1. Grade 8 Math Teachers Guide

The Connected Mathematics Project (CMP) was funded by the to develop a complete middle school mathematics curriculum, Connected Mathematics, for students and teachers. CMP is a problem-centered curriculum promoting an inquiry-based teaching-learning classroom environment. Mathematical ideas are identified and embedded in a sequenced set of tasks and explored in depth to allow students to develop rich mathematical understandings and meaningful skills. The Common Core State Standards for Mathematics (CCSSM) and the Standards for Mathematical Practice are embedded within each problem. The overarching goal of CMP is to help students and teachers develop mathematical knowledge, understanding, and skill along with an awareness of and appreciation for the rich connections among mathematical strands and between mathematics and other disciplines. The CMP curriculum development has been guided by our single mathematical standard: All students should be able to reason and communicate proficiently in mathematics. They should have knowledge of and skill in the use of the vocabulary, forms of representation, materials, tools, techniques, and intellectual methods of the discipline of mathematics, including the ability to define and solve problems with reason, insight, inventiveness, and technical proficiency. Extensive research and evaluation over 27 years show that CMP students out-perform non-CMP students on reasoning, flexibility, problem-solving, and do as well or better on procedural skills.

Recent studies also show that CMP students are more likely to have positive attitudes toward mathematics than non-CMP students at the end of four years of high school mathematics. More details.

AAAS Conference on Developing Textbooks That Promote Science Literacy February 27-March 2, 2001 American Association for the Advancement of Science Washington, D.C. Developing Coherent, High Quality Curricula: The Case of the Connected Mathematics Project A background paper commissioned for the AAAS Project 2061 Science Textbook Conference Washington, D.C. February 27- March 2, 2001 Elizabeth Difanis Phillips Michigan State University Glenda Lappan Michigan State University Susan N. Friel University of North Carolina James T. Fey University of Maryland Table of Contents. Developing Coherent, High Quality Curricula: The Case of the Connected Mathematics Project (CMP) The Connected Mathematics Project is one of the 13 curriculum development projects that were funded by NSF between 1990 and 1993 to help realize the National Council of Teachers of Mathematics’ Curriculum and Evaluation Standards for School Mathematics in classrooms.

Connected Mathematics, the project’s middle grades curriculum, was awarded an “Exemplary” designation by the Mathematics Panel of the U.S. Department of Education and was the most highly ranked middle grades curriculum material in an evaluation of mathematics textbooks conducted by the American Association for the Advancement of Science. Connected Mathematics is a complete curriculum for teachers and students in grades six, seven, and eight. The curriculum is published in eight modules for each grade level accompanied by a comprehensive teacher’s guide for each module. While the modules might appear to be interchangeable within a grade level, they are carefully sequenced to provide the most powerful mathematics learning experience for students. Within a module, the mathematics is developed through a series of investigations, each focusing on key mathematical concepts, skills, processes, and ways of thinking.

Investigations involve work on two to five mathematically rich situations that engage students in extended problem-solving activity, often in collaboration with their classmates. About the Project In the Connected Mathematics Project (CMP), the curriculum materials for students and teachers reflect a commitment to the philosophy that teaching and learning are not distinct—“what to teach” and “how to teach it” are inextricably linked. We believe that The circumstances in which students learn affect what is learned.

We have worked to produce a curriculum that helps teachers and those who work to support teachers to examine their expectations for students and analyze the extent to which classroom mathematics tasks and teaching practices are aligned with their goals and expectations. (Lappan & Phillips, 1998) Prior Experiences A critical factor in the development of Connected Mathematics was the collective experience of the authors.

Each of us has had extensive prior experiences with research and with innovative approaches to mathematics curriculum and teaching. In the Middle Grades Mathematics Project (MGMP), Glenda Lappan, Elizabeth Phillips, and William Fitzgerald did extensive research on creative new teaching strategies and professional development of teachers. They developed five MGMP curriculum units ( Mouse and Elephant, Spatial Visualization, Factors and Multiples, Probability, and Similarity) that challenge students to become active learners, both intellectually and physically. These units were built on a teaching model that employed mathematical investigations as the primary locus for mathematics learning. The MGMP materials were used in a variety of research studies. One such study explored teacher professional development through expert coaching (Lappan et al., 1988).

Several studies were conducted on student learning, in proportional reasoning (Lappan & Friedlander, 1987) and spatial visualization (Ben-Chaim et al., 1986, 1988, 1989; Lappan et al., 1985). A study of the learning and teaching of growth relationships was conducted by Fitzgerald and Shroyer in 1979.

