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The evidence behind each concept

20 concepts. For each: what it means, what the child works with, and the sources that named that representation — quoted exactly, with links.

Only sources we have checked ourselves appear here: each one needs a link and the quoted words, or it does not get published. 67 such quotations across the 20 concepts.

Download all three pages as a PDF — generated from the same records, so it says exactly what this page says.

Maths

Multiplying by ten Proportional length scaling

Multiplying by ten makes every quantity ten times bigger. Because our numbers are written in places, and each place is worth ten of the place to its right, a quantity that becomes ten times bigger moves up one place: ones become tens, tens become hundreds. Nothing is added. The zero that appears in the ones column is there because nothing is left in the ones column — it holds the place. It is a consequence of the move, never the move itself.

7 sources for this choice:

Reading and setting a clock Analogue clock face

A clock face shows two different quantities with two hands on the same twelve marks. The same mark means a different thing depending on which hand points at it: the short hand near 3 means the hour is 3; the long hand at 3 means fifteen minutes, not three. The short hand moves continuously and spends almost the whole hour BETWEEN two numbers — where it sits tells you how far through the hour you are, which is why it is not yet the next hour just because it is close to it.

5 sources for this choice:

What each digit is worth Ten ones fused into one ten, then base ten pieces in named positions

TEN ONES BECOME ONE TEN — one thing you can hold, count and move as a unit. That fusing is not a step on the way to place value, it IS place value: everything else follows from it. Once ten ones are one ten, a written number stops being a row of separate digits. It can be taken apart into parts that add back to the whole — 73 is 70 and 3 — and, separately, the POSITION a digit sits in decides what that digit is worth: the 7 in 73 is not seven, it is seven tens. Those are two different things to hold, and the second is much harder than the first: a child can partition 73 into 70 and 3 and still say the 7 means seven.

4 sources for this choice:

A fraction as equal parts of one whole One whole partitioned into congruent parts — not built yet

A fraction names a quantity, not a pair of counts. It says: this whole was shared into a number of EQUAL parts, and we are talking about some of them. The bottom number is not a tally of cuts — it is the claim that the sharing was fair, and it is false if the pieces are different sizes. Equal means equal in AMOUNT, which is not the same as looking identical: a square cut corner to corner and a square cut into strips can both show quarters.

4 sources for this choice:

Which fraction is bigger Two bars of the same length

A fraction is one amount, so two fractions can be put in order the way two amounts can. Neither number on its own decides it: cutting a whole into more pieces makes each piece SMALLER, so a bigger bottom number does not mean a bigger amount. And the two fractions have to be parts of the same size whole, or the comparison means nothing.

3 sources for this choice:

Where a fraction sits on a number line A 0 to 1 line, the whole shown entire

A fraction is a NUMBER, so it has one place on the line — a point, not a piece. Finding it means deciding what one whole is on that line, cutting the space between 0 and 1 into the number of equal parts the bottom asks for, and counting that many along. The line may run to 5, or to 4, or to 1; the fraction does not move when the line changes, because it is a number and the line is only where we drew it.

5 sources for this choice:

Adding and taking away fractions with the same bottom number Two bars cut the same way

When two fractions are made of the SAME SIZE piece, adding them is just counting those pieces. Three eighths and two eighths are five eighths for the same reason three apples and two apples are five apples — the piece is the thing being counted, and the bottom number NAMES that piece rather than counting anything. So the bottom does not change: you have not changed what a piece is, only how many you have.

3 sources for this choice:

Adding and taking away in columns Dienes pieces on a column workmat, regrouped by the child

Columns work because ten ones ARE one ten — the same fusing that place value is made of, now run in both directions. Adding, you gather ones until you have ten of them and swap them for a single ten, which is what the small carried digit records. Taking away, when there are not enough ones you break a ten back into ten ones and carry on. Nothing is created or destroyed in either move: the amount is identical before and after, only how it is BUNDLED changes. That is why the digits may be rewritten and the number is still the number.

5 sources for this choice:

Adding, within twenty Two groups of counters, shown apart, then combined into one

Adding is putting two amounts together to make one amount. Once a child can do that by counting, the work is no longer the answer but the ROUTE to it: 8 + 7 is worth knowing as 8 + 2 + 5, because ten is easy to hold and the pieces either side of it are small. A child who only ever counts every object has the concept and none of the strategy.

2 sources for this choice:

Taking away, within twenty Jumps back along a 0-to-max number line, the equation always beside it

Subtracting is moving back along the numbers by a known amount and seeing where you land. The landing place IS the answer, so it is found rather than recalled. And the same fact can be asked the other way round — "what do I add to 4 to reach 13" is the same picture walked forwards, which is why a child who only ever counts back does not yet have the whole idea.

2 sources for this choice:

What coins are worth Real-denomination coins, acted on directly — never a numeral standing in for a pile

A coin is worth what it SAYS, not what it looks like and not one. This is the first thing a child meets where counting the objects gives the wrong answer: five pennies and one ten are six coins and fifteen, and the pile with more things in it can be worth less. So the question "how many?" has to be replaced by "how much?", and the amount has to be built by adding what each coin is worth rather than by counting how many there are. Different piles can be worth exactly the same, which only makes sense once value has come loose from appearance.

