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The effect of perceived qualities of curriculum materials on mathematical performance

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The effect of perceived qualities of curriculum materials on mathematical performance
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Journal of Authentic Learning
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Rule, Audrey C. ( author )
Sobierajski, Mickey Jo ( author )
Schell, Robert E. ( author )
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A qualitative and quantitative study of the effects of perceived beauty, motivation, distractibility, and legibility upon the mathematical performance of preservice teachers (N=76; 66F, 10M) and fourth graders (N=67; 36F, 31M) using boards made of four materials that varied widely in these characteristics was conducted. Each board had nine numbers arranged in a three by three grid and an additional target number at the top. Students formed equations with three numbers in a row that resulted in the target number. Significant differences in mathematical performance across materials were found with participants writing more equations for boards that were perceived as high in beauty and motivation (a construct called "Enticement") and high in legibility but low in distraction (another construct called "Focus"). Children judged beauty mostly on color and texture, whereas adults included neatness and artistic appeal.
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Submitted by Brian McDonald (bmcdonal@oswego.edu) on 2005-11-07.
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Made available in DSpace on 2005-11-07T17:48:27Z (GMT).

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SUNY Oswego
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1558-7320 ( issn )

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The Effect of Perceived Qualities of Curriculum Materials on Mathemtical Performance 26 The Effect of Perceived Qualities of Curriculum Materials on Mathematical Performance Audrey C. Rule State University of New York at Oswego Mickey Jo Sobierajski Granby Elementary School, Fulton, NY Robert Schell State University of New York at Oswego A qualitative and quantitative study of the effects of perceived beauty, motivation, distractibility, and legibility upon the mathematical performance of preservice teachers (N=76; 66F, 10M) and fourth graders (N=67; 36F, 31M) using boards made of four materials that varied widely in these characteristics was conducted. Each board had nine numbers arranged in a three by three grid and an additional target number at the top. Students formed equations with three numbers in a row that resulted in the target number. Significant differences in mathematical performance across materials were found with participants writing more equations for boards that were perceived as high in beauty and motivation (a construct called "Enticement") and high in legibility but low in distraction (another construct called "Focus"). Children judged beauty mostly on color and texture, whereas adults included neatness and artistic appeal. ______________________________________________________________________________ Manipulative and visual materials are important to most types of learning. Representing ideas with drawings, symbols, charts, graphs, and physical objects, then connecting these representations to mathematical concepts lies at the heart of mathematical understanding (National Council of Teachers of Mathematics, 2000, Chapter 2: The Learning Principle). Alkhateeb (2002) found that preservice teachers who were taught with a hands-on manipulative approach experienced a positive change in their attitudes toward mathematics. Because visual and hands-on curriculum materials are so important to the teaching of mathematics, we wondered if perceived qualities of these materials would have an effect on the mathematical performance of the user. In other words, does it matter whether the visual/ manipulative materials used are "beautiful" or not? Will students perform just as well when using unattractive materials as attractive materials? The current study investigates the effect on mathematical performance of the perceived qualities of curriculum materials. Montessori (1912, 1914) emphasized the importance of a prepared environment with well-thought-out materials and lessons that nurture the student's exploration and creativity. As Standing (1957, p 248) describes, "It goes without saying that we should make this prepared environment as beautiful as possible." Wentworth (1998) gave four criteria for effective Montessori materials: simple, dynamic, self-corrective, and attractive to children. Simple means that the materials are easy to understand and apply, easy to make or assemble, and can be adapted to many uses. Simple may also imply a lack of distraction. Dynamic materials involve motion: the materials are designed for manipulation in solving problems or discovering relationships. The self-corrective nature of effective materials allows students to be self-reliant rather than dependent upon the teacher for determining if answers are correct. Finally, the criterion of attractiveness favors materials that are Journal of Authentic Learning, Volume 2, Number 2, Pages 26-41, 2005

