1. This module was developed by Amy Hillen, Kennesaw State University. Video courtesy Pittsburgh Public Schools and the University of Pittsburgh Institute for Learning. These materials are part of the Principles to Actions Professional Learning Toolkit: Teaching and Learning created by the project team that includes: Margaret Smith (chair), Victoria Bill (co-chair), Melissa Boston, Fredrick Dillon, Amy Hillen, DeAnn Huinker, Stephen Miller, Lynn Raith, and Michael Steele. Principles to Actions Effective Mathematics Teaching Practices The Case of Elizabeth Brovey and the Two Storage Tanks Task Grade 8 Principles to Actions Effective Mathematics Teaching Practices The Case of Elizabeth Brovey and the Two Storage Tanks Task Grade 8

2. Overview of the Session Solve and discuss the Two Storage Tanks Task Watch a video clip of a teacher facilitating small groupwork using this task Discuss ways the teacher supports her students’ learning of mathematics Connect specific teacher actions seen in the video clip to the Effective Mathematics Teaching Practices

3. The Two Storage Tanks Task What are all the ways the task can be solved? Which of these methods do you think your students will use? What misconceptions might students have? What errors might students make? (Smith, Bill, & Hughes, 2008, p. 134)

4. The Two Storage Tanks Task Adapted from NAEP Released Items, 2003-8M10 #13. http://nces.ed.gov/NationsReportCard/nqt

5. Ms. Brovey’s Learning Goals

6. Connections to the CCSSM Standards for Mathematical Practice (SMP) 1.Make sense of problems and persevere in solving them 2.Reason abstractly and quantitatively 3.Construct viable arguments and critique the reasoning of others 4.Model with mathematics 5.Use appropriate tools strategically 6.Attend to precision 7.Look for and make use of structure 8.Look for and express regularity in repeated reasoning National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards For Mathematics. Washington, DC: Authors.

7. Connections to the CCSSM Standards for Mathematical Practice (SMP) 1.Make sense of problems and persevere in solving them 2.Reason abstractly and quantitatively 3.Construct viable arguments and critique the reasoning of others 4.Model with mathematics 5.Use appropriate tools strategically 6.Attend to precision 7.Look for and make use of structure 8.Look for and express regularity in repeated reasoning National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards For Mathematics. Washington, DC: Authors.

8. SMP 6: Attend to Precision Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards For Mathematics. Washington, DC: Authors.

9. Ms. Brovey’s Mathematics Learning Goals Students will understand that: The solution(s) to a system of two linear equations in two variables is represented graphically by the point(s) of intersection of the lines, and is represented by the ordered pair(s) (x, y) that make both equations true statements or satisfy the equations simultaneously Linear relationships… – have a constant rate of change between the quantities – are depicted graphically by a line – can be written symbolically as y = mx + b, where m is the constant rate of change and the slope of the line, and b is the value of the y-quantity when x = 0 (i.e., the y-intercept)

10. Connections to the CCSSM Grade 8 Standards for Mathematical Content Expressions and Equations (EE) Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.C.8 A: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. B: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. C: Solve real-world and mathematical problems leading to two linear equations in two variables. Functions (F) Use functions to model relationships between quantities. 8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards For Mathematics. Washington, DC: Authors.

11. Context of the Video Clip School: Pittsburgh Classical Academy, Pittsburgh, PA District: Pittsburgh Public Schools Principal: Valerie Merlo Teacher: Ms. Elizabeth Brovey, Math Coach Class: 8 th Grade Pre-Algebra Curriculum:Connected Mathematics Project 2 Size:27 students At the time the video was filmed, Elizabeth Brovey was a coach at Classical Academy in the Pittsburgh Public School District. The students are mainstream eighth grade Pre-Algebra students. The lesson occurred in April. (Elizabeth Brovey is currently a coach at Propel Andrew Street High School, a charter school in Pittsburgh, PA.)

12. Lens for Watching the Video Clip: Viewing #1 As you watch the video clip, make note of what Ms. Brovey does to support her students’ learning.

