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The Impact of Origami-Mathematics Lessons on Achievement and Spatial Ability of Middle-School Students



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The Impact of Origami-Mathematics
Lessons on Achievement and Spatial
Ability of Middle-School Students
Norma J. Boakes
1 Introduction
Origami has become a popular instructional method in the mathematics
classroom. Numerous books and practitioner articles cite origami as a use-
ful way to teach mathematics concepts, especially as it relates to geometry
and spatial concepts [6, 16, 22, 27]. In addition, the National Council of
Teachers of Mathematics (NCTM) [20], in its Principles and Standards of
School Mathematics, supports the use of such methods, suggesting that
students engage in active exploration that allows students to study the
construction and deconstruction of two- and three-dimensional figures. An
examination of literature regarding origami as an instructional tool, how-
ever, reveals a lack of studies focusing on the impact of origami instruction
within the mathematics classroom. With a continued need to find effec-
tive instructional methods in mathematics and the substantial support for
origami as such a method, this study was designed and implemented to
explore origami and its effect on student understanding.
472 V. Origami in Education
2 Research Questions
In exploring the impact of origami in the mathematics classroom, the fol-
lowing research questions were formulated:
1. How did students who participated in origami-mathematics lessons
integrate into a traditionally instructed geometry unit compared to
students who were instructed solely through traditional instruction in
terms of (a) spatial visualization skills and (b) mathematical achieve-
ment level?
2. Do the effects of origami-mathematics lessons differ by gender? [3]
3 Spatial Visualization and Related Research
Named within the major topics areas in the K-12 mathematics curricu-
lum is the study of geometry. A key component within geometry is the
development of spatial thought [29]. Generally speaking, spatial thought
deals with a student’s ability to visualize, describe, and critically analyze
spatial aspects of the real world. NCTM recognizes the importance and
need to assist students in developing this ability, often referred to as spa-
tial ability. Within this capability is the concept of spatial visualization.
Though definitions vary from author to author, this term refers generally
to the visualization and mental manipulation of figures in two- and three-
dimensions [21]. Spatial visualization, beyond its importance to geometry,
also has direct connections to defining and quantifying human intelligence
and, more specifically, mathematics ability. As a result, a great deal of
research links to this skill and its development in children.
Research reviewed concerning spatial ability and its attainment fall into
three areas: the connection of spatial ability to gender and age, the con-
nection between spatial ability and overall mathematics ability, and the
potential of improving spatial ability through training. Clearly important
after a review of gender- and age-based research is the awareness that males
and females may differ in terms of spatial ability. Though in some cases
females outperformed males on spatial tasks [17], generally researchers con-
clude that a difference persists, with males’ scores superior to females’ on
spatial tests [5,30]. In terms of age, males and females both improve their
spatial abilities as they mature [18]. However, as children reach middle-
and high-school age, improvements are not necessarily equal, with male
gains often greater than females [18,19].
Many studies sought to research the connection between spatial ability
and mathematics achievement. According to a review of educational re-
search from 1910 to the late 1950s, spatial ability did play a critical role
The Impact of Origami-Mathematics Lessons on Achievement and Spatial . . . 473
in evaluating mathematical ability [23]. However, more recent publications
are not necessarily in agreement. While there are those that found spa-
tial ability to be a predictor of performance [1, 5,11, 12], a meta-analysis
by Friedman questions the claim, noting that “the bulk of correlational
evidence casts doubt on the conjecture that spatial skill is pervasive in
mathematics as mathematics is taught and tested today” [15, p. 29].
A final area reviewed dealt with studies seeking to improve spatial abil-
ity through specialized instruction. Training came in many forms from
computer software [8, 9] to a variety of hands-on metho ds [1, 2, 24]. Many
of these studies found that, in some fashion, students made improvements
in their performance as a result of training [2, 4, 8].
