The tasks students do, and the sequence of these tasks over time, can dramatically shape how students imagine and think about fractions. To consider how students build their understanding of fractions based on their prior understanding of whole numbers, analyze the following two problems from a Japanese textbooks series and discuss the questions that follow the problems.
In grade 2, Japanese students investigate how many liters of water fit in various containers. (The shaded area represents the water.)
How many liters of water are there?
Then, in grade 3, Japanese students encounter the following related problem. (Assume that the 1-liter container is divided into equal parts.)
How much water is shown by each interval? How much water is in the container? How many liters are in the container?
- What are the similarities between the two problems shown above?
- What knowledge about whole numbers might students bring to the fraction problem?
- Some US students struggle to answer the question: “How many 1/5 are in 2/5?" How might the tasks above support their understanding?