January 1st, 2016

*This Knowledge Base article was written collaboratively with contributions from Steven Greenstein and Andee Rubin. This article was migrated from a previous version of the Knowledge Base. The date stamp does not reflect the original publication date.*

## Overview

The proliferation of spaces for digital design and production has generated new opportunities to teach and learn new mathematical things in new ways and to even think in new ways about what teaching and learning mathematics might look like. Not only can new tools be produced to better engage with the mathematical concepts that have historically been taught, but we can also produce new tools that enable the invention and exploration of new mathematical ideas.

The use of â€śmanipulativesâ€ť (Post, 1981) in mathematics education is typically reserved for early elementary school. Their power lies in their capacity to support the construction of abstract mathematical concepts from sensorimotor engagement with concrete tools (Kamii & Housman, 2000; Piaget, 1970). This process of abstraction is grounded in a theory of how all people learn mathematics, which means that manipulatives could be useful in learning any mathematical idea, from arithmetic to algebra to analysis and beyond. Moreover, tools can be seen not only as useful for learning particular mathematical ideas, they can also be seen to mediate more powerful and authentic forms of mathematical activity (Dewey, 1938; Schoenfeld, 1989) than what learners often experience in school.

## Findings from Research and Evaluation

Research has identified the mathematics that people use as they engage in making (Pattison, Rubin, & Wright, 2016), and further research seeks to explore how new environments can be created to support such mathematical making (http://www.informalscience.org/cimble-conference-integrating-math-informal-building-learning-environments-math-making).

The Math in the Making workshop identified the following high-level themes and questions about the relationship between mathematics and making:

- There is a tension between the importance of highlighting the mathematics in making and tinkering experiences and concerns about compromising the essential qualities of making, i.e. an agentive, unconstrained, creative activity.
- The widespread negative perception of mathematics across society is a major challenge (and major reason) for this work.
- A broad conceptualization of mathematics â€“ including mathematical dispositions, habits of mind and identity – is necessary in order to see and support opportunities for math in making. The mathematics most people encounter in school provides only a limited view of mathematicsâ€™ potential richness.
- There is a difference between starting with a making experience and looking for the potential mathematical connections (â€śmath in makingâ€ť) and designing a making experience with a mathematical content in mind (â€śmaking in mathâ€ť). It is worth considering both approaches, as they have different affordances and challenges.
- Making is not a new activity. Historically, many communities have engaged in making activities out of necessity, not as an extra-curricular hobby. Adopting a â€śfunds of knowledgeâ€ť perspective requires that we ask about, and honor, the mathematical and making knowledge and skills that already exist in communities.
- Educators and parents need support for their roles in facilitating mathematical making and tinkering experiences. This may be especially true if they have narrow views of what constitutes mathematics that come from their school experiences.
- Mathematical reasoning may be easier to identify and build on in making activities that occur over an extended period of time and involve multiple iterations of a design than in short drop-in activities.

While this is a strand of research that essentially involves exploring how mathematics can be learned through making experiences (i.e., “learning through making”), other projects seek to explore how new tools can be seen to generate new opportunities for the having of mathematical experiences (i.e., “making for learning”). Research is currently underway (Greenstein & Olmanson, in review; Greenstein & Olmanson, 2016) that uses making as a context in which teachers or facilitators design and produce new tools and corresponding tasks that generate environments for mathematical thinking, reasoning, and learning. This research finds that these experiences have broadened teachersâ€™ conceptions of mathematical learning beyond the acquisition of target concepts to also include the engagement of their students in mathematical thinking and reasoning (e.g., quantitative, relational, geometric, and topological reasoning)â€”so that mathematical activity (e.g., “having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each otherâ€™s work” (Lockhart, 2002)) is privileged over the content. Teachers and facilitators designing for these experiences then begin to think not only think about how new tools can enhance the way we teach math, but also how new tools can engage students in rich and productive forms of mathematical activity.

With the use of 3D printing, making and mathematical thinking are in relayâ€”feeding back and feeding forward into and through each otherâ€”freeing teachers/facilitators to rapidly prototype and change their designs as their ideas evolve over time. Making and doing lead to new ideas and experiments in embodied (Johnson, 2007), networked (Latour, 1996), and tool-centric (Vygotsky, 1978) engagement that lead to powerful innovation in math learning, centered in the direct experience of mathematical activity.

## Directions for Future Research

What new tools and tasks does â€śmakingâ€ť make possible for the teaching and learning of mathematics?

What roles might new access to making spaces have for the preparation and professional development of mathematics teachers? How might these new roles contribute to a model of mathematics teachers as educational designers?

What other theory and design principles are relevant to mathematics education and could be brought to bear on teachersâ€™ educational design work?

## References

Dewey, J. (1938). Experience and education. New York: Kappa Delta Pi.

Greenstein, S. & Olmanson, J. (2016, June) Designing at the Intersection of Theory, Content, & Pedagogical Experience. Presented at the 6th Annual Emerging Learning Design Conference (ELD16), Montclair State University, Montclair, NJ.

Kamii, C., & Housman, L. B. (2000). Young children reinvent arithmetic: Implications of Piaget’s theory: Teachers College Press, Teachers College, Columbia University.

Post, T. (1981). The role of manipulative materials in the learning of mathematical concepts. In National Society for the Study of Education & National Council of Teachers of Mathematics (Eds.), Selected issues in mathematics education. Berkeley, CA: McCutchan Publishing Corporation.