Though research is not fully in agreement, there are general themes that
were important to this study. For one, gender and age are both factors
that could have some effect on performance. There is also the possibility
that there is a direct correlation between spatial ability and mathematics
achievement. A final conclusion is that training does have the potential to
improve student performance. Though these are more themes than absolute
truths, they are valuable in studying the influence of origami instruction
on students’ abilities.
4 Procedures
A quasi-experimental pre-test/post-test design with a control and treat-
ment group was used for this study. A convenience sample of 56 seventh-
grade students with the same mathematics instructor from a southern New
Jersey middle school served as participants. Of these, 31 (based on class
assignment) made up the control group and received strictly traditional
textbook-based instruction during experimentation. The remaining 25 stu-
dents (also based on class assignment) served as the experimental group
receiving treatment. Treatment consisted of 12 origami lessons, taught by
the researcher, interspersed within traditional instruction over a one-month
geometry unit. To determine how this setup impacted performance, a 2 ×2
factorial design was used. Independent variables included gender and the
method of instruction. For dependent variables, mathematics achievement
level and spatial ability were selected.
To determine the mathematics achievement level of students, a 27-
question multiple-choice test was created using released items from the
National Assessment of Educational Progress [20]. All items were from the
geometry/spatial skill strand of the NAEP and geared for middle-school-
age students. A report reviewing a sample of mathematics questions from
NAEP mathematics assessments between the years of 1973 and 1996 calcu-
lated weighted alpha reliability levels of .87 and .85 for middle-school-age
children, establishing fairly strong reliability for NAEP items [28].
474 V. Origami in Education
According to NCTM, spatial visualization refers to a student’s abil-
ity to “[build] and [manipulate] mental representations of two- and three-
dimensional objects and [perceive] an object from different perspectives”
[21, p. 40]. With the close tie spatial visualization has to geometry [1]
and to the act of paper-folding [14, 26], three spatial ability tests were also
selected as instrumentation for this study. Based on a review of existing
research on spatial ability and appropriateness for the age of participants,
the Paper Folding, Surface Development, and Card Rotation Tests were
chosen from the Kit of Factor-Referenced Cognitive Tests [10]. Each of
these tests consisted of two parts and took between three and six minutes
to complete. Due to time constraints, one part from each test was used.
Reliability was established based on a study conducted by Fleishman and
Dusek, who reported test-retest reliability values ranging from .76 to .92
for spatial-based tests [13].
At the start of the study, all students were pre-tested using the selected
mathematics achievement and spatial ability tests. The regular classroom
teacher then began the unit on geometry. During this time, the treat-
ment group participated in origami instruction three times a week. Each
of these origami lessons was conducted by the researcher with no involve-
ment by the regular classroom teacher. While modeling each step to pro-
duce the origami figure, the researcher interspersed relevant mathematics
terminology and encouraged dialogue with students regarding mathemat-
ics concepts and terminology identifiable within the folding process. (See
the appendix for sample dialogue used with the instruction of an origami
During the one-month time period, three days of instruction were ran-
domly selected and videotaped. This was done to assure that the only
difference in instruction between the control and treatment groups was the
addition of the origami lessons within the treatment group. Three read-
ers (including the researcher) reviewed the videotapes using a researcher-
designed checklist to track what objectives and terminology were covered
during each session as well as what teaching delivery method was utilized.
Accumulated information revealed that the regular classroom teacher main-
tained the same instructional methods and covered very similar material
in all classes.
When the one-month unit was complete, students were again given the
mathematics achievement and spatial ability tests. Data was then gath-
ered and analyzed using a statistical analysis software package. Analysis
of Covariance (ANCOVA) was used to determine if significant differences
occurred between adjusted mean post-test scores, with the pre-test score
serving as the covariate (to control for initial differences that may have
existed between groups).
