1. Teacher Quality Workshops for 2010/2011

2. Group Norms Be an active learner Be an attentive listener
Be a reflective participant
Be conscious of your needs and needs of others

3. Year-long Objectives
Strengthen our mathematical knowledge for teaching to foster in our students conceptual understanding and mathematical thinking
Develop activities with high cognitive demand for students to engage
Orchestrate productive math discussion in our classrooms
Build a professional learning community

4. Mathematical Knowledge for Teaching
Subject Matter Knowledge
Pedagogical Content Knowledge
Common Content
Knowledge (CCK)
Knowledge of
Content and Students (KCS)
Specialized Content Knowledge (SCK)
Knowledge of curriculum
Knowledge at
the mathematical horizon
Knowledge of Content and
Teaching (KCT)

5. Cognitive Demand Levels

6. “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 and Briars, 1995
… because …
“Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.”
Stein, Smith, Henningsen, & Silver, 2000
“The level and kind of thinking in which students engage determines what they will learn.”
Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver & Human, 1997

7. Four levels of cognitive demand
Memorization
Procedures without connections to concepts or meaning
e.g., remember a ratio is written as A : B or A/B.
e.g., use a scale-factor to find equivalent ratios
Procedures with connections to concepts or meaning
Doing mathematics
e.g., use diagrams to explain why the scale-factor method works
e.g., the watermelon problem
Stein, Smith, Henningsen, & Silver, 2000

8. Two Lower-Level Cognitive Demands
Memorization
Involve either reproducing previously learned information (facts, rules, formulae, or definitions) OR committing them to memory
Involve exact reproduction of previously-seen material
Have no connection to the concepts or meaning that underlie the information being learned or reproduced
2. Procedures Without Connections
Are algorithmic (specifically called for OR based on prior work)
Has obvious indicator of what needs to be done or how to do it
Have no connection to the concepts or meaning that underlie the procedure being used
Are focused on producing correct answers rather than developing mathematical understanding
Require only “how” explanations, no “why” explanations
For more information on what constitutes the various levels of task, see Chapter 1 of Stein, Smith, Henningsen, & Silver, 2000.

9. Two Higher-Level Cognitive Demands
3. Procedures with connections to concepts or meaning
To deepen student understanding of concepts and ideas
Suggest pathways that are broad general procedures that have close connections to underlying conceptual ideas
Can be represented in multiple ways
Cannot be followed mindlessly (require cognitive effort)
4. Doing Mathematics
Require complex and non-algorithmic thinking (ie. non-routine)
Require students to access relevant knowledge/experiences
Require students to analyze tasks and examine task constraints
Require students to explore and understand relationships
Demand self-monitoring
Require considerable cognitive effort (may lead to frustration)
For more information on what constitutes the various levels of task, see Chapter 1 of Stein, Smith, Henningsen, & Silver, 2000.

10. Let’s Compare These Two Tasks

11. Let’s Compare These Two Tasks
What key understandings can be fostered in each task?

12. Comparing the Two Tasks
Which task involves a higher-level cognitive demand? Why?
Which task is more appropriate for your students?
Which task better prepares students for STAAR?