5 sources for this choice:

Sharing out and making groups Objects moved into containers, with whatever is left over still on the table

Division answers two different questions with the same calculation. "Share 12 between 3" asks what lands in each of three places — the answer is a SIZE. "How many 3s in 12?" asks how many places you can fill — the answer is a COUNT. Both are 12 ÷ 3, and a child fluent in one can be stuck on the other. And the calculation is not the answer. What division hands back is a quotient and sometimes something left over, and what the question wanted has to be read off that — sometimes the quotient, sometimes the next whole number up, sometimes the remainder itself. Twelve soldiers left over still need a bus.

5 sources for this choice:

Multiplying A filled rectangle — rows and columns the child builds, not counts

A product is not a longer sum. 6 × 9 can be COUNTED as six nines added up, and for whole numbers that works — but what it MEANS is a rectangle six along and nine down, or six groups each the same size. Holding it as a rectangle buys two things adding does not: turning it ninety degrees shows without argument that 6 × 9 and 9 × 6 are the same amount, and it keeps working when the multiplier stops being a whole number, where "add it up that many times" has nothing to say. Multiplying by a half is still a rectangle; it is not still a sum.

6 sources for this choice:

Making an equal fraction by doing the same thing to both numbers The equation with both denominators shown

If you cut every piece of a whole into the same number of smaller pieces, you have more pieces and each is smaller — and the amount you are holding has not changed. That is why doing the SAME operation to the top and the bottom leaves the fraction equal: the bottom says how the whole was cut and the top says how many you took, so cutting twice as finely and taking twice as many is the same amount described differently. The rule is not a trick about numbers; it is the arithmetic of re-cutting, and it only works because the operation is the same on both.

3 sources for this choice:

Adding fractions when the pieces are different sizes Two bars re-cut to a shared piece

Two fractions can only be counted together when they are made of the SAME size piece. Halves and thirds cannot be added as they stand, not because a rule forbids it but because there is no single piece to count: one half and one third is not two of anything. So the work is not the adding — it is re-cutting both wholes the same way first, which changes how the amounts are NAMED and not what they ARE. Once both are sixths, it is the same counting as before.

4 sources for this choice:

Two fractions that are the same amount Stacked lengths measured against each other

Two fractions can be written differently and still be the SAME AMOUNT. Cutting a whole into more pieces does not give you more — it gives you more, smaller pieces, and taking proportionally more of them lands you back where you were. So 1/2 and 2/4 are not two amounts that happen to be close: they are one amount with two names, and on a number line they are one POINT. The numbers going up is not the amount going up, which is the whole difficulty.

5 sources for this choice:

Reading, spelling and handwriting

Sounds and the letters that spell them One tile per grapheme — the letters that spell one sound move as a single piece, and say that sound when tapped

A written word is a record of the SOUNDS in a spoken word, and the piece that spells one sound is not always one letter. "sh" is a single sound written with two letters; "tch" is a single sound written with three; the two p's in "unhappy" are one sound. So counting the letters does not count the sounds — "thing" is five letters and three sounds. Knowing which letters spell which sound is what turns a word never seen before into a word that can be said, and a word that can be said into letters that can be written. A letter's NAME is a partial and unreliable route to this: most consonant names do contain their sound, and the letters where that breaks — c in cat, g in gate — are exactly the letters that spell more than one sound.

7 sources for this choice:

Forming letters and writing a word A stroke-ordered letter guide the child writes over — a start dot where the pencil goes, and guide dots running in the taught direction

A letter is a MOVEMENT, not a picture. "b" is a line pulled down from the top and then a bowl pushed round from the middle; a "b" that ends up looking right but was assembled as a circle with a stick beside it was drawn rather than written. The movement is what becomes automatic, and automatic is the whole point: while forming the letters still takes attention, that attention is not available for the word being spelt or the sentence being written. So writing a word by hand is two jobs at once — knowing which letters, and making them — and being fluent at the second is what lets a child spend everything they have on the first.

8 sources for this choice:

Spelling a word from memory One word, met whole — its meaning in a sentence, then its sounds, its position rule and its parts, then written from memory and returned to on a schedule

A spelling is not a row of letters to be remembered in order. Which letters a word takes is decided by three things at once: the SOUNDS in it, the POSITION those sounds sit in, and the PARTS the word is built from. The /ch/ at the end of catch is written tch because it follows a short vowel — as in match and fetch — while the /ch/ in chip is not; the /j/ at the end of bridge is dge for the same reason. Those are not exceptions to be memorised one by one, they are rules about WHERE. And unhappy is not a string, it is un + happy, which is why it keeps one p and why replay keeps the play you already know. A speller working from sound alone writes cach and brij: every sound correct, and the word still wrong.

11 sources for this choice:

Making sense of a passage The passage stays on the page — every question is answered with the text in reach, and the help ladder points back into it rather than around it

Understanding a story is not remembering it. It is building a picture of what happened that holds together — and the text never says everything, so parts of that picture have to be worked out rather than found. "She pulled her coat tighter" is not a sentence about a coat; it tells you it was cold, and nothing in the passage says so. That is why the text stays on the page: the answer to a question you cannot answer is to go BACK, find the line that bears on it, and put it beside what you already knew. A reader who never goes back is not reading carefully from memory — they have no way of noticing that their picture stopped making sense.

7 sources for this choice: