Mathematical Literacy (ML) is a relatively new school subject that learners study in the final 3 years of high school and is examined as a matric subject. An investigation of a 2009 provincial examination written by matric pupils was conducted on both the curriculum elements of the test and learner performance. In this study we supplement the prior qualitative investigation with an application of Rasch measurement theory to review and revise the scoring procedures so as to better reflect scoring intentions. In an application of the Rasch model, checks are made on the test as a whole, the items and the learner responses, to ensure coherence of the instrument for the particular reference group, in this case Mathematical Literacy learners in one high school. In this article, we focus on the scoring of polytomous items, that is, items that are scored 0, 1, 2 … m. We found in some instances indiscriminate mark allocations, which contravened assessment and measurement principles. Through the investigation of each item, the associated scoring logic and the output of the Rasch analysis, rescoring was explored. We report here on the analysis of the test prior to rescoring, the analysis and rescoring of individual items and the post rescore analysis. The purpose of the article is to address the question: How may detailed attention to the scoring of the items in a Mathematical Literacy test, through theoretical investigation and the application of the Rasch model, contribute to a more informative and coherent outcome?
The subject Mathematical Literacy (ML), introduced in 2006 in South Africa, is a compulsory subject for those Grade 10–12 learners who do not study Mathematics. The purpose in ML is not that learners learn more and higher mathematics: the emphasis in ML is on the use of mathematics to explore the meaning and implications of quantitative information presented in many reallife situations.
The Department of Education (2003) defines ML as follows:Mathematical literacy provides learners with an awareness and understanding of the role that mathematics plays in the modern world. Mathematical literacy is a subject driven by liferelated applications of mathematics. It enables learners to develop the ability and confidence to think numerically and spatially in order to interpret and critically analyse everyday solutions and to solve problems. (p. 3) ML differs from Mathematics in purpose and in content. In Mathematics, emphasis is placed on engaging with increasingly more complex and abstract mathematical concepts, the relations between them and some applications to problems. However, in ML the emphasis is specifically on the application of basic mathematics to understand situations in real life. There is some lack of clarity evident in the description of Mathematical Literacy noted above, as being mathematically literate requires a sound mathematical base of algebraic concepts and skills. The debate about the subject content of Mathematics and Mathematical Literacy, though regarded as critical, is not the focus of this article. The juxtaposition of mathematics content with real life contexts has meant that many people are unclear about how competence or proficiency in ML may be demonstrated. It is clear that ML as a subject in its infancy requires much research in respect of teaching, learning and assessment, in order to generate debate and establish some consensus on the many contrasting perspectives within the ML field. In this study the focus is the Grade 12 ML preparatory examination, which was set by a provincial Department of Education and is intended to prepare students for the final examination. We pay attention to one aspect, that is, assessment in ML, by identifying some issues arising from the analysis of the empirical data obtained from learners’ responses to this provincial preparatory assessment. The construct under scrutiny is the notion of proficiency in the subject ML. We apply Rasch measurement theory (RMT) to investigate the validity and accuracy of the test in providing a measurementlike representation of ML proficiency in terms of person proficiency and item difficulty. A valid and reliable test would provide teachers with some indication of the levels of mastery of curricular elements and of developing proficiency in ML. It should also provide the Department of Education with an overview of the entire learner cohort taking ML. An application of the Rasch model will help us to identify anomalies and inconsistencies amongst these assessment items and the accompanying scoring rubrics and working memoranda. In this article we consider the implications of considering the purpose of the test and the construction of rubrics so that they work coherently in the interests of valid measurementlike properties and consequently provide reliable information for teachers. Some concepts underlying the Rasch measurement theory are introduced to clarify the analytic process. The aim of the article is to investigate the domain of ML and offer some observations concerning the assessment of ML.
This study, focused on the scoring of items, is part of a larger study on ML (Debba, 2012). The instrument investigated here comprised 51 items, two of which were dichotomous items, marked either correct or incorrect. Twenty items had a maximum of two marks, 14 items a maximum of three marks, 9 items had four marks and 6 items had five marks. The maximum possible score was 150. The participants in the study were 73 Grade 12 ML learners.^{1}The Grade 12 KZN provincial preparatory ML examination paper is intended to assist Grade 12 learners in their preparation for the final examination. It is set by a team of examiners selected by the education department and written under examination conditions. For the purposes of this study the ML 2009 preparatory test was remarked by the third author to ensure that the final version of the marking was entirely consistent with the marking memorandum supplied by the KZN Department of Education. A Rasch analysis supported the investigation of the test as a whole, the items and the ML learner responses. This analysis was conducted to identify factors that may have affected the coherence of the instrument for this sample of learners. The first requirement for this analysis was to capture the score obtained in each of the 51 items for each of the 73 students.^{2} The Rasch model offers various statistics to help diagnose where the data differs from what is expected by the model. Multiple means are applied to an analysis of this nature, to enable the subject expert to make an informed judgement. The output provides statistics on the test as a whole as well as the individual item statistics, in particular the fit residual statistic and the chi square probability statistic, which provide information on the fit of the items. In addition to these statistics we investigated the item characteristic curves (ICCs) to identify which items were misfitting in the ways to be discussed. The research question directing the study is: How may detailed attention to the scoring of the items in a Mathematical Literacy test through theoretical investigation and the application of the Rasch model contribute to a more consistent outcome?
Assessment and measurement


