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Mathematics · Kindergarten-Grade 12

Eight strands. One argument. Thirteen years.

Mathematics taught as a thing you can argue with, by someone who uses it to make things fly. Personally, every class, one student at a time.

Request an assessment See the thirteen-year plan
a² + b² = c² 9 + 16 = 25
8
strands, opened in every stage, not arithmetic followed by algebra
1:1
live on Zoom, taught personally by a Master’s-qualified aerospace engineer
13
years, three stages, one continuous plan from Kindergarten to Grade 12
15
students in the practice, in total. Never a group. Never a stand-in.

01 The thing that gets lost

Somewhere around seventh grade, most students stop asking why.

Not because they have stopped wondering. Because they have worked out that a “why” question has no line on the mark scheme, and the method does.

Mathematics is the subject where a child is most often told the truth and still does not believe it, because nobody showed them why it has to be true. That gap is where a capable student quietly stops trusting the subject, and it is almost never a gap in ability. It is a gap in what they were allowed to ask.

So the work here is not covering topics faster. It is closing the distance between what a student can do and what a student can justify. A child who can justify a method can rebuild it when they forget it, which is what an examination actually tests once the questions stop being routine, and what every subject after school tests all the time.

There is a second reason, and it is the reason this is taught by an engineer rather than a mathematician: every technique on this page is one I have had to get right with something real depending on the answer. A student is told what a piece of mathematics was invented to do before they are shown how it works, which is the order in which it stops being arbitrary.

A student who can justify a method can rebuild it. A student who can only remember one is a bad week away from losing it. Abhishek Dabas · MSc Astronautics and Space Engineering

02 Where it all points

Thirteen years of this end at one idea, and you can watch it happen.

Below is the curve y = x2. Two points sit on it, joined by a straight line. As you scroll, the second point slides down towards the first, and the line’s gradient closes in on a single number. That number is the derivative. It is the most used idea in engineering, and a student who has watched it arrive never has to be told it.

The derivative Gradient of the chord as h → 0 f′(1) = 2

1 2 1 4 P (1, 1) Q
1.10h, THE GAP
3.41RISE, f(1+h) − f(1)
3.10GRADIENT OF THE CHORD
1.10DISTANCE FROM f′(1)

Nothing in that picture was estimated, and nothing was drawn to fit. The chord’s gradient is exactly 2 + h for every non-zero h, which is why it walks so calmly into 2, and why the answer never depended on how carefully anybody drew it. That is the whole of differentiation, and a Grade 5 student can follow the numbers in it.

They will not meet it as calculus until senior year. They meet it here as a question about a gradient, because the point of a thirteen-year plan is that the hard ideas arrive early enough to become familiar and late enough to be earned.

03 The eight strands

Mathematics is not a queue. It is eight strands, and your child works on all of them every year.

School treats the subject as a queue: arithmetic, then algebra, then the rest, one carriage at a time. It is taught here as eight strands opened together and reopened deeper each stage. Choose a stage and the whole map redraws itself at that depth.

Stage One · Sense

The first stage is about number as a quantity a child can feel, not a symbol they can copy. Two habits are built here because they are cheap to build now and expensive to repair later: saying out loud why an answer is right, and reading the equals sign as a balance rather than as here comes the answer. Almost every algebra difficulty at thirteen traces back to one of those two. Stage One runs alongside the school curriculum rather than repeating it.

8strands, every year
1:1live on Zoom
0rules given without a reason

Stage Two · Structure

The middle stage is where a rule stops being something to remember and starts being something that follows. Letters arrive as a way of saying something about every number at once, not as a harder kind of arithmetic. This is also the stage with a deadline attached: the placement decision that governs which mathematics is reachable by senior year is taken here, and it is taken quietly.

8strands, every year
1:1live on Zoom
1proof written per term

Stage Three · Argument

The final stage is taught at a level a first-year undergraduate would recognise, and maps onto whichever examination the family is sitting: AP Precalculus, AP Calculus AB and BC, AP Statistics, the Digital SAT, IB Mathematics AA and AI at both levels, GCSE and IGCSE, A-Level and Further Mathematics, and the UK admissions papers. The purpose underneath all of them is the same: a student who can construct an argument, defend it under questioning, and tell the difference between a rule and a theorem.

