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STRENGTHENING MATHEMATICS INSTRUCTION Cognitive Complexity and Instructional Practices

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Characteristics of the Workshop 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

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Why? Not analyzing and clarifying situations that have led to anomalies and common errors Not making concepts and content clearer through examples, non- examples and counterexamples = confused students What are some of the causes that lead to students being confused about mathematical concepts and content?

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Cognitively Complex Problems 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 These types of problems require students to

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Causes of Low Proficiency Levels Activity Low proficiency levels ?? Lack of basic skills ??? 1.Think about things that you believe contribute to low proficiency levels in students’ work. 1.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 < x < 1, arrange the terms in increasing numerical order from smallest to largest

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Locating Cognitively Complex Problems 1.Choose a section or chapter in your textbook that you will be teaching in the next few weeks. 2.Use post-it notes to indicate any problems that are cognitively complex. 3.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? Activity

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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 Definition The Context

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It’s Your Turn to Identify Structures! 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? As a Learner 1 2 As a Teacher Partner Up with someone you haven’t worked with before. 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 “AHA’s” with each other.

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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… What teachers said about a pilot workshop

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