From the youngest ages we classify ourselves as either "math people" or "creative types." I have heard students use these labels to make excuses such as "I'm just not good at math…" or "I can't draw" and later to influence important decisions such as what courses to study or which challenges to accept. Of course, people learn in different ways, but these labels are limiting and to some extent untrue. Artists and mathematicians ask many of the same questions.

Two years ago, while discussing our Middle School curricula, we realized that our math and art programs were simultaneously exploring many of the same concepts: how we use space, reference points, distortion, patterning and relative relationships. As our conversation developed, the similarities (and comparative differences) between disciplines emerged. We agreed that our common objective for our students was to have a deep understanding of concepts and to be able to apply their knowledge to the specific disciplines/situations as needed. Initially we hypothesized, and now have shown, that exploring these concepts in both math and art projects, has increased students' understanding and interest in both disciplines.

"Space" is a pivotal concept in math and art: how to best represent it and the relationships between objects. Artists use shading techniques, illusionary angles and points, levels of detail, overlapping, and size to portray relative space. Mathematicians use numbers and symbols to convey similar information. For example, visually, a larger object appears closer than a smaller one. Mathematically, this relationship of relative space is expressed in an equation: a:b = A:B. Art strives to represent universal abstract concepts by depicting the essential elements of specific circumstances; math strives to represent all specifics by depicting universal relationships as abstractions. These are two of the most basic human intellectual drives; and, although they are apparently contradictory pursuits, they share a parallel quest for clarity.

Another example of how artists and mathematicians examine similar concepts using different methodology is how the disciplines look at perspective and proportionality. In art, the decreasing size of individual people or specific objects reflects distance from the point of reference. Automatically, our eyes approximate the relative distance based on the comparative sizes of shapes. For example, if we were to see a photograph of a chair, we infer that the legs are the same size even though the closer legs appear larger. In math, perspective is abstracted into equivalent ratios that represent relative size. In both cases, the relationship between the two parts is communicated to the viewer/reader.

Recognizing that both subjects explore relative relationships (proportions), our students did a project manipulating these ratios. In art, students drew a 2" x 2" square and sketched a simple scene of something they had done during the summer. It was important that students did not know what they would be doing with the drawings. If they knew that they would be reworking them several times, they might not have put in the same level of detail. When they finished the drawing, they lightly drew in a grid on the half-inch lines over their image. Upon completing the square "summer sketches," students made a second rectangle, this one 2" high by 4" wide. They filled in the half-inch marks and lines and marked the horizontal line by inches instead of half-inches. Students copied their initial drawings into the new rectangle using the grid lines as guides. Of course, the end result was an image doubly wide as the original or "stretched out." Students, amused by these funny distortions, completed the exercise again, stretching vertically. Finally, students must stretch their drawings further to 8" x 2". Though students were allowed to choose which way to stretch their images, the results were visually ridiculous and entertaining. These drawings gave visual representation to common algebraic questions such as "What happens when you double one of the variables? What would happen if you doubled both the height and the width at the same time? What does it look like? " We found that the visual model of mathematical concepts dramatically helped students who otherwise had difficulty grasping abstract math concepts. With these drawings, students better understand equivalent ratios and learn the concept of scale factors.

Another successful math/art integrated project is the creation of geometrical piñatas. This project, usually done by pairs of students, involves the design and construction of different geometrical solids to depict an abstraction of an animal or a familiar object such as a guitar, cell phone or piece of pizza. The mathematical requirements include: a) drawing a flat pattern of the piñata that includes tabs (drawn to small scale to fit on notebook paper); b.) finding mathematical formulas that show how the students will calculate the surface area of each solid; and c.) coloring sketch of what the final piñata will look like when assembled. Students operate as if their piñata would be mass-produced; therefore, they can not have extra textures on the final constructions unless they are designated on the flat pattern.

