David Williamson Shaffer
The Media Laboratory • Massachusetts Institute of Technology • Room E15-320p • 20 Ames Street
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David Williamson Shaffer
Learning Mathematics through Design:
The Anatomy of Escher's World
David Williamson Shaffer
Massachusetts Institute of Technology Media Laboratory
To appear in the Journal of Mathematical Behavior, Volume 16 Number 2
For preview purposes only. Please do not quote or cite.
This paper explores one example of an open learning environment created by combining mathematics and design activities in a "mathematics studio." Two iterations of the mathematics studio experiment in a project at the MIT Media Laboratory known as Escher's World suggest that: (a) students can learn about the mathematical concept of symmetry in a studio learning environment, (b) students learn to use visual thinking to solve mathematical problems in a studio learning environment, and (c) students develop a more positive attitude towards mathematics as a result of working in a studio learning environment. This paper uses a qualitative research model to explore the specific characteristics of the mathematics studio that were influential in creating a successful learning environment—in particular, how expressive mathematics activities and expressive computational media give students a sense of control over their learning.
Since the time of John Dewey and Francis W. Parker at the turn of the century, educators have understood that the arts can play a fundamental role in education (Sidelnick 1995). This paper explores one role for arts learning made possible by recent technological advances: helping students learn mathematics by making art in a computer-based "mathematics studio." The paper presents a detailed qualitative study of one successful intervention in which students learned mathematics through design activities. The focus is on three key questions:
• What specific aspects of a "mathematics studio" make it a valuable environment for learning mathematics?
• What role does the computer play in the success of a "mathematics studio"?
• What implications does the idea of a "mathematics studio" have for mathematics pedagogy or for our understanding of the process of learning more generally?
The answers to these questions suggest that computational media in conjunction with arts learning can be used profitably in mathematics education.
Arguments for art in formal education have historically (and logically) have called for two distinct, though not necessarily exclusive, places for art in the curriculum. One role is as a distinct discipline. Citing beneficial effects on students' study skills, use as a set of job skills, or promotion of self-awareness and the understanding of others, educators have argued that students should take art as a formal course of study in addition to the "academic" subjects of mathematics, history, English, science, and foreign languages (Brown & Korzenik 1993). The other position, which goes back at least as far as the turn of the century, is that art as a mode of expression is a key component in the process of all learning. Following the lead of Colonel Parker's Cook County Normal School, proponents of this view argue that arts learning should be integrated into all of the traditional disciplines (Sidelnick 1995).
This second position—that arts learning should be integrated across the curriculum—presents a tantalizing vision for mathematics educators. The idea of "learning math by making art" appeals to those who feel disenfranchised by the "traditional" mathematics curriculum and pedagogy. People get excited by the idea of learning mathematics with the freedom and joy associated with art-making. Since the publication of the National Council of Teachers of Mathematics "Curriculum and Evaluation Standards for School Mathematics" in 1989, key components of this vision have also been sanctioned by a section of the professional community of mathematics educators. The Standards call for the introduction of extended projects, group work, and discussions among students—elements of a learning environment that overall seems more similar to a studio course in design or architecture than to a traditional mathematics class (NCTM Commission on Standards for School Mathematics 1989, NCTM Commission on Standards for School Mathematics 1991).
In recent years, there have been several examples of interventions that demonstrate the power of this vision of a "mathematics studio." In a quantitative study, Leslie Willett demonstrated that mathematics learning is more effective in the context of arts-based lessons than with standard mathematics pedagogy at the elementary school level (Willett 1992). Arthur Loeb's visual mathematics curriculum (Loeb 1993) has not been studied formally, but substantial anecdotal evidence supports his approach to the study of the formal mathematics of symmetry through a design studio as an effective learning environment for undergraduate students.
This paper examines another successful attempt to create a "mathematics studio" where students learn about mathematics and art simultaneously in an art studio-like environment. The Escher's World research project at the Massachusetts Institute of Technology Media Laboratory brought twelve high-school students from public schools in Boston, Massachusetts to the Media Laboratory for brief but intensive workshops during the spring and summer of 1995. In these workshops, students created posters and worked on other design projects using mathematical ideas of mirror and rotational symmetry. Using a qualitative research methodology based on observations of student behavior and structured interviews, this paper documents the process by which mathematics learning took place in the Escher's World design studio: the anatomy, as it were, of a learning environment. The central question is: What aspects of the design studio made it a good venue for learning mathematics?
