1. Developing Mathematics PD Sessions: Planning Conversations and Instructional Decisions that Lead to Improved MKT in District Leaders. National Council of Supervisors of Mathematics Indianapolis, Indiana April 11, 2011 Melissa Hedges, MathematicsTeaching Specialist, MTSD Beth Schefelker, Mathematics Teaching Specialist, MPS Connie Laughlin, Mathematics Instructor, UW-Milwaukee The Milwaukee Mathematics Partnership (MMP), an initiative of the Milwaukee Partnership Academy (MPA), is supported with funding by the National Science Foundation.

2. Session Goals We Are Learning To…build Math Knowledge of Teaching (MKT) about proportional reasoning. We will be successful when we can look at student work that will strengthen teacher content knowledge about proportional reasoning.

3. Questions we ask ourselves What is the critical content idea that leaders needed to understand? How much content can be explored in a 90- minute professional development session? What learning experiences can be used to launch, explore, and summarize the content idea? What connections can be made to current work?

4. A decision-making process Identifying the math to be discussed Connecting to standards Completing the task Looking at student work to push thinking

5. Process of development Identify the need Listening to math leaders Looking at data Reviewing student work with teachers Identify a Domain as outlined in CCSS Decide on the cluster of standards that connects to the need Narrow the focus to specific standard(s).

6. Why focus on proportional reasoning? Proportional reasoning has been referred to as the capstone of the elementary curriculum and the cornerstone of algebra and beyond. Van de Walle,J. (2009). Elementary and middle school teaching developmentally. Boston, MA: Pearson Education.

7. Connect to the CCSS Narratives from grade 6 and grade 7 Cluster statements: Understand ratio concepts and use ratio reasoning to solve problems (6.RP.3) Analyze proportional relationships and use them to solve real world and mathematical problems. (7.RP.2)

8. Never ending question…. What are teachers expected to know and do to make sure students develop proportional reasoning?

9. A Definition of Proportionality When two quantities are related proportionally, the ratio of one quantity to the other is invariant, or the numerical values of both quantities change by the same factor. Developing Essential Understandings of Ratios, Proportions & Proportional Reasoning, Grades 6-8. National Council of Teachers of Mathematics, 2010, pg. 11.

10. Essential Understanding A proportion is a relationship of equality between two ratios. 3 girls to 4 boys is the same ratio as 6 girls to 8 boys

11. Essential Understanding A rate is a set of infinitely many equivalent ratios.

12. Cassandra’s Faucet Cassandra has a leaky faucet in her bathtub. She put a bucket underneath the faucet in the morning and collected data throughout the day to see how much water was in the bucket. Use the data Cassandra collected to determine how fast the faucet was leaking. TimeAmount of Water 7:00 a.m.2 ounces 8:30 a.m.14 ounces 9:30 a.m.22 ounces 11:00 a.m.34 ounces 2:00 p.m.58 ounces 5:30 p.m.86 ounces 9:30 p.m.118 ounces

13. Engaging in a task Complete the task Cassandra’s Faucet Share out your thinking with the person next to you. In what way did you use proportional reasoning in your thinking? Is your reasoning the same as your partner’s reasoning?

14. What should student work look like? Turn and talk: In order to know if students understand proportions, what traits would you want to see demonstrated on their work? Share out ideas with the whole group.

15. Cassandra’s Faucet Student Work How did student’s make sense of the table? What were the various entry points? What conclusions can you make about how students are thinking as they engaged in purposeful struggle to understand rate?

16. Making decisions about using student work Which papers would drive conversations around the big math ideas of proportional reasoning? Why?

17. Student work A

18. Student work B

19. Student work C

20. Student work D

21. Student Work E

22. Purposefully selecting student work Student A – shows the start of proportional reasoning by using additive thinking Student B – multiple equivalent rates and checking more than one time interval Student C – multiple representations to prove an answer Student D – a right answer, but explanation needs clarity Student E – proportional thinking but with some assumptions

23. Think About… Proportional reasoning may at first seem straightforward, but developing an understanding of it is a complex process for students. How do we support our teachers’ understanding of proportional reasoning?

24. Deliberate decision-making process Identifying the math to be discussed Connecting to standards Completing the task Looking at student work to push thinking

25. Why pay attention to MKT? “Knowing mathematics for teaching often entails making sense of methods and solutions different from one’s own and so learning to size up other methods, determine their adequacy and compare them is an essential mathematical skill for teaching…” -D. Ball

26. Thank you for coming! Melissa Hedges, MathematicsTeaching Specialist, Mequon-Thiensville School District mhedges@mtsd.k12.wi.us Beth Schefelker Mathematics Teaching Specialist, Milwaukee Public Schools schefeba@milwaukee.k12.wi.us Connie Laughlin Mathematics Consultant, Milwaukee WI laughlin.connie@gmail.com

27. Thank You for coming! Find the power point for this session on two websites: NCSM conference website Milwaukee Mathematics Partnership website: mmp.uwm.edu