These two developments are fundamentally different. On the one hand, embodiment
gives a global overall picture of a situation while symbolism begins with coordinating
actions, practicing sequences of actions one after another to build up a procedure,
perhaps refining this to give different procedures that are more efficient or more
effective, using symbolism to record the actions as thinkable concepts. The problem
here is that the many different procedures can, for some, seem highly complicated
and so the teacher faces the problem of reducing the complexity, perhaps by
concentrating on a single procedure to show the pupils what to do, without becoming
too involved in the apparent complications. Procedures, however, occur in time and
become routinized so that the learner can perform them, but is less able to think about
them. (Figure 3.)
As an example, consider the teaching of long-multiplication. First children need to
learn their tables for single digit multiplication from 0 ! 0 to 9 ! 9. They also need to
have insight into place value and decimal notation.
The method used by Hideyuki Muramoto in the lesson study at Sapporo City
Maruyama Elementary School on December 6, 2006 can be analysed in terms of an
initial embodiment representing 3 rows of 23. Here the learner can see the full set of
counters: the problem is how to calculate the total. The embodiment can be broken
down in various ways, separating each row into subsets appropriate to be able to
compute the total. In the previous lesson the students had already considered 3 rows
of 20 and had broken this into various sub-combinations, breaking each row into
10+10 or 5+5+5+5, or even 9+9+2, or 9+2+9. Now the problem related to breaking
Figure 3: Developmental framework through embodiment and symbolism