Presentation on theme: "STRENGTHENING MATHEMATICS INSTRUCTION Cognitive Complexity and Instructional Practices Instructor’s Notes: Depending on when this module is presented,"— Presentation transcript:
1 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.
2 Characteristics of the Workshop 18-24 hours of professional development; 8 modules to allow for flexibility in schedulingStandards based and tied to the CSTs and CSU placement standardsIncludes content and activities for teachers of Algebra 1 Geometry, Algebra 2, Pre-CalculusDraws on problems and lessons from the major textbooksDesigned for teacher practice and implementation between workshop sessions based on lesson study modelReflective of the recently adopted national mathematics standardsNo cost to the school(s) for workshop and materialsInstructor’s Notes: minutes total Slide 1 of 3This 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 participantLarge poster of diagram for each groupEngagement/Process StrategiesClosing the WindowFinger MinutesProcess as Given, Process as Understood
3 Workshop OutcomesIdentify instructional strategies that will help students organize and solidify conceptual understandingIdentify characteristics of cognitively complex problemsLocate standards-based cognitively complex problems within participants’ classroom textsModify standards-based textbook problems to increase the level of cognitive complexityPractice writing standards-based cognitively complex problemsExperience the varying roles in the teacher/learner continuumModel a variety of student engagement strategiesInstructor’s Notes: 1 minutesExplain 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.
4 Not providing exposure to cognitively complex problems Why?Not analyzing and clarifying situations that have led to anomalies and common errorsNot making concepts and content clearer through examples, non-examples and counterexamplesWhat are some of the causes that lead to students being confused about mathematical concepts and content?= confused studentsInstructor Notes: 1 minuteThis 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
5 Cognitively Complex Problems These types of problems require students toExtend previously encountered tasksIntegrate several topics and/or conceptsRecognize and use underlying mathematical structuresUse multiple representationsConsider multiple approaches to the problemIdentify patternsBe flexible and strategic in their mathematical thinkingInstructor’s Notes: 10 minutesThis 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 StrategiesBrainstorm
6 Causes of Low Proficiency Levels ActivityThink 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 skillsInstructor’s Notes: minutes total Slide 1 of 3This 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 participantLarge poster of diagram for each groupEngagement/Process StrategiesClosing the WindowFinger MinutesProcess as Given, Process as Understood
7 Example 3 – The Real Numbers Arrange the numbers in increasing order from smallest to largestIf 0 < x < 1, arrange the terms in increasing numerical order from smallest to largestInstructor’s Notes: Slide 3 of 3In 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 algorithmsHigher complexity:Involves abstractions, alternative forms of representationRequires student to make connections between different representationsAsk participants to suggest additional terms that would make this question applicable to Algebra 2/Precal concepts.Suggestions might include ln x, ex e 3lnx
8 Locating Cognitively Complex Problems ActivityChoose 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: minutes Slide 1 of 2After 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 StrategiesProcess Given, Process as UnderstoodYellow LightClosing the WindowFinger MinutesProcess as Given, Process as UnderstoodPaired Squares
9 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 13The 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.
10 Motivating and Making Sense of Definitions The DefinitionInstructor’s Notes: Slide 2 of 4The definition for absolute value is also provided in its geometric representation. Make connections between the two representations.The Context
11 It’s Your Turn to Identify Structures! As aLearner12As aTeacherPartner 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 minutesHave 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 StrategiesAdapt the A/B Each Teach strategyAdapt the Assumptions Inquiry strategyMost Important Point ” strategyFinger Minutes ” strategyProcesses Given, Process as Understood ” strategyAdapt the “Say Something” strategyClosing the Window strategyMaterialsIndentifying structures page
12 What teachers said about a pilot workshop It gave me a starting point to improve instruction…Working with my fellow teachers and having timeto explore complexity was most valuable…Learning about cognitive layering in problems isvery important…I learned to ask more open-ended questions anduse “what if” to explore mathematical ideaswithout fearThis workshop showed me strategies to helpstudents think mathematically…
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