From 1975 to –1990, Fitzgerald, Lappan, and Phillips spent many hours in the classroom doing teaching experiments around drafts of units, conducting research, and working with teachers and students in a variety of settings. In 1988, Glenda Lappan and Perry Lanier were funded through the National Center for Research on Teacher Learning to conduct a multi-year study of the development of a cadre of 24 elementary teacher education students in both mathematics and the teaching of mathematics. We used the MGMP curriculum as a core set of ideas from which to develop a sequence of university mathematics courses for these students. This research project gave us another opportunity to examine carefully the mathematical development of students and then to follow them into their classrooms as elementary teacher over the first two years of teaching. We learned a great deal about curriculum design and effective teaching that has been of use to us in our current work. In the Used Numbers Project, Susan Friel had done similar curriculum development work focusing on making statistics ideas accessible to upper elementary grades students. She followed that curriculum work with the Teach-Stat Project that focused on the professional development of teachers.

Jim Fey’s work on the Computer Intensive Algebra Project (CIA) explored strategies for teaching algebraic ideas by embedding them in authentic problem-solving activities and by employing calculators and computers as learning and problem-solving tools. Fey is also an author of Core-Plus, a high school curriculum project funded by the National Science Foundation (NSF).

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Each of the research and development projects in which the authors participated prior to CMP was related to curriculum, teaching, and/or student learning. Each led to publications in refereed journals, presentations at meetings, and to the development of curriculum materials that were implemented in many schools. The CMP authors also had a deep understanding of the National Council of Teachers of Mathematics (NCTM) standards.

Lappan was Grades 5–8 Chair of the NCTM Curriculum and Evaluation Standards for School Mathematics and Chair of the NCTM Professional Standards for Teaching Mathematics (1989). Friel was Chair of the Professional Development Writing Group for the NCTM Professional Standards for Teaching Mathematics (1991). Phillips was an author of Patterns and Functions for Middle Grades—the NCTM Addenda to the Standards (1991). Fey wrote a chapter, “Quantity,” for On the Shoulders of Giants (National Research Council, 1990) and was a member of the NRC committee that produced curriculum recommendations for Reconstructing School Mathematics (NRC, 1990). Lappan was also NCTM president during the development of NCTM Principles and Standards 2000. Emerging Philosophy: A Problem-Centered Curriculum Formal mathematics begins with undefined terms, axioms, and definitions and deduces important conclusions logically from those starting points. However, mathematics is produced and used in a much more complex combination of exploration, experience-based intuition, and reflection.

The CMP team began its work with some shared assumptions about mathematics, learning, and teaching. Over time and with experience, many of these assumptions have evolved into a philosophical framework to guide our design of curriculum materials. Among these guiding philosophies are those that relate to: Curriculum materials:. An effective curriculum has coherence—it builds and connects from investigation to investigation, unit to unit, and grade to grade. The “big,” or key, mathematical ideas around which the curriculum will be built should be identified.

Each key idea may be related to a number of smaller concepts, skills, or procedures. These need to be identified, elaborated, exemplified, and connected. Ideas must be explored in sufficient depth to allow students to make sense of them. Superficial treatment of an area produces shallow and short-lived understanding and does not support the making of connections among ideas. Mathematical tasks that students work on inside and outside the classroom are the primary vehicle for student engagement with the mathematical concepts to be learned. Posing mathematical tasks in context (i.e., a problem-centered curriculum) provides support both for making sense of the ideas and for processing them so that they can be recalled. Field testing and evaluation:.

Effective curriculum materials development requires careful field-testing, evaluation, and rewriting over several trials. Research—including formative and summative student evaluation as well as other relevant studies—is essential in determining what seems to be working and where revision is needed.

Research is needed on student and teacher learning from the curriculum to provide evidence for school districts that may consider adopting the materials. Teacher support:. Successful implementation of materials depends on teachers enacting the curriculum in a way that supports the philosophy of learning and teaching underlying the materials’ development. In order for this to be the case, teachers need effective, ongoing professional development that focuses on the materials themselves rather than generic teacher enhancement. Teachers need opportunity and support to collaborate with each other in study and planning for teaching the curriculum. Teacher support materials for the curriculum are needed to provide help with mathematics, assessment, and pedagogy.

Administrative and community support:. Superintendents, principals, and other administrators as well as school boards and parents must have access to clear information about the project materials. The development staff needs a well-developed strategy for providing the mechanisms through which such information is made available and kept updated. The validity and importance of these guiding principles were not completely obvious to us at the outset of our work.

