A survey of paper cutting, folding and tearing in mathematics textbooks for prospective elementary school teachers.

Origami or the art of paper folding receives substantial endorsement from current reform initiatives in mathematics education. Particularly, at the elementary school level, the National Council of Teachers of Mathematics in its Principles and Standards for School Mathematics recommends that students use paper folding to perform their initial investigations of symmetry and congruence in geometry (NCTM, 2000). Since prospective elementary school teachers need a solid foundation in the mathematical basis of these reform initiatives, textbooks for these future teachers should contain paper-folding applications.

Although origami spans academic disciplines, paper folding has increasingly been applied to mathematical systems and structures (Hull, 1996; Auckly and Cleveland, 1995). Moreover, origami is a unique mathematical activity because one can take a sheet of paper and transform it into a three-dimensional object. This concrete experience in spatial reasoning can then be transformed into a lesson on symmetry or graph theory (Peterson, 1995). Educators have also applied paper folding to such diverse mathematical objects as logical structures (Cipra, 1998), axiomatic systems (Alperin, 2000), and tessellations with geometrical figures (Stewart, 1999).

Clements and Battista (1992) defined school geometry as the study of spatial objects, relationships, and transformations that have been formalized. As students participate in the physical opportunities afforded by origami and other manipulations with paper, they gain a foundation from which to build the formalizations needed in school geometry. Particularly, students benefit from experimenting and exploring with physical materials and models, and learning opportunities that require students to visualize, draw, and compare figures in various configurations help to develop spatial sense. Since geometry at the elementary and intermediate school levels aims to help students describe, relate, and represent objects in the environment, students can benefit from activities involving paper cutting, folding and tearing.

Put another way, transforming a flat piece of paper into a three-dimensional object is essentially a manifestation of spatial reasoning, a key component of geometrical learning. Thus, origami along with paper cutting and tearing can allow students to create, modify, and investigate specific geometric shapes such as polygons and polyhedra.

The use of origami in school programs is also one of the notable differences between American and Asian educational systems. Given the better performance of Asian children in international comparisons in mathematics, it may be worthwhile to give attention to the potential impacts origami can have on the development of US students' mathematical thinking. Rooted in Asia, origami reflects the creativity and aesthetics of those cultures. By taking part in origami and other paper manipulation activities, American students can gain an appreciation of Oferent cultures and alternate ways of observing and studying geometric figures. Cooney (2001) points out, that in our internal, national zeal to improve mathematics education in the US, we should not overlook the attributes and approaches of other cultures that have demonstrated high student achievement in mathematics. Clearly, Asian societies and their contributions to origami and other manipulations with paper are a case in point.

Silverman and Manzano (1996) further noted that what is accomplished by using origami is no less than the planting and nourishing of the seeds of geometric thinking. Among the geometric concepts that are embedded in origami are similarity, congruence, measurement, and construction. Understanding these and other basic concepts and the developmental ways that children learn allows the teacher to facilitate activities which are rich in exploration, application, representations, communications and mathematical reasoning. Silverman and Manzano conclude that paper folding provides a highly engaging and motivating environment within which children extend their geometric experiences and powers of spatial visualization. In short, well-planned activities involving paper can give a venue for students' creative nature and invite play, problem solving, and problem posing.

Arcidiacono (1992) in his geometry work at the Math Learning Center at Portland State University further noted that paper folding activities can cause students to reflect about the relationships among specialized polygons such as isosceles triangles, rhombi, and kites. However, the endorsement of paper folding extends back even further western education. Friedrich Froebel, a German founder of the Kindergarten Movement, introduced paper folding into nineteenth century kindergartens, and this introduction was further developed and carried on by later reformers (Lister, 2002). Through paper folding, children use their hands to follow a prescribed set of steps, and Froebel believed strongly that successful completion of the steps resulted in objects that children found both pleasing and mathematical in nature. The renowned child development psychologists, Piaget and Inhelder (1956) agreed with this assessment when they posited that motor activity in the form of skilled movements was vital to the development of c hildren's intuitive thought and their mental representations of space.

