ADDITION AND SUBTRACTION WITH NEGATIVES:
ACKNOWLEDGING THE MULTIPLE MEANINGS OF THE MINUS SIGN
Laura Bofferding (lboff@stanford.edu)
Stanford University
In order for students to use negative numbers in operations, they must make sense of the binary,
symmetric, and unary interpretations of the minus sign. However, little is known about the
extent to which children reason about these three interpretations prior to integer instruction.
Through interviews, I explore the strategies 22 second graders use to solve addition and
subtraction problems with negative numbers, the meanings (binary, symmetric, and unary) they
assign to minus signs, and their understanding of the operations. Analysis of students’ solutions
and language suggest that the second graders think comparably to students who have had
integer instruction.
Purposes of the Study
Algebra is increasingly recognized as a gateway to higher mathematics and as a required skill
for employment; consequently, one goal of middle school mathematics is to help students build a
foundation of algebra readiness skills (EdSource, 2009). Even with the intensified focus, one
reason why students still struggle with algebra concepts is they have difficulty understanding and
working with negative numbers (Gallardo, 2002; Vlassis, 2004). Tatsuoka and Baillie identified
89 different rules that students use to solve integer addition and subtraction problems (e.g.,
subtract the smaller absolute value from the larger and add the sign of the first number);
although, these rules have varying levels of success (Tatsuoka, 1983). In particular, students
have trouble making sense of the changing role of the minus sign (Vlassis, 2008); they solve
problems such as 8 - - 4 by ignoring the duplicate sign (Murray, 1985) or ignoring the first
minus, solving, and adding the sign at the end (Fagnant, Vlassis & Crahay, 2005; Schwarz,
Kohn, & Resnick, 1993; Vlassis, 2004); they claim that two minuses in a problem mean a plus
(e.g., -1 + -5 = 6) (Vlassis, 2004); and they answer problems such as 6 + -2 by adding 2 and then
subtracting 2 (Murray, 1985).
Children’s learning of whole numbers and whole number operations relies heavily on the use
of physical manipulatives and having extended experiences with whole numbers from their
toddler to mid-elementary years (Sarama & Clements, 2009). On the other hand, manipulatives
for negative numbers are contrived, and according to the California mathematics content
standards, students are introduced to negative numbers in fourth grade and are expected to add
and subtract them the following year (California Department of Education, 1999). Some of
students’ struggles with negative numbers might arise because a brief introduction to them a year
before learning integer operations is not long enough or because the instruction they receive is
not sufficient. If this is the case, we would expect some of their same misconceptions to also
arise with students who have not had instruction. The purpose of this study was to closely
explore how students solve addition and subtraction problems involving negative numbers and
explain their thinking, prior to instruction. More specifically, the goal was to ascertain to what
extent students before integer instruction and students post-integer instruction exhibit similar
reasoning.
Theoretical Framework
Gallardo and Rojano (1994) proposed three levels of conceptualization of negative numbers
which are further elaborated by Vlassis (2008). These levels include interpreting minuses as
binary, symmetric, and unary functions. The binary nature of the minus sign corresponds to the
subtraction operation (e.g. 5 - 3 = 2). With negative numbers, the binary interpretation is also
associated with the changing meaning of subtraction as addition when subtracting a negative
number. The symmetric nature of the minus sign is associated with “taking the opposite of a
number” (Vlassis, 2008, p. 561-2), so – (3) = –3 and – (–3) = 3. Finally, the unary nature of the
minus sign aligns with the traditional notion of negatives as a new class of numbers to the left of
zero on the horizontal number line, with decreasing quantities as one moves away from zero.
Students’ descriptions of how they solve integer arithmetic problems provide insight into the
functions that they assign to minus signs across problem types. While we know students who
have been exposed to negative integers utilize these three functions (Gallardo & Rojano, 1994;
Vlassis, 2008), we need more information on younger students’ conceptions to determine if
acknowledgement of the three functions arises out of instruction or not.
As Van Oers (2002) points out, “it is highly relevant to study the processes involved in
symbolizing activities in a detailed way, as well as to find out the dynamics…that may influence
the development of this symbolizing capacity” (p. 30). As children learn new signs, they reflect
on the signs and their possible meanings. Based on his framework, students form pseudoconcepts as they take their current understanding of symbols, such as the minus sign, and apply
them to new situations, such as integer problems. They also start to use language introduced by
adults, although they may still attribute old meanings to the new expressions (Van Oers, 2002).
