## Tuesday, April 1, 2008

### How to divide by zero?

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?

1. Dr. B.S.ThandaveswaraApril 2, 2008 at 5:38 AM

Thank you for the way the concept of division is explained, it is wonderful.
In India the stumbling Block is the parent- neither the educationists nor the syllabus writers are responsible. Parents do not allow the children to explore the world ( there could be some exemptions).
Lack of models(physical) and of course the untrained not educated teachers who are employed by the private management for the paltry salary, are also responsible.
Basically the teachers require commitment and are to be 24 hours teachers and not just during the working hours.
Thandaveswara

2. Yes sir, you are right in pointing out that parents are equally responsible.

The paltry salaries offered to teachers just do not motivate the teachers enough.

3. I personally believe that teaching is much more than just cramming up things. These days education is a business and hence people involved in this profession have to be result oriented.

Onus lies with parents and teachers to ensure that children do get the essence of education and just not score marks.

Two movies which show just this theme are: School of Rock and Dead Poets Society :)

Regards,
Sachin

4. excellent way of explaining division and explaining division by zero specifically.Suoerb.thanks

5. Hats off to you ma'am!!!! Even i could not properly understand the concept of dividing by zero, you've cleared my doubts too!!

I wish all teachers were like you, not just emphasising on bookish knowledge, but going deeper and explaining things as they happen in everyday life. Same goes for the process of teaching 'clockwise and anti-clockwise'. Practical and experimental knowledge aid the process of teaching a lot.

6. Impressed !

In fact I too think many a time that education by concepts, education where children begin to relate to things like mathematics with real world is really missing.. I agree in my case too.. I learnt many concepts after my class VIII which I shud have learned in primary education.. This reminded of a home tuition I used to take of a boy of class VI during my graduation.. My responsibility was to teach maths to him.. He was kid of a rich family, good in all subjects but just manages to score pass marks in maths. In a couple of classes, I understood that he is suffering from the same problem -- no conceptual understanding, only mugging and trying to grasp without applying thought.. U won't believe, i started with him from very basics -- way back into his II and III standard... - understanding numbers, addition, subtraction, units of measurements etc... Gradually, he began to love the subject he used to hate.. I used to give him assignments that makes him apply thought and gradualy he began to understand the magic of numbers !! My job was done.. My effort got reduced in his problem solving. He scored above 80% in VI and VII .. I was the most happiest person !!

I think its not one person/party that can be blamed for this. Surely, students are not to blame. I remember my cousin in class X, who finds it tough to convert gm to kg, ml to litres !

Each party/person can play some role. What I can think is:

Curriculum Designers: I think activity based learning is something that's required, which does not happen. Light-weight curriculum focused on catching interest of students as they grasp the study concepts .. playful, activity-full...
Schools Adminstration: Less pressure on students in terms of their assignments/homework, increasing age of entry in nursery is a preliminary step.
Teachers: Teaching with interest and with an aim of delivering quality education is really mandatory. Teaching is not just a job to earn money. I would say its a Nobel profession, like Doctory -- it cannot be just done as a job. I know its difficult to achieve for several reasons -- At times teachers enter into this profession not out of willingness/interest; but out of need, stressful personal life of teachers, stressful life with school activities, teachers themselves not aware of activity based methods of delivering education etc...
Parents: If teachers just focus on teaching curriculum, parents can aid in activity based learning.. It would definitely add value to child.. The aim should be not forcing, just make the child feel that learning is just another play/fun for him. Learning or mugging by force, pressure, scolding must be a NO NO.. This is primary cause of suicides due to bad results.. I really feel pain when I see children ending their lives just becoz of pressure and fear of their parents.

More later...

7. Damn!! its so easy!! this is what i meant by "things we consider trivial". I have been in the same situation with a 9th grader. She NEVER understood the concept of division. And I was stumped as to how to explain it her in a short period of time. I guess this blog will help me a lot.

8. Good work Jaishrie! You made the concepts clearer to me too :-)

Jayanti

9. hi jaishrie ,
this is a really cute episode . To keep the curiosity alive is a very hard and precious task . Teaching with interest , with a desire to make a child a whole individual is what motivates a real teacher or parent (parents being the first teachers ).
But before saying anything about any teacher a parent needs ro sit back and think how many times they make the effort for their child . lata.