The focus of this paper is on a detailed understanding of one specific instance of the "mathematics studio" idea, but the goal of this research is to elaborate a framework for thinking more generally about design activities as a context for learning. In particular, this paper looks at the role of expression in design activities, as well as the role that computational media can play in the process of learning.
The Escher's World project conducted two workshops during the spring and summer of 1995. At each workshop, six ninth- and tenth-grade students from Boston public schools came to the Media Lab for twelve hours over two or three days. The students had volunteered for "a workshop about mathematics and art" that they had heard about from their mathematics teachers or through a local after-school enrichment program. Students came from a range of backgrounds and ability levels. Five of the twelve students in the workshops were foreign-born, and eleven of the twelve students were persons of color, including two Latino students; there were no Asian or Asian-American students. Five of the students were male and seven were female.
Students spent twelve hours at the Media Lab. Workshops were divided into two sections, each lasting approximately six hours. The first section was organized around the concept of mirror symmetry; the second was organized around the concept of rotational symmetry.
Each section of the workshop consisted of a "warm-up" activity, an investigation, and an exploration. Warm-ups lasted approximately 1/2 hour and presented students with a short mathematical game or puzzle to "stretch their minds." The games were chosen to help create a relaxed atmosphere, and to help students meet each other and become more comfortable working together. At the end of each day, there was time for reflection on the day's activities lasting approximately 1/2 hour. Students wrote in their workshop journals in response to specific questions about the content and structure of the workshop. There was also time to discuss as a group any problems or concerns that came up during the day. The emphasis throughout the workshop was on creating an open, studio-like atmosphere for learning. Students were encouraged to sit and work where they liked, to use media of their own choosing, to collaborate or work alone as they wished, to eat, take breaks, go to the bathroom, and change projects at their own discretion.
Investigations lasted approximately one hour, with students working on short problems on their own or in small groups. In the first day of the workshop, students began their investigation of mirror symmetry by making name-tags that read normally when viewed in a mirror. This was followed by a search for words that look the same when viewed in a mirror, and from there to the classification of the letters of the alphabet by their mirror lines. Students worked on each of these problems individually or in small groups at their own discretion, with the whole group discussing the "results" of each problem. Students conducted a similar sequence of investigations involving rotational symmetry using a kaleidoscope in the second section of the workshop.
Based on their investigations, students spent two to three hours working on extended projects in design on their own or with a partner. Students worked on one shorter project (approximately one hour), and then presented their work to the group for discussion, questions, and comments. Following this "peer review," students began a more ambitious project (approximately two hours), integrating ideas about symmetry, principles of design, and feedback from their presentation. In the first day of the workshop, students made a design using mirror symmetry. After discussing their designs, students worked for the remainder of the day creating designs that had mirror symmetry but did not place the focus of the composition in the middle of the design (see figure 1). In the second section of the workshop, students tried to make designs that used rotational symmetry but presented a lopsided or unbalanced composition.
Figure : Student work from Escher's World: explorations of composition and symmetry.
Workshops took place in a conference room at the Media Lab that had been modified to resemble an art studio. Works of art by students and professional artists were placed on the walls, and a variety of artistic media were available for students' use. In addition to the author, who acted as workshop leader for both workshops, there were one or two other adults in the studio as a resource for students. Macintosh computers were provided throughout the workshops, with one computer available for every two or three students. The computers were connected by a network to flatbed scanners, color printers, and a large format color plotter. Computers were equipped with commercial drawing and image-manipulation programs and with the Geometer's Sketchpad (Jackiw 1995). During the investigation portion of the workshops, students were introduced to some of the basic functionality of these programs. Students were able to work on the computers or with traditional materials during their explorations; all of the students chose to use a computer for some portion of their work.
The main source of data for the Escher's World workshops was structured interviews conducted with each of the workshop participants before and after the workshop, as well as a shorter follow-up interview with each student from two to five months after the completion of the workshop. These interviews were supported by videotapes of the workshops and field notes from the workshop leader and other facilitators. Student sketches and designs from the workshops and student journals were preserved for review and analysis. Students in the second workshop were also given a brief written survey about their feelings towards mathematics, art, and computers before and after the workshop.
Pre- and post-interviews were divided into three components. The first part was a series of questions about mathematics and art, focusing particularly on attitudes towards these disciplines. In post-interviews, this first section of the interview also contained questions about the workshop and students' experiences during workshop activities. The second section of each interview was a detailed discussion of four works of art from a set of seven images. The final section of the interviews consisted of two to four mathematics problems from a set of 16 problems. Follow-up interviews (conducted two to five months after the workshop) asked students to describe their attitudes towards mathematics and art, focusing particularly on changes students experienced as a result of the workshop.
