Overview
According to Ron Aharoni, the author of Arithmetic for Parents: A Book for Grownups about Children's Mathematics (6), the meaning of numbers and arithmetical operations is students' link to reality.
What is Addition?
At its most basic, addition is putting things together or joining. There are subtle differences with joining as described by Aharoni. Those differences include dynamic addition and static addition. In dynamic addition, the situation changes over time: one group is added to another. In static addition, a large group is made of two subgroups, but there is no action of combination - the two parts simply coexist in the whole. An example of each is listed below:
Dynamic Addition
3 butterflies were sitting on a limb. 2 butterflies joined them. How many butterflies are there now?
Static Addition
A vase contains 3 red flowers and 2 yellow flowers. How many flowers are there altogether?
The Random House Unabridged Dictionary defines addition as "the process of uniting two or more numbers into one sum, represented by the symbol +." The numbers added are called the addends. As stated earlier, addition focuses on joining. According to Children's Mathematics: Cognitively Guided Instruction (Carpenter et al., 1999), in dynamic addition, three different types of join problems can be created by changing the quantity that is the unknown. The following chart demonstrates the join problem types as described in Carpenter's book.
Unknown
- Result Unknown. Example: Robin had 5 toy cars. Her parents gave her 2 more toys cars for her birthday. How many toy cars did she have then?
- Change Unknown. Example: Robin had 5 to cars. Her parents gave her some more toy cars for her birthday. Then she had 7 toy cars. How many toy cars did Robin's parents give her for her birthday?
- Start Unknown. Example: Robin had some toy cars. Her parents gave her 2 more toy cars for her birthday. Then she had 7 toy cars. How many toy cars did Robin have before her birthday?
What Is Subtraction?
According to Aharoni, subtraction means removal. Very similar to addition, there is a dynamic subtraction which means that the situation changes over time. An example of dynamic subtraction is listed below.
Dynamic Subtraction
There are 7 girls playing in the park. Four of the girls leave the park and go home. How many girls are left in the park?
The number sentence to solve the above problem is 7 - 4 = 3. In this number sentence, the 7 is called the minuend. The 4 s called the subtrahend. The result of the action is called the difference.
In addition to dynamic subtraction or "take away", Children's Mathematics describes two other meanings to subtraction. One is identified as "part-part-whole" or "whole-part." The third meaning of subtraction is comparing. An example of each is listed below.
Whole-Part Subtraction
In a group of 5 children, 2 are girls. How many boys are there?
Comparing
Joseph has 7 cats and Travis has 4 dogs. How many more cats does Joseph have than Travis has dogs?
Relationship between Addition and Subtraction
Children need to understand that there is an inverse relationship between addition and subtraction. Before teaching this unit (and very early in the year), I will ensure that students have mastered number facts and fact families. Students will learn that a number fact is made up of three numbers. Those three numbers can be used to make up other number facts. Knowing one fact can help children with other facts. Look at the number facts that we make with 2, 5, and 7.
Addition Facts | Subtraction Facts
2 + 5 = 7 | 7 - 5 = 2
5 + 2 = 7 | 7 - 2 = 5
Generally, subtraction facts are harder for children to learn than addition facts. If a child knows that 6 + 9 = 15, and he or she sees the subtraction sentence 15 - 9 = __, the child can think, 9 and what are 15? I plan to encourage students to think of the related addition fact when they encounter a subtraction fact they don't know. Children often find themselves either counting up or counting back to solve subtraction. That is inefficient. If children learn the important inverse relationship between addition and subtraction, subtraction facts will become much easier. As I work with the children, I plan to use questions that encourage this strategy of the inverse relationship between addition and subtraction.
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