Teaching

Teaching is a profound passion of mine. I have always considered research and teaching to go hand in hand in the cycle of creating, sharing, and acquiring knowledge. 2025 marked 15 years since I started teaching mathematics — from one-on-one tutoring through high-school gymnasium curricula to large university courses. I started a small collection of student drawings from submissions along the way.

Teaching philosophy

My teaching is centered on fostering deep understanding, critical thinking, and active engagement. In practice it is realized through the following principles:

1. Curriculum as a web of interconnected concepts. Rather than a single linear path, I structure courses as a network of small learning objectives that link to one another — a Zettelkasten-style concept map. I teach students how to structure and digest material themselves rather than pre-digesting it for them. Built and tested in MAT 188, MAT 246, and MATA22.
2. Experiential learning. Every concept starts from concrete examples students can manipulate before any formal definition. The guiding question is always: why would anyone invent this concept, and what problem does it solve? (MAT 102.)
3. Inquiry-based learning. Carefully designed exercises guide students to discover concepts themselves, with the instructor as facilitator. Drawn from the Russian problem-based tradition and my work at the MSU Gymnasium (MAT 188, MAT 246).
4. Teaching students how to learn. I treat meta-cognitive skills as explicit course content: how to read a proof, how to self-question, how to recognize genuine understanding — a gap made more urgent by AI tools and post-COVID disruptions.
5. Continuous feedback loops. Weekly reflections, anonymous channels (Padlet, Google Forms), and dedicated class time to respond publicly. In MAT 188, students explicitly noted that I adapted my teaching based on their input.
6. Clear learning objectives. Specific, measurable statements of what students should be able to do — a roadmap for studying and a bridge to mastery-based grading (MAT 188, MAT 246).
7. Alternative grading. Assessment as feedback rather than performance measurement: standards-based grading in MAT 188, optional assessments in MAT 246.
8. Peer-to-peer learning. Mathematics is deeply social. From discussion-driven office hours in MAT 224 to graded Piazza participation in MAT 246 to oral group presentations in MATA22, I build a learning community in every course.

Courses taught

University of Toronto

Linear Algebra I for Mathematical Sciences (MATA22) — Summer 2026, University of Toronto Scarborough. ~180 students, sole instructor.

• Full course design — lectures, tutorials, and assessments — for a single, non-coordinated section.
• Coordinating 6 tutorial sections and a team of 6 TAs; tutorials structured around collaborative group work and student oral presentations.
• Lecture materials developed as an inquiry-based concept network emphasizing experiential and peer-to-peer learning.

Introduction to Proofs (MAT 102) — Fall 2025, University of Toronto Mississauga. 140 + 90 students; coordinated by Prof. Tyler Holden.

• Two sections of a crucial first-year course building foundations in proof comprehension and writing.
• Active-learning format with pre-class readings and a partial flipped classroom; emphasis on developing student learning skills.

Concepts in Abstract Mathematics (MAT 246) — Winter 2025, University of Toronto. 160 students; non-coordinated.

• Designed a new course structure and curriculum emphasizing active learning and breadth of exposure.
• Peer-to-peer learning, learning objectives as a review tool, assessment optionality, and ungraded assignments.

Linear Algebra for Engineering (MAT 188) — Fall 2024, University of Toronto. 140 students; coordinated by Prof. Camelia Karimian Pour.

• Active-learning course with pre-class readings, mastery/standards-based grading components, and explicit learning objectives.

Linear Algebra II (MAT 224) — Winter 2023 & Winter 2024, University of Toronto. 140 students; coordinated.

• Distinguished by the F.V. Atkinson Teaching Award for Postdoctoral Fellows (2023).

Other university experience

Differential Topology — Higher School of Economics, 2019–2020. 25 students; core course taught with Prof. Alexander Gaifullin. Topology — Independent University of Moscow, 2018–2019. Teaching assistant, 50–100 students. Post-quantum cryptography for non-mathematicians — private teaching, 2020–2023. Higher-mathematics curriculum tailored for a cryptography specialist with no formal math background. Tutoring US-based undergraduates — Alderwood Education, 2019–2021. One-on-one teaching of undergraduate courses.

High-school and competition mathematics

Introduction to Modern Mathematics — Russian School of Mathematics, 2025–2026. Course design and instruction; adapted university-level modern mathematics for advanced high-school students. National Math Competition Program — Russian School of Mathematics, 2020–present. Course design and instruction, 10–15 students per class. AMC 8/10/12 and AIME problem-solving webinars — Russian School of Mathematics, 2023–present. 200–500 attendees per webinar. Algebra, Geometry, Calculus, Higher and Popular Mathematics — MSU Gymnasium, 2016–2018. Curriculum design and instruction.

Grants and awards

• LEAF Grant — Graph-Based Concept Mapping for Introductory Proofs, Learning and Education Advancement Fund, University of Toronto Mississauga, 2026 (approved).
• F.V. Atkinson Teaching Award for Postdoctoral Fellows, University of Toronto, 2023.