The surveys given to students before and after the second workshop asked students to rate how strongly they agreed with a series of statements about their experiences in mathematics, art, and computers. Ratings ranged from 5 (most positive) to 1 (least positive).
Interviews from the workshop were transcribed and broken into excerpts, where each excerpt represented one complete answer to a question, including any follow-up questions or clarification by the student. This was done to preserve the coherence of students' thinking as reflected in their responses, and resulted in somewhat longer excerpts that often were coded in multiple categories. This helped identify ways in which different themes in the students' experience of the workshop were related.
Each section of the interviews (general questions, image descriptions, and word problems) was coded separately. General questions were first coded for key concepts used by students. These concepts were then aggregated into larger analytic categories based on fundamental theoretical concepts in the literature of learning theory, including concepts such as Control, Expression, and Interaction (a more detailed description of these categories is provided in the Results section of the paper). Excerpts were also coded for students' like or dislike of mathematics and art..
A more detailed description of the coding process for image descriptions and word problems is provided in another paper on the Escher's World project (Shaffer in press). Briefly, image descriptions were coded as comments about "Form," "Color," "Symmetry," and "Composition." Within each category, comments were further divided into "General" and "Analytical" comments. One important measure of the success of the project was students' development of mathematical or analytical language for the evaluation of images, reflected in the presence of "Symmetry/Analytical" references such as "it's four time radial symmetric" in students' description of images in interviews.
Word problems were coded for students' use of a visual representation during some portion of the problem-solving process (usually some form of sketch of the problem). Following Rieber, the term visual representation was used broadly to refer to "representations of information consisting of spatial, non-arbitrary (i.e., 'picture-like' qualities resembling actual objects or events), and continuous... characteristics," including both internal and external representations (Rieber 1995). Problems were also coded for correct or incorrect answers, where "correct" answers included answers that fit the stated conditions of the problem even if a student's solution was not the "expected" answer.
The results section that follows presents two analyses of the Escher's World project. The first is a summary of the "effects" of the workshop. These results are presented in more detail in another paper (Shaffer in press). Following this summary is a more detailed analysis of the particular factors in the workshop environment that led to its "success."
Mathematics Learning in Escher's World
During the Escher's World workshops, students developed their understanding of the mathematical concept of symmetry. Students also began to use visual thinking, and began to like mathematics more as a result of the workshop.
The criteria used for "understanding" mathematics in this analysis is "the ability to use ideas in appropriate contexts, to apply ideas to new situations, to explain ideas, and to extend ideas by finding new examples"(Gardner 1991, Gardner 1993, Sierpinska 1994, Shaffer in press). By this definition students were able to develop their understanding of symmetry as a result of the Escher's World workshops. During the workshops all of the students (12/12) were able to make designs using mirror symmetry, and 83% of the students (10/12) were able to make designs using rotational symmetry. Only 1 of 12 students was able to use and explain ideas about symmetry before the workshop, whereas 11 of 12 students were able to do so after completing the workshop. After the workshop students were able to find new examples of symmetry in the world around them: 75% of the students (9/12) reported thinking about symmetry beyond the context of the workshop in post interviews or follow-up interviews. Students reported seeing symmetry in drawings, chairs, wallpaper, rugs, video games, flowers, and clothing.
Students also clearly developed their ability to apply the concept of symmetry to the analysis of images. Before the workshop, students made analytical references to symmetry an average of 0.5 times while looking at 4 images in structured interviews. After the workshop, mean analytical references to symmetry rose to 4.3 references over 4 images (see figure 2; mean change +3.8, p<0.01).
Figure : Students learned to use symmetry to analyze images. Students 1.1–1.6 attended the first workshop; students 2.1–2.6 attended the second workshop. Students have been ordered for clarity of presentation.
This change is statistically significant even with the small sample size of the Escher's World project (n=12). But perhaps more striking is the change in the overall tone of students' responses to the works of art they were looking at during interviews. Students used a richer, more formal, more analytical, and more mathematical vocabulary to describe images after the workshop. In a pre-interview, for example, one student described a quilt:
Mmmm, that's like a square with a square inside of it and another square inside of it. And inside the third square, there's some—like paint blotches or something, I'm not sure what they are. They almost look like chicken wings or chicken legs. And then, there's two—I don't know—gears, I guess, in the middle. That's pretty much what it looks like.
