In organizing this unit I have relied on an idea gleaned from the Ma book. Here Ma compares and contrasts the mathematical knowledge of American and Chinese teachers. Ma refers to a Chinese teacher’s use of the term “knowledge package.”5 This term suggests that there is not one piece of knowledge that the teacher is presenting but a group of related ideas that scaffold the learning of that final concept. Interestingly there is not one correct package for teaching a given idea. This will depend on the teacher and the students. Just what do they need to construct the knowledge they will need to be able to successfully understand subtraction with renaming? The package I would like to suggest for subtraction with renaming would include:
Composing and decomposing numbers
Addition and subtraction within 10
Addition and subtraction as inverse operations
Addition and subtraction within 20
Addition without regrouping
Subtraction without regrouping
Addition with regrouping
Subtraction with regrouping
What is being suggested is that if a child is having trouble with subtraction there is probably some deficiency in another area that is the cause. If he doesn’t see a relationship between addition and subtraction or he cannot subtract numbers to 20 successfully he will probably have trouble with subtraction with renaming. While the path I have chosen to follow with my students is one way to reach the level of understanding for my students, -- the literature consistently acknowledges that there may be different paths and children may not need all of what is in the package to successfully reach an understanding of subtraction with regrouping..
First it is important to discuss the terminology of addition and subtraction which is often confusing and misleading. For many years the term “borrowing” was used for what is now called “subtraction with renaming”. The word “borrowing” suggests that something is being given for a short term use and will later be returned. Mathematically speaking this is not so in subtraction. Ma mentions the terms “composing and decomposing of units” used by the Chinese math teachers to indicate how numbers are constructed and can be broken down by the processes of addition and subtraction. Currently in the United States “renaming” is the most popular term used. It is used in Saxon Math which New Haven uses for its K-4 curriculum. Throughout the paper I will use all three terms. I have become a fan of the composing and decomposing terms because it seems to express what is happening to the numbers in addition and subtraction.