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The roots of traditional teaching mathematics instruction run deep in American history: the first true American school mathematics textbook, Arithmetic, was published in 1788 by Nicolas Pike, who recommended that teachers follow the following script when teaching mathematics:

1. State a rule.

2. Demonstrate the rule with an example.

3. Have students practice the rule.

Pike's approach to mathematics instruction casts a long shadow; it is still the dominant approach in many classrooms across the country. If Pike's method is outdated however, the question remains: what should mathematics instruction look like today? I believe that effective math teaching today shares five critical features:
1. Students Develop Conceptual Understanding and Procedural skills
The history of mathematics education in the United States can be seen as a pendulum swing between an overemphasis on procedural skills at one end, and an overemphasis on conceptual understanding at the other. However, the research is clear: students need both. Students should not only learn how to do something, but also why it works and when to apply their knowledge and skills.
2. Students Communicate with Peers About Mathematics
More traditional mathematics classrooms are often quiet. Today, effective classrooms are filled with purposeful discussions among classmates and the teacher about concepts, skills, and problem solving strategies. Students who communicate with their peers and teachers about their thinking, solution pathways, and insights into how they solved a problem develop a deeper understanding of math. One way for educators to generate discussions is by asking open-ended follow-up questions such as, "Why?" "Can you explain?" And "How do you know?" after students answer questions.
3. Students Develop Perseverance and Practice Mathematics
Anyone who becomes an expert at an activity must practice. Learning mathematics is no different. In effective classrooms, students practice mathematics both in class under the guidance of the teacher and at home, independently. In order to be successful, students also need to practice problems that require them to reason, problem solve, and develop the confidence to know they can solve problems they have never seen before. Teachers in effective classrooms resist the urge to water down expectations in an effort to make work easier for students; instead, they expect them to try and persevere, particularly when it's hard.

4. Students Use Teacher and Peer Feedback to Learn from Mistak es

When learning mathematics, students should make an effort to solve problems, embrace the possibility of making mistakes, and rethink their solution pathway. Mistakes should not be viewed as a setback, but an opportunity to learn, which means the most effective mathematics instruction leverages meaningful feedback and student action on that feedback as part of a formative learning process. In effective classrooms, homework does not count for the bulk of a student's grade because students are encouraged to get feedback on their homework and view any mistakes made as a learning opportunity.
5. Students Use Technology to Support Learning
Technology, including computers and graphing calculators, should be used to help students develop a deeper understanding of mathematics. These tools permit students to see visual representations of mathematical ideas, receive immediate feedback on their work, have access to 24/7 tutorials, and collaborate with peers on homework and projects. Although technology does not and cannot replace an effective classroom teacher, its presence is a critical component of effective mathematics classrooms today.

Although the above are certainly not the only features of effective mathematics classrooms today, I believe they are critical features that, when in place, will better serve student learning today. We are seeing movement away from Pike's outdated method and towards a more engaging and meaningful teaching method, and these five critical features are a step in the right direction.

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To learn more about Nicolas Pike and the history of math instruction in the United States, please refer to Balancing the Equation: A Guide to School Mathematics for Educators and Parents (2016).

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Five Modern Features of Effective Math Instruction

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The roots of traditional teaching mathematics instruction run deep in American history: the first true American school mathematics textbook, Arithmetic, was published in 1788 by Nicolas Pike, who recommended that teachers follow the following script when teaching mathematics: State a rule. Demonstrate the rule with an example. Have students practice the rule.

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Classroom Instruction that Works (p.35-47).

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Classroom Instruction that Works (p.35-47).

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~Positive interdependence and individual accountability are key elements in cooperative learning (p. 37).
~ "Use cooperative learning consistently and systematically" (p. 39).
~ "Design cooperative learning tasks to include strategic use of other instructional strategies that help students deepen their understanding and use knowledge meaningfully" (p. 47).

Dean, C.B., Hubbell, E.R., Pitler, H., and Stone, B.  (2012).  Classroom instruction that works: Research-based strategies for increasing student achievement 2nd edition.  Alexandria, VA.  ASCD.

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Ch 3 - Guiding Principles for Enacting Number Talks in the Classroom.
"Students need to know that no matter the problem, people won't all see or solve it in the same way.  And we want them to learn that when they listen and build on one another's ideas, and look for relationships among our diverse ways of seeing, everyone learns more deeply and understands more clearly" (p. 31).
Humprhreys, C. and Parker, R.  (2015).  Making number talks matter. Portsmouth, NH. Stenhouse Publishers.

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This book is full of reasons to use cooperative learning and ways to implement it in the classroom.

"Students who work in teams feel better about themselves -- not only because their need for inclusion is met, but also because they are more successful academically" (p  4.1).

"Cooperative learning structures are content-free, repeatable instruction sequences that organize the interaction of students to implement the basic principles of cooperative learning" (p. 5.3).

Chapter 12 - Basic Principles

Kagan, S. and Kagan, M. (2017).  Kagan Cooperative Learning. San Clemente, CA. Kagan Publishing

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Teaching Strategies for Improving Algebra Knowledge in Middle School and High School

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Teaching Strategies for Improving Algebra Knowledge in Middle School and High School

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I like this article because it gives a lot of questions that can be asked to students as they are working through problems.

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Cooperative Learning:  Teaching Strategies

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Cooperative Learning:  Teaching Strategies

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Besides giving a break down of what collaborative learning is, I liked this article because it gives several ideas of how to assign jobs or roles for each student in a group. This article also provides a list of collaborative ideas to do within a class.

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Cooperative Learning

What Is It?

Cooperative Learning, sometimes called small-group learning, is an instructional strategy in which small groups of students work together on a common task. The task can be as simple as solving a multi-step math problem together, or as complex as developing a design for a new kind of school. In some cases, each group member is individually accountable for part of the task; in other cases, group members work together without formal role assignments.

According to David Johnson and Roger Johnson (1999), there are five basic elements that allow successful small-group learning:

Positive interdependence: Students feel responsible for their own and the group's effort.

Face-to-face interaction: Students encourage and support one another; the environment encourages discussion and eye contact.

Individual and group accountability: Each student is responsible for doing their part; the group is accountable for meeting its goal.

Group behaviors: Group members gain direct instruction in the interpersonal, social, and collaborative skills needed to work with others occurs.

Group processing: Group members analyze their own and the group's ability to work together.

Cooperative learning changes students' and teachers' roles in classrooms. The ownership of teaching and learning is shared by groups of students, and is no longer the sole responsibility of the teacher. The authority of setting goals, assessing learning, and facilitating learning is shared by all. Students have more opportunities to actively participate in their learning, question and challenge each other, share and discuss their ideas, and internalize their learning. Along with improving academic learning, cooperative learning helps students engage in thoughtful discourse and examine different perspectives, and it has been proven to increase students' self-esteem, motivation, and empathy.

Some challenges of using cooperative learning include releasing the control of learning, managing noise levels, resolving conflicts, and assessing student learning. Carefully structured activities can help students learn the skills to work together successfully, and structured discussion and reflection on group process can help avoid some problems.

Why Is It Important?

The authors of Classroom Instruction that Works cite research showing that organizing students in cooperative learning groups can lead to a gain as high as 28 percentiles in measured student achievement (Marzano, Pickering, and Pollock 2001).

Other researchers report that cooperation typically results in higher group and individual achievement, healthier relationships with peers, more metacognition, and greater psychological health and self-esteem (Johnson and Johnson 1989).

When implemented well, cooperative learning encourages achievement, student discussion, active learning, student confidence, and motivation. The skills students develop while collaborating with others are different from the skills students develop while working independently. As more businesses organize employees into teams and task forces, the skills necessary to be a "team player" (e.g., verbalizing and justifying ideas, handling conflicts, collaborating, building consensus, and disagreeing politely) are becoming more valuable and useful. Using cooperative groups to accomplish academic tasks not only provides opportunities for students to develop interpersonal skills but also gives them authentic experiences that will help them be successful in their future careers.

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Cooperative Learning: Teaching Strategy (Grades K-12)

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Cooperative learning is an instructional strategy in which small groups of students work together on a common task.

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Cooperative Learning Tips and Techniques

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Cooperative Learning Tips and Techniques

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6 Strategies to Use in the Classroom:
1.  Jig-Saw
2. Think-Pair-Share
3. Round Robin
4. Numbered Heads
5. Team-Pair-Solo***
6. Three-Step Review

Also, the article gives classroom management tips to use to help use the collaborative strategies smoothly.

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Cooperative learning is a teaching strategy classroom teachers use to help their students process information more quickly by having them work in small groups to accomplish a common goal. Each member that is in the group is responsible for learning the information given, and also for helping their fellow group members learn the information as well.

In order for Cooperative learning groups to be successful, the teacher and students must all play their part. The teacher's role is to play the part as facilitator and observer, while the students must work together to complete the task.

Use the following guidelines to achieve Cooperative learning success:

Arrange students heterogeneously in groups as few as two and no more than six.

Assign each member of the group a specific role such as: recorder, observer, book keeper, researcher, time keeper, etc.

Monitor each group's progress and teach skills necessary for task completion.

Evaluate each group based upon how well they worked together and completed the task.

1. Noise Control - Use the talking chips strategy to control noise. Whenever a student needs to speak in the group they must place their chip in the middle of the table.

2. Getting Students Attention - Have a signal to get students attention. For example, clap two times, raise your hand, ring a bell, etc.

3. Answering Questions - Create a policy where if a group member has a question they must ask the group first before asking the teacher.

4. Use a Timer - Give students a predetermined time for completing the task. Use a timer or stop watch.

5. Model Instruction - Before handing out the assignment model the instruction of the task and make sure every student understands what is expected.

Here are six common cooperative learning techniques to try in your classroom.

Students are grouped into five or six and each group member is assigned a specific task then must come back to their group and teach them what they learned.

Each member in a group "thinks" about a question they have from what they just learned, then they "pair-up" with a member in the group to discuss their responses. Finally they "share" what they learned with the rest of the class or group.

Students are placed into a group of four to six people. Then one person is assigned to be the recorder of the group. Next, the group is assigned a question that has multiple answers to it. Each student goes around the table and answers the question while the recorder writes down their answers.

Each group member is given a number (1, 2, 3, 4, etc). The teacher then asks the class a question and each group must come together to find an answer. After the time is up the teacher calls a number and only the student with that number may answer the question.

Students work together in a group to solve a problem. Next they work with a partner to solve a problem, and finally, they work by themselves to solve a problem. This strategy uses the theory that students can solve more problems with help then they can alone. Students then progress to the point that they can solve the problem on their own only after first being in a team and then paired with a partner.

The teacher predetermines groups before a lesson. Then, as the lesson progresses, the teacher stops and gives groups three minutes to review what was taught and ask each other any questions they may have.

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Take Two Minutes to Become an Expert in Cooperative Learning

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Cooperative learning is a teaching strategy classroom teachers use to help their students process information more quickly by having them work in small groups to accomplish a common goal. Each member that is in the group is responsible for learning the information given, and also for helping their fellow group members learn the information as well.

