STRENGTHENING MATHEMATICS INSTRUCTION Cognitive Complexity and Instructional Practices

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STRENGTHENING MATHEMATICS INSTRUCTIONCognitive
Complexity and Instructional Practices
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Characteristics of the Workshop

• 18-24 hours of professional development 8
  modules to allow for flexibility in scheduling
• Standards based and tied to the CSTs and CSU
  placement standards
• Includes content and activities for teachers of
  Algebra 1 Geometry, Algebra 2, Pre-Calculus
• Draws on problems and lessons from the major
  textbooks
• Designed for teacher practice and implementation
  between workshop sessions based on lesson study
  model
• Reflective of the recently adopted national
  mathematics standards
• No cost to the school(s) for workshop and
  materials

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Workshop Outcomes

• Identify instructional strategies that will help
  students organize and solidify conceptual
  understanding
• Identify characteristics of cognitively complex
  problems
• Locate standards-based cognitively complex
  problems within participants classroom texts
• Modify standards-based textbook problems to
  increase the level of cognitive complexity
• Practice writing standards-based cognitively
  complex problems
• Experience the varying roles in the
  teacher/learner continuum
• Model a variety of student engagement strategies

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Why?
What are some of the causes that lead to students
being confused about mathematical concepts and
content?
confused students
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Cognitively Complex Problems
These types of problems require students to

• Extend previously encountered tasks
• Integrate several topics and/or concepts
• Recognize and use underlying mathematical
  structures
• Use multiple representations
• Consider multiple approaches to the problem
• Identify patterns
• Be flexible and strategic in their mathematical
  thinking

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Causes of Low Proficiency Levels
Activity

• Think about things that you believe contribute to
  low proficiency levels in students work.
• Write each idea on a separate post-it note.

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Example 3 The Real Numbers
Arrange the numbers in increasing order from
smallest to largest
If 0 lt x lt 1, arrange the terms in increasing
numerical order from smallest to largest
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Locating Cognitively Complex Problems
Activity

• Choose a section or chapter in your textbook that
  you will be teaching in the next few weeks.
• Use post-it notes to indicate any problems that
  are cognitively complex.
• At your table, discuss the following questions
• Where did you find these problems?
• Compare the number of complex problems to the
  number of standard problems in your textbook.
• How often do you assign these problems for
  homework?
• How often do you include these problems in your
  section/chapter assessments?

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Geometry Extension 3 (Problem)

• A square is inscribed in a circle of radius 3
  units. What is the total area enclosed within
  the circle but outside the square?

A circle of radius 3 units is inscribed in an
equilateral triangle. Find the length of the side
of the triangle.
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Motivating and Making Sense of Definitions
The Context
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Its Your Turn to Identify Structures!
Partner Up with someone you havent worked with
before.

• Discuss
• Have I provided my students with these types of
  problems? If not, why?
• How would I begin to incorporate more of these
  types of problems in my teaching?
• What are some challenges I might face in
  developing these types of problems?

• Using the activity page
• Determine the basic structure for each of the
  problems.
• Determine which problems were easier and harder
  for you and why.
• Share your AHAs with each other.

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What teachers said about a pilot workshop

• It gave me a starting point to improve
  instruction
• Working with my fellow teachers and having time
• to explore complexity was most valuable
• Learning about cognitive layering in problems is
• very important
• I learned to ask more open-ended questions and
• use what if to explore mathematical ideas
• without fear
• This workshop showed me strategies to help
• students think mathematically