Cognitively Guided Instruction (CGI) Math (Grades kindergarten through third)
Students are presented math questions in the form of word problems. The students are instructed to think of ways to solve the problem presented. Students use several strategies such as the use of manipulatives, counting on their fingers, and using their knowledge of numbers to solve the problem. Students are then asked to explain their thinking with the class. The students are provided with opportunities to compare multiple strategies and approaches of the same problem. A key aspect of CGI is the ability of students to explain and justify their strategy. The class and teacher, through discussion and exploration, determine whether the problem was solved correctly.
Extending Mathematical Understanding (Grades third through fourth)
This approach is very similar to CGI. The difference being that this process focuses heavily on the the idea of questioning, student exploration, and discussion over possible mathematical outcomes.
These styles of math put more responsibility on the students. Instead of being able to simply solve the same type of equation, students are provided with opportunities to find their own way of solving a problem and are capable of explaining and justifying their thinking.