Abstract
Three experiments report classroom instruction derived from cognitive research and focused on moving children across two developmental transitions: from counting-all to counting-on with entities present for the second addend and from the latter to counting-on as a number-word sequence solution procedure to solve symbolic single-digit addition problems. For the second transition, a new particularly efficient method of keeping track of number words counted-on was taught: counting-on with one-handed finger patterns. Classroom teachers were able to teach both kinds of transitions to most first and second graders, even to first graders below average in mathematics. Many first graders spontaneously transferred counting-on with finger patterns to the solution of addition word problems; some required instruction to do so. Most children were readily able to extend counting-on with finger patterns to counting up with finger patterns for subtraction. Counting-on with finger patterns was procedurally efficient enough to be used in instruction in multi digit addition problems; second graders successfully learned to use counting-on with finger patterns to add 10-digit numbers. Teaching these transitions to children permitted them to learn addition and subtraction topics from 1 to 4 years earlier than is usual in American schools.