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 foregrounding multiple abilities and multiple representations, requiring students to work together in solving complex mathematical problems. These tasks involve sufficient interdependence and challenge; even those students who are perceived as “advanced learners” often experience difficulty completing the 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 in half again, and so on. You need to find out how many layers there are total for a given number of folds. For example, with two folds there are 4 layers. In other words, you are searching for a formula that represents the relationship 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.