# 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.

# How To Build Meaning for Fractions with Word Problems

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 Extending Children’s Mathematics: Fractions and Decimals, Susan Empson and Linda Levi look at the importance of helping students build an authentic understanding of fractions through solving and discussion word problems before introducing them to fractional symbols and notation.

# Linda Levi: Three CGI Books You Should Read Next

Children’s Mathematics, Extending Children’s Mathematics, and Thinking Mathematically are essential reading for teachers who want to understand how children learn mathematics. Grounded in the Cognitively Guided Instruction (CGI) philosophy pioneered by the authors, these books have helped hundreds of thousands of teachers learn more about children’s intuitive mathematical thinking and teach math more confidently. In today's blog, co-author Linda Levi talks about the connections between the three books in the CGI family.

SPECIAL OFFER: From now until November 20, 2015, use the code CGI30 at online checkout to save 30% off the list price of these three Cognitively Guided Instruction books.

# Your Heinemann Link Round-Up for the Week of September 28–October 2

Welcome BACK to the Heinemann Link Round-Up. Your intrepid rounder-upper was on vacation last week, and thus nothing was lassoed, nothing was harangued. Here we are again with a full pen of links.

These links are interviews with educators, posts from our authors' and friends' blogs, and any interesting, newsworthy item from the past seven days. Check back each week for a new round of finds!

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The official Global Teacher Prize blog wrote out "3 Life Changing Lessons from Teacher Prize Winner Nancie Atwell’s Keynote at CGI."

##### Nearly a quarter of American children fail to achieve minimum levels of literary. For Nancie, the solution is books. She says “book reading is just about the best thing about being human and alive on the planet.” For this reason, children cannot be allowed to discover the joys of reading by accident – an enticing collection of literature is central to the children becoming competent, voracious and engaged readers. This collection must include writing at a variety of levels, from a variety of genres and to appeal to every taste.

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NPR's Weekend Edition did some research on homework. Here it is:

##### In 2012, students in three different age groups—9, 13 and 17—were asked, "How much time did you spend on homework yesterday?" The vast majority of 9-year-olds (79 percent) and 13-year-olds (65 percent) and still a majority of 17-year-olds (53 percent) all reported doing an hour or less of homework the day before. Another study from the National Center for Education Statistics found that high school students who reported doing homework outside of school did, on average, about seven hours a week.

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In NCTM’s “Teaching Children Mathematics,” Children’s Mathematics coauthor Susan Empson looks at the strategies used by fifth-graders to solve division-of-fraction problems set in the context of making mugs of hot chocolate.

##### Children in the elementary grades can solve fraction story problems by drawing on their informal understanding of partitioned quantities and whole-number operations (Empson and Levi 2011; Mack 2001). Given the opportunity, children use this understanding to model fractional quantities, such as 1/4 of a quesadilla, and reason about relationships between these quantities, such as how much quesadilla there would be if 1/4 of a quesadilla, 1/4 of a quesadilla, and 1/4 of a quesadilla were combined.

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Both chums in the Chartchums (Marjorie Martinelli & Kristi Mraz) have had busy years with other projects, so it's always great to see them back with a blog. This week: organizing charts.

##### When projects come to an end and before new ones begin, starting off with a fresh clean start helps one move forward. Whether you have taught for one year or twenty, the amount of paper and stuff accumulated can become mountainous. Inspired by the book, The Life-Changing Magic of Tidying Up: The Japanese Art of Decluttering and Organizing (Ten Speed Press 2014)by Marie Kondo, who suggests discarding as the first rule of tidying, we thought about how we could apply this to charts so that we start off the year with a fresh and tidy start. Marie Kondo’s only rule about what to keep is to hold each item in your hands and to ask, “Does this spark joy?” For a teacher to be able to answer this question you need to also ask, “Can I use this again?” “Will this save me time?” “Will this engage my kids?”

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And we have a round-up competitor in Dana Johansen at Two Writing Teachers! It's a great round-up of tweets about writing from September. Click here for it!

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That's it! Be sure to check back next week for another round of links. If you have a link or a blog, be sure to mention them in the comments below. You can also email them to us or tweet at us. We're pretty available over here. Cheers to your weekend!

*Photo by Matt Lee

# 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.