Creat ive Educati on
2011. Vol. 2, No. 1, 22 -26
Copyright © 2011 SciRes. DOI:10.4236/c e.2011.21004
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.
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.
23
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.
24
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
r2
t
B SE
B
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.
25
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.
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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.