CGI_Multiplication_And_Division

CGI: MULTIPLICATION AND DIVISION 1
Cognitively Guided Instruction in Multiplication and Division or: How I Learned to Stop
Worrying and Love Math
Christiana Challoner
Arizona State University

Structure and Intuition
Within the Common Core State Standards (CCSS), there are eight mathematical practices
that educators at all levels should help develop in their students in order to help them become
proficient and fluent mathematicians. The first of these standards is to, “Make sense of problems
and persevere in solving them” so that students develop the ability to explain the meaning of a
problem in order to understand it, look at all its possible solutions, analyze givens, make
conjectures about their solutions, monitor and evaluate their progress, and most importantly,
change their course if necessary in order to find solutions rather than simply give up (Common
Core Standards Initiative, 2012). This first standard sets the precedence for understanding,
explaining, reasoning, and solving complex mathematical problems. One way that educators can
do this is through the use of Cognitively Guided Instruction (CGI), which seeks to use a child’s
intuitive understanding of mathematical concepts and use it as a basis for instruction.
The authors of Children’s Mathematics: Cognitively Guided Instruction (1999) note that
young children have different conceptions about mathematical operations than adults do, but that
does not mean they are wrong. Rather, their conceptions “provide a basis for learning basic
mathematical concepts and skills with understanding” (Carpenter, Fennema , Franke, Levi &
Empson, 1999, p. 1). They argue that children do not actually have to be taught that a certain
strategy goes with a certain problem, but rather that the strategies come naturally to children who
construct and model the action or relationships in mathematical problems, whether they are
addition, subtraction, multiplication, or division (Carpenter et al., 1999, p. 3). It is not the
operation in the problem that dictates which strategy a child chooses when solving it, but the
structure of the problem.

The basis of CGI, according to Carpenter et al. is, “that children enter school with a great
deal of informal or intuitive knowledge of mathematics that can serve as the basis for developing
understanding of the mathematics of the primary school curriculum” and that they have the
ability to construct viable solutions to different problems without formal or direct instruction
because of their intuitive problem-solving processes and skills (1999, p. 5). This claim is also
evident in the results of a study conducted by researchers at State University of New York
Albany (Kouba, 1989). Researchers observed a sample of female students four times between
grades two and three while they solved the same set of 24 word problems and found that students
used three main strategies for multiplication: repeated addition, direct counting, and
multiplicative operation. In division, they used repeated subtraction. Researchers concluded that
students acquire intuitive models that they use on problems to which they assign structure to.
Much like the research found in Children’s Mathematics: Cognitively Guided Instruction (1999),
this study supports the idea that children use the structure of a problem in order to assign
meaning and choose a strategy in which to solve the problem. In the three problems that were
given to children, Kouba notes, “the key distinction among these problems is how clearly the
one-to-many mapping relationship is identified” (Mulligan, p. 148). This “one-to-many mapping
relationship” described is almost identical to the distinctions between the three basic problem
types noted by Carpenter et.al. (1999).
A later study, conducted at Macquarie University, sampled 128 students in grades one
through three who were given two multiplication and four division word problems. Researchers
found that children have intuitive two-step models for multiplication and division, the former of
which is based on subtraction. What is most notable about this article is that it builds off of and
references Kouba’s work in 1989 as well as the original work of Carpenter, Ansell, Franke,

Fennema, and Weisbeck (1993) which found that kindergarteners could learn to solve
multiplicative problems by using the problems structure and their own intuitive problem-solving
and reasoning (Mulligan & Mitcelmore, 1997). The research done at Macquarie University found
that multiplication problems can be classified according to the nature of the quantities involved
and the relationships between them (Mulligan & Mitcelmore, 1997), which is supported once
again by the three basic problem types listed and described in CGI. Something else of note
regarding this article is that the research was done in Australia, which shows that while
instruction may have regional or cultural standards and associations, children’s intuitive thinking
in mathematics is something that can be considered universal.
Ultimately, the content reviewed in the three publications illustrates that children have a
far greater intuitive understanding of mathematics than most would think and that they are able
to construct meaning from the structure of a problem and use that structure to choose or design a
strategy in order to solve the problem.
Part 1: The Three Basic Problem Types
In multiplication and division, the problems and strategies involve either grouping
countable objects into one large group or partitioning countable objects into several smaller
groups. The practice of grouping and partitioning countable objects can be categorized into three
problem types. Within the three problem types and their related problems there are three
quantities: the total number of objects, the number of groups, and the number of objects in each
group. Any of the quantities can be unknown. The unknown quantity is what determines the
problem type.
In a Multiplication problem, the total number of groups and the number of objects in each
group are givens and the unknown is the total number of objects. Carpenter et al. note that it is