We note that Mathematical Literacy and its assessment have been introduced relatively recently into the South African high school curriculum. We agree with Matters (2009) that assessment in the 21st century has a powerful influence, but this influence is only warranted if the assessment is of a sufficient quality to support the inferences, in this case the inferences about the mathematical literacy proficiency of learners that are drawn from the test results. The assessment process involves the theoretical exploration of the construct of mathematical literacy, the operationalisation of the construct in items designed to gauge proficiency, the compilation of a test instrument and the administration and marking.From the classical theory of measurement, and measurement in the physical sciences (Wright, 1997), we note that the property of invariance of comparisons across the scale of measurement is a requirement. The application of the Rasch model enables the calibration of item measures and the estimation of person locations on a common continuum that together fit the criteria of invariance for a particular frame of reference (Rasch, 1960/1980; Humphry, 2005; Humphry & Andrich, 2008). The comparative difficulty of any two items should be constant regardless of the abilities of the persons responding to the items. Where the data do not conform to the measurement principles, the model will highlight the anomalies for further investigation. In the current study, the application of a Rasch analysis highlighted anomalies and inconsistencies that constituted threats to the construction of measures in the sense understood in classical measurement theory. In particular, the allocation of marks was inconsistent with the grading of proficiency along a continuum. In this article, we investigate the outcomes of both the initial scoring memorandum and the revisions, utilising both Rasch analysis and the educational considerations.
The fundamentals of Rasch measurement theory (RMT) are covered in many publications (Andrich, 1988; Rasch, 1960/1980; Wright & Stone, 1979, 1999). Here we note that with RMT there is an assumption that for the construct of interest there exists a latent trait in the learner that may be gauged through the operationalisation of the construct in various items. Both learner ability, denoted by β_{n}, and item difficulty, denoted by δ_{i}, may be represented on the same scale. This explanation is presented as follows by Dunne, Long, Craig and Venter (2012): Each outcome of an interaction between a person and an item is uncertain but has a probability governed only by these two characteristics, that is by person ability (β_{n}) and item difficulty (δ_{i}). The Rasch model avers that the arrays of numbers β_{n} and δ_{i }are on the same linear scale, so that all differences between arbitrary pairs of these numbers such as (β_{n} − δ_{i}) and hence also (β_{n} − β_{m}) and (δ_{i} – δ_{j}) are meaningful. Through these differences we may not only assign probabilities to item outcomes but also measure the contrasts between ability levels of items, and offer stochastic interpretations of these contrasts. (p. 7)
Alignment of item difficulty and person proficiency on same scale
We have noted that a key feature of the Rasch model is that the difficulty of items is located on the same scale as the ability of the persons attempting those items,
precisely because the construct of interest underpins both the design of the items and the proficiency of learners. The focus of the model is on the interaction between
a person and an item and is premised on the probability that a person v with an ability β_{v} will answer correctly, or partially correctly, an
item i of difficulty δ_{i}. The equation that relates the ability of learners and the difficulty of items is given by the logistic function:
This function expresses the probability of a person v with ability β_{v} responding successfully on a dichotomous item i with two ordered categories, designated as 0 and 1. Here P is the probability, X_{vi}_{ }is the item score variable allocated to a response of person v on dichotomous item i, x is the response, either 0 or 1, β_{v}_{ }is the ability of person v and δ_{i }is the difficulty of item i (Dunne et al., 2012). Applying Equation 1, we can see that if a person v is placed at the same location on the scale as an item i, then β_{v} = δ_{i}, that is, β_{v} − δ_{i} = 0, and the probability in Equation 1 is thus equal to 0.5 or 50%. Thus, any person will have a 50% chance of achieving a correct response to an item whose difficulty level is at the same location as the person’s ability level. Similarly, if an item difficulty is above a person’s ability location, then the person has a less than 50% chance of obtaining a correct response on that item, whilst for an item whose difficulty level is below that of the person’s ability the person would have a greater than 50% chance of producing the correct response. In Figure 1, the person location is represented on the horizontal axis, with the probability of a correct response located on the vertical axis.

