Place Value Explained

El método, paso a paso

1. 1

   Name the columns before reading the number

   4 7 0 2 → thousands, hundreds, tens, ones

   Reading a number aloud correctly is the first check that place value is secure. A child who says "four seven zero two" is reading digits; a child who says "four thousand, seven hundred and two" is reading value. The second is the one every later method depends on.

2. 2

   Give each digit its actual worth

   4702 = 4000 + 700 + 0 + 2

   Partitioning a number into what each digit is really worth turns an abstract string into a sum. This is not busywork — it is the exact move that makes column addition make sense later, because carrying is just a partition that got too big for its column.

3. 3

   Treat zero as a place holder, not nothing

   4702 vs 472 — the 0 holds the tens column open

   Zero is the digit that causes the most trouble, because children reasonably read it as "nothing there" and leave it out. It is not nothing; it is the instruction that this column is empty and the digits to its left must stay where they are. Remove it and every digit slides one column right.

4. 4

   Multiply and divide by ten by moving columns

   47 × 10 = 470 · 470 ÷ 10 = 47

   The digits do not change when you multiply by ten — they move one column left. Teaching it as "add a zero" works until decimals arrive, and then it fails badly: 4.7 × 10 is not 4.70. Teaching it as movement works for the whole of the rest of school.

5. 5

   Compare numbers from the left, not the right

   3891 vs 3907 → thousands equal, hundreds 8 < 9, so 3891 is smaller

   To compare two numbers, line up the columns and work left to right until they differ. The first column where they disagree decides it, and nothing to the right of that can change the answer. Children who compare by counting digits get this right by accident until the digit counts match.

The idea in one sentence

Our number system uses ten digits, 0 to 9, and gets every other number by position. That is the whole idea. A 7 is worth seven in the ones column, seventy in the tens column, and seven hundred in the hundreds column, and nothing about the symbol itself tells you which — only where it sits.

This is worth stating plainly because it is genuinely strange. Roman numerals do not work this way: X means ten wherever you put it. A child who has understood that position carries meaning has understood the single most important fact about written arithmetic, and one who has not will keep hitting walls that look like different problems but are all this one.

• Each column is ten times the one to its right
• A digit's value = the digit × its column
• Zero holds a column open so the others stay put
• Reading a number aloud correctly proves the idea is secure

The misconceptions that survive longest

The most persistent is "longer means bigger" — that 1,000,000 must be larger than 999,999 because it has more digits. It happens to be true here, which is why the misconception survives: it gives right answers for years before it gives a wrong one, usually the first time decimals appear and 0.5 has to be larger than 0.25.

The second is treating multiplication by ten as adding a zero. It is a rule that works on whole numbers and breaks on decimals, and because it breaks quietly — producing a plausible number rather than an obvious error — children keep using it for years after it has stopped being correct.

Where it goes next: decimals

Decimals are not a new topic. They are the same columns continued to the right of the decimal point: tenths, hundredths, thousandths, each one ten times smaller than the last, exactly as each column to the left is ten times bigger. A child secure in place value meets decimals as an extension; a child who is not meets them as a new and arbitrary system.

This is why we do not rush past place value in the early grades even when a student can already compute. The national curriculum for England places it at the head of the number strand in every year from 1 to 6 for the same reason — it is the foundation the rest of the strand is standing on.

How we teach place value

We diagnose it backwards. If a Grade 6 student is losing marks on column subtraction or long division, we do not start with the method — we ask them to read 4,702 aloud and to say what the 7 is worth. The answer tells us in thirty seconds whether we are fixing a procedure or a foundation, and those need completely different lessons.

When it is the foundation, we go back and rebuild it with partitioning rather than re-drilling the method that was failing. It feels slower to a parent watching a Grade 6 lesson spent on Grade 3 content. It is considerably faster than another year of the same errors.