There's a new edition of

*Children's Mathematics* coming out soon, and I got my paws on an advance copy of the book. I like

the people at Heinemann (the publisher) and I like the book so I said I'd write a few words about this new edition, and the book more generally.

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Here's my story: I'm a high school teacher that got a job at a school where the high school teachers also teach little kids. Last year I taught 4th, 5th, 9th and 11th graders. This year it's 3rd, 4th, 5th and 9th. I love it.

I came across Cognitively Guided Instruction (the research program) and

*Children's Mathematics *(the book describing it for teachers) when I started teaching elementary school students. I knew that this was new territory to me, so I asked a very knowledgeable friend to help me out. He sent me a

reading list, and CGI was at the top of it.

CGI is fundamentally about arithmetic. It's about how kids learn arithmetic, how their strategies usually (always?) develop, and it's about taking all of the knowledge that teachers have about student thinking and mushing it into a useful system. It's this theory of strategies that makes the whole CGI program so valuable. Anticipating a kid's thinking is the closest thing we have to a teaching superpower, and CGI provides a system of anticipations for the classroom teacher.

One thing the first edition of the book was

*not *about was activities or tasks. Unlike most of the math education books I read, there were no sample lessons or detailing of pedagogical moves. It's really about how kids think about arithmetic. From the 1st Edition Foreword:

"During this conversation, I realized how serious [the researchers] were about respecting teachers' judgments on particular issues. Since they had little evidence about representing these situations, they would see how teachers and children handled it. As they worked with teachers, sharing their research knowledge about students' learning of addition and subtraction, they would continue to learn from teachers and children."

I loved this focus on student and teacher thinking. The constraints of teaching are really so different in different environments. Attempts to change the way teachers act seems misguided -- much better to improve the way we think.

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Enough talky-talk. Here are two lists. The first is a list of reasons why I think high school teachers should check out

*Children's Mathematics*, and the second list describes some differences between the first and second editions.

**Reasons Why CGI Is For High School Teachers:**
**Problem Solving: **CGI is a study in how students think about arithmetic, and it's this sort of deep understanding of student thinking that enables teachers of arithmetic to drive student learning through problem solving. If you're interested in problem solving at high school, here's a model of the sort of knowledge needed to pull that off successfully.
**Modeling: **There's a chapter titled "Problem Solving as Modeling." Cool idea, right?
**The High School Disadvantage: **Reading CGI (and then teaching elementary school) made it clear to me why students have an easier time learning arithmetic than they do learning high school math: time. Learning takes time, and high school teachers are at a huge disadvantage because of the ridiculous flurry of topics that need to be taught. Watching student thinking develop through these pages drives that point.
**We Need More Research: **CGI offers a systematic overview of how students learn arithmetic operations. *Nothing like this exists for most high school topics *and it's a shame. What's the development that kids pass through while they're learning quadratics? Complex numbers? Exponents? We just don't know, and it would be amazing if we did.

**Differences Between the First and Second Editions:**

- The new text is full of links to videos. I was hugely impressed by the quality of the videos. The camera never strays from the kid. A question is asked, and you watch the kid's reaction. One by one, you watch kids use the strategies detailed in the book.
- The first edition didn't really dwell on how and when students might think in writing, but the new edition does a nice job with this.
- The authors are far less cautious about offering classroom recommendations in this new edition than they were in the first. There are two new chapters detailing their advice for teaching through problem solving in elementary classrooms. I'm sure that many will find these chapters helpful, but there's something austere and lovely that I'll miss in the restraint of the first edition.
- Stray observation: they bumped up the magnitude of the numbers in a lot of the examples. Interesting!
- After each chapter there are a whole series of exercises for working through the ideas of the chapter. I was skeptical, but after working through the exercises from one chapter I was impressed by their quality. CGI is accessible, but it's still an interconnected system of strategies and patterns of thought and it took me time to get down. The exercises helped.