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## A Problem-based Curriculum

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**A Problem-based Curriculum**College Preparatory Mathematics Tom Sallee University of California, Davis**Outline**• How do we get more students to learn algebra 1? • Math Goals. Attitude Goals. • Learning Approach. • Examples of how approach was implemented. • How did we approach writing the books? • Lessons learned. • If you want to try this yourself. • Questions Rigorous • Aligned • Balanced • Accessible**CPM Fast Facts**• CPM has developed curriculum for 19 years. • Is a non-profit organization, and has curriculum for grades 6 through 12. • Was started with an Eisenhower grant… not NSF • Written by 6-12 teachers, mostly from California • Was heavily influenced by the 1985 and 1992 California Frameworks and the 1989 NCTM Standards • Has evolved significantly Rigorous • Aligned • Balanced • Accessible**Primary Focus**Getting more students to learn algebra 1, retain their knowledge, and be able to transferit, not just “cover the material”. Originally the first year of a three-year sequence. Rigorous • Aligned • Balanced • Accessible**Central Issue**Difficulties of most students are more about Learning than about Mathematics Rigorous • Aligned • Balanced • Accessible**Math Goals for Students**• Understand the Big Ideas as a connected set of concepts • Be able to move among different representations of the same concept: written, tabular, graphical, symbolic. • Use Problem Solving techniques as both a solution tool and a learning tool Rigorous • Aligned • Balanced • Accessible**Attitude Goals for Students**• I can figure out most problems without being told by the teacher. • I want to learn math. • I want to understand what I learn. Rigorous • Aligned • Balanced • Accessible**Big Ideas**• Representing functions with equations, graphs, tables, and contextual situations, and making connections among these representations. • Writing equations from contexts (word problems) • Solving equations, systems of equations, and inequalities • Symbolic manipulation, using equivalence • Proportionality Rigorous • Aligned • Balanced • Accessible**Learning Approaches**• Math is not a spectator sport Work matters. Engagement matters. • Solving problems is the best way to learn new ideas. • Talking about mathematics with others will help you understand new ideas. • Connecting abstract concepts (like factoring trinomials) withconcrete experiences (like manipulating building rectangles with algebra tiles) helps integrate your knowledge. Rigorous • Aligned • Balanced • Accessible**Learning Approaches**• You will retain ideas better if practice is spaced over weeks or months • It takes a long time to learn a big idea. • There are mathematical ways of thinking (such as generalizing, justifying, connecting) that take time and practice to develop. Rigorous • Aligned • Balanced • Accessible**Multiple Representations**Rigorous • Aligned • Balanced • Accessible**Use of Algebra Tiles**Symbolic manipulation is developed through use of concrete tools • “Legend” reminds students and teachers which tiles are positive and negative • “Minus” region negates the tiles in that region, helping students represent the opposite of a negative. Rigorous • Aligned • Balanced • Accessible**Intro to simultaneous equations**Rigorous • Aligned • Balanced • Accessible**Student tasks for problem**Rigorous • Aligned • Balanced • Accessible**Guidance as necessary**Rigorous • Aligned • Balanced • Accessible**Introduction of a new idea**Rigorous • Aligned • Balanced • Accessible**End of the problem**Rigorous • Aligned • Balanced • Accessible**Setting up equations**A rectangle is 3 cm longer than it is wide and has a perimeter of 54 cm. What are its dimensions? Write an equation that will allow you to solve this problem. Rigorous • Aligned • Balanced • Accessible**Setting up equations**A rectangle is 3 cm longer than it is wide and has a diagonal of 30 cm. What are its dimensions? Write an equation that will allow you to solve this problem. Rigorous • Aligned • Balanced • Accessible**How did we approach writing the book?**• Constrained optimization problem. • Have talked about math goals and attitude goals for students. • What were constraints? Rigorous • Aligned • Balanced • Accessible**Constraints**• Assumptions need to be made about • Students • Teachers • Schools • States • Parents Rigorous • Aligned • Balanced • Accessible**Examples of our Assumptions**Students Most think math is something to be memorized. Many of those we are most anxious to reach will not have a place to do homework. Teachers A course that requires more work will not be kept. Rigorous • Aligned • Balanced • Accessible**Examples of our Assumptions About Constraints**Schools Generally a new program must co-exist with the old. States Standards, frameworks, accountability. Parents (Mostly parents of high-ability students) I need to be able to help my child. There needs to be plenty of practice. Rigorous • Aligned • Balanced • Accessible**Generation 2Algebra Connections**• Major differences • Clearer storylines of the mathematics • More transparent daily structure--eg what is homework? • Made the mathematical goal of problems, lessons, and chapters explicit. • Teacher has the option of presenting problems with less scaffolding, so teamwork is more necessary. • Much more extensive teacher notes Rigorous • Aligned • Balanced • Accessible**Biggest Things We Learned**1. Can’t just write a book with a new approach Need LOTS of professional development to go with it. Current model--eight days (free) inservice for each course plus classroom visits 2. Politics matters. 3. In judging effectiveness, facts matter a lot less than personal prejudices. Rigorous • Aligned • Balanced • Accessible**If you want to do this yourself**• Think very hard about the students for whom you are writing the books and your goals for them. • Trust yourself on the math. • Trust teachers on pedagogy. Don’t ever think you know more about what will work in a classroom than a good teacher. • Go sit in classrooms and find out what the reality is before you begin. • Iterate your efforts. Rigorous • Aligned • Balanced • Accessible**Get Involved in K-12 Math**• There is a need, • There is funding, and • It is the most fun you will ever have. Rigorous • Aligned • Balanced • Accessible