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Article - The Effect of Cooperative Learning on Middle School Math

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Article - The Effect of Cooperative Learning on Middle School Math

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This easy-to-read article gives 20 best practices for teachers using collaborative learning.  "Collaborative learning teams are said to attain higher level thinking and preserve information for longer times than students working individually" (TeachThought Staff, 2018). The following tips stood out to me:
~ Focus on problem solving and critical thinking
~ Include different learning scenarios
~ Use scaffolding until students are comfortable with expectations

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contributed by Miriam Clifford

There is an age-old adage that says, "two heads are better than one."

Consider collaboration in recent history: Watson and Crick or Page and Brin (Founders of Google). But did you know it was a collaborative Computer Club about basic programming at a middle school that brought together two minds that would change the future of computing?

Yes, those two were, of course, Bill Gates and Paul Allen, the founders of Microsoft.

Collaborative learning teams are said to attain higher-level thinking and preserve information for longer times than students working individually. Why?

Groups tend to learn through "discussion, clarification of ideas, and evaluation of other's ideas." Perhaps information that is discussed is retained in long-term memory. Research by Webb suggests that students who worked collaboratively on math computational problems earned significantly higher scores than those who worked alone. Plus, students who demonstrated lower levels of achievement improved when working in diverse groups.

Collaborative learning teams are said to attain higher level thinking and preserve information for longer times than students working individually.

Many consider Vygotsky the father of 'social learning.' Vygotsky was an education rebel in many ways. He controversially argued for educators to assess students' ability to solve problems rather than merely acquire knowledge. The idea of collaborative learning has a lot to do with Vygotsky's idea of the "zone of proximal development." It considers what a student can do if aided by peers and adults. By considering this model for learning, we might consider collaboration to increase students' awareness of other concepts.

What are some ways to include best practices for collaborative learning in our classroom?
20 Collaborative Learning Tips And Strategies For Teachers 1. Establish clear group goals
Effective collaborative learning involves the establishment of group goals, as well as individual accountability. This keeps the group on task and establishes an unambiguous purpose. Before beginning an assignment, it is best to define goals and objectives to save time.
2. Keep groups midsized
Small groups of three or fewer usually lack enough diversity and may not allow divergent thinking to occur. Groups that are too large create 'freeloading' where not all members participate. A moderate size group of four or five is ideal.
3. Establish flexible group norms
Research suggests that collaborative learning is influenced by the quality of interactions. Interactivity and negotiation are important in group learning. In the 1960's studies by Jacobs and Campbell suggested that norms are pervasive, even deviant norms were handed down and not questioned.

If you notice a deviant norm, you can do two things: rotate group members or assist in using outside information to develop a new norm. You may want to establish rules for group interactions for younger students. Older students might create their own norms. But remember, given their durable nature, it is best to have flexible norms. Norms should change with situations so that groups do not become rigid and intolerant or develop sub-groups.
4. Build trust and promote open communication
Successful interpersonal communication must exist in teams. Building trust is essential. Deal with emotional issues that arise immediately and any interpersonal problems before moving on. Assignments should encourage team members to explain concepts thoroughly to each other.Studies found that students who provide and receive intricate explanations gain most from collaborative learning. Open communication is key.
5. For larger tasks, create group roles
Decomposing a difficult task into parts to saves time. You can then assign different roles. A great example in my own classroom was in the science lab, fifth grade students assumed different roles of group leader, recorder, reporter, and fact-checker. The students might have turns to choose their own role and alternate roles by sections of the assignment or classes.
6. Create a pre-test and post-test
A good way to ensure the group learns together would be to engage in a pre and post-test. In fact, many researchers use this method to see if groups are learning. An assessment gives the team a goal to work towards and ensures learning is a priority. It also allows instructors to gauge the effectiveness of the group. Changes can be made if differences are seen in the assessments over time. Plus, you can use Bloom's taxonomy to further hone in on specific skills.

Individuals should also complete surveys evaluating how well the group functioned. 'Debriefing' is an important component of the learning process and allows individuals to reflect on the process of group learning.
7. Consider the learning process itself as part of assessment
Many studies such as those by Robert Slavin at Johns Hopkins have considered how cooperative learning helps children develop social and interpersonal skills. Experts have argued that the social and psychological effects on self-esteem and personal development are just as important as the learning itself.

In terms of assessment, it may be beneficial to grade students on the quality of discussion, student engagement, and adherence to group norms. Praise younger groups for the following (for digital collaborative learning, for example) standards. This type of learning is a process and needs explicit instruction in beginning stages. Assessing the process itself provides motivation for students to learn how to behave in groups. It shows students that you value meaningful group interactions and adhering to norms.
8. Consider using different strategies, like the Jigsaw technique.
The jigsaw strategy is said to improve social interactions in learning and support diversity. The workplace is often like a jigsaw. It involves separating an assignment into subtasks, where individuals research their assigned area. Students with the same topic from different groups might meet together to discuss ideas between groups.

This type of collaboration allows students to become 'experts' in their assigned topic. Students then return to their primary group to educate others. Strategies here include using clusters, buzz groups, round-robin, leaning cells, or fishbowl discussions.
9. Allow groups to reduce anxiety
When tackling difficult concepts, group learning may provide a source of support. Groups often use humor and create a more relaxed learning atmosphere that allows for positive learning experiences. Allow groups to use some stress-reducing strategies as long as they stay on task.
10. Establish group interactions
The quality of discussions is a predictor of the achievement of the group. Instructors should provide a model of how a successful group functions. Shared leadership is often useful here. Students should work together on the task and maintenance functions of a group. Roles are important in group development. Task functions include:

Initiating Discussions

Clarifying points

Summarizing

Challenging assumptions/devil's advocate

Providing or researching information

Reaching a consensus

Maintenance involves the harmony and emotional well-being of a group. Maintenance includes roles such as sensing group feelings, harmonizing, compromising and encouraging, time-keeping, relieving tension, bringing people into the discussion, and more.
11. Use real-world problems
Experts suggest that project-based learning using open-ended questions can be very engaging. Rather than spending a lot of time designing an artificial scenario, use inspiration from everyday problems. Real world problems can be used to facilitate project-based learning and often have the right scope for collaborative learning.
12. Focus on enhancing problem-solving and critical thinking skills
Design assignments that allow room for varied interpretations. Different types of problems might focus on categorizing, planning, taking multiple perspectives, or forming solutions. Try to use a step-by step procedure for problem-solving. Mark Alexander explains one generally accepted problem-solving procedure:

1. Identify the objective

2. Set criteria or goals

3. Gather data

4. Generate options or courses of action

5. Evaluate the options using data and objectives

6. Reach a decision

7. Implement the decision

13. Keep in mind the diversity of groups
Mixed groups that include a range of talents, backgrounds, learning styles, ideas, and experiences are best. Studies have found that mixed aptitude groups tend to learn more from each other and increase achievement of low performers. Rotate groups so students have a chance to learn from others.
14. Consider demographics
Equally, balanced gender groups were found to be most effective.

Some research suggests that boys were more likely to receive and give elaborate explanations and their stances were more easily accepted by the group. In majority male groups girls were ignored. In majority girl groups, girls tended to direct questions to the boy who often ignored them. You may also want to specifically discuss or establish gender equality as a norm. This may seem obvious, but it is often missed. It may be an issue you may want to discuss with older students.

15. Use scaffolding or diminished responsibility as students begin to understand concepts.

At the beginning of a project, you may want to give more direction than the end. Serve as a facilitator, such as by gauging group interactions or at first, providing a list of questions to consider. Allow groups to grow in responsibility as times goes on. In your classroom, this may mean allowing teams to develop their own topics or products as time goes on.

After all, increased responsibility over learning is a goal in collaborative learning.
16. Include different types of learning scenarios
Studies suggests that collaborative learning that focuses on rich contexts and challenging questions produces higher-order reasoning. Assignments can include laboratory work, study teams, debates, writing projects, problem-solving, and collaborative writing.
17. Technology makes collaborative learning easier
Collaboration had the same results via technology as in person, increased learning opportunities. Try incorporating free savvy tools for online collaboration such as Stixy, an online shared whiteboard space, Google groups, or Mikogo for online meetings. Be aware that some research suggests that more exchanges related to planning rather than challenging viewpoints occurred more frequently through online interactions.

This may be because the research used students that did not know one another. If this is your scenario, you may want to start by having students get to know each other's backgrounds and ideas beforehand on a blog or chat-board.
18. Avoid 'bad group work'
As with any learning strategy, it's important to have a balanced approach. Cynics usually have a valid point. A New York Times article, cites some criticism of collaboration for not allowing enough time for individual, creative thinking. You may allow some individual time to write notes before the groups begin. This may be a great way to assess an individual grade.

19. Be wary of 'group think '

While collaborative learning is a great tool, it is always important to consider a balanced approach. At times, group harmony can override the necessity for more critical perspectives. Some research suggests that groups favored the more confident members. Changing up groups can help counter this problem.
20. Value diversity
Collaborative learning relies on some buy-in. Students need to respect and appreciate each other's viewpoints for it to work. For instance, class discussions can emphasize the need for different perspectives. Create a classroom environment that encourages independent thinking. Teach students the value of multiplicity in thought. You may want to give historical or social examples where people working together were able to reach complex solutions.

By definition, learning is social in nature. Using different mediums, whether it be books, discussions, technology or projects we study and develop new ideas. We impart ideas and share perspectives with others. Collaboration is a learned process. If managed correctly, it is a powerful tool that can allow educators to tap into new ideas and information.
This is a cross-post from opencolleges.edu.au; image attribution flickr user flickeringbrad

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20 Collaborative Learning Tips And Strategies For Teachers

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contributed by Miriam Clifford There is an age-old adage that says, "two heads are better than one." Consider collaboration in recent history: Watson and Crick or Page and Brin (Founders of Google). But did you know it was a collaborative Computer Club about basic programming at a middle school that brought together two minds that would change the future of computing?

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"Teachers should design activities to promote math understanding by having students practice, experiment, manipulate, reason, and problem solve. Such math activities may help students make connections across math skills and concepts, and other disciplines"

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"Cooperative learning" (i.e., jigsaw, learning together, group investigation, student teams-achievement divisions, and teams-games-tournaments) is a generic term that is used to describe an instructional arrangement for teaching academic and collaborative skills to small, heterogeneous groups of students (Rich,1993; Sharan,1980). Cooperative learning is deemed highly desirable because of its tendency to reduce peer competition and isolation, and to promote academic achievement and positive interrelationships. A benefit of cooperative learning, therefore, is to provide students with learning disabilities (LD), who have math disabilities and social interaction difficulties, an instructional arrangement that fosters the application and practice of mathematics and collaborative skills within a natural setting (i.e., group activity). Thus, cooperative learning has been used extensively to promote mathematics achievement of students both with and without LD (Slavin, Leavey, & Madden, 1984; Slavin, Madden, & Leavey,1984).

According to the National Council of Teachers of Mathematics (NCTM; 1991), learning environments should be created that promote active learning and teaching; classroom discourse; and individual, small-group, and whole-group learning. Cooperative learning is one example of an instructional arrangement that can be used to foster active student learning, which is an important dimension of mathematics learning and highly endorsed by math educators and researchers. Students can be given tasks to discuss, problem solve, and accomplish.

Cooperative learning activities can be used to supplement textbook instruction by providing students with opportunities to practice newly introduced or to review skills and concepts. Teachers can use cooperative learning activities to help students make connections between the concrete and abstract level of instruction through peer interactions and carefully designed activities.

