Author(s):
Judy Holdener

As mathematicians, most of us believe that our subject is full of beauty and creativity. We use words such as "elegant" and "beautiful" to describe our favorite proofs, and we admire mathematical greats such as Gauss, Euler, and Ramanujan for their creative capacities -- not just their computational abilities.

As teachers of mathematics, however, we are all too aware of the misconceptions that many people seem to share about our discipline. We lament the facts that mathematicians are seen as "number crunchers" and that our subject is seen as a litany of formulae and procedures. This problem is exacerbated by the unfortunate reality that many of our students don't encounter the truly creative side of mathematics until they start proving theorems for themselves in upper level courses.

Reform efforts in recent years have placed a greater emphasis on ideas and concepts and have made some progress in allaying these student misconceptions. However, there is more to be done. Not only should we work harder to reveal the beauty of our field to a wider audience, but we need to assign coursework at the introductory level that will serve as motivation and inspiration to students.

To this end, I have designed and assigned computer-based calculus projects that involve elements of art and design. As a mathematician who started out as an artist -- my mother was an artist -- I have a propensity for the visual arts, and I have found that many of my students share this propensity. In Calculus II, my students design goblets and tiles. In Calculus III, they draw parametric pictures.

Students enjoy these projects, and they are willing to put in enormous amounts of time creating their own masterpieces. While these projects might not capture the same type of creativity involved in proving an elegant theorem, most students find the work rewarding, and they learn a lot of mathematics along the way.

In this article, I discuss the merits of using calculus projects that involve an element of design. I will highlight a few examples of such projects, including the accompanying module (Parametric Plots: A Creative Outlet), and then discuss the reasons why I think these projects work.

### Judy Holdener is an Associate Professor of Mathematics at Kenyon College

Published June, 2004

Author(s):
Judy Holdener

As mathematicians, most of us believe that our subject is full of beauty and creativity. We use words such as "elegant" and "beautiful" to describe our favorite proofs, and we admire mathematical greats such as Gauss, Euler, and Ramanujan for their creative capacities -- not just their computational abilities.

As teachers of mathematics, however, we are all too aware of the misconceptions that many people seem to share about our discipline. We lament the facts that mathematicians are seen as "number crunchers" and that our subject is seen as a litany of formulae and procedures. This problem is exacerbated by the unfortunate reality that many of our students don't encounter the truly creative side of mathematics until they start proving theorems for themselves in upper level courses.

Reform efforts in recent years have placed a greater emphasis on ideas and concepts and have made some progress in allaying these student misconceptions. However, there is more to be done. Not only should we work harder to reveal the beauty of our field to a wider audience, but we need to assign coursework at the introductory level that will serve as motivation and inspiration to students.

To this end, I have designed and assigned computer-based calculus projects that involve elements of art and design. As a mathematician who started out as an artist -- my mother was an artist -- I have a propensity for the visual arts, and I have found that many of my students share this propensity. In Calculus II, my students design goblets and tiles. In Calculus III, they draw parametric pictures.

Students enjoy these projects, and they are willing to put in enormous amounts of time creating their own masterpieces. While these projects might not capture the same type of creativity involved in proving an elegant theorem, most students find the work rewarding, and they learn a lot of mathematics along the way.

In this article, I discuss the merits of using calculus projects that involve an element of design. I will highlight a few examples of such projects, including the accompanying module (Parametric Plots: A Creative Outlet), and then discuss the reasons why I think these projects work.

### Judy Holdener is an Associate Professor of Mathematics at Kenyon College

Published June, 2004

# Art and Design in Mathematics - Sparking Student Interest with Parametric Art

Author(s):
Judy Holdener

In the fall of 1997, during my first year of teaching at Kenyon, I wrote a MAPLE lab for my Multivariable Calculus class that served as an introduction to parametric equations. My idea at the time was that students would gain a deeper understanding of parameterizations if they were encouraged to “get their hands dirty” by experimenting with the equations. I encouraged them to make small changes to various parameterizations -- starting out with circles and ellipses, and moving up to lines and functions -- so they could see for themselves what the effects were.

At the end of the project, I threw in a final exercise asking students “to have some fun” by creating a picture. I included a simple smiley face as an example and told the students that I would offer a gourmet pizza dinner (made by my husband) for the best picture.

At the time, this final exercise was just an afterthought -- I hadn’t anticipated the level of excitement that would emerge. As it turned out, the students saw the initial exercises about circles and lines as merely “warm-up” for the final exercise. The final exercise became the project for them, as they spent amazing amounts of time on the pictures -- sometimes creating hundreds of curves. Indeed, one of my colleagues complained to me about our joint students who had decided not to do their homework in his class in order to allow more time for my project!

Since that first semester at Kenyon, the parametric plots project has continued to evolve. The current version, Parametric Plots: A Creative Outlet , was co-authored with my colleague Keith Howard. Keith wrote MAPLETs for the project, making it more user-friendly and more appropriate for use on the Internet. The current version also includes a larger repertoire of curves.

