The following statement expresses my teaching philosophy and the ideals to which I aspire.  It is not a judgment of or challenge to the values or ideals of others.  Academic freedom is essential to the vitality of any society in which I would wish to live, and I deeply value the diversity of approaches it fosters.

Here is an opinion piece that I recently wrote for the California Learning Lab: Opinion Piece

Curriculum designers, course coordinators, and classroom teachers face recurring questions:

1. What should I teach?

2. How should I teach?

3. How should I interact with my students?

4. How should I grow as a teacher?

There are no universal answers. Effective teaching depends on context, evolving factors, and a coherent narrative that aligns with the educator’s personality.  To address these complexities, I aim to follow a conceptual framework that identifies key balances that effective teaching requires.  My philosophy centers on the focus of instruction, the roles of the instructor, and fostering a mindset of mutual growth.

Focus of Instruction

Mathematics is too precise to express the qualitative, nuanced description of human experience that natural language captures.  Natural language lacks both the precision for mathematical deduction and the information density to efficiently express the full scope of mathematical meaning.  Mathematicians navigate a middle ground between formal reasoning and human intuition, where intuition guides the development of formal concepts, and formalism deepens human intuition.  My focus in designing a curriculum and in teaching classes is to equip students to engage with mathematics at both the intuitive level and at the level of precise logical deduction in order to encourage students to develop the internal recursive process of mathematical conceptualization. My courses encourage students to experiment in order to discover and develop mathematical ideas.

Role of the Instructor

The education process is a path of self-discovery.  As an instructor and mentor, I aim to support students' autonomy, engage them in developing intrinsic motivation, and provide opportunities for them to discover and realize their creative potential.  As a course coordinator, I aim to help instructors excel by providing them with the resources and advice they need and request, and encouraging them to appreciate the impact of their contribution.  As a curriculum developer, I aim to create course materials that foster my own growth as a mathematician, inspire instructors to re-explore their own understanding, and encourage students to pursue further learning.

The consequence of initially forgetting how to approach a specific problem is undoubtedly very different for a freshman calculus student and an experienced mathematician.  The student will typically need to model their work on a similar example or work through the problem with a tutor.  In contrast, the experienced mathematician will typically think about the problem and reimagine a creative solution.  The difference between the ability of the student and the ability of the experienced mathematician is qualitative rather than quantitative, having little to do with the number of facts that the student knows, and far more with the quality and depth of the understanding.  Recognizing this is the first step in redesigning and teaching courses that offer more enriching learning experiences for students.

Principles and Concepts

It is useful to distinguish between the notion of concept and principle.  Concepts are abstractions, or organizational tools, that help people to sort out and organize their experiences.  They provide a framework for understanding the connection between many specific examples, or course topics.  

Principles are abstractions that help in sorting out and arranging different aspects of experiences with concepts.  In a broad sense, principles are concepts, but it is useful to refine both meanings.  Concepts are more concrete and relate more directly to concrete examples, or topics, in a course of study.  Concepts emerge from the interplay of principles of reasoning in a given language framework.  In this way, principles may be thought of as meta-concepts, they are concepts about reasoning that are useful for developing concepts that organize more specific experiences.

Modes of Engagement

There is some utility in classifying knowledge as being procedural, conceptual, or principled.  Students often operate in a hieroglyphic mode of thinking, focusing on symbol and pattern recognition rather than deeper understanding.  Their knowledge is procedural.  This mode, while helpful for rote memorization and basic procedural tasks, is insufficient for success in courses involving mathematics or any field requiring problem-solving or independent thinking. 

A conceptual mode of thinking focuses on using knowledge about concepts to not only perform procedures but also understand the meaning of the procedures, justify them, and use inferential reasoning to extend the procedures in novel ways.  Although it is admirable to train students with the goal that they will operate in a conceptual mode, such a mode still does not reflect a deep enough form of understanding for a university-level education.  There are countless procedures in mathematics, and understanding the concepts behind these procedures is insufficient for transferring knowledge between different courses in a university curriculum. 

