Practicing What We Preach: Demonstrating Effective

Cooperative Learning Practices

Annette Leopard and Karen Wells, Monroe Community College

34th Annual AMATYC Conference

November 21, 2008

Washington, DC

This workshop is an expanded version of a workshop presented at the Seaway Section, MAA and NYSMATYC Region 1 2007 Fall Joint Conference, October 20, 2007, Monroe Community College, Rochester, NY.

Table of Contents

page
Background Notes / 4
Cooperative Learning Methods Descriptions
Round Robin / 5
Numbered Heads / 6
Jigsaw / 7
Think/Pair/Share / 8
Team/Pair/Solo / 9
Three Minute Review / 10
Sample Classroom Activities
Round Robin / 11
Round Robin Activity for Calculus / 12
Numbered Heads Activities / 17
Jigsaw for Review – Solving Systems of Linear Equations / 18
Jigsaw for Learning Pôlya’s Problem Solving Approach / 19
Jigsaw Activity for Finding the Greatest Common Divisor and Least Common
Multiple / 21
Jigsaw Activity for Calculus – Chain Rule / 23
Jigsaw for Learning to Find the Greatest Common Divisor / 36
Think/Pair/Share Activities / 40
Team/Pair/Solo for Discovery Rules of Exponents / 41
Team/Solo Group Quiz on Optimization for Calculus / 49
Three Minute Review Sample Activities / 54
Group Discovery Activities
Cooperative group activity to discover the rules or tests for divisibility / 55
Math for Elementary School Teachers Exploration activity – triangles and angles / 57
Group Exploration Activity to Discover the Rules for Corresponding Angles / 58
Cooperative Learning – Flash Cards / 60
References / 61

Background Notes for

Practicing What We Preach: Demonstrating Effective Cooperative Learning Practices

Annette Leopard and Karen Wells, Monroe Community College

34th Annual AMATYC Conference

November 21, 2008

Washington, D.C.

Cooperative Learning – Our Working Definition

Cooperative learning is a kind of collaborative or peer learning which provides structured tasks for small groups characterized by both interdependence among group members and individual accountability.

Cooperative Learning – Why?

Collaborative/Cooperative Learning is the first strategy listed in the Active Student Learning section of AMATYC’s Beyond Crossroads Chapter 7 on Instruction that calls on faculty to ‘use a variety of teaching strategies that reflect the results of research to enhance student learning.’ (p. 51) It is not our intention here to review the research and literature on cooperative learning in general and college mathematics instruction in particular. The Mathematics Association of America (MAA) through its Project CLUME (Cooperative Learning in Undergraduate Mathematics Education) produced two MAA Notes issues, numbers 44 and 55, which are an excellent place to begin. Instead, we want to list some of the claims about the benefits of this approach supported both in the literature and our experience with using cooperative learning in our mathematics classrooms.

  • Increased academic performance
  • Deeper understanding of course materials
  • Increased retention
  • Increased problem solving ability
  • Increased ability to learn independently
  • Increased student satisfaction
  • More positive attitudes about mathematics
  • Increased self-confidence in the ability to do mathematics
  • Increased communication skills
  • Improved social skills
  • Improved race relations

Characteristics of Cooperative Learning

  1. Interdependence – Group members must be responsible for each other’s learning.
  2. Individual Accountability – Students must be held accountable for their contribution to the group effort; the principal component of a student’s evaluation must be his or her own individual performance.
  3. Structured Learning Activities – Students must be provided with clear directions for procedures and criteria for completion of the group work.

Round Robin Brainstorming

  1. Students are divided into small groups ( no more than 6).
  2. One person is chosen to be the recorder. This person will write down everyone’s responses whether he/she agrees with them.
  3. A question that has multiple answers is given to the students to think about by themselves for a few minutes. Then going around the circle (starting with the person sitting next to the recorder) each person gives a response to the question. This procedure continues until all possibilities are exhausted or until the instructor calls time.
  4. The teacher may also give each group a set of problems to solve simultaneously that involve several steps. The first student writes down the answer to the first step and passes the problem on to the next student who looks over the answer and then solves the second part. This procedure continues until the whole problem is solved.
  5. The group may allow a student to pass on any round but should only allow them to do it once. Otherwise the person may opt to pass on all the rounds, thus not contributing to the group.

Advantages:

  1. Generates many examples and ideas.
  2. Provides an opportunity for students to explore new ideas.

Role of the instructor:

  1. Form groups and establish ground rules.
  2. Provide the question or problem to solve.
  3. At the end of the activity the teacher may collect the group’s work for evaluation or call for groups to share with the class.

