Teaching a CMP Unit
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Planning - Getting to Know a CMP Unit
Teaching a Unit
Reflecting
Collaborating with Colleagues
A Quick Guide to Planning
Planning - Getting to Know a CMP Unit
The first stage in planning to teach a unit is becoming familiar with the key concepts and the way the unit develops concepts, reasoning, and skills. In general, the unit subtitle gives a broad view of the important ideas that will be developed in the unit. For example, the Moving Straight Ahead unit has the subtitle "Linear Relationships," which identifies linear relationships and functions as the central idea. What the title does not reveal is what aspects of linear relationships are developed and how understanding is enhanced. The following suggestions can serve as a guide for getting to know a unit at this more detailed level.
The Mathematical Ideas in a Unit
In the Teacher's Guide, read the introductory material including the Goals of the Unit, Developing Students' Mathematical Habits, Overview, Mathematics Background and Content Connections to Other Units. These will give you a broad view of the mathematical goals and connections to prior and future units.
Read the Summary of Investigations near the beginning of the Teacher's Guide and the Mathematical Reflections in the student books at the end of each Investigation. These outline the development of the mathematics in the unit.
Look over the Assessment Resources for the Unit. They give you an idea of what students are expected to know at various points in the unit, and the level and type of understanding students are expected to develop.
The Development of the Ideas in a Unit
To help you investigate the details of concept and skill development and guide you as you teach each investigation, read the student unit and all of the problems and ACE questions. Then read the Mathematical and Problem-Solving Goals and the Summary of Problems given at the start of each Investigation. Ask yourself questions such as:
What part of the main mathematical goal of the unit is being developed? How does each problem in the Investigation contribute to the development of the mathematics? What level of sophistication do I expect my students to achieve in answering the problems in the Investigation?
How will student responses show development in understanding the big ideas of the unit?
What mathematical ideas will need emphasis?
What connections can be made among the problems in this Investigation, to other Investigations in this unit, and to other units?
How can I structure the writing assignment for the Mathematical Reflections so students get the most from it?
What ACE questions are appropriate for my students to do after each problem?
How long should this Investigation take?
What can I do to assure the amount of time spent in class is appropriate for the problems and the goals of the Investigation?
Guidance in answering these questions can be found throughout the Teacher's Guide in Pacing Charts, Assignment Guides, and sample answers for ACE exercises and Mathematical Reflections.
For more help on planning a unit see A Quick Guide to Planning.
Teaching a Unit
The role of the teacher in a problem-centered curriculum is different from the curriculum in which the teacher explains ideas clearly and demonstrates procedures so students can quickly and accurately duplicate these procedures. A problem-centered curriculum such as Connected Mathematics is best suited to an inquiry model of instruction. As the teacher and students investigate a series of problems, it is through discussion of methods of solutions, embedded mathematics, and appropriate generalizations that students grow in their ability to become reflective learners. Teachers have a critical role to play in establishing the norms and expectations for discussion in the classroom and for orchestrating discourse on a daily basis. It is through the interactions in the classroom that students learn to recognize acceptable mathematical practices, and those needing explanations or justifications.
The CMP materials are designed in ways that help students and teachers build a pattern of interaction in the classroom, as they become a community of mutually supportive learners working together to make sense of the mathematics. This is done through the problems themselves, the justification students are asked to provide on a regular basis, student opportunities to discuss and write about their ideas, and the help provided to the teacher through the assessment package and the embedded problem-centered instructional model. In addition, the following are useful:
To help teachers think about their teaching, the three-phase instructional model contains a launch of the problem, an exploration of the problem, and a summary of the problem. (See a detailed discussion of the instructional model in Teacher Materials)
The teacher is provided with detailed help- Investigation by Investigation, and Problem by Problem. The Teacher's Guide contains a discussion of the Launch, Explore, and Summarize phases for each Problem. These discussions contain specific help on the focus for each Problem, how to build on previous Problem(s) or Investigation(s), what strategies or misconceptions students might have, and connections to other mathematical concepts. Also included are suggestions for specific questions to ask during each phase of instruction. Before you engage your class in a Problem, you will find it helpful to read the detailed teaching notes for it.
The discussion on Organizing the Classroom (see Classroom Environment) contains helpful suggestions for organizing the classroom and encouraging student participation.
Reflecting
The following questions are all part of teacher reflections on the effectiveness of the classroom environment:
Do the tasks engage the students, and are they effective in helping them learn mathematics?
Do the activities stimulate the richness of discussion that helps students to develop mathematical power?
