Some teachers wonder then what their role in the classroom could be if it is not focused on demonstrating and explaining strategies to students and monitoring students' progress in using these strategies.
Susan B. Empson and Linda Levi
I'm late to the math party. I came to my current position as a K-5 specialist from a general elementary education background. I taught all subjects during the 23 years I spent in my grades 2 and 3 classrooms, and math didn't stand out in any significant way. I knew who Marilyn Burns was, but if I had ever met the luminaries that my colleagues in the MTBoS speak of so highly, people like Van de Walle and Fosnot and Kamii and Richardson, for example, it was long ago in teacher school, and I'd certainly forgotten who they were. So I play catch up.
Cognitively Guided Instruction (CGI) is something that, upon hitting the MTBoS, I heard referred to again and again. Glowingly. Reverently. And so this summer I decided to find out what the fuss was all about. I wanted to explore Extending Children's Mathematics: Fractions and Decimals, and put together a small, very informal book study PLC composed of our two grade 4 teachers and my former partner and fellow specialist Theresa. We met twice in July and once in August to discuss our reading assignments and share the work we had done.
|Kudos to my colleagues who gave up some of their summer evenings to participate in this project.|
It was an incredible experience, and we learned so much from the book and from each other. We increased our own content knowledge, and are now armed with some tools that I hope we can employ to better analyze student work.
A few weeks ago I came across a set of photos I had taken late last school year of responses to an item on a grade 5 unit assessment. The problem seemed pretty simple, and I was struck by the many different ways students had attempted a solution. At the time I was ignorant of CGI, but looking at them now through a CGI lens I can better understand and describe what's happening.
|Convoluted, but it worked. This student is also able to use relational thinking to express 2 5/8 as 21/8 and 4 as 32/8. Why did he feel it necessary to have common denominators? |
|No question here. Issues with the identity property of multiplication.|
|This was a common mistake: multiplying 4 x 2 then just adding the 5/8. This student has used an equal sign to separate the expressions, rendering an incorrect equation. 4 x 2 does not equal 8 + 5/8.|
|With no actual work, I deduce that this student multiplied 4 x 2 to get 8, then, in the mixed number, multiplied 2 x 5 and 2 x 8 to get 10/16. He finished by adding 8 and 10/16.|
|At first look it appears as if this student multiplied 4 by every digit he saw in the mixed number. How did he get 160? By multiplying 8 x 20? |
CGI argues for a decrease in the amount of time teachers spend, "demonstrating and explaining computation and problem-solving strategies to students." Instead, teachers are encouraged to allow their students' intuitive strategies to emerge first. "Children have some conceptually sound understanding of fractions, even before instruction," Empson and Levi write, "(but they) can learn to ignore this understanding in favor of models introduced in school that portray fractions in narrow ways."
This is a tremendous shift away from the traditional "I do, we do, you do" model, and entails teachers taking on a different role. "This new role," the authors of CGI explain, "Centers on helping students communicate strategies to other students, directing questions to specific students to help them draw connections between these strategies and more basic strategies, introducing equations to represent students' strategies, and highlighting the fundamental properties of operations and equality that that underpin these strategies."
That's a lot to ask. But I've read the book and I'm a believer. Let's roll up our sleeves and get to work.