Insights Into Algebra 1: Teaching for Learning
Variables and Patterns of Change Teaching Strategies: Cooperative Learning
The process of cooperative learning involves students working together in small groups on a structured activity. The members of the groups learn to work as a team to accomplish a specific goal, to solve a problem, to complete a project, or to develop a product. Teachers hold students accountable individually, but also assess group work. Students are responsible not only for learning the material, but also for ensuring that the other members of the group learn the material too. In an algebra classroom, cooperative learning is a key component in attaining “algebra for all.”
Read how Fran Curcio defines “algebra for all:”
In order to be equitable, to provide equal learning opportunities and job opportunities for all learners, we need to have algebra accessible to all learners. So, “algebra for all” means reaching out, bringing all children closer to the subject of algebra.
Cooperative learning differs from the traditional, teacher-centered classroom in that it is student-centered. In a traditional classroom, the teacher plays the role of “learning disseminator,” giving the students all of the information they need. (An old joke provides the following definition of lecture: “The transfer of ideas from the notes of the teacher to the notes of the student, without passing through the minds of either.”) In a cooperative classroom, however, the teacher takes on the role of “learning facilitator,” helping students figure out what information will be helpful to complete the assigned task.
The structure of cooperative learning provides a place where:
- Students are stakeholders in their own learning.
- All learners are active participants.
- Students learn social skills, such as cooperation and conflict resolution.
- Projects are designed to be interesting, yet challenging.
- Teachers sometimes learn, and students sometimes teach.
- Teachers encourage and value the expression of differing opinions.
- Teachers and students demonstrate mutual respect.
Read what Miriam Leiva has to say about cooperative learning as one aspect of good teaching:
She [teacher Jenny Novak] used several different modes of teaching that I thought were important in this lesson. She had the teacher with whole class … then, the teacher addressing groups … and, within the groups, the teacher addressing individuals. Overriding all of that was students teaching students, which I think is at the heart of cooperative learning.
Cooperative Learning: Benefits
Many studies document the academic benefits of using cooperative learning in the classroom. Among the results are improved student achievement; increased self-esteem and confidence; higher levels of motivation; improved behavior; better attendance; and more positive attitudes toward school, learning, and classmates. Research has consistently shown that when two key elements of cooperative learning – positive interdependence and individual accountability, both of which are discussed later in this session – are present, student achievement improves.
Cooperative learning is especially helpful in developing social skills. Students learn to work with all types of people. During small group activities, they are able to think about and reply to the diverse ideas of their fellow group members, many of whose perspectives reflect cultural differences. Classmates learn to relate to their peers, and students who work with others in cooperative groups tend to like each other. Consequently, there are improved relationships among different ethnic groups, and cooperative learning becomes a celebration of diversity. Further, structured interactions between students can help students who have difficulty in social settings. They can also improve relationships between students with learning disabilities and their peers.
The end results of a cooperative assignment are usually superior to those reached through individual study. When ideas and questions are offered in a group, members provide a variety of responses, and the final product will, therefore, reflect a broad range of perspectives. The project is typically more comprehensive and rewarding when it involves the mutual exchange of ideas.
During cooperative learning, students actively participate. Rather than being passive recipients of information, they are generally enthusiastic about their own learning. Students take ownership and responsibility when working as part of a team – possibly because other group members will be affected by their actions. As a result, students gain a deeper understanding of mathematics and develop some of the social skills, such as cooperation and teamwork, that are valued in today’s workplace.
Cooperative learning provides many more opportunities for students to receive feedback. Whereas a teacher is only able to respond to one student at a time, members of cooperative groups are able to give feedback simultaneously. Students will receive more feedback when they discuss mathematics in cooperative groups than when the teacher alone attempts to respond to everyone’s questions. During large group instruction, one or two students may exchange ideas and dominate a discussion as the rest of the class listens – or, in many cases, while the rest of the class does not listen. On the other hand, if one student in each cooperative learning group asks a question, a number of questions can be answered at the same time, increasing the overall effectiveness of instruction. As the teacher circulates and listens, he or she can help students answer their questions appropriately.
Read what Jenny Novak learned by listening to her students while they worked in groups:
When we were doing the round-robin activity, I could tell that the students were engaged. They were working together. They were talking about [the mathematics]. They were working through the problems together and checking answers. And if they were getting something wrong, they were saying, “Well, let’s look at it, and see what we did wrong.” They were then able to discover their own mistakes rather than waiting for me to say, “Oh, well, here was your mistake,” and I think that’s much more meaningful for the students.
Finally, cooperative learning is easy to do and inexpensive. Consequently, there are numerous benefits of cooperative learning that go beyond learning:
- Instead of requesting materials for 36 students, teachers only need to request materials for nine groups. That helps a lot in times of tight school budgets.
- Reticent and low-achieving students often feel more comfortable asking and answering questions in a small group. It prevents some of the stage fright that arises from speaking in front of the entire class.
- Effective cooperative learning groups actually assist with classroom management. Students who know they will soon have time to work with their partners are less likely to misbehave during a presentation by the teacher.