The Impact of Origami-Mathematics Lessons on Achievement and Spatial . . . 475
Instrument Group Gender (N) Pre-test SD Post-test SD
Mean Mean
Card Rotation
Experimental Male (14) 62.69 17.14 69.00 13.45
Female (11) 49.27 15.65 48.64 11.87
Total (25) 56.56 17.46 60.04 16.22
Control Male (11) 49.45 14.44 55.82 13.38
Female (20) 53.85 15.21 62.30 13.37
Total (31) 52.29 14.85 60.00 13.52
Experimental Male (14) 4.14 2.25 5.36 1.69
Female (11) 3.91 2.55 4.55 2.62
Total (25) 4.04 2.34 5.00 2.14
Control Male (11) 3.00 1.18 3.36 2.01
Female (20) 3.95 1.67 4.85 1.63
Total (31) 3.61 1.56 4.32 1.89
Surface Devel-
opment Test
Experimental Male (14) 10.50 8.30 15.57 9.75
Female (11) 9.36 5.16 12.64 6.07
Total (25) 10.00 6.98 14.28 8.30
Control Male (11) 5.73 2.83 9.91 6.76
Female (20) 12.60 7.64 16.00 8.01
Total (31) 10.16 7.13 13.84 8.04
Experimental Male (14) 14.50 3.80 17.00 3.68
Female (11) 13.36 5.14 16.09 4.30
Total (25) 14.00 4.38 16.60 3.91
Control Male (11) 15.55 3.70 15.91 4.30
Female (20) 14.65 4.00 15.55 3.87
Total (31) 14.97 3.86 15.68 3.96
Table 1. Descriptive statistics for all instruments.
Descriptive statistics for all pre- and post-tests administered are given in
Table 1. Results are further broken down by group and gender.
A2×2 between-groups ANCOVA was conducted for each of the spatial
ability tests. For the first of three spatial tests, the Card Rotation Test,
values calculated revealed a significant interaction effect between group
and gender (F(1,51) = 9.09, p<.005) with a small effect size (∂η2=
.15), while neither of the main effects were statistically significant (group:
F(1,51) = .78, p=.381; gender: F(1,51) = 2.69,p =.107). For the Paper-
Folding Test, calculated ANCOVA values approached significance for the
476 V. Origami in Education
Group Gender (N)Adjusted Mean Combined
Experimental Male (14) 64.62 57.96
Female (11) 51.30
Control Male (11) 58.39 60.44
Female (20) 62.49
Combined Male (25) 61.50
Female (31) 56.90
Table 2. Adjusted means for Card Rotation post-test scores.
interaction between group and gender (F(1,51) = 3.59, p=.064). For
main effects, no significance was found (group: F(1,51) = 1.39, p=.244;
gender: F(1,51) = .05, p=.830). The ANCOVA completed for the final
of three spatial tests, the Surface Development Test, revealed no significant
interaction effect (F(1,51) = .38, p=.540) as well as no main effects by
group or gender (group: F(1,51) = .10, p=.750; gender: F(1,51) = .45,
p=.504). An ANCOVA was also completed for the NAEP Mathematics
Achievement Test. Calculations revealed no significant interaction effect
between group and gender (F(1,51) = .05, p=.817). Similarly, for the
main effects, neither produced significant values (group: F(1,51) = 2.96,
p=.091; gender: F(1,51) = .00, p=.977).
One of the four tests utilized in the study resulted in statistical signifi-
cance. For this test, the Card Rotation Test, further analysis was completed
by reviewing the adjusted means shown in Table 2. Males who received
treatment maintained a higher adjusted mean than females within the same
group. Males in the experimental group also scored higher than males in
the control group while females’ adjusted mean scores within the control
group surpassed their female counterparts experiencing treatment.
6 Conclusions
In terms of spatial ability, analysis of pre- and post-tests reveals that
origami did have some impact on students’ spatial visualization skills. The
Card Rotation Test, selected to evaluate students’ ability to mentally ro-
tate two-dimensional figures, produced interesting results with males and
females responding differently dependent upon the instructional method
experienced. Based on results of the ANCOVA, males seemed to respond
best to origami instruction while females seemed to flourish within a tra-
ditional structure. Additionally, for the Paper Folding Test, interaction ef-
fects approached significance, indicating again that group (experimental or
control) and gender had some bearing on mean scores received. The Paper-
Folding Test also dealt with students’ two-dimensional visualization abili-
The Impact of Origami-Mathematics Lessons on Achievement and Spatial . . . 477
ties. The final of these tests, the Surface Development Test, was selected to
determine students’ ability to visualize in both two- and three-dimensions.
