With My Waterfall

The final course I enrolled in and completed before earning my Bachelor’s degree in biological science was calculus. I was terrified about taking the course because it was my final course before graduating. I would also be student teaching at the same time all day, every day, over the course of the corresponding sixteen-week semester. I was paying for college myself and I had to quit my full time job in order to student teach and take calculus at night to meet my May graduation deadline goal. If I did not successfully complete the class, I would have to reenroll the next semester, thus delaying graduation a half year.

I will never forget the first night of class. The professor told all of us to take out our notebooks and then proceeded to write a plus sign, a minus sign, a multiplication sign, and then a division sign on the overhead projector. He then walked to a whiteboard and drew an enormous equals sign. He asked students to explain what these were. Students randomly volunteered names and answers to explain what each sign meant. No matter how much we tried to explain what each sign was or how it was used he would delve deeper and always end with asking “But why do we use it?” or “Where did it come from?” After a good ten or fifteen minutes of this exercise, he both jokingly and seriously stated to the class, “This is the problem with mathematics education. Everyone in this class is a senior college student and in your early 20s. You have completed sixteen years of education and do not even know basic kindergarten mathematical symbols.”

He started with the equals sign and explained the reason we use that sign in mathematics is because hundreds of years ago, mathematicians and engineers identified that nothing is more equal than two parallel lines. He continued with the plus sign and its origins from the 1500s as shorthand for the Latin “and.” He continued with a variety of other symbols we had been using since childhood and provided historical, cultural, interdisciplinary contexts with respect to the reasons they exist and what they mean. To say the least, I was more than enthused about our class.

Every class we attended included lessons relating to the cultural, historical, scientific, artistic, and engineering relationships between the content we were learning. The calculus content we were working on was difficult but I was interested and motivated to learn, as well as persevere, as a result of the curricular design and instructional delivery of the content. The irony was this was the last course I was completing to earn my Bachelor’s degree but was the first mathematics course I had ever experienced in my life that was creatively delivered and intrinsically motivating.

I found myself wishing I had been taught mathematics utilizing this approach many years earlier. I also began thinking about how I could apply this creative, relevant instructional delivery approach myself when I started teaching. As a result of my authentic experiences, I believe we must first stir the curiosities and passions of students, then the content becomes important. Then the learning begins.

Although I was a biology major, that specific mathematics course remains my most fond memory of all my undergraduate coursework. The creative instructional delivery and use of interdisciplinary instruction employed during that class are lessons learned that remained with me throughout my time as a teacher and throughout my career as an instructional leader. Over the course of many years as an educational leader, I have experienced a great deal of success in affording students with creative ways to approach mathematics.

Oftentimes, I have found students to be intimidated with taking mathematics courses at the high school level and only choosing to enroll in mathematics electives if they are required to do so or only if they are genuinely interested in mathematics. In addition, freshman and sophomore off-track data tends to reflect students struggling the greatest in terms of high school mathematics coursework at the ninth and tenth grade levels as opposed to the other content areas.

These challenges are compounded further when students who fail or earn a “D” (an academic “F”) in freshman algebra are nonetheless automatically enrolled into the next semester or even the next year’s sophomore-level geometry or algebra II course. My concern has been (and continues to be) we are unintentionally creating a conveyor belt, systemic milieu that contributes to an inherent dislike for mathematics in high school students.

Furthermore, this structural setting reinforces negative student perceptions regarding mathematics and contributes to off-track student predictors at the onset of high school impacting the likelihood of high school graduation. As a result, there are very few mathematics electives students will choose to enroll in unless they are forced to do so or are intrinsically motivated to learn mathematics.

Nothing Can Harm Me at All
A course I have successfully offered students as a mathematics STEAM elective is Art in Mathematics. I have offered this course at the high school level as both an Honors and Regulars course credit elective. It is also a course I have designed and taught. I have afforded students the opportunity to enroll in Art in Mathematics in a variety of high school settings, ranging from a selective enrollment college preparatory high school setting to a therapeutic day school learning environment. I have observed students at both course levels across a spectrum of settings flourish and thrive as a result of taking this particular course. As a result, students exit the course with a greater love and appreciation for the world of mathematics surrounding each of us daily.

Employing a creative, innovative approach towards learning mathematics via interdisciplinary teaching and learning results in a deeper appreciation for the subject matter and affords opportunities for teachers to engage in cross-curricular instruction. The following is an overview of content that not only includes STEAM-related curricula in Art in Mathematics but also incorporates social constructs and cultural units of study within this interdisciplinary framework.

Golden Ratio
Students are introduced to Art in Mathematics curricula by developing an understanding of the golden ratio in art and architecture as well as the occurrence of the golden ratio in nature. These approaches include learning patterns in Fibonacci numbers (sequence) and related artwork students observe and create to reflect understanding. The concepts of patterns, proportions, and symmetry are introduced with respect to how they occur in nature and in artistic and architectural design. Students create mathematical sequences for art via Euclid’s study of the golden ratio and its appearance in some patterns in nature (for example the spiral arrangement of leaves and/or plant organs and structures).

