2013. Vol.4, No.12, 767-773
Published Online December 2013 in SciRes (http://www.scirp.org/journal/ce) http://dx.doi.org/10.4236/ce.2013.412109
Open Access 767
The Rasch Model Analysis for Understanding Mathematics
Proficiency—A Case Study: Senior High School
Carlo Andrea Pensavalle1, Giuliana Solinas2
1Department of Science for Nature and Environmental Resources, University of Sassari, Sassari, Italy
2Department of Biomedical Sciences, Laboratory of Epidemiology and Biostatistics, University of Sassari,
Received November 5th, 2013; revised December 5th, 2013; accepted December 13th, 2013
Copyright © 2013 Carlo Andrea Pensavalle, Giuliana Solinas. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited. In accordance of the Creative Commons Attribution Li-
cense all Copyrights © 2013 are reserved for SCIRP and the owner of the intellectual property Carlo Andrea
Pensavalle, Giuliana Solinas. All Copyright © 2013 are guarded by law and by SCIRP as a guardian.
Many students enrolled in Italian Universities don’t have regular careers in their first level of university
education mainly because of the obstacles encountered in their studies. As far as Mathematics proficiency
assessment there are national and international important systematic studies which give evidence for posi-
tive relationships between achievement and varied classroom settings and provide a larger context for
better understanding regional performance, extending and enriching the local picture. This paper presents
some preliminary results aiming to evaluate specific Mathematics abilities in Senior High School Sardin-
ian Students approaching university studies. For this purpose the Rasch model was applied. The informa-
tion obtained by the application of this measurement approach provides clear indication for further analy-
sis to ascertain the causes that influence Mathematics proficiency. The Rasch model was performed on
888 students coming from 28 High Schools located in the central-northern part of the Sardinia region us-
ing a questionnaire to evaluate the level of ability in procedural fluency and a second questionnaire to
evaluate strategic competence. The study provides more evidence in favor of Rasch Model as an appro-
priate way for teachers and researchers to obtain richer interpretations on the relationship between stu-
dents’ proficiency and test items. Based on Infit and Outfit MNSQ, all items are within acceptable range
between 0.7 - 1.3. In light of preliminary results there is a need for local schools and universities to be-
come attuned to the full extent of the Mathematics problem as it affects Senior High School Sardinian
Keywords: Rasch Analysis Model; Mathematics Proficiency; Procedural Fluency; Strategic Competence
This work is part of the wider project STUD.I.O.: “Students
in Orientation”, funded by the Sardinia Region with participa-
tion of the European Social Fund and the Italian Ministry of
Labor and Social Policies. One of the main goals of the project
is to promote a realignment of general skills (Reading/Writing,
Mathematics and Science), aimed at improving the preparation
of students attending the last year of High School (19 years old).
There is evidence of a serious decline in students’ mastery of
basic mathematical skills and level of preparation for mathe-
matics-based degree courses since the late nineties (Hawkes &
Savage, 2000). This decline is well established and affects stu-
dents at all levels. Students’ difficulties in accessing universi-
ties are increasing and it is common to doubt if High Schools
are adequately preparing students for university entrance. The
admission test scores are becoming alarmingly low, anticipating
a trend where students will not be regular with their course
studies. This represents a serious problem in the Italian Univer-
sity system especially after the University reform of 2001
which introduced the “3 + 2” system for most of the curricula.