Finally, cooperative learning can be used to promote classroom discourse and oral language development. Wiig and Semel (1984) described mathematics as "conceptually dense." That is, students must understand the language and symbols of mathematics because contextual clues, like those found in reading, are lacking in mathematics. For example, math vocabulary (e.g., greater-than, denominator, equivalent) and mathematical symbols (e.g., =, ‹, or ›) must be understood to work problems as there are no contextual clues to aid understanding. In a cooperative learning activity, vocabulary and symbolic understanding can be facilitated with peer interactions and modeling.

Research (e.g., Johnson & Johnson, 1986) supports cooperative learning as an effective approach for including students with LD in classroom group work and promoting peer acceptance. However, some researchers (e.g., Andersen, Nelson, Fox, & Gruber, 1988; Rivera, Gillam, Goodwin, & Smith,1996; Slavin, Madden, & Leavey,1984) caution teachers to integrate direct instruction and cooperative learning. Thus, in cooperative learning groups with proper instruction and preparation (i.e., previous direct instruction on skills), students with math disabilities can benefit from peer interactions to learn mathematics skills and concepts. The purpose of this article is to discuss the components of cooperative learning and to present an example of how cooperative learning can be used to teach mathematics skills.

Cooperative learning components

The literature (e.g., Johnson, Johnson, & Holubec,1994) is replete with descriptions of cooperative learning; therefore, only a brief overview of the components of cooperative learning are described to serve as a foundation for the remaining section of this article. Cooperative learning consists of three major components: "lesson preparation," "lesson instruction," and "lesson evaluation." Each component is briefly described.

Lesson preparation

During the "lesson preparation" component, teachers (a) select the mathematics and collaborative objectives to target for instruction and cooperative learning groups, (b) plan the math activity, (c) identify ways to promote the elements of cooperative learning, (d) identify roles, and (e) establish groups. To identify mathematics content area objectives for instruction, teachers can examine a variety of resources, such as curriculum guides, textbooks, the Curriculum and Evaluation Standards (NCTM, 1989), and students' Individualized Education Programs (IEPs). Additionally, teachers can use assessment information obtained from clinical interviews, criterion-referenced tests, and error analyses (Enright, 1995; Rivera & Bryant, 1992) that reflects students' knowledge of prerequisite mathematics skills and concepts (see Bryant's article in this series).

Collaborative objectives, in turn, can be drawn from curriculum guides, IEPs, and other references (e.g., Jackson, Jackson, & Monroe,1983; McGinnis & Goldstein,1984; Walker et al.,1983). Additionally, it is recommended that student group behaviors and interactions be observed to identify those collaborative skills (e.g., listening, sharing, taking turns, asking questions, using self-control, compromising, contributing ideas) that require intervention to enable students to work successfully as a group towards task completion.

Designing math activities for cooperative learning groups requires consideration of both the instructional objectives and the purposes for having children work in a cooperative instructional arrangement. Teachers should design activities to promote math understanding by having students practice, experiment, manipulate, reason, and problem solve. Such math activities may help students make connections across math skills and concepts, and other disciplines. Kagan (1989/90) identified ways to structure group activities to foster group interactions. Table 1 presents four examples of activity structures, their definitions, and applications to math activities.

According to Johnson et al. (1994), there are five basic elements of cooperative learning: positive interdependence, face-to-face interaction, individual accountability, group behaviors, and group processing. Positive interdependence means that students see the importance of working as a team and realize that they are responsible for contributing to the group's effort. face-to-face interaction involves students working in environmental situations that promote eye contact and social space so that students can engage in discussions. Individual accountability suggests that each person is responsible to the group and must be a contributing member- not someone who lets others do all of the work. Group behaviors refer to those interpersonal, social, collaborative skills needed to work with others successfully. Finally, group processing is a time after the cooperative learning task is finished when team members analyze their own and their group's abilities to work collaboratively.

Structure

Definition

Math Example

1. Categorizing

1. Students analyze and classify objects based on specific criteria.

1 a. Categorize based on attributes.
1 b. Categorize numbers in various ways: odd, even, multiples.

2. CO-OP

2. Each student studies a part of a topic and then presents his or her information to group teammates.

2 a. Study classmate preferences on certain topics and construct graphs.
2 b. Learn algebraic formulas and solve equations together.

3. Numbered Heads

3. After each team member numbers off, students discuss the answer to a question. Then, in a large group the teacher calls a specific number and group to answer the questions.

3 a. Discuss the answer to a mental computation problem.
3 b. Apply the definition of a rule previously introduced to problems; explain the application of the rule.

4. Round the Table

4. Students work on problems jointly by passing the problems around the table for each member's response.

4 a. Pass a worksheet with multiplication facts for each member to answer a problem.
4 b. Pass problems for each member to compute the next step of an algorithm.

These five elements can be structured to promote team work and collaborative skills. They can be facilitated in various ways, for example, by (a) asking students to be responsible for certain duties (e.g., record keeper, spokesperson, encourager); (b) providing limited materials thus necessitating sharing; (c) providing bonus points for demonstrating collaborative behaviors; (d) asking students to self-evaluate after-task completion, (e) assigning a group grade to the math activity, and (f) arranging the environment so students interact in small groups (see Johnson et al.,1994 for a thorough discussion of the five elements and activities to promote them).

Roles with specific responsibilities can be assigned to each group member. Examples of roles include materials person, spokesperson, writer, encourages and timekeeper. Roles should be taught and practiced prior to placing students in cooperative groups; students need a good understanding of the responsibilities associated with each role.

Groups should contain various ability levels. By limiting group size to four to six students, each member should be able to have an active role and access materials within a reasonable amount of time.

Lesson instruction

The "lesson instruction" component of cooperative learning refers to the time in which cooperative learning activities occur. Students should engage in cooperative learning activities after they have received direct instruction in the mathematics and collaborative skills objectives targeted for the group activity. Asking students to perform math activities and collaborative skills for which no previous direct instruction has occurred puts students with LD (as well as other students) at risk for failure and group frustration. Inevitably, the lack of direct instruction prior to cooperative learning may result in numerous questions requesting clarification and assistance. Therefore, "lesson instruction" consists first of direct instruction, and then the cooperative learning activity. Cooperative learning can be used as the "guided practice" time when students engage in tasks to practice introduced skills. Cooperative learning can be used at the onset of math instruction as a means of reviewing skills and concepts or after the presentation of subject matter where new material is practiced within the context of previously taught material. For example, if the math objective is to teach students how to solve story problems using a strategy, then the strategy steps should first be taught directly. Students could then work in a cooperative learning activity that requires the use of the strategy to solve story problems.

An important aspect of the "lesson instruction" component is the teacher's role. The teacher must (a) have students transition quickly after direct instruction, (b) have activities and materials ready, (c) monitor student progress in groups, and (d) reinforce the occurrences of collaborative behaviors.

During cooperative learning activities, teachers should circulate among groups monitoring the students' ability to complete the assigned mathematics activity and demonstrate the targeted collaborative skills. The teacher can facilitate group work by asking questions to help students redirect their work, by providing additional instruction to some students who may be struggling with the task, and by reinforcing students' efforts for working collaboratively and seeking solutions to problems.

Lesson evaluation

The purpose of the "lesson evaluation" component is to assess student mastery of the math objectives and the group's ability to work collaboratively. Teachers can conduct such evaluation by (a) observing students during the cooperative learning activity, (b) having students complete individual tasks following cooperative learning activities, and (c) asking students to engage in group processing (self-evaluation).

Teachers can assess students' mathematics abilities during the group activity by addressing evaluation questions, such as those listed in Table 2. Group and/or individual responses and needs can be recorded on a clipboard to determine if additional instruction and group work are necessary for students to achieve mastery. Answers to the evaluation questions may suggest further direct instruction in a math skill with some or all of the students.

When the cooperative learning activity is finished, teachers may want to administer an individual posttest to determine how well each student has mastered the mathematics content. This is a common form of pupil evaluation that typically yields some type of permanent product, which can then can be graded. The purpose of this evaluation is to ascertain whether students are capable of performing the mathematics objectives independently at mastery level.

Students also should be given the opportunity to evaluate their ability to be team players; this is called group processing. Johnson and Johnson (1986) recommended that, following any cooperative learning activity, students should have time to discuss how their group performed in completing the math activities. Their responses could be recorded and discussed with the teacher to determine pupil-teacher agreement on the group's ability to work collaboratively.

With careful planning, implementation, and evaluation cooperative learning activities can be achieved successfully by most students. The next section provides an example of using cooperative learning to teach mathematics.

Teaching mathematics using cooperative learning

Below is an example of using cooperative learning to teach a math lesson based on the three major components of cooperative learning: "lesson preparation," "lesson instruction," and "lesson evaluation." In this example, five students with LD attend a third grade general education classroom for most of the school day and receive special education resource remedial assistance for mathematics skills. The cooperative learning activity in this example is taking place in the general education setting where the general and special education teachers plan and teach cooperative learning math activities collaboratively twice a week.

Lesson preparation

During "preparation" the cooperative learning math activity is designed; a description of "preparation" activities follows.

Establish objectives. In this example, the instructional objective for mathematics is: "Students will solve two-step story problems containing extraneous information with 90% accuracy." The collaborative objective is: "Students will encourage and support teammates and share materials when requested." The objectives are based on (a) school district special education curriculum guides, (b) students' Individualized Education Program goals for mathematics and social skills, (c) curriculum-based assessment of whole number computation, and (d) observations of group behaviors and interactions.

Structure the activity. In whole group instruction, the instructional objective will be addressed by reviewing with all students the steps of a story problem-solving strategy that was learned the previous week. Students will recite the strategy's steps using cue cards. Using the strategy, two story problems will be solved by the teachers who will recite the steps and verbalize their thinking processes as they work through the problems. Then, students will solve two story problems with the teachers. Next, students will review cooperative learning role responsibilities and explain ways to encourage and support each other. Rules about sharing also will be reviewed.

In the cooperative learning group, "numbered heads" will be used as the activity structure. Students will use their strategy cue cards to solve four story problems. Teachers will facilitate group work and interactions. Time will be allowed for group processing and students (when called on by group and number) will explain how their group solved a particular story problem.

Promote the elements of cooperative learning. Student roles will be assigned and bonus points will be distributed intermittently based on each group's demonstration of encouraging and supportive behavior. One strategy cue card will be distributed to each group, thus necessitating sharing of the card. A posttest will be individually administered containing four story problems to determine if students can solve the story problems independently using their cue cards. The reading level of the story problems will be controlled for different ability levels in the classroom.

Identify the roles and groups. Each group will include a timekeeper to monitor the time and keep the group on task, a materials person to manage the cue card, a writer to record the group's problem-solving responses and answers, and a spokesperson to lead the group during group processing time and to share the group's results with the teacher. The groups will consist of four students; only one student with LD will be a member of each group.

1. Language/Vocabulary
- Are students using new vocabulary words properly?
- Do students possess prerequisite vocabulary?
- Can students provide explanations in their own words for cooperative learning math activities, such as solving word problems and algorithms?

2. Rules
- Can Students explain to each other the rules that were taught during direct instruction, which must be applied in the cooperative learning activity?
- Can students apply the rules to the cooperative learning math activity or do they require teacher assistance?
- Can students use manipulatives to demonstrate rules?

