Cognitively Guided Instruction (CGI)

“ In the past I thought children didn’t understand subtraction with regrouping, when what they didn’t understand was how to use the process that I was insisting that they use, rather than really understanding the concept of subtraction that might encompass regrouping.” --- Kerry Burkey, Second grade teacher

“ We have learned to gear our instruction Towards what the children know and What they are thinking rather than trying To push a certain method down the children’s throats as far as how they learn to do math. They come up with their own ideas, and you just branch off from that. I’m a decision maker.” -- Ann Badeau, Second Grade Teacher

“ CGI is a philosophy versus a recipe … You as a teacher have to take the knowledge that CGI is about problem types, about solution strategies, about how children develop cognitively, and you have to apply that to your own teaching style.” -- Mazie Jenkins, 1st Grade Teacher

“ It is only when you build from within that you really understand something. If children don’t build from within and you just try to explain it to a child then it’s not really learned. It is only rote, and that’s not really understanding.” -- Ann Badeau, 2nd Grade teacher

“ A CGI classroom is where you build on the math knowledge of your children according to what they know … You don’t build objects that say they should be doing this, this, and this. You sort of take what they know and build on there from there.” -- Susan Gehn, 1st and 3rd grade teacher

“ They’re learning to think. They’re understanding what they’re doing and not just computing. They’re learning to explain how they’re thinking, which is a lot different from what we use to do.” -- Ruth Steiner, 1st Grade Teacher

“ CGI has been a wonderful addition to both my Kindergarten classroom and my 2nd grade classroom. The children have demonstrated a much higher level of conceptual understanding. Thos children who struggled with math are now “beaming” with confidence. Most importantly my children LOVE MATH.” -- Elizabeth Tate, Kindergarten and 2nd grade teacher

“ The students feel successful in math. They arrive at the answer using that strategy that works for them. They love to share how they arrived at their answers. It is a great boost for self confidence. Everyone feels good about themselves. --Judith White, 1st Grade Teacher

“ CGI gets students excited about math. They are solving more difficult problems than ever before. They understand what they are doing and why. My students are surprised when the bell rings for the end of the day because they are actively involved and engaged.” -- 2nd Grade Teacher (Windom Elementary)

“ CGI has changed the way I teach math in my classroom. My students understand problem solving better and can demonstrate how to solve the problems. CGI has made me a better teacher - I listen to my kids explanations rather than always doing it my way. -- Tracie Ford, 1st Grade Teacher

“ I love this philosophy of math because it takes each child at their developmental level and doesn’t hold them back. They are working at extremely high levels of math problems that are real and meaningful.” -- Marti Dougherty, Kindergarten teacher

Days 1 and 2 By Michelle Flaming Cognitively Guided Instruction (CGI)

Day 1 Agenda

Defining Cognitively Guided Instruction

Defining Direct Modeling

Building our knowledge of Problem Types

Introduction to Problem Solving Strategies

My Family

Children’s Mathematics Cognitively Guided Instruction

Built on the belief that learning occurs as new knowledge is linked to existing knowledge, and teaching is most effective when instruction directly builds on what children already know .

Focus: On “student thinking” and how numbers and problems are perceived.

Clock Buddies

12:00 Partner - Someone who has a birthday the same month as you.

3:00 Partner - Someone who has the same number of children as you.

6:00 Partner - Someone who has the same favorite color as you.

9:00 Partner - Someone who has taught close to the same number of years as you.

How best to teach mathematics depends on one’s concept of what it means to know mathematics and the related question - How is mathematics learned by people?

12:00 Partner Solve Roller Coaster Problem:

“ At the fair, there are 36 children in line to ride the roller coaster. The roller coaster has 10 cars. Each car holds 4 children. How many children can sit 3 to a car, and how many have to sit 4 to a car?”

Solve in 2 different ways.

Solve Roller Coaster Problem:

“At the fair, there are 36 children in line to ride the roller coaster. The roller coaster has 10 cars. Each car holds 4 children. How many children can sit 3 to a car, and how many have to sit 4 to a car?

Defining CGI

Watch Video

Take notes in your journal.

Read Introduction and Chapter 1 from the book.

capture “Big Ideas” on poster. 3:00 Partner

Shoot for 10:30 to have your poster on the wall with your names.

