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
Common
Content
Knowledge
(CCK)
Knowledge at
the mathematical
horizon
Specialized
Content
Knowledge
(SCK)
Knowledge of
Content and
Students (KCS)
Knowledge
of
curriculum
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
1. Memorization
e.g., remember a ratio is written as A : B or A/B.
2. Procedures without connections to concepts or meaning
e.g., use a scale-factor to find equivalent ratios
3. Procedures with connections to concepts or meaning
e.g., use diagrams to explain why the scale-factor method works
4. Doing mathematics
e.g., the watermelon problem
Stein, Smith, Henningsen, & Silver, 2000
Two Lower-Level Cognitive Demands
1. 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
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
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•
•
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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)
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
i. Which task involves a higher-level cognitive demand?
Why?
ii. Which task is more appropriate for your students?
iii. Which task better prepares students for STAAR?