TEACHER'S GUIDE
Cognitively Guided Instruction
A Culturally Sensitive Approach for
Teaching Mathematics to All Children

Cognitively Guided Instruction
Cognitively Guided Instruction is a primary level approach to teaching mathematics that was developed at the Wisconsin Center for Education Research, Madison, Wisconsin. This extensively researched approach strives to provide teachers with opportunities to understand how young children develop mathematical thinking. Possessing this knowledge enables teachers to plan instruction that guides students toward learning important mathematical skills - with understanding.

Developing Mathematical Reasoning and Whole-Number
Knowledge Using Word Problems
Cognitively Guided Instruction uses word problems to develop mathematical thinking. These problems are based on detailed analyses of various problem situations involving number. The word problems are organized within a framework that makes it possible to distinguish among problems in terms of difficulty. This knowledge enables teachers to sequence word problems from easiest to most difficult. Such sequencing promotes the development of mathematical reasoning.

Understanding the Structure of Word Problems
The information presented within The Teacher’s Guide is provided to help teachers understand the problem-difficulty sequence and to decide which problems are appropriate for individual children.

The chart on page 62 provides examples of the14 different types of word problems. Number sentences have been included to help teachers identify the missing quantity and determine the relative difficulty among problems. The problems that are easiest for young children to solve have been indicated with an asterisk.

CGI Resource Information
A text describing this approach, Children’s Mathematics: Cognitively Guided
Instruction, is available through the Heinemann website: http://www.heinemann.com

CGI website: http://www.abacon.com/ie/berk/wlp452a.htm

PROBLEM SOLVING SITUATIONS

Problem Chart based on Cognitively Guided Instruction Problem Types. Carpenter et al.,1996

Building on Children’s Intuitive Understanding

A goal of Cognitively Guided Instruction is that young children become independent problem solvers who are able to approach and solve word problems without having to rely on the teacher’s telling them how to do it. A number of factors influence whether a problem is more or less difficult for a child to solve independently. Understanding these factors helps the teacher decide on the word problems to be used during instruction. These factors include the following:

Whether the Problem Involves an Action Situation that the Child Can Act Out

A problem that can be acted out is easier for a child to solve than one that cannot be acted out. For example, the first of the following two problems is easier. Here the child can actually pretend to give the strawberries away. The second problem is more difficult because the problem cannot be acted out directly.

SRU Problem (action direct):
Marlon had 8 strawberries. He gave 3 of them to Anna.
How many strawberries does Marlon have now?

SRU Problem (action indirect):
Marlon gave 3 strawberries to Anna. He started with 8 strawberries.
How many strawberries does Marlon have now?

Whether a Problem Can Be Modeled or Acted Out in the
Order in Which It Is Heard

Young children, when first learning to solve word problems, approach them in the order in which they hear them. They do not begin at the end of the problem and work backward. The ability to do this develops after children have had many experiences with solving problems and have developed an understanding of the relationships among the numbers within a problem. For example, they understand that combined parts make up the total. For this reason, the first of the following two problems is easier. The wording encourages the child to set out five counters and then add eight more. However, the second problem does not specify a beginning number. The young child who has not yet developed the ability to relate a part of a quantity to the total quantity will respond to the second question with, “Some. Anna had some strawberries.”

JRU Problem (can be solved in order given):
Anna had 5 strawberries. Marlon gave her 8 more.
How many strawberries does Anna have now?

JRU Problem (cannot be solved in order given):
Anna had some strawberries. Marlon gave her 8 more. Then she had 13 strawberries. How many
strawberries did Anna have before Marlon gave her any?

Whether the Problem Can Be Directly Modeled

When all the quantities given in a problem refer to complete sets of physical objects or amounts, the problem can be modeled directly. A word problem that can be directly modeled, that is, represented in some concrete way (on fingers, by manipulating counters, with tally marks or drawings), is easier.

The first of the following two problems is easier because each quantity can be set out or modeled directly. When modeling this problem with counters, a young child will count out the two quantities, lining them up side by side, and match the rows to determine the difference.

