PROCEEDINGS of the HUMAN FACTORS AND ERGONOMICS SOCIETY 52nd ANNUAL MEETING—2008

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Visual Representation of Arithmetic Properties Douglas J. Gillan Department of Psychology North Carolina State University Raleigh, NC [email protected]

Copyright 2008 by Human Factors and Ergonomics Society, Inc. All rights reserved. 10.1518/107118108X350203

Quantitative graphs use a spatial metaphor in which spatial features represent numerical amounts. Tally number systems (used in ancient civilizations) also made use of the spatial feature of length. Arithmetic properties, such as the commutative property of addition, can also be represented spatially. An experiment trained college students about 11 basic arithmetic properties either using numerical examples or spatial/graphical examples. Both groups were tested using arithmetic problems based on the arithmetic properties both prior to and following the training. The Spatial Training Group improved response time more from the pre- to the post-test than did the Numerical Training Group. The discussion focuses on the cognitive representation of numbers and arithmetic operations by means of spatial representation, evidence from neurophysiology, and the historical development of number systems. Early number systems were based in large part, on repetition of the same symbols to represent an amount (Gullberg, 1997). For example, the symbols were sometimes created by pressing an object into a soft piece of clay. This type of number system is called a tally system (Ifrah, 2000) because each item in a set of items is tallied individually. A tally system provides some obvious advantages as well as clear disadvantages. On the plus side, users can visually compare two representations to see which set of items is more numerous. They can also visually determine the ratio of two sets of items (by simply seeing how many times the smaller representations could fit into the larger). The size of the difference between two representations would also be immediately available visually by seeing the spatial difference between two representations. Likewise, users can add sets of items by appending two sets of tallies. In other words, traders in the ancient world and other users could have a permanent and unique representation of amounts and could perform basic arithmetic operations on those represented amounts through spatial manipulations. The advantages for using a tally system occur because the length of the set of tallies represents the magnitude that the number symbolizes (e.g., Gillan, 1995). Note that this lengthmagnitude relation also lies at the heart of modern graphical representations. One clear disadvantage is that a pure tally system requires a substantial amount of display space (on a clay tablet) to represent large numbers. Plus, although it is possible to represent multiplication with tallies (for example, N x M can be represented by creating a matrix with N tallies in the horizontal dimension and M tallies in the vertical dimension), such a representation requires a large area, as well as a significant amount of time and effort to create; in addition, multiplying N x M x O is not possible on a two dimensional spatial display. Evidence for the earliest tally number systems dates back 30,000 years, well before the start of civilization (Ifrah, 2000). Ultimately, advances in trade and farming that might have fostered the need to represent large amounts may have provided the impetus for the development of positional number systems. For example, about 5,500 years ago, the

Sumerians and the Babylonians made important advances in number systems by using different symbols for 10 so that 9 was represented by 9 tallies of the symbol for 1, but 10 had its own symbol. Representing multiple units of ten required repetition of the “ten” symbol – a triangular shape. For example, the symbol for 45 had four “ten” marks and five “one” marks. Also, the Sumerian-Babylonian number system created a base 60 system in which the number 60 was represented by placing the symbol for one in a position to the left. Thus, despite maintaining features of a tally system, this was one of the earliest uses of a positional number system (Ifrah, 2000). The advances in representational efficiency and computational power that accompanied positional number systems led to the ultimate abandonment of tally systems in India and the Middle East as their number systems used unique symbols for the numerals 1 – 9, with ten represented by the symbol for 1 in a tens position. It appears that most of our current numerals 1 to 9 were derived from tallies, but squeezed into the space for a single symbol (Ifrah, 2000). One is represented by a vertical line; two is represented by two horizontal lines connects by a diagonal; three represented by three lines connected at the right; and so on with the symbols containing more lined crowded into the same amount of space [for example, 8 can be decomposed into four vertical lines and four horizontal lines]). Although positional number systems have replaced tally systems, the advantageous parts of the tally system can be found in modern quantitative graphs in which the relation between a spatial feature (e.g., height, length, area, angular size) and the amounts being depicted by the graph. This relation between space and quantity has been referred to as a spatial metaphor (Wainer, 1984; Gillan, 1995). The relation between graphs and numerical amounts might also be considered to be an isomorphism – when two systems have the same deep structure, but with different surface representations (e.g., Hofstadter, 1979). Several researchers have explored the extent of the structural similarity between graphs and numerical amounts (Bricken, 1992; Davis, 2001; Gillan, 1995; Hurts and van Leeuwen, 1998). For example, Bricken (1992)

