Lessons – The Math Cocoon

About the Lesson

Lessons may be conducted in one of two locations: online via Google Meet or in the Tutor’s classroom in Mornington. Both formats can be highly effective. The best choice often hinges on the student’s preferred learning style and circumstances. Lessons go for a minimum of 60 minutes and the duration of a longer lesson must be a multiple of 30 minutes. Students may take as many breaks as they need. Each lesson is designed to be engaging, supportive, and focused on helping students not only understand maths but develop the confidence to apply their knowledge independently.

Georg Cantor
1845 – 1918

Cantor was a Russian mathematician whose work was foundational to the development of set theory. He introduced the concept of one-to-one correspondence between sets, formalized infinite and well-ordered sets, and developed cardinal and ordinal numbers along with their arithmetic. Cantor famously proved that the set of real numbers is strictly larger than the set of natural numbers, a result that implies the existence of multiple levels of infinity—often described as an "infinity of infinities". His ideas are of profound philosophical significance.

Online Lessons

Online Lessons offer flexibility and convenience, allowing students to learn from the comfort of their own home with easy access to all their school resources. They eliminate travel time and related expenses, making them an ideal option for busy families. While online lessons may feel less personal at first, strong rapport can still be built within a few sessions. They also provide a calmer, less intrusive learning environment, which can be especially beneficial for students who experience anxiety. Students will require a stable internet connection, speaker, and microphone to participate effectively. It is also strongly recommended that students purchase a cheap A3 handheld whiteboard, which they can hold up to their webcam to show the Tutor their work. An alternative (more expensive) option would be an electronic writing tablet/pad.

In Person Lessons

In Person Lessons provide valuable face-to-face interaction, making it easier for teachers to build rapport, identify misunderstandings, and keep students engaged. The structured classroom environment can help students stay focused and motivated, particularly those who benefit from routine. Hands-on activities and immediate feedback are also more easily incorporated into this type of learning. The downside is that these lessons require fixed scheduling and travel time, which may be less convenient for some families. Travelling also serves as an additional barrier for students to attend their scheduled lesson, and students will need to remember to bring along any required resources or documents to their lessons.

Paul Erdős
1813 – 1996

Erdős was a prolific 20th-century Hungarian mathematician who made groundbreaking contributions to number theory, combinatorics, graph theory, and probability. Famous for his collaborative spirit, he coauthored over 1,500 papers with mathematicians worldwide, creating the concept of the “Erdős number.” His unconventional lifestyle and relentless focus on problem-solving left a profound and lasting impact on modern mathematics and the global mathematical community.

Archimedes of Syracuse
287 – 212 BC

Archimedes was a Greek mathematician, physicist, and engineer whose discoveries laid the foundations for mechanics, hydrostatics, and geometry. He formulated the fundamental principles of levers, buoyancy, and the Archimedean screw, and made key advances in calculating areas, volumes, and approximating π with remarkable precision. Beyond theory, he applied his insights to invent ingenious machines and war devices, demonstrating a rare combination of abstract reasoning and practical engineering. Renowned for both his theoretical brilliance and inventive skill, Archimedes’ work profoundly influenced science, mathematics, and engineering for centuries, inspiring generations of scholars and innovators.

Which to Choose?

Students are not forced stick with one type of lesson. In fact, they are strongly encouraged to expereince both. This is the best way for them to figure out which one they prefer. Specifically for online lessons, its a good idea for students to expereince using the online learning environment to see just how easy it is to work with and navigate. All too often students are unable to make the trip to the tutors home. Maybe its the weather or the traffic, or maybe the students just too tired and doesnt have the energy to make the trip. Students who have experienced the ease of using the online learning environment are much more likely to use the spare time (that would’ve been used for travelling) to rest up, before hopping on their PC and atteding the lesson online.

The Trial Lesson

The initial lesson is called the Trial Lesson. Its duration is one hour and it’s free of charge. Use this free lesson to assess the Tutor’s teaching style and decide if its the right fit for you. If you enjoyed the trial lesson but feel you would’ve preferred a few things done differently, please provide feedback and future lessons will be customized to your satisfaction. Students who wish to continue tuition after the trial lesson will need to create a free Google account if they do not already have one. This online platform will serve to store lesson notes, share rescources, and facilitate communication with the tutor. Google’s services have been chosen for their accessability, reliability, usability, and security.

Sir Isaac Newton
1643 – 1727

Newton was an English mathematician whose groundbreaking work laid the foundations of modern mathematics. He co-invented calculus, developed methods for infinite series, and formulated the generalized binomial theorem. Newton also made significant contributions to algebra, geometry, and numerical analysis, creating tools that enabled precise calculations in physics and astronomy. Renowned for his logical rigor and inventive methods, his mathematical discoveries profoundly shaped not only the study of mathematics itself but also its applications across science and engineering for centuries.