The extent to which younger students acknowledge the three functions of the minus sign and the
language they use to describe solutions to integer problems have implications on the timeliness
of instruction students should receive on negative numbers.
Research Questions
The following questions guided my research:
1) What do second grade students’ solution strategies indicate about how they make sense of the
changing nature of the minus sign? To what extent do the students acknowledge the three roles
of the minus sign?
2) What do their solutions and language indicate about their understanding of addition and
subtraction with negative numbers?
Methods
Subjects and Site
This exploratory study was conducted at an elementary school in a district in northern
California which has a low percentage of English language learners (only 2.3% of the students
compared to 25% statewide and 9% nationwide) (Education Data Partnership, 2010). It was
important to interview children with a strong grasp of English to get more detailed descriptions
of students’ solution explanations without confounding the difficulty of explaining problem
interpretations with the difficulty of using the language. A total of 22 second graders (8 male, 14
female) from two classrooms agreed to participate in the interviews.
Materials and Data Collection
Individual interviews lasted for 16 minutes on average. During the interviews, students
answered 18 negative number addition and subtraction problems presented one at a time and-except for the first two questions--in random order. Before moving on to the next question,
students had to explain how they came to their answer. Siegler (1996) found that this method is
accurate and reliable.
Based on the position of the signs and numerals in the problems, there are 32 possible integer
addition and subtraction problem types. Two test forms were created in an effort to cover more
of the problem types without overwhelming students with too many questions; students’ first two
questions were the same across forms and four other questions were repeated on both forms.
Aside from the first two problems, the order of the questions was randomized on each test, and
students were randomly assigned to test form. Students taking Form A saw either the first or
second half of their problems with boxes around the numerals to see if this format changed the
ways they talked about the negative signs (see Table 1).
Table 1. Items students completed during interviews
Form A
6-6
4-9
Form B
6-6
4-9
-8 + 4
or
-8 + 4
-8 + 4
-9 + 5
or
-9 + 5
-9 + 5
-3 + 1
or
-3 + 1
-3 + 1
-4 + 2
or
-4 + 2
-4 + 2
9 - -1
or
9 - -1
2 - -4
6 - -4
or
6 - -4
1 - -2
7 - -2
or
7 - -2
3 - -5
5 - -3
or
5 - -3
4 - -6
-4 - -4
or
-4 - -4
-3 + -7
-7 - -7
or
-7 - -7
-2 + -5
-3 - -3
or
-3 - -3
-4 + -8
-8 - -8
or
-8 - -8
-3 + -6
-2 - 5
or
-2 - 5
-3 - 2
-3 - 8
or
-3 - 8
-7 - 4
-1 - 6
or
-1 - 6
-2 - 1
-2 - 7
or
-2 - 7
-6 - 3
Analysis
Students’ solutions were coded for accuracy, and their verbal reports were transcribed and
coded for each question. Solutions were coded for the strategies students used to set up the
problem (e.g., ignored all negatives, reversed the numerals, changed the operation), strategies
students used to compute the answers (e.g., recall, count the distance between numbers, use a
related fact), and manipulations of the answer (e.g., added a negative sign). Student responses
which reflected one of the three meanings of the minus sign (binary, symmetric, and unary) were
identified based on the codes. Students’ responses were also coded for references to their
descriptions of addition and subtraction in comparison to how they solved the problems.
Results
Three Meanings of the Minus Sign
All students correctly solved the first question, 6 - 6, so this problem was dropped from
analysis. Overall, the average number correct score was 3 out of 17 (20%); however individuals’
scores ranged from 0 to 9 out of 17 (0% - 53% correct). Among the group of 22 second graders
and 17 problems, students acknowledged all three roles of the minus sign. Together, they used
the binary function on 198 out of 374 questions (53%), the symmetric function on 74 out of 374
questions (20%), and the unary function on 38 out of 374 questions (10%).
