Begin with something the learner can see, feel or reason about before introducing formal notation.
My approach to mathematics
I didn't always understand maths.
That experience shaped everything I now do. I spent years trying to memorise mathematics that made little sense to me. When I finally stopped trying to remember procedures and started rebuilding the subject from ideas I genuinely understood, my results — and my relationship with mathematics — changed completely.
Where it began
Understanding changed everything.
Like many students, I enjoyed maths when I was younger. Then algebra arrived and it began to feel like manipulating symbols on a page for no obvious reason. By university I was badly lost. I would see equations and formulae without really understanding the language they were written in.
I blamed myself. I tried harder to memorise methods, but memorising things I did not understand simply gave me more things to forget. After two difficult years I realised I had to stop pretending I understood and go back to a point where I genuinely did.
I spent a summer rebuilding my mathematics from there. The difference was dramatic: results that had been around 30% rose to over 90%. I became so enthused by the subject that I moved further into mathematics and began tutoring other students.
The breakthrough wasn't that I had suddenly become better at memorising maths. I finally understood what I was doing — and why.
Those early students became my real education as a teacher. Again and again I saw the same problem: capable people had learned mathematics as a collection of rules and procedures rather than a connected system of ideas. For more than 25 years, I have been trying to find clearer, more intuitive ways to reveal those connections.
The philosophy
A different way
of learning maths.
My aim is not to give students more rules to remember. It is to reduce the number of disconnected ideas, make the underlying mathematics visible, and build from understanding into fluency.
Knowing why a method works makes it easier to remember, adapt and use when a question looks unfamiliar.
A small number of powerful ideas can often replace many apparently unrelated methods, reducing cognitive load and increasing confidence.
The methods
Mathematics can be
simpler than it looks.
I continually look for ways to make methods more efficient, intuitive and connected — without sacrificing the mathematics underneath them.
Fast, intuitive calculation
I developed alternative approaches to arithmetic, including multiplication, that reduce unnecessary written steps and reveal reusable patterns. One strong idea can then support several different areas of mathematics rather than requiring a new algorithm each time.
See it before you symbolise it
Visual and numerical understanding comes first wherever possible. Algebra is then introduced as a language for something the student already understands, rather than as unexplained symbol manipulation.
Advanced mathematics, made intuitive
At A Level and beyond, I focus on the meaning behind differentiation, integration, indices, quadratics, trigonometry and graphs, including where important formulae come from and why they work.
Binomial expansion & powers
I have developed an intuitive approach that unifies ideas normally presented through separate formulae, including a way to navigate Pascal's Triangle and find coefficients of binomial expansions. Elements of the approach have been successfully understood by students as young as 12.
Find the real point of difficulty
A major part of effective tuition is diagnosis. I try to identify exactly where understanding first breaks down rather than repeatedly treating the visible symptom. Once that missing connection is repaired, topics that seemed difficult can begin to fall into place surprisingly quickly.
Advanced doesn't have to mean difficult.
Beyond the syllabus
Potential is often hidden by presentation.
I do not believe an idea is automatically difficult simply because it appears later in a curriculum. Sometimes the difficulty lies in how the idea has traditionally been presented.
When sophisticated mathematics is built from ideas a learner already understands, younger students can often engage with concepts that would normally be considered far beyond their level. The objective is not acceleration for its own sake. It is to let students discover how much they are capable of understanding.
This matters equally for students who are struggling. Very often the problem is not a lack of mathematical ability. It is that one crucial idea never became secure, and everything built on top of it has consequently felt unstable.
The moment a learner thinks “I can do this.” changes everything.
The Feeling of Power
Confidence follows understanding.
The most rewarding moment in teaching is often not the final grade. It is the instant a student who has spent months or years believing they are “bad at maths” suddenly sees something clearly.
Understanding creates confidence. Confidence changes effort. Effort creates further success. That positive cycle is what I call the Feeling of Power.
It is the philosophy behind my tuition and my teaching: mathematics should not leave learners feeling that clever people possess a collection of secrets they somehow missed. It should become something they can understand, predict and ultimately own.
Eventually, I started turning those ideas into software. I designed and built ACE Maths Pro so students could experience the same approach outside a lesson — and I made it completely free so anyone can use it.
About Paul Carson
Mathematician.
Engineer.
Teacher.
My route into teaching has never been particularly conventional. Alongside more than 25 years of private mathematics tuition, I have studied mathematics, physics and engineering, worked in military aviation, written books, lectured for the Open University and developed educational technology.
