CPM is a non-profit organization that has developed curriculum and provided its users with professional development support for 18 years. Our textbooks are written mainly by middle school and high school classroom teachers in collaboration with university professors in math and education. The Algebra Connections course was written with the California standards in hand. Based on a review the released items from the 2003-2006 CST Algebra exam, a student who successfully completes this course should be able to correctly answer any of the questions on this test.
The CPM curriculum is based on contemporary research. We have monitored the progress of teachers and students using CPM materials. Our primary goal is the long-term retention of mathematical knowledge . The research supports three fundamental principles: Social interaction increases the ability of students to learn ideas and integrate them into existing cognitive structures. Hence, CPM lessons use study teams. The integration of knowledge is best supported by engaging in a wide array of problems around a single idea. Hence, CPM lessons are problem-based. Long-term retention and transfer of knowledge are best-supported by spaced practice. Hence, CPM spreads practice with ideas over days, weeks, and months.
Studies of Results Since the standards were adopted, CPM schools have outperformed the state average on the STAR tests. CPM high schools scored 6-10% higher than the state average on the SAT-9, 9-11th grade tests from 1998-2002. The CST scores for 2002-2007 have shown similar results. 8th grade scores in CPM schools are 59% higher than the state average and 31.5% higher in 9th grade. Other studies show that students at both ends of the ability of spectrum have also been well served by CPM materials. Students in both low-performing and high-performing schools score higher than matched groups of comparison schools. These reports are available on this CD. The same reports and others are available at the CPM website, www.cpm.org.
Briefly, the components of the California Edition of the Algebra Connections Program are the student text, the teacher binder, an Extra Practice workbook, a Parent Guide, and free homework help through Hotmath.com. CPM has created a California Edition of the Algebra Connections Program. Several problems in various homework sets have been replaced to add practice with some of the content standards. Four additional lessons have been added to extend the development of a few concepts. The Teacher Edition contains most of the resources needed--lesson plans, resource pages, and assessments--to use the program. Extra practice and parent support are offered in booklets and as free downloads at the CPM web site. Hotmath.com provides an “electronic copy” of the textbook along with tutorial solutions to all homework problems.
To highlight the features of this program, we will examine a typical lesson which helps to develop the content for California Content Standard 9.0: Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables to sketch the solution sets.
The lesson we will examine is lesson 6.2.3, which is in the third lesson of the second section of Chapter 6 . It is one of several lessons that develop the skill of solving systems of linear equations. This lesson is a good example of a typical Algebra Connections lesson. A copy f this lesson is on the CD and in the PDF copy of the Teacher Edition. Note that this slide shows part of the extensive discussion of the lesson for the teacher.
The Lesson Objective clearly states the content objectives for the lesson. For example, the lesson objectives for this lesson is “Students will develop the Elimination Method for solving systems of equations.” Then the length of the lesson, core problems, ways of thinking, and materials are listed. Core problems are the minimum work required to meet the objectives of the lesson. Their designation is especially helpful when the pace of the course needs to be adjusted for individual students or the class.
A key part of the teacher materials is the “Suggested Lesson Activity” portion of the teacher notes. These notes are descriptive so that teachers understand how the problems are intended to support students’ development of the skill or concept. These suggestions outline how the lesson should unfold. Some of the common suggestions found in these notes are: • Advice on how to use any manipulatives, models, or technology, • Questions to ask students during classwork, • Discussions of the mathematical content for the teacher. * Possible strategies for implementing the lesson, like that highlighted on the slide.
Finally, there are suggestions for how to bring closure to the lesson. These notes will often recommend questions to use during a whole-class discussion, or will require students to summarize their understanding of a new concept in their Learning Logs (a structured note-taking tool). Following the teacher notes are the elements from the student text. As you can see here, a header states the math content of the lesson (Solving Systems Using Elimination) and offers a broad mathematical question that is answered with this lesson (Can I solve without substituting?). The puzzle-piece icon contains a design that links it to the other lessons of this section, both to give a visual clue and to represent the connections between mathematical content of the lesson. Each lesson contains a brief lesson introduction which motivates the content and makes connections to previously-learned content. This is followed by the problems that develop the content. In this lesson, students are given a challenging system of equations and asked to use the substitution method. However, converting an equation so that it is solved for a variable creates messy fractions which makes solving more complicated. This problem not only requires students to review the previous method they learned, but also motivates them to seek another method to solve complicated systems of equations.
