Discussion of results regarding Question (c)

161

lower high school, 14 at upper high school, and 18 at both levels—, the five teachers categorized in theme clusters B4 and B9 should be disregarded, as well as all the teachers who provided keywords considered not useful to deal with or solve the given task. Since some teachers 4 teachers belong to more than one of these clusters, at the end 12 teachers must be disregarded from the initial 49 respondents. Therefore, 37 teachers who answered Question (b)—69.8% overall, 13 working at lower high school, 12 at upper high school, an 12 at both levels—appear to have fully met the assessment criteria of Indicator C-2.

162

more variability, supported such choice by arguing that “Distribution B has more variability because is symmetric”. Thus, this teacher seems to think of variability in terms of symmetry: the greater the degree of symmetry of a histogram, the greater is its variability, which is contrary to the common understanding that the histogram with lesser pattern in the heights of the bars or the more asymmetric one has a greater variability of its data set. This way of thinking of variability expresses a misconception that might be found commonly in students and—in a lesser extent—teachers, which has been well-reported in the literature (Meletiou & Lee, 2003, 2005; Cooper & Shore, 2007; González, 2011; Isoda

& González, 2012, González, 2013a, 2013b). Among the remaining 7 teachers, 5 provided answers falling into category A6, and 2 provided answers falling into category A5.

Among the 17 teachers who partially answered Question (c), all the 17 provided assessment on the correctness or incorrectness of Student 1’s response, 12 provided some explanation about why that answers was correct or incorrect, and 11 gave a comment about the most likely reasoning behind such student’ answer, with 8 of them giving their judgment about whether the answer was correct or not, why they think so, and what reasoning might have led student to produce the given answer. Regarding Student 2, 16 teachers provided assessment on the correctness or incorrectness of Student 2’s response, 11 provided some explanation about why that answers was correct or incorrect, and 7 gave a comment about the most likely reasoning behind such student’ answer, with 4 of them giving their judgment about whether the answer was correct or not, why they think so, and what reasoning might have led student to produce the given answer. Finally, regarding

163

Student 3, 15 teachers provided assessment on the correctness or incorrectness of Student 1’s response, 12 provided some explanation about why that answers was correct or incorrect, and 6 gave a comment about the most likely reasoning behind such student’

answer, with just 3 of them giving their judgment about whether the answer was correct or not, why they think so, and what reasoning might have led student to produce the given answer. Table 21 presents a breakdown of the teachers who partially answered Question (c) by school level.

Among the 17 teachers who partially answered Question (c) but provided assessment on the correctness or incorrectness of Student 1’s response, 12 (22.6% overall;

4 working at lower high school, 5 at upper high school, an 3 at both levels) correctly judged the accuracy of the answers given by the fictitious Student 1; that is, 12 teachers assessed Student 1’s response as incorrect. Among this group of teachers, 7 (13.2%

overall; 2 working at lower high school, 3 at upper high school, and 2 at both levels) also provided an explanation about why they considered such answer as incorrect, with all these 7 teachers giving appropriate explanations. Moreover, among these 7 teachers, 6 gave comments about the most likely reasoning behind Student 1’s answer, with all of them providing appropriate explanations. The 5 teachers (9.4% overall; 1 working at lower high school, 1 at upper high school, and 3 at both levels) who assessed Student 1’s response as correct seem to think of variability in terms of symmetry, regarding the histogram with the greater the degree of symmetry as the one with lees variability, which is in line with the common understanding that the histogram with lesser pattern in the

164

Table 21: Breakdown of the teachers who partially answered Question (c) by school level – Frequency of occurrence

Frequency (%)

Student 1 Student 2 Student 3

L.H. School (19 teachers)

U.H. School (15 teachers)

Both Levels (19 teachers)

L.H. School (19 teachers)

U.H. School (15 teachers)

Both Levels (19 teachers)

L.H. School (19 teachers)

U.H. School (15 teachers)

Both Levels (19 teachers)

Teachers’ Answers

Provided

assessment on the accuracy of student’s

response

5 6 6 4 6 6 4 6 5

Provided explanation

about why student’s

response was correct or incorrect

3 4 5 2 5 4 2 5 5

Provided a comment about the most likely reasoning behind student’s response

2 5 4 1 3 3 1 3 2

Provided both an assessment on the accuracy of student’s

response and an explanation about why so

3 4 5 2 5 4 2 5 4

Provided both an assessment on the accuracy of student’s

response and a comment about the most likely reasoning behind such response

2 5 4 0 3 3 0 3 1

Provided an assessment on the accuracy of student’s

response, an explanation

about why so, as well as a comment about the most likely reasoning behind such response

2 3 3 0 2 2 0 2 1

165

heights of the bars or the more asymmetric one has a greater variability of its data set, a misconception that might be found commonly in students and—in a lesser extent—teachers, which has been well-reported in the literature (Meletiou & Lee, 2003, 2005; Cooper & Shore, 2007; González, 2011; Isoda & González, 2012, González, 2013a, 2013b). In fact, by cross-referencing the answers given by these 5 teachers regarding Student 1 with their answers to Question (a), we can see that 1 teacher did not answer it, one gave an answer falling into category A1, one gave an answer falling into category A2, and 2 gave answers falling into category A3, which indicates that most of these teachers seem to be harboring the conception “Variability as visual cues in the graph”, since they disregard the actual data values, their spreads, and the measures of distance or difference, and mistakenly acknowledge variability as unevenness in the frequencies of a histogram or lack of symmetry.