8strands, at depth
1:1live on Zoom
12examination routes covered

Number & quantity

What a number actually is
  • Counting, cardinality and place value
  • Addition and subtraction as one idea, not two
  • Multiplication as equal groups and as area
  • Fractions as parts of a whole, and on a number line
  • Estimation: is this answer even sensible?

Why it comes firstA child who cannot estimate cannot check, and a child who cannot check believes the calculator.

Algebra

The missing number
  • The equals sign as a balance, not an instruction
  • Missing-number problems, in both directions
  • Patterns continued, then described in words
  • Properties of operations, discovered rather than stated

Why it mattersAlgebra begins at six, in the way the equals sign is read. It just is not called algebra yet.

Geometry

Shape, space, and the words for them
  • Naming 2-D and 3-D shapes by their properties
  • Symmetry, tiling and tessellation
  • Position, direction and turning
  • Perimeter and area, by covering and counting

Why it mattersGeometry is where a young child first meets a claim they can check for themselves.

Trigonometry

Turning, angle and the right angle
  • Whole, half and quarter turns
  • Right angles, found in the room
  • Clock angles and compass directions
  • Why a triangle will not bend and a square will

Why it mattersRigidity is the first engineering idea a child can hold in their hands, and it is a geometry fact.

Functions & calculus

Machines that change a number
  • Function machines: in, rule, out
  • Doubling, halving, and undoing an operation
  • Reading a simple graph or chart
  • Growth patterns, drawn before they are calculated

Why it mattersA function machine at seven is the same object as f(x) at fifteen. Meeting it early removes the fear.

Probability & statistics

Collecting, sorting and reading
  • Sorting, tallying and pictograms
  • Bar charts, read and then drawn
  • Likely, unlikely, certain, impossible
  • Fair and unfair games, tested by playing them

Why it matters“Is this game fair?” is a probability question a five-year-old will argue about for an hour.

Reasoning & proof

Saying why you are sure
  • Explaining an answer out loud, every time
  • Finding an example that breaks a claim
  • Sorting by a rule, then guessing the rule
  • Logic puzzles and simple deduction

Why it mattersThis strand is the reason the page exists. It is the one school has the least room for.

Mathematical modelling

Mathematics for a real question
  • Measuring and comparing real objects
  • Money, time, and planning something small
  • Turning a story into a sum, and back again
  • Estimating before calculating, always

Why it matters“Estimate first” is the single habit that separates a student who checks from a student who hopes.

Why all eight, every year

These strands overlap on purpose. Trigonometry is geometry answered with algebra; calculus is a question about a gradient, and then about an area, and both are questions about a graph; a proof is what the reasoning strand does to a result from any of the others. Taught in a queue, a student meets each one as a fresh burden. Taught together, each one keeps explaining the last, and the subject starts to hold together on its own.

04 Three stages, thirteen years

One plan that starts in Kindergarten and ends with a college shortlist.

The same eight strands, three times, each at a depth the previous stage made possible. Here is what each one actually is.

Kindergarten-Grade 4

Sense

Feel the quantity, and say why you are sure.

Number as something a child can picture, not a symbol they can copy, alongside the school curriculum rather than on top of it.

The two things being protected here are small and cost almost nothing at this age: reading the equals sign as a balance, and explaining an answer out loud before writing it down. Both are ordinary habits at seven and hard-won repairs at thirteen, which is the entire argument for starting a mathematics plan in elementary school.

What a child is practising

  • Explaining an answer before writing it
  • Estimating first, then checking against the estimate
  • Saying “I don’t know yet”, then finding out
  • Spotting a pattern, then testing whether it holds
  • Finishing a problem that did not work the first time

At the end of a termA child who reaches for a reason before a rule, and who will tell you when an answer looks wrong.

Grades 5-8

Structure

Find out that the rule had to be that way.

Letters arrive as a way of saying something about every number at once. Fractions, ratio, proportion and early algebra stop being separate topics and start being one.

This is also the stage with a clock on it, for the reason set out further down this page: the middle-school track decides what is still reachable in senior year, and it is chosen on evidence gathered now. A student is prepared for the harder track deliberately, rather than discovering afterwards that a door had shut.

