STRENGTHENING MATHEMATICS INSTRUCTION Cognitive Complexity and Instructional Practices Characteristics of the Workshop  18-24 hours of professional development; 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 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 Workshop Outcomes Why? What are some of the causes that lead to students being confused about mathematical concepts and content? = confused students 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 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 Example – The Real Numbers Arrange the numbers in increasing order from smallest to largest If < x < 1, arrange the terms in increasing numerical order from smallest to largest − −1 −1 x x2 x − x2 − x x 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 you assign these problems for homework? • How often you include these problems in your section/chapter assessments? Geometry – Extension #3 (Problem) A square is inscribed in a circle of radius units What is the total area enclosed within the circle but outside the square? A circle of radius units is inscribed in an equilateral triangle Find the length of the side of the triangle Motivating and Making Sense of Definitions The Definition  x if x ≥ x =  −x if x < The Context It’s Your Turn to Identify Structures! As a Learner 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 As a Teacher 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? 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… [...]... these types of problems in my teaching? • What are some challenges I might face in developing these types of problems? 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