The Japanese series offered only four chapters on fractions from grades 3 to 5. The chapter on
fractions began with an opening problem, followed by a set of lessons. General practice exercises
followed under three categories: practice, check, and challenge. The opening problem
presented a situation that highlighted the need to learn the concept introduced in the chapter.
The Japanese lessons began with a real world problem or a question for students to think
about, followed by the learning objective or question, to focus students’ attention on what
they would learn from the problem. For example, in the third grade, when fractions were
introduced, the problem presented was about measuring a student’s arm. The text
highlighted the need to express fractional parts (Fig. 3). The text then presented a hint to
direct students to a solution to the problem, followed by a summary of the new concept.
The lesson was followed by a set of exercises for practicing the new idea or skill. There was
little variation from this sequence, other than some lessons that omitted the hint or the
summary. This sequence appeared several times in each lesson, depending on the number of
concepts the lesson presented. In most of these lessons, an answer to the problem was
provided. The number of example problems increased by grade, from 66% in the third
grade to 70% in the fourth and 90% in the fifth grade.
What was unique about the Japanese series was the connection between fractions and
real-world problems involving measurement. Two thirds of the opening problems presented
to introduce a concept and all of the problems in the lessons dealt with measures of length
(distance) or the volume of liquids. All of the models in the lessons focused on measuring
units in meters or liters. Such models make a strong connection with students’ lives and
make fractions meaningful for students. The Japanese texts also made use of the number
line. The number line helps students to visualize fractions and compare differences in size
and order. Almost a third of the problems in the lesson from the third to the fifth grade used
the number line. The area model of a fraction as a part of a whole was used only in limited
situations, such as to illustrate the addition and subtraction of mixed numbers.
The US series included 11 chapters on fractions between grades 1 and 5. These chapters
began with some fact related to social studies, science, history, or other subjects. These facts