Results from this test reveal no significance and seem to indicate that the
instructional method had little to no bearing on mean scores earned by
participants. Clearly the lesson to be learned here, of course within the
limitations of this study, is that males and females can respond differently
to origami instruction.
In considering why such gender differences occurred, one must take
into account what might have inadvertently impacted the results found in
this study. For instance, many spatial ability tests have been found to be
predisposed to gender differences due to the nature of the test questions
[19, 30]. Social and environmental factors such as out-of-school activities
and hobbies may also cause males and females to benefit differently from
such a spatially-based instruction technique [2, 11]. Though the cause of
the results found here cannot be identified with certainty, future studies
should consider gender and factors influencing performance on spatially-
related tasks.
A set of samples from the geometry/spatial sense strand of the NAEP
assessment was used to determine students’ mathematical achievement
level. Based on results presented, mathematics achievement gains were
similar regardless of gender or the type of instruction experienced. This
leads one to conclude that both methods were equally beneficial. Though
this is limited by the confines of this study, this result implies that origami
lessons integrated within mathematics instruction can be a valuable expe-
rience for students. A further convincing factor is the fact that by adding
origami instruction into traditional instruction, 20 to 30 minutes of in-
structional time was lost within each meeting. Although the treatment
group spent less time under traditional instruction by the regular class-
room teacher, students still did as well as those within the control group.
This finding stands then to substantiate the numerous claims that origami
is an effective instructional tool in mathematics.
As is always the case, there are factors that may have contributed to
the results found for mathematical achievement. For one, multiple-choice
tests such as the NAEP are quite common and are something students
are comfortable taking. In addition, the specific NAEP questions selected
for use in the assessment may have had some influence. Though each was
chosen because of their relationship to geometry and spatial ability, they
may not have accurately captured the specific skills and concepts gained
from the unit on geometry [24]. A final factor may be the text used by the
instructor. Most mathematics texts now recognize the importance of the
national mathematics standards and include material to fulfill them. With
the classroom instructor in this study using such a text, it may be that
geometry instruction was already geared to develop students’ geometry
478 V. Origami in Education
knowledge and understanding, dampening the effect origami might have
had on students’ skills.
In all, the intent of this study was to determine the impact origami
lessons integrated into a mathematics classroom would have on students’
abilities. Though spatial visualization and mathematics achievement re-
sults differed, it can be said that origami can be beneficial to students.
It is important, though, that others continue to study origami’s true im-
pact on students. There is little formal research currently reported in this
field. Future studies should seek to substantiate other possible benefits of
origami and for a variety of age levels. It is in this way that the mathemat-
ical community can realize the full potential of origami as an instructional
7 Appendix: Sample Dialogue for Instruction of an
Origami Model
The following presents key questions that could accompany the instructions
for the Leaping Frog origami model (Figure 1). Note that in the following
text, italics indicates answers expected to teacher-initiated questions.
1. Before you fold your card, what mathematical terms could you use to
describe it? [Rectangle or plane.] Once you make the creases using
adjacent corners of the card, what kind of line segments were formed?
[Perpendicular line segments.] What kind of angles are formed then?
[Right angles.] What could you say about the measure of two adjacent
right angles? [They’re supplementary.]
2. Once you mountain fold you form a third line segment. Do you
recognize any of the angles formed here? [There are right and acute
angles formed.] Can you find a set that are supplementary? [Have
student show where they are.] Could you find the exact measures of
the angles without a protractor? [Yes, since the last line cut them in
half, the angles are 45 degrees and 90 degrees.]
