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Cognitively-Guided Instruction: Supporting Students to Create Their Own Mathematical Understanding

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This post is by Jeff Feitelberg, 4th Grade Teacher, High Tech Elementary Explorer.

When I was at school, math was about speed and getting the right answer. It was something you did alone and you were either good at it or you weren't. It was not until high school that I had to grapple with complex concepts. When that happened, I didn't have the skills to persevere individually or work collaboratively with others. I felt ashamed and avoided asking questions when I was unsure. When I decided to become a teacher, I knew that I didn't want students to experience math the way I had--to feel ashamed that they answered a problem incorrectly or didn't understand a concept. I wanted my future students to cultivate their mathematical understanding and to be collaborative problem solvers regardless of the difficulty of the task.

Two years ago, I was introduced to the Cognitively-Guided Instruction (CGI) model of mathematics instruction. CGI is a student-centered approach to teaching math. It starts with what children already know and builds on their natural number sense as well as their intuitive approaches to problem solving.

Most teacher training involves instruction that is more teacher-centered -- also known as direct instruction -- in which teachers lead students through the mastery of skills. One of the challenges for me in introducing CGI in my classroom was ceding this control.  How would students learn if I let them discover the strategies rather than taught them directly? Could students really grow as mathematicians just by grappling, sharing, questioning, and discussing their mathematics together?

What I saw led me to challenge all of my assumptions about math instruction. Yes, I was letting the students discover strategies for themselves, but I had to plan much more carefully in order to help them do so. Prior to a CGI lesson, educators collaboratively anticipate a range of complexity in strategies that students may or may not use for each math problem. This allows teachers to think about the intricacies of invented strategies as well as metacognition, or how students think about math. The more we understand about our students' mathematical understanding the better we can help students independently, confidently, and accurately create workable strategies. For example, if a student is trying to model numbers with pictures, we question their process to help them improve the representation of their thinking. This helps to develop students' conceptual understanding of mathematics, improve strategy-sharing abilities, and address potential misunderstandings.

Telling students how to solve a problem does not allow them to develop as mathematicians. We show respect for our unique students through open-ended questioning and targeted strategy sharing. It is truly the most socio-emotional-conscious way of teaching math: putting students in charge of their development. I ask questions to help understand a child's mathematical thinking and to help my class share their strategies with their peers. It is such a pleasure to see them take ownership of the complex mathematical discussions!

The first year I tried CGI, I had a student in my class named Isabella. She spent the first month of the CGI lessons writing next to nothing on her paper. When she started to pay close attention to the strategies that were shared and wrote them furiously on her paper, I was concerned that she was simply copying down information. I spent extra time talking to her, saw she was slowly coming to some realizations, and began using previously created charts to begin a workable strategy. After two months of struggling to complete a story problem, it finally clicked! She began excerpting strategies from multiple students and solving problems with ever more advanced approaches. Respecting Isabella's process allowed her to blossom into a flexible and confident mathematician.

I believe that authentic learning experiences only occur when opportunities for deeper learning, respectful conversation, and rigorous discussion are nurtured. I champion CGI practices because they connect so well with social-emotional learning and project-based learning. It is powerful because of its student-centered approach, which allows students to make their own meaning of the problems they are solving.


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