After the workshop, the same student described the same image:
This is a box with a red border, and then a smaller blue border around inside that, and then a box inside. And then a yellow box. And then it goes back. And then it uses again the red and blue colors and it just starts making little shapes on the inside that are sort of curvy with the blue. And then it makes red shapes that are connecting the blue. And then you go in further. And then there's sort of a blue circle with spikes, and inside of that there's a red circle with spikes, and inside of that there's a yellow dot. And you're sort of drawn to that dot. Meaning your eye is drawn to it. That's like the focus of the picture. And I guess I was going to say that it was, um, symmetrical, but then I noticed these little, little blue lines coming out of these little red designs—whatever they are—and I realized that it was, um, angular symmetry. Well, what it looks like is almost they took—they made— the person whoever made this, or wherever you got this, could have just started out with a block that had—that had, um, on two sides the red and blue base, and then on the rest of it just made the yellow, and these little—these blue dots and red dots and the little line right there. And then made part of the circle, or one fourth of those circles. And then just use like the way we did with the computer and made four versions of it with different angles, and then just moved them together.
Visual Thinking in Mathematics
One of the most surprising results of the Escher's World workshops was that students started to use visual problem solving strategies after the workshop. Before the workshop, only 33% of the students (4/12) used visual representations to solve word problems in interviews. After the workshop 75% of the students (9/12) did so. For example, in figure 3 the student did not use a visual representation to solve the problem: "One day, Julie decides to go for a walk. She leaves her home and walks for 2 miles due north. Then she turns right and walks for 3 miles due east. After Julie turns right again and walks for another 2 miles, she decides to go home. How far does she have to go to get back to her home?" After the workshop, the same student working on a similar problem used a visual representation of the problem situation. Across all students, use of visual representations for word problems after the workshop was correlated with success in problem solving during post interview problems (r=0.83).
Figure 3: One student's notes while solving a problem during interviews. In the pre-interview (left image), the student did not use a visual representation. While solving a similar problem during the post-interview (right image), the student represented the problem visually and produced a correct solution.
Students Like Mathematics More
In post-interviews and follow-up interviews, 67% of students (8/12) reported feeling more positive about mathematics as a result of the workshop. This reported change was supported by survey data (collected for 6/12 participants). In the survey, students responded to 4 prompts about mathematics:
"I like math class/I don’t like math class."
"I like doing math problems/I don’t like doing math problems."
"I like thinking about math/I don’t like thinking about math."
"I understand math/I don’t understand math."
Students marked a scale from 5 (most positive) to 1 (least positive). Total rating for the 4 mathematics questions went up for 67% of students surveyed (4/6). Change for the "I like math class/I don't like math class" prompt (mean +0.67; p<0.01) was particularly striking.
As before, however, these statistics only confirm changes that were obvious to the students themselves and were reflected in their comments about the workshop. In post interviews and follow-up interviews, students said things like: "I love symmetry—whenever I see a drawing with these type of shapes I always look at it, just to see the shapes or see the symmetry in it." Or: "Math doesn't seem as complicated like before... I used to always think that [math] was so hard." Other students reported returning to school to find that mathematics became their favorite class, or that they thought of mathematics as more useful or interesting.
Understanding the Success of Escher's World
Students learned about the mathematical idea of symmetry in the Escher's World workshops, and learned to apply visual thinking skills to mathematical problem solving. At the same time they discovered they liked mathematics and liked this new kind of learning environment. One student said simply: "If school was like this, attendance would be perfect!" Certainly something good must have been going on if students were willing to give up time on their spring break or summer vacation to work for four to six hours at a time learning math with a kind of focus rarely seen in school class rooms. The question is: What made this learning environment "work" for students?
The remainder of the results section describes in some detail students' experiences during the workshop based on their comments in interviews. In particular, the analysis focuses on students' experience of "control" over their learning and learning environment, and on the relationship between control and the expressive nature of the activities in Escher's World.
Terms and Conditions
Students' comments about the Escher's World workshops were coded in five major analytic categories: Control, Expression, Interaction, Novelty, and Computer. Novelty and Computer need little explanation. Comments about computers (hardware, peripherals, or software) were coded accordingly. (Students used tools other than computers during the workshop, but did not particularly talk about other tools during interviews.) Students did not use the word "novelty" in any of the interviews, but they often talked about things being "new" or "different"; these comments were coded for Novelty. The other categories (Control, Expression, and Interaction) have a history in the literature of education theory. There is not space here to describe their etymologies in full, but the sections below outline the context in which these ideas apply in the Escher's World mathematics studio.
