Supporting Rigorous Mathematics Teaching and Learning

Supporting Rigorous Mathematics Teaching and Learning Engaging In and Analyzing Teaching and Learning Tennessee Department of Education High School Mathematics Algebra 1

Rationale Common Core State Standards for Mathematics, 2010 Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? 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….…Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness. By engaging in a task, teachers will have the opportunity to consider the potential of the task and engagement in the task for helping learners develop the facility for expressing a relationship between quantities in different representational forms, and for making connections between those forms.

Session Goals Participants will: • develop a shared understanding of teaching and learning; and • deepen content and pedagogical knowledge of mathematics as it relates to the Common Core State Standards (CCSS) for Mathematics.

Overview of Activities Participants will: • engage in a lesson; and • reflect on learning in relationship to the CCSS.

Looking Over the Standards • Look over the focus cluster standards. • Briefly Turn and Talk with a partner about the meaning of the standards. • We will return to the standards at the end of the lesson and consider: • What focus cluster standards were addressed in the lesson? • What gets “counted” as learning?

Bike and Truck Task Distance from start of road (in feet) Time (in seconds) A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle’s distance and K(t) represent the truck’s distance.

Bike and Truck Task • Label the graphs appropriately with B(t) and K(t). Explain how you made your decision. • Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description. • Which vehicle was first to reach 300 feet from the start of the road? How can you use the domain and/or range to determine which vehicle was the first to reach 300 feet? Explain your reasoning in words. • Jack claims that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel. Explain why you agree or disagree with Jack.

The Structures and Routines of a Lesson MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on: • Different solution paths to the • same task • Different representations • Errors • Misconceptions Set Up of the Task The Explore Phase/Private Work Time Generate Solutions The Explore Phase/Small Group Problem Solving Generate and Compare Solutions Assess and Advance Student Learning SHARE: Students explain their methods, repeat others’ ideas, put ideas into their own words, add on to ideas and ask for clarification. REPEAT THE CYCLE FOR EACH SOLUTION PATH COMPARE: Students discuss similarities and difference between solution paths. FOCUS: Discuss the meaning of mathematical ideas in each representation REFLECT:by engaging students in a quick write or a discussion of the process. Share, Discuss, and Analyze Phase of the Lesson 1. Share and Model 2. Compare Solutions 3. Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write

Solve the Task(Private Think Time and Small Group Time) • Work privately on theBike and Truck Task. • Work with a partner and then others at your table. • Consider the information that can be determined about the two vehicles.

Expectations for Group Discussion • Solution paths will be shared. • Listen with the goals of: • putting the ideas into your own words; • adding on to the ideas of others; • making connections between solution paths; and • asking questions about the ideas shared. • The goal is to understand the mathematics and to make connections among the various solution paths.

Bike and Truck Task Distance from start of road (in feet) Time (in seconds) A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle’s distance and K(t) represent the truck’s distance.

Bike and Truck Task • Label the graphs appropriately with B(t) and K(t). Explain how you made your decision. • Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description. • Which vehicle was first to reach 300 feet from the start of the road? How can you use the domain and/or range to determine which vehicle was the first to reach 300 feet? Explain your reasoning in words. • Jack claims that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel. Explain why you agree or disagree with Jack and why.

Discuss the Task(Whole Group Discussion) How did you describe the movement of the truck, as opposed to that of the bike? What information from the graph did you use to make those decisions? In what ways did you use the information you determined about the two vehicles to determine which vehicle was first to reach 300 feet from the start of the road? When, if ever, is the average rate of change the same for the two vehicles?

Reflecting on Our Learning What supported your learning? Which of the supports listed will EL students benefit from during instruction? Which CCSS for Mathematical Content did we discuss? Which CCSS for Mathematical Practice did you use when solving the task?

Pictures Manipulative Models Written Symbols Real-world Situations Oral Language Linking to Research/LiteratureConnections between Representations Adapted from Lesh, Post, & Behr, 1987

Language Context Table Graph Equation Five Different Representations of a Function Van De Walle, 2004, p. 440

The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra *Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010, p. 65, NGA Center/CCSSO

The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 65, NGA Center/CCSSO

The CCSS for Mathematical ContentCCSS Conceptual Category – Functions Common Core State Standards, 2010, p. 69, NGA Center/CCSSO

What standards for mathematical practice made it possible for us to learn? Common Core State Standards for Mathematics, 2010 • 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.

Research Connection: Findings by Tharp and Gallimore Tharp & Gallimore, 1991 For teaching to have occurred - Teachers must “be aware of the students’ ever-changing relationships to the subject matter.” They [teachers] can assist because, while the learning process is alive and unfolding, they see and feel the student's progression through the zone, as well as the stumbles and errors that call for support. For the development of thinking skills—the [students’] ability to form, express, and exchange ideas in speech and writing—the critical form of assisting learners is dialogue -- the questioning and sharing of ideas and knowledge that happen in conversation.

Underlying Mathematical Ideas Related to the Lesson (Essential Understandings) • The language of change and rate of change (increasing, decreasing, constant, relative maximum or minimum) can be used to describe how two quantities vary together over a range of possible values. • A rate of change describes how one variable quantity changes with respect to another – in other words, a rate of change describes the covariation between two variables (NCTM, EU 2b). • The average rate of change is the change in the dependent variable over a specified interval in the domain. Linear functions are the only family of functions for which the average rate of change is the same on every interval in the domain.

Essential Understandings Cooney, T.J., Beckmann, S., & Lloyd, G.M., & Wilson, P.S. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12 (p. 8-10). Reston, VA: National Council of Teachers of Mathematics.

Supporting Rigorous Mathematics Teaching and Learning