Supporting Rigorous Mathematics Teaching and Learning Tennessee Department of Education High School Mathematics Algebra 2 Illuminating Student Thinking: Assessing and Advancing Questions

Rationale Effective teaching requires being able to support students as they work on challenging tasks without taking over the process of thinking for them (NCTM, 2000). Asking questions that assess student understanding of mathematical ideas, strategies or representations provides teachers with insights into what students know and can do. The insights gained from these questions prepare teachers to then ask questions that advance student understanding of mathematical ideas, strategies or connections to representations. By analyzing students’ written responses, teachers will have the opportunity to develop questions that assess and advance students’ current mathematical understanding and to begin to develop an understanding of the characteristics of such questions.

© 2013 UNIVERSITY OF PITTSBURGH Session Goals Participants will: learn to ask assessing and advancing questions based on what is learned about student thinking from student responses to a mathematical task; and develop characteristics of assessing and advancing questions and be able to distinguish the purpose of each type.

© 2013 UNIVERSITY OF PITTSBURGH Overview of Activities Participants will: analyze student work to determine what the students know and what they can do; develop questions to be asked during the Explore Phase of the lesson; – identify characteristics of questions that assess and advance student learning; – consider ways the questions differ; and discuss the benefits of engaging in this process.

© 2013 UNIVERSITY OF PITTSBURGH The Structures and Routines of a Lesson The Explore Phase/Private Work Time Generate Solutions The Explore Phase/Small Group Problem Solving 1.Generate and Compare Solutions 2.Assess and Advance Student Learning MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on: Different solution paths to the same task Different representations Errors Misconceptions 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. Set Up of the Task 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

© 2013 UNIVERSITY OF PITTSBURGH Missing Function Task If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning. xf(x)

© 2013 UNIVERSITY OF PITTSBURGH The Common Core State Standards (CCSS) for Mathematical Content : The Missing Function Task Which of CCSS for Mathematical Content did we address when solving and discussing the task?

The CCSS for Mathematical Content CCSS Conceptual Category – Number and Quantity The Real Number System N-RN Extend the properties of exponents to rational exponents. N-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal 5. N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Common Core State Standards, 2010, p. 60, NGA Center/CCSSO

The CCSS for Mathematical Content CCSS Conceptual Category – Algebra Seeing Structure in Expressions A–SSE Write expressions in equivalent forms to solve problems. A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. ★ A-SSE.B.3c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. A-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. ★ ★ 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. 64, NGA Center/CCSSO

The CCSS for Mathematical Content CCSS Conceptual Category – Algebra Arithmetic with Polynomials and Rational Expressions A–APR Understand the relationship between zeros and factors of polynomials. A-APR.B.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). A-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Common Core State Standards, 2010, p. 64, NGA Center/CCSSO

The CCSS for Mathematical Content CCSS Conceptual Category – Functions Building Functions F–BF Build a function that models a relationship between two quantities. F-BF.A.1 Write a function that describes a relationship between two quantities. ★ F-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. F-BF.A.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. F-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. ★ ★ 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. 70, NGA Center/CCSSO

© 2013 UNIVERSITY OF PITTSBURGH What Does Each Student Know? Now we will focus on three pieces of student work. Individually examine the three pieces of student work A, B, and C for the Missing Function Task in your Participant Handout. What does each student know? Be prepared to share and justify your conclusions.

Response A 13

Response B 14

Response C 15

© 2013 UNIVERSITY OF PITTSBURGH What Does Each Student Know? Why is it important to make evidence-based comments and not to make inferences when identifying what students know and can do?

© 2013 UNIVERSITY OF PITTSBURGH Supporting Students’ Exploration of a Task through Questioning Imagine that you are walking around the room, observing your students as they work on the Missing Function Task. Consider what you would say to the students who produced responses A, B, C, and D in order to assess and advance their thinking about key mathematical ideas, problem-solving strategies, or representations. Specifically, for each response, indicate what questions you would ask: –to determine what the student knows and understands (ASSESSING QUESTIONS) –to move the student towards the target mathematical goals (ADVANCING QUESTIONS).

© 2013 UNIVERSITY OF PITTSBURGH Cannot Get Started Imagine that you are walking around the room, observing your students as they work on the Missing Function Task. Group D has little or nothing on their papers. Write an assessing question and an advancing question for Group D. Be prepared to share and justify your conclusions. Reminder: You cannot TELL Group D how to start. What questions can you ask them?

© 2013 UNIVERSITY OF PITTSBURGH Discussing Assessing Questions Listen as several assessing questions are read aloud. Consider how the assessing questions are similar to or different from each other. Are there any questions that you believe do not belong in this category and why? What are some general characteristics of the assessing questions?

© 2013 UNIVERSITY OF PITTSBURGH Discussing Advancing Questions Listen as several advancing questions are read aloud. Consider how the advancing questions are similar to or different from each other. Are there any questions that you believe do not belong in this category and why? What are some general characteristics of the advancing questions?

© 2013 UNIVERSITY OF PITTSBURGH Looking for Patterns Look across the different assessing and advancing questions written for the different students. Do you notice any patterns? Why are some students’ assessing questions other students’ advancing questions? Do we ask more content-focused questions or questions related to the mathematical practice standards?

© 2013 UNIVERSITY OF PITTSBURGH Characteristics of Questions that Support Students’ Exploration Assessing Questions Based closely on the work the student has produced. Clarify what the student has done and what the student understands about what s/he has done. Provide information to the teacher about what the student understands. Advancing Questions Use what students have produced as a basis for making progress toward the target goal. Move students beyond their current thinking by pressing students to extend what they know to a new situation. Press students to think about something they are not currently thinking about.

© 2013 UNIVERSITY OF PITTSBURGH Reflection Why is it important to ask students both assessing and advancing questions? What message do you send to students if you ask ONLY assessing questions? Look across the set of both assessing and advancing questions. Do we ask more questions related to content or to mathematical practice standards?

© 2013 UNIVERSITY OF PITTSBURGH Reflection Not all tasks are created equal. Assessing and advancing questions can be asked of some tasks but not others. What are the characteristics of tasks in which it is worthwhile to ask assessing and advancing questions?

© 2013 UNIVERSITY OF PITTSBURGH Preparing to Ask Assessing and Advancing Questions How does a teacher prepare to ask assessing and advancing questions?

© 2013 UNIVERSITY OF PITTSBURGH Supporting Student Thinking and Learning In planning a lesson, what do you think can be gained by considering how students are likely to respond to a task and by developing questions in advance that can assess and advance their learning in a way that depends on the solution path they’ve chosen?

© 2013 UNIVERSITY OF PITTSBURGH Reflection What have you learned about assessing and advancing questions that you can use in your classroom tomorrow? Turn and Talk

© 2013 UNIVERSITY OF PITTSBURGH Bridge to Practice Select a task that is cognitively demanding, based on the TAG. (Be prepared to explain to others why the task is a high-level task. Refer to the TAG and specific characteristics of your task when justifying why the task is a Doing Mathematics Task or a Procedures With Connections Task.) Plan a lesson with colleagues. Anticipate student responses, errors, and misconceptions. Write assessing and advancing questions related to the student responses. Keep copies of your planning notes. Teach the lesson. When you are in the Explore Phase of the lesson, tape your questions and the student responses, or ask a colleague to scribe them. Following the lesson, reflect on the kinds of assessing and advancing questions you asked and how they supported students to learn the mathematics.