Visual Thinking: Art Students Have an Advantage in Geometric Reasoning

doi:10.4236/ce.2011.21004

Paper Menu >>

Journal Menu >>

Creat ive Educati on

2011. Vol. 2, No. 1, 22 -26

Visual Thinking: Art Students Have an Advantage in Geometric

Reasoning

Caren M. W alker1, Ellen Winner2,3, Lois Hetland3,4, Seymour Simmons5, Lynn

Goldsmith6

1University of Ca lifornia, Berkeley, USA;

2Boston College, Boston, USA ;

3Harvard Graduate School of Education, Cambridge, USA;

4Massachusetts Coll ege of Arts and Design, Boston, USA ;

5Winthrop University, Rock Hill, USA;

6Educa tion Development Center, Newton, USA.

Email: car [email protected]

Received August 25th, 2010; revised December 7th, 2010; ac cepted Decemb er 9th, 2010.

We investigated whether individuals with training in the visual arts show superior performance on geometric

reasoni ng tasks, given that b oth art and geometry entail visualization and mental manipulation of images. Two

groups of undergraduates, one majoring in studio art, the other majoring in psychology, were given a set of

geometric reasoning items designed to assess the ability to mentally manipulate geometric shapes in two- and

three-dimensional space. Participants were also given a verbal intelligence test. Both training in the arts and

verbal i ntelligence wer e strong p redictors of geom etric reason ing, but training in the ar ts was a signifi cant pre-

dictor even when the effects of verbal intelligence were removed. These correlational findings lend support to

the hypot hesis that trainin g in the visua l arts may imp rove geometric reasonin g via the learn ed cognit ive ski ll of

visualization.

Keywords: Visualization, Geometry, Spatial Reasoning, Art Education, Mathematics Education

Introduction

The ability to visualize what cannot be seen directly is con-

sidered a critical skill in mathematics and scien ce (Cu nningh am,

2005). For example, when asked to reflect on his thinking,

Einstein wrote that words “do not seem to play any role in my

mechanism of thought.” Instead, he described the primary ele-

ments of his thinking as “certain signs and more or less clear

images” (Had amard, 1 945). Keku lé report ed that he disco vered

the circu lar stru cture of the ben zene molecule after d reaming o f

a snake biting its own tail (Perkins, 1983). And Watson and

Crick’s discovery of DNA’s double-helix structure involved

model building and visualization (Watson, 1968). Edward

Tufte’s work on visualizing statistical data states, “Graphics

reveal data. Indeed graphics can be more precise and revealing

than conventional statistical computations” (Tufte, 2001). Vir-

tually every STEM (Science, Technology, Engineering, and

Mathematics) discipline calls upon visual or spatial thinking:

chemists envision molecular structures and their interactions;

geologists use field observations to envision structures that

cannot be seen; engineers use visual feedback from computer

models as they develop and test designs; topologists and geo-

meters investigate mathematical relationships under various

transformations.

Educational organizations in mathematics and science also

emphasize the importance of visual representation and reason-

ing capacities (National Council of Teachers of Mathematics,

2000). For example, the Principles and Standards for School

Mathematics and the Common Core Standards explicitly de-

scribe visualization as a tool for problem-solving and also rec-

ognize t he essent ial ro le of being ab le to r epresent and interp ret

mathematical ideas and problems in visual forms, including

graphs, sketches, and diagrams. Despite the acknowledged

importance of the role of visualization in mathematics, however,

it is given relatively short shrift in many mathematics curricula

(Hogan, 1993; Lappan, 1999; Goldenberg, 1996). Even geo me-

try, a highly spatial area of mathematics, is generally taught

with a strong symbolic, algebraic focus. Instead, one could

argue that the dual perspectives of formal s ymbolism and visu-

alization constitute complemen ta r y approaches for conceptua-

lizing the same geometric task. Visualization may thus be ap-

plied as a tool for solving a mathematical problem in the same

way that symbolic algebraic expressions can (Whiteley, 2004).

