© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Supporting Rigorous Mathematics Teaching and Learning Enacting Instructional Tasks: Maintaining the Demands of the Tasks Tennessee Department of Education High School Mathematics

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Using the Assessment to Think About Instruction In order for students to perform well on the Constructed Response Assessments (CRAs), what are the implications for instruction? What kinds of instructional tasks will need to be used in the classroom? What will teaching and learning look like and sound like in the classroom?

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). By analyzing the classroom actions and interactions of five teachers enacting the same high- level task, teachers will begin to identify classroom- based factors that are associated with supporting or inhibiting students’ high-level engagement during instruction.

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Session Goals Participants will: learn about characteristics of the written tasks that impact students’ opportunities to think and reason about mathematics; and learn about the factors of implementation that contribute to the maintenance and decline of thinking and reasoning.

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Overview of Activities Participants will: discuss the differences between two written tasks and their relationship to the CCSSM; discuss how tasks are implemented in classrooms and the impact on students’ opportunities to learn; and make connections to what research says about task implementation.

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Comparing Two Mathematical Tasks

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Comparing Two Mathematical Tasks Compare the two tasks. How are they similar and how are they different? The Hexagon Pattern task The Square Tiles Task

The Hexagon Pattern Task Trains 1, 2, 3, and 4 (shown below) are the first 4 trains in the hexagon pattern. The first train in this pattern consists of one regular hexagon. For each subsequent train, one additional hexagon is added. 1.Compute the perimeter for each of the first four trains.. 2.Make some observations that help you describe the perimeter of larger trains. 3.Determine the perimeter of the 25 th train without constructing it. 4. Write a function that can be used to compute the perimeter of any train in the pattern. Explain how you know it will always work. Extension How can you find the perimeter of a train that consisted of triangles? Squares? Pentagons? Can you write a general description that can be used to find the perimeter of a train of any regular polygons? (Adapted from Visual Mathematics Course I, Lessons published by The Math Learning Center. © 1995 by The Math Learning Center, Salem, Oregon.)

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER The Hexagon Pattern Task Below are some of the possible expressions that can be used to compute the perimeter of any train in the pattern if n represents the train number. Explain how each expression relates to the diagrams. a (n – 2)b. 4n + 2 c (n – 1)d. 6n – 2(n – 1)

The Square Tiles Task Using the side of a square pattern tile as a measure, find the perimeter (i.e., distance around) of each train in the pattern block figure shown below. (Adapted from Visual Mathematics Course I, Lessons published by The Math Learning Center. © 1995 by The Math Learning Center, Salem, Oregon.) train 1train 2 train 3

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Comparing Two Mathematical Tasks (Whole Group Discussion) What are the similarities and differences between the two tasks? The Hexagon Pattern task The Square Tiles Task

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Similarities and Differences Similarities Both require prior knowledge of perimeter Both use geometric shapes Differences Way in which the perimeter is used The need to generalize The amount of thinking and reasoning required The number of ways the problem can be solved

The CCSS for Mathematical Content CCSS Functions Conceptual Category 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. Interpreting Functions F-IF Understand the concept of a function and use function notation. F-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. Common Core State Standards for Mathematics, 2010

The CCSS for Mathematical Content CCSS Functions Conceptual Category Interpreting Functions F- IF Interpret functions that arise in applications in terms of the context. F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Analyze functions using different representations. F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F-IF.C.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Common Core State Standards for Mathematics, 2010

The CCSS for Mathematical Content CCSS Algebra Conceptual Category Seeing Structure in Expressions A-SSE Interpret the structure of expressions. A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context. A-SSE.A.1a Interpret parts of an expression, such as terms, factors, and coefficients. A-SSE.A.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r) n as the product of P and a factor not depending on P. A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 – y 4 as (x 2 ) 2 – (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 – y 2 )(x 2 + y 2 ). Common Core State Standards for Mathematics, 2010

The CCSS for Mathematical Content CCSS Algebra Conceptual Category 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 an expression. Creating Equations A-CED Create equations that describe numbers or relationships. A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Common Core State Standards for Mathematics, 2010

The CCSS for Mathematical Practice 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. Common Core State Standards, 2010

Mathematical Tasks: A Critical Starting Point for Instruction There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics. Lappan & Briars, 1995

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER A Glimpse into Students’ Opportunities to Think and Reason about Mathematics

The Mathematical Tasks Framework TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development, p. 4. New York: Teachers College Press.

The Mathematical Tasks Framework TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development, p. 4. New York: Teachers College Press.

The Mathematical Tasks Framework TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development, p. 4. New York: Teachers College Press.

The Mathematical Tasks Framework TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development, p. 4. New York: Teachers College Press.

The Mathematical Tasks Framework TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development, p. 4. New York: Teachers College Press.

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER The Enactment of the Task (Private Think Time) Read the vignettes. Consider the following question: What are students learning in each classroom?

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER The Enactment of the Task (Small Group Discussion) Discuss the following question and cite evidence from the cases: What are students learning in each classroom?

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER The Enactment of the Task (Whole Group Discussion) What opportunities did students have to think and reason in each of the classes?

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Research Findings: The Fate of Tasks

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Linking to Research/Literature: The QUASAR Project How High-Level Tasks Can Evolve During a Lesson: Maintenance of high-level demands. Decline into procedures without connection to meaning. Decline into unsystematic and nonproductive exploration. Decline into no mathematical activity.

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Decline Problematic aspects of the task become routinized. Understanding shifts to correctness, completeness. Insufficient time to wrestle with the demanding aspects of the task. Classroom management problems. Inappropriate task for a given group of students. Accountability for high-level products or processes not expected.

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Maintenance Scaffolds of student thinking and reasoning provided. A means by which students can monitor their own progress is provided. High-level performance is modeled. A press for justifications, explanations through questioning and feedback. Tasks build on students’ prior knowledge. Frequent conceptual connections are made. Sufficient time to explore.

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Maintenance Scaffolds of student thinking and reasoning provided. A means by which students can monitor their own progress is provided. High-level performance is modeled. A press for justifications, explanations through questioning and feedback. Tasks build on students’ prior knowledge. Frequent conceptual connections are made. Sufficient time to explore. Decline Problematic aspects of the task become routinized. Understanding shifts to correctness, completeness. Insufficient time to wrestle with the demanding aspects of the task. Classroom management problems. Inappropriate task for a given group of students. Accountability for high-level products or processes not expected.

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Linking to Research: The QUASAR Project Low-Level Tasks: – Memorization. – Procedures Without Connections. (The Square Tiles Task) High-Level Tasks: – Doing Mathematics. – Procedures With Connections. (The Hexagon Pattern Task)

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER Cognitive Demands at Set-Up

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER The Fate of Tasks Set Up as Doing Mathematics

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER The Fate of Tasks Set Up as Procedures WITH Connections to Meaning

Linking to Research/Literature: The QUASAR Project Stein & Lane, 1996 A. B. C. High Low HighLow Moderate Low High Task Set-UpTask Implementation Student Learning

Mathematical Tasks and Student Learning Students who performed the best on project-based measures of reasoning and problem-solving were in classrooms in which tasks were more likely to be set up and enacted at high levels of cognitive demand (Stein & Lane, 1996; Stein, Lane, & Silver, 1996). Higher-achieving countries implemented a greater percentage of high-level tasks in ways that maintained the demands of the task (Stiegler & Hiebert, 2004). The success of students was due in part to the high cognitive demand of the curriculum and the teachers’ ability to maintain the level of demand during enactment through questioning (Boaler & Staples, 2008).