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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• When they developed independent thinking, such as when trying to think out the
problem or new tasks by themselves;

• When they reduced their negative feelings about mathematics learning, noticing that
they could master the learning content if they tried hard to do so.

• When they showed greater willingness to share their understanding with their peers;

• When they were more positively involved in learning, for example, raising hands to
answer questions, putting forward views in discussion and asking concrete
questions.

5^th grade teachers also seemed to think that their pupils might feel a sense of progress
from concrete results such as better performance in lessons and test settings. Teachers
supposed that pupils felt a sense of progress in the following cases:

• When they began to get higher marks in mathematics tests (N=15, 75%);

• When they began to find answers accurately and quickly (N=10, 50%),

• When they could raise views in discussion and get approval from the teacher and
classmates (N=4, 20%);

• When they could understand content (N=3, 15%).

Most teachers complained that pupils were likely to put too much emphasis on such
tangible results. One 5^th grade teacher mentioned that when children put too much
stress on better results it could lead to a neglect of processes, although he admitted that
results such as approval and praise in discussion might give the child confidence
(ap.6.1.34.5^th).

8^th grade teachers also perceived that understanding of content (N=4, 33%) and
development of mathematical ideas (N=5, 42%) were indicators of pupils’ progress in
mathematics learning. They believed that understanding the content written in a textbook
(ap.6.1.35.8^th) and teachers’ explanation of that content (ap.6.1.36.8^th) was more than
memorisation of formulae. One 8^th grade teacher explained that pupils who have
understood the principles can find out why they have got a wrong answer and amend
their understanding, while pupils memorising the formula believe that their wrong
answers are due to their carelessness in manipulating the formula and normally
progress more slowly (ap.6.1.37.8^th). Teachers seeking a sense of progress in the pupils’
development of mathematical ideas believed that developing pupils’ mathematics ideas
could enhance pupils’ affective attitudes such as interest, motivation, sense of

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