3. Strategies and Algorithms
- Have students learned the strategies and algorithms?
- Can students explain the strategies and algorithms to each other?
- Do students require visual cures for remembering the strategies an algorithms?
- Can students apply strategies and algorithms to a variety of problems?
- Do students require teacher prompting and questions to help remember the strategies and algorithms?

4. Connections
- Can students explain how the new information relates to previously mastered math skills and concepts?
- How do students explain the relevance of learning new math skills and concepts to everyday life?
- How do students apply the new knowledge to activities that involve other disciplines (e.g., science, social studies)?
- Can students depict math information using visuals, graphics, manipulatives, and abstract symbols?
- Can students make connections between concrete-semi-concrete-abstract representations?

Lesson instruction

Implementation of the math lesson, in this example, requires direct instruction followed by the cooperative learning activity. The instructional steps are described below.

Provide an advance organizer. Explain the purpose of the lesson and the instructional and collaborative objectives. Describe the lesson's activities and the teachers' roles in the lesson. Remind students that they worked on a story problem-solving strategy last week and ask for a definition of a strategy.

Present the lesson. Have students refer to their strategy cue cards and repeat the strategy steps. Ask individual students to recite the steps, then ask students to repeat the steps without referring to the cue card, if possible. Next, model solving a story problem using the strategy cue card and verbalizing the steps. Have students imitate this process solving another problem at their desks. Ask for answers and explanations of how the problem was solved.

Explain the cooperative learning activity, using the "numbered heads" structure. Remind students that they can use a cue card to solve their four story problems. Review students' roles and responsibilities and ask for explanations of how students encourage and support one another. Provide directions for transitioning into cooperative learning groups, set a time, distribute materials, and review the task. Once students are in groups, serve as a facilitator by guiding students with questions (e.g., "What are the steps in the strategy?" "What do you do first?" "How do you determine extraneous information?") or providing further instruction if necessary. Reinforce groups for demonstrating appropriate collaborative behaviors. Provide time for group processing, and call on students by number and group to provide answers to the story problems.

Lesson evaluation

Evaluating the students' mastery of the instructional and collaborative objectives is critical. As mentioned earlier in this article, there are three types of evaluation. In this example, the first evaluation can be done during the cooperative learning activity: note evaluative comments that may assist in planning additional lessons or document individual student difficulty. For instance, evaluation questions like those in Table 1 can be used to determine mastery or potential trouble spots solving story problems. The second evaluation is individual and can be done following the group activity by administering a posttest. This can help teachers determine students' ability to solve story problems on their own and to apply the strategy. Finally, have students evaluate themselves during group processing to determine their abilities with the designated collaborative skills. This evaluation should be shared with the teacher to be sure that teacher and student perceptions of abilities match.

Conclusions

Cooperative learning is a popular instructional arrangement for teaching mathematics to students both with and without LD. Coupled with direct instruction, cooperative learning holds great promise as a supplement to textbook instruction by providing students with LD opportunities to practice math skills and concepts, reason and problem solve with peers, use mathematical language to discuss concepts, and make connections to other skills and disciplines. Carefully constructed lessons, using the "lesson preparation," "lesson instruction," and "lesson evaluation" components can offer students with LD rich learning opportunities in mathematics instruction.
References
References

Andersen, M., Nelson, L. R., Fox, R. G., & Gruber, S. E. (1988). Integrating cooperative learning and structured learning: Effective approaches to teaching social skills. Focus on Exceptional Children, 20(9), 18.

Andrini, B. (1991). Cooperative learning and mathematics. San Juan Capistrano, CA: Resources for Teachers, Inc.

Enright, B. (1995 ). Basic mathematics. In J. S. Choate, B. E. Enright, L. J. Miller, J. A. Poteet, and T. A. Rakes, Curriculum-based assessment and programming (3rd ed.) (pp. 286319). Boston: Allyn & Bacon.

Jackson, N. F., Jackson, D. A., & Monroe, C. (1983). Getting along with others: Teaching social effectiveness to children. Champaign, IL: Research Press.

Johnson, D. W., & Johnson, R. T. (1986). Mainstreaming and cooperative learning strategies. Exceptional Children, 52(6), 553561.

Johnson, D., Johnson, R., & Holubec, E. (1994). The new circles of teaming: Cooperation in the classroom and school Alexandria, VA: Association for Supervision and Curriculum Development.

Kagan, S. (1989/90). The structural approach to cooperative learning. Educational Leadership, 47(4), 1215.

McGinnis, E ., & Goldstein, A. P. (1984) . Skillstreaming the elementary school child. Champaign, IL: Research Press.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author.

Rich, Y. (1993). Education and instruction in the heterogeneous class. Springfield, IL: Charles C. Thomas, Publisher.

Rivera, D., & Bryant, B. R. (1992). Mathematics instruction for students with special needs. Intervention in School and Clinic, 28(2), 7186.

Rivera, D. P., Gillam, R., Goodwin, M., & Smith, R. (1996). The effects of cooperative learning on the acquisition of story problem solving skills of students with learning disabilities. Manuscript submitted for publication.

Sharan, S. (1980). Cooperative learning in small groups: Recent methods and effects on achievement, attitudes, and ethnic relations. Review of Educational Research, 50(2), 241 271.

Slavin, R. E., Leavey, M. B., & Madden, N. A. (1984). Combining cooperative learning and individualized instruction: Effects on student mathematics achievement, attitudes, and behaviors. The Elementary School Journal, 84(4), 409422.

Slavin, R. E., Madden, N. A., & Leavey, M. (1984). Effects of team assisted individualization on the mathematics achievement of academically handicapped and non-handicapped students. Journal of Educational Psychology, 76(5), 813819.

Walker, H., McConnell, S., Holmes, D., Todis, B., Walker, J., & Golden, N. (1983). The Walker Social Skills Curriculum. Austin, TX: PROED.

Wiig, E., & Semel, E. (1984). Language assessment and intervention for the learning disabled child(and ed.). Columbus, OH: Merrill.

Diane Pedrotty Rivera, The University of Texas at Austin, LD Forum: Council for Learning Disabilities, Spring 1996

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Using Cooperative Learning to Teach Mathematics to Students with Learning Disabilities

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LD OnLine is the leading website on learning disabilities, learning disorders and differences. Parents and teachers of learning disabled children will find authoritative guidance on attention deficit disorder, ADD, ADHD, dyslexia, dysgraphia, dyscalculia, dysnomia, reading difficulties, speech and related disorders.

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Group work: Using cooperative learning groups effectively

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Group work: Using cooperative learning groups effectively

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I found this article helpful because not only does it explain what cooperative learning should look like, but it also gives evidence and strategies to use.  One strategy that impressed me that was new to me was "Peer Instruction" through the use of clickers.
"In addition to providing feedback on group and individual performance (link to preparation section above), it is also useful to provide a structure for groups to reflect on what worked well in their group and what could be improved" (Brame and Biel, 2015).

"In team-based learning, two approaches promote positive interdependence and individual accountability. First, students take an individual readiness assessment test, and then immediately take the same test again as a group. Their grade is a composite of the two scores. Second, students complete a group project together, and receive a group score on the project. They also, however, distribute points among their group partners, allowing student assessment of members’ contributions to contribute to the final score".

Brame, C.J. and Biel, R. (2015). Setting up and facilitating group work:

Using cooperative learning groups effectively.

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Cite this guide: Brame, C.J. & Biel, R. (2015). Setting up and facilitating group work:
Using cooperative learning groups effectively. Vanderbilt University Center for Teaching. Retrieved [todaysdate] from http://cft.vanderbilt.edu/guides-sub-pages/setting-up-and-facilitating-group-work-using-cooperative-learning-groups-effectively/.

Many instructors from disciplines across the university use group work to enhance their students' learning. Whether the goal is to increase student understanding of content, to build particular transferable skills, or some combination of the two, instructors often turn to small group work to capitalize on the benefits of peer-to-peer instruction. This type of group work is formally termed cooperative learning, and is defined as the instructional use of small groups to promote students working together to maximize their own and each other's learning (Johnson, et al., 2008).

Cooperative learning is characterized by positive interdependence, where students perceive that better performance by individuals produces better performance by the entire group (Johnson, et al., 2014). It can be formal or informal, but often involves specific instructor intervention to maximize student interaction and learning. It is infinitely adaptable, working in small and large classes and across disciplines, and can be one of the most effective teaching approaches available to college instructors.

What can it look like?

Informal cooperative learning groups
In informal cooperative learning, small, temporary, ad-hoc groups of two to four students work together for brief periods in a class, typically up to one class period, to answer questions or respond to prompts posed by the instructor.

Additional examples of ways to structure informal group work
Think-pair-share
The instructor asks a discussion question. Students are instructed to think or write about an answer to the question before turning to a peer to discuss their responses. Groups then share their responses with the class.
Peer Instruction
This modification of the think-pair-share involves personal responses devices (e.g. clickers). The question posted is typically a conceptually based multiple-choice question. Students think about their answer and vote on a response before turning to a neighbor to discuss. Students can change their answers after discussion, and "sharing" is accomplished by the instructor revealing the graph of student response and using this as a stimulus for large class discussion. This approach is particularly well-adapted for large classes.
Jigsaw
In this approach, groups of students work in a team of four to become experts on one segment of new material, while other "expert teams" in the class work on other segments of new material. The class then rearranges, forming new groups that have one member from each expert team. The members of the new team then take turns teaching each other the material on which they are experts.
Formal cooperative learning groups
In formal cooperative learning students work together for one or more class periods to complete a joint task or assignment (Johnson et al., 2014). There are several features that can help these groups work well:

The instructor defines the learning objectives for the activity and assigns students to groups.

The groups are typically heterogeneous, with particular attention to the skills that are needed for success in the task.

Within the groups, students may be assigned specific roles, with the instructor communicating the criteria for success and the types of social skills that will be needed.

Importantly, the instructor continues to play an active role during the groups' work, monitoring the work and evaluating group and individual performance.

Instructors also encourage groups to reflect on their interactions to identify potential improvements for future group work.

This video shows an example of formal cooperative learning groups in David Matthes' class at the University of Minnesota:

There are many more specific types of group work that fall under the general descriptions given here, including team-based learning, problem-based learning, and process-oriented guided inquiry learning.

What's the theoretical underpinning?

The use of cooperative learning groups in instruction is based on the principle of constructivism, with particular attention to the contribution that social interaction can make. In essence, constructivism rests on the idea that individuals learn through building their own knowledge, connecting new ideas and experiences to existing knowledge and experiences to form new or enhanced understanding (Bransford, et al., 1999). The consideration of the role that groups can play in this process is based in social interdependence theory, which grew out of Kurt Koffka's and Kurt Lewin's identification of groups as dynamic entities that could exhibit varied interdependence among members, with group members motivated to achieve common goals. Morton Deutsch conceptualized varied types of interdependence, with positive correlation among group members' goal achievements promoting cooperation.

Lev Vygotsky extended this work by examining the relationship between cognitive processes and social activities, developing the sociocultural theory of development. The sociocultural theory of development suggests that learning takes place when students solve problems beyond their current developmental level with the support of their instructor or their peers. Thus both the idea of a zone of proximal development, supported by positive group interdependence, is the basis of cooperative learning (Davidson and Major, 2014; Johnson, et al., 2014).

