Eleven instructional tasks requiring changing how numbers are represented were used in U.S. classrooms and
21 (How were these numbers obtained?) instructional tasks requiring changing how numbers are represented were
used in Chinese classrooms. For each of the instructional tasks, verbal explanations were always used either alone or combined with other representations. U.S. teachers used only verbal explanations in 3 out of 11 instructional tasks
(27%). For the remaining eight instructional tasks, visual representations or physical manipulatives were used as part
of the pedagogical representations in solving these tasks in the U.S. classrooms. For example, in Ms. T’s second lesson,
she asked her students to use fraction pieces to demonstrate the meaning of 4/3 and use pictures to represent 4/3 as a
mixed number. For four of the instructional tasks (36%), symbolic representations were also used in addition to visual
representations.
In contrast, in 19 out of 21 (90%) instructional tasks requiring changing how numbers are represented, Chinese
teachers used symbolic representations combined with verbal explanations. Only for one of the instructional tasks,
which is related to shading a 100 × 100 chart board to represent 17, 50, and 99%, did the teacher and students use
visual representations. For the remaining task, only verbal explanations were used.