Cognitively Guided Instruction: A Knowledge Base for Reform in Primary Mathematics Instruction Jody MacLean
October 31, 2008
Thomas Carpenter, Elizabeth Fennema and Megan Franke propose in their article Cognitively Guided Instruction: A Knowledge Base for Reform in Primary Mathematics Instruction that teaching is a complex problem-solving activity. To better understand teaching, three kinds of teacher knowledge must be examined: knowledge of the subject matter, knowledge of the pedagogy and knowledge of the pedagogical content. The focus of the article is on one element of pedagogical content, knowledge of student thinking. The authors argue that “understanding students’ mathematical thinking can provide a unifying framework for the development of teachers’ knowledge” (4). This understanding is developed through their teacher development program, Cognitively Guided Instruction (CGI), which is still used in teacher inservicing more than a decade after the publishing date of this article.
In this article, CGI is explained through the use of sample questions in elementary whole number operations and the corresponding student responses. The examples serve to introduce a concept fundamental to CGI: that students “intuitively solve word problems by modeling the action and relations described in them” (6) and that by gaining insight into students’ thinking, teachers can and help them to move forward in a logical sequence from direct modeling to counting to derived facts. As students progress from single digit to multi-digit computations and from addition/subtraction situations to multiplication/division situations, they begin each new concept with direct modeling and move through the steps toward abstraction. “Children initially solve problems with larger numbers using the same modeling strategies they use for problems with smaller numbers” (10).
From the examples, we could see the critical role that effective teacher questioning and student interaction play in the evolution from direct modeling to abstraction. This supports Reinhart’s position, that students should explain things for themselves and to the class to more fully understand their own thinking (Reinhart, 2000). One concern I have is that teachers without a very solid knowledge of the subject matter might find themselves unable to ask the right questions. The “guided” aspect of CGI is one key to its success. For a teacher who doesn’t know what to ask next, I expect that GCI would not be effective. I realize that a part of CGI is training teachers to develop questioning and facilitating techniques, but even with strict adherence to the CGI framework, if a teacher doesn’t have all the mathematical concepts that make up the big picture at her disposal, the struggle and confusion for students might extend beyond what is productive for students’ learning. That said, I think that for the teacher, CGI would be a valuable learning experience no matter what the previous level of subject matter knowledge. Teachers cannot help but learn as they themselves struggle through the mathematics, the pedagogy and the pedagogical content. Would the students involved in this teacher’s learning process be any worse off than if they were learning from more traditional and “less confusing” direct teaching methods? I’m not sure.
After reading the article, I watched the video segments that accompany the book by the same authors on CGI. Seeing the students in action, I was more convinced of the value of this instructional framework. In kindergarten examples, the teacher, who admittedly was working with a small group of advanced students, was getting incredible results. A five or six year-old who can go to a number line to solve a multiplicative word problem and then clearly explain his strategy so that other children and the teacher can understand is the goal we’re all striving to meet.
I just got this book a few weeks ago, and after reading the article (which seems to be a summary of the work that two years later led to the book), I’m inspired to pick up the book again and delve deeper. Perhaps I’ll find some responses to the questions that I have around content knowledge and its effect on student outcomes for teachers using CGI.