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.

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