More Rigorous SOL = More Cognitively Demanding

Teaching and Assessing

Dr. Margie MasonThe College of William and Mary

mmmaso@wm.edu

Adapted from http://www.doe.virginia.gov/instruction/mathematics/professional_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

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What is cognitive demand?

thinking required

Task Sort Activity

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• Sort the provided tasks as high or low cognitive demand.

• List characteristics you use to sort the tasks.

Discussing the Task Sort

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Task A B C D E F G H I J

Low

High

Criteria for a low level task

Criteria for a high level task

Task A

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This line plot shows the number of letters in the names of 7 students.

x x x x x x x 5 6 7 8 9 10 11

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

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.

Determine the value of each expression.

10³ 10² 10¹ 10º 110 210 310

Graph each of these values on the same number line. What do you notice? List three true statements about your graph.

Task C

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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

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Your job is to design plastic containers for ice creamSprinkles. 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

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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

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A box shaped like a rectangular prism has a volume of 360 cubic inches. This box has a width of 6 inchesand 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

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Identify each number that has an absolute value of 4.

16 4 2 ¼ 0 -2 -4 -16

Task H

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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

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

Task J

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A rectangle as shown has a length of 0.9 centimeters and a length of 0.4 centimeters. A circle is drawn inside that touchesthe 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

25Adapted 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 GuideHigher-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

26Adapted 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.

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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.

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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

32Adapted 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

33Adapted 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)

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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 2009

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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