Division
4 Models of Division
by Prof. Mahesh Sharma of The Center for Teaching & Learning Mathematics
Repeated Subtraction - How many groups/sets/shares of 3 are in 12? A: 4
Partitioning - If 12 is divided into 3 equal parts, what is the size of each? A: 4
Array - If 12 objects are placed evenly into 3 rows, how many columns or objects will be in each row? A: 4
Area of a Rectangle - If a rectangle has an area of 12 and a vertical side of 3, what is the size of the horizontal side?
A: 4 3 x 4 = 12
3 high x 4 wide = 12 area
3 high by 4 wide = 12 area
Division is Whole to Part
Division begins with a whole thing.
As you divide it into parts. the whole disappears.
Losing sight of the whole, makes division difficult.
3
4
12
Area of a Rectangle is the origin of
Long Division.
3
3
4
4
1. Standard Long Division
127
6
0
0
1
2
2
2
1
0
1
7
6
1
r 1
6 x 1
6 x 2
6 x 0
2. Short Division
6
1
2
7
0
1
2
1
r 1
4th Grade Focus on Division
Don't start division until students understand the 4 models of division. Introduce one model per day.
Understand why we start with ones on the right when doing multi-digit addition, subtraction, multiplication, and division, but start from the left when doing long division.
Long division is a very important invention.
Didactic: What the student does not know.
Questioning: Questions instigate language. Language instigates models. Models instigate thinking. Thinking instigates meaningful understanding. Understanding produces competent performance. Competent performance produces long-lasting self-esteem. Self-esteem produces life-long learning.
Teacher is Coach: What is the standard of performance? Help practice to the standard.
Outcomes: What is the desired outcome? What is the proof of mastery? What behavior is expected? What does competence look like?
Japan has a National Treasure award. The NT teacher travels anywhere with government support. A good question produces good answers.
Variety of Methods
A teacher said, "I only want to teach one method of division because more than that will confuse my special ed students."
Professor Sharma replied, "Do you only teach one way to spell /ay/ and forget about the rest because it may be confusing?" The teacher said, "No! We teach all the ways." Professor Sharma said, "Then, of course, you must teach all of the different ways to solve problems in mathematics! You cannot create terminal students incapable of functioning and learning outside of your classroom!"
