also use a gateway exam to make sure the students are computationally ready for the next course.

Grading and Assessment:

In my calculus 1 syllabus you will notice that I give a high percentage for meaningful participation (40%) because this is most important to me.
You can read the blog about homework stories to see how students write up in detail what makes sense to them. Here is an example of a homework story in calculus 3 . Additional problems from the book ensure that students also practice computations and strategies.

While it is fairly easy as an instructor to get a sense during class of what the students understand and what they don't understand, it is surprisingly hard to write questions on an exam that show these results. Mairead Greene and Paula Shorter are putting currently a lot of effort into analyzing (exam) questions. They distinguish between skills, methods and conceptual reasoning. This distinction is reflected deeply in their IBL materials and how they guide the students to think more independently. See here for more information about their framework.

Student Reflections/Journals:

Once or twice a semester I let my students write a journal about their learning experience. This really helps me talk to the students more that still struggle and see if there are ways in which I need to change my teaching. While some students complain that they believe they learn “better” using lecture almost all students express how much they are learning or have learned in the class. And that’s what counts for me. Sample final journal calculus 1.

Before this class, I was a dependent learner who needed a teacher or lecturer to show me how to do a problem. Now, I have grown more independent and do not require help as often as I did. Now, I am not saying that I do not need help with math anymore, because that is the furthest from the truth. I am saying that now I do not need to be babied and I can actually solve things by myself or with a small group of people. This class has helped me grow not only as a mathematician, but as a learner as well.

Calculus Student

Large Classes

My calculus classes are kept at 25 students so I never had to teach any large classes. Angie Hodge has successfully taught calculus using IBL in large classes, so use her as a resource. Angie's website .

Differentiated Instruction and IBL (and My own Learning Edge)

So what’s different from teaching calculus using IBL versus a math for liberal arts class or an upper level math course? At Westfield the students coming into calculus have very varied backgrounds in mathematics: some have not even taken pre-calculus, while others took AP calculus in high school. One of my big tasks as an instructor is to differentiate in a way that all students have a chance to learn at their own learning edge. This means that not all students will learn the same material equally well; and that (ideally) I measure their progress on different levels.

Ok, so how can this be done? First of all I change groups around for the first two weeks (imagine calculus speed dating ☺) until I have groups as homogenous as possible in terms of speed, ability and work ethic. I frequently check in with students if they have found partners or groups that match their own learning. It may be surprising that the students don’t mind being in a “slower group” or not working with the “fastest” students. I think all students know how they learn best and we need to give them a chance to access that knowledge. Sometimes I also suggest specific groups or work partners based on what I observe during class. See also our blog about grouping students .

In my perfect world, any exploration we do in class would have some base problems that all students need to understand and then enough extension questions to let the faster students dive deeper and connect farther. In reality I only do this every few weeks, simply because it is a lot of work to always find good extension questions that don’t already move ahead in the material.

This is what I am still working on as a facilitator: creating in class and homework questions that are based on the different expectations for different groups of students. Right now my faster groups are often moving faster through the material, which leads to fewer opportunities for class discussions and doesn’t challenge my faster students as much as I want.