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The Effect of Perceived Qualities of Curriculum Materials on Mathemtical Performance 27 well-finished, colorful, pleasant to handle, and address topics of interest to students. Montessori had an intuitive understanding of the effect of the beauty of instructional materials on student performance, but there has been a lack of formal research addressing this idea. Therefore, an investigation was conducted to determine whether such a relationship exists. In our study, we examined two groups of students: preservice teachers in a mathematics methods course, and fourth graders at an elementary school. We wanted to determine if their mathematical performance would vary with the perceived beauty (and other perceived qualities) of the materials used in a problem-solving activity, and if there were any differences between the two populations in their performance with different activity stimuli (Bingo-like boards) and their reported criteria for beauty judgments. Investigations of Perceptions of Beauty Other investigators have conducted inquiry concerning human perceptions of beauty. Green (1995) reviewed psychological research on the golden section. The golden section refers to a line segment of special proportions. This line segment is divided into two parts with the point dividing these parts positioned such that the ratio of the short segment to the long segment is equal to the ratio of the long segment to the whole. Green concluded that there are real psychological effects associated with rectangles with sides of this proportion. Magro (1997) investigated people's conceptions of human beauty finding that Barbie dolls are considered attractive because the doll's anatomical proportions represent highly evolved human traits rather than primitive ones. Magro (1999) found that nonhuman designs were also judged as beautiful when they showed proportions that mirrored the proportions of derived (more highly evolved) human forms. Forsythe, Presley, and Caton (1996) investigated shoppers' judgments of men's dress shirts. They found that styling, design, and overall appearance combined with durability were the major components on which consumers judged quality. Stamps (1994) evaluated subjects preferences for environments, finding they preferred natural over built scenes and old buildings over new buildings. Jacobson and Hofel (2002) and Hofel and Jacobsen (2003) studied adult university students' judgments of graphic patterns, finding symmetry and complexity of design as the most common criteria in personal definitions of beauty. They noted, however, that some students consistently chose other criteria; for example, some thought nonsymmetric patterns were more beautiful. They questioned whether a nomothetic approach to aesthetic judgments in which individual differences are treated as error variance was appropriate. Jacobson and Hofel advocate taking two perspectives, both nomothetic (general, rule-based judgments) and idiographic (individual judgments), in an investigation of beauty judgments. If the results indicate sufficient agreement, then idiographic accounts may be abandoned. The current study was devised to test this hypothesis: preservice teachers and fourth grade students perform better mathematically when using materials that they perceive as beautiful, motivating, legible and nondistracting. The perceived qualities have been expanded beyond beauty because of preliminary conversations and trials with a class of college students antecedent to the current study. At that time, students noted that there might be other important material qualities besides beauty that affect performance, such as distractibility of highly decorated materials, Journal of Authentic Learning, Volume 2, Number 2, Pages 26-41, 2005

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The Effect of Perceived Qualities of Curriculum Materials on Mathemtical Performance 28 legibility of numerals or symbols, or the motivating effect of the materials. Participants' individual ratings of beauty and other qualities are used for data analysis. Mathematical performance in our study is defined as the number of correct solutions that participants generate for the problems presented to them on boards (described later) differing widely in these perceived qualities. Factors that Affect Mathematical Performance Cognitive Factors. The Board on Behavioral, Cognitive, and Sensory Sciences and Education of the National Research Council, in their book, How students learn: History, mathematics, and science in the Classroom (2005), state three major learning principles for mathematics: 1) elicit, build upon, and connect student knowledge; 2) build learning paths and networks of knowledge; and 3) build resourceful, self-regulating mathematical thinkers and problem solvers. These ideas address cognitive aspects of learning, but what about affective aspects? Mathematics anxiety. Mathematics anxiety is rampant in today's population: over two-thirds of American adults fear and/or loathe mathematics (Burns, 1998). Mathematics anxiety, "an emotional avoidance reaction to situations requiring numerical or mathematical conceptual tasks" (Hadfield, Martin, & Wooden, 1992, p. 171), is related to poor performance on mathematics achievement tests. It relates inversely to positive attitudes toward mathematics and is bound directly to avoidance of the subject" (Hembree, 1990, p. 33). Mathematics anxiety can prevent a student from engaging in mathematics and may even make that person physically ill. There are many suggested solutions to mathematics anxiety. Family mathematics activities, sensitive teachers, peer tutoring, and teaching for understanding with a variety of approaches may help (Fotoples, 2000). Cooperative learning may increase enjoyment of mathematics classes and self-esteem (Bernero, 2000). Hackworth (1992) suggests using active learning strategies and study techniques, writing about feelings associated with mathematics, and developing positive ways to calm oneself and deal with fear. Hadfield, Martin, and Wooden (1992) advise manipulatives, real-life mathematical problems, a non-threatening teacher, and step-by-step explanations. Humanizing Mathematics. Many students find mathematics and science cold, abstract, difficult and unappealing (Alber, 2001). Watts (2001) calls for humanizing science and mathematics to make these subjects more appealing; he suggests poetry. Ufuktepe and Ozel (2002) take a dramatic approach, using theatre to change student attitudes toward mathematics. A meta-analysis of six studies (Vaughn, 2000) indicated a significant causal relationship between music study and mathematics achievement. Hetland (2000) in two large meta-analyses, found evidence for improved spatial-geometric performance, specifically, mental rotation of figures, of students listening to music. Our study asks whether the beauty of materials used in a mathematics task affects achievement. Method Subjects Data were collected from two different groups of participants. The first group consisted of seventy-six preservice teachers (66 female, 10 male; 55 traditional college age (20-22 years), 21 nontraditional age; 75 Euro-American, 1 African American) enrolled in a mathematics methods class at a four-year college in Journal of Authentic Learning, Volume 2, Number 2, Pages 26-41, 2005