What a student is practising

  • Writing a line of algebra that is actually true
  • Proving a small number fact, in symbols
  • Killing a claim with a single counterexample
  • Reading a graph for its gradient, not its shape
  • Defending a method when it is questioned

At the end of a termA written proof, short and correct, that the student can talk through without notes.

Grades 9-12

Argument

Prove it, model it, and defend it under questioning.

Taught at a level a first-year undergraduate would recognise, and mapped onto whichever examination the family is sitting.

Examinations are taken seriously here, and prepared for the way an examiner would recognise, because the constraint is real. But they are the floor, not the ceiling. Every written note is scored against the same criteria, term after term, so four years read as a trajectory rather than a pile of grades, and a reference has something specific to say.

What a student leaves with

  • Calculus understood from first principles, not memorised
  • Proof by induction, contradiction and contrapositive
  • A modelling piece: IB internal assessment, EPQ or coursework
  • The examination results the family came for
  • A shortlist of courses chosen on evidence

At the endA student who can construct an argument, find the weak step in someone else’s, and tell a rule from a theorem.

The eight strands are fixed. The route through them is not.

There is no downloadable syllabus on this page because there is no fixed syllabus in the practice. Two students in the same grade, in the same week, are usually working on different things. What does not change is the map above: the eight strands, and the depth each stage reaches. What is written after the assessment is the route across it, for one child. It is shared with the family before the first class, and revised every term.

Fixed

Eight strands, three stages, and the standard each one has to reach.

Written for one child

The order, the pace, the entry point, and what gets skipped because it is already secure.

Revised every term

Against what the written notes actually show, not against what was planned in September.

05 How it is taught

Why is a negative times a negative a positive?

One question, answered three times. This is not a syllabus. It is what the same idea looks like at three different ages, and the clearest way to show you how your child would be taught rather than tell you.

Kindergarten to Grade 4

Follow the pattern until it tells you.

We write out a times table and walk it downwards, one step at a time, and the child reads the pattern out loud.

3 × 3 = 9
3 × 2 = 6
3 × 1 = 3
3 × 0 = 0
3 × (−1) = −3 Every line goes down by three. The child says the next one before I write it.

Then we do the same with a negative on the left, and let them predict.

(−3) × 2 = −6
(−3) × 1 = −3
(−3) × 0 = 0
(−3) × (−1) = ? Every line here goes up by three. The child says “plus three” before I do.
What has happenedNobody has been told a rule and nobody has been asked to remember one. The pattern did the work, and the child was the one who finished it.

Grades 5 to 8

The rule is forced. It could not have been anything else.

By this age a pattern is not enough, and it should not be. So we write down only the things a student already uses without thinking — that anything times zero is zero, that multiplying by one changes nothing, and that brackets multiply out the way they always have — and see what those three force.

0 = (−1) × 0
  = (−1) × (1 + (−1))
  = (−1)(1) + (−1)(−1)
  = −1 + (−1)(−1)

The left-hand side is zero. So (−1)(−1) has to be the number you add to −1 to get zero, and there is only one of those.

(−1)(−1) = 1 Not a convention. A consequence.
What has happenedFor many students this is the first time a rule they were handed in an earlier grade turns out not to have been somebody’s decision. It was the only option available.

Grades 9 to 12

It is not a rule at all. It is a theorem.

At this level the question changes shape. We stop asking what the answer is and start asking what we assumed in order to get it. The argument above used exactly four properties: that multiplying by zero gives zero, that multiplying by one changes nothing, that the distributive law holds, and that additive inverses are unique.

All four hold in any ring — any system with addition, subtraction, and a multiplication that distributes over them — so the result holds in every one of those too, including systems that look nothing like the integers.

In any ring R,  for all a, b ∈ R:
(−a)(−b) = ab The same argument, with the numbers taken out.
What has happenedWhat a student takes away is not the identity. It is that some of what they were told were rules are consequences, and that they are allowed to ask which is which. That habit is the difference between a student who is good at mathematics and a student who is still good at it at university.

Argue with it yourself.

