Cognitively Guided Instruction in Mathematics (CGI)

1. Cognitively Guided Instruction in Mathematics(CGI) Tricia Essick Randolph County Schools October 7, 2009

2. CGI Sessions Session 1- October 7 What CGI is and is not Getting started Introducing problems, problem types When and how to do CGI Developing problems for students Video segments

3. CGI Sessions Session 2- November 4 Developing problems Problem collections Differentiation/ Modifications Student sharing Video segments

4. CGI Sessions Session 3-January 13 Analyzing student work Developing the next lesson Moving children to higher levels/strategies Tips for success Video segments

5. What is CGI? Cognitively Guided Instruction (CGI) is an approach to teaching and learning mathematics in which knowledge of children’s thinking is central to instructional decision making. • Adapted from materials prepared by The Wisconsin Center for Education Resource, The University of Wisconsin-Madison.

6. How Do Teachers Use CGI? Teachers use research-based knowledge about children’s mathematical thinking to help them learn specifics about individual students and then to adjust instruction to match students’ performance. Adapted from materials prepared by The Wisconsin Center for Education Resource, The University of Wisconsin-Madison.

7. How do Teachers Use CGI? In implementing CGI, teachers assess students’ thinking and use that knowledge to plan instruction. Adapted from materials prepared by The Wisconsin Center for Education Resource, The University of Wisconsin-Madison.

8. What is CGI? • a philosophy • a developmental approach to problem solving • teacher facilitated approach • child-led instruction

9. CGI is Not: • textbooks and worksheets • a program that requires expensive materials • a “here’s the right way to solve it” approach

10. Problem Task Read the problems on the sheet, Cognitively Guided Instruction Word Problems vs. Grade Level. At your table, talk together and decide at what grade level each problem would be appropriate. Justify your answer.

11. CGI Classrooms Teachers help students develop their mathematical understanding by making curricular decisions based on what students know and understand. As curricular decisions are implemented, a unique classroom emerges, structured to fit the teacher’s teaching style, knowledge, beliefs, and students.

12. CGI Classrooms-Common Components • Problem solving is the focus on instruction, with children deciding how they should solve each problem. • Many strategies are used to solve problems. “It's better to solve one problem five different ways, than to solve five problems one way.” -George Polya

13. Children communicate to their teachers and peers how they solved their problems. • Each person’s thinking is important and respected by peers and teachers. • Teachers understand children’s problem-solving strategies and use that knowledge to plan their instruction.

14. Planning the Classroom “It is not simple to describe a typical CGI classroom because each one is unique and can appear to be quite different from other CGI classrooms. “ Grouping children- • Whole group vs. small group Be prepared- • Have materials ready. • Plan ahead.

15. Planning the Classroom Student-centered • Make students responsible • Encourage cooperation and sharing • Establish routines and expectations

16. Planning the Classroom Classroom Climate- • High expectations • Acceptance • Praise • Listening • Questioning

17. CGI Teacher Tools • Log for teacher observations • Problems generated by teacher • Chart paper/tablets • Number line posted in classroom • Rubric/expectations posted • Overhead projector, whiteboards, SmartBoard, document camera • Transparencies, markers, overhead manipulative kit • Extra manipulatives

18. CGI Student Tools • Student journals • Whiteboards, markers • Tool boxes with manipulatives • Crayons, pencils, glue, scissors • Extra paper

19. Tool Boxes

20. Planning the Classroom “There is no optimal way to organize a CGI class. Whatever organization enables a teacher to get the children to solve problems and to listen to the students’ problem-solving strategies is the optimal organization for that teacher.” Children’s Mathematics, Thomas P. Carpenter

21. CGI- Problem Solving “Most, if not all, important mathematics concepts and procedures can best be taught through problem solving.” John Van de Walle

22. Classification of Word Problems Join

23. Classification of Word Problems Separate

24. Classification of Word Problems Part-Part-Whole

25. Classification of Word Problems Compare

26. Classification of Word Problems Activity: Work with a partner. Cut out the problems on the sheet given. Looking at the blank problem type chart, decide where each problem fits.

27. Writing CGI Problems • A lesson should revolve around 1-2 problems. • Problems should be connected to children, their lives, experiences, field trips, or the curriculum. • Numbers in problems should align with children’s number sense development. • Make actions/ relationships as clear as possible.