John Dewey wrote in great detail about the role of freedom and social control in students' development, suggesting, in particular, that "freedom" is a necessary (though not sufficient) condition for the development of self-control. By "freedom" Dewey meant not only the physical freedom to move in space, but also the more important freedom to make decisions, to "frame purposes" and to exercise judgment (Dewey 1938). Other theorists similarly emphasize the extent to which learners must control their learning experiences (Sizer 1984, Papert 1991, Gardner 1993, Prawat 1995). For purposes of this analysis, excerpts were coded for Control when students referred to freedom of physical as well as intellectual movement, when they talked about making their own choices, judgments, or decisions—in short, when they described in a positive or negative way the effects of their own control (or lack of it) in their learning experience.
In his "Talks on Pedagogics," Francis Parker argued that "expression" is a process fundamental across many disciplines of thought (Parker 1894). Rather than emphasizing individual modes of expression such as singing, writing, dance, or painting, Parker suggested that all of the means of expression are essential vehicles through which ideas in any domain are learned. In Parker's vision, the arts were integrated throughout the curriculum, much as writing is used in many different disciplines today. The key component of Parker's idea of expression is that it represents "the manifestation of thought and emotion" (Sidelnick 1995). That is, expression is the process of taking some part of ones internal being (an idea, a feeling, an impression) and representing it externally. Student comments were coded for Expression when students talked about making things that reflected their own ideas or preferences—or to times and places when they are prevented from doing so.
One of the major themes of Lev Vygotsky's work is the idea that social interaction is a critical component in cognitive development. In particular, Vygotsky argued that the immediate potential for cognitive development (what he called the zone of proximal development) could only be fully realized with adequate adult guidance or peer collaboration (Vygotsky 1978). Other theorists similarly argue that an essential part of learning to think is learning to think with others (Bruner 1986, Bruner 1990, Pea 1993). For purposes of this analysis, excerpts were coded for Interaction when students referred to ways in which their learning experience was affected by the active participation of others (or lack thereof). This includes descriptions of help given to or received from adults or peers, collaborative work, public presentations of ideas or work, conversations or other "purely social" interactions—in short, Interaction refers to the range of students' relations to other people as it connects to their learning experiences.
A Framework for Thinking about the Mathematics Studio
The question, then, is how to understand the particular combination of control, interaction, expression, novelty, and use of computers that made the Escher's World a powerful learning environment for the twelve students who participated in the workshops.
Control and Interaction
The relative frequency of student comments in the various categories shows that Control and Interaction were the most important aspects of the Escher's World mathematics studio for these students (see figure 4). Interestingly, comments about these aspects of the workshop showed significant overlaps, with 36 joint references to control and interaction. Student comments about interaction were correlated with comments about control with r=0.79.
Figure 4: Frequencies of student comments in interviews about the workshop, by analytic category.
Students talked about their ability to decide for themselves when to work alone, when to work with a peer, and when to consult with an adult: "[In the workshop] if I don’t know something, I just ask you or other friends to sit by me. In class [at school] you can’t talk." Similar sentiments were echoed in two-thirds of the comments where students talked about both control and interaction. In almost all of the comments about working with peers (16/19) and about getting help from adults (16/18) students talked about the fact that in the workshop they were in control of how and when these interactions took place.
Students also talked about control and interaction as independent categories. Most of the comments about interaction that did not have to do with control focused on public discussions about students' work or about ideas in the workshop. The majority of comments about control that did not deal with interaction were about expression.
Student comments about expression were strongly linked to the issue of control. Two thirds (67% or 19/28) of the comments about expression reflect the fact that the expressive arts-based activities of the workshop put students in control of their own learning. But perhaps more remarkable, the frequency with which students refer to expression is directly correlated to the frequency of comments about control overall—with r=0.94!
This control-through-expression was manifested in two ways: control over goal and control over effects. Students were able to adjust, for example, the level of difficulty of the projects they took on (control over goal). "You gave us an idea," one student said, "but we basically made up [the specific problem]. So it could be as hard as you want it to be or as simple as you want it to be." Students were able to "decide what we wanted [and] make it any which way we wanted." Or, as another student put it: "If I didn't like the way I was going I could stop and change it around [or] start over.... You said, 'I want something with angular symmetry.' That still leaves a lot of different ways you can go."
Students also had a sense of control over the emerging products of their work (control over effects). As one student said: "I can change where people's eyes go, and where the focus is, and even change how they look at it.... Before I would draw something, and if I didn't like it, I would just forget it and try something totally new. Whereas now, I know if I change something minor and change it drastically then it can become a whole new picture, but still very similar."