Crockett and Marcroft (1993) contributed another recent example of the value of origami in developing mathematical understanding. Crockett, a student in Marcroft's high school class, became so excited about origami that he developed his own unit. for constructing polyhedra. The resulting collaboration between student and teacher resulted in the development of a series of original origami activities configured on a basic paper-folding assembly that Crockett and Marcroft named the "Crockett Corner". The creators found that constructions made with the Crockett Corner were strong in compression, but tended to spring apart when pulled. Additionally, the Crockett Corner enabled some unique approaches for constructing cubes, octahedra, and other polyhedra (Figure 1).

Based on the preceding literature and results it is clear that origami is both an art form and a study that can enhance elementary school programs in mathematics. However, a review of the literature shows that there is no recent or systematic survey of paper folding applications in mathematics textbooks for prospective teachers. Consequently, a survey was undertaken to determine the nature and extent of origami applications in textbooks intended for mathematics content courses for future primary and intermediate school teachers. Such textbooks are often used in courses that are prerequisite for elementary school mathematics methods courses.

Procedures

To conduct the survey, ten popular mathematics content textbooks for prospective elementary school teachers were selected (Bassarear, 2001; Bennett and Nelson, 2001 a; Bennett and Nelson, 2001b; Billstein, Libeskind, and Lott, 2001; Long and DeTemple, 2000; Masingila and Lester, 1998; Miller, Heeren, and Hornsby, 2001; Musser, Burger, and Peterson, 2001; O'Daffer, Charles, Cooney, Dossey, and Schielack, 1998; Wheeler and Wheeler, 1998). These texts (hereafter cited by the first author's last name) were chosen based on an informal poll of the researchers' colleagues who also taught mathematics content courses for prospective teachers. The textbooks were all published between 1998 and 2001 and only the latest edition of each text was used in this study. To survey the textbooks for instances of origami and to describe the nature of these instances, four definitions of paper folding were established. These definitions facilitated the categorizing of the occurrences of origami that were found in the textbooks. Ta ble I lists these definitions as they were used in the survey and in this report.

After the ten textbooks were collected, a panel of two graduate students and two Ph.D.s in mathematics education was used to classify the instances of paper folding in the textbooks. To complete this classification task, the panel met on two separate days for approximately four hours per day. During each of these sessions panel members reviewed the textbooks and noted any instances of origami and the nature of those instances. Also during each of the sessions, a consensus was reached about the paper-folding opportunities according to their type and nature. At the end of the two sessions there were no unresolved differences among the panel members with respect to the extent and nature of the occurrences of origami in the textbooks. Early in the panel's work, it became evident that the texts included activities that involved paper cutting or tearing as well as paper folding. Since definitions of origami often stipulate that cutting, gluing, or drawing on the paper is to be avoided, and only paper folding is use d to create the desired result (Gross, 1995), the panel members were reluctant to classify as origami any instances that did not strictly involve paper folding. As a result, the panel members decided for informational purposes to classify and report separately any occurrences of paper cutting or paper tearing that were found in the texts.

In all, the classifications employed in this study included: paper-folding activities, paper-folding examples, paper-folding exercises, and a generic term that included all of these instances of paper-folding opportunities. Additionally, paper-cutting and paper-tearing opportunities were also counted and the nature of these opportunities were noted and recorded as well.

Figures 2, 3 and 4 give some basic illustrations of paper cutting, folding, and tearing as they appeared in activities, examples, and exercises in a number of the texts used for this study. Figure 2 shows a basic paper-cutting procedure to construct a square from a rectangular sheet of paper, while figure 3 illustrates the use of an angle drawn on paper to construct the angle's bisector by folding one side of the angle onto the other. Figure 4 displays a before-and-after view of using paper tearing to demonstrate that the sum of the angle measures in a triangle is 180[degrees].

