More Rigorous SOL = More Cognitively Demanding Teaching and Assessing Dr. Margie Mason The College of William and Mary Adapted from l_development/index.shtml

Sorting Mathematical Tasks Examine the four tasks on your handout. In your group, discuss the following questions: What do students need to know to solve each task? How are the tasks similar? How are the tasks different?

What are the decimal and percent equivalents for the fractions and ?

Memorization What are the decimal and percent equivalents for the fractions and ?

Convert the fraction to a decimal and a percent.

Procedures without Connections Convert the fraction to a decimal and a percent.

Using a 10 x 10 grid, identify the decimal and percent equivalents of.

Procedures with Connections Using a 10 x 10 grid, identify the decimal and percent equivalents of.

Shade 6 small squares in a 4 x 10 rectangle. Using the rectangle, explain how to determine each of the following: (a)The percent of area that is shaded, (b)The decimal part of area that is shaded, and (c)The fractional part of area that is shaded.

Doing Mathematics Shade 6 small squares in a 4 x 10 rectangle. Using the rectangle, explain how to determine each of the following: (a)The percent of area that is shaded, (b)The decimal part of area that is shaded, and (c)The fractional part of area that is shaded.

Lower Level Demands Memorization Procedures without connections Higher Level Demands Procedures without connections Doing mathematics

Examining Differences between Tasks 12 What is cognitive demand? thinking required

Task Sort Activity 13 Sort the provided tasks as high or low cognitive demand. List characteristics you use to sort the tasks.

Discussing the Task Sort 14 TaskABCDEFGHIJ Low High Criteria for a low level task Criteria for a high level task

Task A 15 This line plot shows the number of letters in the names of 7 students. x x x x x x x Determine the balance point for this set of data and explain how you arrived at this answer. Adapted from EPAT practice items, VDOE, Grade 6, 2010.

Task B 16. Determine the value of each expression. 10³ 10² 10¹ 10º Graph each of these values on the same number line. What do you notice? List three true statements about your graph.

Task C 17 Jordan and Paul were comparing two numbers. Jordan said, “My number is greater than your number.” Paul said, “That may be true, but the absolute value of my number is greater than your number.” Locate Jordan’s and Paul’s number on a number line and explain your reasoning. Compare your answers with other classmates. What do you notice?

Task D 18 Your job is to design plastic containers for ice cream Sprinkles. Design and sketch containers in the shape of a right triangular prism, a rectangular prism, and a right circular cylinder. Each must fit on a shelf space that is 12 cm tall, 6 cm wide, and 6 cm deep. Sketch each and label the dimensions. Explain which container will hold the most sprinkles for the given shelf space. Which container design would save money by using less plastic? Explain your reasoning.

Task E 19 Hannah made 54 cupcakes for Erin’s birthday party. She made half of the cupcakes chocolate and half of the cupcakes yellow. She put sprinkles on 1/3 of the chocolate ones. She put one candle on each of the 2/3 cupcakes that did not have sprinkles. How many candles did Erin have to blow out?

Task F 20 A box shaped like a rectangular prism has a volume of 360 cubic inches. This box has a width of 6 inches and a length of 10 inches. A. What is the height of the box? B. If you doubled the length, what would be the new volume? Explain how you found each. Adapted from MCAS, Grade 8, 2011

Task G 21 Identify each number that has an absolute value of ¼

Task H 22 Cindy surveyed 60 students about their favorite type of movie. This circle graph represents the results of the survey. Construct a bar graph that could represent the same set of data. Adapted from EPAT practice items, VDOE, Grade 6, 2010.

Task I 23 What is the value of 2x² + 5(x³ - 4) when x = 4?

Task J 24 A rectangle as shown has a length of 0.9 centimeters and a length of 0.4 centimeters. A circle is drawn inside that touches the rectangle at two points. 0.9 cm 0.4 cm What is the total area of the unshaded region in the rectangle?

Task Analysis Guide Lower-level Demands Involve recall or memory of facts, rules, formulae, or definitions Involve exact reproduction of previously seen material No connection of facts, rules, formulae, or definitions to concepts or underlying understandings. Focused on producing correct answers rather than developing mathematical understandings Require no explanations or explanations that focus only on describing the procedure used to solve 25 Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

Task Analysis Guide Higher-level Demands Focus on developing deeper understanding of concepts Use multiple representations to develop understanding and connections Require complex, non-algorithmic thinking and considerable cognitive effort Require exploration of concepts, processes, or relationships Require accessing and applying prior knowledge and relevant experiences Require critical analysis of the task and solutions 26 Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

Characteristics of Rich Mathematical Tasks High cognitive demand (Stein et. al, 1996; Boaler & Staples, 2008) Significant content (Heibert et. al, 1997) Require Justification or explanation (Boaler & Staples, in press) Make connections between two or more representations (Lesh, Post & Behr, 1988) Open-ended (Lotan, 2003; Borasi &Fonzi, 2002) Allow entry to students with a range of skills and abilities Multiple ways to show competence (Lotan, 2003) 27

Now try to identify the cognitive demand of the items for your grade level.

Once we have identified the items requiring low cognitive demand, work as a team and try to rewrite each low item to make it more demanding.

Thinking About Implementation A mathematical task can be described according to the kinds of thinking it requires of students, it’s level of cognitive demand. In order for students to reason about and communicate mathematical ideas, they must be engaged with high cognitive demand tasks that enable practice of these skills. 30

The Challenge of Implementation BUT! … simply selecting and using high-level tasks is not enough. Teachers need to be vigilant during the lesson to ensure that students’ engagement with the task continues to be at a high level. 31

Factors Associated with Lowering High-level Demands Shifting emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer Providing insufficient or too much time to wrestle with the mathematical task Letting classroom management problems interfere with engagement in mathematical tasks Providing inappropriate tasks to a given group of students Failing to hold students accountable for high-level products or processes 32 Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

Factors Associated with Promoting Higher-level Demands Scaffolding of student thinking and reasoning Providing ways/means by which students can monitor/guide their own progress Modeling high-level performance Requiring justification and explanation through questioning and feedback Selecting tasks that build on students’ prior knowledge and provide multiple access points Providing sufficient time to explore tasks 33 Adapted from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing standars-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press

Lesson Structure To foster reasoning and communication focused on a rich mathematical task, a 3-part lesson structure is recommended: 1.Individual thinking (preliminary brainstorming) 2.Small group discussion (idea development) 3.Whole class discussion (idea refinement) 34

Organizing High-Level Discussions: 5 Habits Prior to the lesson, 1.Anticipate student strategies and responses to the task More on 5 Habits can be found in: “Orchestrating Discussions” by Smith, Hughes, Engle, & Stein in Mathematics Teaching in the Middle School, May

Cognitive Demand …students who performed best on a project assessment designed to measure thinking and reasoning processes were more often in classrooms in which tasks were enacted at high levels of cognitive demand (Stein and Lane 1996), that is, classrooms characterized by sustained engagement of students in active inquiry and sense making (Stein, Grover, and Henningsen 1996). For students in these classrooms, having the opportunity to work on challenging mathematical tasks in a supportive classroom environment translated into substantial learning gains. ---Stein & Smith, 2010