Building Teachers Capacity to Create and Enact Tasks that Engage Students in Challenging Mathematical Activity Peg Smith University of Pittsburgh Teachers’ Development Group Leadership Seminar on Mathematics Professional Development February 16, 2007

Overview  Solve and discuss a series of multiplication tasks  Consider the role of representation in developing students’ understanding of mathematics and increasing the cognitive demands of a task  Read and discuss the Case of Monique Butler  Discuss the factors that support or inhibit students engagement in high level thinking and reasoning  Discuss the knowledge needed for teaching that teachers might learn from these experiences

Task1: Multiplying Binomials Directions: Find the product of each expression in its simplest form: 1. 2x (x – 1) 2. (x + 1) (x + 2) 3. (x - 2) (3x + 3) 4. (x – 3) (x + 3) 5. (2x + 2) (2x – 2) 6. (x + 3) (x + 3)

Making Sense of Multiplication Using any of the tools available at your table: Model 3 x 4 Model 3 x n

Making Sense of Multiplication Using any of the tools available at your table: Model 6 x 13 Model 6(n + 3)

Making Sense of Multiplication Using any of the tools available at your table: Model 26 x 12 Model (2x + 6)(x + 2)

Multiplying Binomials: Task 2 1.(x + 3) (x + 3) 3. (x - 3) (x + 3) 2. 2x (x - 1) 4. (x - 2) (3x + 3) Directions: Use algebra tiles to show the product of these binomials. Make a sketch of your model. (Note that these four problems also appeared in Task 1.)

Comparing Task 1 & Task 2 How are Tasks 1 and 2 the same and how are they different?

Comparing Task 1 & Task 2 In what ways did we “ make connections to meaning ” in our work on Task 2?

Ways of Representing Mathematical Ideas Pictures Written Symbols Manipulative Models Real-World Situations Oral Language Van De Walle, 2004, p. 30

Linking Multiple Representations The learner who can, for a particular mathematical problem, move fluidly among different mathematical representations has access to a perspective on the mathematics in the problem that is greater than the perspective any one representation can provide. Driscoll, 1999

Linking Multiple Representations Research has shown that children who have difficultly translating a concept from one representation to another are the same children who have difficulty solving problems and understanding computations. Strengthening the ability to move between and among these representations improves the growth of children ’ s concepts. Lesh, Post, Behr, 1987

Linking to Research/Literature If we want students to develop the capacity to think, reason, and problem solve then we need to start with high- level, cognitively complex tasks. Stein and Lane, 1996

Consider… What if the textbook teachers are using doesn’t have many high-level, cognitively complex tasks? How can you do help teacher learn to modify tasks so that they will provide opportunities for students to develop the capacity to think, reason, and problem solve?

General Techniques for Modifying Tasks  Ask students to create real world stories for “naked number” problems  Use an additional representation and make connections between two (or more) representations  Solve an “algebrafied” version of the task  Use a task “out of sequence” before students have memorized a rule or have practiced a procedure that can be routinely applied  Eliminate components of the task that provide too much scaffolding

General Techniques for Modifying Tasks  Ask students to create real world stories for “naked number” problems  Use an additional representation and make connections between two (or more) representations  Solve an “algebrafied” version of the task  Use a task “out of sequence” before students have memorized a rule or have practiced a procedure that can be routinely applied  Eliminate components of the task that provide too much scaffolding

Task A Larson, R., Boswell, L., Kanold, T., & Stiff, L. (2001). CA Algebra 1: Concepts and Skills, McDougal Littell, 213.

Task A Adaptations For the following equations: 1.Graph on a coordinate plane. 2.Identify 3 solutions for each equation. 3.Are your solutions unique? Explain your reasoning. a. y= 3x - 5 b. 5x + 2y = 10 c.-5x -3y = 12 3x - 5y = 15 (i)Find 3 solutions to the equation (ii)Plot the points and describe all the patterns you see. (Group extension: plot all points on group graph) (iii)How many solutions do you believe exist for this function? Explain your reasoning.

Task B Smith, S., Charles, R., Dossey, J., & Bittinger, M. (2001). Algebra 1, California Edition, Prentice Hall, 321.

Task B Adaptations Find the slope of the lines containing these points using a graph. 11. (4, 0) (5, 7) 13. (0,8) (-3,10) 17, (0,0) (-3,-9) Can you write a rule that will always work for finding the slope? (Or explain why the rule you already know WORKS?) Original Problem + How can you use the slope to find other points on the same line? Use a graph to explain why the rule you used to find the slope works.