Numbered Heads

  1. Students are divided into small groups and within their individual groups number themselves from 1to the number of people in the group. Groups need to be the same size otherwise someone within the group will need to have two numbers.
  2. The teacher gives each group a problem, issue or question to think about solve, or discuss together.
  3. The students are told ahead of time that one number will be called at the end of the group time and that person will be asked to give their group’s response. Therefore everyone in the group must be able to discuss their answer.
  4. The teacher calls out a number and on each team the student whose number is called must give their response either verbally or in writing.

Advantages:

1. Promotes positive effect interpersonal relationships.

2. Enhances motivation towards learning because everyone is held accountable.

Role of the instructor:

  1. Form groups and establish ground rules.
  2. Provide the question or problem to solve.
  3. Choose the number which determines who will give the group’s response.
  4. Evaluate the responses according to some criterion.

Jigsaw

The Jigsaw method can be used as a review or teaching strategy. Students begin by dividing into small groups (3 – 6). Each member of the group is assigned one part of the task to become an “expert” in. To learn or review their assigned task, “expert groups” are formed for each task. Within their “experts groups” the students learn or review the material and develop teaching strategies together to use when they go back to their original group. This part can take anywhere from 15 minutes to a couple of class periods depending on whether you are using it as a review or a teaching strategy.

When the students go back to their original groups, it is their responsibility to make sure everyone in the group masters the material they will be teaching. As a follow up, quizzes can be given to each person to assess how well they learned the different components.

Advantages:

  1. Involves all the students in your classroom.
  2. Builds a sense of positive interdependence because in order for everyone to succeed, they must work together and help each other out.
  3. Everyone is “accountable”.

Role of the instructor:

  1. Prepare instructions for the groups and instructional materials for each expert group.
  2. Circulate around the room, listening to discussions and answering questions only when needed.
  3. Decide when to call students back into their original groups.
  4. Assess learning outcomes.

Think/Pair/Share Activity

This is an activity in three stages:

  1. Think: The instructor poses a question or problem and gives students time to think about how to answer the question or solve the problem individually. The amount of time would vary, and could even extend over more than one class.
  2. Pair: Students pair up and exchange ideas and/or solutions, coming to consensus on an answer or articulating their differences.
  3. Share: The instructor may choose to have one, some or all of the pairs share their results with the entire group. Another option is to have pairs share with pairs.

Advantages:

  1. Involves all the students in your classroom.
  1. Builds a sense of confidence since students don’t have to share their ideas/solutions with the whole group until they have had an opportunity to share with another student.

Role of the instructor:

  1. Pose thought-provoking questions.
  2. Circulate around the room, listening to discussions and answering questions only when needed.
  3. Decide when and how to share results.
  4. Assess learning outcomes.

Team/Pair/Solo Activity

This activity is a strategy for giving students the confidence to tackle difficult problems on their own by first working similar problems with other students. Students are given a problem to solve within a small group of 4 – 6 students. After the small groups solve the problem, two pairs or three pairs (or a triad and a pair if need be) are formed and these new teams solve a problem similar to the problem solved in the larger group. Lastly, individual students are given another similar problem to solve, and this time must solve it on their own.

This activity lends itself well to group quizzes, having both group and individual accountability. Some part of the student’s grade can be based on preparatory homework done outside of class, some part on the group’s problem solution, a part on the pair’s problem solution, and a part on the individual’s problem solution. A bonus could be given to the scores of team members if every member of the team scores at a given criterion level. Alternatively, the group portion of the score could be based part on the group’s solution to the problem and part on the average score of the group members on the pair and individual problem solutions. Research shows that the individual accountability should be a larger portion of the grade than the group grade.

Advantages:

  1. Involves all the students in your classroom.
  2. Builds a sense of positive interdependence because in order for everyone to succeed, they must work together and help each other out.
  3. Gives students the confidence to tackle problems they might not otherwise have attempted as they see their peers successes and learn from them.
  4. Everyone is “accountable”.

Role of the instructor:

  1. Prepare a set of related problems.
  2. Circulate around the room, listening to discussions and answering questions only when needed.
  3. Decide when to move from groups to pairs, from pairs to solo efforts.
  4. Evaluate results in a way that holds everyone accountable both individually and as a group.
  5. Assess learning outcomes.

Three Minute Review Activity

A quick way to ‘test for understanding’ during a lecture or presentation is to pause to ask for questions, or to ask students to review the main points of what you have been saying. Doing this in pairs or small teams provides an opportunity for those who are too shy to ask questions in the larger group a chance to speak. You might have standing groups, form the groups from those sitting near each other, or use it as an opportunity to have students move around by forming groups of students who are not sitting together. Questions which the group cannot answer can be asked by a spokesperson for the group at the end of three minutes. The teacher might call upon one of the groups to give a synopsis of the material and invite others to add to it, or disagree with it.