FIGURE 1: Category probability curves (Item 2.2.3).


Dichotomous and polytomous item responses
The Rasch model was initially developed for the analysis of dichotomously scored test items.^{3} However, in many cases, tests require items that are scored at graded levels of performance. Rasch (1960/1980) extended the model for dichotomously scored items to include a model for test items with more than two response categories, with possible scores of 0, 1, 2, … m^{4}. These items are termed polytomous items. Figure 1 models the conditional probability of a score of 0, 1 or 2 for a polytomous item (Item 2.2.3a) with three categories. As the person ability increases, the conditional probability of a score of 0 decreases. By contrast, as ability increases the probability of obtaining a maximum score of 2 increases. Also on this graph is the curve that shows the probability of a score of 1. In summary, this curve shows that when a person has very low ability relative to the item’s location, then the probability of a response score of 0 is most likely; when a person is of moderate ability relative to the item’s location, then the most likely score is 1 and when a person has an ability much greater than the item’s location, then the most likely response score is 2 (see also Van Wyke & Andrich, 2006, p. 14).
In Figure 1, the thresholds,^{5} and the categories they define, are naturally ordered in the sense that the threshold defining the two higher categories of achievement is of greater difficulty than the threshold defining the two lower categories of achievement. The first threshold (τ_{1}), which represents the point where a score of 1 becomes more likely than a score of 0, is about –1.10 logits. The second threshold, where a score of 2 becomes more likely than a score of 1, is approximately 1.25 logits. These thresholds show that progressively more ability is required to score a 0, 1 or 2 respectively on this item (Van Wyke & Andrich, 2006, pp. 13–14).
Requirements of the model


We have noted that the central proposition for the Rasch model^{6} is that the response of a learner to a dichotomous item is a function of both the item difficulty and the person ability and nothing else. The probability of a person achieving success on a particular item is entirely determined by the difference between the difficulty of the item and the learner’s ability.<
The principle underlying the Rasch model is: [A] person having a greater ability than another person should have the greater probability of solving any item of the type in question, and similarly, one item being more difficult than another means that for any person the probability of solving the second item is the greater one. (Rasch, 1960/1980, p. 117) In RMT it is expected that the data will accord well with the model. The notion of ‘fit’, that is, accord with the model, is defined as ‘the correspondence between a data set and a statistical model’ (Douglas, 1982, p. 32). The model provides indicators that alert researchers to where this principle of invariance of comparisons is not being met, which may result in item misfit. The fit residual is a measure of the difference between the observed response of each person to each item and that predicted by the model. The analysis process, whether showing a degree of conformity with the model or not, inevitably leads to greater understanding of the construct in question.
Item misfit
As noted, it may be observed that the items are working well and are a good indicator of the learners’ proficiency. It may also be the case however that some items are highlighted as problematic. In subsequent sections we refer to particular examples where we focus on item functioning and the scoring rubrics. In some of the examples, the Rasch model analysis confirms that the scoring rubric is working as required by the model. In other items, the analysis discloses that the scoring rubric is not working in an ordinal way.In a Rasch analysis test of fit, the learners are placed into class intervals of approximately equal size. We have used four groups. The mean ability of the four groups becomes the horizontal coordinate of points in the diagrams, depicting the probability of answering correctly. Where the data conform to the model, the theoretical curve (the expected frequencies) and the observed proportions (the empirically established average of the actual item scores in the four chosen groups) are in alignment.
Figure 2 shows the theoretical curve as expected by the model and the observed proportions, represented by black dots.

FIGURE 2: Item characteristic curve for Item 2.1.3, indicating fairly good fit.


Where the theoretical curve and the observed proportions are in alignment we assume fit to the model, but where the theoretical curve and the observed proportions deviate substantially we are alerted to some kind of misfit between the data and the model. There are four broad categorisations that describe how the observed proportions might relate to the theoretical expectation. In this section we describe a selection of items that fall into the categories of fairly good fit, underdiscrimination, overdiscrimination and haphazard misfit. Firstly, the observed proportions may align with the theoretical curve, in which case there is a good fit to the theoretical requirement. Figure 2 shows the item characteristic curve (ICC) for Item 2.1.3, in which the observed proportions are aligned fairly well with the theoretical curve. Note that the fit residual, 0.601, is relatively small tending towards zero and within an acceptable range of good fit (–2.5 to +2.5). This relatively small residual means that the difference between the observed response of each person to each item and the expected response is small. A second phenomenon may be that the observed proportions are flatter than the theoretical curve, in which case the item does not discriminate enough. The pattern is labelled underdiscrimination or underfit and is illustrated in
Figure 3. This unexpected pattern indicates that learners of lower proficiency appear to perform better than expected on this item and consequently, because of the
interactive nature of item difficulty and learner ability, the high proficiency learners are falsely estimated to respond to the item as if the item was easier than it
really was. The qualitative analysis suggests that a possible explanation lies with the marking rubric, which gives scores between 0 and 3. The scoring rubric allocates
an arbitrary method mark and an additional mark for presenting information provided in the instruction. The allocation of marks appears to be more generous for the
poorly performing learners and too constrained for the higher performing learners.