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The Effect of Perceived Qualities of Curriculum Materials on Mathemtical Performance 29 upstate New York. The students were childhood (elementary) education majors in their junior year. Their demographics were representative of the general population of students majoring in childhood education at this college. The second group consisted of sixty-seven Euro-American fourth grade students (31 male, 36 female) of mixed ability levels from a suburban Title I elementary school in upstate New York. This population was representative of much of the service area of the college at which the study was conducted. Both preservice teachers and elementary students were the subjects of this study because we wanted to see if both groups had similar perceptions of beauty and whether their mathematical performance would be affected in the same ways. In elementary mathematics education, we are interested in improving the performance of elementary students; therefore, the students perceptions and performance need to be assessed to determine effective teaching strategies. However, the perceptions of preservice teachers are also important because they are the persons who will be creating or choosing the visuals and manipulatives for the elementary students. A comparison of the reactions of both groups to this experiment may provide important insights. Description and Preparation of the Manipulative Boards Thirty-two different boards were produced for the study from four different stimulus materials: eight cardboard, eight construction paper, eight dough, and eight velvet. Two boards of each "target number" (see explanation below in activity section) were produced in each material. Materials were chosen with the intent of producing boards that would receive a variety of beauty ratings. All numbers were drawn, formed of clay, or embroidered by the same person (the first author) so that they would be consistently legible and of the same handwriting "font". Photographs of representative boards are shown in Figure 1. Stimulus B ( B oard). The cardboard boards were cut from discarded cardboard backings of paper pads with unevenly cut, jagged edges. Wavering grid lines were hand-drawn in ballpoint pen ink with well-formed numerals. Corners of the board were delaminated, some lines were creased, while light grease stains and pinholes marked some empty areas. Each card bore two glued-on irregularly shaped patches of a different shade of tan cardboard that covered and replaced two incorrect numbers. Stimulus C ( C onstruction paper). The construction paper boards were made with faded blue construction paper cut neatly but not perfectly with scissors. A fine, red, water-based marker was used to quickly draw the lines that sometimes extended beyond the grid. A black waterproof marker was used to hand-letter the numbers. Each board had been stepped on with wet, muddy shoes and bore two faint rings from a drippy coffee cup. Some of the red marker lines ran where the card became wet. Two small scraps of torn construction paper were stapled to the top of each card. The two boards described above were similar in construction and substance to learning materials the first author encountered in other public school teachers' classrooms when she taught public school or supervised student teachers and visited colleagues' classrooms, or when students turned in materials as college class assignments. Stimulus D ( D ough). The polymer dough cards were made with four coordinated colors of dough that harden into a permanent plastic when baked in the oven. The rectangular base was made of one color; Journal of Authentic Learning, Volume 2, Number 2, Pages 26-41, 2005

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Figure 1. Photographs of the four board types: cardboard at top left; construction paper at top right; polymer dough at bottom left; and velvet at bottom right.

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a rolled border was ap plied on top of the base to represent the "grid squares", and an oval was placed for the top target number. Numbers were formed of another highly contrasting color and applied on top. This produced a stiff, colorful, three-dimensional board. Nine small round disks of another color were added on top of the base to represent the "grid squa res" and an oval was placed for the top target number. Numbers were formed of another highly contrasting color and applied on top. This produced a stiff, colorful, three-dimensional board. Stimulus V ( V elvet) The velvet boards were made by machine-embroidering the grid of nine squares and top rectangle in light blue satin thre ad on black velvet. Numbers were hand-embroidered in light, contrasting thread colors: yellow, pink, orange, blue, or a comb ination of these. Two thread colors were used on each board. Each board was carefully backed with black quilt fabric with all seams hidden. The Mathematics Activity Participants were asked to write as many equations as possible for each board. An equation consisted of a set of three numbers in a row (horizontal row, vertical column, or diagonal line) that were combined in any order using one or two operations (addition, subtraction, multiplication, or division) to result in the top number on the board, the "target number." The four target numbers and other numbers used on the boards are shown in Table 1. The activity was introduced and explained by drawing an example board with a target number of 10 on the blackboard. Students volunteered to suggest equations that resulted in 10. Boards were each marked with an identification number on tape on the reverse and grouped into sets of four. Each set contained a board of each material and each target number, with different sets having different combinations of target numbers on materials. Therefore, each target number appeared on each board type an equal number of times in the classroom set. Table 1. The numbers on the boards 12 15 36 4 3 60 6 10 2 3 7 15 2 3 6 16 8 3 7 2 18 24 3 7 3 8 5 2 18 6 15 30 5 1 12 5 1 14 1 11 Participants worked individually, but sat in groups of four and rotated the four boards of a set, so that each participant used each board of the four boards in a different order. Therefore, each material appeared with each target number and was solved first, second, third, or fourth with equal frequency in the data pool. Preservice teachers were given two minutes to write equations for each board (a total of eight minutes), and fourth grade students were given three minutes to write equations for each board (a total of twelve minutes) because of differences in experience. Preservice teachers participated in the activity as a model for using spreadsheets in mathematics class during a mathematics methods class for elementary teachers. They were asked to participate in the activity so that performance data could be collected, placed on a spreadsheet, and sorted to obtain answers to questions that the preservice teachers late r developed. Some of the questions that preservice teachers asked and used the spreadsheet function to obtain were: What is the average number of equations we wrote? Did males or females write more equations? Did students write more equations for certain target numbers?