The next problem introduces the main content of this lesson: The Elimination Method. Algebra tiles are used to help students understand why two sides of two different equations can be added together. This builds on work with algebra tiles and Equation Mats from Chapters 2, 3, and 5 previously. Notice that this lesson is not a free-form or unguided exploration, but instead A STRUCTURED, PROBLEM-BASED LEARNING APPROACH, WITH A CLEAR, EXPECTED MATHEMATICAL OUTCOME.
The structured developmental problems provide contexts and questions that require students to be actively involved in learning the content of the lesson. They are essentially a Socratic outline of the lesson. Note that the student text contains questions that teachers can use to guide a whole-class discussion. The teacher text also includes these questions. In addition, each part of the student problems is followed by the answer in bold.
After the core content is developed, the remainder of the lesson regularly provides problems for students to further develop as well as practice the idea. In this lesson, problem 6-58 offers a context which can be solved by the Elimination Method. Problem 6-59 presents a situation where adding the equations does not immediately eliminate a variable. It asks students to consider what else they could do with the equations to eliminate the x-terms. Then problem 6-90 provides additional practice for students to solve in class.
Once the students have been introduced to an idea and worked on the developmental problems toward the expected outcome in the structured lesson, the idea is formalized in a “Math Notes” box. These resources use accepted mathematical language to state definitions, properties, and theorems, as well as provide examples and explanations.
Study teams foster the learning of mathematics. They have additional benefits, but we use the research-based team learning as a major part of how students learn mathematics. Students develop skills and concepts using structured problems with clear objectives. Study teams create vibrant classrooms where students are engaged in doing and talking about mathematics.
Homework assignments are designed to: Offer practice with the day’s topic, Include spaced practice to reinforce and deepen the knowledge of previous topics, Provide extensions and enrichments of some topics, and Occasionally contain a pre-problem that anticipates an upcoming topic.
Students have free access to tutorial help from any computer with internet access. All of the problems and math notes boxes are available at www.hotmath.org, which provides everyone with an electronic textbook. All of the “Review and Preview” problems have step-by-step solutions there. Students may ask for a hint to get started or the first step of a solution. They use this resource as necessary until they can complete the problem themselves or use the solution as a learning aid for the skill or concept in the problem.
The structure of the lessons offers teachers several ways to interact with students. During the lesson, they circulate among the study teams. When they notice that several or most of the students need assistance with a specific skill or idea, they pull the class together for a targeted lecture*. As students work on problems, teacher check for understanding by asking questions, oftentimes those that are provided in the teacher materials. Teachers test student results , requiring students to form logical arguments to support their reasoning. These observations allow teachers to catch and correct student errors , as well as decide what action is necessary for the lesson closure. Teachers can also question the processes used by the students in their solutions, requiring them to reflect on the best strategies to use for different types of problems. Finally, teachers conduct closure activities for the lessons as well as the chapters as a whole. *Targeted lectures means that a teacher addresses the entire class when the observation of student work indicates a need to do so, in contrast to assuming that everyone needs to hear the same thing and giving a general lecture without reference to student need.
The assessment sections of the Teacher Edition provide comprehensive resources for various types of formal assessment. The Assessment Handbook discusses several formats for assessing the progress of students as listed on the slide. There is an assessment plan for each chapter. Each plan suggests where the five strategies can be used during the chapter as well as the content to include on the individual test. The test bank contains some pre-made tests and numerous problems for teachers to use to construct their own tests. In addition, teachers are able to monitor student progress on an informal, daily basis while interacting with the study teams. They gather information about student understanding, their difficulties, and issues that they may need to address with the class as a whole.
In short, the lessons are a blend of student engagement with rich developmental problems that allow thinking and talking about mathematics, targeted lectures based on observed students needs, teacher interaction with students in their study teams on a regular basis, class discussions, and student presentations. The course teaches the required standards. The lesson plans suggest and their structure allows various paths through the lessons so that the objectives are accessible to a broad range of students. There are several forms of intervention available for the teacher. Much of this is addressed in the “Universal Access” tab of the teacher edition and in the lesson notes. One resource for intervention is the Extra Practice booklet, which contains straight-forward, direct instruction of most topics, along with solved examples. It offers an alternative explanation of a topic after students have some experience with it by doing the developmental lesson.