important to distinguish the difference in what the two given numbers represent in these
problems as it is reflected in the strategies students will use to solve the problem. In a
Measurement Division problem, it is the total number of objects and the number of objects in
each group that is given. The unknown is the number of groups. In order to solve these problems,
children would solve for the number of groups by taking the total number of objects
(measurement) and grouping them into the number of objects in each group (division). In a
Partitive Division problem, the total number of objects and the number of groups are given and
the number of objects in each group is unknown. In order to solve this problem, children would
partition the total number of objects into the number of groups in order to find the number of
objects in each group (Carpenter et al., 1999)
In addition to these three problem types, there are four related problems with similar
structures. These are Grouping/Partitioning, Rate, Price, and Multiplicative Comparison.
According to Carpenter et al., while there are differences between these four problem types, they
are minor and it is not crucial to distinguish where a problem falls within these four categories.
The importance lies in distinguishing the differences between the three basic problem types
described above because that is where children’s thinking is truly reflected. For example, while
Partitive and Measurement Division appear to be similar, it is important to note the difference
between grouping objects, which involves dividing the known total number of objects into the
known number of groups, and partitioning objects, which involves separating the known number
of objects into a known number of groups. This distinction may seem nuanced, but it is important
to understand in order to truly listen to a child’s mathematical thinking and use it as a basis for
instruction.
Part 2: Clinical Interview

The student I selected for the clinical interview, JSB, is an 8-year-old male who presents
with signs of Autism. In class, he rarely focuses long enough to complete assignments and will
turn in blank pages. However, on his Benchmark assessments, he always exceeds the standards.
Because of this, he was recently accepted to represent our classroom in the district’s Math
Challenge. JSB loves puzzles, so I thought that by choosing him for the clinical interview, I
could both gain insight into how he works best so that he can complete assignments in class as
well as give him something engaging to do during Math Centers, something he rarely
participates in. Prior to the interview, I prepared by setting out paper, pencils, colored markers,
and two-color counters the students use as a manipulative in class. I have never seen JSB use
manipulatives in class, but I left them out just in case. JSB rarely shows his work either, so I was
interested to see what he would do to solve the problems. The problems that I gave JSB are the
problems in Table 1.1 and Table 1.2. I wrote the problems on index cards and shuffled them so
that he would not be solving problems from the same context twice in a row.
The interview itself was difficult. I asked JSB to join me at the horseshoe table in the
back of the class because I had some special math problems for him to solve. JSB was eager to
complete the task and readily joined me at the table. I pointed out each of the materials and told
him that he could use anything there that he wanted. He immediately grabbed a sheet of paper
and the markers and started drawing. After explaining to him that I needed him to focus on the
problem I was about to give him, I turned one of the notecards over and JSB read the problem to
himself.
The first problem was a Price/Multiplication problem: Thor’s favorite Pop Tarts cost 4
dollars a box. How much do 8 boxes cost? JSB answered $32 without hesitating. I asked if he
was sure and he nodded. I asked if he wanted to show his work to prove to me that it was $32.

JSB’s response was, “Ms. C, I know 4 x 8 is 32. And you know 4 x 8 is 32. I don’t need to prove
my answer to you because you know.” After that, I did not encourage JSB to show his work, but
instead asked him for another fact that he could use to solve the problem. Typically, he answered
with either another multiplication problem (for example, 6 x 4 = 24, and 4 x 6 = 24) or a division
problem (such as 3 x 5 = 15 so 15/5 = 3). When the problem required division, he proved his
answer by stating the inverse multiplication fact. This pattern of using known facts continued in
the Multiplication, Measurement Division, and Partitive Division problems for Price, Rate, and
Grouping/Partitioning. I asked JSB where he had learned his facts so quickly. He responded that
he did “these kinds of problems” in second grade.
The problems that JSB did appear to struggle with were the Multiplicative Comparison
problems. For these problems, JSB used the paper and pens I had provided. For the question,
“When shrunk, Wasp is 2 feet tall. Captain America is 6 feet tall. How many times taller is
Captain America than Wasp?” JSB started by drawing a horizontal line across his paper and
labeling it 2 feet. He then drew a longer line and labeled it 6 feet. I asked why he chose to draw it
this way. He explained that the shorter line was Wasp, who is 2 feet, and the longer line is
Captain America, who is 6 feet. I asked if there was a reason he drew it horizontally instead of
vertically. His reason was that he was using a number line to solve the problem and then
proceeded to subtract 2 feet from 6 feet for an answer of 4 feet. When I asked how he got his
answer, he explained that he subtracted 2 from 6 and got 4, so Captain America was 4 feet taller
than Wasp. I prompted JSB to read the problem again and double check to make sure he
answered the question the problem was asking. He re-read the problem, out loud at my behest,
and said that his answer was correct before asking for another question.
Four problems later, JSB solved the problem “Captain America is 6 feet tall. He is 3 times as
tall as Wasp when she’s shrunk. How tall is Wasp?” by dividing 6 by 3. When I asked why he chose to