PROCEEDINGS of the HUMAN FACTORS AND ERGONOMICS SOCIETY 52nd ANNUAL MEETING—2008

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PROCEEDINGS of the HUMAN FACTORS AND ERGONOMICS SOCIETY 52nd ANNUAL MEETING—2008

Following training, participants completed a 15-minute intervening task that involved drawing pictures that were part of an unrelated experiment on object perception. Finally, participants received a post-test that was identical to the pretest except that the specific numbers used in the arithmetic equations were different. The assignment of specific equations to pre- or post-test was counterbalanced across participants within each group. Following the post-test, participants received a debriefing. Two participants in each group did not complete the entire procedure, so their data were not included in the analysis. All other participants completed the entire session, including the intervening task, in less than 60 minutes. R"S$%TS Problem Solution Time Overall, participants responded more quickly on the posttest (mean G 3I58 msec) than on the pre-test (mean G 625I msec), ! (1, 15) G 27I.07, " N .0001. The improvement in response time was greater for the Spatial Training Group than for the Numerical Training Group, as can be seen in Figure 1, Group x Test interaction, !(1, 15) G 5.5I, " N .05. Post-hoc tests showed that the two groups’ mean time to solve the arithmetic problems did not differ significantly in the pre-test, # (30) G 1.07, " T .05, whereas the groups’ mean problemsolving time differed significantly in the post-test, # (30) G 3.34, " N .05.

Numerical Training Group

Spatial Training Group

Figure 1. Mean response time for the Numerical and Spatial Training Groups on pre- and post-tests. Accuracy Participants in both groups had high accuracy on both pre- and post-tests. In the pre-test, participants responded correctly on I0X (Numerical Group) and I1X (Spatial Group) of the trialsY in the post-test, they responded correctly on I3X (Numerical) and I4X (Spatial) of the trials. The higher accuracy on the post-test resulted in a significant main effect of test, !(1, 15) G 4.74, " N .05. The small difference in accuracy between groups was not significant, nor was the group by test interaction. The percent of correct responses varied across the type of arithmetic operation involved in the

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trial (I4X for addition, I3X for subtraction, I2X for division, and I0X for multiplication), !(3, 45) G 4.00, " N .05Y but the difference among the type of operations did not vary significantly across tests, Operator x Test interaction !(3, 45) N1.00. 7"N"RA% D:SC$SS:o,ever, given that the participants solved the problems ?uickly, a manipulation that reduces the problem solution time by more than 2AB ,ould seem to be ?uite po,erful. These spatial methods might be especially helpful in teaching younger children ,ho are first learning arithmetic rules and properties. :dditional research ,ill be needed to test this hypothesis. The results of this experiment, combined ,ith previous ,ork by Gillan ;1FFG= and >urts and van Heeu,en ;1FFI= suggest that by constraining our use of graphics to the display of ?uantitative information, ,e are limiting their usefulness. Graphical representations of magnitude can be useful for rapid arithmetic calculations and for training people ,hen they are faced ,ith ne, problems concerning number and arithmetic operations and properties. Finally, these data may point to an improved understanding of ho, number systems developed in ancient cultures. Kne can speculate that tally number systems gre, out of enumerating obLects by pointingM for example, ,ith a set of obLects to count, a preNSumerian trader might have pointed at one item, then stuck his finger into a soft clay tablet that he had located next to the obLects, thereby creating a hole to represent that first itemM then he might have pointed to the next item and stuck his finger into the clay next to the first hole, and so on do,n the set of obLects. The anatomy of the brain also lends some plausibility to this speculation about the development of number systems. Pot only are numerical and spatial representations of magnitude located in the inferior parietal cortex ;e.g., Qehaene, et al., 2AA3M Salsh, 2AA3=, neural systems implicated in goal directed sensorimotor behaviors, like pointing are also found in this area ;e.g., Freund, 2AA1=. Hike,ise, young children may ac?uire number concepts best ,hen they activate spatial and motor systems in concert. Terhaps older children ;even college students= ,ho have arithmetic and other mathematical deficits could be helped by going back to simpler spatial and motor representations of number before trying to associate those representations ,ith the mathematically more po,erful but cognitively more complex numericalUverbal representations.