13. Effective Mathematics Teaching Practices 1.Establish mathematics goals to focus learning. 2.Implement tasks that promote reasoning and problem solving. 3.Use and connect mathematical representations. 4.Facilitate meaningful mathematical discourse. 5.Pose purposeful questions. 6.Build procedural fluency from conceptual understanding. 7.Support productive struggle in learning mathematics. 8.Elicit and use evidence of student thinking. (NCTM, 2014)

14. Effective Mathematics Teaching Practices 1.Establish mathematics goals to focus learning. 2.Implement tasks that promote reasoning and problem solving. 3.Use and connect mathematical representations. 4.Facilitate meaningful mathematical discourse. 5.Pose purposeful questions. 6.Build procedural fluency from conceptual understanding. 7.Support productive struggle in learning mathematics. 8.Elicit and use evidence of student thinking. (NCTM, 2014)

15. Different representations should: Be introduced, discussed, and connected; Focus students’ attention on the structure or essential features of mathematical ideas; and Support students’ ability to justify and explain their reasoning. Strengthening the ability to move between and among these representations improves the growth of children’s concepts. (Lesh, Post, & Behr, 1987) Use and Connect Mathematical Representations

16. Students should be able to approach a problem from several points of view and be encouraged to switch among representations until they are able to understand the situation and proceed along a path that will lead them to a solution. This implies that students view representations as tools that they can use to help them solve problems, rather than as an end in themselves. (NCTM, 2014, p. 26)

17. Connecting Mathematical Representations (NCTM, 2014, p. 25)

18. Connecting Mathematical Representations (Van de Walle, 2007, p. 280) Language TableEquationGraphContext

19. Pose Purposeful Questions Effective questions should: Reveal students’ current understandings; Encourage students to explain, elaborate, or clarify their thinking; and Make the mathematics more visible and accessible for student examination and discussion. Teachers’ questions are crucial in helping students make connections and learn important mathematics concepts. Teachers need to know how students typically think about particular concepts, how to determine what a particular student or group of students thinks about those ideas, and how to help students deepen their understanding. (Weiss & Pasley, 2004)

20. Evidence should: Provide a window into students’ thinking; Help the teacher determine the extent to which students are reaching the math learning goals; and Be used to make instructional decisions during the lesson and to prepare for subsequent lessons. Formative assessment is an essentially interactive process, in which the teacher can find out whether what has been taught has been learned, and if not, to do something about it. Day- to-day formative assessment is one of the most powerful ways of improving learning in the mathematics classroom. (Wiliam, 2007, pp. 1054; 1091) Elicit and Use Evidence of Student Thinking

21. The gathering of evidence should neither be left to change nor occur sporadically…Waiting until the quiz on Friday or the unit test to find out whether students are making adequate progress is too late. Rather it is important to identify and address potential learning gaps and misconceptions when it matters most to students, which is during instruction, before errors or faulty reasoning becomes consolidated and more difficult to remediate. (NCTM, 2014, p. 53)

22. Lens for Watching the Video Clip: Viewing #2 As you re-watch the video clip, pay particular attention to: The ways in which Ms. Brovey supported (or could have supported) students’ use of and connections between representations The ways in which Ms. Brovey elicited students’ thinking and used (or could she have used) this evidence to inform her instruction Be prepared to give examples and to cite line numbers from the transcript to support your observations.

23. Reflecting on the Effective Mathematics Teaching Practices As you reflect on the Effective Mathematics Teaching Practices examined in this session, identify a concept that you want your students to understand deeply. – What representations could be used to model this concept? How could these representations be connected so as to improve students’ understanding? What questions could you ask to help students make these connections? – What questions could you ask to elicit students’ thinking? In what ways might students respond to these questions? How will their responses influence your subsequent instruction?

25. References National Center for Educational Statistics. (2003). NAEP Released Item 2003-8M10 #13 (“Solve problem involving two linear relationships”). http://nces.ed.gov/NationsReportCard/nqt http://nces.ed.gov/NationsReportCard/nqt National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors. Smith, M. S., Bill, V., & Hughes, E. K. (2008). Thinking through a lesson: Successfully implementing high-level tasks. Mathematics Teaching in the Middle School, 14(3), 132-138. Van de Walle, J. A. (2007). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson Education, Inc.