Insights into Mathematics Courses

Covering Spaces and 2-Oriented Graphs - Algebraic Topology

Uses the wedge of two circles to illustrate covering spaces as directed, labeled 4-valent graphs, then introduces the universal cover.

• YouTube
• 31 minutes
• Self-Paced
• Free Video

Free Abelian Groups and Non-Commutative Groups - Algebraic Topology Review 4

Explore how free abelian groups resemble vector spaces, then see how non-commutativity changes cosets and leads to normal subgroups and quotient groups.

• YouTube
• 51 minutes
• Self-Paced
• Free Video

Introduction to Group Theory

See how finite commutative groups decompose into cyclic groups, starting from the basic axioms and concrete examples.

• YouTube
• 47 minutes
• Self-Paced
• Free Video

What Is a Sequence? - Real Numbers and Limits in Mathematics

A precise mathematical definition of sequences emerges through examples from natural numbers, primes, and rationals.

• YouTube
• 27 minutes
• Self-Paced
• Free Video

Mechanics and Curves in Mathematical History

Trace how mechanics shaped the study of curves, from the catenary and cycloid to Euler’s elastica and Bézier control points.

• YouTube
• 58 minutes
• Self-Paced
• Free Video

Solving Quadratics and Cubics Approximately - Real Numbers and Limits - Math Foundations 85

Compare standard quadratic and cubic formulas with Newton’s method, showing how each approach produces approximate solutions.

• YouTube
• 36 minutes
• Self-Paced
• Free Video

Newton's Method for Finding Zeros - Real Numbers and Limits

Newton’s method turns tangent-line intersections into a recursive process for approximating square roots, illustrated with x² and √2.

• YouTube
• 26 minutes
• Self-Paced
• Free Video

Addition and Number Names - Elementary Mathematics Explained

Explore addition with small numbers, then compare English number names with an alternate system and organize the results in addition tables.

• YouTube
• 36 minutes
• Self-Paced
• Free Video

Fun with Polyominoes - Elementary Mathematics Explained - Lecture 6

See how polyominoes are counted and classified, then use checkerboard coloring to explore when trominoes and tetrominoes can tile a grid.

• YouTube
• 30 minutes
• Self-Paced
• Free Video

Laws of Arithmetic Using Geometry - Elementary Mathematics Explained - Lecture 5

See how translations and reflections in the grid plane reveal the commutative laws for addition and multiplication.

• YouTube
• 41 minutes
• Self-Paced
• Free Video

Fundamental Group of Surfaces - Multiplication and Properties

See how loop classes acquire group structure through multiplication, identity, inverses, and associativity, with examples from the torus and projective plane.

• YouTube
• 35 minutes
• Self-Paced
• Free Video

Knots and Surfaces - Seifert Surfaces in Algebraic Topology - Lecture 2

See how Seifert surfaces are built for knots and how adding boundary cuffs changes a surface’s Euler number.

• YouTube
• 38 minutes
• Self-Paced
• Free Video

Classification of Combinatorial Surfaces - Lecture 2

Classify surfaces by cutting and pasting their polygons into standard forms with handles and crosscaps.

• YouTube
• 1 hour
• Self-Paced
• Free Video

Duality for Polygons and the Fundamental Theorem of Algebra - Algebraic Topology Lecture 12

Uses winding numbers of polynomial images of concentric circles to give a geometric proof of the Fundamental Theorem of Algebra.

• YouTube
• 46 minutes
• Self-Paced
• Free Video

Rational Curvature, Winding, and Turning in Algebraic Topology - Lecture 11

Introduces a rational turn-angle measure for polygons, then uses winding and turning numbers to describe planar curves.

• YouTube
• 48 minutes
• Self-Paced
• Free Video