In the second problem the 3 does not refer to a quantity that exists in the situation, That is, neither Marlon or Anna has 3 strawberries. So, the 3 cannot be set out or modeled directly when setting up the problem situation. To solve this type of problem, the child must have developed the ability to mentally visualize and analyze the problem.

CDU Problem (direct modeling situation):
Marlon has 8 strawberries. Anna has 5 strawberries.
How many more strawberries does Marlon have?

CRU Problem (cannot be modeled, directly analysis required):
Anna has 5 strawberries. She has 3 fewer strawberries than Marlon.
How many strawberries does Marlon have?

Whether the Unknown Quantity Is Located at the End, Middle, or Beginning

Because young children solve problems in the sequence that they hear them, problems that are worded in such a way so that the unknown quantity is located at the end (first example below) are easier to solve. Problems with the missing quantity in the middle (second example below) or at the beginning (third example below) are more difficult.

As the child’s understanding of quantity and relationships among quantities develops, the child will be able to make sense of the entire problem, represent the situation, and plan a solution. When a child is able to do these steps, the child will not need to use manipulatives. Rather, the child will mentally manipulate quantities.

SRU Problem (location of unknown at end of problem):
Anna had 7 strawberries. She gave 4 to Marlon. How many strawberries does Anna have now?

SCU Problem (location of unknown in middle of problem):
Marlon had 8 strawberries. He gave some to Anna. Now he has 5 strawberries.
How many strawberries did Marlon give to Anna?

SSU Problem (location of unknown at start of problem):
Marlon had some strawberries. He gave 3 strawberries to Anna. Then he had 5 strawberries left. How many strawberries did Marlon have before sharing with Anna?

Simple Multiplication or Division Problems that Can Be Acted Out or Modeled

Very young children can solve small number multiplication and division problems because such problems can be easily modeled. However, it is important that children first solve many problems involving joining and separating situations. These experiences will allow them to develop the ability to think about numerical quantities within the context of situations and to make sense of the question being asked.

Multiplication problem:
Anna has 4 piles of strawberries. There are 3 strawberries in each pile. How many strawberries does Anna have?

Measurement Division problem:
Anna gave12 strawberries to some children. She gave each child 3 strawberries. How many children got strawberries?

Partitive Division problem:
If Marlon gives 12 strawberries to 3 friends and each friend is given the same number, how many strawberries will each friend get?

Developmental Levels of Children's Thinking
Revealed in Intuitive Solution Strategies

The extensive research on which Cognitively Guided Instruction is based documented developmental thinking processes that children go through when learning to solve word problems. It is important to emphasize that these processes are intuitive, ones that are not
taught to the student by a teacher.

To effectively promote students’ learning, a teacher must clearly understand the relationships among the different types of word problems discussed in the previous sections and the developmental levels of children’s thinking. Most children pass through three levels when acquiring problem-solving skills. Initially, they solve problems using modeling strategies. Over time, modeling strategies are replaced by counting on or back strategies. Finally, most children come to rely on number facts. Detailed descriptions of how children’s solutions vary depending on the problem that they are solving and their developmental ability are provided in the following sections.

Relating Solution Strategies to Developmental Stages

Level 1: Examples Involving Direct Modeling

A child using a Direct Modeling strategy represents each set of numbers with concrete objects. In the following examples, the child models with counters.

Level 2: Examples Involving Counting Strategies

A child using a counting strategy holds a number in her/his mind and counts on or back from that number keeping track of the quantity of objects added on or removed with fingers or tally marks.
A child at this level must be able to immediately recognize amounts modeled on fingers without having to recount the fingers (group recognition). Group recognition is required so that the child knows when to stop counting on or back.

Level 3: Examples Involving Deriving Strategies

A child possessing reasoning ability and good number sense is able to solve problems in flexible ways, often breaking numbers down and recombining them by using known facts. This child frequently visualizes the quantities and solves the problem with mental math.

