How counting tells how many

Why this exists

When a child counts a group, the last word said is not the name of the last thing touched: it is the size of the whole group. That number belongs to the group, so it stays the same when the things are moved, bunched up or spread out, and it comes out the same whichever thing is counted first. These four days take a child who can already count a set of objects and make that last word mean “how many”, for groups up to 20 in lines, rings and rows.

Each day takes about ten to fifteen minutes. Ask “how many?” often, and ask “how do you know?” even more.

Day 1: The last word is all of them

You need: about 15 pom-poms, a paper bag, a hand puppet (a sock works).

Have the child drop pom-poms into the bag one at a time, saying a counting word with each drop. Stop at 5 and fold the bag shut. Ask, “How many pom-poms are in the bag?” The child cannot see them or count them again, so the only way to answer is to remember the last word. That is the point: “five” is now a fact about what is in the bag.

Next, pour out a new group and let the child count it. Then say, “Show me the five.” A child who points at one pom-pom is treating “five” as that pom-pom’s name. Sweep your hand around the whole group and say, “These are five. All of them.” Have the child copy the sweep.

Last, let the puppet count and get it wrong on purpose. It counts a group correctly and then says a different number, or it starts over from one when asked “how many”. The child explains what the puppet should have said.

Practice.

1. The child drops 3 pom-poms into the bag, counting. How many are in the bag? Answer: 3, the last word said.
2. The puppet counts “one, two, three, four, five, six” and then says, “There are two.” What should it say? Answer: six, because “six” was the last word.
3. The child counts 11 pom-poms. You say, “Show me the eleven.” Answer: the child sweeps a hand around the whole group, not one pom-pom.
4. The puppet counts 4 pom-poms, you ask “How many?”, and it starts counting again. What could it say instead? Answer: “Four”, straight away.

Day 2: Moving them does not change how many

You need: 10 wooden clothespins, a paper plate.

Lay the clothespins in a straight line and have the child count them: 10. Now push them into a ring on the plate while the child watches, adding nothing and taking nothing away. Before anyone counts, ask, “How many clothespins now?” Some children will want to count again; let them, and the count comes out 10. Then ask, “Did we need to count? Why not?” The answer to aim for: we only moved them.

Then do the one change that does matter. Add one clothespin to the ring and ask, “Are there still 10?” Moving keeps the number; adding or taking away changes it.

Practice.

1. Count 10 clothespins in a line, then clip them all around the rim of the plate. How many? Answer: 10; they were only moved.
2. Count 7 clothespins, then sweep them into a heap. How many? Answer: 7.
3. Split 12 clothespins into two heaps, then push the heaps back together. How many? Answer: 12.
4. Mia counts 15 clothespins, then clips them all onto the edge of her shirt. How many are on her shirt? Answer: 15.
5. Count 8 clothespins in a ring, then add one more. How many now? Answer: 9, because one was added, not just moved.

Day 3: Start with any one

You need: 6 different things from around the room, such as a spoon, a key, a sock, a ball, a toy horse and a glue stick.

Set the things out and have the child count them, starting with the spoon: 6. Then ask, “Can you count them again, but start with the sock?” The words land on different things, and the last word is still “six”. Try once more starting from the thing in the middle.

Now play “make it number one”. Say, “Make the key be number one.” The child has to start the count at the key. Then, “Make the key be number six,” so the key has to be counted last. A thing does not own a counting word; it just gets whichever word comes next. The total does not move, as long as each thing gets exactly one word.

Practice.

1. Count the 6 things starting with the ball. Answer: 6.
2. Make the key be number one. How many things are there? Answer: still 6.
3. Make the key be number three. Answer: the child counts two other things first, then says “three” on the key, and finishes at 6.
4. The puppet says, “If I start with the horse, I’ll get more.” Is it right? Answer: no. Any start gives 6, because each thing still gets one word.

Day 4: Big groups keep their number too

You need: 20 pom-poms, a small sticky note, an empty ice cube tray with 2 rows of 8 cups.

The child already knows how to count rows and rings from counting a set of objects: go one row at a time, and in a ring, mark where the count starts. Today the groups are bigger, and the question is whether the number still belongs to the group. Set out 15 pom-poms in three rows of five and have the child count them: 15. Now sweep them into a ring and ask, “How many now?” The answer to aim for is “fifteen, we only moved them.” Let the child check: stick the note beside one pom-pom, count round from it, and stop just before reaching it again. Then move the note to a different pom-pom and ask, “Will you get a different number this time?”

Ask, “Did we need to count the ring? Why not?”

Practice.

1. How many objects are in the ring in the picture? Answer: 12, counting from the marked one and stopping just before it comes round again.
2. How many objects are in the rows in the picture? Answer: 15, five in each of three rows.
3. If those 15 were pushed into a ring, how many would the ring have? Answer: 15, with no counting needed; they were only moved.
4. Put one pom-pom in each cup of the ice cube tray and count them. Then tip them into a heap. How many in the heap? Answer: 16 both times.
5. Set 18 stuffed animals in a ring. Count them starting with any one, then again starting with a different one. Answer: 18 both times; each animal still gets one word.

Exercises

1. How many are in the ring? No start is marked, so watch how the child decides where to begin and end. Then ask, “If you started with a different one, would you get a different number?”
2. Sam counts 13 toy dinosaurs in a row. His sister mixes them into a heap and says, “Count again, but start with the green one.” How many will he get? Does he need to count to know?

Where this goes next

Knowing the last word names the group lets a child count out a given number from a bigger pile and stop on time, and match that number to its written numeral. Seeing that adding one moves the count one word along is the start of one more and one less, and keeping a number in mind instead of starting over leads to counting on from a number. It also builds on the matching in One-to-one correspondence.