High Cognitive vs. Low Cognitive 1

An effective mathematical task is needed to challenge and engage students intellectually. 2

Read an excerpt from the article:

Comparing Two Mathematical Tasks Solve Two Tasks: Martha’s Carpeting Task The Fencing Task 4

How are Martha’s Carpeting Task and the Fencing Task the same and how are they different? Comparing Two Mathematical Tasks 5

Similarities and Differences Similarities Both are “area” problems Both require prior knowledge of area Differences The amount of thinking and reasoning required The number of ways the problem can be solved Way in which the area formula is used The need to generalize The range of ways to enter the problem 6

Do the differences between the Fencing Task and Martha’s Carpeting Task matter? Why or Why not? Comparing Two Mathematical Tasks 7

Does Maintaining Cognitive Demand Matter? YES

Criteria for low cognitive demand tasks Recall Memorization Low on Bloom’s Taxonomy

Developing Mathematical Practices Review each of the “higher demand” tasks: What mathematical practices do they engage students in using ?

Criteria for high cognitive demand tasks Requires generalizations Requires creativity Requires multiple representations Requires explanations (must be “worth explaining”)

Lower-Level Tasks Memorization What are the decimal equivalents for the fractions ½ and ¼? Procedures without connections Convert the fraction 3/8 to a decimal

Higher-Level Tasks Doing mathematics 1. Shade 6 small squares in a 4 X 10 rectangle. Using the rectangle, explain how to determine: A. The decimal part of area that is that is shaded; B. The fractional part of area that is shaded

Higher-Level Tasks  Procedures with connections Using a 10 x 10 grid, identify the decimal and percent equivalents of 3/5.

What causes high- level cognitive demand tasks to decline?

Stein & Lane, 2012 A. B. C. High Low HighLow Moderate High Low Task Set UpTask ImplementationStudent Learning Patterns of Set up, Implementation, and Student Learning 17

Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands 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 (Stein, Grover & Henningsen, 2012) 18

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, Grover & Henningsen, 2012) Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands 19

“Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.” (Stein, Smith, Henningsen, & Silver, 2011) “The level and kind of thinking in which students engage determines what they will learn.” (Hiebert et al., 2011)

Characteristics of good questions  Require more than remembering a fact or reproducing a skill  Students can learn from answering the questions; teachers can learn about the students  May have several acceptable answers Sullivan and Lilburn

Making questions open ended  Method 1: Working backwards Identify a mathematical topic or concept. Think of a closed question and write down the answer. Make up a new question that includes (or addresses) the answer. 22

Method 1: Working backwards What is the product of 90 and 4? can become …. Two numbers multiply to make 360. One of them has a zero on the end. What might the two numbers be? 23

Making questions open ended  Method 2: Adapting a standard question Identify a mathematical topic or concept. Think of a standard question Adapt it to make an open ended question. 24

Method 2: Adapting a standard question I bought a notebook that cost $1.65. How much change did I receive from $2.00? Can become… I bought an item at a shop and got 35 cents change. What did I buy and how much did it cost? 25