One day, while I was taking a mathematics class of grade 6, a student gets up and asks me in a curious and innocent tone, "Miss, till today we have divided numbers by non-zero numbers. I was told once by a teacher in 4th grade, that we cannot divide by zero. But why cannot we divide by zero? I do not understand."
I was surprised that this child had been thinking of divide by zero since 4th grade. At that time, I had to take a decision whether to discuss with the whole class or explain only him. Since, it was the last few minutes of the period, I promised him that next class I will discuss this the first thing. But before leaving, I asked the other students to ponder over the problem and discuss their finding amongst themselves before we meet in the next class. He had twinkle in his eyes that his question is finally going to get answered. Another thing I observed that he was happy that his question got acknowledged and given importance!
I had my next period with his class after lunch break. I kept thinking as to how best explain him as well as the whole class without confusing any one of them. The bell rang to mark the end of the lunch break. I went to the 6th grade and found all the students seated on their seats quietly and eagerly waiting for my arrival. I got a grand welcome by these eager young eyes. I thought that today I have a great responsibility to fulfill.
I first asked them whether they had spent time discussing amongst each other. They said that they finished their lunch in 10 minutes and spent the rest of the lunch break in discussion. Many gave the response that the answer will be zero. I asked the child who had posed the question to explain his thoughts. He said that, "zero times any number is zero. But that way I get remainder as the same number, so I get confused."
So I started from the same thread.
I asked students to tell what is 6 divided by 2. All of them in one voice said 3. So now I asked them what does 6 divided by 2 actually mean? There was silence.
I drew 6 apples on the board. Asked them to divide them into groups of 2. One child came up to the board and drew boxes around the apples to represent the group. Then she counted the boxes and found that there were 3 boxes. Slowly, they started understanding the meaning of division.
Next, I asked them to divide 10 by 5. Some students shouted the answer as 2. I was surprised to see that the others were thinking on grouping objects.
This time I asked one of the child to describe how he grouped. He drew 10 cones on the board. And then drew two boxes around 5 cones and wrote there are two groups.
Quickly, the whole class was able to grasp the concept. Then we tried some more division problems mathematically and by grouping.
Now, divide 14 by 6. They drew 14 objects, draw boxes for every group of 6, and found 2 were left out. Hence they understood the meaning of remainder.
Then, I wrote 4 divided by 0.
I drew 4 trees. Now asked the students to form groups of zero trees.
They spent some time thinking and how to form groups containing zero trees. I observed their work on their respective sheets. All were trying all combinations to draw a box that will contain zero trees.
Then finally, one child stood up and came to the board and drew an empty box. He said that, "Miss, this box contains zero trees. But how do I explain from this 4 divided by 0?"
I asked them if any one was able to form groups. Many looked puzzled. I asked them if they understood the meaning of an empty box that was drawn on the board by one student. They all understood that zero trees means no trees. Thus, the box is empty.
So, I told them that how many such boxes can you draw which contains zero trees?
Some said 10, some 20 and some said many. I asked them, I can keep on drawing these empty boxes and still have groups of zero trees. They all agreed. Does that mean I can draw infinite boxes and still have groups of zero trees? They all agreed.
I asked them what they understood by infinite. Immediately one child got up and said infinite is lots of many... Then one child answered something which cannot be counted. Bingo!!
"That is your answer to 4 divided by 0. You cannot count the groups. Hence any number divided by zero is infinity. Your remainder is 4."
One child asks, "Will that mean that 28 divided by 0 is also infinity?" I asked him to draw 28 objects and group them again with zero objects. He understood.
Just 2 minutes before the bell rang, some children come to me and said, "Miss, we never understood division. It is the first time we really understood division. But we are still not confident about divide by zero." I asked them to just focus on the division part that they have understood, the divide by zero concept we will discuss at a later stage.
After the class, I went into a pensive mood. Only if students at primary level understand the concepts then their secondary and higher level mathematics will be more seamless. Immediate reaction will be to blame the primary teachers who did a sloppy job and kept promoting confused students. But on further thinking, I do not feel that it is entirely the teachers' fault as well.
Then who is at fault? Are the curriculum designers to be blamed, who have made a packed curriculum that leaves no scope for creative and effective teaching? Is the school management, who has packed the timetable with so many activities that the teachers are stressed out planning? Are the teachers entirely to be blamed for producing ill-informed students at the end of each academic year?
I guess all of the education community has to share the blame. What do you feel?