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The Effect of Perceived Qualities of Curriculum Materials on Mathemtical Performance 32 Did students write more equations for certain board types? Fourth grade students participated in the activity as a novel way to practice basic skills in operations. Permission was obtained from the college's Human Subjects Committee, the school's principal, preservice teachers, and parents of fourth grade students for inclusion of data in this study. Rating of Materials To prevent bias, attention was not drawn to the beauty, distractibility, legibility, or motivation of the materials until after students had completed the equation-writing part of the activity. After the papers on which equations had been written were collected, students were given a form to rate each board on a one to ten point scale with "10" labeled "most beautiful," "most distracting," "most easy to read," or "most motivating," and "1" labeled "least". Participants were allowed to reexamine the four boards they had used while rating them. The professor or teacher helped students understand the rating scheme by asking a question each time a board was rated: "How beautiful do you think the board is?"; "How much did the board distract you from working on the problem?"; "How easily can you read and recognize the numbers?"; and "How motivated are you to do the activity by the appearance or feel of the board?" Both preservice teachers and fourth grade students were asked to describe in writing the criteria they used for determining if materials were beautiful or not beautiful. Preservice teachers were also asked to discuss in their groups of four whether the beauty of the materials mattered in mathematics teaching and learning. Quantitative Analysis Results There were three steps in the quantitative analysis of the data. First, the nature of the perceptions of the respondents to the four stimuli with a principal component analysis of the four judgments were examined and described. Second, the relationship between the stimuli and the scores on the descriptive principal components was described. Then, the components' scores were used to determine the predictive relationship between the perceptions and the ability to generate solutions to the equation-writing problem across stimuli. Two separate analyses were undertaken. One analysis was done for preservice teachers and another for fourth grade students. Principal Components of Perceptions The correlations between the four ratings of the perceptions of the stimuli (Beauty rating, Distractibility, Legibility, and Motivation) were subjected to a principal components analysis. Two components with eigenvalues in excess of 1.0, accounting for 73.8% of the variance, were retained and rotated to the varimax criterion. Scores were calculated for each rotated component as shown in Table 2. Table 2: Rotated principal component of perceptions Component 1 2 Beauty Rating .892 Motivation .877 Distractibility -.858 Legibility .325 .707 Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization. Journal of Authentic Learning, Volume 2, Number 2, Pages 26-41, 2005

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The Effect of Perceived Qualities of Curriculum Materials on Mathemtical Performance 33 The first rotated component had high loadings on Beauty (0.89) and Motivation (0.88). This component was labeled Enticement, because respondents judged a stimulus with a high component score to be beautiful and it motivated them to seek solutions to the target number problem. The second rotated principal component had a high negative loading on distractibility (-0.86) and a positive load on Legibility (0.71). It was labeled Focus, because respondents indicated that the stimulus with a high component score was legible and they were not distracted. Component scores for each participant were calculated and used in subsequent analyses as summary judgments. Differences between the Stimuli The main effects of stimuli, group membership, and their interaction, on the two component scores were tests with a multivariate analysis of variance. Both main effects and their interactions were statistically significant ( < 0.05). The significant interaction occurred on the Enticement score (F (3, 554) = 4.265, p < 0.005), as did the significant main effect of stimulus (F (3, 554) = 398.11, p < 0.001). With the exception of stimulus the effects were small but detectable. Stimulus had a large impact on Enticement (partial beta squared = 0.691), but not on Focus (partial beta squared = 0.072). The significant group difference occurred on the Focus score (F (1, 554) = 35.75, p < 0.001) as shown in Table 3. In terms of the interaction, fourth graders had higher Enticement scores on stimulus B than did preservice teachers, and lower Enticement scores on stimuli D and V (Table 4). In terms of the main effect of group, fourth graders had higher Focus scores regardless of the stimulus than did preservice teachers. Perhaps most important, the main effect of stimulus was characterized by higher Enticement scores for D and V than for stimuli B and C, and a higher Enticement score for stimulus V than D, irrespective of group. Table 3: Tests showing differences between the stimuli Tests of Between-Subjects Effects Source Dependent Variable Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared stimulus Enticement 374.232 3 124.744 398.11 2 .000 .683 Focus 3.672 3 1.224 1.303 .273 .007 group Enticement .235 1 .235 .751 .387 .001 Focus 33.583 1 33.583 35.749 .000 .061 stimulus group Enticement 4.009 3 1.336 4.265 .005 .023 Focus 2.864 3 .955 1.016 .385 .005 Error Enticement 173.590 554 .313 Focus 520.438 554 .939 Total Enticement 561.000 562 Focus 561.000 562 a R Squared = .691 (Adjusted R Squared = .687) b R Squared = .072 (Adjusted R Squared = .061) Predicting Problem Solving Success from Perceptions Because the group by stimulus interaction of the perception was significant, it seemed reasonable to suppose that the effect of perception of success (number of total equations) would differ between the preservice teachers and the fourth grade students; consequently, two separate regression analyses were undertaken to predict total equations from the Enticement and Focus scores. Journal of Authentic Learning, Volume 2, Number 2, Pages 26-41, 2005