Cooperative learning follows this idea as groups work together to learn or solve a problem, with each individual responsible for understanding all aspects. The small groups are essential to this process because students are able to both be heard and to hear their peers, while in a traditional classroom setting students may spend more time listening to what the instructor says.

Cooperative learning uses both goal interdependence and resource interdependence to ensure interaction and communication among group members. Changing the role of the instructor from lecturing to facilitating the groups helps foster this social environment for students to learn through interaction.

Is there evidence that it works?

David Johnson, Roger Johnson, and Karl Smith performed a meta-analysis of 168 studies comparing cooperative learning to competitive learning and individualistic learning in college students (Johnson et al., 2006). They found that cooperative learning produced greater academic achievement than both competitive learning and individualistic learning across the studies, exhibiting a mean weighted effect size of 0.54 when comparing cooperation and competition and 0.51 when comparing cooperation and individualistic learning. In essence, these results indicate that cooperative learning increases student academic performance by approximately one-half of a standard deviation when compared to non-cooperative learning models, an effect that is considered moderate. Importantly, the academic achievement measures were defined in each study, and ranged from lower-level cognitive tasks (e.g., knowledge acquisition and retention) to higher level cognitive activity (e.g., creative problem solving), and from verbal tasks to mathematical tasks to procedural tasks. The meta-analysis also showed substantial effects on other metrics, including self-esteem and positive attitudes about learning. George Kuh and colleagues also conclude that cooperative group learning promotes student engagement and academic performance (Kuh et al., 2007).

Springer, Stanne, and Donovan (1999) confirmed these results in their meta-analysis of 39 studies in university STEM classrooms. They found that students who participated in various types of small-group learning, ranging from extended formal interactions to brief informal interactions, had greater academic achievement, exhibited more favorable attitudes towards learning, and had increased persistence through STEM courses than students who did not participate in STEM small-group learning.

The box below summarizes three individual studies examining the effects of cooperative learning groups.

What are approaches that can help make group work effective?
Preparation Articulate your goals for the group work, including both the academic objectives you want the students to achieve and the social skills you want them to develop. Determine the group conformation that will help meet your goals.
In informal group learning, groups often form ad hoc from near neighbors in a class.

In formal group learning, it is helpful for the instructor to form groups that are heterogeneous with regard to particular skills or abilities relevant to group tasks. For example, groups may be heterogeneous with regard to academic skill in the discipline or with regard to other skills related to the group task (e.g., design capabilities, programming skills, writing skills, organizational skills) (Johnson et al, 2006).

Groups from 2-6 are generally recommended, with groups that consist of three members exhibiting the best performance in some problem-solving tasks (Johnson et al., 2006; Heller and Hollabaugh, 1992).

To avoid common problems in group work, such as dominance by a single student or conflict avoidance, it can be useful to assign roles to group members (e.g., manager, skeptic, educator, conciliator) and to rotate them on a regular basis (Heller and Hollabaugh, 1992). Assigning these roles is not necessary in well-functioning groups, but can be useful for students who are unfamiliar with or unskilled at group work.
Choose an assessment method that will promote positive group interdependence as well as individual accountability.
In team-based learning, two approaches promote positive interdependence and individual accountability. First, students take an individual readiness assessment test, and then immediately take the same test again as a group. Their grade is a composite of the two scores. Second, students complete a group project together, and receive a group score on the project. They also, however, distribute points among their group partners, allowing student assessment of members' contributions to contribute to the final score.

Heller and Hollabaugh (1992) describe an approach in which they incorporated group problem-solving into a class. Students regularly solved problems in small groups, turning in a single solution. In addition, tests were structured such that 25% of the points derived from a group problem, where only those individuals who attended the group problem-solving sessions could participate in the group test problem. This approach can help prevent the "free rider" problem that can plague group work.

The University of New South Wales describes a variety of ways to assess group work, ranging from shared group grades, to grades that are averages of individual grades, to strictly individual grades, to a combination of these. They also suggest ways to assess not only the product of the group work but also the process. Again, having a portion of a grade that derives from individual contribution helps combat the free rider problem.
Helping groups get started Explain the group's task, including your goals for their academic achievement and social interaction. Explain how the task involves both positive interdependence and individual accountability, and how you will be assessing each.
Assign group roles or give groups prompts to help them articulate effective ways for interaction. The University of New South Wales provides a valuable set of tools to help groups establish good practices when first meeting. The site also provides some exercises for building group dynamics; these may be particularly valuable for groups that will be working on larger projects.

Regularly observe group interactions and progress, either by circulating during group work, collecting in-process documents, or both. When you observe problems, intervene to help students move forward on the task and work together effectively. The University of New South Wales provides handouts that instructors can use to promote effective group interactions, such as a handout to help students listen reflectively or give constructive feedback, or to help groups identify particular problems that they may be encountering.

Assessing and reflecting

In addition to providing feedback on group and individual performance (link to preparation section above), it is also useful to provide a structure for groups to reflect on what worked well in their group and what could be improved. Graham Gibbs (1994) suggests using the checklists shown below.

The University of New South Wales provides other reflective activities that may help students identify effective group practices and avoid ineffective practices in future cooperative learning experiences.

References

Bransford, J.D., Brown, A.L., and Cocking, R.R. (Eds.) (1999). How people learn: Brain, mind, experience, and school. Washington, D.C.: National Academy Press.

Bruffee, K. A. (1993). Collaborative learning: Higher education, interdependence, and the authority of knowledge. Baltimore, MD: Johns Hopkins University Press.

Cabrera, A. F., Crissman, J. L., Bernal, E. M., Nora, A., Terenzini, P. T., & Pascarella, E. T. (2002). Collaborative learning: Its impact on college students' development and diversity. Journal of College Student Development, 43(1), 20-34.

Davidson, N., & Major, C. H. (2014). Boundary crossing: Cooperative learning, collaborative learning, and problem-based learning. Journal on Excellence in College Teaching, 25 (3&4), 7-55.

Dees, R. L. (1991). The role of cooperative leaning in increasing problem-solving ability in a college remedial course. Journal for Research in Mathematics Education, 22(5), 409-21.

Gokhale, A. A. (1995). Collaborative Learning enhances critical thinking. Journal of Technology Education, 7(1).

Heller, P., and Hollabaugh, M. (1992) Teaching problem solving through cooperative grouping. Part 2: Designing problems and structuring groups. American Journal of Physics 60, 637-644.

Johnson, D.W., Johnson, R.T., and Smith, K.A. (2006). Active learning: Cooperation in the university classroom (3 rd edition). Edina, MN: Interaction.

Johnson, D.W., Johnson, R.T., and Holubec, E.J. (2008). Cooperation in the classroom (8 th edition). Edina, MN: Interaction.

Johnson, D.W., Johnson, R.T., and Smith, K.A. (2014). Cooperative learning: Improving university instruction by basing practice on validated theory. Journl on Excellence in College Teaching 25, 85-118.

Jones, D. J., & Brickner, D. (1996). Implementation of cooperative learning in a large-enrollment basic mechanics course. American Society for Engineering Education Annual Conference Proceedings.

Kuh, G.D., Kinzie, J., Buckley, J., Bridges, B., and Hayek, J.C. (2007). Piecing together the student success puzzle: Research, propositions, and recommendations (ASHE Higher Education Report, No. 32). San Francisco, CA: Jossey-Bass.

Love, A. G., Dietrich, A., Fitzgerald, J., & Gordon, D. (2014). Integrating collaborative learning inside and outside the classroom. Journal on Excellence in College Teaching, 25(3&4), 177-196.

Smith, M. E., Hinckley, C. C., & Volk, G. L. (1991). Cooperative learning in the undergraduate laboratory. Journal of Chemical Education 68(5), 413-415.

Springer, L., Stanne, M. E., & Donovan, S. S. (1999). Effects of small-group learning on undergraduates in science, mathematics, engineering, and technology: A meta-analysis. Review of Educational Research, 96(1), 21-51.

Uribe, D., Klein, J. D., & Sullivan, H. (2003). The effect of computer-mediated collaborative learning on solving ill-defined problems. Educational Technology Research and Development, 51(1), 5-19.

Vygotsky, L. S. (1962). Thought and Language. Cambridge, MA: MIT Press.

Vygotsky, L. S. (1978). Mind in society. Cambridge, MA: Harvard University Press.

This teaching guide is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

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Group work: Using cooperative learning groups effectively

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by Cynthia J. Brame and Rachel Biel Print Version Cite this guide: Brame, C.J. & Biel, R. (2015). Setting up and facilitating group work: Using cooperative learning groups effectively. Vanderbilt University Center for Teaching. Retrieved [todaysdate] from http://cft.vanderbilt.edu/guides-sub-pages/setting-up-and-facilitating-group-work-using-cooperative-learning-groups-effectively/. Many instructors from disciplines across the university use group work to enhance their students' learning.

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Article - Cooperative Learning Strategies

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Article - Cooperative Learning Strategies

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"Cooperative learning provides opportunities for productive struggle, in which students learn from their mistakes through explanations from their peers and teacher. The classroom environment must be such that students feel safe to make mistakes" (Joyce, 2016).

Joyce encourages teachers to set rules and norms before students work in groups.  He also highly suggests a participatory quiz.  I am still a little confused exactly how this works.

Strategies to try in class:
Red Light Green Light
Pair Shares
Hot Potato

Joyce, M. (2016, December 5). Cooperative learning strategies. National Council of Teachers of

Mathematics.

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By Martin Joyce, posted December 5, 2016 -

Cooperative learning is a learning and teaching style that contrasts greatly with traditional direct instruction. In direct instruction, the teacher completes a math example for the class, then works with the class, and finally the students try it on their own. With cooperative learning, the students start the problem and work it out together. The teacher then provides closure, after students have presented their ideas and shown how they have connected the ideas and added academic vocabulary. According to my review of the research, fewer students can access the content using direct instruction and usually forget it quickly. Cooperative learning provides opportunities for productive struggle, in which students learn from their mistakes through explanations from their peers and teacher. The classroom environment must be such that students feel safe to make mistakes.

Group work takes commitment. For many teachers (myself included), it is difficult at the beginning. You need to fight the urge to take back control. When students are in control of their learning, they have time to try out ideas, listen to one another, and gain confidence among their peers and the whole class. It can and will be frustrating at times. Remember, we aren't all naturally great at group work. I had to teach my accelerated class the word "tact" after hearing what was said among classmates. It takes practice. I hope that the following suggestions of establishing study team norms and using study team support strategies will help your students improve and that you will see positive results.

Each year, I introduce study team norms and reinforce them with participation quizzes. The following are some of the rules: No talking outside your team; keep the conversations on math; ask questions, do not give answers; wait for everyone to finish; don't work ahead; keep your desk clear of clutter; justify your answer; and ask your team before the teacher. A colleague of mine, Aristotle Ou, developed these study team norms with each class. This idea was suggested to him by the Week of Inspirational Math on youcubed.org, which asked students to finish two statements: "When working in groups, I like when. . . ." and "When working in groups, I don't like when. . . ." Student answers contain such gems as "I don't like when people are off topic, give up, say the answer before you tried it," and so on.

