Left of Zero: Representing Negative Numbers on the Mental Number line
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Left of Zero: Representing Negative Numbers on the Mental Number line Richard W Prather ([email protected]) University of Wisconsin-Madison Department of Psychology 1202 W. Johnson St., Madison, WI 53706 USA Lera Boroditsky ([email protected]) Department of Brain & Cognitive Sciences, MIT NE20-456, MIT, 77 Massachusetts Ave Cambridge, MA 02139 USA Introduction How do people represent and use abstract mathematical notions that have few or no real-world counterparts? For example, how are we able to think of imaginary numbers, surreal numbers, or even negative numbers? One proposal is that abstract knowledge domains (such as number, time, ideas, ect.) are represented through analogical extensions from more concrete or more experience based domains (e.g., Lakoff & Johnson, 1980, Boroditsky 2000). Lakoff and Nunez (2001) have suggested that the same kinds of experience based may be used for all manner of mathematical concepts. This paper focuses on the domain of negative numbers. Previous studies of positive number representations have found evidence for the psychological reality of a mental number-line (e.g., Dehaene, Bossini, & Giraux, 1993; Reynvoet & Brysbaert 1999). In these studies subjects were shown positive integers one at a time and asked to indicate whether the number was odd or even. The subjects made their response by pressing the ‘odd’ and ‘even’ buttons that were spaced far apart on the keyboard. These studies found that people made an implicit association between spatial location and number magnitude called the Spatial Numerical Association Response Codes or SNARC effect. We consider the new case of negative numbers. How will people represent numbers less than zero on the odd/even task described above? Does the SNARC effect hold only for positive numbers that can correspond to actual numbers of objects in the world? If negative numbers are represented on the mental number-line, will they be laid out according to magnitude or absolute value? Experiment Participants made speeded odd/even judgments on numbers –9 to 9. Numbers appeared in the center of a computer screen one at a time and in random order. For each number subjects were instructed to indicate whether the number was odd or even by pressing the corresponding key. Subjects were equally fast to respond to negative numbers as positive numbers. Overall, the SNARC effect held, in fact was slightly stronger for negative numbers than positive numbers. The results show that participants were faster to make an odd/even judgment for large numbers when the response key was on the right, but faster for smaller and negative numbers when the response key was on the left. The equivalent point was around zero It is especially surprising that the SNARC effect could be found for negative numbers even in a task that included positive numbers since participants did not at all need to consider whether a number was positive or negative in order to determine if it was odd or even. It is quite surprising that the mental number-line representation intruded even in this task. The results suggest that negative numbers, like positive numbers, are represented with the help of a spatial mental number-line that is laid out left to right (at least for people who read left to right) according to magnitude that extends left of zero. Acknowledgments This work was funded by a grant from the Searle Scholarship Foundation to the second author. The authors would like to thank the citizens of Cognation for all the helpful discussions of this research References Boroditsky, L. (2000). Metaphoric Structuring: Understanding time through spatial metaphors. Cognition, 75(1), 1-28. Dehaene, S. Bossini, S., & Giraux, P. (1993) The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122, pg371-396 Lakoff, G. & Johnson, M. (1980) Metaphors We Live by. University of Chicago Press, Chicago. Lakoff, G. & Nunez, R. (2001). Where Mathematics Comes from: How the Embodied Ming Brings Mathematics into Being. Persus Books. Reynvoet B., Brysbaert M. (1999) Single-digit and two-digit Arabic numerals address the same semantic number line. Cognition 72 pg191-201 1394
Transcript of Left of Zero: Representing Negative Numbers on the Mental Number line
Left of Zero: Representing Negative Numbers on the Mental Number line
Richard W Prather ([email protected])University of Wisconsin-Madison
Department of Psychology1202 W. Johnson St., Madison, WI 53706 USA
Lera Boroditsky ([email protected])Department of Brain & Cognitive Sciences, MIT
NE20-456, MIT, 77 Massachusetts AveCambridge, MA 02139 USA
IntroductionHow do people represent and use abstract mathematicalnotions that have few or no real-world counterparts? Forexample, how are we able to think of imaginary numbers,surreal numbers, or even negative numbers? One proposalis that abstract knowledge domains (such as number, time,ideas, ect.) are represented through analogical extensionsfrom more concrete or more experience based domains (e.g.,Lakoff & Johnson, 1980, Boroditsky 2000). Lakoff andNunez (2001) have suggested that the same kinds ofexperience based may be used for all manner ofmathematical concepts. This paper focuses on the domainof negative numbers.
Previous studies of positive number representations havefound evidence for the psychological reality of a mentalnumber-line (e.g., Dehaene, Bossini, & Giraux, 1993;Reynvoet & Brysbaert 1999). In these studies subjects wereshown positive integers one at a time and asked to indicatewhether the number was odd or even. The subjects madetheir response by pressing the ‘odd’ and ‘even’ buttons thatwere spaced far apart on the keyboard.
These studies found that people made an implicitassociation between spatial location and number magnitudecalled the Spatial Numerical Association Response Codes orSNARC effect.
We consider the new case of negative numbers. How willpeople represent numbers less than zero on the odd/eventask described above? Does the SNARC effect hold onlyfor positive numbers that can correspond to actual numbersof objects in the world? If negative numbers are representedon the mental number-line, will they be laid out according tomagnitude or absolute value?
ExperimentParticipants made speeded odd/even judgments on numbers–9 to 9. Numbers appeared in the center of a computerscreen one at a time and in random order. For each numbersubjects were instructed to indicate whether the number wasodd or even by pressing the corresponding key. Subjectswere equally fast to respond to negative numbers as positivenumbers.
Overall, the SNARC effect held, in fact was slightlystronger for negative numbers than positive numbers. Theresults show that participants were faster to make anodd/even judgment for large numbers when the responsekey was on the right, but faster for smaller and negativenumbers when the response key was on the left. Theequivalent point was around zero
It is especially surprising that the SNARC effect could befound for negative numbers even in a task that includedpositive numbers since participants did not at all need toconsider whether a number was positive or negative in orderto determine if it was odd or even. It is quite surprising thatthe mental number-line representation intruded even in thistask.
The results suggest that negative numbers, like positivenumbers, are represented with the help of a spatial mentalnumber-line that is laid out left to right (at least for peoplewho read left to right) according to magnitude that extendsleft of zero.
AcknowledgmentsThis work was funded by a grant from the SearleScholarship Foundation to the second author. The authorswould like to thank the citizens of Cognation for all thehelpful discussions of this research
ReferencesBoroditsky, L. (2000). Metaphoric Structuring:
Understanding time through spatial metaphors. Cognition,75(1), 1-28.
Dehaene, S. Bossini, S., & Giraux, P. (1993) The mentalrepresentation of parity and number magnitude. Journalof Experimental Psychology: General, 122, pg371-396
Lakoff, G. & Johnson, M. (1980) Metaphors We Live by.University of Chicago Press, Chicago.
Lakoff, G. & Nunez, R. (2001). Where Mathematics Comesfrom: How the Embodied Ming Brings Mathematics intoBeing. Persus Books.
Reynvoet B., Brysbaert M. (1999) Single-digit and two-digitArabic numerals address the same semantic number line.Cognition 72 pg191-201
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