FIGURE 3: Item characteristic curve for Item 3.2.2, indicating underdiscrimination.


Figure 3 presents the ICC for Item 3.2.2, which shows the observed proportions to be flatter than the expected theoretical curve. The fit residual indicating difference between observed response and that expected by the model is relatively high at 2.410.A third general category occurs when the observed proportions are steeper than the theoretical curves, in which case the discrimination is greater than expected, as shown
in Figure 4. Overdiscrimination in an item may unduly advantage high proficiency learners, whilst disadvantaging learners of lower proficiency. Whilst traditional
test theory asserts the greater the item discrimination the better, the case in RMT is that highly discriminating items provoke a concern that there is a marked dependence
amongst responses in one form or another. An example of such a misfit is that of Item 1.3.2, shown in Figure 4. Again we note that the fit residual is relatively high
at –2.613. Both a high negative fit residual and a high positive fit residual signal poor fit to the model.

FIGURE 4: Item characteristic curve for Item 1.3.2, indicating overdiscrimination.


The fourth general category occurs when the observed proportions are haphazardly but substantially different from the theoretical requirement, as in Figure 5. This pattern demands specific investigation of the construct, an examination of the suitability of the item or the identification of another educational explanation. After analysis, Item 4.1.1 was deleted from the test on the grounds of its misfit. This excision is discussed in the section Refinement of the instrument. Note that here the fit residual is 3.321, indicating a fairly large deviation from the model that should be investigated.

FIGURE 5: Item characteristic curve for Item 4.1.1, indicating haphazard misfit.


Results from initial analysis


From the initial Rasch analysis, the summary statistics (Table 1), personitem location distribution (Figure 6) and personitem threshold distribution (Figure 7)
were generated. Table 1 presents the initial summary statistics, which report the item mean as 0 (as set by the model) and the person mean as –0.2537. The
standard deviation for the item location is 1.1378, whilst the standard deviation of the person location is just 0.3988. This contrast suggests that the spread of
the items is high whilst the person locations are clustered together. Cronbach’s alpha and the person separation index both indicate internal consistency
reliability.
TABLE 1: Summary statistics prior to rescoring.


FIGURE 6: The personitem location distribution prior to rescoring.



FIGURE 7: The personitem location distribution prior to rescoring.


Figure 6 illustrates the personitem location distribution (PILD). The item location mean is set at zero; the person location mean is calibrated at –0.254. The item locations range from –2.2035 to 4.565 logits. The person locations are estimated between –1.414 and 0.441. The fact that the person location mean is lower than the item location mean suggests that the test was difficult for this particular learner cohort. Reasons for the mismatch^{7} may be posited, for example the test questions might have been more easily answered if the cohort had been afforded more experience in basic algebra. Discussion of explanatory conditions and factors may be found in Debba (2012).
The personitem threshold distribution (PITD) (Figure 7) shows that the spread of the item threshold location ranges from –22 logits up to 22 logits, whilst the person location is from –1.4 to 0.4. The PITD representation indicates the distribution of the various categories in the items, for example an item that was weighted at 3 marks will have three thresholds. This wide distribution suggests that some of the 51 items may have been awarded too high a score. This is supported by the fact that for many items several of their possible score values between 0, 1, 2 … m, were not observed in this class of 73 learners or observed only once. There are at least 40 problematic thresholds. See the frequency chart (Appendix 3).Clearly this odd arrangement of thresholds and persons is out of alignment with what is expected of a balanced assessment. The items as originally conceptualised are not distinguishing the intended range of proficiency levels in this particular set of learners. The detailed discussion of thresholds, and the disordering of thresholds is not presented here. See Andrich (2012) for a detailed discussion. Both the summary statistics (see Table 1) and the PITD (see Figure 7) indicate some disjuncture between the items and the persons suggesting that there are some anomalies in the data. Further investigation is required for both items and persons in terms of fit to the model^{8}. In this article, and in the next section, we focus only on the possible anomalies that arise due to the scoring of items.
Individual item analysis
In this section we present a short discussion about problems identified in particular items. We describe how the problems were highlighted and how the qualitative verification of measurement problems prompted rescoring. Three items are discussed, firstly Item 3.2.3, an example of an item that showed haphazard misfit (see Figure 5), Item 4.1.4, which shows how the allocation of two marks is not warranted, and Item 1.3.2, which shows disordered thresholds.
Item 3.2.3
Item 3.2.3a forms part of a question with a farming context (see Appendix 2). The task is to determine whether doubling the dimensions of the bucket will double the volume
of the bucket. The question requires a yes or no answer, and in requiring this response may not gauge the understanding of dimensions or of volume:
The ICC for Item 3.2.3a (Figure 8) shows a haphazard misfit, with learners of lowest proficiency on the test as a whole having an almost equal chance of obtaining a
correct answer as learners of high proficiency on the test as a whole. Information provided here and the qualitative investigation suggests a revision of wording of this
question. It is possible that learners at the lower end of the proficiency scale could have randomly chosen yes or no.

