What comes to your mind when you think of zero, apart from the numeral ‘0’?
Nothing, right? There’s just a blank space with nothing in it. Now just think about how this concept was taught to you in school and the struggle you had as a child to understand this concept of “nothingness” and its numerical significance. Zero is a very fundamental concept in mathematics but is quite difficult to understand for a six-year-old student.
Counting is an important step in children’s numerical development. Between 2 and 3 years of age, children usually start learning the number sequence (one, two, three, etc.). At this stage, children may not yet understand that the number name ‘two’ refers to the numerical value of a set of two objects. One-to-one correspondence, along with the correct order of number words is necessary for successful counting. Finally, to understand that the number word named last actually represents the number of items in the set counted, children need to understand the concept of cardinality. Interestingly, there is a difference in the way students acquire the concept of zero and other small cardinal numbers. For small cardinal numbers, after number four, the students are able to understand the concept of cardinality and hence the fact that each numeral signifies a number of objects. However, students face many difficulties in understanding the term zero arising from the difficulties in understanding the concept of nothing and that it is also a part of the number set.
The data gathered from a question attempt by the students of the government schools of Rajasthan shows that only 30% of students in grade 1 were able to answer the following question correctly.
John has no balloons. We can say this as ‘John has 0 balloons.’
You have 3 balloons. They all went POP! How many balloons are left with you?
- A) 3
- B) 2
- C) 1
- D) 0
This shows that even after reading that “0” signifies “no object”, most of the students are not able to answer this question. It may be because the students are not able to understand the question statement or that they don’t consider “0” as a number. My experience from a visit to Uttarakhand will help to explain this last part further.
On a visit to a government primary school in Uttarakhand, I wanted to understand how students of grade 1 and 2 do single digit addition which also includes an addition with zero. I had come up with a set of simple addition questions that I wanted to ask the students as a part of the student interview. The questions are given below:
|1) 3 + 0 = ____||2) 5 + 3 = ____||3) 2 + 1 + 3 = _____|
|4) 3 + 3 = ____||5) 0 + 5 = ____||6) 4 + 0 + 1 = _____|
|7) 20 + 6 = ____||8) 55 + 28 = _____|
Now let us look at the data of the questions involving operations with zero and compare it with that of questions involving operations without zero:
|Sl. No.||Question||Accuracy||Most common wrong answer (% of students)||Second most common wrong answer (% of students)|
|1)||3 + 0 = ____||45%||30 (80%)||0 (4%)|
|2)||5 + 3 = ____||54%||53 (5%)||4 (4%)|
|3)||0 + 5 = ____||44%||50 (3%)||–|
|4)||4 + 0 + 1 = _____||22%||4 (11%)||40 (4%)|
This chart shows that there is a lot of difference (9-10%) in the accuracy between the addition of two positive numbers and the addition of a positive number and zero. But there is an even more interesting observation to be made here. If we look at question number 1, we can see that around 8% of the students have answered 3 + 0 as 30. This means that 8% of the students in that set were not aware of the meaning of “0”. They were not able to understand that if we add “0” to any number, it will be the same number because “0” is effectively not contributing anything to the sum. They wrote it on the right side of the addend because that is how it was there in the question. Or maybe they don’t consider it as a number because of which they just write it next to the other number (like a drawing). For the same question, 4% of the students answered 0. This also points to the fact the students are not able to clearly understand what does “0” stand for.
However, if you look at the data for question 3, you will see that 3% of students have answered 0 + 5 as 50. Now you must be thinking why is this percentage so less as compared to question 1 (8%) which is of a similar nature. I thought so too. And so, I tried to get some more insights by looking at the individual student responses for question 3. That’s when I saw that a significantly large number of students have answered 0 + 5 as 05. So, I modified my table a bit to add 1 more column (Accuracy if we remove 0x type answers) which means what is the % of students who have answered correctly if we consider 0x as a wrong answer.
|Sl. No.||Question||Accuracy||Accuracy if we remove 0x type answers||Most common wrong answer (% of students)||Second most common wrong answer (% of students)|
|1)||3 + 0 = ____||45%||44%||30 (8%)||0 (4%)|
|2)||5 + 3 = ____||54%||–||53 (4.5%)||4 (4%)|
|3)||0 + 5 = ____||44%||37.5%||05 (6.5%)||50 (3%)|
|4)||4 + 0 + 1 = ____||22%||–||4 (11%)||40 (4%)|
Now, looking at the data, do you think that those who answered 05 instead of 5 actually knew that they are equal, or did they answer it 05 because 0 was on the left side in the addition operation? This is a more difficult case because even though the answer looks correct, the students’ thought behind it is not. An interesting observation here is that those students who have answered question 3 as 05 are the same students who have answered question 1 as 30 in most cases.
Similarly, if we look at the data from question number 4, we can see that only 22% of students are able to answer it correctly, which can also be because it is a difficult question involving 3 addends. But here also, a few have given 40 as the answer which means they have just written 0 to the right of 4.
It is also interesting to note that when I was in the classroom asking these questions to the students, I could also observe that when the students were solving the additional questions, they were not finding it difficult to count positive numbers on their fingers. But when it came to zero, they were not sure what to do and how to count it on their fingers. These responses and the observation given above show that students don’t have a clear understanding of the addition of numbers with zero. It can also mean that they don’t properly understand the meaning of “0” or they don’t consider “0” as a number even in grade 2 and hence treat it differently from other numbers.