Addition and Subtraction Concepts, Development, Research, Activities, and Assessment

Overview
  • Big idea
  • Research information
  • Development notes
  • Addition concepts
  • Subtraction concepts
  • Addition and subtraction related properties
  • Addition and subtraction algorithms

Big idea (generalization) for addition and subtraction

Addition and subtraction are two ways to operate on two or more numbers to create a third number of equivalent value. The different ways are 1. combination of number values, 2. separation of number values, 3. part-part-whole relationships of number values, and 4. comparing or equalizing number values. See examples for each and analysis of the differences.

Research bits:

Development notes

Children learn addition and subtraction based on their understanding of number value. For example: as students learn number value, they will develop the understanding of one more, one less, and the sequence of numbers. For example on activity to develop this is the use of dot plates to subitize cardinality and learn one more and one less, begin to memorize addition facts and understand hierarchical inclusion. Once these are achieved students will be better able to construct and deconstruct numbers with an understanding of how different number values are related so they can accurately use the operations of addition and subtraction.

Students, who are discouraged from counting or using touch points, will naturally begin to decompose and compose numbers. Students will naturally use decomposition and composition to invent their own algorithms - if they are provided an enriched environment and plenty of encouragement.

The environment needs to include problems and activities which will enable students to naturally incorporate the following ideas when solving problem.


At the beginning of this discussion addition and subtraction was described as having four different ways to be represented, which are analyzed at this page. To keep the above list less complicated addition was referenced as joining and subtraction as separating. However, doing this is a dangerous idea since it is important students learn all four ways addition and subtraction is represented. Additionally teachers should know any addition and subtraction problem can be solved by by either adding or subtracting. Review the example on this page and think about how interchangeable addition and subtraction really are when operating on numbers.

This should raise an important question for every math teacher. Do curriculum developers or text book authors take similar short cuts? How many of the four ways and the subcategories of subtraction and addition are included in your math curriculum or text book? You can bet the ones that are not represented have been discovered as good to include in normative testing, because they will efficiently sort students into different levels.

If students are presented with problems and encouragement in developmentally appropriate ways to understand, they will, usually by fourth grade, invent a traditional addition or subtraction algorithm and flexibility in selecting from a variety of ways to add and subtract efficiently.

Historically we should recognize Constance Kamii who first published ideas on how students' reinvent algorithms. She built on Piaget's development of understanding.

Addition concepts

Scoring rubric

Subtraction concepts

Addition, subtraction and related properties

Scoring rubric for regrouping

Addition and subtraction algorithms

Scoring rubric for addition and subtraction algorithms

Dr. Robert Sweetland's notes