Counting On
When a group grows, hold the count you already trust and count on from there, keeping track of how many more you add. · 9 min
Yesterday you counted the crayons in your box: six. Today two new crayons arrive. You could tip everything out and start again at one. You do not have to. There is a faster way, and you can trust it.
Guess before you learn
Six crayons in the box, and you drop in two more. What is the quickest honest way to find how many now?
Seven, eight — the answer is eight. Starting over works too, and if you chose it, that is the careful choice, not a wrong one. But the first six words would come out exactly the same as yesterday, so you may skip them. That skip has a name: counting on.
K–2
3–5
Counting on has two parts. First, trust the number you already have — do not count those things again. Second, keep track of the new things: raise one finger for each one, saying the next number word as you do. Six... seven, eight. Two fingers stand up, so you added two, and the count is eight.
A habit worth keeping: start from the bigger group. To join 2 and 9, do not hold two and count on nine times. Hold nine and count on twice: ten, eleven. Same answer, much shorter trip.
6–8
Counting on works because counting is cumulative: the count so far already contains every earlier word. Recount the six crayons and you would say one through six and arrive exactly where you started. Counting on refuses to redo finished work — the answer to how many now simply continues from the answer to how many before.
Notice what you are quietly doing. Six, and two more, gives eight — that is the addition fact 6 + 2 = 8, discovered with your voice before anyone writes a plus sign. Counting on is where adding comes from.
9–12
Each step of a count applies one operation: successor, the function carrying any number n to n + 1. Counting to six is applying successor six times, starting from zero. Counting on from six is applying the same function again — the process resumed, not restarted.
Addition is this resumption made official: 6 + 2 means take the successor of 6, twice. Every sum within 20 that you will ever compute is iterated successor wearing a shorter name.
K–2
You already counted six. Keep that six in your head. A new crayon drops in — say seven. Another new crayon — say eight. Eight crayons in all.
You did not start over at one. You started from six and counted up. One new crayon, one new word.
Undergrad
The Peano axioms build arithmetic from almost nothing: a first element 0; a successor function S that is injective and never returns to 0; and induction. The numbers are 0, S0, SS0, and so on. Addition is then defined by recursion: a + 0 = a, and a + S(b) = S(a + b).
Unwind that recursion for 6 + 2: 6 + SS0 = S(6 + S0) = SS(6 + 0) = SS6 = 8. Each unwinding step is one spoken word of a child counting on. The formal definition and the playground procedure are the same computation.
Postgrad
The recursive definition is licensed by Dedekind's recursion theorem: on a Peano system, a function specified by its value at 0 and its behavior under S exists and is unique. Second-order Peano arithmetic is categorical (Dedekind, 1888) — all models isomorphic — so the numbers a child counts on are, up to isomorphism, the only ones there are.
Counting on also presupposes a theorem: yesterday's count of the untouched six must survive today's recount — order-invariance, provable by induction. And addition-as-iterated-successor is the first rung of primitive recursion; multiplication, exponentiation, and the rest of the hierarchy rise by the same move.
counting on
Starting from a number you trust and counting up from there: six... seven, eight.
Count on: 8 books on the shelf, then 3 more arrive — the steps fade as you master them
eight
nine (one finger up)
ten (two fingers up)
eleven (three fingers up) — 11 books
One warning. The number you count on from must be one you trust. If the six might be wrong, recount it once, carefully. Counting on saves work; it cannot repair a broken start.
You can now join the counting words at any point. Start from a number you trust and count up from it. Soon that move gets a name of its own: adding.
Practice — new ink and old, interleaved
1.What are the two parts of counting on?
Trust the count you already have, and give each new thing one new word while keeping track of how many you add.
How close were you? Grade yourself honestly — it sets your review date.
2.Five ducks on the pond; three more land. Count on to find how many.
3.A box for teddy bears stands empty. How many bears are in the box?
4.While counting shells, you touch the same shell two times by accident. What happens to your count?
5.Counting on from ten, you add one apple. What do you say?
6.You hold a count of seven marbles, and zero new marbles arrive. What is the count now?
7.A careful count of your books ends on the word twelve. How many books do you have?