Among the 16 teachers who partially answered Question (c) but provided assessment on the correctness or incorrectness of Student 2’s response, 10 (18.9% overall;

3 working at lower high school, 4 at upper high school, an 3 at both levels) correctly judged the accuracy of the answers given by the fictitious Student 2; that is, 10 teachers assessed Student 2’s response as incorrect. Among this group of teachers, 8 (15.1%

overall; 2 working at lower high school, 3 at upper high school, and 3 at both levels) also provided an explanation about why they considered such answer as incorrect, with all these 7 teachers giving appropriate explanations. The one who did not provide an appropriate explanation—a teacher working at both levels—wrote that “Variability is not

166

so much about the distribution limits, as about the shape how the data are distributed”.

This explanation is in line with the choices made regarding Student 1, in which the same teacher assessed that answer as correct, as well as in Question (a), in which this teacher chose Distribution A as the one with more variability, which appears to be evidence of the conception “Variability as visual cues in the graph”.

Among these 7 teachers who gave appropriate explanations about the accuracy of Student 2’s response, 2 gave comments about the most likely reasoning behind Student 2’s answer, with all of them providing appropriate explanations. Moreover, the 6 teachers (11.3% overall; 1 working at lower high school, 2 at upper high school, and 3 at both levels) who assessed Student 1’s response as correct seem to think of variability in terms of the span in the vertical axis—i.e., judging the variability of the data displayed in a histogram by the largest difference in height of its bars—, instead of looking at the horizontal spread of data around a measure of central tendency. This is a common mistake made by many students, and even by some of their teachers, at any school level (Isoda &

Gonzalez, 2012). In fact, by cross-referencing the answers given by these 6 teachers regarding Student 2 with their answers to Question (a), we can see that 3 gave answers falling into category A4, 2 gave answers falling into category A6, and 1 gave an answer falling into category A2, which indicates that most of these teachers considered Distribution B as the one with more variability, but 3 of them evidencing in Question (a) misinterpretation of how to appropriately measure variability.

167

Among the 15 teachers who partially answered Question (c) but provided assessment on the correctness or incorrectness of Student 3’s response, 8 (15.1% overall; 2 working at lower high school, 4 at upper high school, an 2 at both levels) correctly judged the accuracy of the answers given by the fictitious Student 3; that is, 8 teachers assessed Student 3’s response as correct. Among this group of teachers, 5 (9.4% overall; 1 working at lower high school, 3 at upper high school, and 1 at both levels) also provided an explanation about why they considered such answer as correct, with all of them giving appropriate explanations. Among these 5 teachers who gave appropriate explanations about the accuracy of Student 3’s response, just 2 gave comments about the most likely reasoning behind Student 3’s answer, with all of them providing appropriate explanations.

All the 7 teachers (13.2% overall; 2 working at lower high school, 2 at upper high school, and 3 at both levels) who assessed Student 3’s response as incorrect, provided explanations about why they thought so. From such explanations, it seems that these teachers think of variability in terms of neither the degree of clustering nor dispersion in data. In fact, these teachers provided explanations such as “the definition of variability doesn’t support this statement”, “this student doesn’t know that symmetry qualifies as variability”, “this fact doesn’t constitute sufficient reason to make any conclusion”, and

“it’s like the same argument given by Student 1”. This lack of understanding was evidenced by the teachers’ inability to adequately choose the distribution with more variability in Question (a). Indeed, all these 7 teachers were unable to answer correctly Question (a): 2 of these teachers gave answers falling into category A4, 2 gave answers

168

falling into category A3, 1 gave an answer falling into category A2, 1 gave an answer falling into category A1, and 1 did not answer. This indicates that these 7 teachers have problems understanding variability as the degree of clustering or deviation from an expected value or a measure of central tendency.

5.9.3.1 Regarding the fulfillment of Indicator B-1

In order to fully meet this indicator, teachers must show evidence of being able to examining the work done by three fictitious students in their absence, and correctly determine whether their answers to the given task are right or not, based just on the provided numerical data and written explanations. Since only 8 teachers (15.1% overall; 4 working at lower high school, 2 at upper high school, an 2 at both levels) consistently exhibited ability to correctly judge the accuracy of the answers given by the fictitious Students 1, 2 and 3, as well as made appropriate observations for each case about why they thought so, Indicator B-1 seems to be fully satisfied only by these eight teachers. Those teachers who mixed correct and incorrect assessments to the answers given by Students 1, 2 and 3 were not regarded as meeting the assessment criteria of this indicator.

5.9.3.2 Regarding the fulfillment of Indicator B-2

In order to fully meet this indicator, teachers must show evidence of being able to explain how students arrived at their solutions by providing the likely reasoning that led

169

them to their answers. This explanation is done in students’ absence, and is based just on the provided numerical data and written explanations. All the 8 teachers who fully answered Question (c) and appeared to fully satisfy Indicator B-1, also provided appropriate comments about the most likely reasoning behind each student’ answer. Thus, Indicator B-2 seems to be fully met—again—only by these eigth teachers (15.1% overall;

4 working at lower high school, 2 at upper high school, an 2 at both levels).