The artistic requirements for the project are divided into two parts. The first is that the piñatas are an assessment of students' art vocabulary. They are required to use complementary colors (colors from across the color wheel), analogous colors (three colors next to each other on a color wheel), a tint (when white is added to a color), and a shade (when black is added to a color). Upon completion of the piñata, students must label where they used these colors. The second artistic requirement is that the final project must demonstrate creative idea formation with careful attention to details and presentation.

After the students complete these preliminary plans, they receive further peer and teacher guidance before the construction begins. To achieve this, they must put the shapes together mentally or using geometric solid models. This helps students recognize which sides will be attached to one another and thereby disappear, and which surfaces will remain exposed in the final construction. Ultimately, the students are required to draw the flat patterns on poster board and calculate the surface area. (We allow them to excluding the tabs from their calculations.) At this point they can paint the surfaces or wait until after the final assembly. They must then construct the piñatas and calculate the total volume. We require each pair of students to take photographs of their project from several different angles and attach these to all their design efforts and calculations. Finally, they fill the piñatas with candy and throw a party for some younger students, who inevitably love the process of breaking the piñatas! This has become a favorite tradition, and each year the designs become more interesting visually and more challenging mathematically.

Another example of a fine intersection between art and math is our tessellation project. As students are investigating tessellations in math class, M.C. Escher's work is introduced in art. Students are then challenged to create tessellations involving three or more transformations including translations, rotations, and reflections. Then they are asked to demonstrate each step of the transformation along a "story board." Finally, using both geometric and color transformations, the students repeat their transformed figures to cover a larger area. Students who are more advanced, artistically and/or mathematically are guided towards the creation of 3-dimensional tessellations.

"Summer squares," piñatas and tessellations are just a few of the many ways art and math can be integrated. Some others include: the design of games and game boards involving probability; the search for examples of the golden ratio in famous paintings; the study of fractals and finding and/or creating fractal patterns such as those seen in African and Asian designs; creating solids with origami and learning some of the algorithms of the folds involved with regular and stellar shapes; studying the six basic transformations by examining Islamic designs; or investigating Kolams and learning basic logical languages with which to describe these visual and number patterns.

As teachers we challenge ourselves to differentiate teaching methods for students with a variety of learning styles. Learning key concepts and manipulating them in both math and art has allowed all students to grasp material in multiple ways. In an informal study, we found that students showed greater mastery of certain math concepts after having done the "Summer Shapes" art project. In addition to the test scores being higher, students who struggle with math, showed particularly high growth on that concept's test and had a more positive attitude towards math class.

When discussing our collaborative planning and projects with art and math teachers at other schools, we realize that we have several factors working towards the success of our integrated work. Most significantly, we spend time talking and sharing our classroom objectives and frustrations. We share our aspirations for individual students and we try to design integrated projects imagining specific students' participation in them. Another factor that facilitates our collaboration is the close proximity of our classrooms. Our workspaces are close enough that either of us can have impromptu visits to observe and interact with our students while working on these projects.

Our students know that we value both disciplines. Students see us in each other's classrooms answering math and art questions. When unsure, we respectfully seek the expertise of the other teacher. Our project rubrics include clear directions and expectations in both the math and the art aspects of a project.

Learning to ask the important questions is essential to a meaningful education. Mathematicians and artists ask many of the same questions, so our students should be encouraged to use both disciplines in their explorations to understand and in their search for answers. Hopefully this understanding will allow students to communicate their ideas with the logical abstractions of math and the visual representation in art. These different perspectives provide students with greater understanding of both disciplines and give them alternate ways to view their world.

Lisa Jacobson teaches art to middle and high school students at Noble and Greenough School in Dedham, MA. Previously, she taught middle school students (with Anne Ray) at Wilmington Friends School in Wilmington, DE. Lisa is also a practicing sculptor.

Anne Ray has taught in schools around the world and currently works in Quito, Ecuador. She has taught math for many years to students of all ages. Before she taught, Anne was a potter.

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