On three Sundays in March 2015, educators came together for a series of collaborative design workshops. The goal: to create a professional development program weaving together art and mathematics for use in informal learning settings. Of special interest: Mathematical Practices of the Common Core State Standards.
4 classroom teachers + 2 science center educators + 3 research staff + paper strips + paper squares + straws + pipe cleaners + food + talk + …
Week One: Active Hands
A mix of educators worked with materials, exploring forms and ideas through action, perception, and many modes of talk.
Participants chose among three activities, which provided starting points for exploration. Some curled, cut, and taped colored strips of paper to make “curly birds” and curvilinear sculptures (left, foreground). Intersecting and concentric circles raised themes of pattern, measurement and precision. Another pair of participants folded origami ninja stars (left, background), accompanied by talk of forms in nature and biomimicry. A tetrahedron fractal model provoked conversation and exploration of repeated structure (center), while a math teacher and science educator engaged in spatial modeling of fractals with straws (right). They wrote mathematical equations describing recursive relationships, and talked about shape, length, sides, geometric proof, precision, tools, prediction, mathematical practices, classroom and field trip pragmatics.
They talked while they worked and played. In this case, there was no distinction. Experimenting, noticing, problem solving, and reflecting, sometimes with grand overarching relevance, “Math isn’t about getting the right answer that the teacher somehow magically knows, it’s about discovering something and proving that it’s true.”
A science center educator asked, “What do teachers look for in informal activities?” “Teachers look for content…what can I do in my classroom next week?” They continued making and talking. They worked in anticipation of STEAM Family Day, when two weeks later, they would field test their activities and engagement strategies. The workshop organizer expressed a focus for next time—incorporating Standards for Mathematical Practice (SMPs)—suggesting attention to precision, use of appropriate tools, measurement and prediction. Concluding the session, she reinforced the collaborative aspiration of the workshop series, “We want to learn from you.”
Week Two: Zooming In
The wide-open exploration and divergent flow of ideas in week one converged into focus areas in week two. The group focused on two SMPs*—attention to precision and appropriate use of tools—and two activities.
*Standards for Mathematical Practice
“What is the goal for STEAM Day?” “I have trouble playing without a goal.” They decided to create a challenge, a problem for participants to solve, a purpose to drive the activity as a way into the open-endedness. Discussion ensued about defining a goal, setting parameters, establishing and acknowledging constraints on materials and time.
They narrowed the field of activity options, yet with more detailed definitions came another flowering of ideas. The goal could address “real problems,” such as structure and function, stability and strength. Pretend to be an architect. Build an assembly of straws, pipe cleaners, glue. Meet a height requirement. How much weight can it hold? Predict, build, test. Attend to aesthetics. Is it beautiful or ugly? How does it make you feel?
They generated words associated with construction, “to provoke thinking,” to expand the vocabulary for imagination and action: build, design, create, engineer, explore, discover, imagine, invent, geometry, challenge, structure, function. Questions came in flurries, “What are they doing with the SMPs?” “Is there a mathematical end to building a structure?” “How much do we want to force them to use math?” “…give them graph paper, scales…?” “and other tools, protractors, rulers, scissors, pencils…” “This is just a test, an experiment.”
Questions remained unanswered as they circled in on a plan, often talking in broad terms, negotiating the structure, content, and affordances of the activities. “We want them to make sense of problems, to persevere, to analyze constraints, and their relationships with goals.” “The SMPs are a lens to do math.” “But we’re not really doing math.” “Why don’t we see what we’re doing as math?”
“Math & the Arts Common Vocabulary” used in K-12, added to and shaped the conversation. What words resonate for you? Symmetry, part, whole, pattern, vertical, diagonal, parallel, perpendicular…
“We can create questions that get them to math concepts.” “How can we promote meaningful conversation with STEAM Day visitors?” “We can ask, what are you trying to do? What strategies did you use? And highlight repeated patterns.” “How might you create a variation on that theme?” “We can also use sentence frames such as ‘I notice…’ and ‘I wonder…’”
They made a game plan for the next week, defining materials, tools, constraints, questions, and an objective: design and test.
