Non-lecture math education seems to be getting its advocates in the math communities at all levels. Every alternative to the traditional math education involves some sort of class activity, where students are given tasks to complete in groups or pairs. These tasks should be carefully devised by instructor(s) to achieve certain goals. One usually decides on mini goals and designs activities and tasks that will lead the students to accomplishing those goals. (Yes, a backward design.) In a previous post, I talked about setting **SMART** goals that are, among other criteria, **M**easurable (progress could be assessed) and **A**chievable (within abilities of the class).

In this post I will talk about the basic components of a class activity, and illustrate them with an extended example. (*I am borrowing these from the context of English teaching! As an ESL instructor I used to look for ways to improve my teaching skills, which lead me to Task-Based Learning (TBL). The category below is from “Designing Tasks for the Communicative Classroom” by David Nunan. It is fascinating to be able to apply it to math education!*)

There are 5 ingredients that go into a purposefully crafted class activity;

**1. Goal**. We already discussed this above. Of course, you must have a desired result in your mind for the activity. It can be anything from getting the students to know each other and begin to communicate math among themselves, preparing them for future work, to ambitious aspirations such as leading them to discover the fundamental theorem of calculus! Whatever your goals are, they must be there to begin with.

**2. Input. **The raw material/data that class will be provided. In a math class we can expect copies of exercises and instructions, but also ropes and scissors and card boxes, say in a topology class. As an instructor, you must prepare the equipment beforehand.

**3. Setting. **Just like a movie set. Is everyone at the board or should they be sitting in round tables of three? Do groups talk to each other? Is the teacher at the board?

**4. Roles. **Talking about movies, we have to assign roles. Will each student in a group be assigned certain jobs, such as the calculator expert, the presenter, etc? What is the teacher’s role? Will they be the central figure, or try to minimize their influence? Deciding on these roles reduces the risk for confusion and increases efficiency.

**5. Activities.** With the set prepared, let the action begin. Make sure the instructions are clear and students know exactly what to do. Do not start with vague tasks that basically say “OK, go on and prove that every graph has a maximal sub-tree!” Break the job into smaller tasks that increase in difficulty. This will give the students a sense of success each time they fulfill one of the steps.

As the promised example, here is a copy of one of the worksheets I have designed for my ordinary differential equations class this summer:

**Goal**: The goal of this worksheet is to review the concept of the derivative of a function from calculus. *By the end of this worksheet you will be able to:*

*define derivative both algebraically and with graphs;
*

*use the sign of $ f’$ to draw conclusions about the graph of $ y=f(x)$;*

*find intervals of increase and decrease;*

*make quantitative estimates about nearby values of $f$ given its derivative at a point;*

**Input**: Copies of these sheets for each student. A calculator per group.

**Setting**: Isolated groups of three.

**Roles**: Once a strategy is determined, one student uses the calculator, one records the numbers, and the third syncs them and corrects mismatches. The teacher circulates to observe progress and provide hints, but not answers.

**Activities**: Follow the instructions in the order written, and complete the tasks.

I may omit some of the above from the actual copy that the students receive. I usually make an instructor and a student version. The activities begin with:

*Section 1: Flashback to Calculus 1; the Derivative*

*Recall that*…

Then later comes the first task:

*Task 1. Let $ f(x)=1+x-2x^{\frac{2}{3}}\ .$ We have $ f(1)=0$. Which do you expect to be true? Circle your choice:*

*a. $f(1.1)>0$*

*b. $f(1.1)<0.$*

And the activities go on…

If the obscurity of writing up class activities was one factor keeping you from experimenting with the amazing world of non-lecture based math education, I hope this will be the little nudge and nod you were waiting for. I really urge you to try it at least for one topic in a semester. Good luck! And don’t forget to share your experience here.

With the set prepared, let the action begin. Make sure the instructions are clear and students know exactly what to do

Regarding group work, I do wish professors would recognize and understand that group work was primarily developed and implemented by k-12 teachers, the vast majority of whom (some 85%) are EXTROVERTS and so personally thrive in such group environments.

As an introverted high school math teacher, I can attest that my introverted (and sometimes disabled) students are miserable working in groups.

How do you adapt such activities for introverted students?

Most introverts just cannot think in a room of fifty of so students all gabbing at the same time. Also, they need to ‘turn inwards’ to think and process material, and so they generally cannot make good use of group activity time.

Personally, I never force students into group work.

As a student, I found such groups activities a total waste of time and a major headache, and so I eventually just learned to walk out of sessions when group activities were announced. Also, I dropped every course with frequent group work listed on the syllabus and re-enrolled in a traditional lecture or fully online course.