admin Mathematics is not only about learning formulas and applying standard procedures. At a higher level, mathematics is about recognising structure, making connections, choosing an appropriate approach, and solving problems that may not immediately reveal how they should be attacked.
This is one of the central ideas behind the new TMUA Mathematics video series from Mathematics Elevate Academy.
I am pleased to announce the release of the first two lessons of the TMUA Mathematics Masterclass, designed to help students strengthen their mathematical foundations while developing the reasoning and problem-solving skills required for challenging university mathematics admissions examinations.
Whether you are specifically preparing for the Test of Mathematics for University Admission (TMUA), working towards another competitive mathematics examination, or simply looking to develop stronger mathematical problem-solving skills, these lessons are designed to provide a valuable foundation.
Why TMUA Mathematics?
The TMUA requires more than familiarity with the mathematics syllabus.
Students need to be comfortable working with mathematical ideas, interpreting information, manipulating expressions, identifying patterns, and making deductions. Many problems require students to move beyond a familiar textbook procedure and decide for themselves which mathematical idea is most useful.
This makes TMUA preparation particularly valuable for developing broader mathematical ability.
The goal of this series is therefore not simply to prepare students to answer TMUA questions. It is to help students become more confident and flexible mathematical problem solvers.
A strong foundation in algebra, functions, equations, inequalities, geometry, trigonometry, calculus, probability, and other areas of mathematics can significantly improve a student’s ability to approach unfamiliar problems.
The first two lessons begin building that foundation.
Lesson 1: TMUA Mathematics Exam Introduction and Basic Algebra
The first lesson introduces the TMUA and establishes the mathematical approach that will underpin the rest of the series.
We begin by discussing the nature of the examination and the type of mathematical thinking students should develop when preparing for university mathematics admissions.
The lesson then moves into basic algebra.
Algebra is one of the most important foundations of advanced problem solving. A student may understand an advanced mathematical concept but still struggle with a problem if the underlying algebra is not sufficiently fluent.
The lesson therefore focuses on developing both technical accuracy and mathematical reasoning.
Topics covered
• Introduction to the TMUA examination
• Understanding the nature of TMUA mathematics
• Mathematical thinking and problem-solving
• Basic algebraic manipulation
• Algebraic identities
• Factorisation techniques
• Quadratic equations
• Roots of quadratic equations
• Relationships between roots and coefficients
• Common algebraic traps and mistakes
• Efficient techniques for TMUA problem solving
Rather than treating algebra as a collection of mechanical rules, the lesson encourages students to look at the structure of expressions and equations.
For example, factorisation should not simply be viewed as a procedure to memorise. Students should learn to recognise when an expression has a useful structure and how that structure can simplify a problem.
Similarly, quadratic equations provide an opportunity to develop relationships between coefficients, roots, graphs, and algebraic structure.
These connections become increasingly important as students progress towards more challenging mathematics.
Lesson 2: Algebra and Functions
The second lesson continues the development of algebraic thinking and introduces functions as a central mathematical idea.
Functions appear throughout advanced mathematics. They connect algebra, graphs, equations, transformations, calculus, and mathematical modelling.
For TMUA preparation, it is particularly important that students understand functions conceptually rather than treating function notation as simply another form of algebraic notation.
Topics covered
• Algebraic manipulation and simplification
• Factorisation and algebraic identities
• Algebraic equations and inequalities
• Functions and function notation
• Domain and range
• Restrictions on functions
• Composition of functions
• Inverse functions
• Functional reasoning
• Graphical interpretation
• Structural problem solving
• Common mistakes and inefficient approaches
• TMUA-style problem-solving techniques
The lesson focuses on the connections between different representations of mathematics.
A function can be represented algebraically, graphically, numerically, or through a mathematical relationship. Being able to move between these representations is an important part of mathematical maturity.
Students are encouraged to ask questions such as:
- What information does the function definition actually provide?
- What restrictions exist on the variables?
- What does the graph tell us?
- Can the expression be transformed into a more useful form?
- Is there a symmetry or structural property that can be exploited?
- Is there a more efficient way to reach the required result?
These questions are useful far beyond TMUA.
Beyond Exam Preparation
One of the main objectives of this series is to demonstrate that preparation for a demanding mathematics examination can also become an opportunity to strengthen mathematics more broadly.
A student who develops the habit of looking for structure, testing assumptions, connecting representations, and searching for efficient solutions is developing skills that are useful across mathematics.
For this reason, these lessons can be valuable even if TMUA is not the student’s only objective.
The same mathematical foundations can support preparation for:
• MAT
• STEP
• Olympiad Mathematics
• JEE Advanced
• A-Level Mathematics
• A-Level Further Mathematics
• IB Mathematics AA HL
• AP Calculus
• Other competitive mathematics examinations
The depth and style of questions may differ between examinations, but the underlying mathematical skills often overlap.
Strong algebraic fluency, logical reasoning, problem-solving ability, and mathematical communication provide a foundation upon which more advanced mathematics can be built.
Watch, Practise, and Elevate Your Understanding
Watching a mathematics lesson is only the beginning.
