Students and Negative Numbers
Running head: STUDENTS AND NEGATIVE NUMBERS
Down and to the Left: Students’ Movement Toward Negative Numbers
Samuel Otten
Michigan State University
SME 840
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Students and Negative Numbers
Down and to the Left: Students’ Movement Toward Negative Numbers
Many children are able to count even before they enter formal schooling. Having
learned from family members, daycare workers, educational television programs,
educational software or elsewhere, preschool-aged children often have experiences
counting people, toys, or bites of vegetables. This counting, however, is typically in the
realm of natural numbers (with or without zero). Most parents and television programs
leave the challenge of teaching negative numbers to the school system. The topic of
negative numbers is especially challenging for several reasons (Kilpatrick, Swafford, &
Findell, 2001; Stacey, Helme, & Steinle, 2001). To cite just two of these reasons,
negative numbers do not make sense if numbers are conceived as the quantity of a
collection of objects (more on this below) and the operations of multiplication and
division with negative numbers are not easily linked with intuition.
Although such challenges exist, this is not to say that negative numbers are
divorced from our experiences with the world. Lakoff and Núñez (2000) present a
cognitive theory of the foundations of mathematical knowledge that is built upon
conceptual metaphors, a mechanism by which abstract ideas are comprehended via a
mapping to concrete concepts grounded in sensory-motor experience. Lakoff and Núñez
identify four grounding metaphors for notions of number and arithmetic—object
collection, object construction, measuring stick, and motion along a path. They contend
that pervasive experiences with, respectively, groups of objects, wholes made up of parts,
length measurements, and continuous spatial motion lay the concrete groundwork on
which abstract concepts of arithmetic can be built.
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Students and Negative Numbers
Negative numbers, however, do not naturally arise from the object collection,
object construction, or measuring stick grounding metaphors. It is the grounding
metaphor of motion along a path, where zero is identified as a point-location separating
the path into two distinct sides, that provides a concrete interpretation of negative
numbers. But even in this case, the operations of multiplication and division of and by
negative numbers force the metaphor to be extended. While it is possible to conceptualize
some multiplication, such as -3 multiplied by 2, as moving 3 units to the left two times,
multiplication such as -3 multiplied by -2 cannot be interpreted as repeated motion to the
left 3 units because there is no concept of performing an action -2 times. Thus the
conceptual metaphor must, according to Lakoff and Núñez, be stretched according to the
previously established laws of arithmetic (e.g., commutativity).
Such metaphorical extensions, moving along formal pathways from grounded
experience to more abstract concepts, are characteristic of mathematics as a discipline. In
this light, we see that negative numbers provide students with an early and powerful
opportunity to engage in mathematical thought. It is therefore possible that the experience
students have with the move to negative numbers—be it intellectually fulfilling or
frustrating, sensible or senseless—will have implications for the way in which students
meet future mathematical extensions (of which there are many). I intend to use this paper
to investigate a handful of articles on students and their relationship with negative
number. I begin with a bit of historical context and a pair of research articles, one which
attempted to provide concrete experiences of negative numbers and another which
uncovered a conflation of conceptual metaphors around negative numbers. I then turn to a
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Students and Negative Numbers
pair of practitioner articles, both of which share classroom activities involving negative
numbers and insights regarding student thinking.
An Examination of Two Empirical Studies
The problems that children face with regard to negative number, some of which
are indicated above, are not unlike the problems that mathematicians faced throughout
the history of the subject. The motion-along-a-path metaphor, the grounding metaphor for
negative numbers, was not accepted in Europe until the sixteenth century, according to
Lakoff and Núñez (2000, p. 73). There is evidence, both before and after this time, of
mathematicians struggling with the concept of negative numbers. As Gallardo (2002)
reported, mathematicians of high regard, from Descartes to Euler to Cauchy, had to
grapple with negative numbers and were not always successful. Diophantus, Fibonacci,
D’Alembert, and others dealt with negative numbers that arose as solutions to equations
by disregarding them as impossible (in much the same way as students of today deal with
complex solutions) or by declaring the problem ill-posed. In other words, upon reaching a
negative solution they stated that the original problem had been phrased improperly and,
with the appropriate adjustment, would result in the “correct” positive solution. For
example, the question “What number when added to twenty results in fifteen?” would be
rephrased as “What number when subtracted from twenty results in fifteen?”
Such an adjustment avoids the notion of negative numbers as mathematical
objects in their own right and shifts the focus back to the (accepted) process of
subtraction. Sfard (1991) explicated this transformation in which mathematical processes
are constructed, encapsulated, and then reified as objects, and she linked it with various
conceptual obstacles in mathematics education. In the case of negative numbers, students
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Students and Negative Numbers
move from the process of subtracting to the object of a negative number. In terms of the
grounding metaphors, the shift is from the process of subtracting—that is, removing
objects from a collection, reducing the length of something being measured, or moving
along a path to the left—to a notion of negative numbers as point-locations on the
number line in their own right.
