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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.

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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

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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].

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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

model.)

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).

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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

Test

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

Paper-Folding

Test

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

Mathematical

Achievement

Test

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.

5Results

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

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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-

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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

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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

tool.

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.]

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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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