Freund, >N]. ;2AA1=. The parietal lobe as a sensorimotor interface[ a perspective from clinical and neuroimaging data. A%5"#/86)%, 14, S1`2 Y S1`X. Gillan, Q. ]. ;1FFG=. bisual arithmetic, computational graphics, and the spatial metaphor. G586( H6$-#"*, 37, aXX Y aIA. Gullberg, ]. ;1FFa=. J6-.%86-'$*K H"#8 -.% L'"-. #, (58L%"*. Pe, cork[ S. S. Porton. >ofstadter, Q. R. ;1FaF=, uman Factors and Ergonomics Society. Wfrah, G. ;2AAA=. P.% 5('4%"*67 .'*-#"B #, (58L%"*K H"#8 3"%.'*-#"B -# -.% '(4%(-'#( #, -.% $#835-%". Pe, cork[ Siley. Singley, _. e., and :nderson, ]. R. ;1FIF=. P.% -"6(*,%" #, $#)('-'4% *2'77. Zambridge, _:[ >arvard fniversity Tress. Sainer, >. ;1FI`=. >o, to display data badly. 98%"'$6( :-6-'*-'$'6(, 3Q, 13a Y 1`a. Salsh, b. ;2AA3=. : theory of magnitude[ common cortical metrics of time, space, and ?uantity. P"%(&* '( @#)('-'4% :$'%($%, 7 , `I3 Y `II. Appendix 1 0 Properties and their Definitions Property :ddition[ :dding A :ddition[ Zommutation

:ddition[ :ssociative Troperty

Subtraction[ Subtracting A Subtraction[ Zommutation

REFERENCES Bricken, S. ;1FF2=. Spatial representation of elementary algebra. Wn !"#$%%&'()* #, -.% /000 1#"2*.#3 #( 4'*567 76()56)%* ;pp. GX Y X2=. Hos :lamitos, Z:[ WEEE Zomputer Society Tress. Qavis, :. ]. ;2AA1=. Hearning from graphs in the business ,orld. !"#$%%&'()* #, -.% 6((567 8%%-'() #, -.% 98%"'$6( :-6-'*-'$67 9**#$'6-'#(; 9-76(-6; ?. Qehaene, S., Tia^^a, _., Tinel, T., and Zohen, H. ;2AA3=. Three parietal circuits for number processing. @#)('-'4% A%5"#3*B$.#7#)B, 3D, `Ia Y GAX.

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_ultiplication[ _ultiplying by 1 _ultiplication[ Zommutative Troperty _ultiplication[ Qistributive Troperty

Qivision[ Qividing by 1 Qivision[ Qividing by A

Definition :dding A to any number leaves that number unchanged The order in ,hich you add numbers doesngt matter Y yougll end up ,ith the same sum no matter ,hat order you add t,o numbers The order in ,hich you add a long series of numbers ,ith parentheses doesngt matter Y yougll end up ,ith the same sum no matter the order of the number in the series Subtracting A from any number leaves that number unchanged Wn subtraction, you end up ,ith a different number depending on the order that you subtract t,o numbers. So, subtraction does not have a commutative property _ultiplying any number by 1 leaves that number unchanged The order in ,hich you multiply numbers doesngt matter Y yougll end up ,ith the same amount no matter the order :dding t,o numbers, then multiplying that sum by a third number produces the same result as if you first multiply the t,o numbers separately by the third, then add those t,o products Qividing any number by one leaves that number unchanged Qividing any number by ^ero produces an undefined amount ;or infinity=

