For Question 3, the situation is a bit different because the conversions are necessary because we need to choose ‘the register in which the necessary treatments can be carried out most economically or most powerfully’ (Duval, 2006, p. 127) which permitted the process of unstandardising of the z-score. One could not access the z-score without doing a conversion operation which would allow movement from the percentage value to the z-score. It is necessary to distinguish between direct and inverse problems (Groetsch, 1999) in this study. A direct problem is one that asks for an output when students have the input and the process. In an inverse problem, students have the output and the problem could ask for the input or the process that led to the output. One can regard Question 1 as a direct problem and Question 2 as a two–step direct problem. One can regard Question 3 as an inverse problem because it consists of a conversion that takes a p-value and converts it to a z-score. The z-table is organised according to the z-scores. For a given z-value, students can read off a corresponding p-value. In Question 3, the students had a probability value and had to scan the tables until they identified a suitable z-score that corresponded to the given probability. Secondly, the formula in the formula sheet was the standardisation formula 3EQN3_132_PYTHAGORAS.jpg . In Question 1 and Question 2, the students used the formula in the form presented. The value x was the input and the output was z. However, for Question 3, the students had output z and they had to calculate the input. Therefore, one can regard Question 3 as a combination of two inverse problems and as an inverse problem in the way that Groetsch (1999) described. In order to present the analysis, the students’ responses are labelled to serve as references, for example, S17 which means the response was that of student 17. Students’ responses are labelled from S1 to S290. The students’ responses are verbatim, although the layout has been changed because of limited space. Findings for Question 1 × Blank or unrelated algorithm Here responses were coded blank if students made no attempt. A response was coded as unrelated algorithms if students wrote a formula where the algorithms did not relate to the standardisation procedure. Two examples follow: 4EQN4_132_PYTHAGORAS.jpg ○ Partial treatments (PT) Here responses were coded as partial treatments (PT) if students wrote the appropriate standardisation formula but did not substitute the correct values or substituted the correct values but did not compute the result correctly, for example: 5QN5_132_PYTHAGORAS.jpg ● Complete or full treatments (FT) Here responses were coded as complete or full treatments (FT) if students completed the standardisation and arrived at the correct figure of -0.945 or if they wrote the value 0.945 as the value they would read off from the z-table. If they went on to other steps that were incorrect, then the responses were coded as FT. For example, some students (12) did not read off a p-value from the z-table and interpreted the z-score as a probability. An example follows: 6EQN6_132_PYTHAGORAS.jpg Some students continued and used the resulting ‘probability value’ (obtained as for S1) to determine a z-score in the z-table. An example follows: 7EQN7_132_PYTHAGORAS.jpg Here the student used the z-score (-0.946) as a probability value, found the z-score that corresponded to the ‘probability value’ and presented the z-score (1.83) as a probability (even though it was greater than 1). □ Partial conversions (PC) Responses were coded as partial conversions (PC) if students determined a p-value from the z-table that corresponded to the z-score even if the value was not accurate as long as there was a reading of a p-value from a related z-score. An example follows: 8EQN8_132_PYTHAGORAS.jpg ■ Complete or full conversions (FC) Here responses were coded as complete or full conversions (FC) if students interpreted the p-values of the z-table in terms of the area under the curve to provide correct (or nearly correct) answers. Each step depends on the previous step. Therefore, a student who completed an FC, would have done the PC, FT and PT steps. Table 4 shows that, of the 290 students, 223 (77%) were able to recognise the correct standardisation formula. Only 199 (69%) were able to complete the standardisation procedure correctly. Fifty-five (19%) performed partial conversions and 79 (27%) completed the conversions and the question. 4Results for Question 1. TABLE4_132_PYTHAGORAS.jpg In order to get a clearer idea of how the students progressed from the treatment steps to the conversion steps, we can consider the cumulative totals: the number of students who managed partial treatments will include those who completed the treatments those who completed the treatments will include those who managed partial conversions those who managed partial conversions will include those who completed full conversions. The bar graph in Figure 5 gives these numbers. Of the 290 students, 223 (77%) students began the appropriate standardisation procedure. Of these 223 students, 199 (89%) completed the standardisation treatments and of these, 134 (67%) were able to complete the first part of the conversions. Seventy-nine (59%) of the last group were able to complete the conversions correctly. Findings for Question 2 The following codes were used for Question 2. It is not necessary to give examples of responses in all categories because they are similar to those for Question 1 except that there are two sets of treatments and conversions. × Blank or unrelated algorithm ○ Partial treatments (PT), where students chose the appropriate standardisation formula (in one or in both cases) but did not complete both.