8. Karina had 20 cupcakes. She put them into 4 boxes so that there were the same number of cupcakes in each box. How many cupcakes did Karina put in each box? 7. Rodney is having some kids over for jelly donuts. 7 donuts can fit on one plate. How many plates will Rodney need for 28 donuts? 6. 11 children were playing in the sandbox. Some children went home. There were 3 children still playing in the sandbox. How many children went home? 5. Will has 12 crayons. Lucy has 7 crayons. How many more crayons does Willy have then Lucy? 4. Max had some money. He spent $9.00 on a video game. Now he has $7.00 left. How much money did Max have to start with? 3. Janelle has 7 trolls in her collection. How many more does she have to buy to have 11 trolls? 2. TJ had 13 chocolate chip cookies. At lunch she ate 5 of them. How many cookies did TJ have left? 1. Lucy has 8 fish. She wants to buy 5 more fish. How many fish would Lucy have then? Rating #2 Notes: Rating #1 Problem:

Rachel’s Problems – Problem Type Difficulty Rate each of the following problems from 1 (easy) to 5 (most difficult)

1. Using a scale - Rate each problem in comparison with each other. - Record under Rating #1.

2. Working with a partner - directly model each problem, discuss the difficulty if the problem is directly modeled

3. At your table, discuss a mutual ranking. Think about the rationale - Rating #2.

Direct Modeling

Act out the problem.

Follow sequence of the action in a problem.

Represent sets with materials (unifix cubes, tally marks, etc.)

Rachel’s Problems – Problem Type Difficulty Rate each of the following problems from 1 (easy) to 5 (most difficult) 4+ 4 4 8. 4+ 5 5 7. 2+ 3 2+ 6. 3 4 3 5. 3 3 3+ 4. 2 1 2 3. 1+ 1 1 2. 1 1 1 1. Group 6 Group 5 Group 4 Group 3 Group 2 Group 1 Problem:

Rachel’s Problems – Problem Type Difficulty Rate each of the following problems from 1 (easy) to 5 (most difficult)

4. Watch Rachel solve the problems. How difficult of a problem is it for Rachel?

Does Rachel have a plan?

Does Rachel know how to solve the problem?

The amount of “time” to solve a problem doesn’t correlate with difficulty. In fact, to be a TRUE problem, a person should not know how to solve immediately.

“ Justifying and explaining ideas improves students reasoning skills and their conceptual understanding.”

(Maher and Martino 1996)

Viewing Direct Modeling

Watch Ms. Yttri Classroom (Video 5)

Journal Writing:

What ran through your mind as you watched the kindergartners solve the problems?

Problem Type Chart ACTION -- LOOK FOR ACTION VERBS (Start, Unknown) (SSU) ____ - 3 = 9 #4 (Change, Unknown) (SCU) 8 - ____ = 3 #6 (Result, Unknown) (SRU) 12 - 4 = ______ #2 SEPARATE - (Start, Unknown) (JSU) ___ + 3 = 12 (Change, Unknown) (JCU) 7 + __ = 11 #3 (Result, Unknown) JRU 5 + 3 = _____ #1 JOIN +

Part 2 of Problem Type Chart. NO ACTION Compare © (Part, unknown) (PPW-PU) (Whole, unknown) (PPW-WU) Part-Part Whole

Part 3 of Problem Type Chart. Division (Partitive) 12 pumpkins and I want to share them with 4 students. How many pumpkins does each student get? Division Measurement 15 pumpkins and I’ll put 3 pumpkins in each basket. How many baskets do I need? Multiplication 5 x 2 = ____ Multiplication/ Division

“Big Ideas” on Problem Types

Stated action and the relationships form the structure of addition/subtraction word problems and distinguish among them.

The structure of the problem reflects how children solve it.

Both the structure and the location of the unknown quantity determine the relative difficulty of the different problems.

The wording and the number size also determines the relative difficulty of the problem.

Introduction to Strategies

Direct Modeling

Act out the problem.

Follow sequence of the action in a problem.

Represent sets with materials (unifix cubes, tally marks, etc.)

Counting Strategies

Hold one number in the head.