Advantages:

  1. Involves all the students in your classroom.
  2. Gives the instructor immediate feedback of the students’ level of understanding.
  3. Is an efficient way to be sure that students’ questions are answered.

Role of the instructor:

  1. Circulate around the room, listening to discussions and answering questions only when needed.
  2. Decide when to initiate and when to terminate the activity.
  3. Decide when and how to share results with the entire class.

Sample Round Robin Activities

The opening activity in our presentation which solicited from participants the benefits of using Cooperative Learning methods was an example of round robin brainstorming.

For the following activities, I usually divide my class into 6 groups and give them about 5 minutes. Then for the next 5 minutes, (using the round robin strategy), I go around the class and each group gives me one example from their list to write down on the board.

Reviewing Subsets:

Given : A = { 2, 4, 6, 7, 8,} in your group list all the possible subsets.

Finding Additive Inverses:

1. Recall that a + (-a) = 0.

2. In your group list examples of additive inverses.

Reviewing Problem Solving Techniques:

In class we have discusses various problem-solving strategies. Discuss within your groups the different strategies and give an example of each.

(working backwards, solve a similar problem, elimination, make a table, use a variable, draw a picture, guess and check, find a pattern etc)

Reviewing Statistics:

1. Problem: Your group believes more people enjoy vanilla ice cream than chocolate ice cream. To prove this, your group decides to do an experiment. List the different ways you can collect your data along with the pros and cons of each method.

2. Given the following data: 2, 4, 5, 5, 6, 7, 8, 10 use various descriptive statistics to find the center of this data.

Round Robin Activity for Calculus

An example of the way in which I use Round Robin Brainstorming in my Calculus classes is to break the ice while starting student out on task on the very first day. The activity is important review to connect what students already know about lines and slopes to the material we will soon be covering in Calculus and to emphasize slopes as rates of change. It works as an ice breaker to make students comfortable working with each other in class because the material, going back to their first algebra course, is very familiar to them and all students will be able to contribute.

Forming the Groups:

I do not form permanent groups in my small classes (maximum size 30). As the course goes on, I form groups as I need them ‘randomly’ in various ways. I do not take much time forming groups on the first day. I will usually quickly put them in groups of three, with some groups of four when I need to.

Time Allotment for Round Robin:

I distribute the attached handout, Round Robin Review of Lines. I give students about five minutes for each question. I tell them when the first five minutes are up and ask them to go on to question two if they haven’t already (most will have).

Follow Up:

I ask for a recorder to volunteer to read the group list for question one and write the forms of equations of lines on the board, inviting other to add others and supplying the names of the forms. I then usually explain that in Calculus we will often want to find an equation of a line given its slope and a point and so I circle the point-slope form of the equation of the line.

I ask for another volunteer to read the group responses to question two. I explain that it is important for them to realize that what makes a line straight is a constant rate of change between any two points. I tell them that we will build on what we know about slopes of lines to be study rates of change of functions for which those rates of change are themselves changing.

I then give the students the attached handout, Review of Lines, and go over that material for their class notes. Depending on how much time is left in class, I will have students find an equation of the line graphed, talk about the signs of the slopes of the lines graphed, and find and interpret the slope in the example problem in their groups, or do it with them as a whole class. I usually take the time to talk about increasing, decreasing and constant functions and the fact that the vertical line does not give us y as a function of x while I am going over the signs of the slopes of the lines graphed. I remind students that they are responsible for more material on lines in the review I have assigned from their text.

MTH 210

Calculus I

Leopard

Round Robin Review of Lines

Instructions to the Group:

  1. Introduce yourselves to each other.
  2. The person whose name is first in alphabetical order is the recorder.

Question One: List different forms of an equation of a (straight) line in the x-y coordinate plane.

  1. Beginning with the person to the left of the recorder, give one form of an equation of a line.
  2. The next person to the left gives another form of an equation of a line and so on until the group can think of no more forms or I say that time is up. Then move on to question two.
  3. Recorder: Write down the responses as given below:

Question Two: List everything you know about slopes of lines – different ways to find them, what information they give about the graph of the line, what they tell us about the way that y depends on x, etc.

  1. Beginning with the person to the left of the recorder, give one fact about slopes of lines.
  2. The next person to the left gives another fact and so on until the group can think of no more facts or I say that time is up.
  3. Recorder: Write down the responses as given below:

Review of Lines