A deep mode involves the understanding of a handful of basic principles that are as relevant to a freshman calculus student as they are to a professional mathematician.  The concepts in a course of study emerge from the interaction of the principles of reasoning together with ample experience with basic examples. 

Principle-Based Curricula and Teaching

Typical topic-based courses focus on developing and reinforcing students’ procedural knowledge and conceptual understanding of a linear collection of related topics.  The goal is to ensure student mastery of various topics and skills important for future courses.  Testing student learning outcomes is a necessity that forces instructors of such courses to focus on, review, and reinforce material that is easy or practical to test.  For this reason, among others, although topic-based courses may attempt to engage students in the conceptual mode, they struggle to move students beyond the hieroglyphic mode.  Example problems are rather limited because there is limited scope for creative approaches to solving problems since taught techniques are rather fragile and narrow in scope of application.

A principle-based course directly engages students in a deep mode of learning so that teaching students how to reason becomes intentional rather than osmotic.  The consequence is that students develop a framework for reasoning and the ability to utilize this framework in both routine and novel ways. There are still topics, but the presentation may be non-linear.  Topics become examples of the interaction between principles rather than the focus of the course.  A principle-based course efficiently scaffolds ideas to provide students with a powerful organizational structure.  It also supports students in developing concrete ways to leverage their innate spatial reasoning ability and to connect meaning to symbolic manipulation. 

The goal of a principle-based course is:

Students become problem solvers by learning to reason using principles in the framework of a sufficiently expressive language.

Challenges and Advantages

A principle-based course naturally provides an efficient scaffolding of ideas and offers students a powerful organizational structure.  It fosters  independent growth and supports student autonomy by helping them develop the tools that they need to become independent problem solvers.  A principle-based course de-emphasizes prerequisite knowledge, is flexible and easily tailored to individual students’ needs, promotes a great deal of learning reinforcement, and offers students a memorable and profound learning opportunity.  By focusing on principles rather than isolated topics, such courses can provide a more meaningful and transferable understanding of mathematics. 

The challenge of constructing a principle-based course is that the course designer must think carefully about the required language, and the instructor must reinforce language acquisition. I focus on teaching students to leverage spatial thinking as a cognitive tool to support their mathematical language acquisition. I also encourage students to develop their spatial reasoning to communicate and consolidate mathematical meaning. Direct engagement with spatial thinking and the development of spatial thinking is critical for the successful teaching of a principle-based course.

Points of Balance

A critical task of the instructor is identifying points of balance between teaching principles versus procedures, rote practice versus deep exploration, and black box learning versus ground-up learning.

Principles versus procedures.  Although it is vital for students to engage with mathematics by developing their reasoning and enriching their language skills, it is still useful for students to be fluent with procedures.  In some courses, it may even be preferable for students to develop procedural knowledge rather than deeper thinking.  Educational institutions offer a wide range of classes and courses of study, from research to vocational, and from elementary to advanced.  In any class, there will be certain parts that require different modes of engagement. The instructor must find a balance between teaching procedure and teaching reasoning.  The point of balance depends on the purpose of the course, the goals of the students, programmatic requirements, and the current capacity of the students to operate in the different modes.

Rote practice versus deep exploration.  Although deep explorations are critical for developing reasoning, the inability to perform rote calculations and recall basic facts, even without deeper understanding, can be an insurmountable obstacle.  A certain amount of rote practice is essential.  The instructor must find a balance between rote practice and deep exploration.  The point of balance depends on factors that include students' computational facility, capacity for focused engagement, working memory, and tolerance for failure.

Black box learning versus ground-up learning.  Balancing black box learning and ground-up learning in a principle-based course presents a tremendous challenge to instructors.  It is tempting to develop all content from the ground up.  The advantage is that this approach allows students to deeply explore the power of reasoning with principles.  The disadvantage is that it takes a long time, perhaps even too much time, to present the basic ideas that students need for subsequent courses.  Following a ground up-approach also prevents students from developing some necessary learning skills.  It is important for students to learn how to quickly develop a basic understanding of useful tools, even when the tools may beyond their ability to immediately develop.  It is critical that instructors help student to develop both types of learning skills, but with some appropriate balance between them. Identifying an appropriate balance is complicated, and presents a real challenge.