Derived Facts/Facts

Doubles +1, -1, +2, and -2

Sums of 10 (8 + 3 = 8 + 2 is 10 plus 1 more = 11)

Invented Approaches

Introduction to Student Strategies

Watch Video II - Students Addition and Subtraction Strategies.

What is the problem type?

Which strategy did the student use?

D2, #4 - How Would Children Solve These Problems?

Big Ideas on Children’s Solution Strategies

At the most basic level, children use physical objects to directly model the action or relationship in each problem.

As children mature, their strategies become more abstract and efficient. DM is replaced by counting, which in turn are replaced with derived fact and recall of number facts.

Direct Modeling provides a basis for learning of the other strategies.

Children in any class will be at different levels of understanding and will use different strategies to solve the same problems.

Children naturally progress through the levels without direct instruction if children have the opportunity to explain their mathematical understanding.

Common Components of CGI Classrooms

Problem solving is the focus on instruction; teachers pose a variety of problems.

Many problem solving strategies are used to solve problems. Children decide how they should solve each problem.

Children communicate to their teachers and peers how they solved the problems.

Teachers understand children’s problem solving strategies and use that knowledge to plan instruction.

Reflection Question

“ Big Idea” poster - add new knowledge

What is one think that has become clear to you? What is something that still confuses you?

How does CGI fit into the other goals from your district?

Assignment

Read Chapter 2 and 3 in Children’s Mathematics.

Write a set of problems based on a theme in your classroom.

Day 2 Agenda

Sharing written problems.

Multiplication/Division Problems

Multi-digit Addition and Subtraction

Video IV (Children’s Invented Approaches)

Classroom Implementation

State Standards Connection

Multiplication/Division Types

Introduction - Video (7) of Craig Meyer - pg 16

Which of these problems are alike in terms of how they would be directly modeled? (#11)

Multiplication

Division Partitive

Division Measurement

1. Megan has 45 stickers. 10 stickers fit on a page. How many pages can she fill with her stickers?

2. There are 3 pockets on my dress. I have 10 buttons on each pocket. How many buttons do I have altogether.

3. Kevin has 44 trucks. He can carry 10 trucks at a time. How many trips does he need to take to carry all of his trucks from the front to the back yard?

4. Abubu has 50 cookies. He will share them equally with 10 friends. How many cookies will each friend get?

5. Annie has 70 pencils. She uses 10 pencils a week. How many weeks will her pencils last?

6. 40 kids want to play team tag. They will make 10 teams. How many kids will be on each team?

7. I each 10 candies each day. How many candies will I eat in 6 days?

Multiplication/Division

Ginny Koberstein Video (6) - 1st Grade

Problem Identification and Strategies - Base Ten

Read pages 44-45 from Children’s Mathematics

Read pages 59-63 from Children’s Mathematics

Big Ideas on Multiplication/Division

Multiplication/division problems are appropriate for beginning problem solvers.

The progression of use of various strategies for solving multiplication/division problems is similar to that seen with addition/subtraction.

Children deal with remainders by considering the context of the problem.

Understanding multiplication/division problem types and solution strategies is the basis for understanding our base-ten system, which is vital for understanding multi-digit addition.

Multi-digit Number Concepts

In order to understand and work with multi-digit numbers, children must learn that collections of tens (or hundreds or thousands) can be counted and talked about just like individual units.

Learning about groups of ten can be done in the context of multiplication and measurement division problems and by working directly with objects grouped by tens.

Children invent algorithms for solving multidigit number problems after they have had many experiences solving problems with single and multidigit numbers and working with groups of ten.

Children should have many experiences solving problems that involve multidigit numbers even if their understanding of working with groups of ten is limited.

Teachers can facilitate children’s invention of multi-digit algorithms by providing many experiences with grouping by ten, by asking children to solve many problems that involve groups of ten, and by helping children recognize the ease of using tens in solving problems.

Solution strategies constructed by children to solve multidigit number problems have direct parallels with and are natural extensions of the strategies they construct for single-digit numbers.

Watching Children Solve Problems - Video 4

Children can and should solve problems with large numbers even if their understanding of base-ten is limited.

Students can practice counting as they solve the problems.

Children need not and should not be deprived of problem-solving experiences just because their counting skills are not yet well developed.

Work with Partner

Solve the following problem: 74 + 39

Solve and record using:

Combining Strategy

Incremental Strategy

Compensation Strategy

Exploring Different Strategies

Work in a small group.