The greatest benefit comes from actively engaging with the mathematics.
When working through these lessons, I encourage students to pause before a solution is presented and attempt the problem independently.
Ask yourself:
- What information is given?
- What exactly am I being asked to find?
- What mathematical structure can I identify?
- Which methods could potentially work?
- Is there a shorter or more elegant approach?
- Can I verify my answer independently?
- What can I learn from the problem beyond its final answer?
This approach gradually changes the way you interact with mathematics.
Instead of asking only:
“Which formula should I use?”
you begin asking:
“What is this problem really telling me?”
That shift is fundamental to mathematical problem solving.
Building Mathematical Fluency
Mathematical fluency is developed through repeated exposure to different types of problems.
It is not enough to solve ten questions that all use exactly the same method.
A stronger approach is to encounter problems where:
- the relevant method is not immediately obvious;
- familiar concepts are combined in unfamiliar ways;
- algebraic manipulation is required before the main idea becomes visible;
- multiple approaches are possible;
- a seemingly complicated expression has hidden structure;
- a graphical interpretation provides a simpler solution;
- an apparently difficult problem can be reduced to a familiar mathematical idea.
This is the type of mathematical thinking that the TMUA Mathematics Masterclass aims to cultivate.
A Structured Journey Through TMUA Mathematics
These first two lessons are the beginning of a larger mathematics series.
The course will progressively develop the mathematical tools and problem-solving techniques needed to tackle increasingly challenging problems.
The emphasis will remain on:
Understanding
Developing a clear understanding of mathematical concepts rather than relying solely on memorisation.
Reasoning
Learning how to make deductions and connect different pieces of mathematical information.
Problem Solving
Developing strategies for unfamiliar and non-routine problems.
Efficiency
Learning to recognise when a particular method, identity, substitution, representation, or observation can simplify a problem.
Accuracy
Identifying common mistakes, checking solutions, and developing reliable mathematical habits.
Mathematical Maturity
Moving from routine procedural mathematics towards deeper mathematical thinking.
Who Should Watch These Lessons?
The lessons are particularly useful for students preparing for the TMUA, especially those who want to strengthen their foundations before progressing to more advanced problem solving.
They may also be useful for students who:
• Want to improve their algebraic fluency
• Find unfamiliar mathematics problems challenging
• Want to develop stronger mathematical reasoning
• Are preparing for competitive mathematics examinations
• Are transitioning towards university-level mathematics
• Want to move beyond routine textbook exercises
• Want to understand mathematics more deeply rather than simply memorise techniques
Students at different stages can use the lessons differently.
A student who is already comfortable with the fundamentals can use them as revision and focus on the problem-solving perspective.
A student who finds some of these topics difficult can use the lessons as an opportunity to strengthen their foundations before moving further into advanced mathematics.
Mathematics Elevate Academy
The TMUA Mathematics Masterclass is part of the broader educational work of Mathematics Elevate Academy, with a focus on mathematics education, advanced problem solving, and personalised mentorship.
At Mathematics Elevate Academy, I work with students preparing for a range of demanding mathematics curricula and examinations, including:
• TMUA
• MAT
• STEP
• Olympiad Mathematics
• JEE Advanced
• IB Mathematics AA/AI HL
• A-Level Mathematics and Further Mathematics
• AP Calculus
• SAT/ACT Mathematics
The broader objective is to help students develop not only examination technique, but also the mathematical understanding and problem-solving habits that support long-term progress.
You can explore the academy and its mathematics resources at:
Start With the First Two Lessons
The first two lessons are now available as part of the TMUA Mathematics series.
Start with:
Lesson 1
TMUA Mathematics Exam Introduction and Basic Algebra
An introduction to the TMUA together with the algebraic foundations needed for more advanced problem solving.
Lesson 2
TMUA Mathematics: Algebra and Functions
A deeper exploration of algebra and functions, with an emphasis on mathematical structure, reasoning, and TMUA-style problem solving.
Watch the lessons actively. Pause, attempt the problems yourself, compare different approaches, and revisit concepts that require further thought.
The objective is not simply to finish another mathematics video.
The objective is to come away thinking about mathematics differently.
Watch. Practise. Think. Elevate.
Strong mathematics begins with strong foundations, but it grows through curiosity, reasoning, and the willingness to tackle problems that do not immediately reveal their solutions.
These first two lessons are the beginning of that journey.
I hope they help you strengthen your mathematics, develop greater confidence with unfamiliar problems, and elevate your understanding.
About the Instructor
Rishabh Kumar is a mathematics educator, author, and founder of Mathematics Elevate Academy. He has taught mathematics across international curricula and competitive examinations, with a particular focus on advanced problem solving and personalised mathematics mentorship.
He is an alumnus of IIT Guwahati and the Indian Statistical Institute.
Through Mathematics Elevate Academy and the Math By Rishabh series, he develops mathematics courses, books, problem-solving resources, and advanced examination-preparation materials for ambitious students.
Explore More
Mathematics Elevate Academy
https://www.mathematicselevateacademy.com/
Math By Rishabh
https://mathbyrishabh.com/
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