The Calculus of (Instructional) Variation

As a professor, I take a lot of professional pride in my teaching. As part of that professionalism, I am always looking for ways to improve the learning experience for my students. In this post I am going to describe how some small changes have made a really noticeable impact in one of my classes. (That's where this post title comes from: a little variation has added up to a big change.)

Before I can describe what I did, I should give a little background about what happens in my classes. For now, I am going to focus on my Transition to Proof course. I teach via inquiry-based learning (IBL). As part of that approach, I use student presenters a lot. This means that students come to class having worked on problems (mostly proofs) at home, and they come to class knowing that for most of the problems, someone will have an opportunity to present their proof attempt in front of the class.

This semester, one of my goals is to improve the quality of the discussions that follow a student's presentation. To achieve this goal, I made a couple of changes. In the past, I collected work from everyone at the beginning of class. Then, a presentation proceeded through the following steps:

1. A student wrote their work on the board 
2. The student explained their work.
3. The class proceeded through a Think-Pair-Share: They were asked to look at the work in silence, then share ideas with a partner, and finally ask questions or make comments to the presenter. 
4. During this entire time, the presenter remained standing to answer questions about their work. 

This term, I started in a small classroom with a small chalkboard and a projector screen fixed in place in front of the chalkboard, so that using (most of) the chalkboard was only possible if I unhooked the screen from the wall and set it on the floor. This was part of the inspiration for a new presentation procedure:

1. I photograph student work and upload it to NotesPlus, and project the student work via iPad. 
2. The student explains their work, but sits down immediately, rather than waiting for questions. 
3. The class proceeds through a Think-Pair-Share: They look at the work in silence, then share ideas with a partner, and finally ask questions or make comments, BUT now the presenter is not on the spot during the discussion, as he or she is sitting down.

In addition, I have changed from collecting work at the beginning of class to collecting at the end of class. During class, students used colored pens (Thanks Clark Dollard and Dana Ernst!) to annotate their work, so that I know what was completed before class. Therefore, students are able to compare their work to the work being presented.

These changes are minor, just changing the medium of the presentation, letting the presenter sit during Q&A, and letting students keep their work in front of them for comparison. But the discussions have been stronger for the four weeks of this semester than in years past. My hypothesis is that the students feel more comfortable asking questions with me at the front, even though I am still directing questions back to the class or to the presenter. The class no longer feels like it is putting the presenter on the spot when they raise issues.  Moreover, they are able to ask questions based on their own efforts that they now have in front of them. I am sure there are other factors involved in the improved discussions, including the fact that cohorts of students vary, and this group seems to have a number of people willing to share. Still, it is amazing how small changes can have such a visible impact. 

A Critical Examination of my Transition to Higher Mathematics course, inspired by Grant Wiggins

People who follow my posts to G+ and my tweets may have noticed that I am a reader of Grant Wiggins' blog. Not long ago, he had a post, What is a course? I thought it would be fun to play along. With that in mind, I picked my Transition to Higher Mathematics/Introduction to Proof course. Here are Wiggins' prompts and my responses:

By the end of Transition to Higher Mathematics, students should be able to write proofs and grasp the role of proof as a formal mathematical explanation.

The course builds toward having students prove more logically complex statements and gaining facility with different kinds of proof. The recurring big ideas surround how to attack a proof. We go into depth on key tools like using examples, applying the forward-backward method, using the logical structure of the statement, using definitions.

All of the chapters support these main goals. Students are first introduced to the idea of proof through familiar ideas of number theory and divisibility. They then gain some initial background in logic, and apply it to some number theory proofs before moving on to sets. They are then asked to apply set ideas to sets of real numbers. Then they move to the critical mathematical idea of a function and write proofs about functions and their properties. Finally they are briefly introduced to equivalence relations before moving on to looking at other techniques of proof, and applying these ideas to concepts already seen in class.

Given my priority goals, assessments need to determine whether students are able to demonstrate an understanding of the key approaches to proof, the structure of a logical argument, and to explain the key mathematical concepts of set, open set, function, and properties of a function.

Given my goals, I have exercises and follow-up tasks that should help students gain insight into how the main ideas in key proofs are put together.

If I have been successful, students will be able to transfer their learning to upper division mathematics courses that follow, by attacking proofs with confidence and awareness of the tools available to them, and will persist through difficult courses in the major. If I have been successful, students should avoid such common issues as waiting for the professor to tell them what to do, believing that only others can write proofs, and being uncertain of the role of examples in generating proof ideas (vs. using examples as proof). 

I enjoyed completing this exercise for this course. The course is built as a coherent whole, telling the story of proof and its role in mathematics. To make this work, I have done some tinkering with the emphasis areas of the course, reducing time spent on truth tables (which have a role, but it need not be multiple weeks of a course), and trying not to spend too much time proving things that are too basic. 

What about your courses? Does anyone want to take this up with one of their math courses?