de Freitas, E., & Sinclair, N. (2013). New materialist ontologies in mathematics education: The body in/of mathematics. Educational Studies in Mathematics, 83, 453-470. https://doi.org/10.1007/s10649-012-9465-z

Encountering

Through their work, Barad presented new possibilities for the understanding of boundaries, knowledge, interactions, and science. As well as many other scholars after them, Barad discussed how “agential intra-action” modify “boundaries and properties of the components of phenomena” (2007, p. 139) generating matter and meaning in an onto-epistem-ological manner.

This text will connect some of these ideas, such as agency, intra-acting, encountering, and fluidity, in order to clarify new approaches to mathematics’ teaching and learning. De Freitas & Sinclair (2013) stated that a “mathematical concept is always fluid and, in some important sense, unfinished” (p. 468). Mathematics is not linear, bounded, or finite; on the contrary, it is epistemologically and ontologically, temporally and physically, broadly and narrowly infinite.

We are “constantly encountering, engaging and indeed amalgamating with other objects” (de Freitas & Sinclair, 2013, p. 458), extending the limits of our bodies through these encounters. We are different pieces of one single puzzle: connecting, becoming, and composing; the world is the consequence as well as the cause of our—human and non-human—encounters: “We need to accept one plane of being, where difference is creative, positive and productive. … This might enable us to produce a different kind of knowing.” (Hultman & Lenz Taguchi, 2010, p.526).

Becoming and Performing

Hultman and Lenz Taguchi (2010) wrote about the girl and the sand—and how they were “becoming with”(p. 530) each other; de Freitas & Sinclair (2013) used Roth’s experiment with the student holding a cube to present the not-only-human encounter in which the cube was “becoming and performing” and should be recognized as a “material agent” (p. 457). Students connect to both internal and external matter in order to become a unique entity that creates—and is affected by—exclusive meanings. It does not matter with which matter (it could be a pencil, a computer, stones, or their own fingers) by which students extend their own matter. Rather, what is important is to make any matter matter: “Learners’ bodies are always in the process of becoming assemblages of diverse and dynamic materialities” (de Freitas & Sinclair, 2013, p. 454).

Barad (2007) spoke about Stephen Hawking, his wheelchair, and his computer as only one “phenomenon”: “Where does he stop? Where are his edges?” (p. 159). Positive¹ teachers encourage students to give up their ‘edges’ and unfinished concepts allow positive teachers to encourage students to believe they do not have edges. Education is about destroying edges instead of building walls. During Math classes, the teacher must break down these walls in order to let students take account of a “performative understanding of scientific practices”, internalizing the fact that “knowing does not come from standing at a distance and representing but rather from a direct material engagement with the world” (Barad, 2007, p. 49).

Meaning and Matter

Mathematical concepts have always been “considered immaterial and inert abstractions acquired after a series of ‘concrete’ activities”; however, this posits mathematics as a disembodied part of the world as it “fails to recognize the materiality and agency of [some] non-humans” matter (de Freitas & Sinclair, 2013, p. 468). Mathematics is not only a discourse, it is not a theory asking to be applied, and it is not even a tool or a ladder. Mathematics is matter and meaning: “Neither discursive practices nor material phenomena are ontologically or epistemologically prior. Neither can be explained in terms of the other. … Neither is articulated or articulable in the absence of the other; matter and meaning are mutually articulated” (de Freitas & Sinclair, 2013, p. 152).

Despite the anthropocentric attempts of privileging discourse as a way to transform matter, as well as the attempts to weaken matter’s agency in transforming discourse, Barad stated that “meaning and matter are bound and woven together in a kind of onto-epistem-ology” (p. 460). According to Hultman and Lenz Taguchi (2010), those attempts “reduce[d] our world to a social world, consisting only of humans and neglecting all other non-human forces that are at play” (p. 526). Actually, onto-epistem-ology spotlights “matter [as well as mathematics] [a]s produced and productive, generated and generative. Matter [/mathematics] is agentive, not a fixed essence or property of things” (Barad, 2007, p. 137).

Mathematics’ Agency

Mathematics is certainly a beautiful encounter between matter and meaning, learner and content, curiosity and creation, fluidity and novelty. Throughout my years working with math, I found out that learners (including myself) do not need someone’s help to understand theories or concepts; rather, they need support to get rid of educational traumas and feel free to re-engaging with mathematics. With this edgeless freedom, student and content get back to the process of becoming with each other, as we can see in the de Freitas and Sinclair (2013) number line’s example: “the body of the number line engages with the body of the student, and a new kind of body-assemblage comes into being”. In addition, “[it] becomes a highly animate concept made vibrant and creative through the indeterminacy buried in it” (p. 466).

According to Barad, agency is the “ongoing reconfigurings of the world”, and agency produce different phenomena through different intra-actions (Barad, 2007). That is, the teacher, the students, the classroom, the objects, the nature’s intra-actions are capable of (re)creating Math every day. Besides, what is eventually created in one classroom is surely different from what is created in any other school. Different fish imply a different river. A different river implies different fish.

