ideas about measurement in terms of point and set paradigms

10 A. BUFFLER ET AL.
plotted points were classified according to the set paradigm. Since least squares
fitting is not introduced at school, responses associated with the set paradigm
before instruction were exemplified by:
The line should have more or less equal numbers of dots (representing measurements)
above and below it, because in that case we get best average value.
(SLG response)
I have tried to make the line lie so that all points lie at a minimum distance from the
line.
(SLG response)
After instruction, students could explain their set action according to the procedure
of least squares fitting, for example:
I just took the path that is in between all points plotted. Not to pass through them all,
but to decrease the space between all points and the line.
(SLG response)
I have drawn a line which approximates a least squares fit to the data. The distance
between the line and each point has been minimized (approximately).
(SLG response)
Table 2 displays the frequencies of student responses for the UR and SLG probes
classified according to the point and set paradigms before instruction against
responses by the same students after the laboratory course. Table 2 shows that
43% of the students used the set paradigm consistently after instruction, while a
similar percentage were inconsistent in their use of the two paradigms. The proportion
of consistent point data processors decreased after instruction from 77% to
13%, while the proportion of consistent set data processors has increased from 7%
to 43%. However, caution needs to be exercised before asserting that these students
may be categorized as consistently using the set paradigm, since it is possible that
the formalistic procedures of finding the mean and fitting a straight line may have
been used by rote. It is necessary to include the analysis of the probes dealing with
the comparison of data sets in order to assess the consistency of the paradigms used
by the students.
Table 2.
Students’ use of paradigms when processing data sets (UR and
SLG probes).
Paradigm after instruction
Consistent point Mixed Consistent set
paradigm paradigm paradigm Total
Paradigm Consistent point 9 26 19 54
before paradigm (13%) (37%) (27%) (77%)
instruction Mixed 0 5 6 11
paradigms (0%) (7%) (9%) (16%)
Consistent set 0 0 5 5
paradigms (0%) (0%) (7%) (7%)
Total 9 31 30 70
(13%) (44%) (43%) (100%)