Results

Table II shows the number of paper-folding activities that were found in the ten textbooks that were surveyed.

Since activities encouraged the reader to actively participate in learning concepts, it was anticipated that most of the texts would contain instances of paper folding under this classification. However, Table II shows that three texts (Long, Bennett(a), and Masingila) contained three-fourths of the paper-folding activities. About 40% of the paper-folding activities were in the Long textbook, and four of the surveyed texts contained no paper-folding activities as they were defined in this study. Four texts (Long, Bennett (a), Bennett (b), and Masingila) contained nearly 88% of the paper-folding activities, but only the Long textbook also contained examples (2) of paper folding, while the Bennett (a), Bennett (b), and Long textbooks also contained exercises in paper folding. More information about examples and exercises related to folding paper is contained in Tables III and IV.

Table III shows the number of paper-folding examples found in the textbooks that were surveyed.

The number of examples in Billstein (7), Musser (8), and Wheeler (6) accounted for all but three of the paper-folding examples found in the ten textbooks. With respect to paper-folding exercises, there were some contrasts between the Billstein, Musser, and Wheeler texts. Table IV shows that Billstein followed up examples of folding paper with five exercises in paper folding, whereas the Musser text contained an equal number of exercises and examples (8), and the Bassarear contained eight exercises but no examples.

Also, the Wheeler text contained more than a third of the total number of paper-folding exercises in the ten textbooks.

Table V shows there were 137 instances of paper folding or paper-folding opportunities in the ten texts surveyed. This produced an overall average of 13.7 and a median value of 13 paper-folding opportunities per textbook. However, one of the texts (Miller) contained no instances of paper folding, while O'Daffer contained only two paper-folding opportunities. On the other hand, the textbook that contained the most opportunities (Long with 34) contained more than twice the average number of opportunities found in the ten textbooks.

There were also contrasts evident in the nature of the paper-folding opportunities. Table VI shows that angles, properties of triangles, and three-dimensional figures, which appeared in four of the texts, were the most popular types of paper-folding opportunities in the ten texts.

Additionally, perpendicular segments, polygons, and symmetry each appeared in three of the surveyed textbooks. There were also paper-folding opportunities that rarely appeared in the textbooks. These included fractions and linear measurement that were evident in two books and parallel lines, the Pythagorean theorem, segments, and spatial visualization which each were found in only one of the textbooks. These rarely occurring paper-folding opportunities occurred in six of the textbooks surveyed: Bennett(a), Bennett(b), Bilistein, Long, Masingila, and Musser.

Table VII shows that the number of paper-cutting opportunities (118) nearly equaled the total number of paper-folding opportunities in the ten textbooks.

Specifically, five texts (Bennett(a), Billstein, Long, Masingila, and Musser) accounted for 104 or more than 88% of the instances of paper cutting. Although all the texts contained at least one paper-cuffing opportunity, the three texts with exactly one were Bennett(b), O'Daffer, and Wheeler.

Table VIII shows that instances of paper tearing were the least common of the classifications found in the survey. Specifically, only ten paper-tearing opportunities were found in the ten textbooks surveyed. The Billstein text contained four, while the remaining six opportunities were contained in four other texts: Bassarear, Bennett(b), Long, and O'Daffer. Four of the textbooks surveyed revealed no instances of paper tearing.

The nature of the paper-cutting/-tearing opportunities like the paper-folding opportunities revealed a number of contrasts. Table IX shows that six of the texts contained occurrences of paper cutting/tearing related to finding the sum of the vertex angles in a triangle. The next most frequent occurrences involved spatial visualization and topology. These instances of paper cutting/tearing were found in four of the ten textbooks. Three of the texts contained the three topics that were the next most frequent: polygons, symmetry, and linear measurement. Two textbooks contained instances involving the cutting or tearing of paper that illustrated tessellations or three-dimensional figures. The remaining paper-cutting or paper-tearing topics each appeared in only one textbook. These most rarely occurring topics were: angles in a quadrilateral, fractions, fraction multiplication, properties of triangles, and the Pythagorean theorem.