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Table 4. Means and standard deviations of perceptions by stimulus type and group Stimulus type Group Mean Std. Deviation N Fourth graders -.76 .70 65 Preservice Teachers -.95 .49 76 B Total -.87 .60 141 Fourth graders -.74 .67 65 Preservice Teachers -.79 .49 76 C Total -.77 .58 141 Fourth graders .60 .68 65 Preservice Teachers .79 .50 76 D Total .70 .60 141 Fourth graders .83 .59 63 Preservice Teachers 1.04 .33 76 V Total .94 .48 139 Fourth graders -.03 .99 258 Preservice Teachers .02 1.01 304 Enticement Total Total .00 1.00 562 Fourth graders .26 1.04 65 Preservice Teachers -.37 .97 76 B Total -.08 1.05 141 Fourth graders .38 1.07 65 Preservice Teachers -.13 .96 76 C Total .11 1.04 141 Fourth graders .22 .96 65 Preservice Teachers -.34 .91 76 D Total -.08 .97 141 Fourth graders .20 .96 63 Preservice Teachers -.06 .91 76 V Total .06 .93 139 Fourth graders .27 1.00 258 Preservice Teachers -.23 .94 304 Focus Total Total .00 1.00 562

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The Effect of Perceived Qualities of Curriculum Materials on Mathemtical Performance 35 Fourth Grade Students The multiple regression analysis for fourth grade students indicated that there was a statistically significant multiple correlation (F 2,257 = 6.31; p = 0.002) between the total number of equations and two perception scores (R = 0.22). The regression equation for fourth graders is: Y = 0.25 X 1 + 0.11 X 2 + 2.65, where Y = total equations, X 1 = Enticement score, and X 2 = Focus score. While the coefficient associated with Enticement was statistically significant, the coefficient associated with Focus was not (Table 5). Since components scores are by definition uncorrelated (orthogonal), it is reasonable to interpret this result as, among fourth graders, success in the target number equation-writing squares task is, in part, a matter of the respondents perceived level of Enticement. Table 5. Prediction of total equations from perceptions scores by group Unstandardized Coefficients Standardized Coefficients Group Model B Std. Error Beta t Sig. (Constant) 2.659 .075 35.303 .000 Enticement .250 .074 .207 3.365 .001 Fourth graders 1 Focus .111 .073 .094 1.521 .130 (Constant) 4.243 .081 52.534 .000 Enticement .218 .078 .157 2.781 .006 Preservice Teachers 1 Focus .222 .084 .149 2.644 .009 Dependent Variable: Total Equations Preservice Teachers The result is somewhat different among the preservice teacher sample. Again, the multiple regression coefficient was statistically significant (F 2, 301 = 8.30; p < 0.001) between the total number of equations and the two perception scores (R = 0.23). The regression equation for preservice teachers is: Y = 0.22 X 1 + 0.22 X 2 + 4.24, where Y = total equations, X 1 = Enticement score, and X 2 = Focus score. In this case, the coefficients associated with both perception scores were statistically significant (Table 5). Again, the orthogonal property of the perception scores suggests that the coefficient can be directly interpreted. Among preservice teachers, both Enticement and Focus play an important role in the prediction of success in the target square problem. Summary of Quantitative Results In summary, these analyses provided evidence that: 1. The respondents made two separate, independent judgments about each of the board stimuli. They rated their perception of its Enticement (Beauty and Motivation) and of its Focus (Legibility and lack of Distractibility). 2. There were differences in the perceptions of the stimuli and between the two groups of respondents. The fourth grade students were more likely to rate all of the Journal of Authentic Learning, Volume 2, Number 2, Pages 26-41, 2005

