Teaching & Learning Trajectories: Building Coherence, Connections, and Retention Across Grades Session 3 May 10, 2012 Oakland Schools Gerri Devine Dana Gosen Valerie Mills University of Michigan Edward Silver Adele Sobania

Focusing on Reasoning and Proving Year 3 1.What is reasoning and proving? 2.How do students benefit from engaging in reasoning and proving while studying algebra? 3.How can teachers support the development of students’ capacity to reason-and-prove across grades ? DELTA Connecting Mathematical Ideas and Practices

Why Focus on Connections & Coherence? “When students can connect mathematical ideas, their understanding is deeper and more lasting.” (NCTM, 2000 p.64) “When students understand the interrelatedness of mathematics, they often have many more strategies available to them when solving problems and insights into mathematical relationships.” (Hiebert, 1997) “Through instruction that emphasizes the interrelatedness of mathematical ideas, students not only learn mathematics they also learn about the utility of mathematics.” (NCTM, 2000, p.64) “Toward greater focus and coherence...” (CCSS, p. 5)

Working to build coherence & connections with these tools Anchor Tasks with Corresponding Analysis Protocols Teaching & Learning Trajectories (TLT)

Working to build coherence & connections at two levels and with these tools…. and others Mathematical Tasks Framework (MTF) Teacher Analysis of Student Knowledge (TASK) Mathematical Representation Star Stein, Grover & Henningsen (1996) Smith & Stein (1998) Stein, Smith, Henningsen & Silver (2000) Tasks as set-up by teachers Tasks as they appear in curricular materials Tasks as enacted by teacher and students Student learning

Session 3 Agenda 8:00 – 8:15 Welcome and Introductions 8:15 – 9:45 Common Core Projects Introduction Streams 9:45 – 10:00 Break 10:00 – 11:15 Common Core Projects Streams (cont.) Illustrative Mathematics 11:15 – 11:45 Debrief of the Morning 11:45 – 12:30 Lunch 12:30 – 1:15 DELTA Data Collection 1:15 -2:15 Kitten Task Analysis 15 min. 2:15 – 2:30 Reflections, Homework, and Closing

Ellen Whitesides Director of Common Core Projects, Institute for Mathematics & Education (IM&E), University of Arizona Teaching Fellow at John F. Kennedy School of Government, Harvard University

Lunch We will reconvene at 12:30.

DELTA Data Collection Please take about 45 minutes to complete the toothpick task and answer the corresponding pedagogical questions.

Kitten Task Work out whether this number of descendants is realistic. Here are some facts that you will need:

Review the student work and write comments attending to the following features. What has a particular student done correctly? What assumptions has he or she made while working to solve the task? What does the student need to improve/learn? What might you do to support this student? Using Student Work to Guide Instruction: Classroom Level

The student has selected the important facts and used them to solve the problem. The student is aware of the assumptions he or she has made and the effect these assumptions have on the result. The student has used more than one method The student appears to have checked whether his or her results make sense and have improved upon a method if need be. The student has presented his or her results in a way that will make sense to others. Using Student Work to Guide Instruction: Classroom Level

Working to build coherence & connections at two levels Work at the classroom level: −Task selection −Assessment for learning −Identifying misconceptions −Connecting to prior knowledge and future lessons (mathematical language, tasks, strategies) Work at the systems level: –Course curriculum design within and across grades/courses –Course offerings (tracking, acceleration, integrated content) –District Assessment Systems –Collegial conversations within and across grades/courses (District Dialogues)

In small groups, analyze student work using the following sorting schema: Sort 1: Argument is attentive to nearly all parameters Sort 2: Mathematically productive solution strategy is attempted based on analysis of situations/cases Sort 3: Reasoning used to support the conclusion is logical Using Student Work to Guide Instruction: Systems Level

Please be specific as you summarize your across grade level findings: What do your students seem to understand? What do they have yet to understand? What implications do these findings have at the systems level?

Student Work Analysis: Reflection Compare this sorting activity to prior data analysis you have done at the classroom and systems levels. In what ways has this process prompted you to consider how you attend to students’understanding and use of mathematics?

Driving Questions for Teachers How do we “cover” so much content in each secondary grade/course with deep understanding? What does it mean to teach algebra in 7th grade, 8th grade, Algebra I and Algebra II? What portion should I be teaching? Are there algebraic connections to lessons in earlier grades that I could be drawing on to help teach my content? Are there ways to support my students’ understanding and use of the Standards for Mathematical Practice?

Shifting Focus from Content to Practices Content-Focused Teaching and Learning Trajectories – Years 1 and 2 Rate of Change Solving Linear Solving Quadratic Data Analysis Equivalence Practice-Focused Teaching and Learning Trajectories – Year 3 Reasoning-and-Proving

District Dialogues

1.Individually, take a few minutes to reflect on your team’s learning throughout the project. 2.As a team, discuss what you have accomplished and in what ways you intend to work together as you continue to grow professionally. Please keep a record of your group’s ideas to reference in the future. 3.Be prepared to share a couple of key points from your conversation. Team Reflection

End-of-Day Reflection 1.Are there any aspects of your own thinking and/or practice that our work throughout the project has caused you to consider or reconsider? Explain. 2.Are there any aspects of your students’ mathematical learning that our work throughout the project has caused you to consider or reconsider? Explain.