Conclusions

Although ten texts were surveyed in the current study and 137 instances of paper folding were identified, nearly one quarter or 34 of these opportunities to fold paper were in the Long textbook. The Wheeler text contained the next largest number of paper-folding opportunities (23), and this accounted for approximately 17% of the total number of opportunities. Thus, more than 40% of the paper-folding opportunities were contained in two of the textbooks surveyed. The third most frequent number ofpaper-folding opportunities was contained in the Billstein text. These accounted for 18 or 13% of the paper-folding opportunities. The fourth most frequent number ofpaper-folding opportunities (16 or 12%) was in the Musser text. Thus, the four textbooks (Long, Wheeler, Bilistein, and Musser) that contained the largest number of paper-folding opportunities, collectively accounted for more than two-thirds of the total number of paper-folding opportunities in the ten textbooks. Table X summarizes this information for the t exts with the most paper-folding opportunities.

With respect to the nature of the paper-folding opportunities, the most popular topics were angles, properties of triangles, and three-dimensional figures along with perpendicular segments, polygons, and symmetry. The most rarely appearing topics that were addressed by paper folding included fractions and linear measurement as well as parallel lines, the Pythagorean theorem, segments, and spatial visualization. Table XI summarizes the nature of the paper-folding opportunities with respect to their frequency of occurrence.

Three of the texts surveyed contained 20 or more paper-cutting opportunities. These texts were Long (24), Muster (24), and Masingila, (20). However, two texts (Bennett(a) and Billstein) also each contained 18 instances of paper cutting. Thus, the remaining five textbooks (Bassarear, Bennett(b), Miller, Q'Daffer, and Wheeler) contained about 12% of the total number of paper-cutting opportunities. Table XII illustrates this information for the five textbooks that contained the most paper-cutting/-tearing opportunities.

Paper-tearing opportunities, of which there were 10 in the ten texts, accounted for the fewest number of classifications in the survey. Notably, the Bilistein textbook contained four of these ten opportunities. Four of the texts (Bennett(a), Masingila, Musser, and Wheeler) contained no instances of paper tearing.

The largest number of paper-cutting/-tearing opportunities related to Ending the sum of the interior angles in a triangle along with spatial visualization and topology. Another group of mathematical topics appeared at an intermediate level of frequency among the topics addressed by paper cutting/tearing. These topics included polygons, symmetry, and linear measurement as well as with tessellations and three-dimensional figures. The most rarely occurring topics developed through paper cutting/tearing were angles in a quadrilateral, fractions, fraction multiplication, properties of triangles, and the Pythagorean theorem. Table XIII summarizes the frequency of occurrence of each topic.

Based on the findings of this study, two textbooks (Long and Wheeler) contained 25% and 17%, respectively, of the total number of paper-folding opportunities. These percentages exceeded the expected percentage of paper-folding opportunities (10%) that would be the case in the event that these opportunities were randomly distributed among the ten textbooks. Two other texts (Billstein and Musser) contained 13% and 12%, respectively, of the total number of paper-folding opportunities, thereby providing somewhat less convincing evidence that these texts also exceeded the expected 10% level.

The evidence from this study regarding the nature of the paper-folding opportunities revealed that topics in informal geometry, geometric constructions, and symmetry received coverage in some, but not a majority of the textbooks surveyed. Particularly, between three and four of the texts surveyed, included the topics from informal geometry, geometric constructions, and symmetry as opportunities for folding paper. Notable among the topics that received less attention in origami were fractions, linear measures, parallel lines, the Pythagorean theorem, and segments. Fewer than three of the textbooks contained instances wherein paper folding was applied to these topics. Given the importance of fractions in school mathematics and the difficulty students frequently associate with the topic, it is remarkable that mathematics textbooks for prospective elementary school teachers contain so few of the newer methods for teaching fractions by means of origami (Camblin, 1998; Sinicrope and Mick, 1992).