Mathematics—as well as the whole universe—is an “agential intra-activity in its becoming” (Barad, 2007, p. 141). Its concepts are not rigid, definitive, or inert. De Freitas and Sinclair (2013) argued that mathematical concepts change over time not only at the epistemological level, but also at the ontological one. Moreover, they used Châtelet and Leibniz’s ideas to speak about flexibility and relativeness, which reminded me of an important part of the History of Mathematics.

Mathematicians, over the past three millennia, have studied possible ways of rethinking its epistemology by using multiple interpretations to better understand its possibilities. Euclid was a Greek mathematician who, more than two thousand years ago, produced 13 books about Geometry. Nowadays, the Euclidean Geometry is still the core of any geometry content discussed in every elementary and secondary classroom in the world. Right in his first book, Euclid proposed five postulates about straight lines, angles, and intersections. However, until the 18th century, great mathematicians tried to formally prove one of the Euclid’s assumptions. That is, they tried to prove Euclid was wrong using theoretical Math concepts. Fortunately, two mathematicians—Lobachevsky and Bolyai—looked at this situation from another angle: instead of trying to demonstrate that postulate, they used an ontological version of Euclid’s ideas to create a new one. This was the beginning of the Non-Euclidean Geometry, which has many practical applications today, such as airplane routes design and astronomy studies. Einstein’s general theory of relativity is also a good example of the Non-Euclidean Geometry influence.

This story represents that, despite the common understanding of Mathematics as inert, immaterial, and finished, Lobachevsky and Bolyai believed that Geometry was still open and flexible. Thus, they were much more successful than other mathematicians because they did not studied Geometry only through its epistemological side; they rethought Geometry looking for its ontological open doors, though.

Rethinking the Nature of Learning

The mobility of concepts grants creative power (de Freitas & Sinclair, 2013) indeed, which leads us to realize that we should switch from usual understanding of curriculum as a final rigid desirable product to a vibrant and animate “rhizomorphus curriculum of becoming” (Eaton & Hendry, 2019, p. 21). Through the question “how might new materialisms allow us to rethink learning as an indeterminate act of assembling various kinds of agencies rather than a trajectory that ends in the acquiring of fixed objects of knowledge?” (p. 464), Eaton and Hendry also tried to remind us that a child that enters the preschool should not have a predetermined educational pathway; actually, children should “emerge through … their entangled intra-actions with everything else” (Hultman & Lenz Taguchi, 2010, p. 531).

As educational scholars, de Freitas & Sinclair utilized Barad and Deleuze’s ideas to affirm that “novelty, genesis and creativity (rather than conditions of possibility) are fundamental concepts in a theory of virtuality”. They extended their argument by emphasizing that virtuality represents mobility and that “matter is animated by the virtual” (2013, p. 463). As an example of this, they brought the number line to the discussion, as it is a crucial point within any Mathematics course when it comes to perceiving a reality we cannot see. The notion of infinity is not easy for students to understand; even more when you need to compare different kinds of infinity. Briefly, we can speak about the countable infinity (such as the Natural numbers 0, 1, 2, 3…) and we can speak about the uncountable infinity as well (such as the number line), in which you can find another infinite quantity of numbers between any two numbers. De Freitas and Sinclair (2013) appealed to our imagination by using virtuality to make this matter matter to students:

The virtual is a kind of intensity that deforms the linearity of extension. The virtual invites in(ter)vention because it is precisely what makes extension plastic and elastic through its intensity. … Students can carve out the virtual real numbers embedded between whole numbers by grabbing and stretching the number line so that it brings forth an infinitude of numbers that were imperceptible a moment earlier. (pp. 465-466)

To wrap up this article, I will adapt one excerpt from Hultman and Lenz Taguchi (2010). Although words that kept reference to ‘researching’ were removed in order to a build a ‘teaching’ approach, the ideas of intra-actions, becoming and transforming, as well as Barad’s (2007) “appreciation of the intertwining of ethics, knowing, and being” (p. 185)—the ethico-onto-epistem-ology—were preserved indeed:

What we do as [teachers] intervenes with the world and creates new possibilities but also evokes responsibilities. If we think in this way, we might not just live differently. … but do our [teaching] and [learning] differently, in order to perhaps make it possible for others (humans and non-humans) to live differently in realities yet to come. (p. 540)

¹The idea of ‘positive’ is applied here as not needing to define things/people by comparing opposite meanings by denying one of them in order to produce the other. (Hultman & Lenz Taguchi, 2010)

References

Barad, K. (2007). Meeting the universe halfway: Quantum physics and the entanglement of matter and meaning. Duke University Press.

de Freitas, E., & Sinclair, N. (2013). New materialist ontologies in mathematics education: The body in/of mathematics. Educational Studies in Mathematics, 83, 453-470. https://doi.org/10.1007/s10649-012-9465-z

Eaton, P. W., & Hendry, P. M. (2019). Mapping curricular assemblages. Teachers College Record, #22804. https://www.tcrecord.org/Content.asp?ContentId=22804

Hultman, K., & Lenz Taguchi, H. (2010). Challenging anthropocentric analysis of visual data: A relational materialist methodological approach to educational research. International Journal of Qualitative Studies in Education, 23(5), 525-542. https://doi.org/10.1080/09518398.2010.500628