The cumulative effect of this sense of control-through-expression was obvious when students tried to make images that had rotational symmetry but were not balanced in their overall composition. Students were able to produce designs that fit the constraints of the problem after a short time, but they continued to work despite growing frustration with the difficulty of the problem. They were looking for a solution that they liked. One student explained: "When you do it for yourself, you can take any time, or you could do it any way you want to, because it's yourself, your own stuff." It may also be worth noting that the frequency of student comments about expression shows some correlation to students performance on word problems (r=0.56 for the correlation of frequency of comments about expression and percentage of correct answers on post-interview word problems).
From a practical point of view, the "novelty" of the Escher's World workshop and its effect on students is problematic. Clearly some portion of students' positive response to Escher's World can be attributed to the fact that the workshop was a very different experience from learning in school—if only because the workshop took place in a different space, for only a few days, using powerful computers, with video cameras, interviews, and all of the other trappings of research. Escher's World would surely be at least somewhat less exciting if it were a regular part of every students' education.
It is clear from interviews that for these students, the most novel aspect of the Escher's World workshops was the use of new media technology, for which students used the generic term "computer." Students who used computers extensively during the workshop made an average of 0.8 comments about the novelty of the workshop, compared to an average of 0.4 comments for students who did not use computers extensively (p<0.01). The number of students' comments about computers overall is correlated with the number of comments about novelty (r=0.55).
At the same time, a close look at students' comments about novelty shows that while there was a significant "novelty" effect in students' reactions to the workshop related to computers, there were also more "deep structure" effects of media technology in the workshop environment. For example, only a third of the comments about computers overall referred to their novelty (35% or 6/17), and only three comments about computers referred to their novelty alone.
Not surprisingly, students' experiences of computers in the Escher's World workshops reflected to some extent the particular software they were using. Most students who used the computers to any great extent during the workshop used the Geometer's Sketchpad program. The Geometer's Sketchpad allows students to create basic geometric figures such as circles, lines, and arbitrary polygons, and to change their size, orientation, and color. More important, students can define mathematical relationships between theses objects: ratios, angles, and geometric transformations. So, for example, a student could create a line and a polygon, and then create the reflection of the polygon in the line.
When objects are moved on the screen in the Geometer's Sketchpad, mathematical relationships are preserved. The display is updated in real time as students "drag" points, lines, and figures on the screen. In this way, students can explore the effects of various mathematical constraints and relations quickly and easily, looking for solutions to mathematical problems that have aesthetic appeal. Sketchpad also preserves a record of a student's actions during a given session with the program. This lets students "undo" their actions; they can step back to and through previous states in their exploration rapidly.
Students commented on the ease with which they could play with designs on the computer using the Geometer's Sketchpad. They described the program's ability to hold an image constant, to let them make very precise changes, and to let them explore the consequences of those changes. In other words, the computer helped students control their explorations:
The computer just made everything easy. You didn't have to hold everything right—[the computer] just it did it for you, so... you could concentrate on actually what you were seeing instead of just [thinking:] 'Well, I think I saw that, let me try that again and see if I see the same thing.'
You drew that dog, and then when you got the mirror on the screen you [moved] it around so that you could get a duplicate of it.... When we did it on the computer... I could actually move the mirror around the screen, move it in closer, and make like one picture out of the two, and move further apart.
The infinite undo feature of the Geometer's Sketchpad also gave students a sense of control over their exploration. As one student said: "The computers helped because it was like easier [than working] on paper [if] you'd have to erase it, or start again. You could just undo it, and then try something differently. That was easier because it was much quicker." Overall six of the seven students who worked extensively with computers commented about one or more of these ways in which the computer increased their ability to control their explorations of the mathematical problems of design.
Differences among students in Escher's World
In collecting qualitative data from interviews and close observations of students at work, one develops a picture of the students as individual learners, rather than as faceless "subjects" or identical elementary particles in a high-energy physics experiment. So for the Escher's World researchers, "student 1.3" is a sophomore in high school in a low mathematics track. But she says that she loves mathematics because she is good at it and because she knows it will help her become a doctor some day. For student 1.3, the workshop was eye-opening—something she and her friends from school who were also in the workshop think about often. They joke about symmetry in the T-shirts they wear to school. Student 1.3 was one of the students whose descriptions of images changed the most as a result of the workshop. Before the workshop she thought the pictures in the interview were "weird," and after the pre-interview it wasn't clear that she wanted to come to a workshop that was about strange things like that! After the workshop, she laughed at the same images, happily talking about symmetry, delighted to have a way to "understand" what had seemed so strange only days before. Student 1.3 was also one of the students who began to use visual representations in her problem solving as a result of the workshop, finding the interview problems less frustrating and more fun as a result.