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The Effect of Perceived Qualities of Curriculum Materials on Mathemtical Performance 36 stimuli as more highly Focused than preservice teachers. All subjects rated the level of the Enticement of the stimuli in the following manner: stimulus V had the highest level of Enticement, followed by stimulus D, with stimuli B and C having the lowest level of Enticement. The interaction of group by stimulus suggested that fourth graders were somewhat more extreme in the perception of the Enticement of the stimuli than were preservice teachers. 3. The perception of the problem stimuli was clearly predictive of the respondents success in solving the equation-writing problem, although the form of the relationship was different for each group of respondents. Among fourth graders, the number of equations was correlated with the level of perceived Enticement; however, among preservice teachers the total number of equations was a correlated with both the perceived level of Enticement and the perceived Focus. Qualitative Analysis Results Reported Criteria for Beauty and Nonbeauty Judgments Both preservice teachers and fourth grade students provided written criteria that explained their judgments of the beauty or nonbeauty of materials. These criteria are listed in the words of the respondents and grouped into categories based upon similarities in Table 6 and Table 7. There are interesting differences between what preservice teachers and elementary students value that support the quantitative findings. Both groups mentioned many of the same criteria; color and texture were highly important to both groups. However, elementary students focused mostly on color, texture, brightness, type of material, and design, while preservice teachers reported more criteria. Preservice teachers valued neatness, legibility, cleanliness, straight lines, even edges, good condition, and durability much more than elementary students. Many preservice teachers also considered visual appeal, creativity, and the time spent in preparing the materials, whereas few elementary students noted these criteria. This finding parallels the indications from the quantitative analysis that the perceived level of Enticement rather than Focus primarily influenced elementary student performance. In contrast, the performance of preservice teachers, who noted neatness and legibility as being important criteria, was affected by both Enticement and Focus. Criteria for nonbeauty of materials reveal similar trends. Fourth graders mentioned colors that were dull, dark, or lacking as the primary reason for nonbeauty, followed by hard texture, rough texture, messiness, cardboard or ordinary material, and overall dull or boring materials. Again, preservice teachers noted messiness and poor condition as the most important criteria for nonbeauty. Main Ideas from Preservice Teacher Discussions Preservice teachers were asked to discuss, in small groups, their ideas of any effects of the beauty of materials on students in mathematics. Each group provided notes from their discourse. A summary of preservice teacher ideas follows. 1. Performance. None of the preservice teachers thought that the beauty of the materials would affect mathematical performance, as shown by these representative comments: "Performance depends on the numbers on the board, not the way it looks." "We found that the numbers on the board are what made it Journal of Authentic Learning, Volume 2, Number 2, Pages 26-41, 2005

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Table 6. Preservice teacher and fourth grad e student criteria for Beauty of materials Number of responses out of 71 adults Preservice teacher criteria for deciding that a mathematics material is beautiful Number of responses out of 51 children Fourth Grade Student criteria for deciding that a mathematics material is beautiful 63 Color, colorful, color contrasts, color range, color variety, pretty colors 47 Color, colorful, colors, pretty colors, not black, a lot of color, different colors, color catches eye, blue 47 Neat 4 Neat 42 Appeal, eye catching, visually pleasing appearance, attractive, attention-grabbing, interesting 7 Caught your eye, fun to look at, way it looks, pretty 38 Texture smooth, soft, interesting texture, 3D, nice to touch 48 Soft, smooth, texture, 3-D, pillow-like, squishy, velvety, bumpy, numbers pop out, not flat more character 29 Creative, artistic 2 Creative 20 Legibility, clarity of numbers, easy to read, recognize, large numbers 4 Easy to read, not hard to read, nice writing 18 Time spent on making, effort, care put into making it 3 Time spent making, effort 13 Materials are interesting 11 Cloth, fabric, stitched, materials, clay 14 Clean 1 Clean 10 Durable, sturdy, well-made, carefully put together, quality materials 1 Tough 10 Lines straight, even, attention to detail 7 Cut evenly, edges straight, neat 7 Good condition: No stains, not ripped, torn or broken 1 New 6 Bright 12 Bright 5 Unique, different 4 Uniformity of parts, areas, and pieces, no stapled-on parts 7 Symmetrical, alignment of numbers, good design, organized, well-presented 9 Design, style, background, dark and light, outside frame 3 Fun to use 3 Correct: No mistakes 2 Lettering/numbering neat, not hand-written 2 Different shapes used 3 Have a lot of stuff, many patterns, interesting figures 1 Real objects 1 Easy to grasp 1 Lightweight 1 Cute numbers 1 Numbers look like frosting 1 Abstract 2 Shiny

