Grounding Negative Numbers
Running head: GROUNDING NEGATIVE NUMBERS
Grounding Negative Numbers: A Third Grade Student
Plays Two Mathematical Games
Samuel Otten
Michigan State University
SME 840
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Grounding Negative Numbers
Grounding Negative Numbers: A Third Grade Student
Plays Two Mathematical Games
Many children learn to count even before they enter formal schooling. This
counting, however, is typically with natural numbers (with or without zero) and parents
and television programs often 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); for
example, negative numbers do not make sense if numbers are conceived as the quantity
of a collection of objects and the operations of multiplication and division with negative
numbers are not easily linked with intuition.
Although such challenges exist, it does not mean 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 metaphor, 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 form the concrete groundwork on
which abstract concepts of arithmetic can be built.
Negative numbers, however, do not naturally arise from the grounding metaphors
of object collection (you cannot collect negatively many things), object construction
(again, you cannot collect negatively many things), or measuring stick (you cannot have a
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Grounding Negative Numbers
negative measure). 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. The motion-along-a-path metaphor, as it
makes room for negative numbers, does not lose the numbers from the other grounding
metaphors (e.g., the number 5 can be associated with a group of 5 objects, an object with
length 5, or a motion of 5 units in a positive direction). Such metaphorical extensions,
where ideas are broadened while maintaining the abstract structure of prior concepts, are
characteristic of mathematics as a discipline. In this light, we see that negative numbers
give students 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—may have implications
for the way in which students meet future mathematical extensions.
This mini-study focuses on the following question: How does a particular
elementary student reason about negative number situations in mathematical games and
how does this reasoning relate to the grounding metaphors promoted by the games?
Method
I interviewed Steve, a student described by a school employee as “typical”, in the
spring of his third grade year (9 years old). I chose to work with a third grade student
because I wanted to avoid extensive prior experience with negative numbers (since I am
interested in how he spontaneously makes sense of negative number situations, not how
he has been taught to handle them), but I did want there to be some existing proficiency
with arithmetic and board games (something a younger student may not have).
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Grounding Negative Numbers
Steve’s elementary school is located in a rural town in the Great Lakes region of
the United States. The interview was conducted in his classroom immediately following a
school day. Steve was a convenience selection and a self selection—the former because
he attended a school at which I had familial connections and entrée, the latter because I
arranged the interview with two students but he was the only one who participated.
The interview lasted for approximately 20 minutes and consisted primarily of two
activities. After general introductions and an explanation of the purpose of the interview,
we played the money game (adapted from Linchevski & Williams (1999)). During the
money game, I drew cards that represented monetary happenings for Steve’s parents
(e.g., “Spent $1 on coffee” or “Got $3 from a friend”) and read them aloud while Steve
worked to keep track of his parents’ finances for the day. At the end of a day, I asked
Steve to report the total for that day. We played the money game four times; the first was
a trial to verify that Steve understood the game, the second and third involved red and
blue counters that were available for Steve to use as aides, and the fourth was played with
the counters being replaced by pen and paper. The second activity was the left-right game
(adapted from Wilcox (2008) and Linchevski & Williams (1999)). Steve and I played the
left-right game together on a number line game board (see Figure 1). We each had a
game piece and took turns drawing cards that indicated our moves (e.g., “Right 3” or
“Left 2”). The object of the game was not to reach the end of the game board but to
achieve a position at least ten spaces to the right of the opposing player. We played one
complete left-right game, after which I asked Steve how we might refer to or label the
spaces to the left of “0” on the game board. I also asked him to talk about any similarities,
aside from superficial features such as the drawing of cards, he noticed between the left-
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Grounding Negative Numbers
right game and the money game. Furthermore, I asked him if the left-right game board
would be useful as an aide in keeping track of his parents’ finances.
Left
Right
Left
0
1
2
3
4
5
6
7
8
9 10 11 12
Figure 1. A winning configuration within the left-right game.
