I was trained to implement Treisman-style collaborative learning workshops in my last year as a graduate student at UCLA, when I worked  as a workshop facilitator in the PEERS program, a program designed to improve retention and academic success among underrepresented students in STEM fields.  Here is a link to the program: PEERS.  This experience in my formative years as an instructor had a deep impact on my pedagogical approach.  Throughout my career, I have used the combination of worksheet construction and collaborative learning workshops as a cornerstone for both teaching and learning at all levels.  I integrate worksheet construction into my own scholarship, using it as a tool for organizing ideas and advancing my research inquiries.

I use worksheets not only to help students explore concepts but also to prepare them for constructing their own worsksheets for independent learning.  Worksheet construction is a powerful approach for developing a research mindset in students and can be implemented at any level.  Constructing worksheets helps students to become independent learners, encourages their creative self-expression, and promotes their personal ownership of their learning process.  It reinforces their communication skills by encouraging clear and structured expression of ideas and promotes writing across disciplines.  

By formulating questions and exploring their answers in order to achieve a concrete learning outcome, students learn to approach problems like researchers—identifying key issues, organizing ideas, and following through on the inquiry process in an intentional way.  By continuously engaging with worksheet construction, students learn how to 'dig' into a subject by producing initial explorations, identifying meaningful stopping points for reflection, and then refining and expanding their understanding in an iterative and reflective process.

For more information, see the following resources: A worksheet on constructing worksheets with a formulaic method, and sample worksheets from Principles of Calculus I, II, and III.  Here is a worksheet on constructing worksheets for learning that modifies the approach for constructing worksheets for teaching.   

The worksheet method not only enhances student learning but also equips educators with a flexible and dynamic tool for curriculum development.

Distance learning has a rich history, and understanding that history is crucial for framing online teaching in its proper context. As early as 1858, the University of London, which later became UCL, began offering distance learning degrees, becoming the first university to do so. In the 1890s, William Rainey Harper, founder and first president of the University of Chicago, promoted the concept of extended education, establishing satellite colleges and encouraging correspondence courses to broaden access to education. By the 1920s, institutions like Chicago, Wisconsin, Columbia, and many others were offering correspondence courses.

Education was a central priority during the Progressive Era (1890s-1920s), with American high schools and colleges expanding significantly. Correspondence schools and night schools opened to serve men who were older or had family responsibilities. One such institution, the YMCA school in Boston, later became Northeastern University. By 1920, less than half of Americans lived in urban areas, leaving many in rural regions with limited access to education. The demand for broader educational opportunities in these areas led to the development of distance learning as a practical solution.

Today, there is often an emphasis on using online education to lower costs, but I believe this perspective is misguided. The original purpose of distance learning was to provide access to education for those otherwise unable to attain it. Modern internet technology has revolutionized distance learning, effectively blurring the line between in-person and remote education. The internet, as the most powerful communication technology ever developed, offers an unparalleled opportunity to enhance student learning through online platforms.

The first decade of online teaching focused on asynchronous lectures, automated grading, and chat rooms for office hours. While this was a vast improvement over traditional correspondence courses, offering students high-quality lectures and timely feedback, the last decade has seen even more dramatic technological advancements. Today, online platforms allow for face-to-face, fully synchronous teaching that offers greater flexibility and, in many cases, superior outcomes compared to traditional classroom instruction. As evidence, my students now universally prefer my online office hours over in-person ones, and I have achieved better teaching outcomes online than in-person.

In Spring 2016, I received a $110,000 ILTI grant to develop UCR’s old Math 5 course into a fully online course. This experience taught me productive methods for online teaching. In my Math 5A course, I offer online office hours, coordinating them with TA hours to provide students with over 25 hours of instructor access each week during the fall. This level of access to instructor support would be impossible without online technology due to physical space constraints. Additionally, the VITAL outreach program, which is entirely online, scales instruction to provide broad access.

While online teaching will play a pivotal role in the future of education, it also presents challenges that we must address. Understanding the cultural effects, mitigating negative consequences, and solving platform-specific weaknesses will be critical in avoiding misuse of the technology. Despite potential pitfalls, it is exciting to realize that, for the first time in history, educators can provide high-quality, personalized education to every person on the planet. This technology has the power to change the world, and it is essential to stay at the forefront of its development and implementation.