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The Effect of Perceived Qualities of Curriculum Materials on Mathemtical Performance 38 Table 7. Preservice teacher and fourth grade student criteria for nonbeauty of materials. Number of responses out of 71 adults Preservice teacher criteria for deciding that a mathematics material is not beautiful Number of responses out of 51 children Fourth grade student criteria for deciding that a mathematics material is not beautiful 30 Messy, not neat, runny marker, sloppy, smeared, smudged 17 Messy, not neat, runny marker, sloppy, smeared, blurry, water spilled 27 Poor condition, beat-up, crinkled, damaged, hole-punched, ripped, stained, tattered, torn, water-marked, worn 8 Dry, has holes, ripped, stained, stapled, stale 23 Bland, dull, boring, uninteresting, unattractive, no appeal, no cool design, no imagination, no creativity, ugly 15 Dull, boring, not pretty, ugly, not cool, doesn't catch eye 14 Color lacking, blah colors, ugly colors, dull colors 35 Dull colors, dark colors, too much black, no color, not colorful, not many colors, not good colors, brown 14 Little time spent in creating, looks thrown together, made quickly, effort in making the material lacking, little care taken 1 Quick and easy 8 Old, old-looking, faded colors 4 Faded 10 Dirty 2 Dirty 8 Made of cardboard material 16 Cardboard, ordinary cardboard 6 Poor durability, falling apart, flimsy, not sturdy, cheap-looking 8 Too light, bendy, thin, papery, bendable, weak 6 Patched with cut-out piece of crooked cardboard, things glued at different heights, added chunks 7 Choppy, rough, jagged edges 17 Rough, rustic, rough texture, crude, pointy, bumpy, 6 Plain 14 Too plain, doesn't catch eye, not much design, doesn't have much stuff, not many patterns 4 Handwritten, written in pen 5 Written in pen, written in pencil 4 Lines are crooked, runny, not drawn neatly, drawn in color 3 Mistakes in production not fixed well, missing pieces 1 Mistakes 3 Made out of ordinary material, common paper 16 Ordinary material, common paper, made of paper, common and seen every day, made of everyday materials 3 Does not match, asymmetric, nonproportional 2 The way it is set up, don't like design 2 Illegible numbers, difficult to read 2 Hard to read 2 Difficult number problem 1 Flat 3 Not physical, 2-D, not 3-D 18 Hard, not soft, hard texture 2 Looks weird, girl colors 2 Not shiny Journal of Authentic Learning, Volume 2, Number 2, Pages 26-41, 2005

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easier or harder, not beauty." "Kids will want the pretty ones, but performance will not be affected." However, one group noted that "Beauty doesn't matter with performance, but it makes you feel different." Another group stated, "Organization made more of an impact on performance than beauty," alluding to preservice teachers' valuing of Focus. 2. Motivation. Preservice teachers did, however, recognize the motivating effect of beautiful materials. One group conceded, "As far as mo tivating them to do the activity, perhaps it could make a difference. Beauty does matter because it could affect the students' attitudes." Other groups noted: "Beautiful materials keep kids interested. Beauty counts because a kid would be attracted to a beautiful board and want to work with it." "Beauty makes children want to do math more, to touch, feel, and use the materials." "Children like to see colorful objects and materials that stand out. If they are appealing, students are more likely to want to use them." Other preservice teachers obs erved, "Everyone in our group wanted the velvet or dough boards the appearance of the manipulative may make children want to learn." Another group had this insight regarding academic engagement: "Kids will focus more if the board is interesting to look at. When things are bright and colorful, children are more apt to be engaged." 3. Distraction. As was indicated in the quantitative analys is, preservice teachers were concerned with Focus. "Having things on the boards is distracting. If a manipulative is too decorated, it may distract learning; so we think, make it attractive, but moderately." Another group noted, "Some people found it hard to work with the more attractive boards, even if they didn't have many decorations. They kept gazing at the board and thinking about how nice it looked. The appearance of the manipulative may distract kids. The board shouldn't be too distracting and take their attention away from learning." 4. Communication of Importance of Mathematics. Several groups recognized another message communicated by beautiful materials. "Beauty and creativity would make a student feel like the teacher cares about learning." "If some thing is made well, it is important and has value to students." "Students know that if the materials are well crafted, then the teacher truly cares about the concepts being taught." 5. Respect. Several preservice teachers identified another effect of beautiful materials. "Beautiful materials show that the teacher cares." "Children will think the teacher respects them more if the teacher makes beautiful materials." "If the material is appealing to the eye, it makes the person feel special to use it." Conclusions The results of this study indicate that the perceived Beauty and Motivation (Enticement) of a set of materials have a significant effect upon performance for elementary school students and that both Enticement and Focus (perceived Legibility and lack of Distraction) have an effect on the performance of preservice teachers. Because of the important role of manipulatives and other instructional materials in fostering a deep understanding of mathematical concepts, instructors at both college and elementary levels should consider the beauty of materials used in class. This study found differences between the criteria fourth graders and preservice teachers used to discern beauty, indicating that teachers cannot rely solely on their own likes and dislikes when choosing instructional materials to appeal to elementary students. Fourth graders in the

