Connected Mathematics provides opportunities for students to tackle mathematical problems individually, in pairs, in small groups, and as a whole class. Each of these arrangements of students enhances learning. The way you group your students will depend on the size, nature, and difficulty of the task. For a particular Problem, students might do individual work as part of the Launch phase or the launch may be a whole-class discussion. In the Explore phase, students may work individually, in pairs, or in small groups. They may take part in a whole-class discussion of a problem during the Summarize phase. The rationale for each of these grouping decisions is the nature of the problem and the goals of the lesson. The Teacher's Guide for each unit offers specific suggestions for grouping.
The Launch of a CMP lesson is typically done as a whole class, yet during this launch phase of instruction, students are sometimes asked to think about a question individually before discussing their ideas as a whole class. However, it is during the Summarize phase, when individuals and groups share their results, that substantive whole-class discussion most often occurs. Led by the teacher's questions, the students investigate ideas and strategies and discuss their thoughts. Whole-class discussion allows a variety of ideas to be presented and the mathematical validity of solutions to be tested. Questioning by other students and the teacher challenges students' ideas, allowing important concepts to be developed more fully. Working together, the students synthesize information, look for generalities, and extract the strategies and skills involved in solving the Problem. Since the goal of the summarize phase is to make the mathematics in the problem more explicit, teachers often pose, toward the end of the summary, a quick problem or two to be done individually as a check on how the students are progressing. Moving flexibly between whole-class and individual work keeps the whole class focused, but allows each student to test his or her understanding of the ideas being discussed.
The teacher's notes often suggest that students spend some time working on a question individually before working with their partner or group. Asking students to first think about and try a question on their own gives them time to sort out their own ideas and assess what makes sense to them and what causes them difficulty.
For an occasional question it is suggested that students work entirely on their own. Such questions may be less demanding than questions for which group work is suggested, or they may provide an opportunity for teachers to assess each student's understanding or skill at an important stage in the development of key mathematical ideas in the unit.
The ACE exercises at the end of an Investigation are intended to be solved individually, outside of class. These exercises give students a chance to practice and make sense of ideas developed in class. These exercises are narrower in scope and demand than are the Problems in the Investigations.
Pairs and Small-Group Work
Working collaboratively allows students to tackle more complicated and more conceptually difficult problems. Carefully managed, collaborative learning can be a powerful tool for teachers to use during classroom instruction. Connected Mathematics suggests two types of collaborative-learning groupings: partner work and small-group work.
Many of the problems in Connected Mathematics are mathematically demanding, requiring students to gather data, consider ideas, look for patterns, make conjectures, and use problem-solving strategies to reach a solution. For this reason, the Teacher's Guide often suggests that students work on the exploration of a problem collaboratively. Group work supports the generation of a variety of ideas and strategies to be discussed and considered, and it enhances the perseverance of students in tackling more complicated multi-step and multipart problems.
It is appropriate to ask students to think about a problem individually before moving into groups, allowing them to formulate their own ideas and questions to bring to the group. These multiple perspectives often lead to interesting and diverse strategies for solving a problem.
Group work is also suggested for some of the Unit Projects. These projects tend to be large, complicated tasks. Working in a group allows students to consider a variety of ideas and helps them complete the task in a reasonable amount of time.
You will want to determine group configurations in an efficient manner so class time is not wasted. You may find it easiest to decide before class how students will be grouped. There are various methods you can use to establish groups, such as assigning students to a group for a whole unit of study or randomly drawing for group assignments on a more frequent basis. You might also want to arrange the seating in the room to minimize movement during the transition from individual to group to whole-class settings.
It is important that you clearly communicate your expectations about group work to your students and then hold them to those expectations. You may want to hand out or post a set of guidelines so students understand their responsibilities. Below is a suggested set of guidelines.
Student Guidelines for Group Work
Move into your groups quickly and get right to work.
Read the instructions aloud or recap what the teacher has challenged you to find out. Be sure every group member knows what the challenge is.
Part of group work is learning to listen to each other. Don't interrupt your classmates. Make sure each person's ideas are heard and that the group answers each person's questions.
If you are confused, ask your group to explain. If no one in the group can answer the question, and it is an important question, raise your hand for the teacher.
If someone in your group uses a word or an idea you do not understand, ask for an explanation. You are responsible for learning all you can from your group. You are also responsible for contributing to the work of your group. Your attempts to explain to others will help you to understand even better.
Give everyone in the group a chance to talk about his or her ideas. Talking out loud about your thinking will help you learn to express your arguments and clarify your ideas.
If your group gets stuck, go over what the problem is asking and what you know so far. If this does not give you a new idea, raise your hand for the teacher.
Be prepared to share your group's ideas, solutions, and strategies and to explain why you think you are correct. Make sure you look back at the original problem and check that your solutions make sense.
You are responsible for recording your group's ideas and solutions in your notes.
Suggestions for Encouraging Participation
When students work in groups, there is always a possibility that some students will dominate, while others will not participate. Making sure the size of the group is appropriate for the size of the task can help ensure that all students play a role. You can also facilitate participation by requiring that each group member be given the opportunity to share his or her thoughts and ideas before the group discussion begins. It is also helpful to give students some time to think about or work on the Problem individually before discussing it with their groups.
In the Summarize phase of instruction, groups share their findings with the class. It is important that all students have an opportunity to participate in this phase. To make sure all group members are prepared, you can randomly choose the presenter from each group or employ questioning techniques that involve all group members. Teachers have found these strategies to be useful:
Have students assign numbers to each student in their group. Then, have them roll a number cube or draw to determine who will present the group's findings.
Write each student's name on a craft stick, store the sticks in a cup at the front of the room, and choose one stick at random to determine who will present.
Have the students choose the presenter for their group, but ask each of the other students a question related to the work.
The classroom conversation that occurs during the Summarize phase provides an important opportunity to push students' mathematical thinking. By examining and testing ideas, students can learn mathematical skills and strategies and make connections and generalizations. You might use the following suggestions to increase interaction and participation.
Encourage students to respond to another group's or student's presentation, conjectures, strategies, or questions.
Have students summarize the essence of a group's or student's presentation.
After a group or student presents, have others in the class ask questions to challenge the group's or student's thinking.
Ask a student to create and post an incorrect solution to stimulate the thinking of the class and generate a conversation.
If you have a student who struggles, find opportunities for him or her to present when you know he or she has a correct answer.
If there is repetition among strategies, have students discuss the similarities or contribute new thoughts, rather than just repeat ideas.
Encourage students to look for common ideas in their strategies and representations.