# What is Cognitively Guided Instruction?

What is Cognitively Guided Instruction? Why do we do it?

Early on a Saturday morning a few weeks ago, I had a conversation with more than 200 teachers and administrators (and a few school board members) about Cognitively Guided Instruction (CGI). The conversation started when I posed the questions, “What is CGI and why do we do it?”

The response was inspiring, thought-provoking, and humbling.

• Inspiring because the ideas shared highlighted the wisdom and commitment to young people.

• Thought-provoking because the response pushed me to reconsider my own ideas of CGI.

• Humbling because it reminded me about the power of collective work and how even in the most challenging times for education, together we can push back and work to change the status quo.

Before sharing what the group came up with, I want to explain why I began this conversation. Over the last year I have found myself needing to define or position CGI in particular ways. As I considered how I might do this, I recognized that CGI is not mine to define. CGI is not mine. It’s not even Tom Carpenter and Eliz Fennema’s. And it never has been.

# Supporting Development of the Cardinal Principle

By: Thomas P. Carpenter, Megan L. Franke, Nicholas C. Johnson, Angela Chan Turrou, and Anita A. Wager

Capturing a child’s understanding of the cardinal principle while they are counting can be challenging, as children don't necessarily end the process of counting by explicitly stating the total amount that they have in their collection. A child may know that counting objects involves reciting a sequence of numbers, but not that the outcome of this process is a number that represents the total quantity. A child may say “1,2,3,4” as they count a collection of four, but this does not necessarily mean that the child understands that there is a quantity of four objects. Applying the cardinal principle requires that children name the set according to the last number used in their count. In this case, that last number used was four, so there are four objects in the collection. Because the process of counting and what the count tells you are not necessarily the same thing, figuring out what a child knows about the cardinal principle often requires waiting for a child to complete their count and then asking a question like, “So, how many do you have in your collection?” Other ways to get at the cardinal principle could include saying to the child: “Here are some blocks. How many are there?” Or “Do you have enough to give me 4?” Asking children to make a group of counters of a given size rather than counting a given collection also can focus them on the cardinal principle.

# Supporting the Development of One-to-one Correspondence

Adapted from Young Children's Mathematics: Cognitively Guided Instruction in Early Childhood Education

by Thomas P. Carpenter, Megan L. Franke, Nicholas C. Johnson, Angela Chan Turrou, and Anita A. Wager

Supporting the Development of One-to-One Correspondence

Developing one-to-one correspondence occurs as children work to coordinate the counting of each object with one and only one number word. You can support the development of one-to-one correspondence by counting together with children, emphasizing matching the action of pointing, touching, or moving each object with the saying of a single number word. Or you might pose a question to the student such as, “How will you keep track of which ones you’ve counted and which ones you haven’t counted yet?” Some children benefit from putting the objects into a container as they count (such as during clean up), as it slows down their count and focuses them on one object at a time. Other children find that spreading out their collection or working with a partner is helpful. However, supporting children to use strategies to keep track of the objects counted will not immediately move children to use the one-to-one principle. Developing understanding of and the ability to use the one-to-one principle takes time and a range of experiences.

# Supporting the Development of Counting

The following post is adapted from the newest book in the Cognitively Guided Instruction family: Young Children's Mathematics: Cognitively Guided Instruction in Early Childhood Education.

Supporting the Development of Counting

By Thomas Carpenter, Megan Franke, Nicholas Johnson, Angela Chan Turrou, and Anita Wager

When thinking about how children’s counting develops, there are four big ideas to keep in mind:

• Details matter when we look for what students know about counting.
• Students can have more counting understanding than we see in a given moment.
• Counting principles do not emerge in a set sequence.
• Counting principles emerge concurrently

# CGI: Integrating Arithmetic and Algebra in Elementary School

The philosophy of Cognitively Guided Instruction (CGI) has helped hundreds of thousands of teachers learn more about children’s intuitive mathematical thinking and teach math more confidently. In today’s blog, which is adapted from Thinking Mathematically, Tom Carpenter, Megan Franke, and Linda Levi talk about the value of viewing arithmetic in elementary school as an essential foundation for algebra.

# Children’s Mathematics: Why Every Math Teacher Should Know About Cognitively Guided Instruction

In anticipation of the new edition of Children’s Mathematics, Christopher Danielson writes about what he wishes all secondary teachers knew about Cognitively Guided Instruction (CGI).

Christopher taught in the Saint Paul Public Schools for six years, then earned a Ph.D. at Michigan State University. While a graduate assistant at Michigan State, he helped revise Connected Mathematics materials. Today he teaches in the math department at Normandale Community College, in Bloomington, MN, where he teaches college algebra, Calculus, and math content courses for future elementary teachers. He blogs at Overthinking My Teaching and Talking Math with Your Kids.

Why Every Math Teacher Should Know About Cognitively Guided Instruction
By Christopher Danielson

CGI, which is summarized in Children's Mathematics, tops my list of things I wish all secondary math teachers (and college math instructors) knew. There are two reasons—the content and the principle.

The Content
CGI lays out the ideas children bring to school about addition, subtraction, multiplication, division, and numbers more generally. It also identifies the four basic categories and eleven subcategories of addition and subtraction problems. All secondary math teachers should know this content.

Secondary math teachers teach algebra, and algebra is—in part—generalized arithmetic. If secondary teachers are fluently aware of the ideas children have about arithmetic, they are better positioned to help older students expand these ideas to accommodate learning algebra.

Here are two examples of how middle and high school instruction can be informed if the teacher knows how students think about basic arithmetic.

Subtraction

Many students, when they begin learning algebra, think of subtraction as taking away (the CGI term is separating). This idea makes it difficult to conceptualize subtracting negative numbers. How do you take away less than nothing? If this question isn’t answered, students revert to memorizing rules.

Teachers who know that taking away is how their students are likely to think about subtraction and also know that comparing is another powerful way of thinking about subtraction can help their students see 3 – -2 as a matter of the distance between 3 and -2 on the number line: how much bigger is 3 than -2? The calculation can be made by comparing the numbers.

Multiplication

Sometimes the expression 3x means I have 3 groups of an unknown size and sometimes it means I have an unknown number of groups of 3. To an algebra teacher, these are equivalent concepts, because an algebra teacher has an abstract understanding of multiplication. We want students to develop this abstract understanding too, to understand multiplication as an operation, so that they can use it to study and understand exponentiation, calculus, and modern algebra. But not all students achieve this.

If all algebra teachers know that their students often think very literally about multiplication and find literal and operational interpretations very different, they can design their instruction to address this misconception.

Maybe some students are better served if they write x3 instead of 3x when they first try to represent an unknown number of groups of 3. The point is that knowing how students think about these ideas helps a teacher make instructional decisions and see new instructional possibilities.

The principle
Students arrive in our classrooms with ideas; they are not blank slates. Cognitively Guided Instruction is based on the ideas students already have. This principle should inform all mathematics instruction: let’s teach algebra based on the ideas students have about variables, geometry based on their ideas about shapes, and so on. Children's Mathematics shows us what this looks like in the classroom.