"Student 2.6," on the other hand, is a junior in high school who likes mathematics and has always been confident in his mathematical abilities. For student 2.6, the workshop felt like it was more about art than about mathematics. He used concepts of symmetry to describe interview images and used visual representations in his problem solving both before and after the workshop, and felt like the workshop was interesting, but mostly review. "I do think it's a good workshop and it could help," he said at one point, "just not me." At the same time, student 2.6 did get some things out of the workshop. He talked in post-interviews about seeing symmetry in his work at the Computer Museum, and about how thinking visually helps him in his advanced physics course. He was also clearly affected by his explorations of the compositional concept of balance, using the idea to talk about all kinds of images and situations in the world around him, including the solution to mathematics problems.
Still, for all this sense of workshop participants as distinct individuals, 12 hours of workshop and three interviews are not enough to determine with certainty why some students responded with such enthusiasm while others were less forthcoming in their praise of the workshop. There was, however, at least one pattern among the students worth noting as a direction for further research. Overall, female students seemed to respond more to the workshop than males. Females used symmetry to analyze interview images substantially more in post interviews than males (females added an average of 5.4 references to symmetry between pre and post interviews, males added an average of 1.4 references, p<0.01). And in post interviews females used visual representations more often than males (females used visual representations in 57% of problems, males used visual representations in 30% of problems, p<0.01).
The data described above in the Results section suggest several areas worth hilighting in the Escher's World project thus far. This section of the paper discusses the role that expression plays in empowering students, and the affective and expressive role of computers in learning.
Empowerment and Expression
The idea of "empowering" students is the topic of much discussion in educational circles, but there is not a consensus as to exactly what student empowerment means and how to achieve it. The results from Escher's World suggest a framework for thinking about empowerment in terms of students' sense of control over their learning environment and sense of control over their learning activities. Students could control their learning environment in the Escher's World workshops by getting help from adults and peers in the amount, manner, and time of their own choosing. Students were able to work collaboratively with peers and adults if and when they chose to. And students had the freedom to organize their activities in space and time—"freedom of movement" as Dewey described, but also the freedom to regulate the pace and timing of their work.
Escher's World also gave students control over the learning activities they were engaged in. The expressive nature of the activities meant that students were able to decide for themselves how to address the mathematical and artistic problems presented in the workshop. They were able to direct their problem solving process, and to decide what constituted a desirable solution. But this control did not mean that students were simply "doing whatever they wanted to do." Adults and the community of peers formed the context in which students evaluated their efforts, and the public presentation and critique of student work thus created a mechanism through which students could learn to regulate themselves.
Other authors since Dewey have described a similar relationship between freedom and control on the part of learners and the power of a learning environment. What the Escher's World research emphasizes is the role that personal expression and expressive activities play in creating a sense of responsible control for learners.
One concrete way to see how expressive activities help students feel empowered is to look at the role of mistakes in Escher's World. Despite reassurances from teachers, students know that in a traditional class room, mistakes are bad. As one student said: "If you make a mistake in school they made it seem like you didn't know what you were doing, and everybody else did.... If you made a mistake you were singled out, and the class had to slow down so you could catch up." In contrast, student's in Escher's World felt that mistakes were an opportunity rather than a liability—as one student put it, "you can learn from your wrongs." Students' control over their learning environment meant that "it was okay to make mistakes because you could always get help." The expressive nature of the activities of Escher's World helped students see that mistakes are not necessarily "wrong"; they can also be steps on the road to finding a better answer. "We could learn from our mistakes," said one student, "do it totally different if we wanted to.... There wasn't any one way we could do it. We could do it differently. We could make a whole bunch of the same thing, or different things.... We could change [a design], and it didn't have to look exactly like a house, or exactly like a dog, or something like that." Similarly, the computers helped create a more constructive role for mistakes: "The computers helped because it was [easier to deal with mistakes]. On paper you'd have to erase it, or start again. [On the computer] you could just undo it, and then try something different."
Students' feelings of control over their learning environment and learning activities can create a powerful context for learning. The results from Escher's World show, moreover, that expressive activities provide a particularly powerful vehicle for empowering students in this way.