3. Once you do the squash fold, what kind of shapes are formed? [Right
triangle and a rectangle.] Can you identify the angle measures of each
of them? [Yes, the rectangle has all 90 degree angles and the right
triangle has two 45 degree angles and one 90 degree angle.]Istherea
more specific name you can give to the triangle? [Yes, it’s isosceles
right!] What special terms are associated with an isosceles triangle?
[Vertex angle, base angles, legs, and base.]
The Impact of Origami-Mathematics Lessons on Achievement and Spatial . . . 479
Figure 1. Accompanying model: Leaping Frog [25].
4. When you fold the base angles of the isosceles right triangle up, what
have you formed? [Two new, smaller triangles.] What can you say
about them? [There are four of them that are all congruent. They’re
all isosceles right like the other larger one.]Howdoestheareaof
the small triangles compare to the one from the previous step? [It’s
exactly a fourth of the original one.]
5. When you fold the sides into the middle, what new shapes do you
have? [Trapezoids.] How do they compare in size? [They’re congru-
ent.] If you ignore all the folds and look at the figure as a whole,
what is it? [It’s a pentagon.]
6. When you’re all done with your frog, what kind of mathematical
terms can you identify with it? [Pentagon, triangles, rectangle, par-
allel lines, perpendicular lines, ....]
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... Improvement of geometrical understanding by origami have been reported (Boakes, 2009(Boakes, , 2011Arıcı and Aslan-Tutak, 2015;Burte et al., 2017). Origami has been considered to stimulate and train spatial ability while folding paper, that is, geometrically transforming a paper's shape and making geometrical figures, which can contribute to the development of geometrical concepts (Boakes, 2009(Boakes, , 2011Taylor and Hutton, 2013;Cakmak et al., 2014;Burte et al., 2020). ...
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The aim of this study is to do a limited trial on the hypothetical learning strategy for early childhood mathematics concepts using origami construction. Using the training devices designed for the limited trial, 17 early childhood education teachers in Bandung and its surrounding areas were trained on the learning strategy. The trial pre-test and post-test scores and it shows an improvement in the knowledge of participating teachers of early childhood mathematics learning using origami. The significance of the improvement had been proven using t-paired test. The improvement might be caused by the fact that most of the participants only have high school diploma as education background with lack of training on early childhood mathematics learning, and also with the fact that origami is mainly used for fine motor skills improvement in early childhood education. Thus, the origami construction can be used as an alternative strategy to learn mathematics concept in early childhood.
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School geometry is the study of those spatial objects, relationships, and transformations that have been formalized (or mathematized) and the axiomatic mathematical systems that have been constructed to represent them. Spatial reasoning, on the other hand, consists of the set of cognitive processes by which mental representations for spatial objects, relationships, and transformations are constructed and manipulated. Clearly, geometry and spatial reasoning are strongly interrelated, and most mathematics educators seem to include spatial reasoning as part of the geometry curriculum. Usiskin (Z. Usiskin, 1987), for instance, has described four dimensions of geometry: (a) visualization, drawing, and construction of figures; (b) study of the spatial aspects of the physical world; (c) use as a vehicle for representing nonvisual mathematical concepts and relationships; and (d) representation as a formal mathematical system. The first three of these dimensions require the use of spatial reasoning.
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This study was conducted to investigate differences in spatial visualization abilities and effects of instruction on spatial visualization skills of fifth through eighth grade students by grade, sex, and site. About 1,000 students from three sites, representing a wide range of socioeconomic status, participated in the study. The spatial visualization unit engaged students in concrete activities, building and drawing solids made of cubes. The instrument used was the MGMP Spatial Visualization Test, with a test-retest reliability of .79, and Cronbach's reliability coefficients for various groups of students ranged from .72 to .86 on the pretest and from .82 to .88 on the posttest. Before instruction, there were significant differences in spatial visualization performance by grade (increasing with age), by sex (favoring boys), and by site (increasing with socioeconomic status). After instruction, fifth through eighth grade students profited considerably from instruction, and the gain was similar for boys and girls despite initial sex differences. Retention of effects persisted after a 4-week period and after 1 year.