PROCEEDINGS of the HUMAN FACTORS AND ERGONOMICS SOCIETY 52nd ANNUAL MEETING—2008

Appendix 2 – Examples of Selected Training Trials Property Addition: Adding 0

Description of Training Numerical: 7 + 0 = 7 Spatial: A stack of seven blocks and a line sit on an x-axis line. The line with an arrowhead moves to the top of the blocks where it adds no additional height Addition: Commutation Numerical: 6 + 3 = 9 3+6=9 Spatial: On the left, a stack of three blocks moves on top of a stack of six blocks. Then to the right, a stack of six blocks moves on top of a stack of three blocks. A line moves across from the top of the righthand stack to the top of the left hand stack to indicate that their heights are equal Addition: Associative Numerical: (3 + 2) + 1 = 6 Property 1 + (2 + 3) = 6 Spatial: On the left, a stack of three blocks moves on top of a stack of two blocks; that stack moves on top of one block. Then to the right, a stack of two blocks moves on top of a stack of three blocks; a single block moves on top of that stack of five blocks. A line with an arrowhead moves across from the top of the righthand stack of six blocks to the top of the left hand stack to indicate that their heights are equal Subtraction: Subtracting 0 Numerical: 5- 0 = 5 Spatial: A stack of five blocks and a line sit on an x-axis line. The line moves to the top of the blocks where it removes no height Subtraction: Commutation Numerical: 6 – 3 = 3 3 – 6 = -3 Spatial: A stack of six blue blocks and a stack of three red blocks sit on a line to the left. The stack of three blocks moves so that it is superimposed over the top three of the six blocks such that the tops of the two stacks match and the top three blocks are purple. Then the purple blocks dissolve leaving only three blocks. On the right, the six blue blocks move so that they are superimposed over three red blocks, resulting in three purple blocks above the line and three blue blocks below the line. Then the purple blocks dissolve leaving only the three blue blocks extending below the line. An arrowhead tipped line moves from the top of the blue stack above the zero line to the bottom of the blue stack below the zero line Multiplication: Numerical: 8 x 1 = 8 Multiplying by 1 Spatial: A stack of eight blocks sits on top of a line on the left and a stack of one block sits to the right. The one block moves to be superimposed on the top of the eight-block stack and there is no change to the eight-block stack Multiplication: Numerical: 7 x 0 = 0 Multiplying by 0 Spatial: A stack of seven blocks sits next to a line. The line moves to the top of the seven block stack, then moves down it erasing the seven blocks as it goes. Multiplication: Numerical: 5 x 2 = 10 Commutative Property 2 x 5 = 10 Spatial: A stack of five blocks sits next to a stack of two blocks on the left, and a similar pair is on the right side of the display. On the left, the stack of two blocks moves near the top of the five blocks, pivots so that it becomes horizontal, then attaches itself so that the leftmost horizontal block covers the topmost vertical block, The two horizontal blocks move down the five block stack, with blocks filling in along the way so that there is a set of blocks 5 by 2. On the right, the stack of five blocks moves near the two blocks and pivots to a horizontal orientation, then moves so that the rightmost block from the horizontal stack of 5 covers the topmost block from the stack of 2. The five horizontal blocks move down the two block stack, with blocks filling in along the way so that there is a set of blocks 2 by 5. The rightmost stack pivots so that it has five vertical blocks by 2 horizontal blocks like the leftmost stack. Division: Division by 1 Numerical: 8/1 = 8 Spatial: A stack of eight blue blocks sits with one red block to the right. The one block moves to the top of the 8block stack so that it covers the top block which turns purple. A line comes out of the purple block pointing to the right, and a blue block appears at the end of the line. The red block moves down the 8-block stack, producing a blue block to its right, until it has gone all the way down the eight block stack, leaving two equivalent stacks of eight blue blocks.

