HIS pre-kindergarten through fourth-grade teachers had the opportunity to participate in Cognitively Guided Instruction (CGI) trainings to help us learn more about how students conceptually construct and understand math. We just completed Day Two this past Friday and will finish up with Day Three in January.
The CGI approach provides teachers with information on how children think about and learn math, and how they move through the various steps and stages to reach math proficiency. CGI is inquiry-based and encourages students to teach each other by sharing their solution strategies rather than adults giving students one prescribed way to solve math problems. The “one prescribed way” system of teaching that most of us grew up with tends to lead to too many struggling math students who get lost with higher-level math and end up hating it.
There are four stages that students move through in solving math problems. Here are the stages with an example of how a student would solve a problem such as 6 + 5:
Direct Modeling: Students typically start solving problems using direct modeling. The student would either pull or draw six cubes and then five cubes. Then, the student would count all of the cubes, one by one, and reach the sum of 11.
Counting: When students count, they still may use the cubes, but instead of counting each cube individually, they can now count forward or count backward to solve the problem. For 6 +5 they would start at 6 and count “7, 8, 9, 10, 11.” They have number conservation, the knowledge that the first amount is six and that they can then “count on” the remaining five being added. They no longer need to count each individual cube at this stage.
Invented Algorithms/Derived Facts: For 6 + 5, students may say, “I know 6 + 4 = 10, plus 1 more would equal 11.” Or they may say, “I know 5 + 5 is 10 and 6 is one more than 5, so one more would be 11.”
Fact Mastery: The last stage is fact mastery where students automatically know that 6 + 5 = 11.
One of the most important teacher strategies with CGI is letting the kids move through the stages on their own. We all want them to reach mastery of the facts and be mathematically proficient, with computational fluency, but it’s important not to rush them through the stages. It’s also important to let kids explore with different strategies – this builds their conceptual understanding and mathematical reasoning. There is more than one way to get to a correct answer, and kids need to have the freedom to be able to do so. We adults often can learn a lot about how math works from students when we allow them that freedom.
-Jordan Johnson, 2nd grade teacher
-John Heffron, Lower School Principal