The system is articulated in two parts: a first three year course
degree corresponding to a first level of university education in a
specific field of interest, followed by a two year specialized
course degree in the same field. This system which was sup-
posed to cope with the gap in students’ preparation entering
university education has generated instead a great level of dis-
tress in students who try to adapt effectively to the standards
required by university studies and prompted a central role for
universities to promote specific support to students in difficulty,
creating instruments and opportunities to enable them to finish
their courses within the allotted time-span. (Chiandotto et al.,
2005; Cingano & Cipollone, 2007; Delvecchio & D’Ovidio,
2002; Solinas et al., 2012). Many students enrolled in Italian
universities don’t have regular careers in their first level of
university education, mainly because of the obstacles encoun-
tered in their studies due to lack of preparation in general skills,
or because of a wrong choice. The withdrawals from university
generally occur in the first year of study, representing about
C. A. PENSAVALLE, G. SOLINAS
18% of the total students enrolled (MIUR/CNVSU, 2011). Re-
liable measures of students’ performance are needed in order to
identify risk factors and predictors of success, also to evaluate
the quality and efficiency of curricula. The typical approach in
analyzing students’ career, centered on the grades and credits
acquired is affected by the inherent characteristics of any higher
education system setting (Mouw & Khanna, 1983; Rohde &
Thompson, 2007). Therefore, it is necessary to provide stan-
dardized indicators to make instructional decisions and to
evaluate individual student achievement in order to develop
strategies to better understand causes and find solutions. As far
as Mathematics proficiency assessment, there are national and
international important systematic studies which give evidence
for positive relationships between achievement and varied
classroom settings (Dunne et al., 2012; Mossi et al., 2012) and
provide a larger context for better understanding regional per-
formance, extending and enriching the local picture (Samuels-
son, 2010; Teddlie & Reynolds, 2000). Among the most im-
portant cognitive tests, we would like to mention the interna-
tional one conducted by the Organization for Economic Co-
operation and Development (OECD), started in 1997, known as
Programme for International Student Assessment (PISA), and
the national one conducted by the Italian Institute for the Edu-
cational Evaluation (INVALSI), started in 2007. Both studies
generate theoretical reflections and empirical studies related to
issues involved in the evaluation of educational programs, edu-
cational institutions, educational personnel and student assess-
ment. They involve groups of students enrolled in the second
and fifth year of primary school, in the first and third year of
Junior High School and in the second year of High School. The
statistics are available worldwide and allow different kinds of
evaluations to identify and test promising approaches and cur-
ricula able to improve student outcomes. Recent indications
show that the average level of Mathematic proficiency in a
fifteen year old Italian student is lower (score 483) than the
OECD (score 496) with a statistically significant difference
(PISA, 2009). Based on these premises and under the circum-
stances of the project STUD.I.O., we decided to extend the
study to Senior High School Students (19 years old), who vol-
untarily participated to the following two evaluations: the first
one aimed at measuring if the average level in procedural flu-
ency established by PISA regarding fifteen year old students,
was well achieved by the sample under study. In order to do so,
we administered a questionnaire composed of specific items
similar in difficulty to the one administered by PISA surveys.
The second one aimed at measuring the level of strategic com-
petence required to freshman entering any Italian university
scientific degree program by CINECA, a non-profit consortium
established in 1969, composed of 54 Italian universities, two
National Research Centers, and the Ministry of University and
The questionnaire administered composed of specific items
was constructed in a similar way to the one used in the last past
years. In this study, the choice of two questionnaires provide
evidence to measure two distinct aspects of Mathematics profi-
ciency in the same group of nineteen year old students graduat-
ing from High School, who aim to pursue scientific academic
studies. For this purpose the Rasch model was applied. The
information obtained by the application of this measurement
approach provides clear indication for further analysis to ascer-
tain the causes that influence Mathematics proficiency.
The sample consists of 1233 Senior High School Students
coming from 28 High Schools located in the central-northern
part of the Sardinia region (Figure 1), representing about 10%
of the entire population of regional nineteen year old High
The first test instrument used in this study was a self-devel-
oped 20 item questionnaire, constructed according to guidelines
provided by OECD curriculum specifications, aiming to evalu-
ate the level of ability in procedural fluency, that is performing
mathematical procedures appropriately and efficiently, includ-
ing those that require decisions in sequence and interpreting
and using representations based on different information
sources (Kilpatrick et al., 2001). The second test instrument is a
self-developed 25 item questionnaire, constructed according to
guidelines provided by CINECA, aiming to evaluate the level
of strategic competence, connected to operational skills and
problem solving strategies. In particular, the ability to recognize
and set a problem properly, to select appropriate information, to
identify and organize the most appropriate tools to be used and
to represent data and situations in an appropriate manner
(Kilpatrick et al., 2001). The questionnaires were administered
online through the Moodle platform. The first questionnaire
was administered in February 2013, the second in April during
Methods of Statistical Analy si s
In this paper instead of Classical Test Theory methods, we
used the probabilistic approach of Item Response Theory (IRT)
Map of the Sardinian High Schools par-
ticipating in the study.