The instructor—whether textbook author, curriculum developer, course designer, or classroom teacher—should promote student learning of mathematics by simultaneously taking on the roles of language teacher, fellow traveler, and pathfinder. The language teacher helps students develop a necessary linguistic framework. The fellow traveller inspires students with a mutual experience of learning.  The pathfinder helps students find meaning in the context of underlying principles and a natural flow of ideas. The classroom teacher must additionally provide an environment that promotes student learning and personal growth, thereby taking on a fourth role: that of a coach.

The Language Teacher

Communicating complicated ideas in mathematics requires the use of human language. Every language contains primitive words—words that speakers mutually understand through shared experience.  Lengthy explanations are difficult to parse, while concise explanations require complex primitive words that are intrinsically difficult to understand due to their high information content.  The mathematics instructor is, in large part, a language teacher who must construct an immersive language learning environment and provide novel experiences centered around well-chosen explorations.

The intrinsic complexity of a concept and the necessity of concision together determine the required information content of the primitive words needed to describe the concept.  For example, one can succinctly define a smooth manifold as a locally Euclidean, second-countable, Hausdorff topological space with a maximal smooth atlas. While this is a simple definition for students who understand the terminology, it is useless for those unfamiliar with topological spaces.  Such students have not yet developed a language rich enough to encode the meaning of "smooth manifold" in a way that is both concise and precise.

Primitive words are impossible to understand without contextual examples and sufficient motivation.  The instructor must introduce complicated definitions by first discussing only those aspects or instances of the definitions that the students are currently prepared to grasp.  Preparing students for a complicated definition involves developing their understanding of the various elements of the definition well enough so that the words describing these elements become part of the students' experience.  A principal challenge in teaching a concept lies in finding a balance between simplicity and concision—between the complexity of the primitive words and the length of the expression required to describe the concept.

The Fellow Traveller

Teaching involves the mutual growth of both students and instructors.  Regardless of its perceived level, audience, or place in the curriculum, every course can and should be interesting in its own right.  Learning should be a shared experience between the student and the teacher, whether in the context of a remedial course, or in the context of a graduate research seminar.  A mindset of mutual growth is critical for both teacher and student.

Teaching a course gives the teacher the opportunity to practice basic skills.  Creating course materials gives an instructor the opportunity to revisit interesting questions and further explore their own understanding.  Constructing a new curriculum gives an author the opportunity to reimagine foundations.  Each activity is a meaningful growth opportunity that challenges a mathematician to grow as a scholar and a researcher.  The excitement that emerges from interactions that involve mutual growth offer students more meaningful learning opportunities than what would be possible in any unidirectional exchange.

It is very easy for instructors to become so wrapped up in their own learning that they forget about their responsibilities to their students.   It is critical for instructors to find a balance between their personal explorations and the excitement that such explorations can generate, and their responsibility to gauge their students' current ability to engage with these explorations. 

The Pathfinder

A handwritten proof of the rank-nullity theorem, a fundamental theorem in a linear algebra course, requires less than one letter-sized page.  Although undergraduate linear algebra students typically find it difficult to recall this proof after studying it, they can easily provide a detailed description of the last few episodes of their favorite television show.  This disparity in recall ability arises because humans relate to stories, but not to strings of symbols or collections of disparate facts.  The storyteller must weave the information in a course into a compelling narrative, so that lines of reasoning become meaningful.

While it may be tempting to construct a historical narrative when teaching mathematics, such an approach is both challenging and limiting.  The history of our subject is extremely complicated and requires a great deal of mathematical maturity to appreciate. Furthermore, learning a subject from a historical perspective requires understanding many aspects that are no longer relevant.  Therefore, it is incumbent upon the instructor to carefully think through the material before teaching the course and construct a flow of ideas—a progression of concepts guided by meaningful questions that tie together the course of study.