Two quantities start at 100 and take an identical first step. One keeps adding that same amount. The other keeps multiplying by the same factor. Move either slider and watch what happens to a straight line when it has to race an exponential.

f(n) = a(1+r)n vs g(n) = a(1 + rn)

Both quantities start at 100 and are given the same first step on purpose, so any gap that opens afterwards is compounding, and nothing else. The bars are drawn on a square-root scale, because at the higher growth rates a linear scale flattens the straight line into the axis and leaves nothing to compare it with.

340ADDING, AFTER n STEPS
1,006MULTIPLYING, AFTER n STEPS
3.0×HOW MANY TIMES AHEAD
9.0STEPS FOR MULTIPLYING TO DOUBLE, ln2 / ln(1+r)

06 What is actually produced

Mathematics that exists on paper afterwards, not only in the hour.

Not a certificate. Not a folder of worksheets. Written reasoning, kept, so that a year later there is something to look back at and measure.

K-4 · every week

An explanation, in the child’s own words

  • A method written out as the child would say it aloud
  • An estimate made before the calculation, and checked against it
  • A pattern found, then tested to see whether it holds
  • A problem they got wrong, redone from the beginning
  • A written note to the family after every class
5-8 · one proof per term

A short proof, and a defence of it

  • Writing a small proof out in full, in symbols
  • A counterexample that kills a plausible claim
  • A real-world situation turned into equations, assumptions stated
  • A graph read for its gradient and what that gradient means
  • A method defended out loud when I argue against it
9-12 · examination plus one extended piece

Work a university would recognise

  • Calculus derived from first principles, not quoted
  • Proof by induction, contradiction and contrapositive
  • A modelling piece: IB internal assessment, EPQ or coursework
  • Whole papers worked under timing, then marked line by line
  • A critique of a published statistical claim

07 The timing

The middle-school decision nobody announces.

Whether a student reaches calculus by senior year is largely settled by the mathematics track they are placed on in middle school. That placement is rarely presented to a family as a decision, and it quietly shapes which courses, which colleges, and therefore which careers stay reachable. Stage Two exists to make a student ready for the harder track before the placement is taken, rather than to appeal it afterwards.

Grades 5 to 7

While the placement is still open

Fractions, ratio and proportional reasoning made secure, and the first real algebra met early enough that it is familiar rather than new.

Grade 8

The year the track is set

Algebra taken on the accelerated track where it is offered, which is what puts calculus, and a second science subject, within reach by senior year.

Grades 9 to 12

What that track makes possible

AP Calculus BC or IB HL rather than a terminal course, a competitive Digital SAT Math score, and a transcript that reads as intent.

08 Where this leads

Every examination a family is likely to sit, and the thing underneath them.

Examinations are taken seriously, prepared for properly, and are still not the point. They are the floor. The ceiling is a student who can build an argument and hold it.

Examinations covered

AP PrecalculusAP Calculus ABAP Calculus BC AP StatisticsDigital SAT Math IB Mathematics AA, SL & HLIB Mathematics AI, SL & HL IGCSE, CIE 0580 & 0606GCSE, Edexcel · AQA · OCR A-Level MathematicsA-Level Further Mathematics MAT · TMUA · STEP

Twelve routes. MAT, TMUA and STEP are the UK university admissions papers, sat alongside A-Level. Work is aligned to Common Core where a school follows it, and school work at any level from Kindergarten to Grade 12 is covered whether or not an examination is involved.

Immediately

The grade the family came for

Past papers, mark schemes and examiner reports, used properly rather than endlessly.

By senior year

Courses that stay open

Calculus, statistics and physics taken at the higher level rather than at whichever level was on offer.

At application

A transcript that reads as intent

Engineering, physics, computer science, economics: every one of them gated by this subject.

After that

The part that does not expire

Knowing what you have assumed, and being able to say so out loud.

Why an engineer, and not a mathematician

Almost nothing in school mathematics was invented for its own sake. Logarithms exist because someone had to multiply enormous numbers by hand. Trigonometry exists because someone had to find a position at sea. Calculus exists because someone needed to know how fast a thing was changing at an instant, and could not get it from an average. Taught in that order, with the problem before the machinery, the subject stops being arbitrary, and the question of what it is all for answers itself.

