Undergraduate and graduate student presentations from the Department of Mathematics, 2021 Online University Research Symposium, Illinois State University

The Effect Of Research Experiences On Teachers' Belifes And Instruction
Seyedehkhadijeh Azimi Asmaroud
Discovery learning is a common teaching method used in schools today (Kistian et al., 2017), and research has found many benefits to using this method in teaching mathematics (Herdiana et al., 2017). The question then becomes, how do we prepare teachers to use discovery learning? One potential avenue is to provide them opportunities to experience mathematical discovery themselves. The Conference Board of Mathematical Sciences stated that “teachers need opportunities for the full range of mathematical experience themselves: struggling with hard problems, discovering their own solutions, reasoning mathematically, modeling with mathematics, and developing mathematical habits of mind” (p. 54). Research has shown professional development influences teachers’ beliefs (Polly et al., 2013), which in turn influences their instruction in the mathematics classroom (Hart, 2002). In fact, “significant transformations of teaching practice are unlikely to occur if related beliefs and theories about teaching and learning do not change” (Hawley & Valli, 2000, p. 4). This study analyzed the influence of a research experience for teachers on their beliefs and instruction. A qualitative approach was used to analyze the responses of 11 participants. This analysis revealed changes in the teachers’ beliefs about teaching and doing mathematics but did not provide sufficient data to connect particular beliefs and subsequent changes in practice. The findings showed that this program changed teachers’ beliefs about students' ability to do highlevel mathematical tasks and their students' expectations. Also, participants stated that they used more explorations and discovery learning in their classroom after participation in this program.

Squigonometry
Andrew Hatfield, Riley Klette, Christopher Moore, and Beth Warden
Trigonometry is the study of circular functions  functions defined on the unit circle where distances are measured with respect to the Euclidean norm. In our research, we develop a parallel theory of trigonometric and inverse trigonometric functions for the pnorm. This is called squigonometry because the resulting functions are defined on a squircle. This approach leads to new transcendental periods, formulas, and identities. It also extends to exponential, hyperbolic, and logarithmic functions in the pnorm.

Examining Middle School Students' Methods Of Justification
Leslie Reyes Hernandez
Researching students’ thinking is imperative to improving the education system throughout the world. From extensive research, it is noted that students are unaccustomed and struggle with providing valid mathematical justifications (e.g. Inglis & Alcock 2012). The National Council of Teachers of Mathematics (NCTM, 2000) and Common Core State Standards of Mathematics (CCSSM, 2010) suggest that students should have several opportunities to construct mathematical arguments across all grade levels. To take a closer look at this educational phenomenon, we prompt fifth to eighthgrade students with nine mathematical tasks. Within our research, we focus on tasks based on number properties, algebraic thinking, and geometric thinking. Furthermore, our research examines students’ methods of justifications as well as the mathematical quality of the justifications. Overall, the research demonstrates that most students’ justifications are not mathematically complete. This research is of high value to educators, parents, school administrators, and students throughout the world as it provides a more beneficial method of student learning. Educational research is highly important as the future of the world lies within our classrooms today.