One of the most important and effective study team strategies suggested by CPM (College Preparatory Math) is the participation quiz. Place a grid on the board and/or your clipboard of the groups' seating arrangements. Before the students start the lesson, highlight one of the study team norms. Then, you update the board as the lesson progresses, with positive and negative quotes that are evidence of sharing ideas, critiquing ideas, and checking if teammates understand it. Students enjoy getting the instant feedback and are not distracted by it.

Another strategy that I've used is called red light green light. Students worked on three tasks in which they had to decide if a situation was proportional or not and justify their answer with a table, a graph, and an explanation. Then a representative from their group goes up to the board and checks the answer. If the student is wrong, that's a red light to stop and discuss the mistake with the group and fix it. Then the student has a green light. If the student representative got it right the first time, then it is a green light. This sounds like a simple strategy, but students are motivated when they don't need the teacher to confirm whether they are right or not. It also empowers students to be responsible for their own learning.

Rally Coach or Pairs Check is a partner and group strategy that I also like. Students work on a problem with a classmate. The catch is, one person talks and explains the problem while the other writes and says nothing. If the writer disagrees, they switch roles. Then the pairs check with the pair across the table from them to see if they got the same answer. This strategy increases accountability for students who hide in a group of 4.

The final strategy I'd like to share is called Hot Potato. A group of 4 students uses 1 piece of paper, and each student uses a different colored pencil. They then do one step of a problem, pass the paper to the next person who completes a step, and so on. I (and the students) can look at the paper and instantly see if everyone is contributing.

How have you gotten students to work efficiently in groups?

Martin Joyce, martyjoyce84@gmail.com, is a middle school math teacher at Taylor Middle School in Millbrae, California. He blogs at http://joyceh1.blogspot.com and Tweets from@martinsean. He has taught every level of middle school from sixth-grade math support to eighth-grade accelerated algebra 1. His passion is developing each student's math identity with cooperative learning, Desmos lessons, and peer feedback. He regularly reads books and blogs to refine his craft.

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By Martin Joyce, posted December 5, 2016 - Cooperative learning is a learning and teaching style that contrasts greatly with traditional direct instruction. In direct instruction, the teacher completes a math example for the class, then works with the class, and finally the students try it on their own.

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Article - Deepening Student Understanding with Collaborative Discourse

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Article - Deepening Student Understanding with Collaborative Discourse

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The article discusses how we as educators should want our students to develop a deeper understanding through rigor.  The author of the article lists ways to observe deep understanding in students.

The author also uses a Collaborative Inquiry Plan to guide groups in investigation and discourse in completing a complex task. "Before students can begin their investigation, they must develop a plan: clarify the task, determine how they will approach the task, identify resources needed, assign a role to each member specifically describing what each will contribute, and restate the success criteria" (Hess, 2019).

Hess (2019) tells teachers to do the following:
1.  Start with a challenging, worth-while task
2. Make sure everyone can have a job
3. Develop and share criteria for success.

Hess, Karin K.  (2019, April 4).  Deepening student understanding with collaborative discourse.  Collaborative classrooms.  14(22).

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Most educators can agree that asking students to engage deeply with the content they are learning is quite different from asking them to recall surface-level information. Knowing what deeper understanding looks and sounds like will help teachers employ collaborative discourse as a go-to strategy for almost any lesson.

This approach requires first knowing what rigor is and dispelling some common misconceptions ("All students cannot think deeply"; "Rigor just means doing more or harder work"; "Deeper thinking should not require scaffolding"). When adults engage with complex tasks, they don't simply do more or harder work. They develop a deeper understanding of the work, benefiting greatly from scaffolding (such as seeing exemplar models) and collaborative discourse with peers. There is no reason to think that children cannot do the same.

In my work with Depth of Knowledge and cognitive rigor, I've uncovered six ways we can move students' thinking to deeper understanding. I've applied these ideas to the design of teacher-friendly tools that facilitate classroom implementation. The Collaborative Inquiry Plan is one such tool.

We can observe evidence of deeper thinking when:

1. Students make connections to consolidate their learning.

New information has to a) connect to something you already know, even if it challenges something you think you know; and b) be meaningful, relevant, or personally useful (Sousa, 2015).

2. Students apply what they've learned to novel situations.

The ability to transfer knowledge and skills effectively involves the capacity to take what we know and creatively, flexibly, and fluently apply it in different situations or to new problems (Wiggins & McTighe, 2005).

3. Students have "just right" support and can productively struggle.

Strategic scaffolding is the purposeful use of supports to achieve a balance between cognitive complexity and student autonomy as a task becomes more challenging (Hess, 2018).

4. Students apply higher-order thinking (analyzing, evaluating, and creating) to interact with content.

We frequently use verbs ( analyze, synthesize) to talk about rigor without considering students' depth of interaction with, or the complexity of, the content. For example, analyzing the data in a graph is not the same as analyzing potential flaws in the experimental design that generated the data. To determine task complexity, we should always ask, "What comes after the verb?" (Hess, 2018)

5. Students engage in collaborative discourse that makes thinking visible.

Collaboration and teamwork can be powerful instructional vehicles for learning and support creative thinking (Hess & Gong, 2014). When small, collaborative groups take on a challenging task, they can understand more complex material, engage with content more deeply, and learn at a faster pace (Hess, 2018).

6. Students ask questions that indicate a shift in teacher-student roles as they learn to construct their own meaning.

After you give an assignment, your students shouldn't only be asking procedural or clarifying questions ( Where do we get the sign-out sheet for the microscopes? How long does this have to be?). They should be asking more rigorous questions that address the quality of their work ( Can we use both primary and secondary sources? Do we have choices about the form of our final product?). Rigorous questions spark deeper understanding and lead to both metacognitive thinking (strategizing while engaged in a task) and reflective thinking (evaluating effectiveness of learning processes afterwards). Student-generated questions are clear indicators of who is directing the learning (Hess, 2018).

From the early work of researchers through today, we find compelling support for thoughtfully designed small-group work. Robert Slavin's research (1991) on "team learning" identified two essential elements of collaboration: shared group goals and individual accountability of all group members. The work of Nottingham, Nottingham, and Renton (2017) stresses the power of group dialogue to move student thinking from surface-level knowledge to deeper understanding of concepts.

It is the structure of small-group investigation combined with collaborative discourse that teaches students how to think, how to express ideas clearly, how to respectfully listen for divergent viewpoints and give critical feedback, and most of all, how to formulate their own reasoning.

The research is clear: Collaboration activities that are intentionally structured for shared goals, individual accountability, and focused discourse drive deeper thinking within groups and can have a major positive effect on each student's learning.

The Collaborative Inquiry Plan (see table) is designed to assist teams in thinking through the first phase of completing a complex task. Thinking through the task requirements together helps students to clarify and take responsibility for monitoring their own progress. When students learn to organize how they will work to accomplish their goals, they become more independent learners.

So, how do we thoughtfully design small-group work for collaborative discourse and deeper thinking? An activity I call "Picture Search" can illustrate how a teacher might introduce use of the Collaborative Inquiry Plan to students. Small groups are presented with a photo or historical artifact and given a few questions to answer. Unlike yes-no or fact-based questions, the answers require investigation and must be supported with evidence from multiple credible sources.

Before students can begin their investigation, they must develop a plan: clarify the task, determine how they will approach the task, identify resources needed, assign a role to each member specifically describing what each will contribute, and restate the success criteria.

[[[[[ **** MISSING TABLE **** ]]]]]

Students first need to draw upon their collective prior knowledge ( Does anybody know what this is?), use investigative skills to determine which clues in the picture might lead them to the answers ( What monuments are there/not there? What kinds of transportation are depicted/not depicted?), and determine how to check credibility of sources.

Then, students must develop their inquiry plan and get approval from the teacher. Some teachers might want to codesign a plan with students before giving students more control over developing their own plans.
STEP 1: Start with a task worth doing.
Most students can do many familiar routine tasks (recall facts, apply basic skills) on their own. You want groups to do something more challenging than what individual students can accomplish independently. This is an opportunity to allow them to take risks, struggle through a problem, and learn together as they construct their own meaning.

To teach students how to use different text structures in their writing, have teams select a topic of interest and create a magazine with short articles that apply different structures to convey information (compare-contrast, description, cause-effect, chronology, proposition-support). Another team challenge might be developing an infographic to show relationships among historical, literary, scientific, or cultural influences on an event or time period. A special education teacher told me she uses the collaborative inquiry planning tool with her students to help them decide what snack to make and sell in their school store.
STEP 2: Be sure this is a task where everyone has a job.
We maximize simultaneous engagement when we structure group work that integrates multiple skills and concepts. To determine group size, answer two questions: 1) What is the purpose of the learning task? and 2) How many people should it take to complete the task? Consider the specific roles needed and how you will hold students individually accountable. For the Picture Search example, if each student researches a different aspect of the photo (types of transportation, monuments, land use), the teacher can ask for evidence from each individual about how they completed their role.

At first, you may want to assign and model each role. Later, shift this responsibility to students, asking them to plan how they will accomplish the task and who is taking responsibility for each aspect. Roles that are more task-specific lead to simultaneous engagement and positive interdependence: Researcher #1 will find out when steamboats and trains were the most common modes of transportation, Researcher #2 will develop a timeline of when each monument was constructed, and Researcher #3 will compare current land use with earlier maps reflecting what is depicted in the painting. Students should discuss what questions they are trying to answer and what each person will do to contribute.
STEP 3: Establish parameters and success criteria for completing the task.
Sometimes we give students too much time and too little direction. Saying, "Discuss this with your partner" is not as clear as, "You have one minute to discuss and write three reasons why___." Whether it is short, informal group sharing or a longer, project-based task over several days, giving students a set amount of time and clear success criteria holds everyone accountable for monitoring progress. You can always give groups more time if they need it, but giving them too much time initially is usually a recipe for impending chaos.

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Deepening Student Understanding with Collaborative Discourse - ASCD

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Most educators can agree that asking students to engage deeply with the content they are learning is quite different from asking them to recall surface-level information. Knowing what deeper understanding looks and sounds like will help teachers employ collaborative discourse as a go-to strategy for

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Article - How to Develop Participatory Listeners

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Article - How to Develop Participatory Listeners

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In this article, the author discusses how to teach students to engage in math conversation.  The author states "listening is an active process that students must consciously undertake for math conversations to be meaningful" (Sammons, 2018).  The author also states "using these strategies, you can make the most of math conversations in your classroom by teaching your students to not only express their mathematical ideas more clearly, but also become more effective listeners" (Sammons, 2018).  The author encourages teachers to teach students to be "participatory listeners" by teaching what it means to listen, modeling and practicing listening, setting up the classroom for listening to take place, and encourage to students to be good listeners all the time.

Sammons, Laney.  (2018, July 26).  How to develop participatory listeners in math class.  Talking in Math.  13(22).

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I found this article and thought that it may be relevant to my research by using peer tutoring as a means to have students work collaboratively.

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December 2014/January 2015 | Volume 72 | Number 4
STEM for All

Erica N. Walker

Dwayne Wayne is one of the few representations in television or film of a black person who excelled at mathematics. He was also one of the few popular culture representations of a "math person" (Boaler & Greeno, 2000) who wasn't socially inept or psychologically stunted; his personality and humanity weren't at odds with his mathematical abilities.