It is th erefore ju st as fund amental to educat e stud ents to en gage

in visual representation of geometric principles as it is to teach

them to generate coherent symbolic arguments using formal

algebraic notation. Support for this position comes from the

work of mathematician William Thurston, famous for demon-

strating the power of visual representations to communicate

abstract mathematical ideas and winner of the Fields Medal in

1982. He argued that formal proofs are less appealing than the

more intuitive tool of graphics to communicate abstract ma-

thematical ideas, and he tried to develop ways to teach basic

geometry through visual arguments (Hogan, 1993).

How can students develop the kinds of visualization abilities

that will help them reason geometrically? We have begun to

explore the possibility that the development (and exercise) of

visualization skills in non-mathematical do main s—for example,

C. M. WALKER ET AL.

the visual arts—may support students’ geometric thinking. As

Gardner (2007) has argued, the visual arts are a domain that

relies h eavily on vi sual-spatial thinking. It is possible, therefore,

that students who acquire training in the visual arts may be able

to apply their visualization skills to support their mathematical

and scientific thinking as well.

Visualization seems to be a fundamental habit of the artistic

mind. Artists do not just magically “see” in their mind’s eye,

but deliberately and systematically analyze shape and space

into familiar simple forms, construction lines, angles, and size

ratios (Kozbelt, 1991). This process is essential for depicting

three-dimensional objects on a two-dimensional sur face. Visu a-

lization is also of value when creating three-dimensional ob-

jects, which often must be ‘pictured’ as a whole befor e they are

built. An ethnographic study of intensive high school arts pro-

grams by Hetland, et al. (2007) found that 'envisioning' (visua-

lization) is one of eight habits of mind that are taught in visual

arts studio classes. Visualization (envisioning) involves the

formation of images (often mental) which can then guide ac-

tions and problem solving and can even lead to problem finding

(Getzles & Csikszentmihalyi, 1975) The art teachers studied

provided their students with continual practice in imagining

space, line, color, and shape, regularly asking their students

such questions as, “What would this look like if you extended

this line?” “What is the underlying structure of this composi-

tion?” “Where would the shadow fall if the light were coming

from that window?” Such questions prompt students to envi-

sion what is not there. Visual art students also study skeletal

and muscular anatomy to help them envision the underlying

structure of the human figure and the forces at work within

various poses.

Whether based on training, innate ability, or some combina-

tion of the two, visual artists have been shown to possess supe-

rior visual-spatial capacity when compared to non-artists.

Visual artists excel in mental rotation and visual memory (Ca-

sey, et al., 1990; Hermelin & O’Connor, 1986; Rosenblatt &

Winner, 1988), visual attention and visual analysis of objects

(Kozbelt & Seeley, 2007), and form recognition (Kozbelt, 1991;

Cohen & Bennett, 1997; Mitchell, et al., 2005). In a functional

MRI study, Solso (2001) examined the n eural activit y of artist s

and non-artists while th ey drew f aces in sid e th e scan ner . Artist s

showed more right prefrontal activation (an area associated

with the manipulation of visual forms) and less activation in the

fusifor m face region (an area associated with duplicat ing visual

forms). Solso interpreted this latter finding as showing that the

artist was focused less on copying the face and more on an

abstract analysis of its s hapes, feat ures, and o rganization.

It has already been demonstrat ed that geometry learnin g fos-

ters improvements in visualization tasks (Ben-Chaim, Lappan,

& Houang, 1988; Clements, 1997). In this study, we examined

this relationship from the opposite direction, asking whether the

development of visualization skills in a non-mathematical con-

text may confer an advantage for geometric reasoning. The

study presented here represents a first step in a program of re-

search exami nin g whether tr aining in the visu al arts t ransfers to

geometric reasoning. The possibility of transfer is plausible

because both domains rely on mental manipulation of forms in

space, thereby allowing for the possibility of “near transfer”

among domains that are related (Salomon & Perkins, 1989).