use division to solve this problem, he explained that if Captain America is 3 times as tall as Wasp, then
that means it was a multiplication problem first and he needed to use division to solve it. His answer was
2 feet. I asked him if there was another way he could show his work to prove that his answer was correct.
He drew a picture showing Captain America was 6 feet tall and Wasp was 3 feet tall. He then drew Wasp
again, stacked on top of his last drawing, until there were 3 Wasps standing on top of one another. This
took him 15 minutes and he drew several different versions in the process, but they all illustrated the same
idea that 3 Wasps would equal 1 Captain America. His answer and reasoning were confusing because the
wording was similar to that of the first problem, but his answer and reasoning were different.
The final Multiplicative Comparison problem that JSB solved asked, “Captain America is 3
times as tall as Wasp when she’s shrunk. Wasp is 2 feet tall. How tall is Captain America?” In this
problem, JSB multiplied 3 times 2 in his head and gave me the answer of 6 feet. When I asked why he
chose multiplication, he said it was because Wasp is 2 feet tall and Captain America is 3 times as tall as
Wasp, so he would have to multiply to get the answer. He did not seem inclined to explain his reasoning
any further.
At the end of the interview, I gave JSB the first Multiplicative Comparison problem again, asking
him how many times taller Captain America was than Wasp. JSB once again subtracted 6 from 2, despite
the fact that the problem uses language that implies multiplication, and division, are necessary to solve the
problem. The only logical conclusion that I can come up with is that rather than solving for the unknown
height of one of the characters, this problem asks students to find how many times taller Captain America
is than Wasp. I believe JSB took this to mean the differences between their heights, although subtraction
was not really implied in the problem. Once he’d solved the problem again, JSB asked if I had any more
math problems for him to solve and was disappointed that I did not.
Part 3: Next Steps
Based on the works of Carpenter et al. (1999), I was able to determine that JSB best fits
into the “number facts” stage of problem solving. For a majority of the problems, JSB used

multiplication and related division facts that he already knew in order to solve the problem.
There was some use of modeling in the Multiplicative Comparison that asked how many times
taller Captain America is than Wasp, but even then, two out of three times, he relied on known
facts. There are very few problems in class that he will solve without knowing the fact
beforehand. After the Clinical Interview, I asked JSB to solve for the product in the following
expression: 12 x 4. Much like in the Clinical Interview, I provided paper, pens, pencils, and
counters.
The first thing he did was decompose 12, then he used the distributive property to solve
(3 x 4) + (4 x 4). When I asked why he approached the problem this way, and chose the
distributive property, he explained that he knew what 3 x 4 was and what 4 x 4 was so he could
add the two smaller products to make the larger one. I asked if he could think of another way to
solve it and he argued that there was no other way to solve it because, “you have to know your
multiplication facts, Ms. C. They’re facts.” This confirmed what I observed in the Clinical
Interview: JSB only wants to use known facts to solve multiplication and division problems.
I would like to work more with JSB to both encourage the use of modeling and exploring
other ways of solving multiplication and division problems, such as skip counting forwards and
backwards on a number line. There is no doubt that memorizing multiplication and related
division facts is beneficial, as students will become faster at recall and gain the ability to solve
more difficult or lengthier problems faster, but I believe JSB would benefit from developing
strategies that would enable him to look at a problem from multiple approaches or solve a
problem in multiple ways in order to better understand the why of mathematical procedures
rather than just the how.

