Part 3: The Case for Changing How We Teach Math
Breaking the Inertia of Procedural Thinking: The Case for Principle-Based Calculus
In 2021, David Weisbart, Ph.D, associate professor of teaching in mathematics at UC Riverside, co-led a team that won a California Education Learning Lab grant to reconceptualize the role of and approach to teaching introductory calculus. Over the last few years, Weisbart and Dr. Yat Sun Poon, a fellow PI and professor of mathematics at UCR, and community college faculty collaborators, Bryan Carrillo (Saddleback College) and Dylan Noack (Yuba College), remade the introductory calculus series to be principle-based, which has eliminated the need for students to take a one-quarter precalculus prerequisite course. Here’s what they did and why:
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Guest Commentary by Dr. David Weisbart:
Procedural thinking remains the dominant default in mathematics education, particularly in freshman calculus courses. Students study algebraic manipulations, differentiation rules, and integration techniques as sequences of steps to memorize rather than as expressions of deeper reasoning. As a result, most students equate success in learning with the correct execution of a list of instructions rather than with understanding meaning, enriching intuition, or developing productive experimentation strategies.
Likewise, many courses, even at the upper-division level, fail to convey the rich, interconnected structure of mathematics. Instead, they butcher the subject, fragmenting it into isolated topics, each taught with as little dependence on broader knowledge as possible in an attempt to maximize short-term learning outcomes and inflate scores on exams with minimal subject coverage. This fragmentation sacrifices depth for convenience and obscures the inherent unity of the discipline. The interconnectedness of mathematics, however, is not an obstacle to be managed; it is the primary opportunity through which to break down barriers to entry, reinforce learning, and develop more efficient and meaningful teaching strategies.
Unfortunately, the procedural approach to mathematics education has created systemic educational inertia: students expect formulas and templates, instructors cater to these expectations by emphasizing procedural proficiency, and institutions reinforce the cycle by evaluating instructors primarily through student satisfaction metrics. While once passable, this model even thirty years ago was already failing to produce true mathematical thinkers—though it still equipped students with some broadly useful skills. Today, the rise of powerful technologies has rendered these limited skills largely obsolete. Continuing to rely on outdated models of procedural instruction is no longer merely inadequate; it is actively counterproductive.
The Persistence and Cost of Procedural Learning
Many students enter calculus expecting to memorize formulas and apply techniques, as they did in earlier mathematics courses. When faced with open-ended reasoning tasks, they panic or resist. This expectation pressures instructors, who in turn default to teaching procedures rather than fostering deeper understanding: it allows for straightforward assessment and grading but does little to cultivate mathematical intuition. Traditional grading methods with automated multiple choice questions reinforce this pattern, favoring speed and mechanical accuracy over conceptual reasoning.
But procedural learning without structural understanding produces serious educational deficiencies. Students cannot generalize methods to unfamiliar problems or adapt their knowledge flexibly. Many lose confidence in their ability to "do mathematics" when rote techniques fail them. Critical mathematical habits—such as recognizing functional structures and reasoning about invariants—are never developed. Higher-level mathematical reasoning, which depends on abstraction and generalization, becomes inaccessible to students trained only in rote procedures. Although a small subset of students eventually develop deeper understanding, this development occurs osmotically—dependent on external factors that are exclusive rather than inclusive—and fosters social divides and artificial barriers within mathematical education.
One troubling symptom of these entrenched practices is the instructor pushback against initiatives designed to eliminate ineffective courses, such as precalculus. Despite clear evidence of their shortcomings, resistance persists, driven by attachment to traditional models and by institutional structures that incentivize maintaining the status quo.
The Development of a Principle-Based Calculus Program
Breaking the inertia of procedural thinking requires a fundamental reorientation of mathematics instruction from rote execution to structural reasoning, from procedural-based learning to principle-based learning. As a collaboration funded by the California Education Learning Lab between the University of California, Riverside, and our California Community College partners at Saddleback College and Yuba College, we have developed a new gateway mathematics course that develops the idea of principle-based instruction and takes the perspective that mathematics education should create real opportunities for students who engage with curiosity, open-mindedness, and effort. Principle-based courses develop transferable reasoning skills and build the capacity for deep mathematical thinking directly, rather than osmotically. Principle-based courses do not eschew basic skills in favor of conceptual learning; rather, they use basic skills as a litmus test for deeper understanding, cultivating them through a natural process of creative experimentation that reinforces and reconstructs prerequisite knowledge. Here are our core strategies for making this shift to a principle-based approach, where we believe nothing is lost, but much is gained:
Our strategy: Restructure rather than review.
Instructors often think they must review topics when introducing new ideas. Topic-based courses artificially separate concepts, creating the need for review. By structuring the curriculum around core reasoning principles, we naturally build in looking backward and forward throughout the curriculum, so that reinforcement happens within the learning process itself.
Our strategy: Unify rather than balance.
Too often, instructors think a course must balance theoretical rigor and concrete application as if they are competing forces. Our approach has been to develop language as a framework for reasoning, teaching students to use physical intuition to develop mathematical concepts and to use mathematical structure to deepen physical intuition. In this way, application and rigorous development are not at odds—they are one and the same.
Our strategy: Engage with ideas, not prerequisites.
A course should not serve merely as a stepping stone to another course, as oftentimes prerequisites do. This mindset undermines the depth of learning possible in the present and reduces education to a series of obstacles rather than a meaningful pursuit. Every idea in mathematics is rich and worth exploring on its own terms, and valuable research questions exist at every level of the discipline.
Our strategy: Redesign the learning experience.
Our course emphasizes spatial reasoning, reading and writing across disciplines, transferable skills, responsible use of technology, and the belief that each student should experience learning that adapts to their interests. Already in its inaugural year, Principles of Calculus increased student retention in the subsequent math course by a factor of 2.5 and in the subsequent physics course by a factor of 1.7. These results reflect more than academic success—they demonstrate the power of a redesigned learning experience.
The Principles of Calculus program builds an equitable pathway to success by teaching reasoning processes directly rather than indirectly, reinforcing core ideas naturally, and making entry possible from any level of preparation—all while enhancing rather than diluting expectations for student achievement. Instead of requiring students to "catch up," Principles of Calculus redesigns the starting point and the flow of ideas to increase accessibility intrinsically. The barrier is no longer preparation; it is only a student's willingness to engage, and instructors’ willingness to engage students in a new way.
Learning Lab note: Poon and Weisbart are actively developing ways to share the Principles of Calculus curriculum and approach with interested faculty in Fall 2025. Fill out this interest form to learn more, or visit https://cllmath.ucr.edu/.
This is part of our ongoing series about Math Instruction. Parts 1 and 2 can be found at the following links.
Read Part 1: A Brief History of Math Instruction and How AI Could Change Everything.
Read Part 2: The Case for Changing the Math We Teach.
If you’d like to submit a commentary on this or other topics related to higher ed teaching and learning, please email info@calearninglab.org.