Generate as many strategies as you can for your problem.

Kevin has 294 baseball cards. He wants to have 723 baseball cards. How many more cards does he need to collect?

The zookeeper gave the elephant 723 peanuts. She ate 294 of them. How many peanuts were left?

The giant pumpkin had 723 pumpkin seeds in it. The medium pumpkin had 294 pumpkin seeds in it. How many more pumpkin seeds were in the giant pumpkin than in the medium one?

Select a spokesperson. This person will stay at your poster to discuss the strategies.

The other members will go to another poster to listen to the strategies.

723 - 294 =

Many Ways to Get to the Same Place:

MAPPS Activity - Number Sense. 28 + 29

Step 1: Become familiar with your procedure by trying it out. Making up some more problems for yourself in order to develop facility with this approach.

Step 2: Discuss with each others in the group why the method works. You may want to use words, manipulatives, diagrams, or any combination of these.

Beginning to Use CGI Instruction

What kind of problems did the teacher ask the children to solve?

What did the teachers learn about the children as the reported their solution strategies?

What questions did the teacher ask to elicit children’s solution strategies, and what did he/she find out?

How did the teacher ensure that all children were solving appropriate problems?

State Standard Connection

How does CGI support the indicators from the state standards?

Look at the indicators for your grade level and the grade level below. Which indicators can be addressed using the CGI instructional strategy? Highlight.

Read the “Teacher Note” S1, B4

Remember ….

You do not have to master the entire research-based framework to begin.

Children can and will learn many important math ideas when many problems are solved and strategies are shared.

Try some CGI problems with your class.

Listen to students share with each other and the class.

Look for different strategies and listen for mathematical understanding.

Day 3 Agenda

Review of Problem Types and Strategies

Your Classroom Experience

Judging Relative Difficulty of problems

pg 50

Multi-digit Addition and Subtraction

Classroom Components - Viewing

Review of Problem Types

Find the person in the room with the same problem type as you.

Recreate the problem type chart with your partner, without the help of notes.

Identify Problem to be Identified: Sample Set - page 12 in handouts

Review of Strategies

Find a Problem for a Strategy - pg 10.

Determine the strategy the student is using.

Direct Modeling, Counting, Derived Fact (sums of 10 or doubles)

Decide which problem each child is solving (JCU - Join Change Unknown)

Determine the numbers in the problem.

Example JCU (4,16)

Review Strategies

Charades in small group

Two at a time:

1. Pull a question to ask.

2. Pull a strategy to model.

The rest of your group determine the problem type and the strategy.

Your Experience with CGI

What do you know about your students?

What is their mathematical understanding?

What did you try?

How did your expectations compare to what happened?

Did you experience any surprises? If so, what do you think contributed to this result?

Judging Relative Difficulty

For each pair of problems, indicate if you believe Problem A or Problem B is more difficult for 1st graders. Circle the more difficult one.

If you believe they are approximately the same difficulty, Circle E.

Assume the problems are read outloud and reread as often as necessary.

Manipulatives are available and don’t worry about the amount of time.

Classroom Components

What kind of problems did the teacher ask the children to solve? How did the teacher ensure that all children were solving appropriate problems? How does the teacher pose the problem?

What did the teachers learn about the children as the reported their solution strategies? What questions did the teacher ask to elicit children’s solution strategies, and what did he/she find out?

What does an active CGI teacher do? Who’s doing the math?

What are the similarities and differences amongst the teachers?

Sharing

With your group create a visual/auditory way to share with the group what you found out about your question from the classrooms.

Be ready by 1:40.

State Standard Connection

Look at the indicators for your grade level and the grade level below. Which indicators can be addressed using this instructional strategy.

State Standard/Assessment Connection with CGI

Look at the indicators for 1st, 2nd and 3rd grades. Which indicators can be addressed using the CGI instructional strategy? Highlight.

Read the “Teacher Note” S1, B4

What might be some of the common values between the state and CGI?

Day 4 Agenda

Principles and Standards for School Mathematics Connections.

Building a Base Ten Knowledge

Recording Observation Examples

Textbook Connection

Your CGI Journey

Base Ten Understanding

Emerges slowly.

Algorithms are often misunderstood if taught before base-ten concepts are developed.

