You don’t want to lecture for hours on end to passive students; you want to stimulate discussion, interaction, active engagement in developing the ideas. But how? I suggest using a system such as WeBWorK to translate the brilliant insights and illuminating trains of thought from your lectures into a worksheet-style format. This gives the discussion a clear structure and purpose, and incentivises students to participate wholeheartedly since answers are automatically graded and count toward their final grade. This works on many levels and to many ends that go beyond the advantages of using WeBWorK for computational practice problems.
At the level of lecture discussion it makes students eager to attend class and attentively follow your reasoning since this will give them “free answers.” For instance, you are introducing multivariable functions and want to convey the idea that a great way of analysing them are by their cross-sections with horizontal and vertical planes. So you pose this problem and work out at least part of it on the board. Students are primed to appreciate your point since it answers a direct need of theirs. And by stopping short of giving away the final answer you force students to pay attention to the underlying method since they will need it to complete the problem.
You also want to create substantive students discussions in pairs or small groups. To this end it is nice to have conceptual questions that allow for multiple reasonable standpoints. An example is this “paradox” on how one integral can have several “different” answers. You can ask students to work in pairs, one checking one method, the other the other, and then try to convince each other that they are right. Heated discussion ensues, ultimately leading to some reflection on the meaning of the answer––a lesson that cannot be taught often enough.
Full-class discussion or group work is especially stimulating for problems that involve more open-ended conceptual thinking, interpretation, and reflection, rather than single-track computation. This and this are examples that work very well.
Exam-oriented thinking is a plague that prevents students from learning and teachers from teaching. Many a traditional course shoots itself in the foot already before it starts by being structured around the idea of a final exam consisting of a fixed number of highly standardised, computational problems. This corrupts the teacher, who in this mindset asks questions that are “good practice for the exam” instead of asking what lines of inquiry are best for actually learning mathematics in a meaningful way. It also corrupts the students, who quickly conclude that rote computations is all they “really need to know” and hence zone out at any attempts by the teacher to explain underlying reasoning.
The worksheet model frees us from this tyranny. Teachers are no longer crippled by the straightjacket of having to ask only “exam-type” questions, and students find that a large part of their grade comes from a variety of questions involving genuine thought rather than a restrictive set of archetype calculations. We are free to pursue interesting “one-off” problems that make you think, instead of having to discard them as “unexaminable” just because they are not replicable ad infinitum with different formulas and numbers. Since a large part of the graded work takes part in a formative, discussion-oriented setting, we are not constrained to ask self-contained, unambiguous questions of a fixed level of difficulty, as a traditional high-stakes exam requires. Instead of designing our course with the exam in mind, we can design it with mathematical thinking and learning in mind.
Here are a number of examples of problems in this vein. These questions make you think about the material from various vantage points: the “why” behind certain formulas; visual, intuitive, qualitative interpretation of what you are doing; and at the end even some “cultural interest” connections.
These types of problems can be incorporated in a class in various ways depending on the format of the class. In a larger lecture setting they can be used as the basis for the lecture, in which case the students have an extra incentive to follow along since they need the answers. They can also be used to break up the lecture for a few minutes of reflection and discussion among students. In some settings the boundary between class discussion material and exercise assignments need not be sharp: one can assign a number of these kinds of problems and let student requests determine which get discussed in class and which are left as homework. A small class could even be entirely student-driven thanks to the structure that a well-thought-out sequence of questions affords.
The worksheet format is also suited for longer “story” problems such as these, which allow us to work out substantial problems from first principles, such as setting up a differential equation before solving it. These problems too can be incorporated in various ways, from making homework more interesting to making extended discussions of applications viable in a lecture (since it is now truly part of the [graded] course content rather than “enrichment” material as in a traditional course). In classes of moderate size one can also assign such problems to individual students to present to the class. Since everyone needs to enter the answers in the online system, it’s everyone’s points on the line and the class will listen attentively and try to catch any errors. Assigning such problems based on individual student interests is also a way of drawing on existing expertise and connecting the course with other parts of their study program.
The worksheet format also allows us to break down the traditional division between “theory” and “practice.” Again, this very unfortunate and harmful aspect of conventional teaching is in large part a product of examination needs: having students run through computation problems that can be multiplied at will is very convenient for examination purposes, whereas asking for explanations and conceptual reasoning is very messy. But with the WeBWorK worksheet model we can make the latter realistically implementable.
One useful way of getting students engaged with proof-oriented thinking is asking them to evaluate purported proofs, like this. In a traditional course, the teacher and textbook may model many examples of good proofs, but students are seldom confronted with erroneous reasoning. Therefore they often come to associate proofs more with superficial aspects such as phraseology than with actual content. Reasoning-evaluation problems like this forces them to look deeper and cultivates a healthy critical mindset for reading proofs in general.
Here’s another theory example: I introduce the fundamental theorem of calculus, give an intuitive proof, and then ask some follow-up questions that should be easy if you followed the proof, but often prove not so easy since students have so little experience with this type of mathematical reasoning––which is exactly why we need these kinds of questions. In a classroom I might go through the given proof on half a board and then ask the students to complete the proof of the follow-up case in parallel on the other half of the board, mimicking the steps of the first proof with minor adjustments as needed. I might ask for a volunteer student to come to the board to carry this out with the help of suggestions from the class and maybe some leading questions from me if needed. If I refuse to do it any other way (i.e., explain it myself) the students will be pushed to go along with it: after all, if they don’t they will have to do this as a homework problem, which will be much harder and more work.
The second follow-up problem asks for much greater conceptual insight. I marked it with a dagger , signalling that it is a challenge problem. I like to include some problems like this for ambitious students to puzzle about, while others do not need to worry about them since the grading scale will reflect that these kinds of problems are for those aspiring to the very highest grades.
Much other theoretical material will be of this “ type”: it is not at all required for average students in introductory courses, but on the other hand you want to encourage students with substantial mathematical aspirations to start reflecting on more theoretical aspects as early as possible. One way of doing this is to include some of the more theoretical material as -marked readings with various interspersed comprehension questions. Here are a number of examples of how this can be implemented in WeBWorK. With such an “interactive textbook” type of presentation, ambitious students are rewarded for reading the theory and given a “training wheels” guide to reflective reading of mathematical texts. These are some examples.
All of the above problems I have written myself. WeBWorK comes with a large library of standard practice problems (which I also use), but to reach all the goals I highlighted above we must go beyond this restricted notion of what an online homework system is for.