The Extra Practice booklet contains many additional problems on the specific skill for use in class or at home. Teachers may use a section of this resource with the entire class, give it to one or a few students who need additional work with part or all of the topic, or use it for tutorial sessions outside of the regular class period. This can be done with the booklet, a copy of the topic, or online. Answers are provided so that the student receives immediate feedback on whether he or she is correctly solving the problems.
Likewise, the Parent Guide presents the ideas for each chapter in the same direct instruction mode as the Extra Practice booklet so that parents can quickly review topics to help their child. Both of these resources are available in print form or as a free download from the CPM website. We want to stress that our field-testing and subsequent usage show that students who complete the lessons as intended, including homework, have adequate practice and master the material. However, should students need additional practice or an alternative explanation of the topic, teachers can integrate these resources to differentiate instruction for students. If teachers need help with the mathematics of the lesson, the Extra Practice and Parent Guide booklets augment the support found in the lesson plan notes.
CPM writers are careful to write so that students can read the textbook. Our Algebra Connections text has a reading level of 7.4 and a score of 64.0 on the reading ease scale. The latter number indicates “easy reading.” This means that its language is targeted for a student enrolled in the first half of the seventh grade. Anecdotally, teachers in some CPM schools have told us that their English departments have attributed improved reading scores to the use of CPM in the math classes.
The features of the Algebra Connections course are listed on the slide.
Requires students to formulate logical arguments (Standards 24 and 25) Making Connections--page 64 of the Teacher Edition (end of Chapter 1) Justifying--pages 166-67 of the Teacher Edition (end of Chapter 2) Generalizing--pages 255-56 of the Teacher Edition (end of Chapter 3) Reversing thinking -- pages 350-51 of the Teacher Edition (end of Chapter4) Applying and Extending--pages 426-27 of the Teacher Edition (end of Chapter 5)
See the section after the last lesson in each chapter of the Teacher Edition.
Algebra Connections uses an advanced type of algebra tile. In addition to 1, x and x^2 tiles, thee tiles include y, xy, and y^2 so that the representation of algebraic expressions and equations can be extended to more complex polynomials.
The larger activities in the developmental lessons usually offer teachers and students the option to read the problem statement devise a solution strategy, and then solve the problem or to use the next few problems for a structured, guided approach to solving the problem that still requires mathematical thinking and dialogue among the students.
CPM Algebra Overview
Algebra Connections California Edition A Presentation to Teachers for the California Mathematics Adoption College Preparatory Mathematics Rigorous • Aligned • Balanced • Accessible
Background of CPM <ul><li>Supported by research in methodology and studies of student results, available at </li></ul><ul><li>www.cpm.org </li></ul><ul><li>CPM has developed curriculum for 18 years </li></ul><ul><li>Written with the help of California classroom teachers based on their experience of what effectively develops both procedural skills and conceptual understanding. </li></ul>Rigorous • Aligned • Balanced • Accessible
Research Base Rigorous • Aligned • Balanced • Accessible <ul><li>Research Support: </li></ul><ul><li>Social interaction increases the ability of students to learn ideas </li></ul><ul><li>The integration of knowledge is best supported by engaging in a wide array of problems around a single idea. </li></ul><ul><li>Long-term retention and transfer of knowledge are best-supported by spaced learning. </li></ul>
Research Summary Rigorous • Aligned • Balanced • Accessible <ul><li>Examples of California data: </li></ul><ul><li>Students in California CPM schools scored 6 - 10 percentage points above the state average for grades 9-11 on the SAT-9 test for 1998 - 2002. </li></ul><ul><li>8th Grade CST scores for 2002-2007 in CPM schools are 59% above the state average. </li></ul><ul><li>Students in both low-performing and high performing schools score higher than matched groups of comparison schools. </li></ul><ul><li>Reports are posted at www.cpm.org. </li></ul>
Program Components <ul><li>The Algebra Connections Program for this review includes: </li></ul><ul><ul><li>CA Edition Student text </li></ul></ul><ul><ul><li>Teacher binder with assessment </li></ul></ul><ul><ul><li>Extra Practice workbook </li></ul></ul><ul><ul><li>Parent Guide </li></ul></ul><ul><ul><li>Homework help (hotmath.com) </li></ul></ul>Rigorous • Aligned • Balanced • Accessible
Examining a Lesson The Structure of the Teacher Materials We will examine a lesson developing the content for Standard 9.0: Rigorous • Aligned • Balanced • Accessible 9.0: Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables to sketch the solution sets.