That mixture matters to my teaching. Engineering continually asks what is actually happening, why a system behaves as it does, and whether there is a clearer or more efficient way to solve a problem. I bring the same questions to mathematics.
A teacher who still learns.
I deliberately continue to study. Being a learner keeps me close to the experience of confusion, persistence and eventual understanding — the same process I ask students to go through. My continuing interests include advanced mathematics and language learning.
25+ yearsPrivate mathematics tuition across school, A Level and university mathematics.
BScUniversity of St Andrews — studies in physics and mathematics.
BEng (Hons)Engineering degree from the Open University, awarded 2:1.
Open UniversityAssociate Lecturer in Mathematics & Engineering.
RAF aviationFormer aircraft/avionics engineer with experience in military aviation.
2 booksNaked Numbers and Guarantee A Top Grade in GCSE Maths.
ACE Maths ProDesigned and built by me to turn mathematical understanding into fluency and confidence — completely free to use.
And yes…I have flown in a Tornado GR4 — and I have also appeared on Netflix.
25 years later, I'm still asking the same question:
Can I make this easier to understand?
Continue exploring
See the philosophy
in practice.
Whether through one-to-one tuition, ACE Maths Pro or the wider philosophy behind my teaching, the objective remains the same: understanding first, then confidence and capability.
Quick maths tool
Prime Factoriser
Choose a whole number and watch it break down into prime factors. Move the slider and the factor tree changes instantly.
Factor tree move the slider to explore
Prime numbers are highlighted using ACE blue and green.
Explore the maths
More visual tools.
Small interactive explorers designed to make the structure visible — move a slider, change a value and see the mathematics respond.
HCF & LCM Explorer
Compare two numbers through their prime factors. Shared factors are highlighted so you can see where the HCF comes from, while the LCM collects everything needed.
36
48
HCF12
LCM144
36
48
Gradient Explorer
Move two points and watch the line, rise, run and gradient update together.
4
3
m = 3/4height 3 ÷ width 4
Circle Explorer
Change the radius and watch diameter, circumference and area update from the same circle.
5
Diameter = 10 cmCircumference = 10π cm ≈ 31.42 cm
Area = 25π cm² ≈ 78.54 cm²Same radius, two different measurements: around the circle and inside it.
Trig Triangle Explorer
Slide the angle and watch opposite, adjacent and hypotenuse change together. The ratios update live.
35°
1
Height = 0.57 · Width = 0.82Hypotenuse = 1
sin θ0.574
cos θ0.819
tan θ0.700
Standard Form Explorer
Enter a number, see the power-of-10 position of every digit, then see the same number written clearly in standard form.
live
Each digit sits in a power-of-10 position
STANDARD FORM
3.451 × 10³
3,451
3×10³ + 4×10² + 5×10¹ + 1×10⁰
Number Systems Explorer
Decimal, binary and hexadecimal are the same place-value idea with different bases. Enter a value in any system and see the powers behind every digit.
Input systemDEC
0–65535
Binary display8 bit
Decimal45
Binary00101101
Hex2D
Decimalpowers of 10
Binarypowers of 2
Hexadecimalpowers of 16
A = 10 · B = 11 · C = 12 · D = 13 · E = 14 · F = 15
x² Area Explorer
See x² as an actual square. Increase the side by 1 or by dx, then make dx smaller and watch how the extra area changes.
4
Increase side by1
1
Slide right to make dx smaller: 1 → 0.1 → 0.01 → 0.001
Original area16
New area25
Change in area
Rectangle 14x × 1
Rectangle 24x × 1
Corner square11 × 1
Total change in area = 9
Change in area without the corner square = 8
Change in area ÷ side change9
With a side change of 1, the extra area is two rectangles plus the 1 × 1 corner square.
x³ Volume Explorer
Start with a cube of side x, increase the side by 1 or by dx, and see how the extra volume is built from larger slabs, thinner edge pieces and one tiny corner cube.
3
Increase side by1
1
Slide right to make dx smaller: 1 → 0.1 → 0.01 → 0.001
Original volume27
New volume64
Change in volume
3 face slabs273 slabs of 3 × 3 × 1
3 edge pieces93 pieces of 3 × 1 × 1
Corner cube11 × 1 × 1
Total change in volume = 37
Change from the 3 large slabs only = 27
Change in volume ÷ side change37
With a side change of 1, the extra volume comes from 3 large slabs, 3 thin edge pieces and 1 small corner cube.
Large slab
x × x × 1
×3
Thin cuboid
x × 1 × 1
×3
Small cube
1 × 1 × 1
×1
Ready to see maths differently?
Explore tuition, ACE Maths Pro, or get in touch to discuss how I can help.