The main feature of lessons in the US series was the use of pictorial
representations and manipulatives. For the first three grades, all of the examples
used models to represent fractions. In the fourth grade, 80% of the examples used
models, but only 30% of the examples in the fifth-grade text used models. Area
models were used in the first and second-grade texts; the third-, fourth-, and fifthgrade
texts used the fraction bar (55%) and the number line (20%).
All of these textbook series move to teach fractions in an abstract way in fifth grade,
where fewer models are used in the worked examples (Table 3). The Japanese text often
presented the solution to problems by giving students two different procedures and asking
them to explain and compare the two procedures. The US and Japanese series presented a
summary of the procedures as early as the third grade. The Kuwaiti series did not present
these procedural summaries until the fifth grade.