Strategies that Foster Collaborative Learning
“How can I stretch this graph horizontally?” a student asked. I turned to direct him to previous examples but realized that he was not talking to me. His groupmate then explained how she was able to change her equation to stretch the graph.
It was October in a regular PreCalculus course. It usually takes my students about a month to internalize my expectations for collaborative learning. All of my students know that I respect their voice and thinking, and I expect the same from their group members. In this article, I offer specific strategies that foster collaborative learning and student discourse during group work. None of these strategies are my own. I collected these over the years to form the core of student group work behaviors that I found useful. In this article, I share these strategies, along with logistical considerations for effective group development so other teachers can expand their list of collaborative learning strategies.
This article offers group work strategies and grouping logistics in response to NCTM President Robert Berry’s message and call to action, “Positioning Students as Mathematically Competent” (2018). Group work promotes equity by assigning competence to each student and supporting students as they develop ownership of their learning. Group work teaches students collaboration, a 21st century workplace skill. Additionally, in an age when teenagers communicate digitally more often than in person, group work promotes emotional well-being by allowing students to interact face to face in order to practice working on teams. Jo Boaler and Megan Staples’ article (2008), “Creating Mathematical Futures Through an Equitable Teaching Approach: The Case of Railside School,” outlines how Railside School’s use of group work improved student learning, enjoyment of mathematics, and progression to higher level mathematics courses. The evolving challenges in today’s high school mathematics classrooms must draw on these research-based strategies to build communities of collaborative learners.
Evaluating Effectiveness of Groups in Your Classroom.
When evaluating the effectiveness of group work in your classroom, start by defining your goals. Goals may include one or some of the following: improving student understanding, giving students ownership and voice in their learning, promoting equity, growing a sense of community, helping students feel capable of approaching new problems, helping students retain information, creating a friendly and interactive learning environment and improving students’ ability to collaborate. Some of these goals can be measured through formal assessments and others by using informal assessments such as your own observation. However, to get a full picture of how each student views the classroom, ask for student feedback regularly through surveys.
My own goal is to use collaborative learning to increase equity by creating a culture where students feel respected and comfortable taking risks. In a recent survey, 51.3% of my students preferred working in groups while 13.5% preferred working alone. The rest, 35.2%, liked both working alone and in groups. Eighty-nine percent of my students felt that group learning helped them understand the mathematics better and 100% of my students felt that group learning helped other students understand the mathematics better. Answers to a free response question, “Write a couple sentences about your general level of well-being in my classroom and your level of stress in the room,” indicated that all of my students felt that my classroom did not create stress for them.
My Favorite Group Activities:
Placemats are a practice activity for a group of four. The activity could be adjusted for groups of different sizes, but four works very well. Placemats increase equity by building a sense of community in which each student is seen as a valuable contributor and each student takes responsibility not only for their own understanding but also for that of their groupmates.
To create a placemat, make a text box for each corner and then rotate each box to face out. There is also a text box in the center of the page. Each of the four corner boxes contains a different math problem. A student will solve the problem in his or her own corner. If completed early, the student can help another student or check another student’s solution. When all four students are done, they sum up all of their answers and write the sum in the middle box on the paper. The teacher checks only the sum. If the answer is right, the group solves a new set of problems. If the answer is wrong, the students turn the paper 90 degrees and each student checks their group members work. They can be checked as many times as necessary by different group members by rotating the paper 90 degrees in the same direction.
Figure 1. Placemat activity example.
Not all solutions need to be numbers. For example, when graphing absolute value, I asked the students to graph all four answers in the center to form two kites. On another example, I listed eight possible equations in the center box and asked students to circle their groups’ answers.
Depending on the number of problems, you could have between three and eight rounds. However, after eight rounds, students tire of the activity. This activity resembles levels found in many video games. At times, students feel they are competing against other groups which you can encourage or discourage depending on your class.
The name Jigsaw comes from the grouping and regrouping to fit pieces together. In this activity, students first learn their area of expertise in one group and then regroup with other students to teach them what they learned. Jigsaws increase equity in the classroom by assigning competence to marginalized students who may not feel that they have a voice in the classroom.
Since a teacher cannot teach five things at once, the first part of this activity takes some planning. Students learn their area of expertise through discovery learning, through video notes, or through a series of clues. The main idea is they become a group of experts. No student teaches until everyone is ready to teach.
Figure 2. An example of a jigsaw activity in which every student becomes an expert.
In the second part of the activity, students regroup and facilitate the learning of a new group of students. Every student, even the struggling students, has something new to teach the others. By distributing the authority, a teacher elevates every student to a prominent position.
Mystery Solvers is based on the United We Solve strategy by Tim Erickson (2015). For any word problem, you can break up the information and give each student a different piece of information. If there are four clues, you can label them A-D. After handing out the clues, tell students to group themselves so they have one person with each clue and then each group attempts to solve the problem. Groups can share their solutions on white boards or on chart paper.