Methods

Participants

A total of 36 (mostly female) college undergraduates partic-

pated. There were 18 studio art majors (mean age = 22;4, range

= 19;5 – 27;7, 14 females) and 18 psychology majors (mean

age = 22;10, range = 19;8 – 26;5, 17 females). Two additional

non-artists were tested but excluded because they were over

two standard deviations above the mean age range (these par-

ticipants were 34;4 and 41;1 years old). Both groups were se-

niors at the same small public university that attracts predomi-

nantly low- to mid-income students (average combined SAT

score: 1045, average GPA: 3.62). There were 71% White

(non-Hispanic) participants, 26% African American partici-

pants, and < 3% Hispanic, Asian, and Native-American partic-

ipants. Both groups of participants had completed an average of

6.5 semesters of undergraduate work (first semester seniors) .

The art majors who participated in the study had a consider-

able amount of art training, having taken an average of 16 un-

dergraduate art courses. Psychology majors had taken no more

than one undergraduate art course. All participants had com-

pleted approximately the same number of undergraduate math

courses at the time of the study: psychology majors had com-

pleted an average of 1.7 undergraduate and 4 high school math

cours es, and art majors had compl eted an average of 1.4 un der-

graduate and 3.8 high school math courses.

Materials & Procedu re

With the help of a group of geometers and mathematics edu-

cators, we adapted a set of items originally developed by Calla-

han (1999) to create a 27-item geometric visualization/reasoning

inventory. The geometric measure that we developed is not de-

pendent upon formal geometry knowledge such as equations or

definitions, but instead focuses on geometric thinking. (See Ap-

pendix A for sample geometric reasoning items used for the test.

The full version of the geometric reasoning items can be found at

http://www2.bc.edu/~winner/current.html). These items requ-

ired participants to rely upon visual working memory and the

ability to engage in various spatial transformations. Participants

were not allowed to make drawings to help them solve the

problems, because we wanted to assess their capacity to solve

the problems using mental visualization, rather than the mani-

pulation of external representations.

Most of the items required the ability to visualize both two-

and three-dimensional space. In some items participants were

asked to imagine building a shape, step-by-step, then to mani-

pulate that shape mentally (e.g., to slice the shape into pieces;

see sample item 4), and to describe th e result ant shap e. In other

items participants were asked to mentally perform a three-

dimensional transformation such as a rotation on a complex

shape and describe the result (see sample item 3). Finally, in

some items, participants were asked to imagine a known shape

(e.g., a cube, a pyramid), to perform some additional mental

manipulation on that shape (e.g., to combine the shape with

other shapes), and then to describe the result (see sample item

5). All items were piloted with undergraduate students and were

found to be solvable, though difficult.

Participants were tested in small groups by one researcher,

and the testing session lasted approximately 1.5 hours. Partici-

C. M. WALKER ET AL.

pant s were first asked to co mpl ete the geometric reas oning t as k.

The 27-item task was administered in written form, but each

item was read aloud as students read along silently. After the

researcher r eviewed ea ch of the items aloud to the group, clari-

fying any confusion about wording, all participants were in-

structed to complete the task at their own pace and to record

their answers on their individual packets. Researchers were

available throughout the testing session to answer any individ-

ual questions as th ey arose.

To control for verbal intelligence, we administered the two

verbal sections of th e Kaufman Brief Intelligence Test (KBI T).

The first sectio n assesses vocabu lary: partici pant s are shown 15

pictures and asked to name each one. Items increase in diffi-

culty (e.g., item 1 is a cash register and item 15 is a thermostat).

