# 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

# Beginning to Use Cognitively Guided Instruction

The new edition of Children’s Mathematics builds on the legacy of the highly influential first edition. It explores the philosophy of Cognitively Guided Instruction (CGI) for helping children develop deep mathematical understanding. In this blog, which is adapted from the book, the authors discuss how to start using CGI in your classroom.

# Beginning to Use Cognitively Guided Instruction

By Thomas Carpenter, Elizabeth Fennema, Megan Loef Franke, Linda Levi, and Susan Empson

“My journey as a CGI teacher has been exciting, not only for me but for my students and colleagues. My students have taught me about math through their amazing thinking. I really enjoy learning, listening, and developing mathematical understanding with my students and colleagues.”

—Kathleen Bird, teacher

Becoming a CGI teacher takes time. Changing your classroom practice so that you can draw on knowledge of the development of children’s mathematical thinking can be challenging. Yet, teachers consistently report that although they felt uneasy as they started CGI, the rewards were worth the struggle for them and their students. We have found that one way to alleviate the uneasiness and to begin the journey that Ms. Bird describes is just to start.

Most teachers start to use CGI by asking students to solve problems like those discussed in Children’s Mathematics. They choose word problems set in contexts with which their students are familiar. When teachers pose these problems, they do not demonstrate a strategy for solving a given problem type, although they may engage students in a discussion of the context in which the problem is set to be sure that students understand what the problem is about. Students are provided a variety of tools and encouraged to use a strategy that makes sense to them. As students solve the problem, teachers find it useful to move around and observe various students asking them about what they are doing, as a way to understand students’ strategies and provide support when necessary.

### Finding an organizational structure that works for you and your students may require you to experiment.

There is no optimal way to organize a CGI class. Some teachers prefer to start by working with a small group of students. In other classes, the entire class is given a problem to solve and discuss. Sometimes teachers adapt the numbers in a problem or provide different problems for different students. Whatever organization enables you to support the students to solve problems using their own strategies and provides you opportunities to listen to the students’ problem-solving strategies is an appropriate organization for you. Finding an organizational structure that works for you and your students may require you to experiment with how you pose problems and support students to share their ideas; and in turn, this experimentation and reflection will lead to ongoing learning for you and your students.

In CGI classes, students learn most new content by engaging in problem solving. Solving and discussing a variety of problems supports students to develop both concepts and skills. Problems can be posed as word problems or symbolic problems or in other formats. The critical consideration is that students are engaged in deciding how to solve problems using what makes sense to them.

Unlike traditional instruction in which the content to be learned is clearly sequenced (addition before subtraction) and where students learn skills before they use them to solve problems, the curriculum in CGI classes is integrated. For example, students do not learn number facts as isolated bits of information. Rather they learn them as they repeatedly solve problems, so that they begin to see relationships between the various facts. Both word problems and symbolic problems are the vehicles through which students learn mathematical concepts and skills. Teachers choose problems so that they will enhance students’ development, but in most cases do not provide explicit instruction on problem-solving strategies. Instead students develop and share their own problem-solving strategies, which become more efficient and abstract over time. Skills and number facts are learned in the process of problem solving and are thus learned with understanding rather than as isolated pieces of information.

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Discover more about Children’s Mathematics, Second Edition and Cognitively Guided Instruction at Heinemann.com/ChildrensMath, and preview the new edition by downloading a sample chapter.

# Learning With Understanding is a Matter of Equity

The new edition of Children's Mathematics explores the philosophy of Cognitively Guided Instruction (CGI) for helping children develop deep mathematical understanding. In today's post, adapted from the book, the authors discuss matters of equity and justice for students as they relate to teaching mathematics.

Written by Thomas P. Carpenter, Elizabeth Fennema, Megan Loef Franke, Linda Levi, and Susan B. Empson

All students benefit from and deserve to be in classes in which teaching for understanding is the norm. Opportunity to learn with understanding is first and foremost a matter of equity. There is no compelling evidence that there are large numbers of students who cannot learn with understanding, and denying any student opportunity to learn with understanding is an injustice.