● Full or complete treatments (FT), where students completed the standardisation procedure in one or in both cases but completed no further correct steps. □ Partial conversions (PC), where students read off a p-value from the z-table in one or in both cases, but did not combine the two p-values correctly, for example: 9EQN9_132_PYTHAGORAS.jpg ■ Full or complete conversions (FC), where students interpreted the p-values of the z-table in terms of the area under curve to provide correct (or nearly correct) answers. Table 5 shows that, of the 290 students, 174 (60%) started one or both standardisation procedures, whilst only 156 (54%) were able to complete one or both standardisation procedures correctly. Only 40 students (14%) completed the questions correctly (two of whom had a final answer that differed slightly from the expected one). 5Results for Question 2. TABLE5_132_PYTHAGORAS.jpg

In order to get a clearer idea of how the students progressed from the treatment steps to the conversion steps, I considered the cumulative totals from right to left: the number of students who managed partial treatments will include those who completed treatments those who completed treatments will include those who managed partial conversions those who managed partial conversions will include those who completed full conversions. The bar graph in Figure 5 gives these numbers. Of the 290 students, 174 (60%) were able to recognise the correct standardisation formula, whilst only 156 (90%) of these student were able to complete it correctly once or twice. Of these 156 students, 96 (62%) completed only the first part of the conversions once or twice (they read off the p-value for the corresponding z-score). Only 40 (42%) of these were able to complete the conversions and arrive at the correct result. Findings for Question 3 The following codes were used for Question 3: × Blank or unrelated algorithm □ Partial conversions (PC), where students interpreted the percentage value given as a p-value, which was the correct one (p = 0.4), but did not carry out any further correct steps or could have interpreted the percentage as an incorrect p-value. 10EQN10_132_PYTHAGORAS.jpg ■ Full or complete conversions (FC), where students read off p-values in a z-table to generate a z-score which was correct or incorrect; students who completed full conversions all continued. ○ Partial treatments (PT), where students chose the appropriate formula for unstandardising a z-score. 11EQN11_132_PYTHAGORAS.jpg ● Full or complete treatments (FT), where students completed the procedure for unstandardisation correctly or nearly correctly. 12EQN12_132_PYTHAGORAS.jpg 6Results for Question 3 TABLE6_132_PYTHAGORAS.jpg The response of S133’s was coded almost correct compared to that of S135, where the final answer was not close to the expected one. Table 6 shows that, of the 290 students, 108 students did not respond and 34 used an irrelevant algorithm. Therefore, 142 (49%) did not even begin partial conversions. Seventy-eight (27%) tried but did not generate the correct p-value whilst 20 (7%) students completed partial conversions by correctly extracting the p-value from the information the students had. Three (1%) students completed the conversions and started the unstandardising treatments, whilst 47 (15%) students managed complete treatments and obtained a correct or almost correct solution (the final answer that 26 students reached differed slightly from the expected answer). In order to get a clearer idea of how the students progressed from the conversion steps to the treatment steps, I considered the cumulative totals from right to left: the number of students who completed partial conversions will include those who completed full conversions those who completed full conversions will include those who completed partial treatments those who completed partial treatments will include those who completed full treatments. The bar graph in Figure 6 gives these figures. There were 148 (51%) students who started the conversions (obtained p-values). Of these 148 students, 50 were able to complete the conversions by reading off p-values and chose the correct formula for unstandardising. That is, 34% completed the conversions (read off the p-values for the corresponding z-score) and started treatments whilst 47 (94%) of the 50 students were able to complete the treatments and solve the problem (the final answers of 26 students differed slightly from the expected one). Performance on the three questions Students clearly found that Question 2 was more challenging than Question 1 was. Only 40 students