5th and 8th grade pupils’ and teachers’ perceptions of the relationships between teaching methods, classroom ethos, and positive affective attitudes towards learning mathematics in Japan

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Overall, 5^th grade teachers indicated that they felt that their pupils made progress in the
following cases, although some teachers cited multiple factors:

• When pupils developed their mathematical ideas (N=14, 70%);

• When pupils understood the principles of the content (N=5, 25%);

• When pupils were able to manage the task and calculate accurately and quickly
(N=3, 15%);

• When pupils showed more positive attitudes towards mathematics learning (N=5,
25%).

5^th grade teachers believed that the development of mathematical ideas and
understanding the principles of the content were more important indicators of pupils’
progress than improved memorisation and manipulation of formulas. 5^th grade teachers
thought that pupils’ analogy, reasoning, deductive thinking, inductive thinking,
expression and examination should be promoted. Some teachers valued developing
pupils’ deductive thinking (e.g. ap.6.1.30.5^th); these teachers emphasised the importance
of developing pupils’ understanding of the principles of content rather than the
acquisition of means of manipulation (ap.6.1.31.5^th). Other teachers favoured developing
pupils’ inductive rather than deductive thinking. One 5^th grade teacher ∞mmented that
inductive thinking processes, whereby pupils collected as much information as possible
to lead to a conclusion, could avoid the memorisation and manipulation of formulas and
develop children’s inspiration (ap.6.1.32.5^th). 5^th grade teachers seemed to think that
developing mathematical ideas was especially important when studying shape. One 5^th
grade teacher mentioned that when teaching about shape he attempted in various ways
to promote pupils’ mathematical ideas, such as competencies to examine a problem,
while he gave priority to the acquisition of fundamental knowledge and skills to solve
problems accurately and quickly when learning about number (ap.6.1.33.5^th). These
teachers, at the same time, suggested the development of pupils’ mathematical ideas or
the improvement of pupils’ attitudes as indicators of pupils’ progress. No teachers
believed that managing tasks and calculations was the only indicator of pupils’ progress
in mathematics learning. Some teachers thought that positive learning attitudes might be
indicators of pupils’ progress, even if they did not necessarily link to better performance.
The teachers stated that they felt that pupils made progress in mathematics learning in
the following cases:

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