The money game, and especially the presence of the counters, was designed to
correspond with the object collection grounding metaphor (Lakoff & Núñez, 2000). Thus,
the interactions around this game may serve to illuminate both Steve’s reasoning within
the object collection metaphor and the theoretical framework’s validity in asserting the
limit of the metaphor with respect to negative numbers. In particular, there was the
possibility of the notion of negative numbers arising within the money game in at least
three ways: (1) as a characterization of total expenses for the day, (2) as a
characterization of the net effect for the day (on the days in which more money was spent
than received), and (3) as an intermediary within the course of a day (i.e., at a point
before the end of the day when more has been spent than received). The left-right game,
on the other hand, was designed to correspond with the motion-along-a-path grounding
metaphor. The notion of negative numbers can arise in the left-right game in several
ways: (1) when a game piece is sitting in an unmarked position to the left of zero, (2) as a
conceptualization of one’s position behind the other player, and (3) as a response to my
explicit encouragement at the conclusion of the game to attend to the positions left of “0”.
Results
Money Game
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Grounding Negative Numbers
Steve was immediately comfortable with the concept of the money game, and by
the end of the trial day (in which negative numbers were not directly encountered) he
seemed to have decided on a technique for using the red and blue counters to keep track
of his parents’ money. He piled the counters in front of himself then slid pieces to his left
when his parents received money, creating what I call his “remaining pile,” and moved
pieces from the remaining pile to a third pile in the center of the table when they spent
money (see Figure 2). This was not his initial strategy for using the counters, which was
to have the blue counters represent whole dollars and the red counters represent change
such as “fifty cents.” He abandoned this strategy when, after a few whole dollar cards
were drawn, he asked whether or not change would be used; I told him that it would not.
At the end of the trial day, Steve reported that his parents “spent three dollars and they
got six dollars.” (His use of “got” meant that they had $6 left at the end of the day, not
that they “got” $6 throughout the day; that is, it corresponded with his remaining pile.)
Steve
counters
spent
pile
remaining
pile
Figure 2. Steve’s setup during the money game.
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Grounding Negative Numbers
Day one began with the parents making two dollars and then spending three
dollars for lunch. Steve moved two blue counters to his remaining pile and, upon hearing
the next card, immediately stopped, looked at me, and said, “They don’t have three
dollars.” He then asked, “They spent how much?” I reread the card and Steve responded,
with his fingers on the two blue counters, that “they only have two dollars.” I stopped to
think about the dilemma with Steve. I asked him if he could think of any way in which
they could still pay for the lunch, even though they had only made $2 that day. Steve’s
first idea was for his parents to eat less food for lunch. Pushing him further, and
appealing to the non-negotiability of the card (which I showed to him), Steve thought for
several more seconds. A possibility would be for his parents to use money that existed
from previous days—the trial day, for instance—to pay for the lunch. Steve did not think
of this, however, and finally shrugged his shoulders and said, “I don’t know.” After a few
seconds more Steve said that his parents could “not have lunch,” thus skipping the card
altogether rather than modifying it. Not wanting to frustrate Steve so early in the
interview, I accepted this approach and we continued. With the next card, his parents
made some money and Steve said that they could now have lunch; he moved three
counters from the remaining pile into the spent pile. The final report for day one was that
they “spent three dollars and they got two dollars left.”
Day two went through several cards before his parents tried to go to the movies
without having made enough money that day to pay for it. Steve quickly said, “They
don’t have enough money, so…skip the movie.” This was the last event for the day, so I
asked Steve for the report and he said that they have $1 left and, after counting his spent
pile, he added that they spent $5 that day.
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Grounding Negative Numbers
For day three, we put away the counters and I replaced them with pen and paper.