got Question 2 correct whilst 79 students managed to complete Question 1 correctly – almost twice as many. Furthermore, there were 67 blank or incorrect algorithms for Question 1 compared to 116 for Question 2. This showed that more students did not attempt to solve Question 2 than those who failed to attempt Question 1. It is clear that Question 2 was more complex than Question 1 because it involves regions bounded by two given x-scores. Therefore, there were two sets of treatments as well as two sets of partial conversions and completing the conversions meant that students had to take a global view of the two areas and decide how they would use them to generate the required percentages. Consequently, solving Question 2 would have been more demanding than just carrying out treatments followed by conversions, as Question 1 required. Question 3 was challenging for the 142 (49%) students who did not start correctly. Forty-seven completed the whole question correctly or almost correctly. This was more than the 40 who completed Question 2 correctly or almost correctly but fewer than the 69 who completed Question 1 correctly or almost correctly. If one compares performance on Question 3 with that on Question 1, 67 students did not start Question 1 correctly. On the other hand, there were more than twice as many (142) students who did not begin Question 3 correctly. There are two possible reasons for this. Firstly, the inverse nature of the question meant that the steps to the solution were reversed, which made it more complex (Bansilal, Mkhwanazi & Mahlabela, in press; Groetsch, 1999; Nathan & Koedinger, 2000). Secondly, students had to complete the conversions before the treatments. This created a bigger first barrier than the situation where the first barrier was not as great as the second was. Duval’s (2006) theory maintains that conversion transformations are more difficult than treatment transformations are because they require crossing into another register of representation. Conversions are more complex because they involve movement in each of the two registers and movement across them, whilst treatments require movement in one register only. Success rates in conversion transformations and treatment transformations The bar graph in Figure 5 provides a visual representation of the progress of students through the stages for Question 1 and Question 2. It shows the number of students who did a PT, FT, FT PC and FT FC respectively and excludes the students who made no response or used a wrong formula. Note that, in this graph, the first set includes the second, which includes the third, which includes the fourth and derives from the figures Tables 4 and Table 5 provide. The cumulative picture for Question 3 (see Figure 6) shows the number of students who completed a PC, FC, FC PT and FC FT respectively. The first set includes the second, which includes the third, which includes the fourth. These figures derive from the information Table 6 provides. There are clear trends in performance on Question 1 and Question 2. Of the 290 students, 223 (77%) performed a PT on Question 1. Of these, 199 (89%) completed the treatments. Of this group, 134 (67%) went on to complete a PC and 79 (59%) of this group were successful. For Question 2, the numbers from Table 2 are 290 (original), 174 (PT), 156 (FT), 96 (PC) and 40 (FC). The flow diagrams below show these figures: Question 1: 100% → (PT) 77% → (FT) 89% → (PC) 67% → (FC) 59% Question 2: 100% → (PT) 60% → (FT) 90% → (PC) 62% → (FC) 42% The attrition rate at each stage of Question 2 was higher than that for Question 1, except for the progression from partial treatments to full treatments, where 90%of students who managed partial treatments for Question 2 completed the treatments. The corresponding percentage for Question 1 was 89%. However, for all other stages, the progression rate from one to the next was higher for Question 1 than it was for Question 2. On both questions, the highest attrition rate was in the progress from PC to FC. It showed that only 59% of students who started conversions for Question 1 completed them, whilst for Question 2 only 42% of students who started the conversions were able to complete them. When one considers the performance on Question 3, the numbers from Table 6 are 290, 148 (PC), 50 (FC), 50 (PT), 47 (FT). The flow diagram below shows the figures: Question 3: 100% → (PC) 51% → (FC) 34% → (PT) 100% → (FT) 94% Here, as for Question 1 and Question 2, the highest attrition rate was in the movement from PC to FC. Only 34% of the group who started conversions were able to complete them and all of these students went on to start treatments. Thereafter, there were few challenges for this group and only three students did not complete the procedure. The treatment procedure for Question 3 was not a problem for those students who completed their conversions. Forty-seven of the 50 students (94%) who completed conversions were able to complete treatments. The conversions were problems in Question 1 and Question 2. They were insurmountable for many, because only 79 of the 199 (39%) and 40 of the 156 (25%) of the students who completed treatments were successful