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Visual Representation of Arithmetic Properties Douglas J. Gillan Department of Psychology North Carolina State University Raleigh, NC [email protected]

Copyright 2008 by Human Factors and Ergonomics Society, Inc. All rights reserved. 10.1518/107118108X350203

Quantitative graphs use a spatial metaphor in which spatial features represent numerical amounts. Tally number systems (used in ancient civilizations) also made use of the spatial feature of length. Arithmetic properties, such as the commutative property of addition, can also be represented spatially. An experiment trained college students about 11 basic arithmetic properties either using numerical examples or spatial/graphical examples. Both groups were tested using arithmetic problems based on the arithmetic properties both prior to and following the training. The Spatial Training Group improved response time more from the pre- to the post-test than did the Numerical Training Group. The discussion focuses on the cognitive representation of numbers and arithmetic operations by means of spatial representation, evidence from neurophysiology, and the historical development of number systems. Early number systems were based in large part, on repetition of the same symbols to represent an amount (Gullberg, 1997). For example, the symbols were sometimes created by pressing an object into a soft piece of clay. This type of number system is called a tally system (Ifrah, 2000) because each item in a set of items is tallied individually. A tally system provides some obvious advantages as well as clear disadvantages. On the plus side, users can visually compare two representations to see which set of items is more numerous. They can also visually determine the ratio of two sets of items (by simply seeing how many times the smaller representations could fit into the larger). The size of the difference between two representations would also be immediately available visually by seeing the spatial difference between two representations. Likewise, users can add sets of items by appending two sets of tallies. In other words, traders in the ancient world and other users could have a permanent and unique representation of amounts and could perform basic arithmetic operations on those represented amounts through spatial manipulations. The advantages for using a tally system occur because the length of the set of tallies represents the magnitude that the number symbolizes (e.g., Gillan, 1995). Note that this lengthmagnitude relation also lies at the heart of modern graphical representations. One clear disadvantage is that a pure tally system requires a substantial amount of display space (on a clay tablet) to represent large numbers. Plus, although it is possible to represent multiplication with tallies (for example, N x M can be represented by creating a matrix with N tallies in the horizontal dimension and M tallies in the vertical dimension), such a representation requires a large area, as well as a significant amount of time and effort to create; in addition, multiplying N x M x O is not possible on a two dimensional spatial display. Evidence for the earliest tally number systems dates back 30,000 years, well before the start of civilization (Ifrah, 2000). Ultimately, advances in trade and farming that might have fostered the need to represent large amounts may have provided the impetus for the development of positional number systems. For example, about 5,500 years ago, the

Sumerians and the Babylonians made important advances in number systems by using different symbols for 10 so that 9 was represented by 9 tallies of the symbol for 1, but 10 had its own symbol. Representing multiple units of ten required repetition of the “ten” symbol – a triangular shape. For example, the symbol for 45 had four “ten” marks and five “one” marks. Also, the Sumerian-Babylonian number system created a base 60 system in which the number 60 was represented by placing the symbol for one in a position to the left. Thus, despite maintaining features of a tally system, this was one of the earliest uses of a positional number system (Ifrah, 2000). The advances in representational efficiency and computational power that accompanied positional number systems led to the ultimate abandonment of tally systems in India and the Middle East as their number systems used unique symbols for the numerals 1 – 9, with ten represented by the symbol for 1 in a tens position. It appears that most of our current numerals 1 to 9 were derived from tallies, but squeezed into the space for a single symbol (Ifrah, 2000). One is represented by a vertical line; two is represented by two horizontal lines connects by a diagonal; three represented by three lines connected at the right; and so on with the symbols containing more lined crowded into the same amount of space [for example, 8 can be decomposed into four vertical lines and four horizontal lines]). Although positional number systems have replaced tally systems, the advantageous parts of the tally system can be found in modern quantitative graphs in which the relation between a spatial feature (e.g., height, length, area, angular size) and the amounts being depicted by the graph. This relation between space and quantity has been referred to as a spatial metaphor (Wainer, 1984; Gillan, 1995). The relation between graphs and numerical amounts might also be considered to be an isomorphism – when two systems have the same deep structure, but with different surface representations (e.g., Hofstadter, 1979). Several researchers have explored the extent of the structural similarity between graphs and numerical amounts (Bricken, 1992; Davis, 2001; Gillan, 1995; Hurts and van Leeuwen, 1998). For example, Bricken (1992)

