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Session 4. Number Sense, Counting and Place Value
Objective. Check whether children really understand counting, and teach place value with concrete materials, ten frames and bundles of ten.
Counting looks simple to adults, but children first learn number words as a song and only later learn that each word tells how many. Place value brings a second surprise: the same digit means something different depending on where it stands. If these ideas are missing, children can still produce correct answers for a while by memory, and then fail suddenly in Grade 3 with larger numbers. This session shows how to check the ideas, and how to build them with materials that cost almost nothing.
Key ideas
Counting is more than reciting
- Many children recite 1 to 20 like a song but cannot count objects. Counting needs several ideas together (Gelman and Gallistel): a stable order of number words, one number word for each object, the last number tells how many, order does not change the total, and anything can be counted.
- A quick check you can do with one child in a minute: 'Give me seven buttons.' Then: 'How many are there?' See whether she points once to each button and whether she repeats the last number as the answer without recounting.
- Count real things children care about: pistachio shells, bottle caps, steps to the school gate, rows of cotton. Let them move objects as they count, which prevents skipping and double counting.
Number sense: seeing numbers as parts
- Subitising is seeing a small amount at a glance, like the dots on a die. Practise it with quick dot cards flashed for two seconds; this supports later counting and adding.
- Ten frames (two rows of five boxes) show numbers as parts of ten: 7 is five and two, and three empty boxes show that 7 needs 3 more to make ten. Draw them on paper if you have no printed frames.
- Ask 'more or fewer, and by how many?' all the time. Comparing and breaking numbers into parts (7 is 3 and 4) is the real foundation of addition and subtraction.
Place value: bundle, then write
- Give each pair 30-40 sticks (matches without heads, straws, or paper strips) and rubber bands or string. Children bundle ten sticks into one ten. A bundle is one object that means ten, which is the big idea of place value.
- Move in three steps: build the number with bundles and single sticks, then draw it (a line for a ten, a dot for a one), and only then write the digits in a tens and ones table. Always go back to bundles when a child is stuck.
- Use the number names. In Uzbek, o'n bir means ten and one and yigirma besh means twenty and five, so the name already shows the tens and ones. In Russian, одиннадцать hides the ten, so show it: один-на-дцать is one on ten. Count by tens aloud: o'n, yigirma, o'ttiz, qirq, ellik; десять, двадцать, тридцать, сорок, пятьдесят.
Zero and the usual mistakes
- Zero holds a place: 30 means three tens and no ones. Children who write twenty-four as 204 are writing the words in order (20 and 4) without seeing that the 4 sits in the ones place.
- Another mistake is comparing 39 and 41 by looking at the ones (9 is bigger than 1). Teach 'tens first': 3 tens or 4 tens decides. Let bundles show it.
- Exchange is the key move: ten single sticks can be bundled into one ten, and one ten can be opened into ten ones. Practise both directions before you teach regrouping in addition and subtraction.
From objects to numerals
What counting really needs
Children can recite numbers and still miscount objects. The Erikson Early Math Collaborative names five counting principles; the first three matter most in Grade 1: stable order, one number word for each object, and the last number tells how many. Scattered objects are harder to count than a row, because the child must also keep track of what has been counted. Teach to move each object across a line as it is counted.
A simple routine is a bag of 15 to 30 small objects for each pair. Children count them and record the total in their own way, with a drawing, marks or numerals. Walk around and ask 'How do you know there are 17?' and 'How did you keep track?'. The answers tell you more than a worksheet, and children enjoy the task.
Seeing numbers as parts
Children move through stages. First they count everything from 1. Then they count on from the larger number. Then they see numbers as made of parts: 7 is 5 and 2, and 6 and 4 make 10. The EEF guidance on early maths recommends concrete objects and building on what children already know, and ten frames do both. A child who sees 8 as a nearly full frame with two empty boxes can see 8 + 2 = 10 at a glance.
Move from objects to pictures to numerals, and go back whenever a child is stuck. Number facts then come from understanding, with memory as a help rather than the only support. Allow ten minutes a day for number sense: dot flashes, ten-frame games and 'how many more to make ten?'
Examples from the classroom
A bag of pistachio shells
Scene. Malika teaches Grade 1 in Samarkand. Each pair gets a bag of 20 pistachio shells and a sheet of paper. Children count and show the total in any way. Malika asks one pair 'How do you know it is 17?'. A boy had been touching shells without moving them and counted one twice; his partner shows him how to slide each shell into a pile.
Insight. A real counting task lets you see missing counting principles that recitation hides.
Ten frames from bottle caps
Scene. Aziz teaches Grade 2 in Bukhara. He draws a ten frame on the board and covers seven squares with bottle caps for two seconds. Children whisper 'five and two'. Then he asks, 'How many more to make ten?' and they hold up fingers. Next he does the same with nine and six. Within a week, the class answers number-bond questions without counting from 1.
Insight. Short, daily ten-frame flashes move children from counting all to seeing parts.
Eleven means ten and one
Scene. Dilnoza teaches Grade 1 in a Russian-medium school in Fergana. Many pupils write 11 as 101. She shows одиннадцать as один-на-дцать, one on ten, and builds it with one bundle and one stick. Pupils then build 12, 13 and 14 and say each name aloud. The following day, she asks them to build the number she writes, not the number she says.
Insight. Linking the number name to tens and ones, and building before writing, repairs a place-value confusion at its source.
Common pitfalls
- Judging counting only by hearing children recite 1 to 20 or 1 to 100.
- Teaching the numbers from 11 to 19 by memory alone.
- Moving to digits before children can build the number with bundles.
- Changing the material every day, so children spend their attention on the material and not on the idea.
Bundles and Boards
- In pairs, take 35 sticks, rubber bands and the place-value board from the Activity Kit. Do the 60-second counting check on your partner with 7 buttons and note what you saw.
- Work through the task cards: build the number with bundles, draw it, write it in the board. Say each number aloud in your own language and explain what the name shows about tens and ones.
- Now look at three sample pupil mistakes on the Mistake cards. For each, decide what the child understood, what is missing, and which bundle move would help.
- Plan a 15-minute place-value lesson for your own class: what you will build, what you will ask, and one quick check at the end for every child.
Your facilitator will guide you through this activity in the session.
Questions to discuss
- What did you see in the counting check that you would not notice by listening to a child recite numbers?
- Which step was hardest to explain to a partner: the exchange of ten ones for one ten, or the role of zero?
- How can you run bundles with 34 children and limited materials?
Reflect
- Which pupils in your class can recite to 50 but have not shown you they can count out 8 objects?
- What materials do you already have in the classroom that could become bundles?
- Which number name in your language already shows tens and ones, and how could you point it out?
Try it tomorrow
- Do the 60-second counting check ('Give me seven buttons', then 'How many?') with five children.
- Draw a ten frame on the board and flash 3, 7 and 9 bottle caps for two seconds each.
- Give each pair 20 sticks and two rubber bands and let them make and name three numbers between 11 and 29.
Further reading and links
Check counting by watching objects, not by listening to recitation, and teach place value by bundling tens before writing digits.