with conversions. A comparison between trends in responses across the questions supports Duval’s assertion that conversion transformations can be more complex than treatments. For Question 1 and Question 2, the percentage of students who proceeded from full treatments to full conversions was 39% and 25% respectively, whilst for Question 3 the percentage of students who proceeded from full conversions to full treatments was 94%. It is clear that, for the group as a whole, the students’ success rates in conversion transformations were lower than in treatment transformations. However, not all the students would have experienced conversions as more difficult than treatments. The movement between the two registers was not a problem for some students. Direction of conversions The direction of conversions is another factor that Duval contends affects the complexity of mathematical activities. Duval maintains that a ‘conversion in one direction can be without any cognitive link with this in the reverse direction’ (Duval, 2008, p. 47), suggesting that the direction of the conversions is important. Duval has shown that, when the original and destination registers of conversions change, students’ performances vary considerably. In one case of linear algebra, 83% of students were able to move successfully between a two-dimensional table representation of a vector to a two-dimensional graphical representation, whereas only 34% of students were able to move in the opposite direction. The direction of the conversions seems to have been a factor that influenced the students’ success rates. Sixty-nine students completed Question 1 correctly, whilst only 40 students did so on Question 3. Of the students who started conversions for Question 1, 59% were able to complete them, whilst only 34% of the students who started conversions for Question 3 were able to do so. The reason for the lower completion rate for the conversions for Question 3 could lie in the fact that the conversion transformation of Question 1 involved moving from the z-scores to the probability value (or area) that travelled in the opposite direction to the conversion in Question 3 (moving from the probability value to the z-score). In addition, 89% of the students who completed conversions for Question 3 went on to complete the treatments. Therefore, the conversions were bigger hurdles. The percentage of students who proceeded from full treatments to full conversions in Question 1 was 39%. One of the factors that made Question 3 more challenging was the direction of the conversions, which was different in the two cases. Duval’s own observations about linear algebra (2008) support this. However, we need further research to help us understand why conversions in one direction were more challenging to complete than were conversions in another. SummaryThis article presented an analysis of 290 students’ responses to a three-part task using applications of the normal distribution curve. Duval’s framework was used to explain the students’ difficulties with solving the task. Question 1 and Question 2 of the task are ‘unknown percentage problems’ and Question 3 is an example of an ‘unknown value problem’ (Watkins et al., 2004) and one can regard it as an inverse problem (Groetsch, 1999). Different parts of the solutions to the questions were categorised into conversions and treatments, depending on whether the operation required students to move across a register or stay within the same register. The students’ responses were coded according to whether they performed partial treatments, complete treatments, partial conversions or complete conversions. The findings show that Question 2 was more difficult than Question 1: twice as many students completed Question 1 correctly compared to Question 2. It was argued that Question 2 was more challenging because students had to complete two sets of conversions and two sets of treatments. The results of these transformations had to be synthesised together to produce an answer. It was also found that Question 3 was more challenging than Question 1 was. Seventy-nine students obtained correct answers for Question 1 and only 47 obtained correct, or close to correct, answers for Question 3. It was argued that one factor could be the inverse nature of Question 3, whilst Question 1 was a direct problem. The other factor could be that students needed to complete the conversion transformations for Question 3 before the treatment transformations. Furthermore, because the conversions were bigger hurdles, more students could not progress further. The students encountered the treatment transformations first in Question 1. More students succeeded with this hurdle than with the first hurdle in Question 3, allowing them to progress. Duval’s theory that conversions are more challenging than treatments is supported by the findings in this study. When the attrition rate is examined at each stage in each of the three questions, there were clear patterns in the performance of the students. On Question 1 and Question 2, 59% and 42%, respectively, of the group that started conversions were able to complete them. This compares to approximately 90% of the