Cognitively guided instruction is a very fancy term for a simple, yet incredibly effective teaching practice. I read students word problems (using a variety of types and structures), students solve them on white boards or paper using unifix cubes, pictures, number sentences...(depending on what strategy they are comfortable solving the problem with), several students share their strategy, students solve the next problem with, hopefully, a more sophisticated strategy than the one they were using before after having been inspired by the students that just shared their strategies. Obviously, students learn at different rates, and it may take some students seeing a strategy several times before they feel comfortable enough to use it.
Benefits for me: I get to see strategies students are using like a window into their brain as well as see the growth over time on the strategies they are using. I really experience joy when students use sophisticated strategies that are different than I thought of (almost daily).
Benefits for students: They get to experience the idea that there is a correct answer, but how we get there may be different because we all have different brains and that all strategies are valued. Most of all students learn that math is more than learning a set procedure. It involves understanding the operation and choosing the best strategy for the student and for the specific problem. They get to see and "borrow" strategies from their peers and are always moving to more sophisticated strategies.
A side note: Students love when I name strategies after them. I record strategies on a chart. Then, minutes or weeks later I may say "Oh, you used the Ophelia strategy, so clever." or "Why don't you try the Ryan strategy for that problem?" It really leads to the feeling of being a serious mathematician and making discoveries about math.
example- 8 kids got on the bus, then 7 more got on the bus. How many kids are on the bus now?
Students are always moving at their own pace to more sophisticated strategies like directly modeling all the parts of the problem (taking out 7 manipulatives, then 8 manipulatives, then counting all of them. Counting up or counting from students just think or say "8", then count up 7 more on their fingers or in their head. Derived fact- students use a fact the know to find a fact they don't know (I know 8+8=16, so 8+7=15).
I can make a problem more difficult for the class by changing the numbers or using another problem type. (8 kids were on the bus, then some more got on at the next stop and there were a total of 15 students on the bus. How many students got on the bus?)
Here are some links for CGI:
Great set of CGI Problems to use- written by Mary Alice Hatchett