Addition and Subtraction 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 these operations are represented in real life can be classified into four groups:

  1. Combination of number values
  2. Separation of number values
  3. Part-part-whole relationships of number values
  4. Comparing or equalizing number values

See examples, sample problems, and analysis of each group.

Addition and subtraction concepts

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 in four different ways they are represented in real life. To keep the categories 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 can be used for both addition and subtraction. In fact it is impossible sometimes to know for sure if a person is using one or the other to arrive at an answer. For example. If I ask you the difference between 18 and 15? Did you think of three before you even thought to use addition or subtraction? Thus, making it difficult if not impossible to know which operation was used. So be it.

Teachers should understand that any addition and subtraction problem can be solved with both addition and subtraction. Review these examples and think about how interchangeable addition and subtraction 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 and subtraction algorithm along with flexibility for selecting from a variety of ways to add and subtract efficiently. Book Cover

Historically we should recognize Constance Kamii who first published ideas on how students' reinvent algorithms in her book Young Children Reinvent Arithmetic (1985. second edition 2000). She built on Piaget's development of understanding.

Dr. Robert Sweetland's notes