The second section consists of 32 words, each of which has

several missing letters. Participants are given a verbal clue and

asked to fill in the complete word. For example, participants see

BR_W_ along with the clue “a dark color;” here the correct

answer is BROWN. Items increase in difficulty (e.g., item 1

was “Santa’s entrance” /_ _ IM_EY [answer: CHIMNEY] and

item 32 was “Due to chance or fate” / _ _ RE_ _I_I_Y [answer:

SERENDIPITY]). Both sections began with the researcher

modeling how to solve a sample problem. The KBIT items

were presented on PowerPoint slides one item at a time, and

participants filled in the words on their answer sheets. Follow-

ing the testing session, participants completed a brief demo-

graphic questionnaire that included questions about the num-

bers of art and mathematics courses taken.

Results

As shown in Figur e 1, perfor mance o n the geometric reaso n-

ing skills task was higher for the art majors (M = 11.46, SD =

4.12) than for the psychology majors (M = 8.12, SD = 2.89),

Cohen’s d = .93. Inspection of the individual items revealed

that the art majors performed better than the psychology majors

on all but two of the items. To determine whether visual arts

training significantly predicted geometry performance, a linear

regression analysis was conducted, with the total score on the

geometric reasoning task included as the dependent variable

and group membership (art major vs. psychology major) and

scores on the two sections of the KBIT included as independent

variables.

Table 1 shows the squared bivariate correlation coefficients

Figure 1.

Mean geometry scores for psychology majors and art majors (out

of a poss ible s core of 27 ) .

Table 1.

Reg ression Analysi s Showing Rela tionship of Arts Training and Verbal

Intelligence Scores on the Geometric Reasoning Skills Task.

Variable

Regression Analysis Results

B SE

Bet a

Art Training

KBIT 1

KBIT 2

.402**

.369*

2.447**

2.310

.777

2.205

2.810 1.219 .340*

.391 .504 .128

.252 .114 .356*

between each independent variable and geometry performance.

Training in the arts and scores on the KBIT 2 (i.e., filling in

missing letters to complete a word) both were strong predictors

of geometry performance, by themselves and in the context of

all three independent variables. Most importantly, training in

the arts was a significant p redictor of geometry perfor mance (b

= .340, t ( 34) = 2.31, p = .027), even when the e ffects o f verbal

intelligence (as measured by KBIT 1 and 2) were removed.

Gender was n ot included as a variable due to the small number

of male participants.

Discussion

This study provides initial evidence that individuals with

college level training in visual art perform better in geometric

reasoning than do individuals without such training. However,

the quasi-experimental design of the research does not allow

causal in ferences ab o ut the direct ion of th is relati onsh ip; we see

three possible explanations for the findings. First, it is possible

that t raining in th e visual arts do es have a causal ef fect on visu-

al-spatial skills (which could be detected with another experi-

mental design): visual arts training may strengthen such skills

and thereby lead to improve ments in geo metry perfor mance. A

second possible explanation is that students self-select into the

visual art s because of stron g visual-spatial skills: these students

may perfor m well in geometr y as a functi on of their preexi sting

strong spatial skills, not as a function of arts training. The third

explanation is that both of these possibilities are true: students

with abo ve average visual -spatial skills may opt into the visual

arts and these skills may grow stronger with training. Thus,

visual arts training would still lead to improved geometry per-

formance through the mechanism of stronger spatial visualiza-

tion skills. Two studies are currently underway using quasi-

experimental pre-post test designs to examine these possibili-

ties.

If transfer of visualization skill from visual arts to geometry

can be demonstrated, then perhaps the visual arts will no longer

be seen as a competitor for time spent in core subjects such as

mathematics. In addition to their intrinsic importance, the visual

arts might then be seen as an entry point for the learning of

geometry or as a novel (and possibly “real world”) context to

deepen geometric understanding. This could especially be the

case where th e focus of th e geometry instr uction is on d ynamic

reason ing (e.g., S eago, Driscol l, & Jacobs, in press) rather th an

the memorization and application of static rules and relation-

ships (as is required in the generation of symbolic proofs). Fur-

thermore, art teachers may come to see themselves as collabo-

rators with their fellows in academic areas, intentionally focus-

ing attention on skills like geometric reasoning, and also inten-

tionally teaching for transfer. This collaboration may provide a

C. M. WALKER ET AL.

more comprehensive education for students in both art and

academics, preparing them for a future in which synthesizing

knowledge from diverse domains will be essential skills

(Gardner, 2007).

