Counting On, Counting Back

Learning progression of proficiencies for addition and subtraction

Adding and subtracting begins with basic understanding of number relationships:

Counting on, counting back

Making five, making ten (knowing combinations to the anchor numbers of 5 and 10)

Problems with identifiable underlying structures are used extensively to introduce the concepts of adding and subtracting with single digits. Children solve these problems by using strategies that eventually grow into fluency with number combinations:

Solving simple joining and separating problems, first by counting objects, then by using strategies such as counting on, making ten, using doubles, etc., then developing automaticity.

Writing number sentences to represent problems, including ones with missing addends or subtrahends, and associating “adding” with the plus sign and “subtracting” with the minus sign.

Fluently adding and subtracting within 5 (kindergarten), 10 (1st grade), 20 (2nd grade).

Adding and subtracting with two or more digits is based on an understanding of place value: Knowing how the digits of numbers from 0-99 represent tens and ones.

Children continue to use objects, drawings and strategies (mental math) to solve multi-digit problems as they develop proficiency with the symbolic procedures:

Solving joining, separating and comparing problems with two digit numbers, using objects, drawings and strategies based on place value or incrementing by tens.

Writing two-digit addition and subtraction problems using symbols, both horizontally and vertically. Giving a reason for each step in the solution of two-digit addition and subtraction problems when using an algorithm.

Fluently adding and subtracting within 100 (2nd grade) and 1000 (3rd grade).

Recognizing and explaining errors in the formal algorithm for two- or three-digit addition and subtraction with regrouping.