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The Effect of Perceived Qualities of Curriculum Materials on Mathemtical Performance 40 study were attracted to materials because of color, texture, and brightness, placing much less emphasis on neatness, an important criterion for preservice teachers. Results of this study indicate that fourth grade students are better able to cope with features that preservice teachers rate as distracting. References Alber, M. (2001). Creative writing and chemistry. Journal of Chemical Education, 78, 748-80. Alkhateeb, H. M. (2002). Attitudes of undergraduate majors in elementary education toward mathematics through a hands-on manipulative approach. Perceptual and Motor Skills, 94, 55-58. Bernero, J. (2000). Motivating students in math using cooperative learning. Master of Arts Action Research Project, St. Xavier University and Skylight Professional Development. (ERIC Document Reproduction Service No. ED446999) Board on Behavioral, Cognitive, and Sensory Sciences and Education of the National Research Council. (2005). How students learn: History, mathematics, and science in the classroom. Washington, DC: National Academies Press. Burns, M. (1998). Math: Facing an American phobia. Sausalito, CA: Math Solutions Publications. Forsythe, S., Presley, A. B., & Caton, K. W. (1996). Dimensions of apparel quality influencing consumers' perceptions. Perceptual and Motor Skills, 83, 299-305. Fotoples, R. M. (2000). Overcoming math anxiety. Kappa Delta Pi Record, 36, 149-151. Green, C. D. (1995). All that glitters: A review of psychological research on the aesthetics of the golden section. Perception, 24(8), 937-968. Hackworth, R. D. (1992). Math anxiety reduction. Clearwater, FL: H & H Publishing Company. Hadfield, O. D., Martin, J. V., & Wooden, S. (1992). Mathematics anxiety and learning style of the Navajo middle school student. School Science and Mathematics, 92, 171-176. Hembree, R. (1990). The nature, effects, and relief of mathematics anxiety. Journal for Research in Mathematics Education, 21, 33-46. Hetland, L. (2000). Listening to music enhances spatial-temporal reasoning: Evidence for the "Mozart effect." Journal of Aesthetic Education, 34, 105-148. Hofel, L., & Jacobsen, T. (2003) Temporal stability and consistency of aesthetic judgments of beauty of formal graphic patterns. Perceptual and Motor Skills, 96(1), 30-32. Jacobsen, T., & Hofel, L. (2002) Aesthetic judgments of novel graphic patterns: analyses of individual judgments. Perceptual and Motor Skills, 95, 755-766. Magro, A. M. (1997). Why Barbie is perceived as beautiful. Perceptual and Motor Skills, 85, 363-374. Magro, A. M., (1999). Evolutionary-derived anatomical characteristics and universal attractiveness. Perceptual and Motor Skills, 88, 147-166. Montessori, M. (1912). The Montessori method. New York: F. A. Stokes, Inc. (reprinted in New York: Schocken Books, 1964). Montessori, M. (1914). Dr. Montessoris own handbook. (reprinted in New York: Schocken Books, 1965). National Council of Teachers of Mathematics. (2000). Principles and Journal of Authentic Learning, Volume 2, Number 2, Pages 26-41, 2005

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The Effect of Perceived Qualities of Curriculum Materials on Mathemtical Performance 41 standards for school mathematics. Reston, VA: NCTM. Stamps, A. E. (1994). Formal and nonformal stimulus factors in environmental preference. Perceptual and Motor Skills, 79, 3-9. Standing, E. M. (1959). Maria Montessori: her life and work. Fresno, CA: Academy Library Guild. Torrance, E. P., Baker, F. B., & Bowers, J. E. (1959). Explorations in creative thinking in the early school years: IV Manipulation of objects and inventiveness. Minneapolis, MN: Bureau of Educational Research, University of Minnesota. Ufuktepe, U., & Ozel, C. T. (2002). Avoiding mathematics trauma: Alternative teaching methods. Paper presented at the International Conference on the Teaching of Mathematics, Crete, Greece, July, 2002. (ERIC Document Reproduction Service No. ED477833) Vaughn, K. (2000). Music and mathematics: Modest support for the oft-claimed relationship. Journal of Aesthetic Education, 34, 149-166. Watts, M. (2001). Science and poetry: Passion versus prescription in school science? International Journal of Science Education, 23, 197-208. Wentworth, R. A. L. (1998). Montessori for the new millennium. Mahwah, NJ: L. Erlbaum Associates. Journal of Authentic Learning, Volume 2, Number 2, Pages 26-41, 2005