Matching Solution Strategies to Problem Situations

The actions and decisions that a child makes when solving a problem are determined by the problem posed to the child and the child’s level of development. Each of the 14 different problem situations requires different reasoning processes. Examples are given below.

Strategies Used by Level 1 Children (Direct Modelers) for Specific Problem Situations

At this level, the child concretely represents (using counters, fingers, tally marks, drawings) all numerical quantities within the problem.

JRU Problem

Marlon had 3 strawberries. Anna gave him 5 more. How many strawberries does Marlon have now?

Joining-All Solution:
The child constructs (with manipulatives) a set of three objects and a set of five objects. The child pushes the sets together and the union of the two combined sets is counted.

JCU Problem

Anna has 5 strawberries. Marlon gave her some more strawberries.
Now Anna has 8 strawberries.
How many strawberries did Marlon give her?

Joining-To Solution:
The child constructs (with manipulatives) a set of three objects. Objects are added to this set until there is a total of eight objects. The child counts the number of objects that were added to find the answer.

SRU Problem

Marlon had 8 strawberries. He gave 3 to Anna.
How many strawberries does Marlon have now?

Separating-From Solution:
The child constructs (with manipilatives) a set of eight objects. Three objects are removed. The answer is the number of remaining objects.

SCU Problem

Anna had 8 strawberries. She gave some to Marlon.
Now Anna has 3 strawberries.
How many strawberries did he give to Marlon?

Joining-To Solution:
A set of eight objects is counted out. Objects are removed from it until the number of objects remaining is equal to three. The answer is the number of objects removed.

CDU Problem

Marlon has 3 strawberries. Anna has 5 strawberries.
How many more strawberries does Anna have than Marlon?

Matching Solution:
A set of three objects and a set of eight objects are matched one to one until one set is used up. The answer is the number of unmatched objects remaining in the larger set.

JSU Problem

Anna had some strawberries. Marlon gave him 3 more.
Now he has 8 strawberries.
How many strawberries did Anna have to start with?

Trial-and-Error Solution:
A set of objects is constructed. A set of three objects is added to or removed, and the resulting set is counted. If the final count is eight, then the number of objects in the initial set is the answer. If it is not right, then a different initial set is tried.

Strategies Used by Level 2 Children (Counting On/Back) for Specific Problem Situations

At this level, the child does not have to represent all quantities in the problem concretely. She or he has learned that a number names a quantity, that is, that a number can be stated rather than represented concretely. These strategies will develop intuitively over time. If a child is not able to make sense of counting strategies, the child is not developmentally ready to use them and needs many more experiences with modeling solutions.

JRU Problem

Marlon had 3 strawberries. Anna gave him 5 more strawberries.
How many strawberries does Marlon have now?

Counting-On-From-First Solution:
The counting sequence begins with three and continues on for five more counts.
The answer is the last number in the counting sequence.

JRU Problem

Marlon had 3 strawberries. Anna gave him 5 more strawberries.
How many strawberries does Marlon have now?

Counting-On-From-Larger Solution:
The counting sequence begins with “five” and continues on for three more counts.
The answer is the last number in the counting sequence.

JCU Problem

Anna had 3 strawberries. Marlon gave her some more
strawberries. Now Anna has 8 strawberries.
How many strawberries did Marlon give her?

Counting-On-To Solution:
A forward counting sequence starts from “four” and continues until “eight” is reached (keeping track on fingers or tally marks). The answer is the number of counting
words in the sequence represented by the number of fingers or tally marks.

SRU Problem

Marlon had 8 strawberries. He gave 3 to Anna.
How many strawberries does Marlon have now?

Counting-Down Solution:
A backward counting sequence is initiated from eight. The sequence continues
for three counts . . . eight, seven, six . . . the answer is the next number.

SCU Problem

Anna had 8 strawberries. She gave some to Marlon.
Now she has 3 strawberries.
How many strawberries did she give to Marlon?

Counting Down To Solution:
A backward counting sequence starts from “eight” and continues until three is
reached., eight, seven, six, five, four. . . .” The answer is the total of number
words in the counting sequence, not including the number word three.