FIGURE 8: Item characteristic curve (Item 3.2.3a).


Item 4.1.4c
Item 4.1.4c requires that the learner give two causes for the observed relative change in a child’s weight. The scoring rubric allocated two marks per reason. It was found that no learner obtained 1 mark without obtaining 2 marks and similarly no learner obtained 3 marks without obtaining 4 marks. The flat category curves in Figure 9 show that the categories 1 and 3 are not functioning at all and category 2 is not functioning well. This outcome reflects the fact that the learners either got 0 marks (for providing no reason) or 4 marks (for providing two reasons). Only rarely did a learner offer only one reason.

FIGURE 9: Category probability curves for Item 4.1.4.c.


Item 4.1.4
Item 1.3.2
In Item 1.3.2 the learner is required to calculate the deduction to his wages. This item was part of a broader problem context (Item 1.3), which required calculating
John’s wages, where the hourly pay was provided, the number of hours worked per day and the percentage deducted.
The item itself is problematic as it depends on the learner answering the previous item (1.3.1) correctly. In addition, the scoring of part marks is odd, as once the learner identifies the 2,2% total deduction, they are likely with the help of a calculator to obtain a correct answer. The category probability curves for Item 1.3.2 (Figure 10) are used to illustrate disordered thresholds. Figure 10 shows that the location of the first
threshold (τ_{1}) (the intersection of the curves for score 0 and score 1) has a difficulty of approximately 0.83 logits. However, the location of
the second threshold (τ_{2}) (the intersection of the curves for a score 1 and score 2) has a difficulty of approximately –0.15 logits. The
location of the third threshold (τ_{3})_{ }(at the intersection of the curves of scores 2 and score 3) is approximately –0.68 logits.

FIGURE 10: Initial category probably curves for Item 1.3.2.


The problem is that the location of the first threshold is greater than the location of the second threshold, whilst the location of the second threshold is also greater than the location of the third threshold. These reversed thresholds are due to the failure of the two middle categories, corresponding to scores of 1 and 2, to function as intended. At no point on the horizontal axis is a score of 1 most likely; neither is there an interval or point where a score of 2 is most likely. Although persons with low ability relative to the item’s difficulty are still most likely to respond incorrectly and score 0 and persons with high ability relative to the same item’s difficulty are still most likely to respond correctly and score 3, persons with ability in the range –0.68 logits to 0.83 logits, where a score of 1 or 2 should be most likely, are more likely to score either 0 or 3. This disordering is evident where the high middle group is more likely to score 1 and where the low middle group is more likely to score 2.The disordering of the thresholds confounds the idea that thresholds between higher level categories are more difficult to attain than thresholds between lower order categories. These initial analyses help us to identify possible anomalies and inconsistencies in the scoring rubrics, which can alert us to possibilities that should be considered when devising scoring rubrics. There are strategies we can use post hoc to adjust the item scoring in order to exhibit ordered thresholds as shown in Item 2.2.3a (Figure 1). The verification that scoring rubrics are functioning as expected, and subsequent revision where necessary, contribute to the reliability of the ML examination paper in the context of this set of 73 learners.
Refinement of the instrument


