The development represented in Table 3 was typical of the case study students. Where
there was not a “reasonable” or “substantial” gain in modes of thinking, there was at least
an increase in the quantity and variety of responses. Similar gains in modes of thinking
were noted when case study student responses for the Pre-Project and Post-Project
Interviews, and responses to the first and final Mathematical Searches, were compared. All
case study students displayed both mathematical and contextual thinking during the first
interview and all three modes of thinking, mathematical, contextual, and strategic, during
the second interview. For six students, this represented a “substantial” gain, for one a “very
substantial” gain, and for one, a “reasonable” gain. A summary is contained in Table 4
where “M” represents mathematical thinking, “C” represents contextual thinking, “S”
represents strategic thinking, and (em) represents emerging thinking.