**Using Digital Argumentation Tools to Support Mathematical Argumentation**

Mathematical argumentation is one of the primary skills that we hope to teach K-12 teachers. It therefore is one of the main components that I incorporate into my “math methods” courses, which are courses for preservice teachers that teach them how to teach mathematics.

**Goal of the Lesson**

The goal of this lesson is to teach preservice teachers (teachers-in-training) how to engage in structured mathematical arguments using the Toulmin Argument Pattern (TAP) and to give them opportunities to critique and improve each others’ arguments using LASAD, a web-based tool for collaborative argumentation.

**Background on LASAD**

LASAD is a microworld that students use (under the guidance of a teacher) to build structured arguments. According to Dragon, McLaren, Mavrikis, and Geraniou (2011), “LASAD is a collaborative, shared workspace containing a graphical argumentation environment and a chat tool. Students use this space to share ideas and organize their thoughts as they learn new concepts, and discuss or argue.” (p. 20). LASAD comes pre-programmed with a variety of argument structures and provides tools for students to collaborate and for the instructor to monitor student work. LASAD has traditionally been used in conjunction with other mathematical microworlds (Dragon, et al.) but can also be used as a standalone system for argumentation of all types, including mathematical argumentation.

**Background on TAP**

One such structure that is available to use in LASAD is the Toulmin Argument Pattern (TAP). The Toulmin Argument Pattern (TAP) proposes that a general argument can be written in a particular form, summarized by Inglis, Mejia-Ramos, and Simpson (2007) as “D, and since W (given B) we can Q conclude C, unless R” (p. 4) or in other words, if one wishes to prove a conclusion (C) given some data (D), one must connect it to the conclusion with a warrant (W) which appeals to a rule, appeals to a definition, or makes an analogy (p. 4). This warrant is supported by a backing (B) that provides additional evidence, the qualifier Q describes the degree of confidence (p.4) and the rebuttal (R) states the conditions in which it might not hold. In this lesson, students will be using TAP to build mathematical arguments.

Inglis et al. (2007) offered a case for using the full Tolumin Argument Pattern when working with mathematicians. TAP has been successful in working with preservice teachers as well; McDonald (2010) demonstrated that explicit instruction in argumentation, such as teaching preservice teachers about the TAP argument structure once appropriate scaffolding has been provided, improved both preservice teachers’ argumentation abilities as well as their understanding of the nature of the discipline.

**Description of Lesson**

In this lesson, the preservice teachers, after being introduced to the Toulmin Argument Pattern, will take existing informal mathematical arguments that they have made earlier in the course and use LASAD to construct a formal argument. When preservice teachers get stuck, LASAD (along with the teacher) will encourage students to work together to help them construct their arguments. The preservice teachers will then use LASAD to critique each others’ arguments, under close guidance from the teacher.

**Constructivist Rationale for Lesson**

**Developmental constructivism**

Microworlds such as LASAD differ from computer guided instruction in that the microworld itself does not teach or provide exercises for the students to engage in. Instead of instructional environments, they offer exploratory environments in which students (or in this case, preservice teachers) can explore either by themselves or under the guidance of a teacher (Greenstein, 2017). Harasmin (n.d.) argued that under developmental constructivism, “the learner is not an empty vessel to be filled with the knowledge of the teacher, but is an active organism creating meaning through contact and interaction with the external world” (p. 64). This applies as well to the simulated environment of the LASAD microworld, in which the preservice teachers practice making arguments and build their knowledge of what makes for a good mathematical argument.

**Social constructivism**

One important consideration when applying social constructivism to this lesson is how it engages preservice teachers within their zone of proximal development. The zone of proximal development is “determined through problems that children [or in this case, preservice teachers] cannot solve independently but only with assistance” (Vygotsky, 1930-1934/1978). One way in which the preservice teachers can move beyond their individual abilities is by the use of tools, such as the argumentation microworld.

In a social constructivist framework, tools (both intellectual and physical) represent collections of accumulated knowledge and experience, and thus are part of the social environment of the student. Using tools also helps students work within their zone of proximal development. Healy and Kynigos (2010) contend that “since the mediational means built in microworlds are designed with the learners’ probable zone of proximal development in mind, they intend to permit learners to do things with a computer that would be impossible without it (in a form analogous to interacting with a more knowledgeable other)” (p. 65). Thus interaction with the tool itself (in this case, LASAD) allows students to achieve more than they could without the tool, and the interaction with the tool itself (a priori of the ways in which the tool allows them to interact with others)

LASAD, in addition to providing interaction with a tool, is also designed to provide support within this zone of proximal development by allowing preservice teachers to discuss and critique each others’ arguments and to allow teachers to monitor the preservice teachers’ agreement or disagreement with each other. Once a disagreement is identified, the methods teacher can encourage them to resolve disagreements using the tools available within the argumentation environment (Dragon, et al., 2011). When the system alerts the methods teacher to a disagreement, the teacher can examine both the preservice teachers’ work as well as the discussions between the preservice teachers before deciding when and how to intervene. Tools help to provide a framework in which the preservice teachers can understand their own dialogue; as Chiappini, Pedemonte, & Robotti (2003) wrote: “the tools are seen as mediator of the individual’s actions and of the communication that takes place between participants in the activity; the teacher is seen as a co-participant in the activity who gives assistance to the students’ performance” (p. 218).

**References**

Chiappini, G., Pedemonte, B., & Robotti, E. (2003). Mathematical Teaching and Learning Environment Mediated by ICT. In C. Dowling & K.-W. Lai (Eds.), *Information and Communication Technology and the Teacher of the Future* (pp. 217–228). Springer US. https://doi.org/10.1007/978-0-387-35701-0_24

Dragon, T., McLaren, B. M., Mavrikis, M., & Geraniou, E. (2011). Scaffolding Collaborative Learning Opportunities: Integrating Microworld Use and Argumentation. In Advances in User Modeling (pp. 18–30). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28509-7_3

Greenstein, S. (2017). Designing a Microworld for Topological Equivalence. *Digital Experiences in Mathematics Education*, 1–19. https://doi.org/10.1007/s40751-017-0035-y

Harasim, L. (n.d). *Learning Theory and Online Technologies.*

Healy, L., & Kynigos, C. (2010). Charting the microworld territory over time: design and construction in mathematics education. *ZDM*, *42*(1), 63–76. https://doi.org/10.1007/s11858-009-0193-5

Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: the importance of qualification. *Educational Studies in Mathematics*, *66*(1), 3–21. https://doi.org/10.1007/s10649-006-9059-8

McDonald, C. V. (2010). The influence of explicit nature of science and argumentation instruction on preservice primary teachers’ views of nature of science. *Journal of Research in Science Teaching*, *47*(9), 1137–1164. https://doi.org/10.1002/tea.20377

Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes (M. Cole, V. John-Steiner, S. Scribner & E. Souberman., Eds.) (A. R. Luria, M. Lopez-Morillas & M. Cole [with J. V. Wertsch], Trans.) Cambridge, Mass.: Harvard University Press. (Original manuscripts [ca. 1930-1934])

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