Materials on Mathematical Performance

The Effect of Perceived Qualities of Curriculum Materials on
Mathematical Performance

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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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 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, 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 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 (Board). 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 (Construction 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 (Dough). 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;

Figure 1. Photographs of the four board types: cardboard at top left; construction paper at top right; polym

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.