C. A. PENSAVALLE, G. SOLINAS
Open Access 769
named Rasch Model (Rasch, 1960). We defined Mathematics
proficiency as a latent trait yielding a score that locates student
ability and question difficulty on a common scale, measured in
logits (Fisher, 1995). In this model the probability Pi of an-
swering correctly to the i-th item is expressed in function of the
difference between the respondent proficiency level θ and the
item difficulty level bi as follows:
This brings to the definition of logit:
Values are placed along a continuum enabling the item dif-
ficulty level to be compared with the respondent proficiency
level. The Rasch model allows the estimation of these two pa-
rameters, with the advantage that despite different test items
with different data collection periods, students’ progress is es-
timated and reported on a common scale regardless of fluctua-
tion in test difficulty. The model was implemented in the Acer
ConQuest software, version 3.1 (Adams et al., 2012). The Mar-
ginal Maximum Likelihood Estimation method was considered
in order to estimate the parameters above, through iterative
calibration of both person and item (Bock, 1970). Information
about the model quality was gained from the fit statistics on
item level. These statistics show whether single test items cor-
respond with the model assumptions. Conquest software gener-
ates Infit Mean Square (MNSQ) and Outfit MNSQ statistics,
which provide indications on the differences between the data
and model’s expectations (Bond & Fox, 2007; Wright & Mas-
ters, 1982). In this study we adopted the range of acceptable fit
between 0.7 and 1.3, for both fit indices (Bond & Fox, 2007),
because Infit values greater than 1.30 and smaller than 0.70 are
labeled problematic (Adams, 2002). Reliability and validity of
measures meaning and interpretation are provided within the
framework of Rash measurement model. In addition, a Wright
Map (Boone & Scantlebury, 2006) is used to visually display
the simultaneous distributions (or “performances”) of items and
respondents related to the instruments used and the sample
considered. Redundant items appear on the same or nearly the
same point on the Wright map scale. The Wright map also il-
lustrates regions of the scale in which items are absent, identi-
fying where new items are needed.
The 72% of the total sample (n = 1233) matched the specifi-
cations required to be part of the statistical analysis: respect of
time administration and answer to all of the items of the ques-
tionnaire. Table 1, displays the main characteristics of the
study sample: 55% of the students come from Lyceum, 28%
Technical and 17% Vocational Schools.
As presented in Table 2, the Rasch Analysis applied to the
first instrument finds both means of Infit MNSQ and Outfit
MNSQ close to the expected value of 1.00.
Inspection with individual items shows Infit MNSQ values
ranging from 0.82 to 1.21 while Outfit MNSQ values range
from 0.93 to 1.15. Evaluation of the Infit statistics reveals that
all cut off points of the items are compliant with the model. The
unidimensionality of the construct was assessed by using prin-
Sample characteristics and school typology.
Gender Total %
Males 414 47
Females 474 53
School Typolo gy Total %
Scientific Studies 288 32
Classical Studies 205 23
Technical 246 28
Vocational 149 17
First instrument: Item statistics.
1 0.144 0.070 0.99 0.99
2 0.205 0.070 0.92 0.93
3 −0.433 0.072 0.99 0.99
4 0.810 0.072 1.06 1.04
5 0.444 0.070 1.01 1.01
6 0.661 0.071 1.21 1.15
7 1.142 0.075 0.99 0.99
8 0.762 0.072 1.10 1.06
9 0.920 0.073 1.06 1.04
10 −0.038 0.070 0.95 0.97
11 −0.219 0.071 0.91 0.93
12 −0.471 0.073 1.10 1.08
13 0.656 0.071 0.97 0.97
14 −0.819 0.076 0.99 1.00
15 0.757 0.072 1.01 0.99
16 −0.018 0.070 1.00 1.00
17 −1.585 0.091 1.01 0.99
18 −2.266 0.114 0.89 0.97
19 −1.724 0.095 0.82 0.94
20 1.072 0.074 0.93 0.94
cipal components analysis of Rasch residuals and item fit sta-
tistics (Brentari & Golia, 2010). Because the scores demon-
strate little variation from model expectation there is evidence
of consistency between students’ responses and items on the
scale and the model expectations.
Reliability of item difficulty measures is 0.99 suggesting that
the ordering of item difficulty is replicable with other compara-
ble sample of students. Consistency of student measures (KR20),
equivalent to Cronbach’s alpha (Cronbach, 1951), is 0.72 indi-
cating that the ordering of the student proficiency can be likely
replicated since most of the variance is attributed to the true
variance of the Mathematics proficiency construct.
Based on the following Figure 2, where the student ability
(to the left) and the item difficulty (to the right) are displayed
graphically with the most able students and most difficult items
C. A. PENSAVALLE, G. SOLINAS
Wright map of the first instrument.
at the top, it is possible to note that the majority of the students
(each x represent 1.3 students) are represented in the center of
the graph (the area included between −1 and +1 logits) as most
of the items (represented by their labels). The items and the
students are quite well approximated by normal distribution.
