What is Cognitively Guided Instruction-Handout
by Bill Huggett
What is Cognitively Guided Instruction?
A CGI classroom is where you build on the math knowledge of your children
according to what they know…You don’t build objectives that say they should be
doing this, this, this, and this. You sort of take what they know and build from
there.
(Susan Gehn, First Grade Teacher)
Cognitively Guided Instruction (CGI) is captured in the above statement of an experienced CGI
teacher; it is about teachers making instructional decisions based on their knowledge of
individual children’s thinking. CGI was created by Elizabeth Fennema, Thomas Carpenter,
Penelope Peterson, and Megan Franke.
What CGI Is
CGI is not a traditional primary school mathematics program. Children in CGI classrooms spend most of their time solving problems, usually problems that are related to a book the teacher has to read to them, a unit they may be studying outside of mathematics class, or something going on in their lives. Various physical materials are available to children to assist them in solving the problems. Each child decides how and when to use the materials, fingers, paper and pencil; or to solve the problem mentally. Children are not shown how to solve the problems. Instead each child solves them in any way that they can, sometimes in more than one way, and reports how the problem was solved to peers and teacher. The teacher and peers listen and question until they understand the problem solutions, and then the entire process is repeated. Using information from each child’s reporting of problem solutions, teachers make decisions about what each child knows and how instruction should be structured to enable that child to learn. Starting at the kindergarten level, CGI teachers ask children to solve a large variety of problems involving addition, subtraction, multiplication, or division. Children learn place value as they invent procedures to solve problems that require
regrouping and counting by 10s. Problems are selected carefully so that children count by 1s, 10s, or 100s depending on the child; discuss relationships between basic number facts; and invent procedures to solve problems involving two-and three-digit numbers.
Above information taken from Children’s Mathematics: Cognitively Guided Instruction
Typical Problem Based Progression for Solving Problems:
Direct Modeling – student acts out the situation using manipulatives or
diagrams.
Counting Strategies – student utilizes strategies such as counting up to
solve the problem. They use aids such as number lines or hundreds charts
to help keep track of their counting. Many students get “stuck” at this level,
never really becoming automatic with their facts, instead relying on a variety
of counting strategies or technology such as calculators or spreadsheets.
Number Facts – student utilizes known number facts to solve problems.
One way to develop computational fluency is to have students frequently solve
problems and then have the opportunity to evaluate if the answer makes sense.
I would suggest posing problems based on the Cognitively Guided Instruction
(CGI) problem types. The child solves the problem by any method that makes
sense to them. Note the main focus at the K-2 level is addition and subtraction,
but do not be afraid to use the multiplication and division situations on the
second CGI problem type chart. It has been shown students as early as
kindergarten can successfully work with division (even reporting their solutions
using fractions) in a problem based situation such as fair sharing of cookies.
A problem and related solutions might look like this:
Sally has 35 rocks. John has 17 rocks. How many more rocks does Sally have than John?
Student draws a diagram or acts out solution using counters:
Sally has 35 rocks. 10 10 rocks 10 rocks
John has 17, so Sally has 18, 19, 20 (that’s 3), 30 (that’s 13), 31, 32, 33, 34, 35 (that’s 18 more). Sally has 18 more rocks than John does.
Another diagram we have seen is an open number line. 17 20 30 35 +3 +10 +5 = + 18
Equation method: Either 17 + 18 = 35 or 35 – 17 = 18
This connection between addition and subtraction may not be obvious to the children.
Discussion of the problem and how people show their thinking will help the child internalize the process.
The goal at this point is to understand the action of addition, subtraction, multiplication and division. We will be focusing on how to record that action using math notation only when students can do so with understanding. It is through daily discussions of problem solving that we hope to move students from direct modeling to counting strategies to using number facts to solve problems, but if we rush this process the students may be mimicking our instruction with no real understanding.
There are 4 Problem Types for Addition and Subtraction
Problem Types
Join Problems (Result Unknown)
Sally has 4 rocks.