In the media, many young people are portrayed as uninterested in school, but portrayals of black and Latino urban students are even more negative; they're rarely shown in ways that highlight their intellectual strengths (Ferguson, 2002; Pimentel, 2010). Such portrayals might contribute to many teachers' low expectations of black and Latino students in subjects like mathematics. Research suggests that some teachers believe that students' cultural backgrounds have a negative impact on their academic behaviors and that socioeconomic status is related to students' motivation, despite evidence to the contrary (Ferguson, 2002; Leonard & Martin, 2013). The emphasis on the achievement gap in math between black or Latino students and others, without a corresponding analysis of the opportunity gap, contributes to these beliefs.

The Reality: Students' Communities Support Achievement

Despite negative media portrayals of black and Latino students' academic abilities, the reality is that, on average, these students have very positive attitudes toward education, school, and mathematics (see, for example, Strutchens, Lubienski, McGraw, & Westbrook, 2004; Walker, 2007). On the 2011 National Assessment of Educational Progress, 48 percent of black 8th graders and 43 percent of Latino 8th graders reported that they "often" or "always/almost always" found their math work engaging and interesting, compared with 37 percent of their white and 50 percent of their Asian counterparts (Snyder & Dillow, 2013).

But young people subjected to daily messages about their supposed lack of potential and interest in mathematics need support. Many of them gain such support through what I call academic communities -informal networks of parents, extended family members, neighbors, classmates, friends, teachers, and other school adults who encourage, teach, and guide young people in mathematics.

Black and Latino youths' interest in math work makes sense within the larger legacy of achievement orientation in underserved communities. Historically, these communities have placed a tremendously high value on education as a means of social mobility and uplift. A large body of research (for example, Leonard & Martin, 2013; Perry, Steele, & Hilliard, 2003) highlights this heritage and the role of community traditions in extending educational opportunities for these students.

My research has explored the various ways that communities-large and small-contribute to the mathematics success of students from underserved backgrounds. My 2012 study of black and Latino students at an East Harlem high school I'll call Lowell High explored how students navigated the worlds of school, home, and peer culture and the roles these worlds played in students' math achievement. High-achieving black students at this school had complex, informal webs of mathematics support-academic communities-of which their teachers were initially not aware. Academic communities appear to also be at work for postsecondary students (Walker, 2014) and even elementary learners (Alexander & Walker, 2012).

Educators who have a particular view of what it means to do mathematics may be surprised to learn that some members of these networks who didn't have a solid academic background themselves-such as parents who didn't finish high school-promoted mathematics success. Fellow students often inspire, collaborate with, or compete with students who have high achievement in math.

Peer Tutoring Power

In 2005, I developed a peer tutoring model at Lowell High that was based on my own experiences as a high school teacher and tapped into this support system. Lowell-a small, predominantly black and Latino high school serving about 300 students-had experienced low math achievement, and according to its math teachers and principal, it needed to jump-start students' interest in math. The model I developed made clear that black and Latino students' peers and classmates could be assets to their academic work, rather than potential-or actual-hindrances.

This program used existing student networks to improve students' education outcomes and exposed math teachers to these networks. Trained high school students became math tutors, and classroom teachers and preservice teachers in graduate school observed the tutors with their tutees and provided support when necessary. Both tutors and tutees were primarily black and Latino.

Students seeking tutoring could drop in; teachers also encouraged students to come for tutoring sessions. Sessions usually focused on homework and test preparation. In the first month of the program, six tutors worked with an average of 8-10 tutees each week; by the program's end, 10-12 tutors worked with approximately 20 tutees each week.

Participating in this tutoring program strengthened students' math achievement. Students' math grades improved, and pass rates on the state's exams increased. Students who attended more peer-tutoring sessions had higher math grades than students who attended fewer sessions. On average, participants gained eight points between fall and spring terms the year they participated.

My interviews with participants and observations of three-times-a-week tutoring sessions over two academic years also revealed that the high school tutors used new, effective instructional approaches with their peers. Participating in the program also increased teachers' knowledge of mathematics instructional strategies. We found that the effectiveness of tutoring sessions depended on the quality of relationships among tutors, tutees, teachers, and graduate student advisors. Most important, tutoring sessions created a space beyond the traditional classroom for students to do math with their friends. High-achieving students were already drawing on peer networks to do math; the program created room for struggling students to benefit from those networks as well.

Four insights from this peer tutoring experience can help teachers and school leaders develop strong academic communities, and build on existing ones, to support math learning.

Listen and Look

As we planned the peer tutoring program at Lowell, math teachers initially had a difficult time identifying top students who could serve as tutors. So I asked the students. They were able to identify more top students (as verified by math grades and test scores) than teachers did-and also identified peers who were good in math, but not necessarily the best students.

Teachers agreed with students' nominations, but it was striking how long it took them to think of top students. They initially chose quiet straight- A students. Students' nominations gave the teachers a chance to reflect on what it means to be good at math. Students selected peers who were A, B, or C students; were good at explaining homework; and who would provide help, when asked, outside class.

Perhaps educators' notions of who is good at math should be more expansive. I often tell teachers the story of Thomas Fuller, an enslaved contemporary of the better-known Benjamin Banneker. Little is known about Fuller, save that he was an illiterate "arithmetical prodigy" who had scant chance to contribute his talents. His obituary, published in 1790, stated, "Had his opportunities of improvement been equal to those of thousands of his fellow-men [not] even a Newton himself, need [have been ashamed] to acknowledge him a brother in science" (Walker, 2014, p. 70). I ask teachers to think, whenever they walk into classrooms, Are there Fullers here-students who may not exhibit traditional characteristics of successful math students but need an opportunity to show what they know?

Get Out of the Way

Tutoring sessions were held in math teachers' classrooms after school, but I encouraged teachers not to be involved in tutoring itself (only to observe and supervise). It was important that student participants took ownership of the tutoring. This enabled teachers to see their students as knowledgeable about math and as agents for their own learning.

Some teachers and administrators were initially surprised at the demand for the program. Students wanted to move almost immediately from a program that met twice a week to one that met three times. They also wanted the program to operate at other times during the school day. Students hoped to hold peer tutoring sessions during lunch. They proposed a "Math Genius Bar" set up in the cafeteria, to which kids could walk up and get help (inspired by the Apple Store model). To my dismay, administrators, concerned about who would supervise the Math Bar, quashed this idea. When the students proposed locating it in the library, administrators also rejected this idea. Sadly, the students wanted to support their own learning, but administrators-most likely unintentionally-thwarted their plans.

Learn and Adapt

An unintended outcome of the peer tutoring program was that teachers ended up learning themselves. They saw struggling students develop into confident, competent students; they saw tutors develop pedagogical strategies to help their peers without spoon-feeding them. Occasionally, teachers and peer tutors worked together with struggling students, and although the teacher-student hierarchy was sometimes present, in most cases teachers allowed tutors to take the lead. I heard one math student helper admonish the teacher to let a tutee "learn it for himself, and not just copy."

The school's mathematics classrooms had generally been traditional environments in which teachers lectured and students sat silently working on problems. As the peer tutoring program developed, students began to collaborate more in the classroom, discussing their answers and helping one another when they got stuck, instead of waiting for the teacher. Teachers referenced peer tutoring interactions when they explained concepts in class, and they cited peer tutors' analogies and examples. The classroom dynamics began to shift from a teacher-dominant space to one where a mathematics community was developing.

Offer More

As at many urban schools serving low-income youth, the opportunities to learn mathematics at Lowell were sometimes limited. For example, there was no precalculus course offered, despite student interest and its importance for college admissions. Teachers and administrators should reflect critically on their policies and practices to make sure they are expanding students' math opportunities rather than limiting them. Where needed, augment curriculum to include rigorous, engaging mathematics problems and courses.

My research indicated that many black and Latino students are interested in mathematics but have been exposed to a limited kind of math both in and out of school. Extracurricular activities should expose students to math beyond the classroom as well as to mathematics-related fields. Teachers should work hard to tap into students' initial interests and seek to engage them further. Such actions raise teachers' and students' expectations-teachers begin to see underserved students as capable contributors to the mathematics enterprise in class, and students begin to see that math is meaningful and worthwhile.

More Engaged Students of Color in Mathematics

These insights could guide similar efforts to support and develop academic communities for black and Latino students learning math. My model is just one of many initiatives that promote student achievement in math by attending to the social aspects of learning, recognizing that collaboration promotes understanding for many students.

In our efforts to improve mathematics teaching and learning for all students, educators should create and sustain communities that actively engage students' learning. Approaches like peer tutoring will help develop cadres of black and Latino students who have strong mathematics identities, are excited about mathematics, and see themselves (and are seen by others) as talented doers of math.

References

Alexander, N. N., & Walker, E. N. (2012, April). Urban elementary school students' academic communities in mathematics. Paper presented at the American Educational Research Association Annual Meeting, Vancouver, BC, Canada.

Boaler, J., & Greeno, J. (2000). Identity, agency, and knowing in mathematical worlds. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 171-200). Stamford, CT: Ablex.

Ferguson, R. F. (2002). What doesn't meet the eye: Understanding and addressing racial disparities in high-achieving suburban schools. Cambridge, MA: Harvard University Press.

Leonard, J., & Martin, D. B. (2013). The brilliance of black children in mathematics. Charlotte, NC: Information Age Publishing.

Perry, T., Steele, C., & Hilliard, A. G. (2003). Young, gifted and black: Promoting high achievement among African American students. Boston: Beacon Press.

Pimentel, C., (2010). Critical race talk in teacher education through movie analysis. Multicultural Education, 17(3), 51-56.

Snyder, T. D., & Dillow, S. A. (2013). Digest of Education Statistics 2012 (NCES 2014-015). National Center for Education Statistics, Institute of Education Sciences, U.S. Department of Education. Washington, DC.

Strutchens, M., Lubienski, S. T., McGraw, R., & Westbrook, S. K. (2004). NAEP findings regarding race and ethnicity: Students' performance, school experiences, attitudes and beliefs, and family influences. In P. Kloosterman & F. Lester (Eds.), Results and interpretations of the 1990-2000 mathematics assessments of the National Assessment of Educational Progress (pp. 269-305). Reston, VA: National Council of Teachers of Mathematics.

Walker, E. N. (2007). Why aren't more minorities taking advanced math? Educational Leadership, 65(3), 48-53.

Walker, E. N. (2012). Building mathematics learning communities: Improving outcomes in urban high schools. New York: Teachers College Press.

Walker, E. N. (2014). Beyond Banneker: Black mathematicians and the paths to excellence. Albany: State University of New York Press.

Endnotes

For details about the design and organization of this tutoring program, see my book Building Mathematics Learning Communities: Improving Outcomes in Urban High Schools (Teachers College Press, 2012).

For a detailed discussion of Banneker and Fuller, see my book Beyond Banneker: Black Mathematicians and the Paths to Excellence (State University of New York Press, 2014).

Erica N. Walker is a professor at Teachers College, Columbia University, in New York.

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How to Engage More Students of Color in Math

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December 2014/January 2015 | Volume 72 | Number 4 STEM for All Erica N. Walker Dwayne Wayne is one of the few representations in television or film of a black person who excelled at mathematics.

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Article - Making Cooperative Learning Powerful

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Article - Making Cooperative Learning Powerful

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This article describes five different strategies that teachers should implement when they are using cooperative learning.