Binary
On average, students used the minus sign as an operator or binary function on 9 of the 17
problems (53%), but the range was large: two students used the binary interpretation on only 2 of
17 problems (12%) and two used it on all 17 problems (100%). However, students frequently
ignored negative signs in order to subtract or interpreted negative signs as minuses in interesting
ways. Table 2 provides examples of students’ binary responses and their corresponding codes.
Notes in “< >” describe students’ actions during the interview and those in “( )” describe my
interpretation of their solutions.
Table 2. Second grade students' binary interpretations of the minus sign
Pseudonym Solution
Gretchen
-9 + 5 = 5
Explanation
Take away the nine and then you add a five.
Code
Subtract: Self
(Student solved 9 - 9 + 5)
Paul
Nicole
Mollie
-9 + 5 = 14
9 - -1 = 7
7 - -2 = 9
It says zero minus nine, nine minus zero, and so it
Subtract:
would be nine…and then I did nine plus five.
New Number;
(Student solved 0 - 9 as 9 - 0, then did 9 + 5)
Reverse Order
It could be seven because minus minus <points to
Subtract:
both “minuses”>. (Student solved 9 - 1 - 1)
Again
I have a master and he taught me you just cross that
Equivalent
<points to the two signs> and do plus. (Student
solved 7 + 2)
Bryan
-4 + 2 = -2
I was subtracting two, so I subtract two of these
Equivalent
<points to the negative four>.
(Student solved -4 - -2)
Monica
5 - -3 = 2
It says five minus three, is two. I did that on
Ignore
fingers. (Student solved 5 – 3)
The problem 4 - 9 presents an interesting case because all students used the binary interpretation
of the minus sign, but 15 of the 22 students (68%) reversed the numerals and solved 9 - 4 = 5.
Surprisingly, none of these fifteen students explicitly stated that you cannot subtract a larger
number from a smaller one, a claim which often persists through middle school regardless of
whether students solve other negative problems correctly (Ball, 1993; Murray, 1985; Vlassis,
2004). Instead, they commented that it was easier to think about the problem with the larger
number first or said they just knew the answer. Three additional students (14%) said there were
not enough to subtract and produced answers of zero (Peled, Mukhopadhyay, & Resnick, 1989),
and four students (18%) crossed the zero point and reached negative answers.
Symmetric
While all students acknowledged the binary nature of the minus sign, only 8 out of 22
students (36%) used the symmetric format of the minus sign. However, when they did use the
symmetric format, they did so for an average of nine problems. Students’ symmetric responses
fall into one of two categories: 1) solving problems as positive and adding a negative sign to the
answer because the problem contained negatives and 2) drawing comparisons between the
positive and negative forms of the problem, calculating using positive numbers, and adding a
negative sign. This distinction may be subtle but represents advancement in students’ thinking.
Amelia belongs to the first category; after solving -7 - -7 = -0, she explained, “It has negatives in
it, and seven minus seven is zero.” In contrast, Bryan changed the original problem, 9 - -1, into a
new problem, -9 + -1, and solved it by reasoning, “Nine plus one equals ten, so if it was minus
nine plus minus one, it would equal minus ten,” which illustrates category 2.
Unary
Finally, 8 out of 22 students (36%) used the unary interpretation of the minus sign and talked
about and operated on negative numbers as separate from positive numbers. These students used
the unary interpretation for an average of five problems. For instance, when solving -2 - 1, one
student, Emily said, “It’s one less than negative two, so it’d be negative three.” While the unary
interpretation was not used as often when students solved problems, 14 of the 22 students (64%)
read negative numbers using language that distinguished them from positive numbers, such as
“minus five,” “below zero five,” or “negative five.”
Overall, 7 out of 22 students (32%) used all three meanings of the minus sign, 1 out of 22
students (5%) used the binary and symmetric formats, 1 out of 22 students (5%) used the binary
and unary formats, and 13 out of 22 students (59%) used only the binary format. Additionally,
use of the boxes around the numbers on Form A did not make a significant difference on their
strategies.
Interpretation of Addition and Subtraction
Especially for students who acknowledged the unary or symmetric interpretations of the
minus sign, the language they used to describe their solution processes reveals conflicting
conceptions of addition and subtraction. When operating with natural numbers, addition results
in a number that is larger than either addend and that is further to the right on the number line.
