CGI = Cognitively Guided Instruction
Thomas Carpenter and Elizabeth Fennema
defined different problem types.
“If we want to give children the
opportunity to build their understanding
from within, we need to understand how
children think about math.”
-Carpenter, et. al. (1999)
3 strategy types for problem solvers:
1. Modeling strategies
2. Counting strategies
3. Facts strategies
Carpenter, et. al. (1999)
Join & Separate Problems
*Both involve an action that causes a change.
*Natural starting point for children.
*Use direct modeling.
1. Join Result Unknown
John had 5 cookies. Maria gave him 5 more. How many does he have now?
2. Join Change Unknown
John has 5 cookies. How many more cookies does he need to have 15 cookies?
3. Join Start Unknown
John had some cookies. Maria gave him 10 more cookies. Now he has 15.
How many cookies did John start with?
1. Separate Result Unknown
John had 15 cookies. He gave 5 cookies to Maria. How many cookies does he
2. Separate Change Unknown
John had 15 cookies. He gave some to Maria. Now he has 5 cookies. How
many cookies did John give to Mary?
3. Separate Start Unknown
John had some cookies. He gave 5 to Maria. Now he has 10 cookies. How
many cookies did John start with?
*The whole is the sum of the parts
and the parts make up the whole.
1. Part-Part-Whole, Whole Unknown
John has 5 sugar cookies and 10 chocolate chip cookies. How many
cookies does John have?
2. Part-Part-Whole, Part Unknown
John has 15 cookies. Five are sugar and the rest are chocolate
chip. How many chocolate chip cookies does John have?
*Often the most difficult type,
because the problem solver must infer
an action to solve the problem.
1. Compare Difference Unknown
John has 15 cookies. Maria has 5 cookies. How many more cookies does John
have than Maria?
2. Compare Quantity Unknown
Maria has 5 cookies. John has 10 cookies more than Maria. How many cookies
does John have?
3. Compare Referent Unknown
4. John has 15 cookies. He has 10 more cookies than Maria. How many
cookies does Maria have?
“I now realize that I must be very
patient, because the growth of young
children as problem solvers is anything but
steady and continuous. Beginning problem
solvers seem to ‘bump along,’ and then one
day they ‘jump’ to a much higher level of