In an attempt to use the insights from Sfard (1991) to guide instruction,
Linchevski and Williams (1999) developed an instructional method for the purpose of
extending students’ conception of number to include negative numbers. They conducted
two instructional programs with groups of upper-elementary students. In the first
program, students simulated a situation in which they had to monitor the number of
people entering (always through the front door) and exiting (always through the back
door) a dance club. To keep track of the changes, the students used a blue abacus, adding
a bead for every entry, and a yellow abacus, adding a bead for every exit. The researchers
then worked to draw the students’ attention to the net effect of various changes and to
push the students to develop strategies for keeping track of the changes even when some
of the beads have ran out (e.g., compensating by removing three yellow beads instead of
adding three blue ones). In a sense, this presented the students with a realistically
motivated situation in which the object collection metaphor was extended to dual
collections, with the idea being that one collection could serve as a conceptual grounding
for positive integers and the other for negative integers. Linchevski and Williams found
that this first program was fairly successful in fostering a general concept of negative
numbers in their students, but there was not a simultaneous development of mathematical
language around negative numbers (i.e., students talked about “ins” and “outs” instead of
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Students and Negative Numbers
“positives” or “negatives”). Furthermore, conceptions of subtraction with negative
numbers were much thinner than the conceptions of addition, even though the researchers
felt they should be on the same plane.
In their second instructional program, Linchevski and Williams (1999) introduced
a dice game to the groups of students. The game started with a blue die and a yellow die
corresponding to blue and yellow teams, and the goal of the game was for one team to
gain a lead of eight points over the other team. Score was kept with the same abacuses as
before. Students were able to quickly create shortcuts and alternate strategies for scorekeeping (e.g., a roll of a blue 5 and a yellow 4 could be scored with 1 blue bead or by
taking 1 bead away from yellow), all falling under the guidance of game “fairness.” Over
time, the researchers modified the game, replacing blue and yellow with the markings
“+” and “–” then moving to a single die with three plus scores and three minus scores and
a card that is drawn simultaneously that says “add” or “subtract.” So, in this final version
of the dice game, if the die came up “+2” and the card read “subtract” then the “+” team
would lose two points (which is equivalent to the “–” team gaining two points). If the die
came up “-2” and the card read “add” then the “–” team would gain two points (which is
equivalent to the “+” team losing two points). In this case, Linchevski and Williams
found that students developed notions of negative number addition and subtraction as
well as mathematically accurate language around them. They recognized, however, the
limit of this model with regard to multiplication and division of negative numbers as well
as the limit of any model with regard to complete comprehensiveness of the concepts.
Rather than developing an instructional method for negative numbers, other
researchers have worked to better understand the manner in which negative numbers are
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Students and Negative Numbers
currently taught and the student difficulties that may be associated with them. For
example, Stacey, Helme, and Steinle (2001) looked closely at the way in which some
lower elementary students conceive of negative numbers, decimals, and unit fractions.
They found that the conceptual metaphor of a mirror is employed by some students in all
three cases. This is most common (and perhaps most natural) with negative numbers as
they are conceptually reflection symmetric about zero. Some students also think of
decimal places as reflective about the ones place (e.g., hundreds, tens, ones, tenths,
hundredths) or erroneously around the decimal place. This, for some students, is linked to
the mirror metaphor of the natural numbers and their inverses, with 1, 2, 3, 4 and so forth
going off to the right of 1 and 1/2, 1/3, 1/4 and so forth going off to the left of 1. Stacey
and her colleagues (2001) documented several errors that they tied back to a conflation of
these metaphors, such as a confusion between 0 and 1 (the two reflection points) and the
misconception that decimals such as 0.5 or fractions such as 1/3 were less than 0.
Just as Lakoff and Núñez (2000) highlighted the role of metaphor in mathematical
knowledge, Stacey and her colleagues highlighted the importance of carefully selected
metaphor in mathematical learning. If the same or similar metaphors are used for more
than one concept, mathematics educators need to be prepared for students who conflate
them. This points to the importance of thinking carefully about the conceptual metaphors
that may be elicited or formed by instructional practices. We turn now to two pieces of
literature focusing on such practice.