Here is a link to the slides for an invited talk on online teaching that I gave to faculty at the American University of Cairo, in 2020: AUC Talk.

Spatial thinking enables students to offload complex computational and language processing tasks into visual processing, enhancing problem-solving by reducing cognitive load and providing a structured framework for numerical tasks. Research shows that visuospatial working memory and strong spatial skills, such as mental rotation, are closely linked to math achievement across educational levels. These skills help students visualize mathematical concepts, transforming abstract ideas into concrete spatial representations like number lines, diagrams, and models.

Despite their proven benefits, formal education often underemphasizes spatial skills. Most students develop these abilities through informal activities, such as playing with blocks, puzzles, or video games, rather than structured learning experiences. Integrating spatial thinking into math curricula can bridge this gap, helping students better understand challenging concepts and improving overall math performance.

I use technology to assist students in building their spatial reasoning skills by constructing simulations for students using a variety of visualization tools, including physical models and Desmos simulations.  I also regularly have students build their own simulations in class and for homework.  It is important for students to recognize which aspects of a problem are challenging, which are creative, and which are routine.  Students often become lost in the mire of a complicated calculation.  Having students create their own simulations helps them learn how to comparmentalize information in ways that they can more easily process, and to efficiently and appropriately utilize technology resources.

In the Principles of Calculus course, we have incorporated spatial thinking offloading throughout the curriculum. For example, we designed a series of exercises that train students to use spatial thinking to tackle a variety of computational problems. Students complete these exercises at home, present their solutions to peers in class, and engage in collaborative review sessions. Here are three examples of such exercises: POC-I Example, POC-II Example, POC-III Example.

For more information, visit my collaborator's page: Kinnari Atit Webpage.

The traditional precalculus and calculus sequence presents procedural knowledge in a linear fashion, creating barriers to student learning. Deficiencies at any stage can critically impact performance in subsequent topics. To address this, topics are often artificially separated, but this diminishes the material's intellectual value by limiting the range of questions that can be explored. It prevents instructors from asking truly integrative questions and disrupts the natural flow of ideas, hindering students from developing a cohesive and interconnected understanding of mathematics. Ideally, a mathematics course should highlight the rich and exciting interplay of ideas.

Principle-based teaching offers a solution by emphasizing fundamental principles such as decomposition, transformation, rigidity, and symmetry, which provide a unifying framework that ties the curriculum together. Rather than treating each concept as isolated, this approach helps students see how core principles apply across various mathematical contexts. This integration allows for continuous review and reinforcement of essential topics, making it easier to address knowledge gaps as they arise. In contrast, the traditional linear approach often amplifies gaps in prerequisite knowledge, particularly harming early-career students.

The National Research Council (2000) states that “1. Learning with understanding is facilitated when knowledge is related to and structured around major concepts and principles of a discipline.” Principle-based teaching reduces complex formulas to a few guiding ideas, transforming the way students engage with mathematics. Developing a principle-based calculus sequence represents the evolution of concept-based teaching but requires a complete overhaul of the standard curriculum. It requires curriculum developers with deep expertise in the subject, who can design courses that foster a research-oriented mindset. The creation of the Principles of Calculus course, for example, involved writing nearly 1,000 pages of notes that rigorously develop a first-year calculus sequence from a unique, principle-based perspective.

The benefits of a principle-based course justify the challenge of its creation. This approach promotes equity in STEM education by emphasizing deep understanding over rote memorization, making mathematics more accessible and engaging for all students, especially those from underrepresented backgrounds. It helps level the playing field, reducing dependence on prior procedural knowledge and empowering students to build a solid, interconnected understanding that supports success in advanced courses.

For more details, see these slides on principle-based teaching: Principle-based Calculus Instruction.

Successful implementation of The Principles of Calculus course requires a pathway for the assimilation of reasoning processes through simple, rote practice.  The development of the pathway begins with two questions:

(1) To what degree can the vocabulary and grammar of a constructed language encode and formalize reasoning processes?

(2) To what degree can exposure to rote practice with simple language learning exercises enhance higher-level reasoning?