As the day proceeded, Steve used a row along the top of the paper to keep track of the
money his parents had left for the day, crossing out previous numerals as he went (thus
forming an analog to the remaining pile). In a second row lower on the page he kept track
of the amount spent throughout the day (analogous to the spent pile). He made a mistake,
however, on the second-to-last card for the day when he subtracted 1 from his remaining
pile instead of 2. This meant that the final card, which was the spending of a dollar,
moved him from 1 to 0 in the remaining pile rather than the correct move from 0 to –1.
This error was especially unfortunate because it meant that he avoided a situation in
which his parents would end the day spending more than they received. (The judgment I
made in the moment was not to correct this arithmetic error, thus preserving Steve’s
confidence, because I still had over half of the interview remaining and Steve seemed to
be somewhat intimidated by the presence of the video camera.) Steve reported that his
parents had “zero dollars left” at the end of day three. Interestingly, he did not report how
much was spent that day, as he had on all the previous days (with the counters). Upon
closer inspection, it was revealed that the row with which he was keeping track of
spending began as a floating total with previous numbers crossed out but changed in the
middle of the day to a discrete record of the different expenses (i.e., expenses of $3 and
$2 recorded as “3 2” rather than “3 5”). Because of this midday modification, it would
have taken a concerted effort on Steve’s part to report the total spending for day three
from his written record.
Left-Right Game
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Grounding Negative Numbers
Steve seemed to be engaged with and excited about the left-right game, asking
several clarifying questions as I explained the rules. One of his questions concerned the
blank left side of the board (see Figure 1) and whether or not it was open for game play; I
said that it was. We each drew a trial card to make sure that we understood how the game
would work and then began the actual game.
Several events took place before Steve eventually defeated me in the game. First,
Steve and I both entered the unmarked land to the left of “0”. Second, there was a point at
which Steve had taken a substantial lead and I asked whether he thought he had won. He
counted with his finger the spaces in between our game pieces and, finding eight,
concluded that he had not. (This method of counting with his finger corresponded with
his general game play technique in which he counted out each space with his game piece,
although he said, with prompting, after the game that he thought of right moves as
addition and left moves as subtraction.) Third, I attempted (unsuccessfully) to elicit
addition and subtraction reasoning based on the game board; more on this below. Fourth,
Steve recognized that he had achieved a winning separation from me even though I was
below “0” on an unmarked space.
After the left-right game, I asked Steve if we could come up with names for the
unmarked spaces and he replied with “negative one, negative two,” and so forth. He also
labeled the spaces with a marker in the conventional manner. It seemed as though he had
previously learned such labels on the number line, though he said that he had not thought
of it until I explicitly asked about naming those spaces. Steve reported that there was “not
really” anything similar between the two games. To push this line of reasoning, I asked if
the game board from the left-right game could be used to keep track of his parents’
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Grounding Negative Numbers
money. He responded that the numbers to the right of “0” could be used to keep track of
whole dollars and the numbers to the left of “0” could be used to keep track of cents.
Asking him for an example, he demonstrated how he would represent $2.50 with a
marker at +2 and another at –5. Then, to correspond with $1 being spent, he moved the
first marker from +2 to +1. (It would be interesting to examine whether this behavior was
related to a “mirror metaphor” (Stacey et al., 2001) wherein cents are recognized as less
than $1 and the “0” on the game board stands in place of the decimal point as the mirror.
This, however, is beyond the scope of the current mini-study.)
Discussion
The two mathematical games present in this paper are related to the object
collection (i.e., collections of dollars in the money game) and the motion-along-a-path
(i.e., moving the game pieces in the left-right game) grounding metaphors. Lakoff and
Núñez (2000) maintain that the object collection grounding metaphor does not allow for
the concept of negative numbers, and this notion was supported by Steve’s performance
in the money game. Using a literal collection of objects to keep track of his parents’
spending, he refused to accept the possibility that they would spend more than they had
made at any point in the day and instead suggested that cards requiring as much be
skipped. This modification of mathematical situations so as to avoid the occurrence of
negative numbers parallels eras of mathematical history in which mathematicians would
reframe mathematical problems such as “what number added to 15 results in 10?” to be
“what number subtracted from 15 results in 10?” (Gallardo, 2002). This is related to
Sfard’s (1991) notion of reification in which mathematical processes, such as subtraction,
are reconceived as mathematical objects, such as negative numbers. Mathematicians in
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Grounding Negative Numbers
the past modified the mathematical process in question so as to avoid the “impossible”
mathematical object of negative numbers. In this mini-study, since I was not investigating
reification on the part of Steve, I purposefully kept the processes (represented by the
cards) conceptually separate from the objects (represented by the money and the game
board positions).
