Learning Study Guided by Variation Theory: Exemplified by Children Learning to Halve and Double Whole Numbers

Abstract

This study aims to describe how the learning study model can be used to improve lesson design and children’s learning outcomes by enabling them to perceive and define the critical aspects of the object of learning, guided by variation theory. Three lesson designs were used with three groups of children (A = 24, B = 13, C = 14) from two schools. The results of the first design were analyzed before the second lesson was designed and conducted in a new group, and the results of the second were analyzed before the third was designed. The patterns of variation were based on simultaneous discernment of a base amount and half or double that amount. This was made to prevent the children from understanding doubling as just copying the original amount without including the original amount in the total sum, giving the double of 4 is 4 instead of 8. The use of varied original amounts made the children separate the concepts double and half from the amount, to conceive them as relative, rather than constant, values. The results show increased learning outcomes in all groups; however, differences between pre- and posttest were significant only for Group B (p = .015).

Keywords:

• variation theory
• learning study
• fractions
• double and half
• primary school mathematics