Identifying meaning is a critical part of the teaching and learning process and a principal task of the pathfinder.  For example, a matrix containing 3-tuples (a, b, c) arranged in a 3,000 by 4,000 grid, where a, b, and c are integral values between 0 and 255, presents an overwhelming amount of information for a human to process.  It is meaningless without context, as are nearly any collection of formal rules for manipulating its values.  However, the information and rules for manipulations gain meaning on visualizing the matrix as a 6"x8" color photograph.  Modifying the values might correspond to de-blurring the image, sharpening it, enhancing boundaries, compressing information, or performing other operations.  The pathfinder’s role is to introduce concepts in a way that students engage with the photograph rather than just manipulate 12 million data points.

A principal challenge that faces the pathfinder is that the subject has many interpretations.  Presenting one perspective may hide or disguise others, and so expressing a viewpoint may be both helpful and restricting, enriching and yet opinionated.  The instructor must find some balance between presenting a perspective and encouraging students to develop their own perspective that may be more useful for their own goals.

The Coach

Effective teaching depends on effective listening, creating opportunities for student engagement, motivating students, and counseling them when appropriate.  Students often develop misconceptions due to inconsistencies in presentation, subtle mistakes in course materials, and ambiguous terminology.  Many diagrams that illustrate concepts are prone to misinterpretations that can be corrected with a single comment.  Listening to students is also crucial for authors, curriculum developers, and course designers because students are frequently more sensitive than instructors to unclear presentations and curriculum misalignments.

Learning requires active participation, and the teacher, as a coach, must engage students and break down barriers that prevent them from openly sharing their ideas. Teachers benefit from the innate human need for creative self-expression and building social ties.  By supporting students’ autonomy, respecting their need for creative expression, and fostering a collaborative environment, teachers can create a classroom community that encourages active participation.

Experimentation is critical for making the independent discoveries necessary for deep learning.  Students can only effectively experiment if they are fluent with basic calculations.  The teacher’s role as a coach involves finding effective ways for students to practice skills and motivating them to do so.  It is always better to facilitate students’ internal motivation rather than rely on external motivations such as grades or praise.

Students with personal problems may reach out to an instructor, who can have a profound impact by directing them to appropriate resources.  However, instructors must balance forming emotional connections with maintaining the professional distance necessitated by the uneven power structure of the job. Instructors must always keep these words close to heart:

A teacher is not a person; a teacher is a role that a person plays. 

Understanding this is critical to managing conflict, maintaining objectivity, and creating a dynamic that maximizes student learning.

It is typically possible to inspire even the most disinterested student.  However, the coach must also be honest and seek to understand their students' needs.  The goal of an instructor should be to help students realize their own creative potential, not to make carbon copies of themselves.  The coach must find a balance between forming the authentic connections that students need in order to see themselves as a part of a community, and maintaining the professional distance that the uneven power structure in a student-teacher relationship necessitates.  The coach must find a balance between inspiring students to study the subject, and supporting their independent development that may lead them away from the subject. 

There are many effective approaches to teaching, and no single method fits every instructor.  More important than the specific approach is the instructor’s mindset.  Although it is essential to create a respectful and supportive environment in the classroom, students must also learn to engage with the realities of the world around them.  As John A. Shedd once wrote:

A ship in harbor is safe, but that is not what ships are built for.

To become independent learners, students must learn to be independent people.  As educators, it is our responsibility to provide students with opportunities to succeed and to fail, to support them through their efforts, and to guide them through the inevitable setbacks.  If we teach students that failure is a natural and necessary part of learning and self-discovery, rather than a reflection of their self-worth, they can learn to embrace it as part of their growth.

College life presents many challenges, and shielding students from discomfort or disappointment would be a disservice to their development.  Not every dream will be realized, but every student can learn to admire the subjects we love, enjoy the process of discovery, become thoughtful individuals, and develop a genuine and lasting appreciation for learning.  Our goal is not to mold students into who we think they should be, but to provide them with the opportunity to discover who they want to become.