## CGI - An Approach to Teaching Mathematics

**What is CGI Math?**

Cognitively Guided Instruction (CGI) is an approach to teaching mathematics that uses a student’s own mathematical thinking as the basis for instruction. The method is the result of research conducted by Elizabeth Fennema and Thomas P. Carpenter from the University of Wisconsin-Madison in the late 80's and early 90's.

Its premise is based on the underlying assumption that students already posses an informal knowledge of mathematics. The teachers role, is to build from this prior knowledge so that students can eventually make connections between situational experiences and the abstract symbols typically found to represent them in mathematical equations (+, -, x and so forth.) This approach is quite different from the traditional method of teaching the symbolic computation first, and then expecting students to apply the concepts to problem solving situations.

In CGI, the emphasis is on understanding and application, rather than rote memorization of algorithms, and is strongly aligned with the rigorous expectations of the Common Core Standards for Mathematical Practices.

**Lesson Design**

Children in a CGI math classroom spend most of their time solving story problems and discussing problem solving strategies. A typical problem might look something like this:

**Megan has 5 bags of cookies. There are 7 cookies in each bag. How many cookies are there altogether?**

Students are encouraged to use and develop a variety of self-selected strategies and models to solve the problems. They are also held accountable for explaining exactly how they solved the problem. The teacher then uses this information to guide the student learning and identify any breakdown in understanding that might occur.

Engaging in this type of ongoing student assessment provides teachers with the opportunity to:

- identify what base knowledge a child has.
- understand different strategies employed for solving problems.
- understand what those strategies tell us about the child's thinking.
- understand and demonstrate the ways that this thinking evolves.

## CGI In Action

## Problem Types

Math story problems can be classified into various problem-types using the CGI approach. Read on to learn more about the different types of math story problems. After a brief description of each problem type, there are several math story problems provided for example.

**Addition Problems**

Addition problems involve a direct or implied action in which a quantity is increased by a particular amount. They are sometimes referred to as "Joining" problems in CGI literature. There are three types of basic addition problems:

1. Result Unknown 15+32+___

2. Change Unknown 15+___=47 (a slightly more difficult problem)

3. Start Unknown ___+32=47 (the most difficult of the basic addition problems)

Below are some sample story problems for each type. These problems can be changed to include familiar names, actions or situations with your students. Be creative! The numbers in the problems can also be altered, depending on the child's mathematical abilities.

**Result Unknown:**

- You have 4 books to read. You get 3 more books from the library. How many books do you have now?
- Robin has 98 toy cars. Her parents gave her 45 more toy cars for her birthday. How many toy cars will she have then?
- There are 320 boys in a club. 29 girls join the club. How many children are in the club altogether?
- Jane gave 10 pieces of candy to Sam. Fred gave Sam 6 pieces of gum. Mary gave Sam 12 pieces of gum. How many pieces of gum does Sam have now?

**Change Unknown**

- Kristin had 17 apples. How many more apples will she need to have 39 apples all together?
- Eric has 72 golf balls. He finds some more in the basement. Now Eric has 183 golf balls. How many golf balls did Eric find in the basement?
- Columbus saw 25 whales in the ocean. How many more whales will he need to see to make it 3 dozen whales all together?
- There were 450 kids in the schools. Some more kids came to the school. Now there are 920 kids at the school. How many more kids came to the school?

**Start Unknown**

- Clifford has some bones. Emily gave him 23 more bones. Now Clifford has 46 bones. How many bones did Clifford have to start with?
- Mrs. Smith saw some ducks. Then she saw 156 more ducks. All together, Mrs. Smith saw 234 ducks. How many ducks did she see to begin with?
- Mom had baked some cookies. Then she baked 4 dozen more cookies. All together, mom has baked 58 cookies. How many cookies did mom bake to begin with?
- Jeff had some leaves in a bag. Then he found 16 more leaves, and put those in the bag. Now Jeff has 76 leaves. How many leaves did Jeff find to begin with?

**Subtraction Problems**

Subtraction problems involve action in which the initial quantity is decreased over time. They are sometimes referred to as "Separating" problems in CGI literature. There are three types of basic subtraction problems:

1. Result Unknown 12-6+___

2. Change Unknown 12-___=6 (slightly more difficult problem type)

3. Start Unknown ___-6=6 (the most difficult of the basic subtraction problems)

Below are some CGI sample story problems for each type. These problems can be changed to include familiar names, actions or situations with your students. Be creative! The numbers in the problems can also be altered, depending on the child's mathematical abilities.