Acknowledgement s

The authors thank Patrick Callahan for his critical role in

developing the geometric reasoning task and his comments on

this paper, and Paul Goldenberg and Mark Driscoll for further

assistance in developing the geometry materials used in this

study. We also thank Hiram Brownell and Johannah Nikula for

their co mments o n earlier dr afts o f thi s p aper. Finally, we thank

Gordon David Brown and Kathy Lyon for their help in recr uit-

ing participants. This research was supported by National

Science Foundation award # DRL-0815588. Any opinions,

findings and conclusions or recommendations expressed in this

material are those of the authors and do not necessarily reflect

the views of the Natio nal Science Foundation.

References

Battista, M. T., Clements, D. H., & Wheatley, G. H. (1991). Using

spatial imagery in geometric reasoning. Arithmetic Teacher, 39,

18-21.

Ben-Chaim, D., Lappan, G., & Houang, R. T. (1988). The effect of

instruction on spatial visualization skills of middle school boys and

girls. American Educational Research Journal, 25, 51-71.

Brieske, T. (1984). Visual thinking about rotations and reflections. The

Colle ge Mathematics Journal, 15, 406-410. doi:10.2307 /2686551

Callahan, P. (1999). Visualization workouts from “Geometry & visua-

lization: A Course for high school teachers”. unpublished notes.

Casey, M., Winner, E., Brabeck, M., & Sullivan, K. (1990). Visu-

al-spatial abilities in art, math, and science majors: Effects of sex,

handedness, and spatial experience. In K. Gilhooly, M. Keane, R.

Logie, & G. Erdos (Eds.), Lines of thinking: Reflections on the psy-

chology of thought. New York: Wiley.

Clements, D. H., Bat tista, M. T., Sarama, J., & Swamina than, S. (1997).

Development of students’ spatial thinking. The Elementary School

Journal, 98, 171-186. doi:10.1086/461890

Cohen, D. J., & Bennett, S. (1997). Why can’t most people draw what

they see? Journal of Experimental Psychology: Human Perception

and Performance , 23 , 609-621. doi:10.1037/0096-1523. 23.3.609

Cunningham, S. (2005). Visualization in science education. In Inven-

tion and impact: Building excellence in undergraduate science,

technology, engineering, and mathematics (STEM) education (pp.

127-128). Washington, DC: AAAS Press. Gardner, H. (2007). Five

Minds for the Future. Cambridge, MA: Harvard Business School

Press.

Getzels, J. W., & Csikszentmihalyi, M. (1975). From problem-solving

to problem finding, In I. A. Taylor and J. W. Getzels (Eds.), Pers-

pecti ves in Cre ativi ty (pp. 90-116). Chicago: Aldine.

Goldbenberg, E. P. (1996). “Habits of mind” as an organizer for the

curriculum. Journal of Education. 178, 13-34.

Hadamard, J. (1945). The psychology of invention in the mathematical

field. NY: Dover.

Hermelin, B., & O'Connor, N. (1986). Spatial representations in ma-

thematically and in artistically gifted children. British Journal of

Educational Psychology, 56, 150-157.

Hetland, L., Winner, E., Veenema, S., & Sheridan, K. (2007). Studio

thinking: The real benefits of visual arts education. New York:

Teachers C o llege.

Hogan, J. (1993). The death of proof. Scientific American, 92-103.

Kozbelt, A. (1991). Artists as experts in visual cognition. Visual Cogni-

tion, 8, 705-723.

Kozbelt, A., & Seeley, W. P. (2007). Integrating art historical, psycho-

logical, & neuroscientific explanations of artists’ advantages in

drawing. PACA , 1, 80-90 .

