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Calculicious: Illuminating Standards Video Series

Type

Videos

Grade Level

Discipline

This film features a cross-disciplinary project which resulted from a collaboration between a mathematics teacher and an art teacher, celebrating the beauty of both fields. Although its sub-title is Making Calculus Delicious, the math featured in the book is primarily geometry and algebra, with students creating artistic interpretations of mathematical concepts and formulas. The translation of mathematics to art promoted fresh conceptual discussion and creative thinking, and inspires consideration of the elegance in mathematics.

The Illuminating Standards Project

In the last two decades of the ‘standards movement’ in American public education, many educators have concluded that ‘teaching to the standards’ and project-based learning are incompatible. Ron Berger (Expeditionary Learning) and Steve Seidel (Harvard Graduate School of Education), co-directors of The Illuminating Standards Project, wondered if this conclusion was true. Indeed, they speculated that long-term, interdisciplinary, arts-infused, community-connected projects may well be one of the best ways to actually see what state standards look like when fully realized in the things students make in school—to make the standards visible.

Three questions frame the work of The Illuminating Standards Project:
-What does it look like when state standards are met with integrity, depth, and imagination?
-How can we use standards to open up and enrich curriculum, rather than narrow and constrain it?
-How can we use student work to raise the level of our understanding of standards and our dialogue about them?

The Videos

Collaborating with Berger and Seidel on The Illuminating Standards Project, over 30 students at the Harvard Graduate School of Education have explored these questions by choosing projects from the Student Work Archive in the Center for Student Work and considering the ways in which those projects did—and didn’t—meet specific state standards. Further, they examined how the student work illuminated the standards—and vice versa. Many of those students created short films and 13 of those films are presented here.We invite you to watch these films and we encourage you to use them as the catalyst for discussions with your colleagues about the relationship between your commitment to meet demanding state standards and approaches to designing powerful learning experiences for our students.


Type

Videos

Grade Level

Discipline

This film features a cross-disciplinary project which resulted from a collaboration between a mathematics teacher and an art teacher, celebrating the beauty of both fields. Although its sub-title is Making Calculus Delicious, the math featured in the book is primarily geometry and algebra, with students creating artistic interpretations of mathematical concepts and formulas. The translation of mathematics to art promoted fresh conceptual discussion and creative thinking, and inspires consideration of the elegance in mathematics.

The Illuminating Standards Project

In the last two decades of the ‘standards movement’ in American public education, many educators have concluded that ‘teaching to the standards’ and project-based learning are incompatible. Ron Berger (Expeditionary Learning) and Steve Seidel (Harvard Graduate School of Education), co-directors of The Illuminating Standards Project, wondered if this conclusion was true. Indeed, they speculated that long-term, interdisciplinary, arts-infused, community-connected projects may well be one of the best ways to actually see what state standards look like when fully realized in the things students make in school—to make the standards visible.

Three questions frame the work of The Illuminating Standards Project:
-What does it look like when state standards are met with integrity, depth, and imagination?
-How can we use standards to open up and enrich curriculum, rather than narrow and constrain it?
-How can we use student work to raise the level of our understanding of standards and our dialogue about them?

The Videos

Collaborating with Berger and Seidel on The Illuminating Standards Project, over 30 students at the Harvard Graduate School of Education have explored these questions by choosing projects from the Student Work Archive in the Center for Student Work and considering the ways in which those projects did—and didn’t—meet specific state standards. Further, they examined how the student work illuminated the standards—and vice versa. Many of those students created short films and 13 of those films are presented here.We invite you to watch these films and we encourage you to use them as the catalyst for discussions with your colleagues about the relationship between your commitment to meet demanding state standards and approaches to designing powerful learning experiences for our students.


Transcript

- [Instructor] What happens when a math teacher and an art teacher work together to design a project? These images show some possible results of this type of collaboration. Beautiful works of art that incorporate interesting mathematics. This project also gives us a lens that can bring a new clarity to mathematics standards, inspiring new possibilities for student work. The common core state standards include two types of mathematics standards, content standards and the standards for mathematical practice. The standards for mathematical practice provide broad statements about how students can think like mathematicians, by reasoning abstractly and quantitatively, by attending to precision, and so on. These standards encourage students and teachers to explore the abstract beauty of mathematics, in addition to its usefulness and applicability. Students can use these skills daily in math class, but they can also engage with these standards through longer-term projects. 

Students at High Tech High in San Diego created artwork that included representations of mathematical concepts. Their creations were displayed in the school and archived in a published book called Calculicious. These seniors each created three works of art over the course of a semester of work in both their math and art classes. A watercolor and acrylic painting, and a sculpture which they worked on with one or two other students. They created preliminary sketches and drafts of their works, investigated the mathematical properties and concepts involved, and posted their final products with commentary on a class website.

Three students, Brad, Carl and Sam, created a sculpture by stacking painting blocks and investigating the centers of mass of their structures, and how far they could hang over the edge of the table without toppling. They displayed their piece with a poster behind it, including descriptions of the math they used to create their structures. Their words and the equations that accompanied their explanations shows some of the math they explored in creating this piece. One of their structures involved stacking blocks in multiple directions, which they showed allowed them to overhang the table to a greater extent than the same number of blocks stacked in one direction. Their unidirectional stack of blocks, they found, follows the harmonic series. The top block overhangs the second by one half a unit, the second overhangs the third by one third of a unit, and successive overhangs are one fourth, one fifth, and so on. As a result, the total overhang of the structure is equal to the sum of the harmonic series shown here, which, as they explained, diverges to infinity and has no upper bound. Before we take a closer look at the explanations, let’s revisit a couple of standards for mathematical practice. Standard number seven says students should look for and make use of structure, that they should be able to look closely to discern a pattern or structure, and step back for an overview and shift perspective. Students who are using this ability should be able to look past numbers and calculations to make connections to more abstract concepts and ideas. Standard number eight asks students to look and express regularity in repeated reasoning. They should maintain oversight of the process while attending to the details, and be able to find patterns in their calculations and use these patterns to simplify their work or connect to broader concept.

The sculpture Brad, Carl and Sam made and their explanation of the math involved required them to use these two standards for mathematical practice. They described the interesting relationship between the harmonic series and total overhang they found in their stack, and explained that this means the maximum overhang for a singlewide stack is unbounded. They applied this relationship to see that for a 100 unit overhang, the minimum number of blocks required is greater than the number of particles in the known universe. They included these statements in their description of their sculpture, which is shown in the Calculicious book opposite the image of their structures. These statements show a use of patterns and structure and a willingness to look beyond simple calculations to find and use a deeper concept. These students were not merely doing math. They used this project as an opportunity to study and discover beautiful patterns in mathematics, and to explain those deep patterns to others.

As we as a nation transition to using the common core state standards in our daily learning and teaching, we should ask, how can we invite students to explore and enjoy the deep concepts and patterns in mathematics suggested by the standards for mathematical practice? This type of teaching is not easy. It requires teachers themselves to have a deep understanding of math and to be comfortable taking risks with their students. If we can lean into this challenge, however, the results can be extraordinary. Beautiful, elegant and deep celebrations of mathematics.

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