Final Reflections on Task Adaptation  The one lesson I found particularly enlightening was the modification of tasks. I used this myself in the PTR assignment, and see the benefits of doing so more in the future. The goal of maintaining the original intent of the problem, was an important consideration that I will remember.  …I learned more about how to create tasks with a certain goal in mind and how to improve my questioning…I also felt more inspired to design more tasks.  I learned how to adapt an activity to raise the cognitive demand to challenge my learning support students and still foster achievement. I feel it is possible to do more open-ended tasks even though they struggle with computation and procedures to build their mathematical knowledge.

The Case of Monique Butler

Analyzing the Case of Monique Butler Discuss with colleagues at your table: What did Monique Butler want her students to learn? What did Monique Butler’s students learn? Cite evidence from the case (i.e., paragraph numbers) to support your claims.

Analyzing the Case of Monique Butler What prevented Monique Butler ’ s students from learning what she wanted them to learn? (Cite evidence from the case to support your claims.)

Analyzing the Case of Monique Butler In terms of the Mathematical Tasks Framework, what is Monique Butler a case of?

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, Smith, Henningsen, & Silver, 2000, p. 4

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, Smith, Henningsen, & Silver, 2000, p. 4

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, Smith, Henningsen, & Silver, 2000, p. 4

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, Smith, Henningsen, & Silver, 2000, p. 4

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, Smith, Henningsen, & Silver, 2000, p. 4

Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Decline Maintenance  Routinizing problematic aspects of the task  Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer  Providing insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off-task behavior  Engaging in high-level cognitive activities is prevented due to classroom management problems  Selecting a task that is inappropriate for a given group of students  Failing to hold students accountable for high-level products or processes  Scaffolding of student thinking and reasoning  Providing a means by which students can monitor their own progress  Modeling of high-level performance by teacher or capable students  Pressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedback  Selecting tasks that build on students’ prior knowledge  Drawing frequent conceptual connections  Providing sufficient time to explore Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press.

Analyzing the Case of Monique Butler How was your experience multiplying binomials (Task 2) similar or different from Monique Butler ’ s students ’ experience multiplying binomials?

Analyzing the Case of Monique Butler What lessons can we learn from the Case of Monique Butler?

The Role of the Teacher Worthwhile tasks alone are not sufficient for effective teaching. Teachers must also decide what aspects of a task to highlight, how to organize and orchestrate the work of the students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminating the challenge. NCTM, 2000, p. 19

Considering the Knowledge Needed for Teaching  What knowledge needed for teaching might teachers develop through experience such as those we discussed today?  How will know if teachers learned what you intended for them to learn?

Mathematical Knowledge for Teaching Common Content Knowledge Knowledge at the mathematical horizon Specialized Content Knowledge Knowledge of Content and Students Knowledge of Content and Teaching Knowledge of Curriculum Ball, Bass, Hill, and Thames, 2006

Using an area model to calculate 6 x 13: x 13 = 78 Making Sense of Multiplication

Using an area model to calculate 6 x 13: x 63 x 6 6 x 13 = 6 x (10 + 3) = (6 x 10) + (6 x 3) = = 78 Making Sense of Multiplication

6 6 x n6 x 3 6(n + 3) = (6 x n) + (6 x 3) = 6n + 18 n 3 Making Sense of Multiplication Using an area model for 6(n + 3):

Using an area model to calculate 26 x 12: x 106 x x 2 6 x 2 Making Sense of Multiplication 26 x 12 = (20 + 6) (10 + 2) = (20 x 10) + (20 x 2) + (6 x 10) + (6 x 2) = = 312

Using an area model for (2x + 6)(x + 2): 2x 6 x2x2 4x Making Sense of Multiplication (2x + 6)(x + 2) = (2x)(x) + (2x)(2) + (6)(x) + (6)(2) = 2x 2 + 4x + 6x + 12 = 2x x + 12

“Algebrafying” a Task Existing arithmetic activities and word problems are transformed from problems with a single numerical answer to opportunities for pattern building, conjecturing, generalizing, and justifying mathematical facts and relationships. Blanton & Kaput, 2003, p.71

“Algebrafying” the Sweater Sale Task Original The cost of a sweater at J. C. Penney's was $ At the "Day and Night Sale" it was marked 30% off of the original price. What was the price of the sweater during the sale? Modified The cost of a sweater at J. C. Penney's was $ At the "Day and Night Sale" it was marked 30% off of the original price. What was the price of the sweater during the sale? What would the sale price be if the original price was $46? $47? $48? $100? What is the relationship between the original price of the sweater and the sale price?

Task C Larson, R., Boswell, L., Kanold, T. D., & Stiff, L. (2001). CA Algebra 1: Concepts and Skills, McDougal Littell, 273.