However, we can see from experience over the past decade that each has been significant, particularly those that relate to focusing on key ideas and providing the right context within which students can engage mathematical concepts and skills. Our development team made a commitment to studying a small and select set of important ideas deeply rather than skimming a larger set of ideas in a shallow manner. This means that time is allocated to developing understanding of key ideas in contrast to “covering” the book. The concept of “spiraling” is philosophically appealing; but, too often, not enough time is spent initially with a new concept to build on it at the next stage of the spiral. Without this deeper understanding of concepts and how they are connected, students come to view mathematics as 1,001 different techniques to be memorized. They cannot apply or communicate about mathematical ideas in any way that makes sense.

This commitment to developing key ideas in depth is illustrated in the way that Connected Mathematics treats proportional reasoning, an important topic for middle school mathematics. Conventional treatments of this central topic are often limited to a brief expository presentation of the ideas of ratio and proportion, followed by training in techniques for solving proportions. In contrast, the CMP curriculum materials develop core elements of proportional reasoning in at least four major units:. Stretching and Shrinking introduces proportionality concepts in the context of geometric problems involving similarity. Which Soft Drink Do You Like Better? Bolda Cola or Cola Nola Take the Cola Taste Test Yourself!

To complete the ad, Bolda Cola wants to report the results of a taste test. A copywriter for the advertising department has proposed four possible concluding statements. In a taste test, people who preferred Bolda Cola out-numbered those who preferred Cola Nola by a ratio of 17,139 to 11,426. In a taste test, 5,713 more people preferred Bolda Cola. In a taste test, 60% of the people preferred Bolda Cola.

In a taste test, people who preferred Bolda Cola outnumbered those who preferred Cola Nola by a ratio of 3 to 2. Problem 1.1 A. Which of the proposed statements do you think would be most effective in advertising Bolda Cola? Is it possible that all four advertising claims are based on the same survey data? What other concluding statements could express the same survey data? Based on the survey results used to write the Bolda Cola ad, what data would you expect in a new survey of 1000 cola drinkers? Xx) In the Mathematical Modeling unit, students explore direct and inverse variation by analyzing data from classroom experiments in which they build paper bridges and compare carrying load (in pennies) with length and thickness of the bridges.

They are asked to make some predictions about the relationships involved before experimenting. Then they collect data, display it graphically, and describe the patterns revealed in those displays. Xx) Field trials of the materials in which these investigations were used revealed remarkable instructional power in such context-based teaching. Students with many different prior experiences and interests became engaged in the tasks.

When the same underlying mathematical ideas were encountered in later work, students often connected the new problems to the settings in which those mathematical structures were first studied. However, we also learned that without careful questioning and reflection, it is quite possible for students to leave the classroom with nothing more than a pleasant and lively experience. Furthermore, we found that if a particular context for mathematics is carried on for too long, students lose interest. As a result, we have limited the use of any single problem storyline and introduced a variety of features in student and teacher materials that prompt reflection on problem-solving experiences and encourage students to articulate the underlying mathematical relationships and procedures used to solve the problems. Each investigation includes Application/Connection/Extension exercises that push students to use their new ideas in different but related contexts and to connect new ideas to related familiar ideas.

Each investigation also closes with a set of Mathematical Reflection questions that ask students to articulate (orally and in journal writing) their understanding about the key mathematical ideas of the preceding material. Finally, each unit closes with a series of overall reflection questions and additional problems that test student understanding of the key ideas. When mathematical ideas are embedded in problem-based investigations of rich context, the teacher has a critical responsibility for ensuring that students get the mathematical messages. In a problem-centered classroom, teachers take on new roles—moving from always being the source of knowledge to one of guiding and facilitating the learner in making sense of the mathematics. Teachers become an even more integral part of the learning process. To teach a problem-centered curriculum requires a deep knowledge of mathematics; a broad and coherent view of the subject matter; and an understanding of effective ways to conduct a class based on inquiry.

The teacher support materials must provide these kinds of help for the teacher. Our goal from the beginning of the project was to develop a curriculum from which both teachers and students could learn. Development of Connect ed Mathematics After the release of the National Council of Teachers of Mathematics (NCTM) Standards for School Mathematics in 1989, the National Science Foundation put forth a call for proposals to develop mathematics curricula for grades K-5, 6-8, and 9-12. It was clear from our past experiences, that curriculum materials were an essential part of improving school mathematics.

