1. Promoting Rigorous Outcomes in Mathematics and Science Education PROM/SE Ohio Spring Mathematics Associate Institute April 27, 2005

2. PROM/SE Goal: To expand the capacity of educators to improve student achievement in mathematics through working with teacher associates. Deeper knowledge of mathematics for teaching Enhanced leadership skills Integration of data from multiple sources

3. Today’s Focus: Mathematics for teaching --Curriculum: Content Trajectories --Analysis of Instructional Materials --Curriculum Decisions Leadership --Models for working with others

4. Mathematics for Teaching: Knowing and Understanding Curriculum Intended Curriculum –Benchmarks/Standards Implemented Curriculum – Instructional materials and teachers’ practice Achieved Curriculum – Student performance: test results and in-class work

5. Agenda In Grade-Level Groups: (elementary, middle and high) Content Trajectory of Identified Mathematical topic Analysis of Topics Content Trajectory Within a Set of Instructional Materials Mapping Grade Level Content Indicators to Instructional Materials or Building New Tasks from Existing Ones As a Whole Group Reflecting on the Process and Next Steps

6. Institute Goals for Associates: Gain knowledge and strategies for analyzing the development of a mathematical concept across grade levels in a set of instructional materials and in the state standards; Identify, for a given mathematical topic, at a grade level and across grade levels, the key mathematical knowledge and skills students should have; Examine gaps and overlaps in the development of the mathematical concept and consider whether these are appropriate.

7. Content trajectories are the development of a specific mathematical topic within a unit and grade and across the grades

8. Characteristics of Coherent Content Trajectories Every component of the trajectory has a mathematical reason for being included. Concept sequences are designed with an awareness of students’ understandings and misunderstandings. The sequence is developed with a clear sense of whether a mathematical idea is treated as being introduced, taught for understanding, or is assumed knowledge.

9. Characteristics of Coherent Content Trajectories Mathematical ideas build on one another so that students’ understanding and knowledge deepens and their ability to apply mathematics expands. The mathematical sequence and connections are defensible. The ideas become more sophisticated as students continue along the trajectory. (Schmidt et al, 2002; NCTM, 2000)

10. An Example of a Content Trajectory: Inequalities Ideas for knowing and understanding inequalities build: From number To variables To functions

11. ‘Big Ideas’ in Inequalities (with number) Greater than, less than Compare and order amounts Definition of inequality Location and relation of two amounts on number line Understanding symbolic notation

12. Inequalities (with variables) Mathematical expressions Relation between symbolic notation and graphical representation Compound inequalities Solving inequalities Absolute value inequalities

13. Inequalities (with functions) Inequalities in the coordinate plane Relationships between functions (greater than and less than)

14. Viewing Curriculum Through Topic Development (Trajectory) Limitations: Linking concepts to other mathematics in a more global way (e.g., making connections between fractions and decimals, or between slope and tangent). Advantages: Allows you to focus on a reasonable amount of curriculum and dig deeply into its components.

15. Deepening Knowledge for Teaching Mathematics through Working on Content Trajectories Topics: ones identified as problematic by the PROM/SE achievement data: –Elementary: Fractions –Middle: Measurement: perimeter, area, volume –High: Quadratics

16. Moving to Groups Elementary 3 sections Leaders: Liz Jones and George, Liz Taylor and Tab Middle 1 section Leader: Mary High 1 section Leader: Karen