Tonight -

Read pages 63 - 70 (Invented Algorithms)

Take notes and be ready to discuss with partner in the morning.

Reread pages 59 - 63 if needed.

What is base-ten understanding?

What tasks and activities do you use to facilitate its development?

Using Multiplication/Division Problems as Starting Points

Video 6 - Ginny Koberstein (Grade 1)

Maren, Casey, Emma, Zack, and Clayton

Video 6 - Karen Falkner (Grade 2)

Nick, Angie, Jacob

One person at each table, focus on one of the children in the class on the tape.

What is the child’s base ten understanding?

Building a Base-Ten Understanding

Read and analyze Chapter 6 with partner.

List important ideas that children need to understand about base-ten.

Following this, the entire group will compile one master list.

In small groups:

use the master list to write an interview for assessing a child’s understanding.

choose a trade book from your group to use for content.

Multi-digit Number Concepts -

In order to understand and work with multi-digit numbers, children must learn that collections of tens (or hundreds or thousands) can be counted and talked about just like individual units.

324 (3 hundreds, or 30 tens, or 300 ones)

Learning about groups of ten can be done in the context of multiplication and measurement division problems and by working directly with objects grouped by tens.

Children invent algorithms for solving multidigit number problems after they have had many experiences solving problems with single and multidigit numbers and working with groups of ten.

Children should have many experiences solving problems that involve multidigit numbers even if their understanding of working with groups of ten is limited .

Teachers can facilitate children’s invention of multi-digit algorithms by providing many experiences with grouping by ten, by asking children to solve many problems that involve groups of ten, and by helping children recognize the ease of using tens in solving problems.

Solution strategies constructed by children to solve multidigit number problems have direct parallels with and are natural extensions of the strategies they construct for single-digit numbers.

Watching Children Solve Problems - Video 4

Children can and should solve problems with large numbers even if their understanding of base-ten is limited.

Students can practice counting as they solve the problems.

Children need not and should not be deprived of problem-solving experiences just because their counting skills are not yet well developed.

80 - 20 -- 60 + 4 --64 -4 = 60 - 5 - --55

Use many different models: base-ten blocks, number lines, invented approaches.

Generate as many strategies as you can for your problem.

Kevin has 294 baseball cards. He wants to have 723 baseball cards. How many more cards does he need to collect?

The zookeeper gave the elephant 723 peanuts. She ate 294 of them. How many peanuts were left?

The giant pumpkin had 723 pumpkin seeds in it. The medium pumpkin had 294 pumpkin seeds in it. How many more pumpkin seeds were in the giant pumpkin than in the medium one?

Select a spokesperson. This person will stay at your poster to discuss the strategies.

The other members will go to another poster to listen to the strategies.

“ Justifying and explaining ideas improves students reasoning skills and their conceptual understanding.”

(Maher and Martino 1996)

Recording Options

Individual Student Observations

Class Summary

Current Textbook

How do we integrate our textbook?

Look at your materials.

What parts of the textbook would you use to promote student understanding of numbers?

Which lessons from the textbook align with assessed questions?

How is your textbook similar or different from CGI?

How does your actions (following the lessons in the textbook) compare to your expectations for students?

Several levels of beliefs and practice in becoming a CGI teacher.

Level 1 Teachers -

Believe that children need to be explicitly taught how to do mathematics.

Instruction is usually guided by an adopted text and focuses on the learning of specific skills.

Teachers demonstrate the steps in a procedure as clearly as they can and students imitate.

Children are expected to solve problems using standard procedures, and there is little or no discussion of alternative solutions.

Level 2 Teachers -

Begin to question whether children need explicit instruction in order to solve problems.

Teachers provide opportunities for children to solve problems using their own strategies.

Teachers begin to show the children specific methods.

Level 3 Teachers - A Turning Point

Believe that children can solve problems without having a strategy provided for them.

They do not present procedures for children to imitate.

Children spend most of the class solving and reporting their solutions to a variety of problems.

Classrooms are characterized by students talking about mathematics, both to other students and to the teacher.

Children report a variety of strategies and compare and contrast different strategies.

They know appropriate problems to pose and questions to ask to elicit children’s thinking, and they understand and appreciate the variety of solutions that children construct to solve them.