28. Writing CGI Problems • Keep units the same. There are 5 dogs and 9 cats. How many animals in all? Or Paula had 5 dollars. She spent 55 cents. How much money does she have left? • Use numbers that encourage children to move to more sophisticated solution strategies. (counting on, skip counting, or use of derived facts).

29. Sample Problem Audrey has some books. She bought 3 books. Now she has 7 books altogether. How many books did she start with?

30. Sample Problem Max took 26 toy cars he no longer wanted and gave them to his friend. Now he has 64 toy cars left. How many cars did Max begin with?

31. Left? Altogether? Fewer? Vocabulary What is the danger of teaching children key words? According to Van deWalle in his book, Teaching Student-Centered Mathematics, 1. Key words are misleading. 2. Many problems have no key words, especially with higher level problems. A child taught to rely on key words then has no strategy. 3. Key word strategy sends a wrong message about doing mathematics. The most important approach is to analyze the problem structure.

32. Questions to Help Analyze • What is happening in this problem? • What will the answer tell us? • Do you think it will be a big number or a small number? • About how many ______? (making a good estimation)

33. What’s next? • Start small. • Use what you already know and what you have learned today. • Pay attention to what you reallysee and hear your students doing. • Plan accordingly. • Ask for help and feedback. • Call or e-mail Tricia 318-6378 Don’t get discouraged. It will come!

34. Cognitively Guided Instruction in Mathematics-CGI Tricia Essick Randolph County Schools November 4, 2009

35. CGI- Session 2 Overview • Share problems • Revisit the toolbox • Developing problems • Differentiation/ Modifications • Keeping students engaged • Student sharing • Sample rubric

36. Problem solving is what you do when you don’t know what to do!

37. Your Turn • Share a problem that went well with your class. Why did you choose that problem? Tell about student successes with this problem. • Share a problem that did not go well. Why do you think it didn’t go well? How could you change the problem to make it better?

38. The Toolbox • Use materials you have • Variety of materials • Share materials, no need for each child to have their “own” • Tools change throughout the year • Start small, add or change

39. Toolbox Activity • What kind of items would you include in student CGI toolboxes at the beginning of the year? • What items would you add later? Why? • What items might you remove or use fewer of later in the year? Why?

40. Tool Box Basics • Pencils, crayons, markers • Glue sticks, scissors • Paper or post-it notes • Counting items: bears, cubes, shells,“junk”, seasonal • Items to represent 2 sets, (red/yellow chips or cubes in 2 colors) • Items for counting on/back, or skip counting: hundreds boards, number lines

41. Toolbox Basics- Continued • Place value materials (base 10 blocks, 10 frames, bean sticks, snap cubes • Items for measurement: rulers, tape measures • Money- Caution! • Materials for sharing- paper, overhead transparency… -(Hint: Teacher keeps marker until needed)

42. Common Teaching Strategy Key Words- Example: altogether Does a key word strategy work with this problem? Billy bought toys that totaled $11.90. He paid with a $20 bill. How much change should Billy get altogether?

43. Common Teaching Strategy If you can’t figure it out,… “read it again and… think harder!”

44. The Value of Teaching With Problems • Problem solving focuses student attention on ideas and sense making. Emerging ideas are integrated with existing ones, thereby improving student understanding.

45. The Value of Teaching With Problems • Problem solving develops the belief in students that they are capable of doing mathematics and that mathematics makes sense. “I believe you can do this!”

46. The Value of Teaching With Problems • Problem solving provides ongoing assessment data. • Plan next lesson • Help individual students • Evaluate progress • Communicate with parents

47. The Value of Teaching With Problems • Problem solving is an excellent method for attending to varying student abilities. Each student gets to make sense of the task using his/her own strategies, as well as getting to hear and reflect on the strategies of others.

48. Value of Teaching With Problems • Problem solving engages students so there are fewer discipline problems. • Problem solving develops “mathematical power.” (problem solving, reasoning, communication, connections, and representations.) • It is a lot of fun!

49. Developing Problems • Use student names in problems. (Ownership, connection) • Choose various problem types. • Select numbers carefully. What are you after? • Use action words! • Keep problems in a file by type. • Make multiple copies for students.

50. Differentiation/Modifications • Same problem type and wording, but change the numbers. • Ask higher children to solve problem, as is, then change their numbers as a “challenge problem”. • Change lower performing child’s numbers first, so child does not get frustrated. It’s still a “challenge”.