A CGI Approach to Integers
HELPING TEACHERS STRUCTURE
THEIR INTUITIVE KNOWLEDGE ABOUT
CHILDREN’S UNDERSTANDINGS OF
NEGATIVE NUMBERS
Dr. Randolph Philipp
Dr. Jessica Pierson Bishop
San Diego State University
University of Georgia
This work is supported by the National Science Foundation, grant number DRL-0918780
Let’s Start with a Game
 Write a number. Any number. Circle it.
 Add 10
 Multiply by 2.
 Add 4.
 Add 8.
 Divide by 2.
 Subtract 17.
 Subtract your original circled number.
 Draw a rectangle around your final number.
“What’s in a name? That which we call a rose
by any other name would smell as sweet.”
 Circle the larger, or write “=“ if they are the equal.
4
3
1
More than two thirds of 5th
graders circled 1.
negative
Today, let’s try not to think negatively
about negative numbers.
Solve each of the following and think about how you
reasoned. If you have time, solve another way.
1) 3 – 5 = ___
2) -6 – -2 = ___
3) - 2 + ___= 4
4) ___+ -2 = -10
Project Z: Mapping Developmental Trajectories
of Students’ Conceptions of Integers

Lisa Lamb, Jessica Bishop, & Randolph Philipp,
Principal Investigators

Ian Whitacre, Faculty Researcher

Spencer Bagley, Casey Hawthorne, Graduate Students

Bonnie Schappelle, Mindy Lewis, Candace Cabral,
Project researchers

Kelly Humphrey, Jenn Cumiskey, Danielle Kessler,
Undergraduate Student Assistants
Funded by the National Science Foundation, DRL-0918780
So, Why Negative Numbers?
 Even secondary-school students who can
successfully operate with negatives have trouble
explaining.
Even Successful Students Can Not Explain.
 Keep Change Change
So, Why Negative Numbers?
 Even secondary-school students who can
successfully operate with negatives have trouble
explaining.
 Many middle-school students do not understand
what they are doing with negatives.
Many Middle-School Students Do Not Understand
 Valentina and friends—Does 6 – -2 = 6 + + 2?
 To a student who answered that the two expressions
are not equal, the interviewer said, “That’s kind of
crazy that you are allowed to change the problem
and it gives you a different answer.”
 The student responded, “That’s math.”
The Common Core State Standards (CCSS) of
Mathematical Practice
1 Make sense of problems and persevere in solving them.
2 Reason abstractly and quantitatively.
3 Construct viable arguments and critique the reasoning
of others.
4 Model with mathematics.
5 Use appropriate tools strategically.
6 Attend to precision (in language and mathematics)
7 Look for and make use of structure.
8 Look for and express regularity in repeated reasoning.
The CCSS of Mathematical Practice
1 Make sense of problems and persevere in solving them.
2 Reason abstractly and quantitatively.
3 Construct viable arguments and critique the reasoning
of others.
4 Model with mathematics.
5 Use appropriate tools strategically.
6 Attend to precision (in language and mathematics)
7 Look for and make use of structure.
8 Look for and express regularity in repeated reasoning.
What Is Mathematics to Students?
70
- 23
53
76
- 23
53
“Yes, math is like that sometimes.”
So, Why Negative Numbers?
 Even secondary-school students who can
successfully operate with negatives have trouble
explaining.
 Many middle-school students do not understand
what they are doing with negatives.
 Many young children hold informal knowledge about
negatives on which instruction might be based.
Reynaldo, 4th grade, -3 + 6 = ___
In a video clip, Reynaldo says that 3 of the 6 go to -3
leaving 3. He then uses a context of money, saying
that if he borrowed $3 from a friend, he might give
the friend $3 from $6 his mom gave him, leaving $3.
Noland, 4th grade, -5 + -1 = __
 “Minus 5 plus minus 1 equals minus something …; 5
plus 1 equals 6, so this is minus 6 …. If you add these
two together, it makes it farther from the positive
numbers.”
Liberty, 2nd grade, -2 + __ = 4
 Liberty solves -2 + __ = 4 using a counting strategy:
 “Negative 1, negative 0, 0, 1, 2, 3, 4, so the answer is
7.”
So, Why Negative Numbers?
 Even secondary-school students who can
successfully operate with negatives have trouble
explaining.
 Many middle-school students do not understand
what they are doing with negatives.
 Many young children hold informal knowledge about
negatives on which instruction might be based.
 Understanding negatives is necessary for algebra