group who started treatments who were able to complete at least one treatment. In addition, only 34% of the group who started conversions for Question 3 were able to complete them, whereas 94% of the group who started treatments were able to complete them. This shows that completing the conversions was harder than completing the treatments in all three of the questions. Furthermore, this study supports Duval’s (2006) examples in linear algebra that show that the direction of conversions also plays a role in the difficulty level of questions. He writes that ‘when the roles of source register and target register are inverted within a semiotic representation, the problem is radically changed for students’ and that ‘performances vary according to the pairs (source register, target register)’ (p. 122, brackets added). This was true for Question 1 and Question 3. In Question 1, if one considers the group of 134 who completed the treatments, then 79 of these (or 58%) succeeded in completing the conversions when the movement was from z0 to P(Z < z0). In Question 3, when the movement was from P(Z > z0) to z0, the success rate was 34% (50 of the 148 had identified some sort of p-value). This shows that the students found the second conversion more difficult. If one considers the percentages for the whole group of 290, then 79 of the 290 (or 27%) were able to complete conversions for Question 1 whilst only 50 of the 290 (or 17%) were able to complete conversions for Question 3. Implications of the findings Duval (2006) differentiated between treatments and conversions and commented that ‘we cannot deeply analyse and understand the problem of mathematics comprehension for most learners if we do not start by separating the two types of representation transformation’ (p. 127). This study has also shown that conversions and treatments in this problem offer different levels of challenges to students. Therefore, educators should note the additional challenge of moving between systems of representations. The findings suggest that educators may need to support conversion transformations more than treatment transformations to help learners to overcome the challenges. One aspect that deserves notice is that this group of students did not receive any computer-aided instruction, nor could they work through computer simulations of normal curves, as normally happens in probability and statistics modules nowadays. If they had had some exposure, they might have had a better idea of the visual aspects of the normal distribution curve and may have been able to switch between representations more easily. Applets or other computer simulation activities could allow students to engage with the properties the different representations reveal. They could also help students to explore situations that show links between the changes in the z-scores with the changes in the area values in the different modes of representation. Drawing on Zazkis et al.’s (1996) VA model, perhaps such opportunities will help students move more effortlessly between the different registers, thus reducing the barriers related to carrying out conversion transformations. The solutions to these questions involved coordinating two different registers, which were initially separate. However, Zazkis et al. (1996) suggest, in their VA model, that even though movement between two modes may start as distinct and separate, they eventually merge. Zazkis et al. confine their discussion to the movement between the acts of visualisation and analysis. However, we can apply it to the two registers that we have identified here to suggest that, at some point, the students will regard the combination of these two registers as one that enriches their ‘cognitive architecture’ (Duval, 2006), and which will enable them to move on to further layers of movement between more complicated registers. Finally, this article delved into students’ engagements with the treatment and conversion transformations associated with one particular problem. Readers may want to consider whether one could look at other areas in similar ways and whether they could help to explain the students’ difficulties in those areas. It is hoped that this study will encourage other researchers to look for evidence to support or contradict these findings in other areas. Additionally, it is hoped that such further research would help to illuminate further the challenges that learners experience when they work with problems that involve moving across different registers of representation. AcknowledgementsI acknowledge a grant from the United States Agency for International Development (USAID), administered through the non-governmental organisation Higher Education for Development for research on the different modules in the ACE certification programme. There was no specific grant for this article. I also acknowledges the contribution from Thomas Schroeder (University at Buffalo, State University of New York [SUNY], USA), who assisted with a preliminary report on this project, sketched the normal distribution curves in the article and acted as peer debriefer during the analysis process. 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