A Problem-based Curriculum Tom Sallee University of California, Davis

Outline Rigorous Aligned Balanced Accessible 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

CPM Fast Facts Rigorous Aligned Balanced Accessible 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

Primary Focus Rigorous Aligned Balanced Accessible Getting more students to learn algebra 1, retain their knowledge, and be able to transfer it, not just “cover the material”. Originally the first year of a three-year sequence.

Central Issue Rigorous Aligned Balanced Accessible Difficulties of most students are more about Learning than about Mathematics

Math Goals for Students Rigorous Aligned Balanced Accessible 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

Attitude Goals for Students Rigorous Aligned Balanced Accessible I can figure out most problems without being told by the teacher. I want to learn math. I want to understand what I learn.

Big Ideas Rigorous Aligned Balanced Accessible 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

Learning Approaches Rigorous Aligned Balanced Accessible 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) with concrete experiences (like manipulating building rectangles with algebra tiles) helps integrate your knowledge.

Learning Approaches Rigorous Aligned Balanced Accessible 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.

Multiple Representations Rigorous Aligned Balanced Accessible

Use of Algebra Tiles “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 Symbolic manipulation is developed through use of concrete tools

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 Rigorous Aligned Balanced Accessible 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. Guess sideOther sidePerimeter=60? low high high xx+32x+2(x+3)=60

Setting up equations Rigorous Aligned Balanced Accessible 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. Guess side Other side Diagonal=30? 1013low 1518 low 2023high xx+3=30

How did we approach writing the book? Rigorous Aligned Balanced Accessible Constrained optimization problem. Have talked about math goals and attitude goals for students. What were constraints?

Constraints Rigorous Aligned Balanced Accessible Assumptions need to be made about Students Teachers Schools States Parents

Examples of our Assumptions Rigorous Aligned Balanced Accessible 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.

Examples of our Assumptions About Constraints Rigorous Aligned Balanced Accessible 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.

Generation 2 Algebra Connections Rigorous Aligned Balanced Accessible 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

Biggest Things We Learned Rigorous Aligned Balanced Accessible 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.

If you want to do this yourself Rigorous Aligned Balanced Accessible 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.

Get Involved in K-12 Math Rigorous Aligned Balanced Accessible There is a need, There is funding, and It is the most fun you will ever have.