Cognitively Guided Instruction

What is Cognitively Guided Instruction?

 Cognitively Guided Instruction (CGI) is an approach to teaching mathematics at the elementary level where teachers utilize what they know about their children’s understanding of mathematics to select problems, pose those problems, question students, and facilitate discussion and sharing. The approach is a result of research led by Elizabeth Fennema and Thomas P. Carpenter at the University of Wisconsin at Madison.  CGI focuses on student thinking and analysis through the use of word problems, combined with the math process standards: problem solving, reasoning and proof, communication, connections, and representation.  CGI has fourteen different types of word problems that are sequenced from easiest to most difficult.  This allows teachers to choose which types of word problems are most appropriate to start their students with and then continuing forward in sequential order while promoting the development of mathematical reasoning in their students.

Fundamental Principles of CGI:

  • a change in teachers’ practices can result from teachers developing a basis for understanding and building on their students’ mathematical thinking
  • teachers learn from listening to their students and struggling to understand what they hear
  • It is important to develop conceptual models of student thinking that can be used to engage students in mathematical inquiry at their appropriate level of understanding. Teachers need to be ready and able to differentiate to each individual student and his/her thinking.
  • “By understanding how the child I thinking, I’m able to see where they are, what level they’re at, and what kinds of things I want to give them to move on to.”  - Michelle Garden, a teacher who uses CGI
  • children are to become independent problem solvers without needing to rely on teachers telling them what to do at every step

Children Learn Problem Solving Three Ways:

1. Using modeling strategies

  • Kelsey had 7 cookies.
  • Tanya gave her 6 more.
  • How many cookies does Kelsey have now?
  • Thinking: Kelsey had 7 cookies…one, two, three, four, five, six, seven.  (Child sets out 7 counters.)  Tanya gave her six more...one, two, three, four, five, six.  (Child sets out 6 counters, then pushes both sets together and counts all.)  She has thirteen cookies.

2. Counting on or counting back strategies

  • Kelsey had 7 cookies.
  • Tanya gave her 6 more.
  • How many cookies does Kelsey have now?
  • Thinking: I don’t have to count the seven again.  I just have to add six to it.  I say, eight, nine, ten, eleven, twelve, thirteen.  (Child holds up a finger with each count.)  I have thirteen.

3. Number facts

  • Kelsey had 7 cookies.
  • Tanya gave her 6 more.
  • How many cookies does Kelsey have now?
  • Thinking: I know that six and six is ten.  I took one from the seven to make six so I have to add one back on.  It’s thirteen.

Why should you use it? What are the benefits? 

CGI does not rely on memorization of facts and equations; instead, it focuses on word problems to develop mathematical thinking.  The sequence of math problems are introduced and taught in a specific order in order to help children develop specific mathematical skills and understandings.  Once children have gone through the three phases (1) modeling, (2) counting on or counting back and (3) number facts, they begin to apply their knowledge when solving problems.  This helps children not to simply rely memorized facts, but understand the relationship between two or more quantities that a math problem represents.  They begin to develop a number sense.  It is important that the students create their own process because it allows them to become more intuitive learners instead of doing what is dictated by a teacher.  Using CGI directly correlates to the National Council for Teachers of Mathematics Problem Solving Standard by “building new mathematical knowledge through problem solving, solving problems that arise in mathematics and in other contexts, applying and adapting a variety of appropriate strategies to solve problems, and monitoring and reflecting on the process of mathematical problem solving” (NCTM 2000).

How can it be implemented?

How do I get started?  It is simple!  Most teachers begin to “get their feet wet” in the world of Cognitively Guided Instruction by asking children to solve simple word problems.  CGI requires teachers to pay close attention to the difficulty of the problem they are asking their students to solve.  The difficulty of a problem depends on:

1. Can the problem be acted out?

  1. YES – Kelsey had 8 cookies.  She gave 3 to Tanya.  How many cookies does Kelsey have now?
  2. NO – Kelsey gave 3 cookies to Tanya.  She started with 8 cookies.  How many cookies does Kelsey have now?

2. Can the problem be modeled in the order it is heard?

  1. YES – Tanya had 5 cookies.  Kelsey gave her 8 more.  How many cookies does Tanya have now?
  2. NO – Tanya had some cookies.  Kelsey gave her 8 more.  Then she had 13 cookies.  How many cookies did Tanya have before Kelsey gave her any?

3. Can the problem be modeled directly?

  1. YES – Kelsey has 8 cookies.  Tanya has 5 cookies.  How many more cookies does Kelsey have?
  2. NO – Tanya has 5 cookies.  She has 3 fewer cookies than Kelsey.  How many cookies does Kelsey have?

4. Is the unknown quantity located at the beginning, middle, or end of the problem?

  1. END (easiest) – Tanya had 7 cookies.  She gave 4 to Kelsey.  How many cookies does Tanya have now?
  2. MIDDLE – Kelsey had 8 cookies.  She gave some to Tanya.  How she has 5 cookies.  How many cookies did Kelsey give to Tanya?
  3. BEGINNING (hardest) – Kelsey had some cookies.  She gave 3 cookies to Tanya.  Then she had 5 cookies left.  How many cookies did Kelsey have before sharing with Tanya?

5. Can simple multiplication or division problems be acted out or modeled?

  1. MULTIPLICATION – Tanya has 4 piles of cookies.  There are 3 cookies in each pile.  How many cookies does Tanya have?
  2. PARTITIVE DIVISION – If Kelsey gives 12 cookies to 3 friends and each friend is given the same number, how many cookies will each friend get?

Along with the problem difficulty, CGI has a specific sequence that must be followed when giving students problems to solve, they are:

FIRST
Join: Result Unknown
Separate: Result Unknown
Part-Part-Whole: Whole Unknown

SECOND
Compare: Difference Unknown

THIRD
Join: Change Unknown
Separate: Change Unknown
Part-Part-Whole: Park Unknown

FOURTH
Join: Start Unknown
Separate: Start Unknown

FIFTH
Compare: Compare Quantity Unknown
Compare: Referent Unknown

While giving students problems to solve, teachers should have several different types of manipulatives out on the floor to allow students to use at free-will.  Remember, it is all about how the child chooses to construct his/her learning.  The teacher is then free to walk around and see how the students are coming along.  This is the time to sit down and ask someone to explain how they solved the problem.  Probe them and ask all sorts of question in order to truly see what processes they are using to come up with their answer.  Once the majority of the students have solved their problem, call up a few to share their “strategies” with the class.  If the teacher documents these shared strategies on a poster and hang it in your room, other students will then be able to use them the next time they are asked to solve a CGI problem.  When a child has the opportunity to choose a strategy and is able to explain his/her thinking, that child is showing an understanding for the mathematical process and not just simply giving “the answer”.  Once the teacher has a feel for the different stages of mathematical development, he/she should begin deliberately selecting specific problems for students to work on.  This is how CGI can be differentiated in a classroom.  Constructing math time this way will allow the students to construct their knowledge and understanding for math at their own pace instead of the pace of a text book or the rest of the class.  CGI is not something that you are going to learn and be an expert at over night.  It takes a lot of practice and purposeful thinking and planning; however, with time it will become second-nature to both you and your students.