Our strategy, linguistic mapping, leverages language development to develop deeper thinking in a structured way.

Linguistic mapping is more than just memorizing definitions: It involves integrating words into a meaningful grammar that facilitates reasoning.  This grammar helps offload complex cognitive processes to a formal system of rules that are natural, intuitive, and memorable.  Understanding and assimilating mathematical language changes students' thinking processes, allowing them to more effectively apply principles of reasoning. The language captures physical intuition and the grammar is robust and mathematically sound.  We constructed a toolset that leverages spatial thinking as a cognitive strategy to facilitate linguistic mapping, and are currently studying its effectiveness.  Although I previously used linguistic mapping in my probability course and the accompanying book, Chance and Choice, we expanded and refined this strategy in the Principles of Calculus course.

Early results suggest that this approach significantly improves students' ability to utilize higher level reasoning.  For example, students typically experience a great deal of difficulty in working with piecewise-defined functions.  A linguistic mapping approach has dramatically improved students' ability to work with such functions, and working with such functions helps students to better understand the notion of domain and range of a function, and the geometric conceptualization of a function as a special subset of a Cartesian product.  

Here is a link to a linguistic mapping worksheet for the Principles of Calculus I course that supports students learning about piecewise-defined functions: LM I.6. Here is a link to a linguistic mapping worksheet for the Principles of Calculus I course that supports students learning about idea of a vector: LM II.1.  Please note that these worksheets are inteded for use by an instructor as lecure materials in a course that supports their implementation. 

Just as internet technology revolutionized distance learning, the AI revolution promises to have a profound impact on the future of mathematics education.  I am currently exploring avenues for its use and responsible development in the education domain.

While at UCLA, I developed an innovative approach to teaching undergraduate analysis, focusing on bridging formal logical language with physical intuition.  I wrote a textbook based on this approach and taught from it many times.  I am currently rewriting this textbook to emphasize concision and formality, providing a contrast to an analysis book that supplements the Principles of Calculus program, which focuses on simplicity and intuition.  Both books aim to connect formal mathematics with intuition, but in fundamentally different ways—one through developing mastery of formal language, and the other through the application of reasoning principles using more intuitive language.

At UCLA, I established a Summer outreach program to engage students in mathematics and ran this program for two summers, in 2010 and 2011.  The UCLA Mathematics Department's Mathematics Institute for Young Scholars was a four-week, summer day program for secondary students interested in mathematics. The program focused on mathematical thinking and its application to combinatorics, probability, and mathematical physics. 

In the Institute, students attended my course lectures, problem solving sessions led by a UCLA graduate student, and afternoon plenary seminars on a variety of topics.  Students also participated in a number of special outings, including a visit with undergraduate and industry researchers from around the world at the Institute for Pure and Applied Mathematics' Research in Industrial Projects for Students.

During my time at UCLA, I redeveloped a basic probability course for undergraduate life science majors, which I taught multiple times using my textbook, Chance and Choice.  Since then, I have expanded the textbook into a comprehensive course text for a two-quarter sequence on probability for math majors, which I have used to teach the 149AB classes at UCR.  Currently, I am further developing the text to support a year-long course.

The textbook was designed as a proof of concept, illustrating how a worksheet-driven approach can be employed in both textbook development and curriculum design.

The concept of a principle-based course matured in my mind through my work on developing the Math 5 course at UC Riverside from Summer, 2015 to Fall, 2020.  Before my involvement, Math 5 was a traditional precalculus course, structured around three hours of lecture and two hour-long discussion sections each week.  This setup offered 28 hours of lecture and 20 hours of discussion, with half of the discussion sections focused on teaching assistants delivering content rather than facilitating student interaction and problem-solving.

At the time, there was a strong push at the university level to incorporate ALEKS, an adaptive learning platform, into precalculus instruction.   The proposed model suggested transforming discussion sections into TA-facilitated, self-paced ALEKS sessions in a computer lab.   While I recognized the potential benefits of ALEKS, particularly given the wide range in student preparation, I was concerned about losing the collaborative learning environment that had been central to my teaching success at UCLA.  There, I had replaced traditional discussion sections with graduate-student-led collaborative learning sessions, which significantly enhanced student engagement and understanding.

