ideas about measurement in terms of point and set paradigms

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6 A. BUFFLER ET AL. Students who indicated practice and an increased level of care for measurement skills as the reason for repeating were also classified as using the point paradigm, as illustrated by: We need to roll the ball a few more times. This would confirm if the results obtained conform to each other. Because if there was a mistake in the first reading, it might be noticed from comparing all the results to each other. (RD response) The largest proportion of responses within the point paradigm consisted of those who argued for repeating measurements in order to find a recurring value, for example: Since the ball was released again and two different results were obtained it is important to release it several more times until the equal and/or the same results can be obtained. This will be the exact distance required. (RDA response) On the other hand, responses were classified as indicative of the use of the set paradigm where repeating was aimed at taking an average of a number of readings. Some responses within this sizeable group specifically justified averaging as a way of dealing with the variation in experimental readings, as illustrated by the second quotation below: If you roll the ball several times you will notice that they will not give you the same answer. So the right thing to do is to roll the ball several times and then take the average distance. That average is more accurate than just one result of rolling. (RD response) More than 3 repeats will give a more accurate result as the average is of a bigger number and abnormal measurements will not affect the average greatly. (RDA response) The set paradigm also seemed to underlie the responses which explained the need for repeating specifically as the way of getting a feel for the spread of the readings: They need to repeat several more times to get an idea of the range of the results. (RT response) In some cases responses specified that averaging was needed to approach the ‘true’ value. These responses were also classified within the set paradigm because of the acknowledgement of the imperfection in experimental measurement and the intention of using the set of readings, for example: By rolling some more, taking the average of these values and measuring the uncertainty will give a better understanding as to which values are close to the actual value of d. (RD response) Several of these ‘set’ responses indicated that the spread associated with data collection was not appreciated and that calculating the mean was largely a rote response, as illustrated below: You need a variety of answers from the exact same height to cut out experimental error, e.g. misreading the ruler or error of parallax. Thus you get an average answer for each height and your level of accuracy increases. (RD probe)
FIRST YEAR PHYSICS STUDENTS’ IDEAS 7 Table 1. Students’ use of paradigms when collecting data (RD, RDA and RT probes) Paradigm after instruction (from RD probe) Point Set Not paradigm paradigm codeable Total Paradigm Consistent point 6 28 4 38 before paradigm (9%) (40%) (6%) (54%) instruction Mixed 5 15 1 21 (from the RD, paradigms (7%) (21%) (1%) (30%) RDA and Consistent set 2 3 0 5 RT probes) paradigm (3%) (4%) (0%) (7%) Not 2 4 0 6 codeable (3%) (6%) (0%) (9%) Total 15 50 5 70 (21%) (71%) (7%) (100%) Table 1 presents the frequencies of the student responses for these probes in terms of the use of a point or set paradigm before instruction, against responses by the same students after their laboratory course. The data in table 1 show a distinct shift after instruction towards the use of a set paradigm, i.e. the notion of repeating measurements in order to find a mean. Whereas before teaching, more than half of the students consistently used the point paradigm, only one in five of the students did so after instruction. On the other hand, one in 12 students consistently used the set paradigm before instruction, whereas over two thirds did so after instruction. Deeper analysis of the data shows that the largest shift from the use of a point to set paradigm occurred in the group of students who initially repeated to find the recurring value, but later chose to calculate a mean. The vast majority of students persistently using the point paradigm claimed that they wanted to repeat in order to perfect their measuring skill. The question arises as to whether the students who indicated the intention of repeating in order to calculate a mean have embraced the set paradigm. As shown in a previous study (Lubben et al. in press), many students simply saw calculating a mean as being a requirement of an experiment. Since the necessity of a mean requires generating many different readings, they repeated their measurements. Thus, the remaining probes explored whether the apparent shift away from the point paradigm was superficial (rote or formulaic), or whether the changes were of a deeper epistemological nature. Data processing Two probes were used to solicit students’ ideas about data processing: one probe asked students to represent a set of data by a single quantity, and the other required a series of plotted points to be modelled graphically.
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