Computers and Expressive Learning
A number of authors have written about specific aspects of computational media that make them effective in the process of learning (Perkins & Unger 1994, Jackson et al 1996). One trait commonly described is the dynamic quality of computer representations—that is, their ability to reflect change in a continuous and immediate fashion. As Albert Michotte argued in his monograph "The Perception of Causality" (Michotte 1963), when objects respond directly to each other's motions in time and space we attribute causality as a primary percept. That is, a sufficiently powerful dynamic representation can create direct intuitions about underlying causal relationships. It is not surprising, then that dynamic representations help students understand complex ideas.
What the Escher's World data shows, however, is that dynamic representations also have an affective component for learners. For students in the workshops, the ability to change a design quickly, easily, and in a continuous fashion contributed not only to the development of intuition, but also to a sense of control over their work. Students interpreted representations not only as being dynamic, but also flexible, and the process of continuous and immediate change was important in part because it gave students the ability to manage their own explorations of mathematical and artistic relationships. This sense of control was reinforced by the "infinite undo" feature of the Geometer's Sketchpad. Students could change their minds—and change mistakes—quickly and easily. The computer provided a forgiving environment in which students could explore freely.
This combination of affective and expressive functions for the computer clearly goes beyond the mere "novelty" of the computer equipment used, and it suggests a richer view of the potential role of computers in education than more cognitively-oriented theories suggest. At the same time, data from Escher's World makes it clear that computers were neither necessary nor sufficient for the success of this learning environment. Some students were able to learn without using the computers extensively—just as some students who used computers extensively showed less dramatic changes in their approach to mathematics after the workshop.
A more realistic view of the role of computers in an environment such as Escher's World might be as a "stimulating" (rather than necessary or sufficient) condition. Computers can play a significant role in empowering students, especially in the service of expressive activities. But computers alone are not enough to empower students, and they certainly can be used in ways that do not help students exercise effective control of their own learning process.
Conclusion: Towards an Expressive Mathematics
All of the above suggests that expressive activities are a powerful context for learning, and that one of the benefits of combining art and mathematics education in an environment such as Escher's World is that students are able to think about mathematical ideas in an expressive way. The larger question, however, is how to extend this idea beyond the arts realm. Part of the point of the Escher's World project is to show that mathematics and arts learning can be combined in a way that is productive for both domains. But ultimately the lessons of that "collaboration" should transfer back into each domain. Here the logical question to ask is: What does it mean to have an "expressive" mathematics problem independent of the arts?
Perhaps an example from arts education will be suggestive. When a student learns to draw a human face in a life-drawing studio class, it is quite likely that before starting he or she will look at the work of masters of the craft: a face drawn by Leonardo, one by Van Gogh, by Picasso, by Matisse. A student may look at ten, twenty, or even a hundred such master works. When he or she sits down to draw a face, the problem is richer, but in every respect just as challenging and as genuine as it was before the student looked at the examples. In fact, even if the student were a perfect mimic and could recreate the masters' drawings perfectly, the drawing of a new face would still be a creative process.
This example from the arts suggests a working definition for an "expressive problem" in any domain. An expressive problem is one that: (1) has multiple valid solutions, where (2) the solutions are more or less interesting, pleasing, beautiful, socially, economically, or politically desirable, or otherwise preferable according to some set of personal criteria. In other words, an expressive problem is a problem where one can say: "Both of these answers are correct, but I prefer the first one because...." The results from Escher's World described above show that students saw the problems and activities of the Escher's World workshops as expressive by this definition—and also that these aspects of the problems and activities contributed directly to students' experience of the workshop as an environment where they were able to control their own learning.
This, in turn, suggests two questions for further study. The first question is whether expressive problems from the arts and from other domains could be the focus of a compelling mathematics education. The second, and perhaps more fundamental question, is about the nature of expressive activities in general—and about the relationship of expression to mathematical thinking in particular.
The data collected thus far from Escher's World shows that expressive activities help students control their own learning. But are there other connections between expression and mathematical thinking? Why, for example, do students begin to use visual thinking in their problem solving as a result of working in Escher's World? Is it just increased familiarity with visual representations as a result of art-making? Or are there some deeper, structural connections between expressive activity and problem solving in mathematics?
In other words, the results from Escher's World show that expression and expressive activities have a noticeable affective influence on the process of learning, and that therefore, expression can play a positive role in mathematics education by helping students control their own learning. More speculative evidence from Escher's World, such as the correlation between expression and students' success in problem solving, suggests that expressive activities may have a cognitive effect on learning as well—and that an investigation of the cognitive aspects of expression would be a productive subject for further research.
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