Week Three: Making Math
Educators worked with visitors to the Fleet Tinkering Studio on STEAM Day.
For our third workshop, we went straight to the Tinkering Studio and jumped into activities with visitors to the Fleet Science Center. This day, designated STEAM Day throughout Balboa Park, offered an array of activities that weave together science, technology, engineering, arts, and mathematics. At one table, educators invited participants to build structures with straws, pipe cleaners, and hot glue, and test their ability to bear weight. At another table, they worked with paper strips and tape to make curvilinear sculptures, such as curly birds. After two hours in the Tinkering Studio, we discussed the experience. Educators highlighted lessons learned in response to the questions (in bold) below.
What did you notice about how people engaged with the activities?
- People were willing to work without instruction; they’re willing to jump in and jump out.
- We need to pose a challenge that isn’t too daunting and provide examples that provoke, inspire, and encourage iteration.
- To make it matter, people need to have a purpose.
- They can make their activity matter to them by creating a story, as people did with the curly birds (where does it live, what does it eat, etc.).
- We need to engage the parents; they can make or the break the interaction.
Regarding tool use, attention to precision, and structure. Did you see those practices in play?
- People used tools. (Discussion and disagreement arose about what is mathematical tool use. When and how can rulers be used mathematically?)
- Making a tool comes out of necessity.
- Someone proposed that diagonals could be tools, used to strengthen the structure, raising the question what is a tool?
- How might shapes, with different structures and symmetrical properties, be used as tools in this context?
- Ideas can function as tools for productive exploration. Contrasting examples, such as symmetry and asymmetry, opens up options and consideration of consequences.
- Discussions of structure included themes of symmetry and asymmetry, aesthetics and beauty, and relationships among them.
What would you keep? What would you change?
- Pre-make some structures as examples or things for people to build on, improve, and strengthen.
- Make a specific display space, and invite visitors to make a label for this museum exhibit.
- Explicitly invite comparison of strategies and structures.
- Find ways to explore the limits of what participants know. Ask questions that might expand their perception of possible pathways forward.
- Explore ideas in two- and three-dimensional space, and relationships between them.
Can you describe an experience when you felt integrated art and math?
- Some activities (e.g. curly birds) seemed more artsy-craftsy, focused on making something “pretty.”
- We can find ways to make the activities more “mathy,” such as draw a plan; change the scale; translate 2D into 3D; distort proportions; change ratios.
- Revisit fractals (worked on during the first workshop). There are so many opportunities to integrate art and math with fractals.
- Use architecture as a way to frame the activity, exploring form and function, and naming structures.
- Intentional use vocabulary that bridges art and math brought attention to the union of art and math.
What did we accomplish in bringing together formal and informal educators?
- We focused our thinking about learning for all ages, i.e. life long learning in informal settings outside of school, and Common Core emphasis on skills, practices, and conceptual connections across grade levels in schools.
- Formal educators found it interesting to focus on the experience, not just the content to be learned. Also, of interest, how the Standards for Mathematical Practice can come into play in a less structured setting.
- Together, we explored how to pose questions to stimulate thinking.
- We discovered interesting possibilities when working with everyday objects.
- Intrinsic motivation is essential to high quality engagement in any setting.
- When teachers go to museums, they want something unique and different from school.
- There is tremendous value in spaces for learning that aren’t like school.
- Formal and informal learning can be (should be!) complementary.
- We can co-create learning communities that bridge formal and informal education.
These educators came together to explore and learn together, sharing their diverse expertise about math, art, and education. Some of their questions remain open—What and where is the math in these activities? When and how do we make the math explicit? Open questioning, in a relaxed, playful, and profoundly collaborative process, embodies the spirit of InforMath.
Professional development workshops focused on integrating math and art were organized by Ashanti Davis and Ashley Atwell at the Reuben H. Fleet Science Center in San Diego, with support from the InforMath Collaborative.
Blog post by Nan Renner