The 5 strategies include:

1. Teacher formed teams

2. Assign goals

3. Individual Accountability

4. Develop communication and listening skills

5. Include cooperative learning as a part of the lesson, but not as the whole lesson

One thing that stood out to me in this article is how the author discussed the importance of making sure each student is accountable and is participating to obtain the goal and learn. I found the quotes below very important.

“First, make sure the team goal requires the learning and participation of all team members. [...] Never choose an outcome one student could do alone. In addition, frequently reinforce the idea that the purpose of the team is to make sure that all members are learning, not just to get the right answers or complete the project” (Slavin, 2014).

"Finally, it's important that the teacher lets the work be challenging and doesn't step in too early to do the work for students who struggle. Doing so not only undermines teammates' developing interdependence, but also teaches students that if they feign helplessness, display apathy, or work slowly, the teacher will soon step in and complete the tasks. If kids instead learn to make appropriate levels of effort and to persist, they will build confidence in their ability to improve and learn. They'll realize that asking for help—not the answers—helps them learn. Letting students struggle constructively and safely in their teams enables each student to reflect on his or her thinking, compare it with others' ideas, and refine that thinking before sharing it with the class—or on a test" (Slavin, 2014).

Slavin, R.E. (2014, October).  Making cooperative learning powerful.  Instruction That Sticks. 72(2), 22-26.

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Just about everyone loves the idea of cooperative learning. Think of kids working productively and excitedly in groups, everyone getting along and enthusiastically helping one another learn. Think of kids completing great projects together, encouraging one another, and forming friendships. In this ideal scenario, all students are engaged, active, and minds-on. They are learning cooperation itself, an important life skill. Cooperative learning: how pro-social, how liberating-and how does anyone really accomplish this?

Cooperative learning can be wonderful. Students often love working this way. I've heard comments like, "An explanation is easier to understand if it's coming from another kid" or "My teammates ... make sure I understand the work." But for many teachers who've tried it, cooperative learning is a noisy mess rather than a stairway to heaven. One student might be doing all the work while the others socialize, for example. Kids may ignore or belittle group mates they perceive to be low achievers. Some students may think cooperative learning is party time rather than learning time. This is why studies find that most teachers don't use cooperative learning regularly, despite extensive research supporting it (Gillies, 2014; Roseth, Johnson, &amp; Johnson, 2008; Slavin, 1995, 2013; Webb, 2008).

It is the "learning" in cooperative learning that is too often left out. But it needn't be. Using these five strategies, teachers can get the greatest benefit possible from cooperative learning and ensure that collaboration enhances learning.

An effective cooperative group is not a collection of kids thrown together for a brief activity. It's a team composed of diverse students who care about helping one another learn-and about the success of the team itself. All members must know they can depend on one another for help.

In the forms of cooperative learning we've used for years at Johns Hopkins University and at the Success for All Foundation in elementary and secondary schools (Slavin, 1995), we group students into four-member teams (if the class doesn't divide evenly, we have a few five-member teams). A foursome provides flexibility. Some activities can be done in pairs and some with the whole team. Teams work together daily for 6-8 weeks; after that period, teachers assign new teams.

Team members move their desks together and choose a team name. At the beginning of the team's work, members engage in group bonding activities, like creating a motto or coat of arms to express their team personality.

In middle or high school, groups might investigate careers that interest them and colleges known for majors tied to those careers. Their research often culminates in each team giving a presentation about why they recommend a particular college, and teams sometimes name themselves after the school they recommend. Even if students choose Duke because of basketball fame, they've spent time helping one another learn something new, shared what matters to them, and met a deadline.

It's best if teams are composed of a cross section of the class: high and low achievers, boys and girls, students of different ethnicities. Groups definitely won't have such diversity if students choose their teams, so the teacher should make team assignments.

Too often in cooperative learning, students are put into teams and instructed to "help one another." Helping one another is a good thing, but kids will do a lot more of it if they share a team goal.

A team goal is a target, product, or indicator that shows a team has done a good job of getting every member to perform at his or her personal best. A team goal could be increasing the average score on a quiz that all students take individually after they help one another prepare. It could be one overall product with individual components that each team member clearly contributed to, such as by coding a specific part of a computer program or contributing essential data to a group lab report.

In each case, a teacher both looks at the team average and evaluates the individual products. Teams whose work meets certain criteria are awarded certificates or small privileges. This helps team members see their joint work as achieving something important. Teams that don't have a goal are like sports teams in a game where no one knows, or cares, what the score is-not the sort of game in which all players do their best.

This is the essential element most often left out of cooperative learning-and when it is, teachers lose a lot of cooperative learning's potential.

Individual accountability means that to reach the team goal, all team members must master the targeted content or skills. Team success should depend on the hard work-and therefore the learning-of all members.

To understand why this is important, consider a team working together without individual accountability. Imagine that a team studies together and then takes one quiz, on which all team members can help one another. This arrangement will likely produce two undesirable outcomes: the free rider and the know-it-all.

Free riders do little work. They may chat with others, but they don't try much to learn or help others learn. "The others will do it anyway," they reason. "Why should I try?" Students who have doubts about their own skills may be shy or reluctant to participate if they know others will do the work.

The know-it-all is another story. A know-it-all (or two) may dominate the group and tell others the answers. I once saw a group of middle schoolers trying to solve a complex math problem together. (Their task was to solve the problem and give the answer as a group.) Two kids were doing most of the talking. When one of their (usually quiet) teammates ventured an opinion, both know-it-alls shushed her and told her to wait for them to deliver the answer.

How can teachers ensure individual accountability? First, make sure the team goal requires the learning and participation of all team members. For example, if the team goal is to score an average of 80 percent or better on a quiz, all students need to do well. Or a goal might be to write an outstanding report on a given country, with each chapter signed by a team member. Never choose an outcome one student could do alone. In addition, frequently reinforce the idea that the purpose of the team is to make sure that all members are learning, not just to get the right answers or complete the project. Explain why individual accountability is fair; students will readily understand why no team member should get a free ride.

During lessons, use informal cooperative structures that reinforce the idea of individual accountability. Teachers we work with in Success for All programs, for example, make frequent use of a technique called "random reporter." Each student is given a number from 1 to 5. When teachers ask a question, they direct it to a team and then pick a number at random. The student on that team with that number has to respond for the whole group-and the team can earn points based on the answer.

Practices like random reporter communicate consistently that teams must make sure all their members learn. Potential free riders understand that they will be held accountable and that they can't wait for their teammates to make all the effort. Know-it-alls realize that if they don't engage, teach, and give feedback to every team member, their team won't succeed. One middle school math teacher noted, "Students ... know they need to work hard on math problems and be ready to explain them. If they aren't ready, they can't get off the hook easily."

Finally, it's important that the teacher lets the work be challenging and doesn't step in too early to do the work for students who struggle. Doing so not only undermines teammates' developing interdependence, but also teaches students that if they feign helplessness, display apathy, or work slowly, the teacher will soon step in and complete the tasks. If kids instead learn to make appropriate levels of effort and to persist, they will build confidence in their ability to improve and learn. They'll realize that asking for help-not the answers-helps them learn. Letting students struggle constructively and safely in their teams enables each student to reflect on his or her thinking, compare it with others' ideas, and refine that thinking before sharing it with the class-or on a test.

Setting up structures that promote effective cooperative learning is not enough. Team members need to know how to make good use of the opportunity to work with one another; this means they need to learn about, practice, and refine key interpersonal skills.

Active listening. Active listening skills are essential in good group work-and in life. When students are listening well, their eyes are on the speaker and they occasionally nod. Active listeners avoid interrupting but periodically summarize what they hear and ask for clarification when needed. These practices communicate respect for the speaker and enable the listener to learn as much as possible from that speaker's words.

Explaining ideas and opinions. Team members need this skill to communicate and persuade in cooperative interactions. Explaining must go beyond single-word answers; students must be able to identify sources or reasons for their personal opinions or conclusions. They must explain their ideas to others so that peers can understand them too. This demands metacognition, evidentiary thinking, summarizing, paraphrasing, and listening to others thoughtfully.

Encouraging teammates. Effective team members know how to encourage and support teammates, disagree with dignity, and help maintain a positive, prosocial tone within the group. Tell students you expect them to make sure that all team members are actively participating in the thinking parts of the group's tasks. Teach them how to make that happen.

Explaining your own ideas while also encouraging others is a complex skill that demands that students respect one another. Guide kids to make teamwork time a safe environment in which to speak, receive feedback, and admit what they don't understand. Once students establish a productive working relationship, they can set goals together, monitor their own progress, and solve learning problems together.

Completing tasks. Students should be expected to work on the group's tasks until they are finished to a high standard and to ensure that at the end of a team study session, all participants have learned the class objective.

When students know what constitutes great work within their team, and reach that standard, they can be proud of themselves and of any recognition their group receives.

Cooperative learning should be seen as a key part of each lesson, but not the whole lesson. Effective class lessons might also include teacher instruction, media- or computer-based activities, and individual assessments of various kinds. Informal cooperative-learning activities such as random reporter or think-pair-share may also be used, but these shouldn't be the only cooperative activity.

The best way to use cooperative learning is to replace individual work, which in traditional lesson cycles happens after lessons and before assessments. Individual, isolated practice is boring and ineffective for most students, especially if they struggle. Cooperative learning makes practicing to mastery engaging and social and gives all students "study buddies" to help them when they run into difficulties.

Sometimes team activities may come before teacher instruction, as when teams are conducting experiments in a discovery learning format. But at some point, a teacher needs to explain the essential objectives and give students parameters and guidance for their group work so they can move forward.

I once observed a class of 9th graders who started the school year reading significantly below grade level. Their first-year teacher worked hard to provide a research-based cooperative learning model in which they would read a wide variety of complex texts, discuss them in teams, and prepare one another to participate in class discussions. However, students weren't making the desired progress. As the teacher looked for what was missing, she noticed students were sloppy in their work and slow to complete tasks. Although they often talked to one another, they weren't helping one another.

The first change this educator made was holding each student individually accountable for the work. The second was providing and discussing with the class rubrics for high-quality answers or comments in discussions. Students and teachers practiced scoring sample answers and exchanged feedback.

Then she made the task more interesting, challenging, personal, and relevant. She asked students to read closely and compare several journal articles about the science of emotions. Team members were to help one another answer questions that required more than recitation of facts from the articles, but to individually form their own hypotheses about the author's intent in writing the articles-and support those hypotheses with evidence from the texts.

In a few days, when the students realized that the teacher wasn't going to dilute this work to save time, they dug into the articles. Discussions flourished. Well-structured cooperative learning and recognition of work that met the standards of the rubrics ultimately transformed this class from remedial to advanced. When the students were asked what they thought about how their class was changing, one girl summed it up: "When I work with my teammates and we use the rubrics and our brains, we know we have a good answer. We're proud to show what we know-we are ready for any of us to report out."

Used properly, cooperative learning is an exciting way for all kids to learn. Research finds that if teachers make the five elements discussed here part of group learning, students learn more, feel more successful, love school, and enjoy the subject they're studying-and like and accept one another (Roseth et al., 2008; Slavin, 2013; Webb, 2008). For outcomes like these, it's worth the effort.

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Five key practices bring out the tremendous potential of this approach.

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Do I need consent if I trying new collaborative and cooperative learning strategies to engage students? I would do this in my class anyways.

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Students do not want to work with others