**Result Unknown**

- Jessica had 14 toy cars. She lost 11 of the cars. How many cars does she have left?
- Melissa had 78 pumpkin seeds. She gave 23 of them to her brother. How many pumpkin seeds does Melissa have left?
- There were 156 ornaments on the tree. 120 of the ornaments fell off the tree. How many ornaments are left on the tree?
- Dan has 386 papers in his desk. He threw away 17 of them. How many papers does Dan have left in his desk?

**Change Unknown**

- Mikey had 16 leaves. Some of her leaves blew away. Now she has 7 leaves left. How many of her leaves blew away?
- Sara had 99 goldfish. She gave some away. Now she has 32 goldfish left. How many goldfish did Sara give away?
- 320 children wore their Halloween costumes to school. Some of the children took their costumes off at recess. There were 210 children left wearing costumes. How many children took their costumes off at recess?
- There were 42 cookies on a plate. Dad ate some of the cookies. Now there are 29 cookies left on the plate. How many cookies did Dad eat?

**Start Unknown**

- Ashley had some guppies. She gave 28 guppies to Timmy. Now Ashley has 12 guppies left. How many guppies did Ashley have to start with?
- Ryan had some trick-or-treat candy. He gave 123 pieces of candy to his sister. Now Ryan has 100 pieces of candy left. How many pieces of candy did Ryan have to begin with?
- Mrs. Smith had some pencils. She lost 24 of them. Now she has 6 left. How many pencils did Mrs. Smith have to begin with?
- A clown had some balloons. 56 of the balloons floated away. Now the clown has 45 balloons left. How many balloons did the clown have to start with?

**Multiplication Problems**

Grouping and Partitioning problems involve three distinct quantities, as illustrated by the following examples of math story problems:

*Megan has 5 bags of cookies. There are 3 cookies in each bag. Altogether, Megan has 15 cookies.*

The three quantities in the problems are: the number of bags, the number of cookies in each bag, and the total number of cookies. In a problem, any one of the three quantities can be an unknown. When the total number of cookies Megan has all together is unknown, the problem is a Multiplication problem.

Below are some examples of math story problems for each type. These problems can be changed to include familiar names, actions or situations with your students. Be creative! The numbers in the problems can also be altered, depending on the child's mathematical abilities.

- Mrs. Smith has 4 boxes of candy. There are 10 pieces of candy in each box. How many pieces of candy does she have now?
- Tommy has 4 packages of Pokemon cards. There are 7 cards in each package. How many cards does Tommy have altogether?
- Our classroom has 6 jars. There are 8 butterflies in each jar. How many butter flies are in our room altogether?
- Farmer Ted has 10 hens. There are 6 eggs under each hen. How many eggs are there altogether?

**Division Problems**

As we learned above, Grouping and Partitioning problems involve three distinct quantities. This is illustrated in the example:

*Megan has 5 bags of cookies. There are 3 cookies in each bag. Altogether, Megan has 15 cookies.*

To refresh, the three quantities in the problems are: the number of bags, the number of cookies in each bag, and the total number of cookies. In a problem, any one of the three quantities can be unknown. In a Measurement Division problem, the total number of objects and the number of objects in each group is given. What is unknown, is the number of groups (in the example, this would be the number of bags of cookies.)

Here are some examples of Measurement Division problems:

- Meagan has 15 cookies. She puts 3 cookies in each bag. How many bags can she fill?
- Karen has 18 Gummy Worms. She puts 3 in each Halloween treat bag. How many bags will she fill?
- Room 124 is going on a hayride. There are 27 kids in the class. 9 kids can fit on a wagon. How many wagons must they take?
- The hens laid 20 eggs. There are 4 eggs under each hen. How many nests are there in the hen-house?

In a Partitive Division problem, the total number of objects and the number of groups is given. The number of objects in each group (in our case, the number of cookies per bag) is unknown.

Here are some examples of Partitive Division problems:

- Megan has 15 cookies. She puts the cookies into 5 bags, with the same number of cookies in each bag. How many cookies are in each bag?
- Karen has 16 caterpillars. She puts the caterpillars into 4 jars, with the same number of caterpillars in each jar. How many caterpillars are in each jar?
- Mom has 12 marshmallows. She puts the marshmallows on 4 sticks, to roast. How many marshmallows are on each stick?
- Mrs. Smith has 20 stickers. She gives them to 10 children, so that they each have the same amount. How many stickers did each child get?