success (inverses, even/odd functions, absolute
value, rates of change, etc.): Compare – x and x.
One Last Reason…
 Negative numbers comprise (almost) half of the
reals!
So what?
Looking Back
Joan has 4 tennis balls. Her mother gives her
some more and now she has 9. How many
tennis balls did Joan’s mother give her?
 What type of problem is this?
A) for you?
B) for the parents of your students?
C) for textbook writers?
D) for your students?
CGI Join and Separate Problems
From Carpenter, Fennema, Franke, Levi, and Empson, 1999
Result
unknown
Change Unknown
Start Unknown
Join
Leon has 9
pencils. He
gets 5 more
pencils. How
many pencils
does Leon have
now?
Joan has 4 tennis balls. Her
mother gives her some more
and now she has 9. How many
tennis balls did Joan’s mother
give her?
Mary has some
marbles. She went
to a party and won
5 more, and now
she has 11. How
many marbles did
she have before the
party?
Separate
Lesley baked 9
cookies. Kevin
ate 3 of them.
How many
cookies are
left?
Eleven children were playing
in the sandbox. Some children
went home. There were 3
children left still playing in the
sandbox. How many children
went home?
Mary had some
bookmarks. She
gave 3 to Kevin.
Now she has 9 left.
How many
bookmarks did she
have to start with?
Some Integer History
 Twenty children go on a field trip. If each car can
take 4 children, how many cars are needed?
 Four children fairly share 6 chocolate chip cookies.
How many cookies will each child get?
Children and Mathematicians Agree
Resisting quantities that are “less than nothing” and
the lack of tangible or realistic interpretations.
 “Negative numbers aren’t really numbers because we
don’t count with them. And there’s no negative 1 cube.”
(Violet, 2nd grade)
 “Zero is nothing and negative is more nothing.”
(Rebecca, 2nd grade)
 One “must reject the definition … of -a, that it is less
than nothing” (De Morgan, 1902, p. 72)
Children and Mathematicians Agree
Resisting removing something from nothing or removing
more than one has
 “Three minus 5 is 0 because you have 3 and you can’t
take away 5, so take away the 3 and it leaves you with
0.” (Sam, 1st grade)
 “I know some who cannot understand that to take four
from nothing leaves nothing” (Pascal, 1669/1941, p. 25)
So what?
Ways of Reasoning
 Students who have negative numbers in their
numeric domains typically approach integer tasks
using one of the following five ways of reasoning:
• Order-based reasoning
• Analogically based reasoning
• Computational reasoning
• Formal mathematical reasoning
• Alternative reasoning
Order-based Reasoning
 Students use the sequential and ordered nature of
numbers to reason about a problem. Strategies
include counting strategies and number line/motion.
 Are any of the strategies we shared earlier order
based?
 Did any of the students in the videos shared use an
order-based strategy?
Predictions
 Rosie, a 2nd grader, will use a number line and/or
counting to try to solve each of the following
problems. Predict which problems she will
successfully solve.
• 3 – 5 = __
• __ + 5 = 3
• -5 – 4 = __
• 5 + __ = 2
• -9 + 5 = __
• 6 - -2 = __
Order-based reasoning
Rosie, Grade 2
 The interviews with Rosie occurred over a period of 3
months.
 3-5 = __ Rosie solves 3 – 5 = __ by jumping to zero.
 -9+5 = __ She said -9 minus 1 first. Rosie is leveraging
the idea that subtraction reduces the size of a number and
addition increases the size or magnitude of a number. But
with negative numbers, addition is reducing the magnitude
of the number but not the number itself. Thus her use of
the language minus.
 __ + 5 = 3 Note that Rosie solved it by using a number line
and trial and error.
Rosie, Grade 2
 Think about Rosie who solved, 3 – 5 = __, -9 + 5 =
__, __ + 5 = 3, and -5 – 4 = __ but could not solve 5
+ __ = 2 and insisted that it was not possible to solve
6 + -2.
 What made the last two problems “not possible” for
Rosie?
Order-based reasoning
 Rosie, at the end of our problem-solving
interviews
Analogically (Analogy)-based Reasoning
 Students reason by relating negative numbers to
another idea, concept, or context. Students reason
about negative numbers on the basis of behaviors
observed in this other concept or idea. Negative
numbers may be related to a countable amount or
quantity and tied to ideas about magnitude. They