PROCEEDINGS of the HUMAN FACTORS AND ERGONOMICS SOCIETY 52nd ANNUAL MEETING—2008

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PROCEEDINGS of the HUMAN FACTORS AND ERGONOMICS SOCIETY 52nd ANNUAL MEETING—2008

Following training, participants completed a 15-minute intervening task that involved drawing pictures that were part of an unrelated experiment on object perception. Finally, participants received a post-test that was identical to the pretest except that the specific numbers used in the arithmetic equations were different. The assignment of specific equations to pre- or post-test was counterbalanced across participants within each group. Following the post-test, participants received a debriefing. Two participants in each group did not complete the entire procedure, so their data were not included in the analysis. All other participants completed the entire session, including the intervening task, in less than 60 minutes. R"S$%TS Problem Solution Time Overall, participants responded more quickly on the posttest (mean G 3I58 msec) than on the pre-test (mean G 625I msec), ! (1, 15) G 27I.07, " N .0001. The improvement in response time was greater for the Spatial Training Group than for the Numerical Training Group, as can be seen in Figure 1, Group x Test interaction, !(1, 15) G 5.5I, " N .05. Post-hoc tests showed that the two groups’ mean time to solve the arithmetic problems did not differ significantly in the pre-test, # (30) G 1.07, " T .05, whereas the groups’ mean problemsolving time differed significantly in the post-test, # (30) G 3.34, " N .05.

Numerical Training Group

Spatial Training Group

Figure 1. Mean response time for the Numerical and Spatial Training Groups on pre- and post-tests. Accuracy Participants in both groups had high accuracy on both pre- and post-tests. In the pre-test, participants responded correctly on I0X (Numerical Group) and I1X (Spatial Group) of the trialsY in the post-test, they responded correctly on I3X (Numerical) and I4X (Spatial) of the trials. The higher accuracy on the post-test resulted in a significant main effect of test, !(1, 15) G 4.74, " N .05. The small difference in accuracy between groups was not significant, nor was the group by test interaction. The percent of correct responses varied across the type of arithmetic operation involved in the

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trial (I4X for addition, I3X for subtraction, I2X for division, and I0X for multiplication), !(3, 45) G 4.00, " N .05Y but the difference among the type of operations did not vary significantly across tests, Operator x Test interaction !(3, 45) N1.00. 7"N"RA% D:SC$SS:o,ever, given that the participants solved the problems ?uickly, a manipulation that reduces the problem solution time by more than 2AB ,ould seem to be ?uite po,erful. These spatial methods might be especially helpful in teaching younger children ,ho are first learning arithmetic rules and properties. :dditional research ,ill be needed to test this hypothesis. The results of this experiment, combined ,ith previous ,ork by Gillan ;1FFG= and >urts and van Heeu,en ;1FFI= suggest that by constraining our use of graphics to the display of ?uantitative information, ,e are limiting their usefulness. Graphical representations of magnitude can be useful for rapid arithmetic calculations and for training people ,hen they are faced ,ith ne, problems concerning number and arithmetic operations and properties. Finally, these data may point to an improved understanding of ho, number systems developed in ancient cultures. Kne can speculate that tally number systems gre, out of enumerating obLects by pointingM for example, ,ith a set of obLects to count, a preNSumerian trader might have pointed at one item, then stuck his finger into a soft clay tablet that he had located next to the obLects, thereby creating a hole to represent that first itemM then he might have pointed to the next item and stuck his finger into the clay next to the first hole, and so on do,n the set of obLects. The anatomy of the brain also lends some plausibility to this speculation about the development of number systems. Pot only are numerical and spatial representations of magnitude located in the inferior parietal cortex ;e.g., Qehaene, et al., 2AA3M Salsh, 2AA3=, neural systems implicated in goal directed sensorimotor behaviors, like pointing are also found in this area ;e.g., Freund, 2AA1=. Hike,ise, young children may ac?uire number concepts best ,hen they activate spatial and motor systems in concert. Terhaps older children ;even college students= ,ho have arithmetic and other mathematical deficits could be helped by going back to simpler spatial and motor representations of number before trying to associate those representations ,ith the mathematically more po,erful but cognitively more complex numericalUverbal representations.

