Why 34 is not '3 and 4': helping children understand place value

A child writes the number 34 correctly, reads it aloud as "thirty-four", and still, deep down, sees it as a three next to a four. When you ask what the 3 means, the answer is "three". It is easy to miss, because the writing and the reading are right. But the child has not yet grasped the idea that makes our number system work: the value of a digit depends on where it sits. That idea is called place value, and it is probably the biggest conceptual step in the early primary years. This article explains what it involves, how to spot a shaky understanding, and how to build a solid one at home.

Written for children aged 6 to 8.

What place value really says

In the number 34, the 4 is worth four ones and the 3 is worth three tens, which is thirty. The number is thirty plus four. Our system uses just ten symbols (0 to 9), and gets all its power from position: the same digit means something different in each place. In 333, the same digit appears three times and means 300, 30 and 3.

This is a clever idea, and a surprisingly modern one in human history. The earliest counting systems, such as Roman numerals, did not use it, which is part of why arithmetic with them is so cumbersome. Position-based notation is what makes written addition, subtraction and multiplication feasible. For a young child, learning it is a genuine intellectual leap.

How to tell whether the idea has landed

Children can look as though they understand place value long before they do. A few gentle checks reveal the difference.

Ask what the digit means. Write 34 and ask, "What does the 3 stand for?" A child with real place value will say "thirty" or "three tens". A child who is still seeing digits as separate numbers says "three".

Ask which is bigger. Which is larger, 29 or 31? Children who look only at the digits sometimes say 29, because 9 is bigger than 1. A child who understands that 3 tens is more than 2 tens will say 31 straight away.

Build it, then say it. Ask your child to make 27 using sticks or blocks. A child with the concept makes two bundles of ten and seven singles. A child without it may count out 27 singles, which is correct but shows they have no notion of tens as a unit.

Ask "what if one more?". If 39 gains one, what happens? Children who understand that ten ones make a ten, and that the units column resets to zero while the tens column goes up by one, can say 40 and explain why. Many children simply recite "forty" because they have learnt the sequence, and they cannot explain.

Mixing up the reading. Children sometimes write "thirty-four" as 304 (thirty, then four) or reverse digits in numbers like 14 and 41, particularly in English where the teens run "fourteen" but you hear the four first. These are common and usually point to place-value understanding that is still forming.

Building the idea with real objects

The key is to let children group real things into tens, and then see the written numeral as a record of the groups.

Bundling. Collect drinking straws, lollipop sticks or dry pasta. Count out ten and fasten them with an elastic band. Count out another ten and do the same. Now make 34 by combining three bundles of ten with four loose items. Counting a pile of 34 singles takes a long time; counting 10, 20, 30, 31, 32, 33, 34 is quick. The child discovers why grouping helps, which is half the point.

Ten-frames and base-ten blocks. If you have access to base-ten blocks (small cubes for ones, long rods for tens), use them. If not, strips of ten squares cut from squared paper work just as well. The important feature is that a ten is one thing, a rod, that is also ten of the small things. Exchange games help: "I have ten small cubes; can I swap them for a rod?"

Coins. Ten pennies for a dime is a real-world version of the same idea in many countries. Playing shop with a pile of one-unit coins and a few ten-unit coins gives an authentic reason to group.

Write it in two places. Put two cups on the table, one labelled "ones" and one "tens". Place counters in the cups to show a number. When the ones cup gets to ten, swap the ten counters for one in the tens cup. This "trading" shows how position and value are linked.

Language and numerals

Number words can help or hinder. English "twenty, thirty, forty" mirror tens well, but the teens do not. French goes its own way with soixante-dix, quatre-vingts and quatre-vingt-dix, which can obscure the tens structure for a while. Whichever language you speak, say "three tens and four ones" alongside "thirty-four" for a while, and read numbers aloud in both ways, so the structure is audible.

Writing digits in a table with columns for tens and ones also helps, and it is a nice moment when children notice that placing a zero in the ones column simply says "no ones".

Games that build the idea

Guess my number. "I have 4 tens and 7 ones. What number am I?" Reverse: "I'm 62. How many tens and ones?"

Greater or smaller. Draw two digits from a bag, such as 3 and 8. Which makes the bigger two-digit number, 38 or 83? Children quickly discover that the digit you place first matters most.

Race to 100. Roll a die; add that many singles to your pile. Whenever you reach ten, swap for a bundle. The first to a hundred wins. The game practises grouping, exchanging and counting in tens.

Number hunt. Look for two-digit numbers at home, on doors, on buses, on book pages, and say each one as "tens and ones".

What to keep in mind

Place value is a big idea, and it takes children months, sometimes a whole school year, to feel at home with it. Let them handle objects for as long as they need. Notice and praise explanations, not only answers. And be patient with mistakes: a child who reads 34 as "three-four" is not careless, they are working out one of the most elegant systems human beings have invented.

When a child can look at 34 and say "that's three tens and four ones, so thirty plus four", they have crossed a significant threshold, and almost every part of later arithmetic, from carrying in addition to long division, will rest on it.

Try it in the app

Practise this at the screen too: Compare two-digit numbers, Partition three-digit numbers, Add and subtract 10 or 100. You may also like our articles on mental maths strategies and number bonds.