A review of the paper-cutting-tearing opportunities in the ten textbooks revealed that half of the texts (Long, Musser, Masingila, Bennett(a), and Billstein) contained the vast majority (88%) of these opportunities. Moreover, each of these five texts substantially exceeded the 10% expected value that would result if these activities were randomly distributed among the ten textbooks. An examination of the topics represented in papercutting/-tearing opportunities revealed that only one (angles in a triangle) was evident in a majority (6) of the textbooks.

Among the other topics addressed by paper cutting/tearing in the texts, a diversified group of seven topics were evident in less than three textbooks, and as a consequence, are good candidates as topics that may be enhanced through origami methods. Recent design and develop work in origami has produced opportunities for folding paper that particularly address a number of the diverse topics: tessellations (Stewart, 1999; Neale and Hull, 1994), investigations with geometric objects and their constructions (Morrow, 2001), and measurement of three-dimensional figures (Higginson and Colgan, 2001).

During the course of reviewing the ten texts for instances of origami, the members of the panel that enumerated and classified the opportunities to fold paper, made several comments concerning the diverse presentations and lack of *~unified approach to origami that was generally evident in the texts. These qualitative findings pointed toward the need for a more systematic treatment of paper folding in mathematics content texts for prospective teachers. Smart (1998) made a valuable contribution toward developing a more systematic treatment by illustrating and describing four basic geometric constructions through paper folding. These constructions are apt to provide prospective elementary school teachers with an opportunity to catch a more unified spirit of paperfolding constructions and to contrast it with other methods.

As research and development in origami continues, future advances are likely to produce educational gains in still other areas. Wet folding, a procedure wherein heavy paper is folded while wet and thus allows the folder to sculpt a model with soft curves and 3-D forms, appears to have considerable potential in educational applications (Wu, 2001). Hypergami, the use of computers to design and construct models for origami, is also a recent application (Eisenberg and Nishioka, 1996) that shows great promise for empowering teachers who use origami to plant and nourish the seeds of mathematical thinking.

Table 1

Definitions Used in the Survey

Definitions

Paper-folding activity - an
opportunity to fold paper that was
not part of an exercise set or an
example, Activities frequently
were located in sidebars or in
special text sections to encourage
active learning

Paper-folding example - a worked-
out rendering of how to fold paper
to illustrate a mathematical
concept

Paper-folding exercise - an item
in an exercise set in which
students were to work either
independently or cooperatively

Paper-folding opportunity - a
generic term that describes any of
the above: paperfolding
activities, examples, or exercises

TABLE II

Number of Paper-Folding Activities

Textbook      Number of Paper-Folding
                    Activities

Bassarear                0
Bennett (a)             13
Bennett (b)              9
Billstein                6
Long                    29
Masingila               12
Miller                   0
Musser                   0
O'Daffer                 0
Wheeler                  3

Total Number            72

TABLE III

Number of Paper-Folding Examples

Textbook      Number of Paper
              Folding Examples

Bassarear            0
Bennett (a)          0
Bennett (b)          0
Billstein            7
Long                 2
Masingila            0
Miller               0
Musser               8
O'Dafter             1
Wheeler              6

Total Number        24

TABLE IV

Number of Paper-Folding Exercises

Textbook      Number of Paper-Folding
                     Exercises

Bassarear                 8
Bennett (a)               1
Bennett (b)               1
Billstein                 5
Long                      3
Masingida                 0
Miller                    0
Musser                    8
O'Daffer                  1
Wheeler                  14

Total Number             41

TABLE V

Total Number of Paper-Folding Opportunities

Textbook      Total Number of Paper
              Folding Opportunities

Bassarear               8
Bennett (a)            14
Bennett (b)            10
Billstein              18
Long                   34
Masingila              12
Miller                  0
Musser                 16
O'Daffer                2
Wheeler                23