Freund, >N]. ;2AA1=. The parietal lobe as a sensorimotor interface[ a perspective from clinical and neuroimaging data. A%5"#/86)%, 14, S1`2 Y S1`X. Gillan, Q. ]. ;1FFG=. bisual arithmetic, computational graphics, and the spatial metaphor. G586( H6$-#"*, 37, aXX Y aIA. Gullberg, ]. ;1FFa=. J6-.%86-'$*K H"#8 -.% L'"-. #, (58L%"*. Pe, cork[ S. S. Porton. >ofstadter, Q. R. ;1FaF=, uman Factors and Ergonomics Society. Wfrah, G. ;2AAA=. P.% 5('4%"*67 .'*-#"B #, (58L%"*K H"#8 3"%.'*-#"B -# -.% '(4%(-'#( #, -.% $#835-%". Pe, cork[ Siley. Singley, _. e., and :nderson, ]. R. ;1FIF=. P.% -"6(*,%" #, $#)('-'4% *2'77. Zambridge, _:[ >arvard fniversity Tress. Sainer, >. ;1FI`=. >o, to display data badly. 98%"'$6( :-6-'*-'$'6(, 3Q, 13a Y 1`a. Salsh, b. ;2AA3=. : theory of magnitude[ common cortical metrics of time, space, and ?uantity. P"%(&* '( @#)('-'4% :$'%($%, 7 , `I3 Y `II. Appendix 1 0 Properties and their Definitions Property :ddition[ :dding A :ddition[ Zommutation

:ddition[ :ssociative Troperty

Subtraction[ Subtracting A Subtraction[ Zommutation

REFERENCES Bricken, S. ;1FF2=. Spatial representation of elementary algebra. Wn !"#$%%&'()* #, -.% /000 1#"2*.#3 #( 4'*567 76()56)%* ;pp. GX Y X2=. Hos :lamitos, Z:[ WEEE Zomputer Society Tress. Qavis, :. ]. ;2AA1=. Hearning from graphs in the business ,orld. !"#$%%&'()* #, -.% 6((567 8%%-'() #, -.% 98%"'$6( :-6-'*-'$67 9**#$'6-'#(; 9-76(-6; ?. Qehaene, S., Tia^^a, _., Tinel, T., and Zohen, H. ;2AA3=. Three parietal circuits for number processing. @#)('-'4% A%5"#3*B$.#7#)B, 3D, `Ia Y GAX.

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_ultiplication[ _ultiplying by 1 _ultiplication[ Zommutative Troperty _ultiplication[ Qistributive Troperty

Qivision[ Qividing by 1 Qivision[ Qividing by A

Definition :dding A to any number leaves that number unchanged The order in ,hich you add numbers doesngt matter Y yougll end up ,ith the same sum no matter ,hat order you add t,o numbers The order in ,hich you add a long series of numbers ,ith parentheses doesngt matter Y yougll end up ,ith the same sum no matter the order of the number in the series Subtracting A from any number leaves that number unchanged Wn subtraction, you end up ,ith a different number depending on the order that you subtract t,o numbers. So, subtraction does not have a commutative property _ultiplying any number by 1 leaves that number unchanged The order in ,hich you multiply numbers doesngt matter Y yougll end up ,ith the same amount no matter the order :dding t,o numbers, then multiplying that sum by a third number produces the same result as if you first multiply the t,o numbers separately by the third, then add those t,o products Qividing any number by one leaves that number unchanged Qividing any number by ^ero produces an undefined amount ;or infinity=