Used, not just taught

Every technique on this page is one I have had to get right in an engineering context.

Honest about difficulty

Some of it is hard. A student is told which parts, and why, rather than left to conclude that the problem is them.

Connected to physics and space

The same student can carry one idea across three subjects in the same week, with one teacher.

09 How it runs

Weekly, one to one, in your time zone, and written down afterwards.

Weekly, one to one

Live on Zoom, taught personally every session by a Master’s-qualified aerospace engineer. Never a group. Never a stand-in tutor. One consistent weekly slot per child, protected.

On U.S. time, recorded

Scheduled to Eastern, Central, Mountain or Pacific, after school or on weekends. Every class is recorded, with a written note on progress after each one.

Written, not just spoken

The written record is what lets a parent watch the reasoning improve rather than take anybody’s word for it. Every note is scored against the same criteria, term after term.

Abhishek Dabas, who teaches every Insight Bay class personally

Taught by

“The students I take are rarely behind. They are usually ahead, and unchallenged.”

  • Abhishek Dabas, MSc Astronautics and Space Engineering
  • Teaching one to one since 2015; every class taught personally
  • A deliberately small practice, by design
  • Mathematics, physics, space exploration, flight theory, Six Sigma and project management, Kindergarten through Grade 12
Read about the practice

10 Before you ask

The questions families actually ask.

Is this tutoring, or is it a curriculum?

Both, and in that order. If your child has a test on Thursday, that is what Tuesday is for, whatever grade and whatever board.

But the plan underneath runs across eight strands and thirteen years, which is what makes this term’s work a step along a route rather than a repair to last week.

There is no fixed syllabus. So what will my child actually study?

The eight strands are the territory, and they do not change. The route across them is written for one child after the assessment, shared with you before the first class, and revised every term.

Keeping the practice small is what makes that possible. You will see the plan, in writing, before you are asked to commit to anything.

My child gets top grades and is bored. Is this for them?

Yes, and that is most of the practice. A student who can already execute the procedure is precisely the student with room to be shown why it is true.

That is also the work that keeps paying later: it is what stands up when the questions stop being routine, and what a university course assumes was already built.

My child is six. What does mathematics even look like at that age?

Counting, comparing, measuring, and being asked to say why. Stage One builds two habits: explaining an answer out loud, and reading the equals sign as a balance rather than as “here comes the answer”.

Almost every algebra difficulty at thirteen traces back to one of those two, which is why a mathematics plan is worth starting in elementary school rather than in the year before an examination.

Do you teach to the examination, or around it?

To it, without teaching only it. Past papers, mark schemes and examiner reports are used properly, because an examination is a real constraint with real rules and a student is entitled to know them.

But a student who can only reproduce a method loses marks the moment a question stops being routine, and there is no quantity of past papers that fixes that. Understanding does.

Why one to one rather than a small group?

Because the plan is written for one child, after watching them work, and revised every term. In a group the pace is set by the median student, so the child who is ahead and unchallenged — which describes most of the children who come here — stays that way.

The practice is deliberately small. Every class is taught personally, never by a stand-in tutor.

What does our family receive after each class?

A recording of the session, and a written note on progress: what was worked on, what the reasoning looked like, and what happens next.

The written record is the point. It is what lets you watch the reasoning improve rather than take my word for it, and it is what makes a reference letter specific four years later.

How does a family begin?

With the seventy-five-minute assessment described at the foot of this page. There is no form to fill in beyond your name and a way to reach you, and no obligation attached to it.

How every family begins

Begin with an assessment.

Seventy-five minutes: forty-five with your child, working real problems, and thirty with you. Within forty-eight hours you receive a written assessment of your child’s reasoning: what they understand, what they only appear to understand, and what I would do about it. Yours to keep, whatever you decide.

Request an assessment Programmes and fees
Kindergarten-Grade 4 Sense

Feel the quantity, and say why you are sure. The equals sign as a balance, and an answer explained before it is written.

Grades 5-8 Structure

Find out that the rule had to be that way. The stage with a clock on it, because the track for senior year is set here.

Grades 9-12 Argument

Prove it, model it, defend it under questioning. The examination results, and the part of this that does not expire.

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