**By Diana Skrzydlo**

When a course has a scheduled tutorial period (a weekly designated hour-long meet time in a classroom, in addition to lecture times), different instructors use them in different ways. Questions and answers, presenting solutions to problems, or closed-book quizzes are some popular choices, but many instructors cancel them or only use them to make up missed lectures. When preparing to teach a third-year probability models course for the first time, I wanted to use a more dynamic approach to promote a deep understanding of the course material. Through the generous support of a LITE (Learning Innovation and Teaching Enhancement) grant, I investigated the effectiveness of interactive tutorial activities. This article will discuss how I structured the tutorials, a sample of my individual activities, the student outcomes, as well as some ideas for how you can use them too, whether or not you have a scheduled tutorial time.

I had a total of 10 tutorials in the term and varied the activities each week. Students completed the activities in randomly-assigned groups, and each one was worth one percent of their final grade. I was there along with a teaching assistant to walk around and actively help groups and answer questions, so most solutions ended up being perfect. The activities were graded and returned, and solutions were posted on the course website immediately afterwards.

The first and last tutorials were used for a pre-test and post-test on the course material. I gave the same test covering the basic concepts in the course both before and after, and had the students report their confidence in their answers as well. At the end, the students and I could see not only how much they had learned but also how much their confidence had improved.

To prepare the students for the midterm I used an activity known as a Jigsaw. I split the class into six groups, each responsible for two things in the first phase: writing down summary notes of the concepts for a specific few lectures, and answering a midterm-like question on that material. In the second phase we shuffled so there were groups of six people (one from each of the original groups.) In the new groups, each member took turns explaining their summary notes and going through the solution to their question. I rotated the pieces of paper with that information around the groups manually. By the end of the class, each person had explained one question and had five others explained to them, providing essentially a complete practice midterm and set of study notes, which I then scanned and posted on the course website afterwards.

For the tutorial after the midterm I "took up" the midterm in a non-traditional way. In groups of three, they received their unmarked tests back as well as a blank test, and had to collaborate on a set of solutions. Unbeknownst to them, each question had been gotten right (or mostly right) by at least one person in each group. Revisiting the material with others helped make the midterm an active learning experience. If people disagreed, they had to debate who was right, which further deepened their understanding of the material. At the end I gave them marked copies of their own original papers, and they already had a perfect set of solutions. I later discovered this method is known as Two-Stage Testing and has many interesting variations.

The other six tutorials were used to do questions and activities relating to important threshold concepts (core concepts that, once understood, transform a student’s perception of the subject) in the course, which many students have struggled with in the past. These included:

- Playing games of chance and calculating probabilities and expected values
- A demonstration of the multinomial distribution and why its marginal, conditional, and convolution distributions are all Binomia
- Several examples of transformations of joint continuous distribution
- The meaning of conditional expectation and double averaging (law of total expectation)
- Exploring the basic properties and definitions of Markov chains (discussed further below)
- The properties of the Poisson process including conditional event times, number of events in a subinterval, and the compound Poisson process

For the Markov chain activity, we used groups of people to physically represent Markov chains. Each person represented a state (1 through 5) and was given instructions for which state the process goes to next based on a coin flip. The students traced the path of the chain several times on a piece of paper until they got a sense of how the chain behaved, and then were given a set of questions to answer. They repeated this for three different chains. Through this activity, I was able to introduce the concepts of the transition matrix, communication/accessibility of states, open/closed classes, periodicity, transient/recurrent states, and the long-term equilibrium distribution, all without any formal definitions. This activity helped make these theoretical definitions more concrete and easy to remember when we then defined them in the next class.

The tutorials were a resounding success. 98 percent of the students reported in an anonymous post-course survey that they found the tutorials helpful in learning the course material, and they were a lot of fun! In addition, I noticed the students’ approaches to problems matured and became more independent over the term. In the beginning they would often ask for help right away without trying the problems, whereas later they would attempt the problems first and only need a little guidance.

Interactive activities can be used in any course, at any level. If you are interested in incorporating these kinds of activities into your own teaching, here are some ideas:

- Simply having students work on problems in class with the instructor available to help (sort of a flipped classroom approach) gets them actively learning.
- A Jigsaw activity for a midterm or final exam review can work well even with large numbers of students.
- Two-stage testing is becoming more popular for the benefits of turning a passive exam into an active learning experience.
- An exploratory activity like I used for Markov chains can help introduce students to new concepts. For example: continuous random variables in a Probability course, multiple-state models in Life Contingencies, Brownian motion in a Finance course, etc.

Even without a formal scheduled tutorial, many of these concepts can be introduced into lectures. Active learning in the classroom can deepen students’ understanding of course material while providing valuable experiences. I hope you’ll give it a try!

Diana Skrzydlo is a continuing lecturer at the University of Waterloo in Waterloo, Ontario, Canada. She can be reached at * dkchisho@uwaterloo.ca*.