STRENGTHENING MATHEMATICS INSTRUCTION Cognitive Complexity and Instructional Practices Instructor’s Notes: Depending on when this module is presented, you may want to revisit the group meeting norms, expectations, etc.

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 Instructor’s Notes: 20 minutes total Slide 1 of 3 This is a two part activity. The first is done individually, the second as a group. Each person is to think about the things that they feel contribute to low proficiency in student work (such as lack of basic skills as indicated on the slide) and write each idea on a post-it note. (Slide 1) Each table groups the individual ideas into broad categories, i.e., clusters and makes a poster with categories on color post-its. (Slide 2) You will facilitate a discussion on commonalities and differences and compare the final product to Slide 3. Materials needed: Small post-it notes for each participant Large poster of diagram for each group Engagement/Process Strategies Closing the Window Finger Minutes Process as Given, Process as Understood

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 Instructor’s Notes: 1 minutes Explain to participants that the highlighted outcomes are the ones that we will be working on in this module. We will always be reviewing instructional and engagement strategies.

Not providing exposure to cognitively complex problems 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 What are some of the causes that lead to students being confused about mathematical concepts and content? = confused students Instructor Notes: 1 minute This slide sets the stage for the module and some of the topics that we will be looking at in future modules. Tie these ideas back to the brainstorm that the participants did during “Setting the Stage.” This is an animated slide. The slide appears without the bottom cog on the left, then the cog blinks four times, and then it stays on the slide. This was done for emphasis. Not providing exposure to cognitively complex problems

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 Instructor’s Notes: 10 minutes This is the first introduction to the module topic. This is not a definitive list. There are other characteristics that teachers might refer to. You may elicit additional responses and place them on chart paper for future reference. Emphasize that “cognitively complex” does not equate with “a word problem,” a problem with very large numbers or a tedious repetition of the same algorithm (such as long division with ten terms). You will need to go in depth for each of these bullets. Ask questions of your participants for clarification, understanding, etc. You will be referring to these characteristics often in all of the modules. Engagement/Process Strategies Brainstorm

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. Low proficiency levels ? Lack of basic skills Instructor’s Notes: 20 minutes total Slide 1 of 3 This is a two part activity. The first is done individually, the second as a group. Each person is to think about the things that they feel contribute to low proficiency in student work (such as lack of basic skills as indicated on the slide) and write each idea on a post-it note. (Slide 1) Each table groups the individual ideas into broad categories, i.e., clusters and makes a poster with categories on color post-its. (Slide 2) You will facilitate a discussion on commonalities and differences and compare the final product to Slide 3. Materials needed: Small post-it notes for each participant Large poster of diagram for each group Engagement/Process Strategies Closing the Window Finger Minutes Process as Given, Process as Understood

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 Instructor’s Notes: Slide 3 of 3 In the following slides, try to elicit suggestions from participants as to what has caused the level of complexity to increase. Refer back to the TIMMS study and why these problems are different. Lower complexity: All procedures are standard algorithms Higher complexity: Involves abstractions, alternative forms of representation Requires student to make connections between different representations Ask participants to suggest additional terms that would make this question applicable to Algebra 2/Precal concepts. Suggestions might include ln x, ex e 3lnx

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? Instructor’s Notes: 10- 15 minutes Slide 1 of 2 After the groups have completed their textbook search, you should show the slide to facilitate a discussion using the next slide to draw out key issues and their implications for instruction. Engagement/Process Strategies Process Given, Process as Understood Yellow Light Closing the Window Finger Minutes Process as Given, Process as Understood Paired Squares

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? Instructor notes: Slide 6 of 13 The top box is the previous extension problem. The second box is the next extension. Give participants time to read and work each problem and think about the changes. A circle of radius 3 units is inscribed in an equilateral triangle. Find the length of the side of the triangle.

Motivating and Making Sense of Definitions The Definition Instructor’s Notes: Slide 2 of 4 The definition for absolute value is also provided in its geometric representation. Make connections between the two representations. The Context

It’s Your Turn to Identify Structures! As a Learner 1 2 As a Teacher Partner Up with someone you haven’t 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 “AHA’s” with each other. Instructor’s Notes: 15 minutes Have participants partner with someone they haven’t worked with before to discuss the questions with their partner. You might want to ask for comments from the whole group as closure to this activity. You should try to keep the conversations positive and focused on how to make this work, rather than “My kids won’t be able to do this” or “It’s too hard” or “I don’t have the time to do this” Engagement/Process Strategies Adapt the A/B Each Teach strategy Adapt the Assumptions Inquiry strategy Most Important Point ” strategy Finger Minutes ” strategy Processes Given, Process as Understood ” strategy Adapt the “Say Something” strategy Closing the Window strategy Materials Indentifying structures page

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…