1. Numerical Reasoning: An Inquiry-Based Course for K-8 Teachers Rachel Cochran, Center for Educational Accountability Jason Fulmore, Center for Educational Accountability John Mayer, University of Alabama at Birmingham Bernadette Mullins, Birmingham-Southern College The Greater Birmingham Mathematics Partnership is funded by NSF awards EHR-0632522 and DUE-0928665

2. GBMP Activities Summer Courses Mathematics Support Teams (Teacher-Leaders) Professional Learning Communities Administrator Sessions Community Mathematics Nights Revised Courses for Pre-service Teachers

3. Summer Courses Over 1700 total enrollment  Patterns: The Foundations of Algebraic Reasoning  Patterns II  Numerical Reasoning  Geometry and Proportional Reasoning  Probability and Data Analysis  Extending Algebraic Reasoning  Extending Algebraic Reasoning II

4. Summer Courses  Challenging nine-day mathematics content courses  Inquiry-based  Menu-driven  Multiple representations  Expandable tasks  Whole class processing  Group work

5. Challenging Courses and Curricula  Big mathematics ideas  Productive disposition  Inquiry and reflection  Communication

6. Numerical Reasoning Pre-Test Item Draw and explain what you would do to model this problem with students. Write a story problem that goes with this problem.

7. Sample Numerical Reasoning Content  Daily Number Talks  Multiple Models for Fractions and Equivalent Fractions  Cuisenaire® Rods  Pattern Blocks  Color Tiles  Finger Fractions  Geometric Problems  Contextual Problems  Making Sense of Operations on Fractions  Reasoning Through Ordering Fractions

8. Sample Numerical Reasoning Tasks The Marriage Problem In a certain town, 3/5 of the women are married to 2/3 of the men. What fraction of the adults in the town are married.

9. Performance Assessment  MEC-developed assessment pre and post  Scored with 5-point Oregon Department of Education Rubric  Two raters; high inter-rater reliability  Median score increased from 2 to 4 on rubric  A Wilcoxon signed ranked test showed statistically significant improvement

10. Participant Surveys  “This course improved my mathematical skills and understanding.” 86% strongly agree; 12% agree  “The Summer course has totally changed the way I feel about myself as a user of mathematics, and therefore, my ability to help my students develop a strong understanding of mathematical concepts.”  “I have looked closely at my questioning techniques as a result of this class. Although I have been teaching for almost 30 years, this was the first model of great questions—set in a class setting so that I could see reactions and results.”

11. Behavioral Checklist  CEA-developed checklist based on CCC dimensions Day 1Day 4Day 8 Mathematical Ideas uses variables to describe unknowns7%27%93% explains why equations make sense geometrically7%27%73% represents linear, quadratic functions in variety of ways0%13%53% Productive Disposition persists when answer is not known0%33%87% asks for guidance but not answers13%27%80% tries variety of strategies for approaching problems13%73%93%

12. Behavioral Checklist Day 1Day 4Day 8 Inquiry and Reflection makes extensions and connections beyond problem0%13%67% explores why it works, whether it will always work0%7%53% confusion and mistakes lead to further exploration20%73%100% Communication explains reasoning fluently0%13%80% asks probing questions20%33%93% shares ideas with class27%47%93%

13. Portfolios  Participant-selected pieces, instructor-selected pieces, reflective writing  Scored with CEA-developed rubric (based on CCC)  Three raters; consensus-reaching Median Score Incomplete Score = 1 Emerging Score = 2 Proficient Score = 3 Expert Score = 4 Problem Translation301127 Mathematical Procedures301136 Productive Disposition301118 Inquiry and Reflection302117 Justification and Communication 302117

14. Classroom Observations RTOP Subscale (maximum of 20)CoursesMedian Lesson Design/Implementation0 1 2 3+ 5 12 13.75 13 Propositional Knowledge0 1 2 3+ 6.5 12 14 14.5  Reformed Teaching Observation Protocol (RTOP)  Two raters; consensus-reaching Sample (N = 116); 0 courses (N=17); 1 course (N=35); 2 courses (N=38); 3+ courses (N=26)

15. Classroom Observations RTOP Subscale (maximum of 20)CoursesMedian Procedural Knowledge0 1 2 3+ 6.5 11 14 12.5 Communicative Interaction0 1 2 3+ 4 10.5 13 Student/Teacher Relationships0 1 2 3+ 6.5 13.5 15 14.5

16. Student Achievement 2007-2008 Implementation Level2007 MeanStd Dev2008 MeanStd DevN High56.623.661.522.1666 Moderate56.721.556.220.71652 Low56.520.755.119.63640 Total56.621.356.120.35958

17. Student Achievement 2007-2008 Implementation Level2007 MeanStd Dev2008 MeanStd DevN High57.121.160.021.03305 Moderate55.120.855.120.96215 Low57.820.856.420.914506 Total57.020.956.521.024026

18. Student Achievement w/o High SES Implementation Level2007 MeanStd Dev2008 MeanStd DevN High54.420.457.120.22886 Moderate54.520.654.520.66070 Low56.620.455.220.413811 Total55.820.555.320.422767

19. Student Achievement 2008-2009 Implementation Level2008 MeanStd Dev2009 MeanStd DevN High59.520.761.621.33620 Moderate54.620.254.820.47217 Low57.920.457.220.78537 Non-Participating57.320.456.920.75498 Total57.120.557.120.82487 2

20. Numerical Reasoning: An Inquiry-Based Course for K-8 Teachers Rachel Cochran, Center for Educational Accountability Jason Fulmore, Center for Educational Accountability John Mayer, University of Alabama at Birmingham Bernadette Mullins, Birmingham-Southern College The Greater Birmingham Mathematics Partnership is funded by NSF awards EHR-0632522 and DUE-0928665