In addition to looking at misfit statistics, we studied the associated category probability curves for each item, to further explore anomalies in the data. When we investigated the category probability curves for each item, we found that in most cases the thresholds were disordered. In the process of refining the instrument, iterative adjustments took place. The first step was to identify items that showed severe misfit according to the fit residual statistics and the chi square probability. In addition the ICCs were investigated. Where there were indications of anomalies we checked the item itself and the scoring rubric to identify any problems from both a mathematics education perspective and an assessment perspective. Where the qualitative investigation confirmed the anomaly and it was deemed proper to adjust the scoring of the item, this step was taken. The item statistics were then reinvestigated by reanalysing and by rechecking the fit to the model with the revised scores. If the fit had not improved we would reverse the change. If there was no theoretical reason to support the rescoring of an item, then no rescoring took place. The details of the various items, together with the marking memorandum provided by the Department of Education, and details of the rescoring appear in Appendix 2.
First round of rescoring
The scoring for the two dichotomous items, Item 2.2.1b and Item 3.2.3a, was retained. All other items were rescored according to the process identified above.
One of the items, Item 4.1.1 (Figure 7), showed extreme misfit when investigating the ICC. In addition the fit residual statistic (3.321) was outside the generally acceptable interval of between –2.5 and 2.5 and the chisquare probability (0.002) showed that the expected and the observed outcome were statistically significantly different. This item warranted further investigation to see whether the problem lay with the scoring and whether rescoring may resolve the misfit. After studying the evidence and finding that there were problems with the item which resulted in the item not contributing to the test, we deleted the item from the test. In Item 4.1.1 learners were given four graphs from a growth chart used by parents and health workers to monitor the weight gain of infants. They were asked to find the normal weight of a baby boy at birth. Two curves represented the weight of boys whilst two represented the weight of girls. The poor quality of the graphs and unclear titles contributed to the difficulties with this item. In addition, it was not clear how these graphs could be used to provide information about a ‘normal weight’. The assessment task did not make sense in the reallife context. This confusion of meanings illustrates a fundamental tension that exists between the intentions of assessment task designers and learner participants in the reallife context. The questions posed in the examination paper may not be the ones that are posed in the context of health workers who use growth charts to identify children whose health is at risk.
Results of first round of rescoring where the rescoring worked well
After refining the scoring, we found that in several cases the rescoring improved the fit, whilst in few other cases it did not. The rescoring of Item 1.1.3 (see Box 1)
resulted in the improved fit, the adjustment more closely approaching measurement principles. We provide the educational rationale for a change in scoring and also show
the category curves both before rescoring and after rescoring to show how the rescoring helped improve the functioning of the categories.
BOX 1: Details for Item 1.1.3.

Figure 11 shows that before the rescoring most of the categories were not working as they should, that is the allocation of scores 2 and 3 appeared somewhat
redundant. Furthermore, the ICC for Item 1.1.3 in Figure 12 shows unduly high discrimination, labelled overfit. This problematic outcome may be explained as
follows: learners obtaining the item’s answer correctly are unduly advantaged by scoring 4 points, whereas those answering incorrectly are unduly
disadvantaged by ‘losing’ 4 possible points.

FIGURE 11: Initial category probability curve for Item 1.1.3.



FIGURE 12: Initial item characteristic curve for item 1.1.3.


The item, was rescored, moving from a fivecategory item (scoring 0, 1, 2, 3, 4) to a fourcategory item (scoring 0, 1, 2, 3). Category 1 remained the same, Category 2 and Category 3 were recoded as 2, whilst category 4 was recoded as 3. A qualitative investigation in this case indicates that scoring 0, 1, 2, 3 can be justified. The details of the item and rescoring appear in Box 1. In each case where the categories were not working according to the model, we applied multiple criteria before revising the scoring, keeping in mind principles of best test design (see Van Wyke & Andrich, 2006; Wright & Stone, 1979). After this rescoring process, both the category curves (Figure 5) and the ICC (Figure 6) indicated better fit according to the model. Comparing Figure 11 and Figure 13, one can see that the categories are now working more appropriately. Figure 14 shows the ICC for Item 1.1.3 after the rescore. When comparing Figure 12 and Figure 14, one can see that the fit between the observed and expected means for the persons in the four class intervals is much improved. A final check of the fit residual statistic and corresponding chi square values indicates that initially the fit residual statistic was –0.794 (chi square probability = 0.4317), whilst the final fit residual statistic is 0.082 (chi square probability = 0.8281), showing that the change affected the fit statistics positively. Also note that with RMT, a pvalue higher than 0.01 for the fit statistics indicates that the difference between the observed and expected is not statistically discernible and hence that the model is working well.

FIGURE 13: Final category probability curve for Item 1.1.3



FIGURE 14: Final item characteristic curve for Item 1.1.3.


Results of first round of rescoring where the rescoring did not work well
As explained earlier, Item 4.1.1 was rejected as the fit was extreme and in addition it was judged to be a very poor indicator of mathematical proficiency. In addition, Item 3.2.2, after rescoring, was now also identified as having extreme fit statistics outside of the acceptable interval. In a second process we investigated each ICC again, and then the category probability curves, and found that the thresholds were disordered for Items 1.2.1b, 1.2.2, 1.3.1, 1.3.2, 3.2.1, 3.2.3b, 4.1.4b, 4.1.4c, 5.1.4 and 5.3. (See Appendix 2 for details of these items.) At this stage, we reexamined the questions to see whether we could justify rescoring these items again. We also examined the fit residual statistics to help us decide whether a rescoring was necessary or not. A summary of these processes and decisions appears in Appendix 1.
After resolving anomalies
After the refinement of the scoring using information from both the qualitative investigation and the Rasch analysis, we now examine the final summary statistics in Table 2.
The figures in Table 2 when compared to those of Table 1 show improvement across most of the statistics. For example, Cronbach’s alpha, a reliability index, has increased from 0.8845 to 0.8887. The standard deviation for the item location has increased to 1.615, showing that the items now have a larger relative spread. We note as well that the differences observed in the new personitem location distribution (PILD) (Figure 15) are an improvement on the initial PILD (Figure 2).

