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

  • 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
  4. Comparing or equalizing number values

Examples, sample problems, and analysis of each.

Research bits:

There are critiques who use emotional words as propaganda techniques; terms like new mathematics, fuzzy math, soft math and claim it is dumbing down students learning of mathematics. However, the mindless use of algorithms is the real dumbing down.

Development notes

Children learn addition and subtraction based on their understanding of number value. Tthey memorize the counting numbers and soon realize they are sequenced with each related in an increasing order. This order develops as an understanding of one more and then one less and the ability to put a number with a set of objects, which eventually becomes cardinality. The relationship of numbers as more or less and cardinality can be decomposed and composed in a hierarchial manner eventually is seen as the operations of additiona and subtraction.

These relationships are developed when students experience activieis such as; dot plates to subitize cardinality and learn one more and one less, begin to memorize addition facts and understand hierarchical inclusion.

Students, who are given activities to quantify groups of objects both before and after combining or separating different groups of objects will naturally compose and decompoe the numbers and invent their own algorithms. If, once they learn to count, they are are pushed away from counting, not taught to use touch points, and encouraged to use skip counting, five as an anchor, ten and more, and eventually decomposing numbers left to right.

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 can be represented.

Teachers should understand that any addition and subtraction problem can be solved with both addition and subtraction. When reviewing the example on this page think about how interchangeable addition and subtraction really can be 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 types of problems to include in normative testing. Why? You ask, because they will efficiently sort students into different levels, whcih is the purpose of all normative tests.

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 along with flexibility for 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 algorithm related concepts

Scoring rubric for addition and subtraction algorithms

Dr. Robert Sweetland's notes