Adults take counting for granted because we cannot remember a time we couldn't do it. For a three-year-old, counting is a five-step skill, and each step has to come online before the whole thing works.
The first step is knowing the number words in order: one, two, three, four, five. Many toddlers can recite this list without understanding it — in the same way they can recite the alphabet song. That is a real accomplishment but it is not yet counting.
The second step is one-to-one correspondence: touching each object exactly once and saying one number for each touch. This is the step that breaks most often. A child who counts "one, two, three" while pointing at the same object twice, or who skips an object, has not yet got it. The fix is patient, shared counting — "let's count together" — while the child's finger moves across the objects.
The third step is cardinality: understanding that the *last* number said is the answer to "how many". A child who counts three cookies and then, when asked "how many?", starts counting again from one has not yet got cardinality. This comes online around age three and a half on average, but the range is wide.
The fourth step is abstraction: realizing that you can count anything. Three cookies is the same number-of as three stars or three jumps. Children figure this out around age four.
The fifth step is order irrelevance: you can count the cookies left-to-right or right-to-left and the answer is the same. Strangely, this is a genuinely difficult idea for a four-year-old. If you see a child start over when you rearrange the cookies, that is what's happening.
At home, the highest-leverage game is "touch and count". Lay out between three and seven objects. Say "let's count them" together, touch each object once in unison, and then ask "how many?". If the child starts counting again instead of answering, that tells you which step hasn't clicked yet — and the fix is simply to do it together again. Counting is the earliest math skill and one of the few that is built purely through repetition with a trusted adult.
One reason "touch and count" works so well is that it makes the difference between counting and reciting concrete. A preschooler who sings "one, two, three, four, five" while their finger lands on the same teddy bear three times is reciting; they are practising a song. A preschooler whose finger lands on a new bear at every word is counting; they are practising a piece of mathematics. The two activities sound the same and feel the same, but only one of them builds the cardinality concept — the idea that the last number you say is the answer to "how many?".
Counting backwards is its own small revolution. Most children learn to count up before they can count down, and that's expected. Around age four or five they often start spontaneously counting down on their own — a rocket launch, a hide-and-seek game, blowing out birthday candles. Lean into it. Counting down is the foundation of subtraction in the same way that counting up is the foundation of addition. A four-year-old who can count down from ten to zero is closer to subtraction-readiness than a four-year-old who can only count up to a hundred.
When you are ready to move beyond pure counting, two activities are particularly powerful. "One more, one less" — show three blocks, ask "what's one more?", add a block, ask the count, ask "what's one less?", remove a block. That game is the seed of the +1/−1 fact families and it almost teaches itself. And "subitizing" — flashing two or three or four objects briefly and asking the child to say how many without counting. By age five most children can subitize up to four; by six, up to five or six. This is the cognitive scaffold that lets later arithmetic happen at speed.