The important topic of Fractions is usually introduced in upper elementary levels. Here we highlight some changes in the way we teach fractions in recent years.
Procedural vs Conceptual Understanding
Traditionally, fraction operations were taught using procedures. Conceptual understanding comes after students use the tools they learn to solve problems. For example, ¾ x 12, students were taught to look for the greatest common factor (GCF) between the numerator (12) and denominator (4) and simplify, i.e.
This leads to common mistakes like these
Without conceptual understanding, the student who made these errors do not have the intuition to realize that something is not right.
Fraction of a Set
Today, for elementary operations in Fractions, we start with conceptual understanding (with the aid of pictures), before introducing procedures. For example, a typical Fraction of a Set problem such as 1/3 of 24 can be visualized as
Students are invited to explore the reasoning behind the procedure of “canceling” numerator and denominator, by showing 2/3 x 24 = 2 x 24/3 = 2 x 8, hence two-thirds of 24 is the same as two groups of 8.
Fraction Addition
Fraction addition (and subtraction) is another topic in Fractions where traditional procedural-focused approach is being replaced by more conceptual understanding in recent years. In particular, the “butterfly” method is a common trick taught to students in elementary schools. Here, a common multiple in the denominators is found by multiplication and the numerators are converted by cross multiplication.
This does not help in understanding fraction addition at all! Moreover, such tricks are often not easy to extend, e.g.
Today, we teach fraction addition using a Concrete-Pictorial-Abstract progression.
Concrete: Start with the fraction wheel as a concrete manipulative, let the students work with simple examples such as 1/2 + 1/4, 3/4 + 1/8 etc.
Pictorial: Transfer the problem to pictorial form. Start with the circle as it is most similar to the fraction wheel,
then translate the problem to the bar model,
Abstract: Finally, present the problems in their abstract form,
With a more solid grounding in conceptual understanding, students are able to understand how the procedures work later on, and more important, are more able to translate fraction word problems into the necessary equations.
Concept of the same whole
In drawing pictures for fractions, we always draw the fractions as part of the same object or the “same whole“. This is a very important concept that students need to understand. If fractions are presented pictorially as referring to different wholes, it is easy for the students to get confused. For example,
For example, the following is a very common mistake where students see 8 equal parts, and therefore, use 8 as the denominator!