- Some students prefer listening to their peers rather than to the teacher, and sometimes their peers provide explanations they find easier to understand.
- Lecturing to your students on a daily basis causes boredom. Cooperative groups will give your class new energy. (But beware: A class can just as easily fall into a rut by using the same cooperative groups every day! Vary the ways in which you organize groups, and mix in other types of lessons to keep your class fresh and your students invigorated.)
During a parent teacher conference, the parent of one of your students questions the use of cooperative learning. The parent believes that time spent working in groups takes away from time the student can be learning. How would your respond to this parent?
Cooperative Learning at the Beginning of the Year
Mastering concepts is a gradual process, and so is becoming an effective member of a group. Through carefully planned projects and well-organized groups, students who are shy and less willing to share at the beginning of the year will often become more participatory as the year progresses. (See Workshop 4, Developing a Community of Learners, to see how teacher Tremain Nelson encourages greater participation as the year progresses.)
Read what Janel Green has to say about her expectations at the beginning of the year:
I felt that the kids are where I would like them to be. They worked well together. It is the beginning of they year, yet they act as if they knew each other for years. They were communicating well together, and they were sharing ideas. I couldn’t ask for anything better. And I think as time goes on, they are going to get more and more comfortable with each other, and they are going to learn how to share ideas more and more.
Read Fran Curcio’s thoughts about the success of Janel’s class:
This particular first experience – [of students] working in small groups – was quite commendable … The students were working very well together, describing their thinking, listening to each other, sharing their ideas. I think that is quite commendable in this early part of their experience in learning algebra.
Just as with rules regarding individual behavior and discipline, it is important to establish rules and norms for working in cooperative groups. A list of simple rules distributed the first time students work collaboratively will go a long way toward ensuring success. Some rules from veteran teachers include:
- Each member of the group is responsible for all work.
- Do not make decisions by voting; instead, try to reach consensus and find a solution with which everyone agrees.
- Everyone’s opinion is valuable, so encourage participation from all group members. Don’t leave anyone out.
Many proponents of cooperative learning believe that using it too early in the school year can be detrimental. Without understanding classroom procedures and behavior expectations, students may view cooperative learning as “free time,” rather than time for learning. To prevent potential discipline problems, allow students to become comfortable with the general expectations of the class before asking them to do a lot of work in groups. It is important that you clarify why cooperative learning is a valued and important activity in the classroom. Students need to know your expectations for cooperative groups, and you need to provide clear evidence that the cooperative learning activities are important by including key aspects of them in your tests and other individual assessments.
When they first decide to use cooperative learning, most teachers assign a relatively simple project. This ensures that students adjust to their partners and learn how to work as a team, rather than being forced to focus immediately on content. One teacher suggests that the first activity ought to be selecting a team name, “because it is cool and because it gives them decision-making power.” (Glosser, Gisele. “Cooperative Learning Techniques,” http://www.mathgoodies.com/articles/coop_learning.html.) By having the team name, as well as the names of the students, placed on index cards, a teacher can randomly select a group, or even a particular student, to present work during later cooperative sessions.
As the old saying goes, there is strength in numbers. Jenny Novak abides by this motto when using cooperative learning in her classroom. Knowing that she cannot possibly attend to every group in the class, she recruits advanced math students from other classes to assist her. Although they were not present for the taping of the video for this workshop, these students are usually available to answer questions the groups have; more importantly, the helpers serve as Jenny’s eyes and ears and report back to her when a group is in need of assistance.
Listen to what Jenny Novak has to say about this process:
|Listen to audio clip of teacher|
What behaviors and attitudes should you expect from your students when you are ready to engage in cooperative learning and group projects?
The Structure of Cooperative Learning
As mentioned earlier, there are two key elements necessary for the success of cooperative learning: positive interdependence and individual accountability. Research shows that the level of improved student achievement depends on the implementation of cooperative learning methods that are characterized by these two essential elements. (Slavin, Robert E. “Research on Cooperative Learning: Consensus and Controversy.” Educational Leadership, December 1989/January 1990; vol. 47, no. 4: pp. 52-54.)
Positive interdependence requires the establishment of an environment in which students must work together; no student is able to succeed on his or her own. Team members must be aware that they need one another to succeed.
Positive interdependence can be attained in many ways. A simplistic method requires that all students in a group must be able to explain a concept to another group. Consequently, each group must continue working until all its members understand the concept.
One of the most common methods of fostering such an environment is known as “reward interdependence,” whereby students within the group receive some type of shared grade. Instead of giving all students the same grade for a project, though, consider offering some reward for the achievement of all group members. For instance, in addition to earning an individual grade on a test, students should earn more points if all members of their group score at or above a certain level.
Other methods include giving only certain pieces of information to each member of the group. The members have to interact and work together to find out what information the other members have. Or each member of the group may have a specific role, such as group leader, scribe, or timekeeper. Rotate roles every so often to ensure that each student gets to experience them all.