may also be related to contexts.
 Are any of the strategies we shared earlier
analogically based?
 Did any of the students we shared video of use an
analogically based strategy?
Analogically based reasoning
 Randy, 1st grade, -8 – -1 = __
 Mathematically speaking, Randy is using the
distributive property to factor out the negative 1,
though of course he does not state it using those
terms!
Visual Representation of Randy’s Strategy
-1
-1
-1
-1
-1
-1
-1
-1
8away
negative
ones
Take
1 negative
one
Leaves
7 negative
ones
So what?
Analogically based reasoning
 Treating Negatives like Positives.
 Multiple students solve
__ + -2 = -10 and -5 + -1 = __.
Formal mathematical reasoning
 Negative numbers are treated as formal objects that
exist in a system and are subject to fundamental
mathematical principles that govern behavior.
Formal strategies often involve comparisons to
other, known, problems so that the logic of the
approach remains consistent and underlying
structural principles are not violated.
 In the last of Rosie’s clips, she used formal
mathematical reasoning.
Formal mathematical reasoning
 Noland, Grade 4, -5 – -3 = __ compared to -5 + -3 = __
 Rosie held the first number and operation constant, varying
the second number—she compared 2 – 4 to 2 – - 4 and
conjectured that subtracting [or adding] a negative does the
opposite than what it normally does. She reasoned that
subtracting -4 would move to the right on the number line.
Noland, instead, held both numbers constant and varied
the operation, leveraging the inverse relationship between
addition and subtraction. He used his earlier observation
that the addition of two negative numbers moves one
farther from the positives to conjecture that subtracting two
negative numbers moves one closer to the positives
Teachers’ Knowledge About Integers
To begin to make sense of how teachers think about
integers, we interviewed 10 seventh-grade teachers
to determine their understanding of integers and
their perspectives about teaching integers and about
students’ thinking. We posed integer tasks, we asked
them about their teaching, and we showed them
video clips of children solving open number
sentences.
Teachers’ Approaches to Integer Tasks
Teachers possess these rich ways of
reasoning (WoR) about integers.
They are
 Able to flexibly apply WoR,
 Able to analyze problems and strategically
combine WoR.
Kalani:
-3 – ☐ = 2
First, Kalani thinks of moving from -3 to 2, as if the number
sentence were -3 + ☐ = 2.
“So I am thinking about the number line …. So I
am starting somewhere, … and what do I do to
end up at positive 2? I am moving 1, 2, 3, 4, 5—
five units to the right.” (Order based)
Second, Kalani accounts for the subtraction sign.
“I am moving the opposite direction [because of
the subtraction sign], so I would write down
negative 5 here.” (Formal)
Interpreting Student Thinking
Difficulty understanding student
reasoning
Kalani: “I think he got confused. There’s no
context involved .… So I don’t see a clear
understanding on his part at all.” (referring to
Noland’s solving -5 + -1 = ☐)
“I don’t quite understand him when he used the
opposite. Opposite of what? …There is no
context.” (referring to Noland’s solving -5 – -3 =
☐)
Pedagogical Goals
 Did we expect teachers to mention ways of reasoning





as goals for instruction?
No!
Did we expect teachers to apply ways of reasoning?
We did not know.
Did they?
Yes!
The Land of Knowledge
for Teaching Mathematics
One Road
A Road Through
Children’s Mathematical
Thinking
So what?
 Children struggle with the same issues about
negative numbers that mathematicians struggled
with historically, but integers provide a rich domain
for reasoning. Furthermore, children come to us
already applying these rich ways, and teachers use
these ways, too.
A Final Thought:
The best approach for teaching integers is….
…..there is no best approach.
We conjecture that flexibility, the ability to select
approaches that are efficient and effective, is the
hallmark of rich integer reasoning.
We suggest that children be given opportunities to
tap into their rich, intuitive understandings of
negative numbers, understandings grounded in
their ordered-based and analogical reasoning, and
then we stand back and let them enjoy the magic of
making sense of wacky but wonderful negative
numbers!
Discussion