Lappan, G. (1999). Geometry: The forgotten strand. NCTM News Bul-

letin, 36, 3.

Mitchell, P., Ropar, D., Ackroyd, K., & Rajendran, G. (2005). How

perception impacts on drawings. Journal of Experimental Psycholo-

gy: H um a n P er cep tion and Per f or manc e , 31 , 996-1003.

doi:10.1037/0096-1523.31 .5.996

National Council of Teachers of Mathematics. (2000). Prin ciples and

standards for school mathematics. Reston, VA: National Council of

Teachers of Mathematics.

Perkins, D. (1983). The mind’s best work. Cambridge: Harvard Uni-

versity Press.

Rosenblatt, E., & Winner, E. (1988). The art of children's drawings.

Journal of Aesthetic Education, 22, 1, 3-15.

Salomon, G. & Perkins, D. N. (1989). Rocky roads to transfer: Re-

thinking mechanisms of a neglected phenomenon. Educational Psy-

chologist, 24, 113-142. doi:10.1207/s15326985ep24 02_1

Seago, N., Driscoll, M., & Jacobs J. Transforming middle school geo-

metry: designing professional development materials that support the

teachin g and learning of similarity. Middle Grades Research Journal,

in press.

Solso, R. L. (20 01). Br ain activi ties in an expert vers us a novi ce artist :

An fMRI study. Leonardo , 34, 31-34.

doi:10.1162/002409401300052479

Sutherland, R., & Mason, J., (1993). Exploiting mental imagery with

computers in mathematics education. New York: Sp ringer-Verlag.

Tuft e, E. R. (2 001). The visu al displ ay of qua nt itat ive in formation (2nd

ed.) CT: Grap hic Pr ess.

Watson, J. (1968). The double helix. New York: New American Li-

brary.

Winner, E., & Casey, M. (1993). Cognitive profiles of artists. In G.

Cupchik & J. Laszlo (Eds.), Emerging visions: Contemporary ap-

proaches to the aesthetic process. London: MacDonald and Jane's

Publishers.

Whiteley, W. (2004). Visualization in mathematics: Claims and ques-

tion s towards a researc h program. Pap er pr es ented at t he 10 Interna-

tional Congress on Mathematics Education, Copenhagen, Denmark,

Cambridge, England: Cambridge University Press.

C. M. WALKER ET AL.

Appendix

Appendix A. Sample Items from the Geometric Reasoning

Task.

Sample Item 1. Below are pictures of “nets.” You can fold

them on the solid lines to make 3-dimensional forms. Circle the

one(s) that can be folded into a closed form (that is, one that has

no holes or ope ning s) .

Sample Item 2. How many different colors do you need to

paint the faces of a cube so that no two faces that touch have th e

same colo r ? Figure out the answer in your head without draw-

ing. Describe your answer in words as best you can.

Sample Item 3. Imagine holding a small square card by the

diagonal corners and spinning it around the diagonal. What

shape would be carved out in the air? Figure out the answer in

your head without drawing. Describe your answer in words as

best you can.

Sample Item 4. Imagine a triangle that has 3 equal sides. In

your mind, mark the sides of this triangle into thirds, and cut

off each o f the triangle’ s corners at th e marks . Describe the

shape you get. Figure out the answer in your head without

drawing. Describe your answer in words as best you can.

Sample Item 5. Imagine five points equally spaced around a

circle. You get a regul ar pentagon (a s hape with five equal sides)

when you connect each point with the one next to it. What

shape do you get if you connect every other (alternating) point?

Try to figure out the answer in your head without drawing.

Describe your answer as b es t you can.

Sample Item 6. Imagine two squares. Both have sides 1 inch

long. Imagine pinning a corner of one square to the center of the

other square. What is the area of the part that overlaps? Try to

figure out the answer in your head, without drawing. Describe

your answer in words as best you can.