Lesson 6.2.3 Rigorous • Aligned • Balanced • Accessible Lesson Objective: Students will develop the Elimination Method for solving systems of equations.
Lesson 6.2.3 Rigorous • Aligned • Balanced • Accessible Suggested Lesson Activity: Description of lesson and strategies for implementation, such as questions for students
Lesson 6.2.3 Rigorous • Aligned • Balanced • Accessible Closure: Every lesson contains closure strategies for teachers. <ul><li>Introduction: </li></ul><ul><li>Motivates learning </li></ul><ul><li>Gets students active </li></ul><ul><li>Challenging </li></ul>
Lesson 6.2.3 Rigorous • Aligned • Balanced • Accessible Conceptual Development: When appropriate, manipulatives (such as algebra tiles) are used to help students move from concrete to abstract understanding. <ul><li>Structured lesson </li></ul><ul><li>Problem-based learning </li></ul>
Lesson 6.2.3 Rigorous • Aligned • Balanced • Accessible Socratic Method: Questions that students answer to develop a new concept <ul><li>Requires students to be actively thinking </li></ul><ul><li>Outlines a process for solving the problem </li></ul>
Lesson 6.2.3 Rigorous • Aligned • Balanced • Accessible Practice: Embedded in classwork Context: Focus on solving word problems.
Lesson 6.2.3 Rigorous • Aligned • Balanced • Accessible Explicit Notes: Following most lessons is a “Math Notes” box containing: <ul><li>Formal definitions </li></ul><ul><li>Solved examples </li></ul><ul><li>Multiple strategies </li></ul><ul><li>Explanation of special cases </li></ul>
Effective Learning Strategy Rigorous • Aligned • Balanced • Accessible Students develop and practice topics in study teams: <ul><li>Research-based </li></ul><ul><li>Structured problems with clear objectives </li></ul><ul><li>Active engagement with the lesson </li></ul><ul><li>Mathematical discourse </li></ul>
Homework Rigorous • Aligned • Balanced • Accessible Review & Preview Icon designates where homework starts. Homework includes: <ul><li>Spaced practice with previous work </li></ul><ul><li>Practice with current work </li></ul><ul><li>Extensions and enrichment </li></ul>
Hotmath Help Rigorous • Aligned • Balanced • Accessible Help with Homework: Students have access to free tutorial help with hints and solutions to all homework questions.
Teacher Intervention Rigorous • Aligned • Balanced • Accessible Extra Practice: Offers multiple practice problems (with answers) for use in class or at home. Answers: Answers are provided so that students can get instant feedback on developing skill.
Parental Support Parent Guide To help parents help their children Rigorous • Aligned • Balanced • Accessible
Reading Level Flesh-Kincaid: Using the Flesh-Kincaid scale of readability, the student Algebra Connections scores are: Rigorous • Aligned • Balanced • Accessible Grade Level Reading Ease 7.4 64.0
Ways of Thinking <ul><li>Justifying </li></ul><ul><li>Generalizing </li></ul><ul><li>Applying and Extending </li></ul><ul><li>Making Connections </li></ul><ul><li>Reversing Thinking </li></ul>Rigorous • Aligned • Balanced • Accessible Algebra Connections focuses attention on five “ways of thinking” mathematically:
Chapter Closure <ul><li>Reflective activities </li></ul><ul><li>Key vocabulary </li></ul><ul><li>Graphic organizers </li></ul><ul><li>Practice problems with answers for self-assessment </li></ul><ul><li>Ways of Thinking </li></ul>Rigorous • Aligned • Balanced • Accessible Each chapter offers extensive closure opportunities, including:
Use of Algebra Tiles <ul><li>“ Legend” reminds students and teachers which tiles are positive and negative </li></ul><ul><li>“ Minus” region negates the tiles in that region, helping students represent the opposite of a negative. </li></ul>Rigorous • Aligned • Balanced • Accessible Symbolic manipulation is developed through use of concrete tools
Further Guidance Rigorous • Aligned • Balanced • Accessible For big activities and key challenging problems, Further Guidance is available for students who need additional assistance.