There has been an ongoing call in mathematics education for students to be engaging in problem solving and collaborative groupwork. Although, many instructors find that when they put students in groups, some students seem disengaged and we may start to worry that groupwork is not nearly as motivating or interesting to students as we might expect. A natural response at this point is to blame the student for their lack of engagement. But, as Alfie Kohn, an author who writes extensively about education and student motivation, often states, “When students are off task, our first response should be to ask: What’s the task?” Indeed, this is one of the key elements to engaging students in the mathematics classroom; we need to design a good task.

What makes for a good task? Rachel Lotan, a teacher educator at Stanford, coined the term *groupworthy task* to describe what we strive for in task design. In a book review (link: http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/view/240/164 ) that I wrote of Mathematics for Equity, I describe groupworthy tasks as follows:

Groupworthy tasks facilitate students’ interdependence by foregroundingmultiple abilities and multiple representations, requiring students to worktogether in solving complex mathematical problems. These tasks involvesufficient interdependence and challenge; even those students who areperceived as “advanced learners” often experience difficulty completingthe tasks on their own.

Let’s take a closer look at interdependence, multiple abilities, and multiple representations:

**Interdependence****:** If we want students to work together, we need to create a task that actually requires working together to be able to solve it. Moreover, we need to *convince* students that they need to work together. If the task is not sufficiently complex and mathematically rich (See: What is a mathematically rich task) then there will be no need to work together. The typical end of chapter exercises in most textbooks are not mathematically rich; merely teaching a skill and having students practice it (so-called “drill and practice” or more derisively, sometimes called “drill and kill”) is not sufficient to satisfy this criterion.

**Multiple abilities and multiple representations:** A groupworthy mathematical task also requires students to use a lot of different academic abilities (verbal, written, spatial, visual) along with intra and interpersonal skills. Going hand in hand with this, a good task also requires the use of multiple representations—the so-called Rule of Four suggests that we need to use graphical, numeric, linguistic, and symbolic ways of representing mathematics.

Let us take a look at a task that I have used with both high school and college algebra students. This task is adapted from a text called Discovering Algebra.

You have a sheet of paper and are folding it in half, and then inhalf again, and so on. You need to find out how many layers thereare total for a given number of folds. For example, with two foldsthere are 4 layers.In other words, you are searching for a formula that represents therelationship between the number of folds and the number of layers.

This task allows for multiple representations; I generally provide physical paper for students to fold and some find the physical folding helps to make things more concrete. The task asks for a formula, but many students record their observations/answers in a table as an intermediate step. There are also multiple abilities needed to solve the task; students need to be able to count, to notice patterns (such as the doubling relationship), to understand operations conceptually (such as how repeated multiplication becomes exponentiation), and to communicate their ideas with each other. The task requires interdependence; there is not a simple procedure for finding this answer and students have to be able to explore and test different ideas. Moreover, the task is mathematically rich; the concepts of multiplication, doubling, exponentiation, geometric series, exponential functions, and recursive functions are all things that have come up when my students have worked on this problem.

In the next part of this series of posts, I will discuss how to adapt problems that you might already have access to in order to make them more groupworthy.

Very nice. I am happy that you’re sharing this now. I am teaching a calculus 1 course this semester, and have been using group work so far. One of my main challenges is to get students to work together. Even when a task is “rich” enough, there seems to be those who give up trying right away, and those who insist on doing the whole part individually.

For example, I had asked them to use their calculators and by experimenting decide how sensitive the area of a circle of radius 10 is to cutting the radius right. (This was phrased in a nicer way for them though.) This required some one to do the calculations, another to record them, and a decision making by the group based on data. No! This did not happen — they just did not work together.

I will recommend sharing practical, real-life examples of success and failure stories from real classes. This can help those who do want to put an end to the lecture-only math education.

There are three things you might try when students are not working together. One is group roles, where you assign one student to be the facilitator, one to be the recorder, etc. I will discuss group roles more in future posts. Another thing is to make sure that tasks use multiple abilities, such that not everyone will be able to do the task on their own. Having some sort of spatial or graphical representation, in addition to numeric and algebraic ones, is a good way to make sure that everyone’s strengths are represented and they will have to work together to compensate for their weaknesses.

Finally, you need to convince students that they need to work together in order to solve the problem. It helps if the task really requires working together, of course. You can explain to them the different kinds of abilities they will need to solve the problem and tell them that although everyone is good at some of these, none of us are good at all of these. This is called a multiple ability orientation (or multiple ability treatment) and i will discuss this more in future blog posts.

Do others have ways in which they encourage students to work together? Feel free to post a reply.