Teaching talks

Aug 5–8, 2026

MAA MathFest 2026, Boston, MA; Contributed Paper Session on Inquiry, Conjecture and Discovery; talk Making Inquiry the Curriculum, Not the Exception: A Graph-Based IBL Architecture

Abstract

Consider organizing a mathematics curriculum as a directed graph: each node introduces a single idea, and each directed edge is a question students would naturally ask next. After establishing that , for instance, a natural edge asks: is addition always commutative, or did we just get lucky? Students navigate by following and contributing such questions rather than a fixed sequence. This structure extends concept maps so that inquiry is the curriculum’s flexible skeleton, not an enrichment layer. The design follows Tao’s three stages of mathematical maturation: informal intuition first, formalism only when students’ own questions demand it, and eventually fluency between the two. We present the graph architecture, a concrete implementation progressing from informal counting to formal logic and set theory, and a framework for course, assessment, and feedback design using the proposed curriculum structure.

June 18, 2026

Tri-Campus Math Education Meeting, University of Toronto; talk Creating the Conditions for Peer Learning in a Proof-Based Course

Abstract

In large courses at U of T, students often have no relationship with the people sitting next to them. In a proof-based course, this is a particular problem: peer learning requires hearing how someone else reasons, and without social bonds or a structured occasion for it, it simply doesn’t happen. This talk describes a tutorial format designed to create that occasion. Students work in groups on parallel but distinct problem variants, then voluntarily present to a section that has worked on structurally similar problems, making every presentation genuinely informative to the room. Presenting a partial or unresolved solution is explicitly worth full credit. I will describe the design subtleties, the features making the structure work in my opinion and some preliminary student feedback of the approach.

Authored teaching materials

A selection of materials I designed; the full set lives on the teaching materials page.

MAT 224

• Solving complex polynomial equations — a direct material page
• Learning-objective review concept map · Orthogonality concept map · High-level course map
• MAT 224 syllabus draft

MAT 188

• A contributed homework problem on nilpotent matrices
• Learning-objective concept maps, weeks 1–2

MAT 246

• Course philosophy page
• Practice sets: predicate logic & proofs · whodunnit · knights and knaves · counting · induction · polynomials · Diophantine tricks

Evaluations

4.5 / 5 on RateMyProfessors across 33 reviews (MAT 224, MAT 188, MAT 246); the most common tags are Caring (17), Amazing lectures (16), and Helpful (15). Course evaluation scores have been consistently above departmental averages.

MAT 224 — course evaluation & selected feedback

Statement	Score (2023 → 2024)	Dept. average
Institutional composite mean	4.0 → 4.2	3.5
Intellectually stimulating	4.1 → 4.1	3.6
Deeper understanding of the subject	4.2 → 4.2	3.6
Atmosphere conducive to learning	4.2 → 4.6	3.7
Overall quality of learning experience	3.7 → 4.0	3.0

“Denis is one of the best lecturers I’ve ever had. His lectures were engaging, flowed well, and included lots of connections to past content and out-of-course related topics.”

“Professor Gorodkov placed emphasis on teaching intuitive understanding rather than rote memorization… had I been a first year, this is the kind of instruction that would tempt me to further pursue math.”

“Denis is the most caring professor I have met in UofT! He gives students confidence to ask questions, and when you are falling behind he will take the time to guide you step-by-step outside of class.”

“His lectures focused more on the intuition and understanding rather than just the rote computations. I don’t think I would remember the ideas as well had I not had this good of an explainer.”

MAT 188 — course evaluation & selected feedback

Statement	Score	Dept. average
Institutional composite mean	3.9	3.4
Intellectually stimulating	4.2	3.2
Deeper understanding of the subject	4.0	3.5
Overall quality of learning experience	3.5	3.3

“The professor is excellent and he explains everything in a way that is very easy to understand. Something remarkable is that he reads all the feedback that students give and constantly adapts to it.”

“Denis Gorodkov is a great instructor that accepts feedback humbly and graciously. Initially his teaching style was more theoretical… he adapted through feedback and dramatically improved.”

“My instructor was without a doubt one of the best teachers and mentors I have had the privilege of having in my academic career. His passion for mathematics is palpable and contagious.”

MAT 246 — course evaluation & selected feedback

Statement	Score	Dept. average
Atmosphere conducive to learning	4.2	4.0
Generated enthusiasm for learning	4.4	4.0
Explained concepts clearly	4.1	4.0
Was approachable	4.4	4.3
Overall quality of instruction	4.0	3.9

“One of the best lecturers I’ve had.”

“The quality of the instruction was phenomenal. The professor was able to find a balance explaining concepts both abstractly and more concretely, and took the time to ensure students’ questions are answered.”

“Denis is an excellent instructor — personable, compassionate, and approachable, with clear and motivated teaching. He is quite sensitive to the level of mathematical experience he teaches, encouraging while remaining fair.”