Julie Fredericks Teachers Development Group

1. How Can (and Why Should) PD Drive Mathematical Justification and Generalization as Classroom Norms? And How Do We Know When It Does? Julie Fredericks Teachers Development Group

2. Session Overview • Definition and Purpose • What is a mathematical justification and what purposes does mathematical justification serve in the K-12 classroom? • Challenges & Strategies • What are the challenges that the teachers we work with face regarding mathematical justification? • Specific examples of tools and strategies we use in our work • Our Wonderings and Future Plans

3. Mathematical Discourse An evidence-based premise of the studio work is the notion that orchestrating productive mathematical discourse increases students’ opportunities to learn and, in turn, raises achievement and participation levels in mathematics. Embracing this premise requires developing teachers’ knowledge, skills, tools, and disposition for building classroom communities of mathematical discourse. Leahy, Lyon, Thompson, and Wiliam, 2005; Yackel & Cobb, 1996; Hufferd-Ackles & Sherin, 2004; Stein, Engle, Hughes & Smith, 2008; Weaver & Dick, 2006

4. Driven by how students learn mathematics Student mathematics achievement will improve if teachers consistently use research-based instructional practices to develop both computational fluency and a deep understanding of mathematics concepts by engaging all students consistently and effectively in the following mathematical practices: • Providing Explanations • Making Justifications • Formulating Conjectures & Generalizations. • Using Multiple Representations • Engaging in Metacognition • Making Connections Bransfordet al, 1999; Cohen, 1994; Donovan & Bransford, 2005; Franke et al, 2007; Kilpatrick, 2001; Lotan, 2003, 2006; Stein et al, 2000; Common Core State Standards Initiative, 2010

5. Let’s Do Some Math Structuring the discussion. • Start with 5 minutes private think time. • Find a partner at your table and switch papers. Review your partners work WITHOUT talking. • First partner shares their interpretation of other partners work. Other partner clarifies. • Second partner shares their interpretation of first partners work. First partner clarifies. • Both discuss compare and contrast their solutions.

6. Let’s Do Some Math Prove that the sum of two odd numbers is even. After you have solved the problem, think about what is mathematical justification? At which points in your work have you provided mathematical justification.

7. Let’s Do Some Math Structuring the discussion. • Start with 5 minutes private think time. • Find a partner at your table and switch papers. Review your partners work WITHOUT talking. • First partner shares their interpretation of other partners work. Other partner clarifies. • Second partner shares their interpretation of first partners work. First partner clarifies. • Both discuss compare and contrast their solutions.

8. Purpose • What are some of the possible purposes that justification could serve in the K-12 mathematics classroom?

9. Common Core Standards “One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from.” -Common Core State Standards for Mathematics

10. Common Core StandardsStandards for Mathematical Practice • Make sense of problems and persevere in solving them. • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics. • Use appropriate tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning.

11. Challenges & Strategies • Teachers recognizing mathematical justification. • Students recognizing mathematical justification. • Choosing when to press for mathematical justification.

12. Challenges & Strategies Teachers recognizing mathematical justification • Teachers often struggle with the differences between mathematical justification, explanation, and verification.

13. Challenges & Strategies Providing opportunities for teachers to discuss their wonderings and questions about mathematical justification. • “[I am] still trying to understand what justify means in the context of 8th grade algebra vs. a general problem solving task vs. high school geometry. Not always sure when students are verifying vs. justifying.” • “First grade has been working on word problems. We want our students to be able to solve a word problem and show how they go their answer instead of just the answer itself, is this justification?”

15. Challenges & Strategies Student Mathematical Discourse Observation Tool • Watch a video clip of a 3rd grader. • As we watch, we are going to code the student discourse. Particularly with an eye for justification, conjectures and generalizations.

16. Allison Video from Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School (Carpenter, Franke, and Levi, 2003)

17. Use Student Discourse Observation Protocol • 5 minutes PTT – Use the provided transcript of the video to code the student mathematical discourse from this episode. • At your table, share one place that you identified a justification, conjecture or generalization.

20. Challenges & Strategies • Students recognizing mathematical justification. • When first pressing for justification in the classroom, students have trouble understanding what you want.

21. A Looks Like/ Sounds Like Chart for Mathematical Justification from a classroom

22. Challenges & Strategies Structured Student Math Talk • Using the structure to press students to share and compare their thinking . • Today we used the Interpret and Compare routine which is outlined on the second page of the tool.

23. Challenges & Strategies • Knowing when to press for justification • Deciding which questions are appropriate for grade level • “I am finding that my students can explain how they solved a math problem, but I am stumbling when it comes to explaining why (justification). Right now we have a standing question in our class: Can you generalize a rule about how diameter and circumference are related and justify why your rule would work with any circle.” • Pressing for justification on every question regardless of mathematical goal of lesson.

24. Challenges & Strategies • Spend explicit time during lesson planning for mathematical justification. • For which questions would pressing for justification be most important? Why? • What does are the acceptable justifications for this task?

25. Planning Questions

27. Challenges & Strategies • “It seems as though this is starting to become one of the norms in our math conversations. I can think of at least 4-5 different occasions in collaborative meetings, with different teachers, that someone said "what would you want kids to do/show for their justification?" This seems to be becoming more of a norm. It is a simple, yet complex, question and really forces the teacher to think ahead of time about what students will need to know and what will be "acceptable justification." Now, if we can all calibrate our own definitions of "acceptable justification" we will really move forward as a school. This is where I think it is necessary for us to continue to work together, refine our ideas in order to come to these shared understandings. This cannot be done in isolation. “

28. Our Wonderings and Future Plans • Continuum of justification • How can we evaluate teachers growth? • How can teachers evaluate their students growth?

29. Thank You!!