Cognitively Guided Instruction in Mathematics- CGI

1. Cognitively Guided Instruction in Mathematics-CGI September 24, 2008 Randolph County Schools

2. CGI Review Sessions Session 1- September 24 What CGI is and is not Getting started Introducing problems When and how to do CGI Developing problems for students Video segments

3. CGI Review Sessions Session 2- October 8 Developing problems Problem collections Differentiation/ Modifications Student sharing Video segments

4. CGI Review Sessions Session 3- November 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. 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. 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.

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

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

8. CGI Classrooms CGI 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.

9. 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. • 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.

10. 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.

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

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

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

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

15. Tool Boxes

16. 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

17. CGI-Modeling Word Problems Activity Model each of the following five word problems with cubes or counters (concrete representation). Afteryou have acted out the problems with manipulatives, write a number sentence for each one.

18. Problems • Lydia has 7 candies. She eats 2 of them. How many does she have left? • Lydia has 7 candies and Juan has 2 candies. How many more candies does Lydia have? • Lydia has 2 dollars. She wants to buy something that costs 7 dollars. How many more dollars does she need? • Yesterday Lydia had 7 balloons. Some of them burst last evening. Today she has 2 left. How many balloons burst? • Yesterday Lydia had some balloons. Today Juan gave her 2 more balloons. Now she has 7 altogether. How many balloons did she have yesterday?

19. Discussion How are these five problems alike? How are they different?

20. 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. • Use numbers in the problems that align with children’s number development. • Make actions/ relationships as clear as possible.

21. 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 counting on, skip counting, or use of derived facts to encourage children to move to more sophisticated solution strategies.

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

23. 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?

24. Vocabulary What is the danger of teaching children key words? According to Van deWalle, 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. Left? Altogether? Fewer?

25. 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 ______? ( doing a good estimation )

26. What’s next? • Start small. • Use what you already know and what you have learned today. • Pay attention to what you really see 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!

27. CGI- Session 2 Overview • Revisit the toolbox • Developing problems • Differentiation/ Modifications • Keeping students engaged • Student sharing • Sample rubric • Looking at your problems

28. The Toolbox • Include materials you already have. • Use a variety of materials in each tool box. • Children share materials, so there is no need for each child to have their “own” except items such as pencils. • Tools change throughout the year. Start small and add or change.

29. 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?

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

31. Toolbox Basics- Continued • Materials that can be used to represent place value, such as base 10 blocks, 10 frames, bean sticks, snap cubes • Items for measurement: rulers, tape measures • Money- Caution! • Materials for sharing such as overhead transparency (1/2 sheet, teacher keep vis-à-vis until needed)

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

33. Common Teaching Strategies 1.Key Words- Example: altogether means add 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?

34. Common Teaching Strategies 2. If you can’t figure it out,… “read it again and think harder!”

35. 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.

36. 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!”

37. The Value of Teaching With Problems • Problem solving provides ongoing assessment data. Students provide a steady stream of valuable information which can be used for planning the next lesson, helping individual students, evaluating progress and communicating with parents.

38. The Value of Teaching With Problems • Problem solving is an excellent method for attending to a breadth of 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.

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

40. Developing Problems • Choose children to include in problem. (ownership, connection) • Choose problem type. (Remember not to stay on same type problem too long, but not to jump around too fast) • Choose numbers carefully. What are you after? • Use action words! • Keep problems in computer folder by type. • Make multiple copies for students.

41. Unpacking the Problem • Meet as a whole group (without tools or journals). • Read and discuss the problem together. • Have children estimate. Is __ a reasonable answer? Why? Why not? • Reread problem. • Send children to their desks to begin work.

42. Differentiation/Modifications • Same problem type and wording, but change the numbers, higher or lower. • 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.

43. Differentiation/Modifications • Allow challenge problem solvers to work together, noting the conversations they have. • Remember that when facing a new type of problem, even the higher achieving student will often move back to lower solution levels, such as direct modeling.

44. The Teachers’ Role • Circulate as children work, making few comments, but offering hints through questioning. • Listen and question actively. • Record specific student responses. • Note children who are to share, based on strategies. • Note children who struggle. Do not try to “teach the right way” at this time.

45. Sharing and Assessing • THIS IS MOST IMPORTANT! Do not omit! • Teachers move, watch, ask, listen, and record! • Ask questions and listen. Remember: They can only do what they understand. • Teacher makes notations. • Choose students to share. • Discuss right answer, but emphasis is on the process. • Later in the year, you may choose quality work samples to post.

46. Sharing and Assessing • Choose child with lowest solution strategy to share first, so they have something to say. • Sharing is the student’s responsibility. Teacher again is facilitator only. • Children then ask for questions and are allowed to repeat their explanations. • Children ask for comments.

47. Sample Rubric • Strategy : -Solve the problem your way. - Be efficient. - Use counting strategies. • Symbols: -Tally marks - Lines or circles - Be neat. • Sentences: –Capitals, spaces, periods - Tell what you did first. - Tell what you did next. • Share: – Share your strategy. - Talk about math. - Use quiet voices.

48. 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?

49. Next Time November 13 Analyzing student work Developing the next lesson Moving children to higher levels of development Homework: Bring 2 student samples to look at, and your documentation about their work.

50. Sample Problem The tooth fairy collected 6 (six) teeth. How many more does she need to collect to have 15 (fifteen) teeth? Direct modeling: Use of manipulatives to represent every part of problem. Counting Strategies: More efficient Counting such as counting on Don’t have to represent everything Derived Facts: Mental Math Inventive strategies Doubles, fives, tens, place value