Total Number          137

TABLE VI

Nature of Paper-Folding Opportunities

Textbook     Nature of Opportunities

Bassarear    symmetry
Bennett (a)  segments
             three-dimensional figures
Bennett (b)  linear measurement
             polygons
             properties of triangles
Billstein    angles
             linear measurement
             parallel lines
             perpendicular segments
             polygons
             properties of triangles
             symmetry
Long         angle trisection
             fractions
             perpendicular segments
             polygons
             properties of triangles
             three-dimensional figures
Masingila,   fractions

Miller

Musser       angles
             perpendicular segments
             properties of triangles
             Pythagorean theorem
             spatial visualization
O'Daffer     symmetry
             three-dimensional figures
Wheeler      angles
             three-dimensional figures

TABLE VII

Number of Paper-Cutting Opportunities

              Number of Paper
Textbook      Cutting Opportunities

Bassarear               8
Bennett (a)            18
Bennett (b)             1
Billstein              18
Long                   24
Masingila.             20
Miller                  3
Musser                 24
O'Daffer                1
Wheeler                 1

Total Number          118

TABLE VIII

Number of Paper-Tearing Opportunities

              Number of Paper-Tearing
Textbook           Opportunities

Bassarear                1
Bennett (a)              0
Bennett (b)              1
Billstein                4
Long                     1
Masingila                0
Miller                   1
Musser                   0
O'Daffer                 2
Wheeler                  0

Total Number            10

TABLE IX

Nature of Paper-Cutting/-Tearing Opportunities


Textbook     Nature of Paper-Cutting
             l-Tearing Opportunities
Bassarear    angles in a triangle polygons
             spatial visualization symmetry
             topology
Bennett (b)  angles in a triangle
             linear measurement
Bilistein    angles in a triangle
             linear measurement
             properties of triangles
             spatial visualization
             symmetry
             tessellations
             topology
Long         angles in a triangle
             linear measurement
             polygons
             Pythagorean theorem
             spatial visualization
             tessellations
             three-dimensional figures
             topology
Masingila    fractions
             fraction multiplication
Miller       angles in a triangle
             topology
Musser       polygons
             spatial visualization
O'Daffer     angles in a triangle
             angles in a quadrilateral.
             three-dimensional figures
Wheeler      symmetry

TABLE X

Textbooks Containing the Largest Number of Paper-Folding Opportunities

Textbook and (Number of       Percentage of Total
Paper-Folding Opportunities)  Number of Paper
                              Folding Opportunities

Long (34)                              25%
Wheeler (23)                           17%
Billstein (18)                         13%
Musser (16)                            12%

TABLE XI

The Nature of Paper-folding Opportunities by Their Frequency of
Occurrence

Topics of Paper-Folding    Frequency of
Opportunities              Occurrence

Angles                          4
Properties of Triangles         4
Three-Dimensional Figures       4
Perpendicular Segments          3
Polygons                        3
Symmetry                        3
Fractions                       2
Linear Measurement              2
Parallel Lines                  1
Pythagorean Theorem             1
Segments                        1

TABLE XII

Textbooks Containing the Largest Number of Paper-Cutting/-Tearing
Opportunities

Textbook and (Number of  Percentage of Total
Paper-Cutting/Tearing    Number of Paper
Opportunities)           Cutting/Tearing
                         Opportunities

Long (24)                        20%
Musser (24)                      20%
Masingila (20)                   17%
Bemett(a) (18)                   15%
Billstein (18)                   15%

TABLE XIII

The Nature of Paper-Cutting/Tearing Opportunities by Their Frequency of
Occurrence

Topics of Paper-Cutting/Tearing  Frequency of
Opportunities                    Occurrence

Angles in a Triangle                  6
Spatial Visualization                 4
Topology                              4
Linear Measurement                    3
Polygons                              3
Symmetry                              3
Tessellations                         2
Three-Dimensional Figures             2
Angles in a Quadrilateral             I
Fractions                             1
Fraction Multiplication               1
Properties of Triangles               1
Pythagorean Theorem                   1

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