John gave her 6 more
rocks. How many
rocks does Sally have
altogether?
(Change Unknown)
Sally had 4 rocks.
How many rocks does
she need to have 10
rocks altogether?
(Start Unknown)
Sally had some rocks.
John gave her 6 more
rocks. Now she has
10 rocks. How many
rocks did Sally have
to start with?
Separate Problems (Result Unknown)
Sally had 10 rocks.
She gave 4 to John.
How many rocks does
Sally have left?
(Change Unknown)
Sally had 10 rocks.
She gave some to
John. Now she has 6
rocks left. How many
rocks did Sally give to
John?
(Start Unknown)
Sally had some rocks.
She gave 4 to John.
Now she has 6 rocks
left. How many rocks
did Sally have to start
with?
Part-Part-Whole
Problems
(Whole Unknown)
Sally has 4 red rocks
and 6 blue rocks.
How many rocks does
she have?
(Part Unknown)
Sally has 10 rocks. 4
are red and the rest
are blue. How many
blue rocks does Sally
have?
Compare Problems (Difference
Unknown)
Sally has 10 rocks.
John has 6 rocks.
How man more rocks
does Sally have than
John?
(Quantity
Unknown)
John has 6 rocks.
Sally has 4 more than
John. How many
rocks does Sally
have?
(Referent
Unknown)
Sally has 10 rocks.
She has 6 more rocks
than John. How many
rocks does John
have?
Note - Many students will solve subtraction problems by thinking of the related
addition fact. For example to solve the fact 10 – 4 = •the student thinks 4 +
•= 10.
Piaget wrote that this is what always happens in our brain but that over
time we can become so adept at it that it happens on an unconscious level. Soeven though these activities are listed as subtraction activities do not be
surprised if the students figure out the answers using addition.
Problem Structures for Multiplication and Division
Problem Types
Multiplication Partition Division Measurement Division
Equal Group Problems (Whole unknown)
Mark has 4 bags of
apples. There are 5
apples in each bag.
How many apples
does Mark have
altogether?
(Size of groups
unknown)
Mark has 20 apples.
He wants to share
them equally among
his 4 friends. How
many apples will each
friend receive?
(Number of groups
unknown)
Mark has 20 apples.
He puts them in bags
with 5 apples in each.
How many bags did
he use?
Equal Group Problems
(rate)
(Whole unknown)
If apples cost 4 cents
each, how much
would 5 apples cost?
(Size of groups
unknown)
Jill paid 20 cents for 5
apples. What is the
cost of 1 apple?
(Number of groups
unknown)
Jill bought apples for
4 cents each. She
spent 20 cents. How
many apples did she
buy?
Equal Group Problems
(rate)
(Whole unknown)
Peter walked for 5
hours at 4 miles per
hour. How far did he
walk?
(Size of groups
unknown)
Peter walked 20 miles
in 5 hours. How fast
was he walking (in
miles per hour)?
(Number of groups
unknown)
Peter walked 20 miles
at a rate of 4 miles
per hour. How long
did he walk for?
Compare Problems
(Product unknown)
Jill picked 4 apples.
Bill picked 5 times as
many. How many
apples did Bill pick?
(Set size unknown)
Mark picked 20
apples. He picked 4
times as many as Jill.
How many apples did
Jill pick?
(Multiplier
Unknown)
Mark Picked 20
apples and Jill picked
only 4. How many
times as many apples
did Mark pick as Jill
did?
Compare Problems
(Product unknown)
This month Mark
saved 5 times as
much money as last
month. Last month
he saved $4. How
much did he save this
month?
(Set size unknown)
This month Mark
saved 5 times as
much as he did last
month. If he saved
$20 last month, how
much did he save last
month?
(Multiplier
Unknown)
This month Mark
saved $20. Last
month he saved $4.
How many times as
much money did he
save this month as
last?
Problems taken from Van de Walle and Lovin page 78 and 79