PROCEEDINGS of the HUMAN FACTORS AND ERGONOMICS SOCIETY 52nd ANNUAL MEETING—2008

Appendix 2 – Examples of Selected Training Trials Property Addition: Adding 0

Description of Training Numerical: 7 + 0 = 7 Spatial: A stack of seven blocks and a line sit on an x-axis line. The line with an arrowhead moves to the top of the blocks where it adds no additional height Addition: Commutation Numerical: 6 + 3 = 9 3+6=9 Spatial: On the left, a stack of three blocks moves on top of a stack of six blocks. Then to the right, a stack of six blocks moves on top of a stack of three blocks. A line moves across from the top of the righthand stack to the top of the left hand stack to indicate that their heights are equal Addition: Associative Numerical: (3 + 2) + 1 = 6 Property 1 + (2 + 3) = 6 Spatial: On the left, a stack of three blocks moves on top of a stack of two blocks; that stack moves on top of one block. Then to the right, a stack of two blocks moves on top of a stack of three blocks; a single block moves on top of that stack of five blocks. A line with an arrowhead moves across from the top of the righthand stack of six blocks to the top of the left hand stack to indicate that their heights are equal Subtraction: Subtracting 0 Numerical: 5- 0 = 5 Spatial: A stack of five blocks and a line sit on an x-axis line. The line moves to the top of the blocks where it removes no height Subtraction: Commutation Numerical: 6 – 3 = 3 3 – 6 = -3 Spatial: A stack of six blue blocks and a stack of three red blocks sit on a line to the left. The stack of three blocks moves so that it is superimposed over the top three of the six blocks such that the tops of the two stacks match and the top three blocks are purple. Then the purple blocks dissolve leaving only three blocks. On the right, the six blue blocks move so that they are superimposed over three red blocks, resulting in three purple blocks above the line and three blue blocks below the line. Then the purple blocks dissolve leaving only the three blue blocks extending below the line. An arrowhead tipped line moves from the top of the blue stack above the zero line to the bottom of the blue stack below the zero line Multiplication: Numerical: 8 x 1 = 8 Multiplying by 1 Spatial: A stack of eight blocks sits on top of a line on the left and a stack of one block sits to the right. The one block moves to be superimposed on the top of the eight-block stack and there is no change to the eight-block stack Multiplication: Numerical: 7 x 0 = 0 Multiplying by 0 Spatial: A stack of seven blocks sits next to a line. The line moves to the top of the seven block stack, then moves down it erasing the seven blocks as it goes. Multiplication: Numerical: 5 x 2 = 10 Commutative Property 2 x 5 = 10 Spatial: A stack of five blocks sits next to a stack of two blocks on the left, and a similar pair is on the right side of the display. On the left, the stack of two blocks moves near the top of the five blocks, pivots so that it becomes horizontal, then attaches itself so that the leftmost horizontal block covers the topmost vertical block, The two horizontal blocks move down the five block stack, with blocks filling in along the way so that there is a set of blocks 5 by 2. On the right, the stack of five blocks moves near the two blocks and pivots to a horizontal orientation, then moves so that the rightmost block from the horizontal stack of 5 covers the topmost block from the stack of 2. The five horizontal blocks move down the two block stack, with blocks filling in along the way so that there is a set of blocks 2 by 5. The rightmost stack pivots so that it has five vertical blocks by 2 horizontal blocks like the leftmost stack. Division: Division by 1 Numerical: 8/1 = 8 Spatial: A stack of eight blue blocks sits with one red block to the right. The one block moves to the top of the 8block stack so that it covers the top block which turns purple. A line comes out of the purple block pointing to the right, and a blue block appears at the end of the line. The red block moves down the 8-block stack, producing a blue block to its right, until it has gone all the way down the eight block stack, leaving two equivalent stacks of eight blue blocks.

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