Strategies Used by Level 3 Children (Deriving and Number Facts)

At this level, the child understands relationships between numbers and solves problems using number facts and derived facts, combining numbers to the problem when a specific fact is not at the recall level. An example of a child using a derived fact would be, “I know that nine plus four is thirteen because nine and one is ten and three more is thirteen.”

Solution Strategies Summary
When children begin to solve problems intuitively, they concretely represent the relationships in the problem. Over time, concrete strategies are abstracted to counting strategies, and eventually, as number facts are learned, children apply this knowledge to solve problems. This developmental approach differs from the practice of rote drill for memorization of facts. Children in traditional classrooms often are able to recite facts but lack understanding that a fact represents a relationship between quantities; they lack number sense. Children who have been allowed to progress through the stages described in this section come to understand these relationships.

Symbolic Procedures
Much of what has been discussed to this point has focused on children’s informal or intuitive problem-solving strategies. Such strategies are often very different from the standard symbolic procedures typically taught in the elementary school. Standard procedures provide powerful problem-solving tools; however, a concern is that many children merely memorize them. They never develop an understanding of the relationships among numbers within procedures. When allowed to progress through the stages described in the preceding section, a child will develop the habit of looking for numerical relationships. When introduced to the standard procedure, this child will understand the numerical relationships and will view the procedure simply as another strategy for solving problems. For this reason, intuitive strategies are emphasized throughout the Anishinabe Teachers for Anishinabe Children Project.

*Note to the Reader
Cognitively Guided Instruction (CGI) is a professional development program in elementary mathematics based on an integrated program of research that focuses on the development of students’ mathematical thinking, instruction that develops that thinking, and teachers’ knowledge and beliefs about student thinking that influence how they teach. In recognition of the breadth and richness of CGI, the editors of this volume strongly recommend teachers who are interested in the approach to participate in a CGI workshop and access the resources listed below.

Cognitively Guided Instruction Resource Information

To schedule a Cognitively Guided Instruction workshop, contact Linda Levi at:

Wisconsin Center for Education Research,
Cognitively Guided Instruction Suite,
Madison, WI 53706
Telephone (920)263-4267.

A text describing this approach, Children's Mathematics: Cognitively Guided Instruction, is available through Heinemann website:
http://www.heinemann.com

CGI website:
http://www.abacon.com/ie/berk/wlp452a.htm

Anishinabe Project information:
http://www.coehs.uwosh.edu/Anishinabe/home.htm

References

Brooks, J. G., & Brooks, M. G. (1993). In search of understanding: The case for constructivist classrooms. Alexandria:VA. Association for Supervision and Curriculum Development.

Carey, D. A., Fennema, E., Carpenter, T. P., & Franke, M. L. (1993). Cognitively guided instruction: Towards equitable classrooms. In W. Secada, E. Fennema, & L. Byrd (Eds.). New directions in equity for mathematics education. New York: Teacher College Press.

Carpenter, T. P., & Fennema, E. (1992). Cognitively guided instruction: Building on the knowledge of students and teachers. In W. Secada (Ed.), Curriculum reform: The case of mathematics in the United States. Special issue of the International Journal of Educational Research (pp. 457-470). Elmswood, NY: Pergamon Press, Inc.

Carpenter, T.P., Fennema, E. Franke, M.L., Levi, L., and Empson, S.B. (1999). Children's Mathematics: Cognitively Guided Instruction. Portsmouth, NH: Heineman.Collins, A.,

Brown, J. S., & Newman, S. (1989). Cognitive apprenticeships: Teaching the craft of reading, writing, and mathematics. In L. B. Resnick (Ed.), Knowing, learning, and instruction: Essays in honor of Robert Glaser. Hillsdale, NJ: Erlbaum.

Fennema, E., Carpenter, T. P., Levi, L., Franke, M. L., & Empson, S. (1997). Cognitively guided instruction: Professional development in primary mathematics. Wisconsin, Madison:Wisconsin Center for Education Research.

National Council of Teachers of Mathematics. (1998). Teaching standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2000). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

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