Proportions and Perspective
Students are provided a review of proportions in the human body and the history of anatomical research throughout time. Students learn about the Renaissance and human proportions reflected in art, proportions in architecture (in both a historical and cultural context), and proportions of the Egyptian pyramids for an introduction to geometric learning. Students explore the historical use of perspective in paintings and the diverse (and complex) employment of perspective during the Renaissance and Chinese art.

Symmetry and Patterns
Introduction to symmetry (beginning with demos on the complex beauty and nature of soap bubbles) and patterns with an introduction to the cultural, artistic and mathematical constructs of Asian, Islamic, and Celtic art. Concepts include symmetrical patterns in Asian art, patterns in Islamic art and architecture, and ornamental symmetry in Singaporean architecture. Students learn the seventeen wallpaper patterns at Alhambra and how knowledge of these seventeen wallpaper patterns may have been learned and employed in Chinese art. Patterns in Celtic knots, textile design, tilings, Penrose tilings (Penrose’s two different rhombi shapes and two different quadrilateral kites and darts) and tangrams (seven flat polygons dissection puzzle) are explored. Students manipulate polygons and examine symmetry via five-fold rotational symmetry, reflection symmetry, and translational symmetry. Tessellations (reflection, rotation, translation, glide reflection) in nature and the art and geometry of Origami paper folding in Japanese culture are investigated.

Geometry and Architecture
Students explore mazes, labyrinths, and the fourth dimension. These explorations segue into mathematical, artistic optical illusions and the engineering of kaleidoscopes (angles for creating stars). Students are provided an integrated STEAM approach for learning about Polyhedrons (geometry and angles) and stellations in nature and as geometric structures in architecture and design engineering (and a review of Origami). Students discover historical geometric design concepts in fortresses and various cultures utilizing pentagon design constructs, the design engineering of arches (especially in terms of their strength), as well as the geometry of war. Students study the architectural, visual illusions in the Parthenon and the architecture of domes and geodesic domes from both a geometric and cultural perspective.

Historical and Cultural Contributions to Mathematics by Artists

The German Renaissance artist Albrecht Dürer’s Melencolia I: Exploration of number appearance and numerology in his work (rows, columns, Dürer’s use of numbers for his second signature, initials, age, etc.). Students learn about historical and cultural beliefs concerning magical/astrological association at the time between magic squares and planets (Saturn and Jupiter 3×3 and 4×4 magic squares).

The mathematically inspired works of the Dutch graphic artist Maurits Cornelis Escher: Exploration of Escher’s woodcuts, lithographs, drawings, and sketches of symmetry and impossible spaces. Review of the geometry of the wall and floor mosaics in the Alhambra and Escher’s fantasy-inspired architectural, perspective illustrations.

German painter Hans Holbein the Younger’s “The Ambassadors” painting with anamorphic perspective: Students review perspective and view the skull in the bottom center of the composition and render the form via an accurate perspective. Students examine the portrait and identify items depicted and their cultural, historical relevance (scientific instruments, terrestrial and celestial globes, a shepherd’s dial, a quadrant, a torquetum, a polyhedral sundial, an oriental carpet, and various textiles including the floor mosaic).

Students complete Art in Mathematics coursework through a capstone project on the contributions of the Italian Renaissance artist and engineer, Leonardo da Vinci: Discussions/explorations include da Vinci’s influence on culture, science, technology, engineering, art, and mathematics. Students explore the many facets of STEAM by Leonardo da Vinci and complete a culminating research project on his contributions to modern day culture and living.

I believe mathematics is an incredibly important foundational content area. It does not have to be a course with merely right or wrong answers or designed and perceived as sterile in aim. Mathematics can be an incredibly exciting course that excites and inspires students to learn more about the world around them.

There are a variety other STEAM-related, cultural, and historical interdisciplinary content areas that can be interwoven into Art in Mathematics (fractal art via computer science and mathematics, architecture of the Taj Mahal, parabolic, ellipsoid and spherical geometry of the Sydney Opera House, etc.). I believe we must strive to design content that embraces and incorporates discovery learning at all grade levels.

If the intent of mathematics is to facilitate logical reasoning in students, the impact of mathematical curricular design must be to provide logical reasons for students to be interested in the learning.

Chris

Referenced links and documents:
Engaging Community Through the Integration of Art and Mathematics, Ellie Balk and Tricia Stanley

Gram Theft Audio by Chris Dignam

May This Be Love by Jimi Hendrix

The Benefits of Fine Art Integration into Mathematics in Primary School, Anja Brezovnik

When Mathematics Meets Art: How Might Art Contribute to the Understanding of Mathematical Concepts?, Liora Nutov