Since the left-right game lent itself to the motion-along-a-path grounding
metaphor, Lakoff and Núñez posit that it provides the conceptual space for negative
numbers. I do not believe, however, that I was successful in encouraging Steve to view
the spaces on the game board as mathematical objects and so could not properly test this
assertion from Lakoff and Núñez. Even though Steve correctly labeled, both verbally and
in writing, the spaces to the left of “0” with negative numbers, it was my impression that
this was done by rote (perhaps he had seen such a number line previously) and without an
accompanying conceptual understanding. The evidence for this claim comes in several
forms. First, Steve seemed to reason exclusively by counting and not by conceiving of the
spaces as numbers. This was the case both when he was moving his game piece and when
I asked him about separation throughout the game (e.g., he would count from 2 to 7
rather than using the number fact 7 – 2 = 5). Second, a person conceiving of the spaces on
the game board as numbers would be expected to recognize the connection between these
and the tracking of money during the money game. Steve, however, did not recognize
connections between the left-right game and money game (or at least he did not report
any). Third, and perhaps most profoundly, when using the game board to keep track of
monetary events, he used the negative numbers as a way to tally cents while the positive
numbers were tallying dollars. This seems to divorce negative numbers from a typical
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Grounding Negative Numbers
conceptual meaning (e.g., negative numbers as expenses) and treat them merely as a
second set of numbers.
There were many limitations to this mini-study beyond the fact that it was a single
interview conducted with a single student. First, there were two points at which Steve
seemed to be fixated on dollars and cents (i.e., when he first used the blue and red
counters and when he used the game board to keep track of money). This affected his
performance on both games and may have been a reflection of recent subject matter in his
third grade mathematics class. Second, I felt as though I failed to push Steve beyond “inschool thinking” (e.g., when thinking about how to buy lunch at the beginning of the
money game), which has been shown to be different for many people than out-of-school
thinking (Lave, 1988). Thus, Steve was limited by his in-school conceptions of negative
numbers, which were underdeveloped, and did not fully access potential out-of-school
means for making sense of the negative number situations. This limitation is likely due in
part to the fact that I conducted the interview immediately after the school day and in
Steve’s actual classroom, firmly within the school context. Finally, there was an instance
when Steve’s arithmetic error prevented him from encountering the negative number
situations in which I was interested (i.e., at the end of day three with pen and paper).
Many of these limitations could have been corrected by piloting the interview or by
expanding the number of students interviewed. For instance, if two students had been
working together as originally intended, the arithmetic error may have been noticed by
the other student and corrected in time to save the upcoming negative number situation.
In summary, this mini-study provided a modest amount of support for the general
ideas of Lakoff and Núñez (2000) with regard to negative numbers. Steve, in particular,
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Grounding Negative Numbers
seemed to reason concretely within the object collection grounding metaphor and avoided
the concept of negative numbers. He also reasoned concretely within the motion-along-apath metaphor and, though he knows the names and labels for negative numbers with
respect to a number line, did not seem to have a conception of negative numbers as
mathematical objects, even though the grounding metaphor promoted by the game made
them conceptually available.
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Grounding Negative Numbers
References
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Lave, J. (1988). Life after school. In Cognition in practice: Mind, mathematics, and culture in
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children's extension of their number concept to include the negative numbers.
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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
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