Teacher Quality Workshops for 2010/2011

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

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

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)

Cognitive Demand Levels

“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

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

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.

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.

Let’s Compare These Two Tasks

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

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?