An Inspection of Two In-Class Techniques
Wilcox (2008) is a mathematics educator who recorded her experiences exploring
negative numbers with her first grade daughter who had never before been instructed in
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Students and Negative Numbers
negative numbers. She and her daughter played a game on a bare number line in which a
card was drawn and the daughter moved a game-piece to the right or to the left. Zero and
the positive integers were marked on the board, but to the left of 0 were simply bare tick
marks. When the daughter was required to move to the left of 0, she hesitated but
eventually moved to the left tick marks and, through prompting from Wilcox, named
these spaces as “zero cousin minus one,” “zero cousin minus two,” and so on. She also,
after playing for awhile, correctly labeled the tick marks to the left of 0. Wilcox then
turned over the number line and played a new game in which her daughter had blocks and
either added or took away blocks based on the drawn cards. Under these circumstances,
the daughter rejected any notion of negative numbers because once the blocks were all
gone there were simply none left (possibly explaining her labeling of negatives as “zero
cousins”). Turning back over the number line, however, the daughter could reinterpret the
negative block situation appropriately.
Putting aside the obvious methodological limitations of this work and taking it for
what it is, the work of Wilcox clearly connects to the grounding metaphors of Lakoff and
Núñez. The game with the number line is a straightforward appeal to the motion-along-apath metaphor, and the daughter was able to make the extension to the negative realm,
even creating appropriate names and labels. The game with the blocks, however, failed to
extend, as Lakoff and Núñez would expect, because the object collection metaphor to
which it was tied does not easily handle negative numbers. It would have been interesting
to see what would have happened if Wilcox had made available a second set of blocks
similar to the double object collections of Linchevski and Williams (1999). In fact, there
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Students and Negative Numbers
is evidence that, more than 2000 years ago, some Chinese used rods of two different
colors in such a way to represent gain and loss or buy and sell (Gallardo, 2002).
Moving from an individual child to an individual class, Behrend and Mohs (2006)
followed an inquiry-based class from first to second grade, focusing their report on the
students’ interactions around negative numbers. Based on a discussion about whether
numbers ever stop in either direction (and the conclusion that they don’t), the teacher and
her students placed negative numbers on the classroom number lines. Later in first grade,
the students spontaneously used negative numbers in number sentences. For example,
when asked to generate number sentences involving 10, students would respond with “-2
+ 12 = 10.” Later, in grade 2, one-third of the class regularly used negative numbers in a
student-generated subtraction algorithm (e.g., 245 – 126 = 100 + 20 + -1 = 119). By the
end of second grade, all but one of the students recognized the difference between “5 – 3”
and “5 – -3,” something that proved problematic for a majority of fifth graders at the
same school.
These remarkable occurrences, which Behrend and Mohs accounted for primarily
with the sense-making that was encouraged to take place in those particular classes,
demonstrate that students have the concrete experiences necessary for the introduction of
negative numbers much earlier than the grade 5 or 6 that is assumed by many curricula.
By building upon the motion-along-a-path metaphor, as evidenced by the number lines at
the beginning, this class was able to largely grasp negative numbers in relation to the
natural numbers and the operations of addition and subtraction, although not
multiplication and division. They also linked this with other areas of mathematical
learning in a sense-making way (e.g., in the multi-digit subtraction algorithm).
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Students and Negative Numbers
Conclusion
In this brief and somewhat rushed paper, I have examined ways in which
instruction can be guided by concrete experience and grounding metaphors (Behrend &
Mohs, 2006; Linchevski & Williams, 1999; Wilcox, 2008) and an example of how
conceptual metaphors can lead to confusion and misconceptions (Stacey et al., 2001).
Though it was not the direct focus of the paper, it also became clear that the operations of
multiplication and division of negative numbers may be even farther removed from
concrete experience and so even more difficult for students to grasp conceptually. The
ideas presented in this paper can be a source for future work attempting to recreate some
of the instructional activities contained herein (e.g., Linchevski’s dice game or Wilcox’s
card game) or future work attempting to extend these ideas into the realm of
multiplication and division with negative numbers.
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Two Pieces by Dewey
References
Behrend, J. L., & Mohs, L. C. (2006). From simple questions to powerful connections: A twoyear conversation about negative numbers. Teaching Children Mathematics, 12, 260-264.
Gallardo, A. (2002). The extension of the natural-number domain to the integers in the transition
from arithmetic to algebra. Educational Studies in Mathematics, 49, 171-192.
Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn
mathematics. Washington, DC: National Academies Press.
Lakoff, G., & Nunez, R. E. (2000). Where mathematics comes from: How the embodied mind
brings mathematics into being. New York: Basic Books.
Linchevski, L., & Williams, J. (1999). Using intuition from everyday life in 'filling' the gap in
children's extension of their number concept to include the negative numbers.
Educational Studies in Mathematics, 39, 131-147.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and
objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1-36.
Stacey, K., Helme, S., & Steinle, V. (2001). Confusions between decimals, fractions and
negative numbers: A consequence of the mirror as a conceptual metaphor in three
different ways. Paper presented at the International Conference for the Psychology of
Mathematics Education, Utrecht, The Netherlands.
Wilcox, V. B. (2008). Questioning zero and negative numbers. Teaching Children Mathematics,
15, 202-206.
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