Even though we had already written five mathematics modules that were widely in use, teachers wanted a more comprehensive curriculum that reflected the NCTM standards. In 1991 CMP submitted a proposal to NSF to write a complete mathematics curriculum for teachers and students in the middle grades, building on and expanding our earlier work. Our original research and development team of Fitzgerald, Lappan, and Phillips now included James Fey, who brought high school experience to the task, and Susan Friel, who provided elementary school experience. Together, we would have the expertise to build on what good elementary curricula were doing, while also preparing students for what was to come in high school. An advisory board consisting of mathematicians, scientists, mathematics educators, teachers, parents, and business representatives was created. They played a critical role in deciding what content to emphasize, in the format of materials, and in planning the research and evaluation conducted throughout the development process. Five professional development centers (PDC) were established, each with a well-known mathematics educator at its head.

Grade 8 Math Teachers Guide

These centers were established in San Diego, CA; Portland, OR; Pittsburgh, PA; Queens, NY; and in a consortium representing Michigan (Portland, Flint, Shepherd, Traverse City, Bloomfield Hills, Sturgis, and Waverly School Districts). The PDCs served as the primary sites for field-testing. Additional sites in Tallahassee, FL; Toledo, OH; Evanston, IL; and Durham, NC were added during the trial phases of the project.

Collectively, the field test sites represented diversity of geographic location, academic ability, ethnicity, and socioeconomic condition. Collaborators A research and evaluation team was formed under the direction of Diana Lambdin at the University of Indiana, along with Judith Zawojewski at Purdue University and Sandra Wilcox at Michigan State University. Their role was also critical to the development of materials. They trained observers at each of the professional development centers; and they collected information on classroom environment, teacher needs and practices, student attitudes and achievement, and management and related school issues. In addition, during the second and third years of the field trials, they conducted a large-scale comparative study of CMP students and students in classrooms using more traditional texts. The Iowa Test of Basic Skills and a problem-solving test developed by the Balanced Assessment Project were used to assess student performance.

This information was critical to the author team as we prepared the final versions of the modules for students and the supporting teacher guides. Classroom teachers also helped to shape Connected Mathematics. A number of collaborating teachers worked with us on all phases of the development and, most important, served as our eyes in the classroom. A full-time middle school teacher headed up the development of the assessment package that was included in each module’s teacher guide. Graduate students in mathematics and mathematics education contributed fresh ideas to the project and learned a great deal from their experiences. As the next generation of curriculum development professionals, these graduate students will have a better understanding of what it takes to develop effective materials. Setting Goals We began with some assumptions about what students would know when they entered grade six.

Then, working with our Advisory Board, our next task was to define the knowledge and skills in each content strand that students should have by the time they leave grade eight. To aid in the process and help us think more broadly about the curriculum, we wrote a set of papers that outlined the exit goals for grade eight in the four mathematical strands that we would develop—number, algebra, probability and statistics, and geometry and measurement. These papers were our touchstone for the development of each strand. Key ideas or clusters of related ideas were identified in each strand and incorporated into units. Initial sequencing of units and appropriate grade level placements were initially identified. Subsequent revisions were based on extensive field tests.

The overarching goal of Connected Mathematics is to help students and teachers develop mathematical knowledge, understanding, and skill along with an awareness and appreciation of the rich connections among mathematical strands and between mathematics and other disciplines. As the CMP materials were developed, the authors synthesized multiple mathematical goals into a single standard that has been a guide for all of our curriculum development: All students should be able to reason and communicate proficiently in mathematics. They should have knowledge of and skill in the use of the vocabulary, forms of representation, materials, tools, techniques, and intellectual methods of the discipline of mathematics, including the ability to define and solve problems with reason, insight, inventiveness, and technical proficiency. (Getting to Know CMP, p. Xx) Another goal for CMP was to align teaching, learning, and assessment with each other as integral parts of the material. To accomplish that broad program goal, development of student and teacher materials was guided by the following five fundamental mathematical and instructional themes:. CMP is organized around a selected number of important mathematical content and process goals, each of which is studied in depth.

CMP emphasizes significant connections among various mathematical topics and between mathematics and problems in other disciplines that are meaningful to students. The instruction in CMP emphasizes inquiry and discovery of mathematical ideas through the investigation of rich problem situations. CMP helps students grow in their ability to reason effectively with information represented in graphic, numeric, symbolic, and verbal forms and to move flexibly among these representations. The goals and teaching of CMP approaches reflect the information-processing capabilities of calculators and computers and the fundamental changes such tools are making in the way people learn mathematics and apply their knowledge of problem-solving tasks. ( Getting To Know CMP, p.