FIGURE 15: Personitem location distribution after refinement to instrument.


The distribution represented in Figure 15 indicates a better balance than the original distribution represented in Figure 2. We can see that the standard deviation for person locations has increased in the personitem location distribution from 0.399 to 0.808 (see Figure 6 and Figure 7). The person location spread was initially between –1.414 and 0.441. After the rescoring the spread of learner locations ranged between –2.460 and 1.176, thus providing greater discrimination of learner proficiencies. We judge the postrescoring estimates of learner proficiencies to be more accurately reflected by the refined scoring.Furthermore, one can see that the most difficult item, Item 5.1.3, was far too difficult for this group of learners. Its estimated difficulty level, calibrated at 5.448 logits, is much higher than the ability level of the most proficient learner in this test, who was estimated just above 1 logit. The items 4.1.2, 4.1.3, 4.2.1, 4.2.2, 5.1.4, 5.2, 2.1.2 and 3.2.3a were also beyond the ability level of the learners, but the difference δ_{i} – β_{v} for this group of items was not as large as it was in the case of Item 5.1.3. Item 5.1.3 required learners to work out the price of an item before VAT, given the final price including VAT. The usual solution involves setting up an equation of the form A + 14% of A = Final price, and they were required to find A. This item required the setup and solution of an equation and involves applying algebraic techniques. The ML curriculum document makes it clear that skill in such algebraic manipulation is not a focus of ML. The document states: ‘As a rule of thumb, if the required calculations cannot be performed using a basic fourfunction calculator, then the calculation is in all likelihood not appropriate for Mathematical Literacy’(Department of Basic Education, 2011, p. 8). Given such a stipulation, and noting that the item was far beyond the ability of this group of learners, it is recommended that a greater use of basic algebraic manipulation skills should be encouraged, to enable them to solve items such as Item 5.1.3. We note that this item could also be solved by informal methods based on reasoning about percentage change or using proportion and perhaps opportunities for such reasoning should also be encouraged. On the other hand, three items, 4.2.3, 3.1.1 and 2.2.1a, were lower than the ability level of all the learners in the class, but the location of the items was less than 2 logits lower than the person with the lowest total score. This difference contrasts with the finding that in the higher ability ranking, one item was almost 5 logits higher than the location of the person with the highest total score, with two items being 2 logits or more higher and six other items being higher than the location of the person with the highest total score. We now present the personitem threshold distribution that was generated after the rescoring process. There is a marked difference between the initial PITD (Figure 7) and the final PITD (Figure 16). The distribution of the thresholds prior to rescoring ranged between –22 and +22 logits (see Figure 7). Now after rescoring the threshold distribution has been limited to between –4 and 8 logits. The number of thresholds has been reduced from 150 to 75 by the rescoring processes (see Appendix 3).

FIGURE 16: Person item threshold distribution postrescoring.


