A Seminar in the Foundations of Mathematical Proof and Reasoning

Bridging the Gap from Computation to Proof

The Foundations of Mathematical Proof and Reasoning Seminar by Arrow15 is a program designed to support community college students in making the challenging transition from computation-based to proof-based mathematics. The seminar offers a transformative experience that sets the stage for future success in mathematics and beyond.

Why this Seminar?

Mathematics is more than just solving problems; it’s about understanding how and why things work. Early on, many students encounter computation-based math, where math is used as a tool for problem-solving and computation. While essential, this stage often focuses on executing steps and formulas without necessarily understanding the deeper principles that underlie them. As students progress in their studies, they transition to proof-based math, a shift from applying math as a tool to studying it as an object of inquiry. This shift involves developing a deeper understanding of abstract concepts, learning how to construct rigorous mathematical proofs, and cultivating the reasoning skills needed for upper-division coursework.

For many students, this transition is not easy. Research has shown that students often struggle as they move from computation-based coursework to the more abstract, proof-oriented coursework that dominates upper-division mathematics. Specifically, students fail to recognize the key idea of a proof (Raman, 2002), exhibit difficulty in detecting logical flaws (Weber 2010), and, more generally, unable to judge whether a proof is valid or not (Hodds et al., 2014). Indeed, K-14 math education often does not adequately prepare students to understand and construct proofs (Stylianou et al., 2015). The lack of sufficient preparation for proof-based learning can lead to frustration, disengagement, and even a change of major. This phenomenon highlights the importance of providing targeted support for students as they navigate this shift.

The Foundations of Mathematical Proof and Reasoning Seminar by Arrow15 is specifically designed to bridge this gap by providing an immersive environment where students can develop the skills and confidence necessary for success in upper-division mathematics courses.

Inquiry-Based Learning (IBL): A More Effective Approach

Our seminar employs inquiry-based learning (IBL), a proven educational approach that emphasizes student-driven exploration and discovery. In contrast to traditional lecture-based teaching, IBL encourages students to actively engage with mathematical ideas, ask questions, and work through problems with their peers. This approach is not only more engaging but also leads to better understanding and retention of material.

Studies have shown that IBL improves mathematical reasoning and conceptual comprehension when compared to traditional lecture-based instruction (Hassi et al., 2010; Laursen et al., 2014). IBL promotes more flexible problem-solving skills (Lewis and Estis, 2020), and has shown to lead to comparable or improved procedural and computational performance relative to lecture-based methods (Chappell and Killpatrick, 2003; Khasawneh et al., 2023; Lewis and Estis, 2020).

Rigor without Exams

The seminar provides a rigorous learning experience without the pressure of grades, homework, or exams. This absence of traditional forms of evaluation allows students to deeply understand the material, engage in intellectual risk-taking, and explore new mathematical ideas without fear of failure. Students are held to high standards through the quality of the work they do and the rigor of the mathematical concepts they explore, not through grades.

Research shows that test anxiety lowers mathematical performance (Bellinger et al., 2015; Jamieson et al., 2020). Moreover, grading reduces students’ motivation and diminishes the quality of their thinking (Kohn, 2011).

In-Person Learning

The seminar is conducted in person, significantly enhancing the learning experience. In a small, face-to-face setting, students benefit from real-time feedback and active collaboration with both instructors and peers. This personalized, interactive approach allows students to ask questions, engage in discussions, and solve problems together in an environment that promotes deep learning and understanding.

No Cost

Our seminar is completely free for all participants. We rely on the generous support of donors who believe in the value of accessible, high-quality math education. This funding covers all aspects of the seminar, including tuition for each student. There is no catch. Our mission is to prepare students for a successful transition to proof-based-mathematics.

Certificate of Completion

Upon completing the seminar, students receive a certificate of completion that can be used in their transfer applications or displayed on their professional profiles, such as LinkedIn. This certificate serves as a testament to the student’s commitment to mastering the skills necessary for success in higher-level mathematics.

In Conclusion

Educators and advisors can confidently refer students to this program, knowing it is built on solid educational practices and supported by research-backed methodologies.

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References

  • Bellinger, D.B., Decaro, M., & Ralston, P.A. (2015). Mindfulness, anxiety, and high-stakes mathematics performance in the laboratory and classroom. Consciousness and Cognition, 37, 123-132. https://doi.org/10.1016/j.concog.2015.09.001
  • Chappell, K.K., & Killpatrick, K. (2003). EFFECTS OF CONCEPT-BASED INSTRUCTION ON STUDENTS’ CONCEPTUAL UNDERSTANDING AND PROCEDURAL KNOWLEDGE OF CALCULUS. PRIMUS, 13, 17 – 37. https://doi.org/10.1080/10511970308984043
  • Hassi, M., Kogan, M., & Laursen, S.L. (2010). Student Outcomes from Inquiry-Based College Mathematics Courses: Benefits of IBL for Students from Under-Served Groups Contributed Research Report.
  • Hodds, M., Alcock, L., & Inglis, M. (2014). Self-explanation training improves proof comprehension. Journal for Research in Mathematics Education, 45, 62-101. https://doi.org/10.5951/JRESEMATHEDUC.45.1.0062
  • Jamieson, J.P., Black, A.E., Pelaia, L.E., & Reis, H.T. (2020). The impact of mathematics anxiety on stress appraisals, neuroendocrine responses, and academic performance in a community college sample. Journal of Educational Psychology. https://doi.org/10.1037/edu0000636
  • Khasawneh, E., Hodge-Zickerman, A., York, C. S., Smith, T. J., & Mayall, H. (2023). Examining the effect of inquiry-based learning versus traditional lecture-based learning on students’ achievement in college algebra. International Electronic Journal of Mathematics Education, 18(1), em0724. https://doi.org/10.29333/iejme/12715
  • Kohn, A. (2011, November). The case against grades. Educational Leadership. Retrieved from https://www.alfiekohn.org/article/case-grades/
  • Laursen, S.L., Hassi, M., Kogan, M., & Weston, T.J. (2014). Benefits for Women and Men of Inquiry-Based Learning in College Mathematics: A Multi-Institution Study. Journal for Research in Mathematics Education, 45, 406-418. https://doi.org/10.5951/JRESEMATHEDUC.45.4.0406
  • Lewis, D., & Estis, J. (2020). Improving Mathematics Content Mastery and Enhancing Flexible Problem Solving through Team-Based Inquiry Learning. Teaching & Learning Inquiry, 8(2). https://doi.org/10.20343/teachlearninqu.8.2.11
  • Raman, M.J. (2002). Proof and Justification in Collegiate Calculus.
  • Stylianides, G.J., & Stylianides, A.J. (2009). Facilitating the Transition from Empirical Arguments to Proof. Journal for Research in Mathematics Education, 40, 314-352.
  • Weber, K. (2010). Mathematics Majors’ Perceptions of Conviction, Validity, and Proof. Mathematical Thinking and Learning, 12, 306 – 336. https://doi.org/10.1080/10986065.2010.495468
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