Henri Poincaré
1854 – 1912

Poincaré was a French mathematician, physicist, and philosopher whose work laid the foundations of modern mathematics and mathematical physics. He made pioneering contributions to topology, celestial mechanics, and the theory of dynamical systems, introducing concepts that would become central to chaos theory. Poincaré advanced number theory, algebraic functions, and the theory of differential equations, and helped formalize the notion of mathematical rigor. Renowned for his deep insight and creative intuition, his discoveries profoundly influenced geometry, analysis, mathematical physics, and the development of modern scientific thought.

Class Structure

Each session is structured to maximize understanding and retention. We begin with a short check-in (~2 minutes) to see what the student has covered at school over the past week and how they are feeling about their progress. Next, we identify any homework or specific topics the student wishes to focus on (~2 minutes). This is followed by a clear overview of key concepts related to these topics (~15 minutes). Finally, we work collaboratively through practice questions, gradually building the student’s confidence and ability to solve problems independently.

Teaching Approach

Students are guided at the outset. The Tutor demonstrates problem-solving strategies and explains key concepts with clarity. Thereafter, students are gradually given space to apply these strategies on their own. Lessons will incorporate mathematical logic with the aid of colours, pictures, graphs, and diagrams, making concepts more accessible and more memorable for the student. Toward the end of each lesson, we review what has been learned, address any tricky areas, and set goals – possibly including some homework..

John Horton Conway
1937 – 2020

Conway was a British mathematician whose creativity and insight reshaped numerous areas of mathematics. He made groundbreaking contributions to group theory, number theory, and knot theory, and invented the surreal numbers, providing a new system extending the real numbers. Conway is perhaps best known for creating the cellular automaton Game of Life, which became a foundational model in complexity and computation. Renowned for his inventive thinking and playful approach to mathematics, his discoveries influenced combinatorics, algebra, geometry, and the study of mathematical structures, leaving a lasting legacy across pure and applied mathematics.

Kurt Gödel
1906 – 1978

Gödel was an Austrian-American mathematician and logician whose work fundamentally transformed the foundations of mathematics. He is best known for his incompleteness theorems, proving that in any consistent formal system of arithmetic, there are true statements that cannot be proved within the system. Gödel made pioneering contributions to set theory, proof theory, and mathematical logic, rigorously analyzing the limits of formal reasoning. Renowned for his extraordinary intellect and precise reasoning, his discoveries reshaped the philosophy of mathematics and influenced generations of mathematicians, logicians, and philosophers, leaving an enduring impact on the field.

Teaching Style

The teaching style is student-centric. Throughout the lesson the student is closely monitored, as the Tutor endeavors to see through the eyes of someone who is encountering these challenging mathematical concepts for the very first time. This helps identify where the student’s understanding is wavering and how to bridge the gap between their current knowledge and a higher-level of understanding. While students will be given the freedom to explore problems independently, guidance and support will be provided when necessary to ensure efficient and effective learning throughout the lesson.

Studnet Participation

Students have the freedom to interact as much or as little as they wish. Participation is encouraged, but never forced. The primary mode of engagement is through discussion and collaborative problem-solving, with questions welcome at any time during the lesson. It is also strongly encouraged that feedback be provided between lessons on a regular basis. This helps to keep lessons tailored to each student’s specific needs and to continuously improve the learning experience.

John von Neumann
1903 – 1957

John von Neumann was a Hungarian-American mathematician whose brilliance transformed mathematics, physics, and computer science. He made foundational contributions to set theory, functional analysis, and quantum mechanics, and developed game theory, which formalized strategic decision-making. Von Neumann also advanced operator theory, ergodic theory, and numerical analysis, laying the mathematical groundwork for modern computing and digital architecture. Renowned for his extraordinary intellect and versatility, his discoveries profoundly influenced pure mathematics and its applications, leaving a legacy that continues to shape multiple scientific fields.

Simple Proven Teaching Strategies

Revising a student’s learning material

Often the biggest issue with teaching math is that the student doesn’t exactly know why they dont understand something. As such, establishing exactly where a student is struggling needs to be the first step. The quickest way to do this is to inspect what they have been learning and identify exactly where they started to lose track.

Guiding a student through problems

Each topic in math tends to come packaged with a variety of new questions. This can be daunting for students, making it seem like there’s no end to what they must learn. But this isn’t true. Most problems can be solved via a very similar logical procedure. Students will begin to experience this repetitiveness firsthand as they work through more and more problems.

Reframing a topic or problem

This can make a topic or problem much more accessible from the student’s perspective. Reframing is an indispensable technique that is applied throughout mathematics. It can give the ugliest problem a beautiful solution and is central to proving theorems. Showing a student how to reframe and training them to do it themselves is a vital part of the learning process.

Illustrating a topic or problem

Humans are visual creatures. Our ability to make determinations based on sight is perhaps our defining characteristic. This is why providing students with good graphical representations is such an effective way to help them learn. This can be done via diagrams or sketches, via computational software, via short online videos, or by using physical objects to demonstrate the mechanics of a problem.