Implications for assessment in Mathematical Literacy


In this study we examined the responses of 73 students to each of 51 items by using tools provided by the RUMM2030 software (Andrich, Sheridan & Luo, 2011). The analysis revealed some important issues, which are discussed below.
Allocation of marks for accuracy and method
We have discussed the case of Item 1.3.2, where marks were allocated for method and for consistent accuracy. This item was just one instance of many where the method and accuracy system did not work well. In fact, it unduly disadvantaged the person who did not address the question correctly. Most learners who identified the method were able to obtain the correct answer because it involved just entering the numbers into the calculator and reading off the answer. The learners who answered correctly were thus unduly advantaged by getting additional marks. There were very few learners (too few as revealed by their respective category probability curves in Figure 10) who achieved the method mark without achieving the accuracy mark. Another related issue was in the case of Item 4.1.4c, which asked for two causes of an observed change in the data. The scoring rubric allocated 2 marks per reason. The category curves in Figure 9 revealed that the categories 1 and 3 were redundant. No learner attained the first mark without getting the second mark and similarly no learner attained 3 marks without getting the fourth mark.
Guessing
Questions that allocated a mark for answering yes or no to a question and then asked for a reason resulted in a haphazard response that could indicate guessing at the lower proficiency level. This pattern was revealed in some items by the ICC where there was a haphazard misfit, such as in Item 3.2.3 (shown in Figure 10) and Item 1.3.3 (not shown). We make a distinction here between the learners who know the answer to the question and the group who guess. It is the latter group who have a 50% chance of answering correctly if they guess.
Item response dependency
In some cases, the answer to a previous item influenced the probability of success of the learner in a following item, as in the case of Item 2.1.2a and Item 2.1.2b as well as Item 1.2.1a and Item 1.2.1b. Those learners who were able to get the first item correct were likely to get the second one correct. None of the learners who answered the first part of Item 2.1.2a incorrectly achieved full marks in the second part of the item, Item 2.1.2b, and none of the learners who answered Item 1.2.1a incorrectly was able to obtain full marks in Item 1.2.1b. The dependency of a subsequent item on an earlier item is regarded as unfavourable test practice. It is the independence of items that offers greater precision.
In this article we illustrated how the Rasch model could be used in conjunction with professional judgment to check the validity of the assessment, by using the responses of 73 Grade 12 ML learners to their preliminary examination. The process described in this article involved identifying items that did not fit the model. We described the items and the original scoring rubric. The Rasch output was provided and the anomalies and inconsistencies were discussed. Before initiating any changes in the scoring we sought educational reasons that warranted rescoring. Thus, the Rasch analysis and rescoring processes were guided at all times by the qualitative analysis that was conducted by an experienced ML teacher.An important advantage of the Rasch analysis is that item difficulty and person proficiency are located on the same scale. Without checking individual item validity, we cannot take the results at face value; instead, we need to verify the item fit. Now that we have subjected the scoring of each item to Rasch analysis and qualitatively investigated the structure of the scoring rubrics, we argue that the new scoring procedures for the instrument allow greater precision than the original instrument scoring. It must be emphasised that the scoring rubrics were not the only threats to validity in this test. The process that was conducted, which involved systematically rescoring each of the assessment items, has revealed the important role of scoring rubrics in contributing to the validity of assessments. Examiners need to ensure that each mark that is allocated can be justified educationally. Marks should not be allocated on the basis of time that is required to be spent during the examination. Neither should marks be allocated for guessing. If these checks are not taken into account, then the total score has diminished meaning. If the total score is indicative of a position on a unidimensional scale, then differences between the score must reflect differences between proficiency levels of learners. A further process that could be followed is to study which items were experienced as more difficult and which items were experienced as easier in order to obtain a description of items on different levels of the scale. It may then be possible to identify different demands of the various questions. A study of a similar nature has been conducted by Long (2011) who presented a comprehensive application of RMT to the multiplicative conceptual field. In her study, Long used the Rasch model to develop clusters of proficiency zones along the continuum representing the alignment of person proficiency and item difficulty. It may be the case that particular items are mathematically demanding because they require sophisticated use of algebraic tools (as in Item 5.1.3). Perhaps the increasing demand of the questions may be related to the contexts that were used. These analyses will be conducted and reported in a followup article.
We would like to acknowledge the valuable insights on the application of the Rasch model to this study provided by Professor Tim Dunne.
Competing interests
We declare that we have no financial or personal relationship(s) that might have inappropriately influenced us in writing this article.
Authors’ contributions
The data was collected by R.D. (University of KwaZuluNatal) as part of his postgraduate studies. C.L. (University of Pretoria) led the process of the Rasch analysis, helped by S.B. (University of KwaZuluNatal) and R.D. The writeup was done by all three authors, led by C.L.
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TABLE 1–A: Decisions after results of first round of rescoring.

Questions, marking guide and rescoring details
Category frequencies for each of the items
1. The small sample size may in some senses present as a limitation, but should not detract from the study’s usefulness in alerting ML educators to the issues identified here. Any teacher of a Grade 12 ML class is likely to be concerned with fewer than 73 learners’ performance on any such test. Larger counts of learners may occur in schools with several ML classes at Grade 12 level. The general rule of thumb for the construction and development of test instruments is that the learner count is about ten times the maximum score count. In the case of this study, the information obtained from the small group is crossreferenced with substantive analysis and therefore generalisable in the sense that the same principles will apply. 2. Missing responses were allocated zeros. 3. See Dunne et al. (2012) for details of the analysis of dichotomous items. 4. In this study we use the Rasch partial credit model, which is the default model in the RUMM 2030 software.
5. The term threshold defines the transition point between two adjacent categories, for example scoring 0 and 1, or scoring 1 and 2.6. The discussion here will concern the dichotomous model. Extensions of the model have been derived from this model for partial credit scoring by Masters (1982) and rating scales by Andrich (1978). 7. This mismatch is in itself not a problem; however, more information could be gleaned from a test situation that is better targeted.
8. In addition, the investigation of factors such as response dependency and differential item functioning is demanded in the interests of valid measurement.