My initial steps included a thorough assessment of our student body and a scholarly study of the existing research on precalculus education.  I had many discussions with mathematics faculty and faculty in other departments in which I tried to determine the essential course components.  I spent several months familiarizing myself with ALEKS and developing modules that aligned with my teaching philosophy.  I approached the redesign of Math 5 with caution, making only small changes each quarter.  Eventually, I eliminated ALEKS use from my course as it did not fit my teaching style. 

Education studies consistently show that traditional precalculus courses are ineffective in preparing students for calculus and can even decrease the likelihood of success in future courses.  With this in mind, I began systematically reworking Math 5, striving to balance the basic learning goals of a precalculus course with a deeper focus on developing reasoning skills.  However, as have many curriculum designers and classroom teachers before me, I quickly realized a tension between the primary learning goals:

1. Procedural Fluency and Topical Knowledge: Ensuring students acquire the procedural skills and topical knowledge required for subsequent courses.

2. Problem-Solving and Application: Developing students' ability to apply procedural skills and topical knowledge in problem-solving contexts.

The prevailing trend was to reduce content in the first category to make more room for the second—a reasonable approach, but one that still felt limited. To better understand this tension, I made a detailed list of all topics covered in the course and asked myself critical questions before eliminating or introducing any content:

1. Purpose: How does learning this topic affect a student's thinking?

2. Redundancy: Do other topics in the course have a similar impact on student thinking?

3. Alignment: Could improved alignment between topics promote the retention of learning?

4. Replacement: Could introducing a new topic make it possible to cover multiple other topics more efficiently?

The idea of identifying and building a course centered around principles of reasoning gradually emerged from a five year iterative process of pruning some topics and introducing others with these questions in mind.  Throughout this process, it became clear to me that the real issue is not a satisfactory resolution to the tension between the two learning goals, but rather that the learning goals are flawed.  Rather than focus on two competing  goals, it is helpful to refocus on these mutually supportive learning goals:

1. Reasoning:  Ensure that students develop a powerful reasoning strategy;

2. Language Assimilation:  Ensure that students assimilate a precise and expressive language that can be used for solving problems across academic domains.  

The course may still expose students to the same topics to which the standard courses expose students, but in a way that does not distinguish between proceedure, topical knowledge, concept building, and application.

With support, encouragement, and guidance from my then department chair Professor Yat Sun Poon, the result of five years of work was a unique reimagining of the role of precalculus in an undergraduate curriculum.  The Math 5 course was an entirely homegrown course that replaced precalculus and focused on the development of key reasoning principles: Decomposition, Transformation, Rigidity, and Symmetry.  By gradually refining content and pedagogy to align with these principles, the course shifted from a focus on procedural teaching to fostering deeper understanding.  

The development of this course was the starting point for my later collaboration with Professor Poon and our team on the development of the Principles of Calculus program.

At the request of the former Vice Provost of Undergraduate Education, I chaired the committee developing the Summer placement, testing, and self-remediation program.  I helped to implement an ALEKS based placement and remediation program.  This  is considered a campus-wide service activity since its effect will be felt by all departments that have a basic math requirement for their students.  Approximately two thirds of all students improved their initial placement through this program.

Math 302 is a graduate teaching seminar that I co-taught with my then-department chair, Professor Yat Sun Poon, in Fall 2016-19, and that I taught as a sole instructor in Fall, 2020 through Fall, 2022.  We collaborated on a complete redesign of this course.  I continued developing the course from Fall, 2020 through Fall, 2022.  My involvement in this course has been important in shaping our later efforts to produce a continuing education program for teachers.  

Our Math 302 class is not a teacher training course, it is a course on teaching philosophy where students and instructors alike can revisit their ideas about teaching mathematics and have a forum for discussing controversial ideas.  I explain to students on the first day: "In a teacher training class, students are taught how to grade. In this class, we discuss why we grade and how assessments can be meaningful."  I believe that the 302 class is valuable because university instruction, at all levels, should be more than just job training.  We must train students to be independent thinkers and independent learners.  Instructors who learn their craft without thoughtful reflection are people who are simply trained to do a job.  A course such as this trains students to question their topical understanding, analyze education literature, and understand that personal growth and learning are essential to developing one's own teaching.  I also used it as a way of teaching graduate students how to learn.  By reflecting on how our students learn, we learn better how to teach ourselves.  I emphasized this point so that even the disinclined students could appreciate the value of learning about teaching.