Individual accountability means assessing whether each group member has accomplished the group goal and determining the contribution of each individual within the group.
One of the most effective means of ensuring individual accountability is to keep the size of the groups relatively small. This allows for greater participation – and hopefully increased learning – by each group member. Another common practice is to allow students to work in cooperative groups and to assign a grade to their collective work. To ensure that each student has learned, the teacher should administer individual tests or assignments to each group member.
Read what Miriam Leiva noticed about cooperative learning and individual accountability in the video for Workshop 1, Part II:
I really appreciated the fact that in this lesson there were several interactions going on. One was [Jenny Novak’s] very good use of grouping. It seemed like [the groups] were carefully put together so that students already at a certain level were together with other students that may not have been at their level. Some students already understood the manipulatives and they understood the symbolic, but they [were] teaching other students … [Jenny] engaged the groups, [and] then, when they reported, showed that all students were responsible for their own individual knowledge. She called on individual students, not on a spokesperson for the group, so students were responsible as individuals.
Individual accountability can also be measured with formative assessments. While circulating among groups and asking questions, the teacher can observe each student’s contributions to the overall success of the group. A formal assessment, such as a test, quiz, or homework assignment, may not be necessary; instead, a few well-posed questions while observing the groups may provide enough evidence of a student’s level of understanding.
Read what Miriam Leiva has to say about assessing group members:
Note that in the lesson, she [Jenny Novak] is calling out the question, pausing, and then calling on an individual student. She’s doing the same thing when she goes around to the groups. Note that she is not just taking a walk around the classroom to keep discipline or to keep them quiet, but quite the opposite … She goes to talk to the groups as groups and then, within the group, to the individual. She is reaching out to all learners.
Cooperative learning can occur in various forms, and groups can be of different sizes. One arrangement that uses groups of just two students is known as Think-Pair-Share. Students first think about a problem on their own; then, as a pair, they discuss their ideas with another student; finally, the pairs share their thoughts with the entire class. In this way, there is joint responsibility for ideas, so students are more likely to share with the class.
Hear how Jenny Novak describes her use of Think-Pair-Share:
|Listen to audio clip of teacher
Whether in Think-Pair-Share groups or groups of three or four, consider a variety of factors when arranging student groups:
Teachers often like to pair students in such a way that those who are more comfortable with the content they’re studying can help those who are having difficulty with it. In a group of four, they might place two students they consider “middle level” with one who is very competent and one who is struggling. Because students have different strengths and competencies, it’s a good idea to change groupings frequently.
Read what Jenny Novak has to say about organizing cooperative groups:
I’m fortunate because I have a number of students who are pretty confident and also are very outgoing and are leaders in their groups … I try to match my groups to the particular students I have, so I’ll have a student that I consider a leader in the class in a group with someone I consider to be a little more shy and reserved. And usually I find that when they work together, the leader will help the other student speak a little more and bring out their ideas a little more. And if I have students who are struggling, they usually are pretty good about working together and working through their problems … I change groups regularly in my class so they are not constantly in the same group. They have exposure to different learners and the learning styles of each other.
Students learn in various ways. Some learn best by listening (auditory), some by reading (visual-textual), some with pictures and diagrams (visual-graphical), and some by acting things out (kinesthetic). (See “Rule of Four” in Workshop 5 for a more complete explanation of learning styles.)
Be sure that each group has a mix of quiet, shy, and outgoing students. More importantly, do not create groups in which all students are shy, as little participation may occur; and do not create groups in which all students are outgoing, as they all may attempt to dominate conversations. The group itself is responsible for ensuring that all members participate. However, the teacher may have to intervene to make sure this happens.
Put simply, each group should include students with various abilities. As much as possible, various intelligences should be represented by students in each group. Education scholar Howard Gardner has identified eight different intelligences: linguistic (“word smart”), logical-mathematical (“number smart”), spatial (“picture smart”), bodily-kinesthetic (“body smart”), musical (“music smart”), interpersonal (“people smart”), intrapersonal (“self smart”), and naturalist (“nature smart”).
Many teachers also find that it’s effective to group students randomly. Along with changing groups frequently, this strategy helps students learn how to work with each of their classmates.
While working cooperatively, different groups will arrive at different conclusions. One group may see immediately how to attain the answer and work in that direction. Another group may struggle and never attain a solution. Yet another group may look at the problem from a completely different perspective and find a solution that is unique. Consequently, it is important for the groups to reconvene and share their discoveries and for the teacher to facilitate clarification of any misunderstandings. While much learning occurs in cooperative groups, students may attain even further understanding by listening to the thoughts and solutions of other groups, as well as sharing their solutions with the class.
Read what Janel Green has to say about bringing the class back together:
Of course, it’s important that the students present their findings. You don’t want to do cooperative learning and not have some type of regrouping and have everybody come together and explain what they found. So I designed a lesson so that we could hear and listen to each other, and learn from each other.
List three ways that you ensure positive interdependence and individual accountability when using cooperative learning in your classroom.
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.