Xx) As materials for students and teachers began to take shape, difficult problems and decisions surfaced at every stage of the process. The major development issues fall into three categories—student materials, teacher support, and project management. Design of Student Materials As we set out to write a complete connected curriculum for grades six, seven, and eight, certain basic questions about the mathematical content and format of student materials quickly surfaced:. What are the overarching mathematical goals in each content strand?.

What size problem is feasible for the teacher and students to explore?. What kind of sequencing or scaffolding is needed to support investigations?. How much help is needed to move from a contextual setting to mathematical understanding that transcends specific situational factors?. What computational skills should be developed and how?. What combination of whole class, small group, and individual work is most effective?. What kinds of practice or homework and reflection are needed to ensure solid understanding of key ideas and automaticity of critical skills?.

What forms of assessment will best inform teachers and students of progress and needed next steps in teaching and learning?. How much help do teachers need in understanding and implementing the content and pedagogy of the curriculum? It took classroom trials and observations, discussions, revisions, more trials, research, and reflections to resolve such issues into a coherent problem-centered curriculum. Although our initial analysis of overarching goals remained sound, our notions of what could be taught in one year changed. To foster students’ deep understanding of the key ideas would require more time. Characteristics of good problems.

To be effective, problems must embody critical concepts and skills and have the potential to engage students in making sense of mathematics. And, since students build understanding by reflecting and communicating, the problems need to encourage them to use these processes. Each problem in Connected Mathematics satisfies some or all of the following criteria:. The problem must have important, useful mathematics embedded in it. Students must be able to approach the problem in multiple ways, using different solution strategies. The problem should allow various solution strategies or lead to alternative decisions that can be taken and defended. The problem should engage students and encourage classroom discourse.

Solution of the problem should require higher-level thinking and problem solving. Investigation of the problem should contribute to students’ conceptual development. The mathematical content of the problem should connect to other important mathematical ideas. Work on the problem should promote skillful use of mathematics and opportunities to practice important skills. The problem should create opportunities for the teacher to assess what students are learning and where they are experiencing difficulty. (Lappan & Phillips, 1998, pp.

87-88) Undoubtedly the most common challenge we faced was deciding how big a problem can be. While rich, open problems encourage students to generate many interesting patterns and conjectures and to solve problems in a variety of ways, it can be difficult to decide which conjectures and problem-solving strategies to follow up on. We struggled to select and sequence problems in a way that (a) would allow both students and teachers a chance to make sense of the mathematics, (b) wouldn’t lead to activity that was irrelevant to the mathematical goals (e.g., mathematics projects that become art projects), and (c) would be manageable for teachers, both mathematically and pedagogically. Our decisions about the grain-size of a problem were generally based on classroom experience. We considered whether the time required to develop an idea fully was time well spent; whether students could grasp the mathematical subtlety of the ideas; and whether students were reaching useful closure on important concepts, strategies, or skills. For example, as we formulated plans for a unit on elementary data analysis and informal inference, we experimented with a student investigation comparing the length of first and last names.

In its most open form, students could be given the simple task of collecting and organizing data to decide whether there seemed to be any significant patterns in such data. We found it useful to allow some initial exploration of this open question (to get a reading of students’ prior knowledge and experiences). However, this sort of open task rarely leads students to utilize useful standard techniques for data display (line plots, box plots, or scatter plots) or standard concepts for data summary (median, mean, and measures of variation). Pilot teachers almost always recommend that such materials need to provide “more structure” for students.

Another challenge was to find engaging contextual problems that would embody the mathematical concepts we wanted to target. Ideally, the context helps make concepts meaningful to students and enables them to retrieve that meaning more readily later on. We often found that when students were eventually presented with mathematical ideas symbolically—without a contextualizing storyline-they were able to go back to the original context in which they had learned those ideas to help them make sense of the symbols. It is important to note, however, that students’ ability to generalize knowledge in this way does not occur without being attended to specifically. (Lappan & Phillips 1998, p.

Organization of the units. Each CMP unit is centered on an important mathematical idea or cluster of related ideas. A unit consists of four to seven investigations, each containing two to five problems for the class to explore. Each investigation ends with a set of problems called Applications, Connections, and Extensions (ACE). The ACE problems provide homework practice with applications of key ideas.

We found a relatively effective mix of skills and applications through our extensive pilots and field-testing (each unit went through a three-year development and testing cycle). Finally, the Mathematical Reflections at the end of each investigation and the assessment package provide opportunities for teachers and students to assess their understanding throughout the unit (Lappan & Phillips, 1998, p.