Furthermore, there is a chance that working to build these strategies with JSB will
provide him with the challenges he seeks in our math class. He is a student who loves math and
is constantly seeking out harder problems or puzzles that he can solve. I believe that in
developing these strategies, he will have a challenge in having to look at the problems we
currently solve from different angles. In the future, I plan on creating differentiated sets of
instructions for JSB, as he is usually one of the first students done with the classwork, that
require him to solve the problem in another way, such as with a drawing, model, or a number
line. Rather than presenting these use of these other methods as another step in the problem, or
another way to solve the problem, I would try and present it as a puzzle for JSB to solve by
phrasing it as a question. For example, in the problem, “When shrunk, Wasp is 2 feet tall.
Captain America is 6 feet tall. How many times taller is Captain America than Wasp?” I could
ask JSB to solve this not by drawing the lines to model as he did, but by using lined up counters
to represent each character’s total height in order to visualize the problem or asked how he could
solve the problem by counting backwards on a number line.
Summary
Ultimately, this assignment has taught me how to stop worrying and learn to love
math. The way I was instructed in elementary school was very much “drill and kill”: we were
expected to memorize our facts and formulas with very little to no conceptual understanding of
what we were doing or why we were doing it. The “drill and kill” method made me hate math,
and that is not what I want for my students. Unfortunately, that was the only way I knew how to
teach. This assignment has given me multiple strategies and activities to use in the classroom in a
way that makes my students enjoy solving problems by using math.

In regards to children’s mathematical thinking, I was surprised to learn that children
actually have a greater intuitive understanding of what they are doing than I originally thought. It
is not the strategy that students focus on, but the structure which leads to their choosing or
inventing a strategy to solve the problem. Already, in my class, I have started to use this in my
instruction. Rather than beginning immediately with instruction, I give students a problem to
solve without giving them the strategy first.
This has already greatly improved both student participation and student achievement in
my room. The students see it as a challenge and they eagerly try and solve it with strategies I
would never have thought of. It has given me insight into their thinking and their own processes
that I can then use to guide my instruction in a way that enables them to better comprehend what
I am teaching them. Another element of CGI that I plan on incorporating in my classroom is the
constant use of problem solving so that students can continue to learn and improve upon their
computational skills (Carpenter et al., 199, p. 96). I look forward to the positive changes this
will make in my classroom and my instruction.

References
Carpenter, T. P., Fennema , E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children's
mathematics: Cognitively guided instruction. Portsmouth, NH : Heinemann
Common Core Standards Initiative. Standards for Mathematical Practice. 5 March 2014.
http://www.corestandards.org/Math/Practice
Kouba, K. L. (1989). Children's solution strategies for equivalent set multiplication and division
word problems. Journal for Research in Mathematics Education, 20(2), 147-158 . doi:
JSTOR
Mulligan, J. T., & Mitchelmore, M. C. (1997). Young children's intuitive models of
multiplication and division. Journal for Research in Mathematics Education, 28(3), 309-
330. doi: JSTOR

Appendix A
Table 1.1 Multiplication, Measurement Division, and Partitive Division Problems
Multiplication Natasha Romanoff is baking cookies. There are
3 trays of cookies and 12 cookies on each tray.
How many cookies does she have all together?
Measurement Division Natasha Romanoff is baking cookies. She has
36 cookies. She puts 12 cookies on each tray.
How many trays can she fill?
Partitive Division Natasha Romanoff is baking cookies. She has
36 cookies. She put the cookies onto 3 trays
with the same number of cookies on each tray.
How many cookies are on each tray?
Table 1.2 Grouping/Partitioning, Rate, Price, and Multiplicative Comparison Problems
Problem Type Multiplication Measurement
Division
Partitive Division
Grouping/Partitioning Bruce Banner has 4 test
tube racks in his lab.
There are 6 test tubes in
each rack. How many
test tubes are there all
together?
Bruce Banner has some
test tube racks in his lab.
There are 6 test tubes in
each rack. All together
there are 24 test tubes.
How many test tube
racks does Bruce have?
Bruce Banner has 4 test
tube racks. There are the
same number of test
tubes in each rack. All
together there are 24 test
tubes. How many test
tubes are there in each
rack?
Rate Pizza Dog runs 5 miles
in an hour. How many
miles can he run in 3
hours?
Pizza Dog runs 5 miles
in an hour. How many
hours will it take for
him to run 15 miles?
Pizza Dog ran 15 miles.
It took him 3 hours. If
he ran the same speed
the whole way, how far
did he run in one hour?
Price Thor’s favorite Pop
Tarts cost 4 dollars a
box. How much do 8
boxes cost?
Thor’s favorite Pop
Tarts cost 4 dollars a
box. How many boxes
could he buy for $32?
Thor bought 8 boxes of
Pop Tarts. He spent a
total of $32. If each box
costs the same amount,
how much did one box
cost?
Multiplicative
Comparison
Captain America is 3
times as tall as Wasp
when she’s shrunk.
Wasp is 2 feet tall. How
tall is Captain America?
When shrunk, Wasp is 2
feet tall. Captain
America is 6 feet tall.
How many times taller
is Captain America than
Wasp?
Captain America is 6
feet tall. He is 3 times as
tall as Wasp when she’s
shrunk. How tall is
Wasp?

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