LEARN FROM OUR MAKERS
Zoe and Maya's No Más Caídas
"Our shared image of No Más Caídas’s simplicity and ease of use is an expression of our desires as current (Maya) and future (Zoe) teachers in making math accessible, active, and embodied to everyone."
HOW THEY STARTED
Maya and Zoe sought out to create a tool for counting. At first, Maya was resistant to explore what her rich cultural background could offer the design process, deeming it irrelevant as she brought with her the limiting yet familiar recognition of the disconnect between out-of-school knowing and in-school learning. But Maya’s partner on the project, Zoe, continued to encourage her to leverage and share her experience learning mathematics from her childhood in the Dominican Republic as they navigated the various design decisions they faced as their design unfolded.
WHAT THEY MADE
After grounding their project in a particularly meaningful social, cultural, and historical context, Zoe and Maya intended for the final design of No Más Caídas to embody how children across countries and cultures “do math” when engaged in “counting as playing.” The Makers drew on the mathematical activities of children in the Dominican Republic engaged in counting games like quien trajo más (who brought more) and el que dice _____ sale! (where a number is picked, and whoever says it as they are counting is out). In the spirit of such games, No Más Caídas aims to provide its users with an invitation to count with concrete objects by sliding, watching, and hearing marbles fall down a 3-foot-tall tower.
WHAT THEY SAID
“Constructing our particular math tool as a way of removing the worry around losing track of these [cubes] gave us something to create that had meaning for both of us and for our students. Our shared image of No Más Caídas’s simplicity and ease of use is an expression of our desires as current (Maya) and future (Zoe) teachers in making math accessible, active, and embodied to everyone. We chose marbles as the objects to be counted as a symbolic token of Maya’s country, the Dominican Republic, and her childhood experiences there.”
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WHAT THIS TELLS US
Our project highlights the possibilities of what can happen when an environment is created where students are able to support each other in articulating their cultural stories and lived histories in the mathematics teacher education classroom.
David’s Prisms with Mismatched Holes
HOW HE STARTED
David was student-teaching in a kindergarten class and brought a lived history with caring teachers to the course. Although David had already decided to work on re-creating an already-existing manipulative with a fellow PMT in the course, David submitted his first video interview with a student with Autism Spectrum Disorder from his kindergarten class, Vincent. Upon watching it, the TE was struck by David and Vincent’s warm interactions and careful calibrations of each other’s movements on the classroom floor and suggested to David the idea of designing a new manipulative centered on Vincent. David, in turn, opens to accepting responsibility for Vincent’s care, sharing and utilizing Vincent’s knowledge and love of diverse shapes to pose a new goal of designing regular polygons for Vincent to tessellate.
WHAT HE MADE
In investigating this goal during the second interview, David realizes Vincent can already tessellate the floor with like and unlike shapes. This prompts David to change his design path away from tessellations, seeking out the TE after the interview to brainstorm. Still honoring Vincent’s love for shapes, they settle on designing triangular, square, and hexagonal prisms with holes and correspondingly shaped inserts intended to create a one-to-one matching task (e.g., which of these shapes fit together?). During a subsequent design session, David notices that multiple printed inserts do not fit into their intended holes. The TE takes advantage of this moment of struggle to support David through his technological anxieties, and recommends including the extra “mis-shapes” in the matching task (e.g., which of the multiple hexagonal inserts can fit into the hexagonal hole?). David reflects on this being a “teachable moment” as his “mis-shapes” can become usable for Vincent’s learning.
WHAT HE SAID
David embraced mistakes as an important part of his learning and celebrated Vincent’s mathematical discoveries, writing:
“Throughout the process of creating my design, I had made numerous changes. Some changes were logistical while others were tweaked based on my interview questions and responses Victor provided. These experiences have helped shape my understanding of mathematics and how learning happens. I have a better understanding of just how versatile mathematics is and that some of the best tasks and learning take place when [an unexpected] problem presents itself.”
WHAT THIS TELLS US
Our study speaks to the inclusivity that caring brings to learning. Vincent, a member of the students with disabilities (SWD) community, approached and demonstrated learning with animated physical enthusiasm. In a typical mathematics classroom, he might be considered a “disturbance” and subjected to repetitive, rote, and explicit instruction (Lambert, 2015). Instead, the TE and David’s caring-centered pedagogies supported opportunities to embrace Vincent’s inclination to learn with his body, and explore open-ended mathematical ideas together. The TE and David cared for and supported each other through technological apprehensions, even recognizing that design “mis-shapes” could become viable learning tools for Vincent, which helped to dissolve their feelings of exclusion from the Maker culture as their attention turned to care.
Erin's Fraction Orange
HOW SHE STARTED
At the time, Erin was student teaching in a fourth grade special education classroom and wanted to make a manipulative that could help one of her students explore fraction division. She felt that giving her student the opportunity to understand the relationships between a whole and its parts might give him a deeper understanding of what it meant to manipulate fractions (i.e. fraction addition, division, etc.) beyond what he was learning in class. Erin drew on her background in fine arts and took inspiration from everyday objects to quickly decide on creating a fraction orange.
WHAT SHE MADE
Erin had experience in design and sculpture which informed her Making process, so she was able to tackle her somewhat challenging design with relative fluency. She began with a sketch and started using methods of addition and subtraction of material and shapes in Tinkercad until she arrived at the fraction orange: a sphere partitioned into two hemispheres, where one hemisphere is further partitioned into fourths, eighths, and sixteenths of the whole, and the other into sixths and eighteenths. The pieces are nested within each other, which was an important part of the design because it offered the possibility of discovery for the user of how a whole can be (de)composed into/of parts.
WHAT SHE SAID
“This design experience turned out to be an exciting and inspiring endeavor for me. Much like my fraction orange, it uncovered multiple layers of my understanding of not only fractions, but of learning in general, and how one makes connections between mathematical thinking and concrete experiences. I also learned that both creative design and teaching processes require an incredible amount of thinking and planning, and also tinkering. A design cannot be successful if all the parts of its whole were not thought out…their relationships and interactions must be considered which requires a deep understanding of the material, the making process, and the execution of the design.”
WHAT THIS TELLS US:
This story highlights the importance of fostering opportunities for self-expression through creative acts in the mathematics classroom. Erin found a lot of value in having the flexibility to create and bring herself to the work, and she was able to bring her confidence in designing and creating to the class. This turned into a foundation of confidence in doing and teaching mathematics that is vital for any preservice teacher entering the field of education. In addition, this story shows the value in creating opportunities for PMTs to share their ideas and manipulatives with real mathematical learners and doers. In this case, the problem-solving interview brought value to actively pursuing what Erin’s student didn’t know (and turns out, what she didn’t know) and nurtured inquiry-driven learning. Her sharing experiences via the interviews ended up being really special and truly revealing, as Erin and her interviewees struggled to navigate between their traditional approach to fraction division (the flip and multiply algorithm) and really exploring what doing fraction division meant with the orange; proving how valuable it is to make space to explore the unknown in mathematics teaching and learning.