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Shahab
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This is my goal: 'Unveiling the Hidden Beauty of Mathematics: Discover the Simplicity and Fascination of Math'
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About Shahab
I earned my Ph.D. in mathematics with a specialization in geometry and topology from Amirkabir University of Technology, Tehran, Iran, in 2015. During my Ph.D. studies, I taught calculus, differential equations, mathematical analysis, and algebraic topology at this university and some other universities in Tehran. I have been working as an assistant professor at Babol Noshirvani University of Technology, Babol, Iran, since completing my Ph.D. Over these more than 8 years, I have taught various graduate and undergraduate courses. I taught “calculus I,” “calculus II”, and “differential equations” for more than 12 semesters, “philosophy of mathematics” for 5 semesters, “topology”, “history of mathematics”, and “fundamentals of geometry” for 3 semesters, “geometry of manifolds” for 4 semesters, and “number theory”, “complex functions”, “preliminary algebraic topology”, “advanced algebraic topology”, and “Riemannian geometry” for one semester each.
Here are brief descriptions of courses that I have taught till now.
I. Undergraduate courses
1. Calculus I: introduction to complex numbers, limits with a focus on definitions, continuity and the Intermediate Value Theorem, derivatives with a focus on definitions and their formulas, The Mean Value Theorem and applications of differentiations, Integrals with a focus on definition and the Fundamental Theorem of Calculus, techniques of integration, applications of integration, improper integrals, infinite sequences and series, power series and Taylor series.
2. Calculus II: a review of vectors and the geometry of space, a concise introduction to linear algebra, vector functions and curves, multiple variable functions and partial derivatives, maximum and minimum values of functions with two variables, multiple integrals, line integrals and Green’s Theorem, surface integrals, Stoke’s Theorem, the Divergence Theorem.
3. Differential equations: some examples of differential equations, first order differential equations, second order differential equations, higher order linear differential equations, series solutions of linear equations, the Laplace transform and its applications in solving differential equations.
4. Philosophy of mathematics: mathematical objects and their existence, certainty of mathematical proofs and their reasons, introduction to main schools of philosophy of mathematics such as Platonism, Logicism, Intuitionism, and Formalism with a focus on Platonism.
5. Topology: basic definition of topology, bases for a topology, examples of topological spaces, continuous maps, metric topologies, and topological properties such as compactness, connectedness, countability axioms and separation axioms.
6. History of mathematics: prehistoric mathematics, ancient mathematics, European mathematics and a brief history of modern mathematics.
7. Fundamentals of geometry: Euclidean geometry, parallel postulate, Non-Euclidean geometries with a focus on hyperbolic geometry.
8. Number theory: induction, divisibility and prime numbers, factorization and the Fundamental Theorem of Arithmetic, congruence and Fermat’s Theorem, Euler’s formula, Euler's Phi Function and the Chinese Remainder Theorem, quadratic reciprocity.
9. Complex functions: introduction to complex numbers, complex functions and their limits, differentiation of complex functions and analytic functions, preliminary complex functions, integrals, series, residues and poles.
10. Preliminary algebraic topology: homotopy, fundamental groups, covering spaces and maps, the fundamental group of the unit circle, some applications like the Fixed Point Theorem and the Fundamental Theorem of Algebra, the Borsuk-Ulam Theorem, Fundamental groups of n-spheres, separation theorems in the plane.
II. Graduate courses
1. Geometry of Manifolds: introduction to manifolds, some examples of manifolds, tangent spaces, tangent and cotangent bundles, differentiation, tensors and forms, manifolds with boundary, integral on manifolds, Stoke’s Theorem.
2. Riemannian geometry: Riemannian metrics, examples of Riemannian metrics, connections, geodesics, curvatures, Riemannian submanifolds, Gauss–Bonnet Theorem.
3. Advanced algebraic topology: review of homotopy theory, the Seifert-van Kampen Theorem, classification of surfaces, simplicial homology theory (briefly), singular homology theory, homology groups of n-spheres.
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Mathematics is often perceived as a complex and challenging subject, but I firmly believe that it has a profound connection to the nature of our minds, making it accessible to all. Socrates once compared teachers to midwives, facilitating the birth of innate learning abilities in students. This unique perspective forms the basis of my teaching philosophy. Let's elaborate on my teaching approach and share my teaching experiences.
1. History of Mathematics. In my experience, students find historical context fascinating. By tracing the historical developments that led to their current learning, students can better appreciate the value of each lesson. Moreover, introducing them to the great mathematicians of the past can inspire students to continue their mathematical journey.
2. Gradual Progression from Familiar Concepts. I prefer to start each topic by engaging students with questions related to concepts they already know or believe they understand. Subsequently, I challenge their responses, encouraging critical thinking and discussion. This gradual progression helps students grasp even complex concepts with ease, ultimately reducing math anxiety.
3. Relevant Examples and Exercises. I always provide a collection of examples related to the topic of the class. These examples shed light on various facets of the concept and empower students to explore further through exercises of varying difficulty levels, from simple to challenging.
4. Respect, Approachability, and Empathy. My interactions with students are grounded in mutual respect and understanding. I aim to be a friendly, reliable presence rather than a strict authoritarian figure. Many former students still seek my guidance, seeking advice on their academic and career paths. I remain accessible through various communication channels, responding promptly to their inquiries. My extensive teaching experience has allowed me to connect with students of diverse backgrounds and scientific aptitudes.
Now, I aim to clarify my ideas regarding a mathematics course to provide an example for the aforementioned points:
My preferred lower-level math course to teach is calculus. I believe that calculus provides one of the earliest opportunities for students to witness the transformation of intuitively obvious concepts into precise mathematical definitions. For instance, the concept of continuity can be intuitively grasped through geometric notions, making it accessible to almost everyone. However, the rigorous mathematical definition of continuity often appears obscure and challenging upon first encounter. Consequently, teaching this foundational and preliminary concept can require considerable effort on the part of instructors. I firmly believe that educators should invest the necessary time to elucidate the strong connection between the intuitive understanding of continuity and its formal mathematical definition. Through this process, the profound importance of theorems like the "Intermediate Value Theorem" becomes evident. Additionally, understanding the process of mathematical modeling is essential for those pursuing careers in science. While continuity can be challenging to mathematically model, it offers valuable insights and serves as a rich source of information. Furthermore, differentiation, which addresses the mathematical solution to the problem of finding the slope of a tangent line, stands as one of the central topics in calculus. It effectively illustrates the process of modeling and problem-solving. It's worth noting that calculus boasts both immediate and long-term applications, making the course engaging and motivating for students. To maintain students' interest, I often incorporate brief historical anecdotes related to calculus. These stories serve to alleviate the potential monotony of the lessons, captivating students with the fascinating history of this mathematical discipline.
Teaching experience:
I earned my Ph.D. in mathematics with a specialization in geometry and topology from Amirkabir University of Technology, Tehran, Iran, in 2015. During my Ph.D. studies, I taught calculus, differential equations, mathematical analysis, and algebraic topology at this university and some other universities in Tehran. I have been working as an assistant professor at Babol Noshirvani University of Technology, Babol, Iran, since completing my Ph.D. Over these more than 8 years, I have taught various graduate and undergraduate courses. I taught “calculus I,” “calculus II”, and “differential equations” for more than 12 semesters, “philosophy of mathematics” for 5 semesters, “topology”, “history of mathematics”, and “fundamentals of geometry” for 3 semesters, “geometry of manifolds” for 4 semesters, and “number theory”, “complex functions”, “preliminary algebraic topology”, “advanced algebraic topology”, and “Riemannian geometry” for one semester each.
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