It’s one of the bugaboos of learning fractions, knowing that 3/8 is a smaller number than 1/2. Learning which fraction is larger or smaller than the other remains a major stumbling block for children in elementary math. Even by Grade 5, children struggle to rank fractions and place them on a graduated number line.
A recent study in Mind, Brain and Education reports that adapting 5 common card games (Memory, War, Old Maid, Treasure Hung and Blackjack) as a teaching activity on fractions improved Grade 4 and Grade 5 students’ conceptual understanding of fractions. However, researchers were surprised to learn that this did not necessarily carry over into better student performance on operations, especially when teachers of control-group children were spending more time teaching procedures.
Conceptual knowledge not enough
“The use of conceptual knowledge as a substitute for procedural knowledge seems to be limited,” write the authors.
“Conceputal knowledge thus appears insufficient to allow pupils to invent appropriate procedures to solve such problems. However, longer and reinforced conceptual learning might perhaps lead to improved procedural performance.”
Experimental-group students improved at additions and subtractions with the same denominator. However they could not do additions and subtractions of fractions with different denominators as well as the control-group students. The experimental group did better at grasping the link between fractions and the concept of unity or of “one” as a unit. They also performed better when placing the unit on a number line. But, overall control-group students outperformed experimental-group students in assessment items on procedures.
“Conceptual knowledge thus appears insufficient to allow pupils to invent appropriate procedures to solve such problems. However, longer and reinforced conceptual learning might perhaps lead to improved procedural performance,” they write. In general, students and adults are subject to the “whole number bias”, a tendency to draw incorrect analogies between whole numbers and fractions, the study reports. Students tend to assume that a fraction with higher numbers, such as 6/8 , is greater in magnitude than a fraction with lower numbers, such as 3/4.
The intervention developed by researchers at the Free University of Belgium and the University of Cambridge invited children to use of wooden “pie” blocks as concrete support as they worked to improve their understanding of fractions and therefore, their card playing. The intervention was based on several educational principles: learning-by-doing, playfulness, collaboration and using concrete support to progressively build more abstract representations.
Participants in the study were 292 Grade 4 and Grade 5 students in 4 schools in the French-speaking Belgium. Two classes for each grade from each of the 4 schools were included in the study, one as a control group and the other as the experimental group. The control group received their regular mathematics lessons on fractions, while the intervention group received special instruction featuring the card games for 10 weeks. The students played different games twice a week for 30 minutes in small groups of three to five. Games increased in difficulty as the study proceeded. The games required that children add fractions, discern the highest value fraction or those of equivalent value and to compute the highest total value of a set of fractions.
Students in both the intervention and control groups received pre- and post-intervention assessments to evaluate both their conceptual and procedural understanding of fractions. Specifically, these tests assessed conceptual comprehension of estimation, comparison, and number lines and procedural comprehension of arithmetic operations and simplification, as well as other related concepts, such as improper fractions and equivalent fractions.
While questions remain about how much improved conceptual understanding translates to operations with fractions, the researchers report that the intervention did visibly increase student motivation.
“Even if we did not explicitly measure motivation levels in the classroom, teachers were positively surprised to see their pupils enjoying learning fractions,” the researchers write.
Gabriel, Florence, Frédéric Coché, Dénes Szucs, Vincent Carette, Bernard Rey, and Alain Content. “Developing Children’s Understanding of Fractions: An Intervention Study.” Mind, Brain, and Education. 2012: Volume VI, 3.