After 14 consecutive semesters of using the project at Kenyon, I am happy to report that the level of student excitement remains high. Students continually try to outdo the work of previous students, and my husband is now quite famous across campus for his gourmet pizza pies. The winning entry for the spring semester of 2004 is shown in Figure 1.

Figure 1. A parametric rendition of Bill Watterson's Calvin and Hobbes comic strip. The cartoon was recreated by Kenyon student Christopher White in the Spring of 2004 using 234 lines of code. Skeptical? Check out the MAPLE file.

Author(s):
Judy Holdener

We have had success at Kenyon with two other projects that I would put in the same category as the parametric plots project, because both involve significant elements of art and design and both are popular among the students.

The first project asks students to design a wineglass (i.e., a solid of revolution) subject to a list of criteria involving volume and center of mass. The second project asks students to design tiles for a swimming pool subject to various constraints dealing with the area of the region between decorative curves on the tiles.

The Goblet Project - Students are asked to design a wine goblet that meets the following mathematical specifications:

 It is a solid of revolution. It holds at least 150 cm3 of liquid. It takes no more than 150 cm3 of glass to manufacture. The ratio of the height of its center of mass to the radius of its base must be no more than 3:1. The radius of its stem must be at least 1/4 cm at its thinnest point.

The goblet project has evolved into its present form after several iterations, and it was difficult for me to trace its origins. However, it appears that Mark Snavely (Carthage College) is responsible for the first iteration. During his first year of teaching at Carthage, he asked his students to design a goblet and a plate and to compute the volume of each. The project was later adapted into its present form by Carol Schumacher (Kenyon College) and Patty McKenna (Metropolitan State College).

An early version of the computer-based calculus course Calculus & Mathematica, written by Jerry Uhl, Bill Davis, and Horatio Porta, also included a problem involving the volume of a wineglass, but it appears that this was written independently and there was no element of design involved.

The Tiling Project – In Problems for Student Investigation, John Ramsay of Wooster College introduced his version of a tile design project adapted from an earlier version appearing in MAPLE: Calculus Workbook Problems and Solutions .

I adapted the project further, making it more appealing for a Kenyon student. Kenyon has a long history of success in swimming and is in the process of constructing a new Fitness and Recreation Center. My version of the tiling project asks students to design a two-colored tile and floor plan for Kenyon’s new Olympic-sized swimming pool, with the constraints that

• each region of a tile must be simple, and
• there may be no more than three regions per tile.

Students then employ integration to compute the percentage of tile devoted to each of the two colors.

I find this project appealing, not only because it is a nice application of integration, but also because it provides a good opportunity to discuss symmetry with the students.

# Art and Design in Mathematics - Why Do These Projects Work?

Author(s):
Judy Holdener

The success of these projects, I believe, rests on the fact that they are open-ended and visual, yet grounded by a significant mathematical component. Students regularly go beyond what is expected of them.

In the case of the parametric plots project, some students produce three-dimensional plots instead of two-dimensional pictures. Some have made short movies, incorporating mathematical models to reflect certain physical realities. Others have learned some linear algebra so that they could rotate a curve to produce exactly the right shape for their picture. I have had students who produce efficient MAPLE code that allows them to translate, rotate, and reflect objects quickly. They often recognize how symmetry can be used to simplify their programming.

Most common, however, is the situation in which a student (or a pair of students) produces 50-100 (even 300) different parametric curves to construct an elaborate picture. The students simply enjoy creating their masterpieces, and they will work hard to make their finished products perfect. Driven by the desire to create a design that they can call their own, they also gain a deeper working knowledge of the relevant mathematical concepts.

As an added bonus, students' finished products provide tangible examples of creative work in mathematics. While it may be difficult for an outsider to see the creativity involved in a proof, the creativity involved in these projects is visible to a non-expert. As such, the results of these projects allow students and spectators to see that mathematical work can be creative.

I have displayed student work in the hallways of our building and in the local coffee shop to better advertise this fact. Recruiting members of the administration and faculty to serve as judges of the contests also helps to make mathematics more visible across campus.

Figure 2. James Sapanski's (Kenyon College, Class of 2007 ) rendition of his dog Bernie.

Author(s):
Judy Holdener

Mathematicians often judge the merits of a lesson by its mathematical content. We are impressed by a well-crafted presentation of an elegant proof, and we enjoy the pathological examples that follow, illustrating the necessity of the hypotheses.

As teachers of introductory level courses, however, we should remember that our students are not (usually) mathematicians. Our students do not always share the same intrigue that we have for the formal aspects of our subject. We should judge the merits of a lesson or project based on the mathematical content learned by the student and the extent to which the lesson or project inspires the student to pursue mathematics further. In my experience, design projects have been very successful in these regards.

While it might take a bit more effort on our part, we can do more to inspire students so that they recognize mathematics as a playground for creative thought. While these design projects might not capture the same type of creative play typically involved in a mathematical proof, they certainly do involve creative play. For students, this creative play results in mathematical understanding, and student understanding is our ultimate goal as teachers.

Figure 3.Tile designs by Katie Capaldi (top), Bethany Taylor and Jack Cerchiara (bottom).