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, are 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.

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).

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).

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.

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.

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.

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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The seminar offers valuable benefits that will help you in your academic journey:

• Develop key abstract reasoning skills necessary for success in upper-division math courses.
• Engage in mathematics in a collaborative, supportive setting that encourages deep thinking and problem-solving.
• Earn a certificate of completion, which can be used in transfer applications or displayed on professional profiles, such as LinkedIn.
• Best of all, this seminar is completely tuition-free, with no exams, grades, or homework.

But, wait, there is more!

De Anza students will have the option of earning college credit by enrolling in Special Projects in Mathematics (MATH 77) and conducting their project in the context of the seminar. This additional commitment is totally optional. Note that while enrolling in the Arrow15 seminar is free, enrolling in MATH 77 isn't.

Who can apply?

There are no formal prerequisites, but the seminar is designed for community college students who are serious about transferring as math or STEM majors within the next year or two. In most cases, you should be on track to complete the full calculus sequence, linear algebra, and ordinary differential equations by the time you transfer. While the seminar is intended for students planning to pursue a math-intensive degree, it welcomes students at all confidence levels, even if you're still building your comfort with the material.

When and where?

The seminar is taking place from Wednesday, July 1, 2026 through Tuesday, August 11, 2026. We meet on Tuesday and Wednesday afternoons (3pm to 4:30pm) in person at De Anza College.

This cohort is co-taught by Prof. Karina Cho and Prof. Rick Taylor.