With negatives, however, adding a smaller positive to a larger negative number (e.g., -8 + 4)
results in a larger number further to the right on the number line but which has a smaller absolute
value. Likewise adding a negative to a negative number (e.g., -2 + -5) results in a number with a
larger absolute value but which is further to the left on the number line.
While all students who took form B of the test got a larger absolute value for problems such
as -4 + -8 (answers were -12 or 12), responses to problems such as -8 + 4 show that several
students were uncertain whether to focus on moving to the right on the number line or on
reaching a larger absolute value. Furthermore, when using the binary meaning of the minus sign,
students either ignored the negative signs and/or reversed numbers so they could do positive
number subtraction or they wavered between moving left on the number line and getting a
number with smaller absolute value. Table 3 provides examples of how students’ answers
changed between number line and absolute value-based answers for addition and subtraction.
Table 3. Students' conflicting explanations for addition and subtraction problems
Pseudonym
Robbie
Solution
-8 + 4 = -12
-4 + 2 = -2
Emily
-7 - 4 = -3
Explanation
Eight plus four equals twelve and
Meaning of Operation
Adding:
negative is just plussing but going lower.
Larger Absolute Value
Negative two…it’s just uh, going up, so
Adding:
negative two is less…is more.
Going Up/Right
You would go from seven, negative
Subtracting:
seven and then minus and take away four
Smaller Absolute
and go from seven, six, five, four, three,
Value
and that would be negative three.
-2 - 1 = -3
It’s one less than negative two, so it’d be
Subtracting:
negative three.
Going Left or Getting
Fewer
Discussion and Conclusions
The results of this study have a few implications toward curricula and task design in the
broader frame of optimizing student understanding. First, merely looking at students’ number
correct scores on integer tests will not provide sufficient information on how students interpret
the negative signs or order and quantity of the negatives. Second graders in this study supplied
incorrect answers on many of the problems but demonstrated insightful and complicated
reasoning about them. For example, when solving -8 - -8, one student, Mollie said, “Well, if that
one wasn’t negative <points to the first negative>, then it would have been sixteen, but it’s
negative, so I think it’s eight,” which is 8 less than the 16 it would have been. Additionally,
some of the students obtained correct answers through incomplete reasoning, which would not be
captured through written tests alone. Therefore, we need to pay more attention to the answers
students provide and select questions which will supply the most contrast and highlight how they
are thinking about the problems. Furthermore, the interviews revealed that students are trying to
make sense of how operations with the negative numbers align with their previous ideas of
addition and subtraction, so future measures should aim to uncover how students think about
these concepts together with the multiple meanings of the minus sign.
Brown and Borko (1992) explain, “Learning involves making connections between new
information and existing systems of knowledge; teaching should facilitate making these
connections by helping students to relate new knowledge to knowledge they have already
developed” (p. 211). Children in second grade are already forming ideas about the multiple roles
of the minus sign or what problems like 4 - 9 mean, and in many cases, they use the same
strategies identified by researchers who have interviewed students after they have had instruction
in negative numbers (e.g., Fagnant, Vlassis & Crahay, 2005; Murray, 1985). Since over half of
the children in the study were willing to explore more than the binary function of the minus sign,
negative numbers can be introduced earlier, and curricula could address their emerging and
conflicting conceptions so that children have more time to become familiar with negatives. One
area which instruction could target is illuminating the lack of the commutative property holding
for subtraction and guiding students to acknowledge the possibility of solving problems like 4 - 9
without feeling the urge to reverse the numerals. Students should also have the opportunity to
explore the multiple meanings of the minus sign. Earlier and extended experiences with the
concepts of negatives may help prepare students for operations with them later.
Aside from instructional implications, these results also suggest future areas of study. In the
case of students who ignore all negative signs and operate with only the binary meaning of the
minus sign, we should explore what experiences influence them to acknowledge the other
functions of the minus sign. Further studies should also compare how students approach the three
meanings of the minus sign and the changing nature of addition and subtraction both before and
after instruction to determine if current instruction helps students progress in their thinking on
these topics. Finally, we should further explore the language teachers use to talk about whole
number addition and subtraction and investigate its impact on students’ reasoning when adding
and subtracting with negatives.
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