Primary school enrollment in India has been a success story, largely due to various programs and drives to increase enrollment even in remote areas. With enrollment reaching at least 96 percent since 2009, and girls making up 56 percent of new students between 2007 and 2013, it is clear that many problems of access to schooling have been addressed. India now has 1.4 million schools and 7.7 million teachers so that 98 percent of habitations have a primary school (class I-V) within one kilometre and 92 percent have an upper primary school (class VI-VIII) within a three-kilometer walking distance.

Despite these improvements, the quality of learning is a major issue and reports show that children are not achieving class-appropriate learning levels. According to Pratham’s Annual Status of Education 2013 report, close to 78 percent of children in Standard III and about 50 percent of children in Standard V cannot yet read Standard II texts. Arithmetic is also a cause for concern as only 26 percent students in grade V can do a Division problem.

The concept of division is one that builds on previous knowledge of addition, subtraction, and multiplication. Often, it is this concept that many students and teachers have problems understanding. One of the misconceptions that students have about multiplication and division is that ‘multiplication makes bigger’ and ‘division makes smaller’. These misunderstandings are frequently demonstrated through their inability to properly explain or interpret their responses.

A team of educational specialists from Large Scale Educational Programs of EI conducted a student interview on the conceptual understanding of Division. The following question was asked to grade 7 government school students.

6279÷ 3

1. 2093
2. 2903
3. 2930
4. 293

Interviewer: Can you read the question stem?

Student: 6279 is to be divided by 3.

Interviewer: Can you explain what does this mean?

Student: Divide(When we asked him to explain the problem, the student was only able to explain the mathematical meaning explaining that the symbol given was division, hence the problem required them to divide. They were not clear about what division means.

Interviewer: Do you know the terms used for division?

Student: No. (Almost the entire class was not aware of the terms used in division)

Interviewer: According to you what is the correct answer?

Student: The correct answer is D, 293.

(The student explains that 6 is divisible by 3, but 2 does not appear in the tables of 3, hence he gets the 7 in the dividend down and thus divides 27 which appears in the tables. And lastly, since 9 is also divisible they put a 3 in the quotient directly without the crucial step of putting a ‘0’ in the quotient.)

Interviewer: What is 3÷ 6279?

Student: Reads the problem correctly.

Interviewer: What do you do in this problem?

Student: We will divide 6279 by 3.

Interviewer: Do 6279÷ 3 and 3÷ 6279 mean the same?

Student: Yes, they are same. ‘A smaller number can never be divided by a larger number’. And since only the digits are interchanged, the problem is the same.

Surprisingly these understandings were common to the entire class. Students seem to mechanically learning but the internalization of the learning (real learning) is missing. It is necessary that teachers work on getting the foundation of these basic concepts right. Students need to be taught strong what ‘Division’ means in the context of mathematics with the contextualization. The conceptual clarity of which number acts as a divisor or dividend should be clearly established. That ‘division is essentially repeated subtraction’ needs to be reinforced in students.

By Chitra Iyengar – Educational Specialist