All students are capable of and benefit from learning mathematics that:

• is organized in a rich network of connections
• provides a basis for ongoing learning
• they can describe, analyze, and justify
• provides support for them to develop an identity as being capable of making sense of mathematics.

Each student brings relevant knowledge to instruction. Some of what students know may be wrong, but often they know more than we give them credit for or even notice. It is our job as teachers to find out what our students do know so that we can build on what they know that is valid and useful. We have consistently found that virtually all young children have informal knowledge of number and problem-solving strategies that they can build upon to develop arithmetic concepts and skills.

We have documented young children’s rich informal knowledge of number operations, which turns out to be remarkably consistent across different demographic groups within the United States and other countries as well as for children demonstrating different levels of achievement. That does not mean that all children have the same knowledge at any given age or grade, but the development of strategies for basic number concepts and skills follow pretty much the same general pattern for all learners. Teachers often find that students who have been identified or presumed to be achieving below grade level know more than anticipated. Building on this knowledge makes learning more efficient. There is less unfamiliar material to learn, and because new ideas are connected to what the students already know, they can make sense of the new ideas and integrate them into a coherent structure.

Students who are presumed to not be successful in mathematics may actually be more capable of learning meaningful networks of ideas than isolated concepts and skills. There is much more to learn when facts and skills are not connected, and what is learned is easily forgotten, subject to errors, and not generalizable to new ideas and solving unfamiliar problems. A great deal of the success of students who show an aptitude for mathematics is due to the fact that they organize their knowledge into rich networks. It is our obligation to make that kind of learning available to each student.

One of the benefits of acquiring knowledge rich in connections is that it can be applied to learn new content and to solve problems. It is not only easier to learn ideas that are interrelated; when knowledge is rich in connections, it is easier to learn new ideas that depend on it. If knowledge is limited to narrow contexts and cannot be applied outside those contexts, it is of little value. Thus, a clear goal of instruction for all students has to be to acquire knowledge that is generative.

The goal that students describe, explain, and justify their mathematical thinking is often presumed to be beyond the capabilities of many students. That has not been the case in our experience within CGI classrooms. We have consistently found that, with support, students can successfully participate in CGI classes in which students routinely describe, explain, and justify their mathematical thinking. When students participate in these ways, we can better understand their mathematical thinking, and we often are surprised by what we learn. Even though students often do not think about mathematics in the same way we might, they have knowledge on which to build.

# Children’s Mathematics: Engaging with each other’s ideas

The newly released second edition of Children’s Mathematics builds on the legacy of the best-selling classic. This highly anticipated book provides new insights about Cognitively Guided Instruction (CGI) based on the authors’ research and experience in CGI classrooms over the last 15 years. In today’s blog, we look at one of the clips from the extensive library of entirely new videos.

Engaging students with each other’s mathematical ideas is an essential part of Cognitively Guided Instruction. We often wonder if sharing multiple approaches and strategies (including approaches and strategies that may have errors) will confuse our classes. But research shows that engaging with another student’s mathematical thinking is productive both for the student sharing and the students listening. In fact, encouraging students to talk with each other about their mathematical ideas is a great way to guide them into developing more sophisticated strategies.

In the clip below, Ms. Barron’s third-grade class is solving the following problem:

Gilberto had 81 stickers. Then he bought some more and now he has 312 stickers. How many did he buy?

After a number of students have shared their strategies on the board, Ms. Barron invites Kelly to share what she has written. While the previous students have used similar strategies, Kelly’s approach is different. Rather than starting at 81 and adding increments until she reached the target of 312, Kelly started at 312 and subtracted increments until she reached 81.

Notice how Ms. Barron doesn’t shy away from engaging Kelly’s mathematical thinking. Instead, she asks questions to help the students analyze how Kelly’s strategy is similar to the strategy another student used. By engaging her class with one another’s thinking, she promotes the development of deeper mathematical thinking.

# 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.