Misconception series: Class 5 Maths, Fractions, Decimals, Ratios and Percentages: Concepts applications

(1) Why was the question asked in ASSET test?

This question was asked to check if students are able to order fractions correctly and place them on a number line accordingly.

Only 19% of students have answered B, correctly. Around 40% have chosen option C, and 31%, option D.

Possible reason for choosing A:Very few students have chosen this option and are probably making a random guess.

Possible reasons for choosing C and D : These students seem to believe that since 3 > 2 , > , and so would be to the right of .

Further, the students answering D, seem to think >1 reasoning that 3>1.

(3) Learnings

The response to this question shows that

• Students don’t seem to understand that proper fractions like , etc are ‘numbers’ with value less than 1.

• Students don’t understand the fractional representation- i.e. numerator/denominator and what number it represents. They tend to treat numbers in the numerator and denominator as separate entities, and don’t think of it as a number or quantity by itself. Students have not been able to grasp that or has a meaning of its own, different from 1, 2 or 3.

So students seem to fall back on their previous experience with whole numbers, and extend the same rules to deal with the new entities also. Possibly, an excessive focus of the curriculum on operations and procedures related to fractions has not allowed students to develop a strong intuitive understanding of what fractions actually represent.

4) How do we handle this?

This can be handled by adopting a teaching strategy that stresses on the understanding of what a fraction represents. In a general class room situation, when the teacher repeats that “2 by 3” is a fraction with 2 as “numerator” and 3 as “denominator”, the children focus on the 2 and 3 separately, and on the names for each. So the idea that “two thirds” is 2 thirds, and that a third is one part of a whole divided into 3 equal parts is lost on the students.

• A fraction represents a part of a whole when it is divided into equal parts. • Fractional parts are equal shares or equal sized portions of the whole thing or the whole set. (You may refer the teacher sheet M5–0908–14.)

• Fractional parts have special names that tell how many parts of that size are needed to make a whole. For example, 3 thirds make a whole. Children can be encouraged to count the number of fractional parts required to make a whole. “How many fourths make a whole?”

• The students realize that 8 eighths are required to make a whole whereas only 4 fourths are needed. This gives the idea that the more parts required to make a whole, the smaller the parts. Consequently 3 eighths are smaller than 3 fourths.

• Now the teacher can introduce the formal notation for the fraction with the understanding that the numerator indicates how many parts are being considered and the denominator indicates the size of parts the numerator counts. • Help students realize that , , etc are all numerically same. (You may refer the teacher sheet M4–1008–15.)

The teacher may use paper strips, cookies, or plain paper as models to demonstrate these ideas, so that children actually see the relationships involved. The links suggested below give many such models. (You may refer to the teacher sheet M6–0608–11.)