Group quiz is a strategy I had traditionally saved for classes having difficulty. This year, I tried it with my regular PreCalculus classes and they loved it too. A group quiz is the ultimate formative assessment. The best way to make it successful for student learning is to make it a low stakes assignment. This means that the grade falls into the classwork category instead of the quiz category. The low stakes help students focus on the learning instead of the grade. The group quiz occurs one or two classes before a major assessment, which requires immediate feedback. Explain to the students that the group quiz is a chance to practice a test to identify what they do not know.
A group quiz length is about half the length of a test. This allows students the time to help each other in a relaxed manner and to discuss errors. I explain to students that by the time they are done, every paper should have the same answers and work supporting those answers. If students have alternate solution methods, the group must agree that all methods used are viable. When a group is finished, choose one quiz randomly and grade it in front of them. By using this group work activity that allows open notes, students are more likely to be successful and they identify their weaknesses.
The feedback on the group quiz activity has been overwhelmingly and enthusiastically positive. All students learn from discussing and debating the solutions. Students feel an ownership of their learning that they can later solve the problems without teacher intervention.
Matching activities are an easy entry point for students who are not accustomed to working in groups or for new groups to get to know each other. Almost any type of function, from quadratic to sinusoidal to logarithmic to polynomial can be described with an equation, a graph, a table of values and a description of the graph’s characteristics. When a student picks up a card, it commits him or her to contributing to the solution. That solution will involve speaking with another student who is holding his or her match by type of equation.
Figure 3. Matching activity example.
Another matching activity is a vocabulary match. Students are tasked with matching a word with its definition. Focusing on the vocabulary especially supports English Language Learners who may struggle with unfamiliar mathematical terms.
In partner problems, students take turns writing steps in the solution of a problem. This is especially helpful when proving trigonometric identities and with geometric proofs. You can allow students to discuss the solution with their partner as they go. Or you can ask students to challenge themselves to complete the problem taking turns writing out the steps without talking. If a student disagrees with a step that his or her partner wrote, the student can redo the step during his or her turn. Although this strategy is especially helpful with proofs, it can help students with any type of procedural focused problem.
For students to do their best collaborative work, they need to sit in groups every day. There are many strategies for grouping students: heterogeneous, homogenous, random or student-chosen. Some will work for some classes on some days and not for other classes or on other days. My rule is to look to the lowest performers. Are these students in their best situation? This does not mean they are placed with the strongest students for peer tutoring. This means these students are placed in a group where their voice is heard and appreciated.
In considering group size, I prefer groups of four to provide multiple perspectives but these four must allow everyone to contribute. I usually allow students to choose their own seats at the beginning of the year to make sure they sit by someone who is familiar to him or her. When I regroup, I try to place each student with one familiar and two new group mates. I usually like to change seats once a quarter to allow students to get to know as many others as possible.
When working with groups, a teacher must be comfortable with a noisy classroom. Sometimes, you will need to interrupt group work to go over a common misunderstanding or allow a student to share a solution method. For this, you will need strategies to bring the class back together. Call and response is a popular strategy. It is changed to reflect the current unit. For example, I call, “Tangent theta equals.” My students respond, “Sine theta over cosine theta.”
If groups are struggling to get started working as a team, you may need to assign roles to each group member. There are many ways to do this and some groups may appreciate this added structure. The Teaching Center at Washington University in St. Louis (2018) recommends the following roles: Facilitator, Recorder, Spokesperson, Reflector, Encourager, Questioner, Checker. For other groups, you may need to scaffold a task and/or break it down into smaller steps.
This article offers teachers choices for group work that I have found successful. Scheduling group work every unit conditions students to working together and working more productively. Just like the young man in the introductory vignette, students get used to asking each other questions and asking for their opinions about their solutions. Collaborative learning alters teaching practices. Instead of asking students to answer a question for the whole class, a teacher will ask students to share their answer with the group first (a type of think, pair, share) before sharing with the class. Likewise, when students come to the teacher to ask questions, the teacher will be more likely to direct the student back to his or her group. Once the students and the teacher are comfortable with group work, it becomes an effective way to promote student engagement. As a result, students feel empowered as mathematical contributors and take ownership in their learning.
I am indebted to the many teachers with whom I have worked in the Collaborative Learning Teams of Fairfax County Public Schools. They have mentored me and inspired me to constantly reflect and adjust my practice. Additionally, I would like to acknowledge the contributions of Terrie Galanti who helped me edit the article.
Berry, Robert. (2018, July 25). Positioning students as mathematically competent. Retrieved from https://www.nctm.org/News-and-Calendar/Messages-from-the- President/Archive/Robert-Q_-Berry-III/Positioning-Students-as-Mathematically-Competent/
Boaler, J. & Staples, M. (2008). Creating mathematical futures through an equitable teaching approach: The Case of Railside School. Teachers College Record (3). 608-645.
Erickson, T. (2015). United we solve: 116 math problems for groups, Grades 5-10. Oakland, CA: eeps media.
The Teaching Center at University of Washington_St. Louis. (2018). Using roles in group work. Retrieved from https://teachingcenter.wustl.edu/resources/active- learning/group-work-in-class/using-roles-in-group-work/
Fairfax County Public Schools