Rigami has been used frequently in teaching geometry to promote the development of spatial sense; to make multicultural connections with mathematical ideas; and to provide students with a visual representation of such geometric concepts as shape, properties of shapes, congruence, similarity, and symmetry. Such activities meet the Geometry Standard (NCTM 2000), which states that students should be engaged in activities that allow them to “analyze characteristics and properties of twoand three-dimensional geometric shapes and develop mathematical arguments about geometric relationships” and to “use visualization, spatial reasoning, and geometric modeling to solve problems” (p. 41). This article begins with an explanation of the importance of communication in the mathematics classroom and then describes a middle school mathematics lesson that uses origami to meet both the Geometry Standard as well as the Communication Standard.
Reliability, practice effects, and factor loadings were investigated for 21 paper-and-pencil tests selected from the French, et al. Kit of Reference Tests for Cognitive Factors. Criteria of brevity and ease of administration were employed for selecting tests to be studied. The results are helpful in selecting those tests of cognitive abilities which may be most useful in studies of the effects of climatic and environmental variables on behavior. Such studies frequently require repeated measurements on a small group of Ss over several days, thus making it extremely important that practice and environmental effects not be confounded. Test-retest reliabilities of the tests were generally quite good; 16 of the tests had a Pearson r of .8 or greater. Eight of the tests showed enough stability in mean performances over six trials to suggest that they may be used repetitively under environmental extremes without serious confounding by practice. The authors caution that factor definitions may not be the same under extreme conditions and under normal environmental conditions.
The relative importance of spatial visualization ability and cognitive development for achievement in a geometry course for preservice elementary teachers was investigated. Both factors correlated significantly with achievement, but in an analysis of variance, only the main effect due to cognitive development was significant. There was no interaction between the factors. The effect of the semester-long geometry course on students' spatial ability was also investigated. It was found that the students' spatial visualization ability was significantly greater at the end of the geometry course than at the beginning.
For 187 Grade 8 students, we compared spatial-mechanical skills with mathematics self-confidence as mediators of gender differences in mathematics. Using items showing the largest male and female advantage, respectively, on the Third International Mathematics and Science Study (TIMSS) U. S. data, we created mathematics Male and Female subtests from items on the 8th-grade TIMSS. Using path-analytic techniques, we decomposed a significant gender/mathematics correlation, favoring males, on the TIMSS-Male subtest into direct and indirect effects. We found only indirect effects. A spatial-mechanical composite accounted for 74% of the total indirect effects, whereas mathematics self-confidence accounted for 26%. By 8th grade, girls' relatively poorer spatial-mechanical skills contribute to lower scores on types of mathematics at which boys typically excel.
Relationships were investigated between mathematics learning, verbal ability, spatial visualization, and eight affective variables. Subjects were 1320 sixth through eighth graders. No sex-related differences over all schools were found for any cognitive variable. Females were significantly less confident of themselves in mathematics, and males stereotyped mathematics as a male domain higher than did females. Results were synthesized with those obtained at the high school level. Significant sex-related differences found in high school areas were not found in the same middle school areas. Where significant differences in achievement were found at both levels, they were accompanied by significant differences in many affective variables.
This study investigated (a) mathematics achievement (Test of Academic Progress) of 589 female and 644 male, predominantly white, 9th-12th grade students enrolled in mathematics courses from four schools, controlling for mathematics background and general ability (Quick Word Test); (b) relationships to mathematics achievement, and to sex-related differences in mathematics achievement, of spatial visualization (Differential Aptitude Test), eight attitudes measured by the Fennema-Sherman Mathematics Attitudes Scales, a measure of Mathematics Activities outside of school, and number of Mathematics Related Courses and Space Related Courses taken. Complex results were obtained. Few sex-related cognitive differences but many attitudinal differences were found. Analyses of variance, covariance, correlation, and principal components analysis techniques were used. The results showed important relationships between socio-cultural factors and sex-related cognitive differences.