We read a variety of texts in this class.  Though the reading list changed from year to year, selected writings of Halmos and Polya became permanent fixtures in the course.  I made sure that all of my students read Treisman's important article "Studying student studying calculus," an article that I read years ago while in my own teacher training program at UCLA as a graduate student.  This article was absolutely critical to my growth as a teacher and so I share it with my students.  

Ultimately, this course was not about teaching students how to teach, but teaching students to become reflective in their teaching and learning.  After seven years of involvement in this course, I decided to hand it off to a colleague, because I think that it is important to have the representation of a variety of perspectives on teaching in a department.  Here is a link to a course website that could be used by anyone interested to teach a similar course: Math 302.

I taught the 149AB courses, upper division courses on probability, successfully in the first ever faculty led education abroad program (FLEAP) offered by the Mathematics Department.  For three weeks in Summer 2018, I gave an intensive probability course to 19 UCR students.  I then traveled with these students to London where we studied probability for another four weeks.  The course was the highest attended of all FLEAP courses in 2018, during its first offering.  

I repeated this course in Summer 2019 with similar success and with an enrollment of 17 students. In both classes, more than half of the students were women and approximately 80 percent were students from groups that are underrepresented in STEM; the majority had never traveled abroad. This course had a major impact on the students involved.  Nine of the students from the 2018 course later worked with me on research projects and presented these projects at local meetings.  One of these research projects has lead to a professional publication with student co-authorship.

Although the Covid-19 pandemic forced cancellation of the Summer 2020 program, I am looking to restart this program in the near future.  Here is flyer for my 2018 program: Flyer. 

Professor Yat Sun Poon was the chair of the department during my first six years as a faculty member, and originally encouraged me to redevelop the Math 5 course.  After observing the success of the redeveloped course, we began our fruitful collaboration in which we developed 'Principles of Calculus', a novel mathematics program aimed at offering a deeper and more enriching learning experience for students while dramatically expanding access.  Our efforts were supported by the California Learning Lab, which funded the development of our program under the grant title 'A New Mathematics Gateway'.  The UCR implementation of our new program is the Math 5ABC sequence.  For more detailed information, please visit: A New Mathematics Gateway.

Already in the first year of our program we eliminated equity gaps in student performance, increased first year student matriculation through multivariable differential calculus by 150%, and increased student matriculation through the first calculus-based physis class by 70%.   It is important to note that our measure of success primarily involves student success rates in critical subsequent courses for STEM majors, and not just the student performance in our courses.

Here is a link to the course website: The Principles of Calculus.

I led three groups of three students from the 2018 FLEAP course in research projects.  All three groups presented their projects at the Mathematics Department's 'Undergraduate Research Presentations' in Spring 2019 and in three poster presentations at the University Student Research Symposium, a University wide event.  One of these research projects has led to a professional research paper with student co-authorship that was accepted in the Journal of Applied Probability.  One of the student authors presented the paper at an AMS sectional meeting.  This student has now graduated and is a graduate student at the University of California, Irvine (UCI).  Two of the students I worked with are community college transfer students.  Two thirds of the students I have worked with are women and most are from groups that are underrepresented in STEM.

I also worked with an undergraduate student on a research project who has twice received support through the California Alliance for Minority Participation (CAMP) program for his work with me.  He attended several courses with me.  He is also a community college transfer student and while he did not attend the FLEAP program, he sought me out because of the program.  He and I studied discrete Feynman-Kac formulas for path measures that approximate p-adic Brownian motion.  He is now a graduate student in physics at UCI and is well on his way to a doctoral degree. 

I have graduated three doctoral students.  All three obtained academic employment.  I currently lead a team of three graduate students, all of whom have already advanced to candidacy.  I have served on 21 oral exam committees or dissertation committees in total, including 14 for which I was not chair.  I was an external oversight member on four of these committees, two in education and two in physics.