Does classroom discussion encourage learner independence? Curiosity? Mathematical thinking? Confidence? Disposition to do mathematics?
Does the classroom environment reach every student and support his or her mathematical development?
What do my students know? What is the evidence? How does this shape what I plan for tomorrow?
It is through reflection that teachers continue to grow and to develop the kind of classroom environment that encourages all students to become independent, confident, and reflective learners. The suggestions below are adapted from those submitted by CMP teachers:
Using Feedback From Class
In their Teacher's Guide or in a separate notebook, many teachers write brief notes or comments on important ideas or suggestions for what worked and what to do differently the next time they teach the unit.
Use the classroom discussions, homework, or Mathematical Reflections as benchmarks for your students' understanding.
Re-evaluate where you and your students are each day.
Reflect on each student's understanding. What do you know about this student? Is this student participating in class discussions? Is he or she completing homework?
Finally, at the end of each day, each Investigation, or each unit ask yourself:
What evidence do I have of what my students learned?
How should this affect my instructional decisions?
Collaborating with Colleagues
Many teachers have found it valuable to plan with a colleague before, during, and after teaching the unit. Very often, student work is a focus for their discussions, as it provides a platform for discussing the mathematics in the Unit, Investigation, or Problem. Discussion can also cover effective teaching strategies and other issues related to teaching. The following sets of summary questions can be useful for working either alone or with colleagues. The Teacher's Guides also contain a wealth of information to help you plan your lessons.
A Quick Guide to Planning
Getting to Know the Unit
It is important to understand the mathematics and how it is being developed. Read the Goals of the Unit, Mathematics of the Unit, and Content Connections to Other Units.
Read the Mathematical Reflections in the Student Unit-they tell the story of the mathematics that is being developed in the unit.
Look over the Assessment Resources.
Work all of the Problems and ACE for each Investigation.
Make use of the help provided in the student and teacher books for teaching.
Use the Launch-Explore-Summarize (LES) as a guide for teaching each Problem.
Keep notes on important ideas or suggestions for the next time you teach the unit.
Use the Mathematical Reflections as benchmarks for your students' understanding.
Reevaluate where you and your students are each day-teacher reflections are an important part in becoming a more effective teacher.
Use the following questions as you plan to teach the Unit, each Investigation, or each Problem.
Questions to Think About Before Teaching the Unit
What are the big mathematical ideas of this unit?
What do I want students to know when this unit is finished?
What mathematical vocabulary does this unit bring out?
What might be conceptually difficult?
What are important connections to other units?
Questions to Think About Investigation by Investigation
What part of the mathematical goal is being developed?
How does each Problem in the Investigation contribute to the development?
What level of sophistication do I expect my students to achieve in answering the questions?
Will their responses show the development in their understanding the goals of the unit?
What ideas will need emphasis?
What are the connections among the Problems, Investigations, and with other Units?
How can I structure the writing assignment for the Mathematical Reflections to get the most from them?
What ACE questions are appropriate for my students to do after the 1st problem, the 2nd problem, etc. in this Investigation?
How long will this Investigation take?
What can I do to assure the time spent in class matches the size of the problems and the goals of the Investigation?
Questions to Think About Problem by Problem
Launch
How will I launch this Problem?
What prior knowledge do my students need to call upon?
What do the students need to know to understand the story and the challenge of the Problem?
What advantages or difficulties can I foresee?
How can I keep from giving away too much of the Problem?
How can I make it personal to them?
Explore
How will I organize the students to explore this Problem? (Individual? Pair? Group? Whole class?)
What materials will students need?
What are different strategies I anticipate them using?
What kinds of questions can I ask:
to prompt their thinking if the level of frustration is high?
to make them probe further into the Problem if the initial question is "answered"?
to encourage student-to-student conversation, thinking, learning, etc.?
Summarize
How can I help the students make sense of and appreciate the variety of methods that may occur?
How can I orchestrate the discussion so students summarize their thinking in the Problem?
What mathematics and processes need to be drawn out?
What needs to be emphasized?
What ideas do not need closure at this time?
What do we need to generalize?
How can we go beyond? What new questions might arise?
What will I do to follow-up, practice, or apply the ideas after the summary?
Teacher's Reflections
At the end of each day, Investigation, or Unit, ask yourself:
What evidence do I have of what my students learned?
How does this affect my instructional decisions?
Finally, it is important to remember that "Rome was not built in one day." It takes time and patience to become the teachers we all aspire to be.