Therefore, the difficulty of the questionnaire is aligned with the
average level of student ability, with Item 7 the most difficult
and Item 18 the easiest (see Appendix 1). At the bottom and at
the top of the chart there are two small groups of students, for
which there are no items able to measure their ability. Alarm-
ingly, it seems that the average level of Mathematics profi-
ciency shown by our sample of Senior High School Students is
comparable with that of fifteen-year-old students. Let’s now
proceed to the analysis of the data collected with the second
questionnaire which was developed to measure in the sample of
Senior High School Students the presence of the strategic com-
petence level required to enter any Italian university scientific
degree program. As presented in the following Table 3, the
Rasch Analysis finds both means of Infit MNSQ and Outfit
MNSQ close to the expected value of 1.00. Inspection with in-
dividual items shows Infit MNSQ values ranging from 0.92 to
1.11 while Outfit MNSQ values range from 0.94 to 1.02. As
before having assessed the unidimensionality assumption of the
construct and the acceptable variation of the scores from model
expectation, there is evidence of consistency between students’
responses and items on the scale and the model expectations.
Reliability of item difficulty measures is 0.82 suggesting that
the ordering of item difficulty is quite replicable with other
comparable sample of students. Consistency of student meas-
ures is moderate (KR20 = 0.51) and mainly this questionnaire
resulted for the students’ sample composed of items whose
Second instrument: Item statistics.
1 −0.126 0.189 1.03 1.02
2 −0.013 0.192 0.98 0.98
3 0.312 0.204 1.03 1.02
4 −0.856 0.181 0.96 0.96
5 −0.755 0.181 1.01 1.01
6 0.446 0.211 0.97 0.99
7 0.446 0.211 1.11 1.06
8 −0.015 0.192 1.02 1.01
9 −1.236 0.185 1.01 1.00
10 0.355 0.206 1.01 1.01
11 0.311 0.204 1.03 1.02
12 0.400 0.208 1.05 1.03
13 0.446 0.211 0.96 0.96
14 0.144 0.197 0.92 0.94
15 0.401 0.208 0.99 1.00
16 0.065 0.195 1.05 1.03
17 −0.342 0.184 0.96 0.96
18 −0.199 0.187 0.97 0.98
19 0.186 0.199 0.96 0.98
20 −0.412 0.183 0.99 0.99
21 0.145 0.197 1.03 1.02
22 −0.481 0.183 1.00 1.00
23 0.105 0.196 1.00 0.99
24 0.402 0.208 0.96 0.97
25 0.270 0.202 1.04 1.02
difficulty levels were higher than the students’ ability levels, as
shown in the following Figure 3.
Item 6, 7, 13 resulted to be the most difficult and Item 9 the
easiest (see Appendix 2).
This paper presents some preliminary results of the project
STUD.I.O., aiming to evaluate specific Mathematics abilities in
Senior High School Sardinian Students approaching university
studies. The study provides more evidence in favor of Rasch
Model as an appropriate way for teachers and researchers to
obtain richer interpretations on the relationship between stu-
dents’ proficiency and test items. Based on Infit and Outfit
MNSQ, all items are within acceptable range between 0.7 - 1.3.
Both statistics show that there is enough evidence that the
data obtained fits the model expectations. With regard to stu-
dents’ procedural fluency investigated with the first instrument
most of the students show responses that are within the expec-
tation of the model. The results give suggestions that the items’
questionnaire can discriminate students with different Mathe-
matics proficiency levels. Since the measures are in interval
scale, one important observation is that the most difficult item
C. A. PENSAVALLE, G. SOLINAS
Open Access 771
Wright map of the second instrument.
of the first questionnaire, Item 7 (1.142 logits), was higher in
difficulty as compared to Item 18 that was almost twice as easy
(−2.266 logits). For the second questionnaire, it is difficult to
define the students’ strategic competence because of large gaps
of the scale in which items are absent. These results provide
evidence for construct validity of the second questionnaire. In
particular, the item difficulty measures show that this question-
naire is composed of items whose level of difficulty did not
correspond to the level of Mathematics proficiency of the stu-
dents’ sample. Therefore, in light of these preliminary results,
there is a need for local schools and universities to become
attuned to the full extent of the Mathematics problem as it af-
fects Senior High School Sardinian Students. The diagnostic
testing of new undergraduates is recommended as an effective
means of investigating levels of Mathematics proficiency and
identifying actions to improve quality, equity and efficiency of
Sardinian educational systems.
The authors would like to thank the Sardinia Region, the
European Social Fund and the Italian Ministry of Labor and
Social Policies whose funding made this study possible.
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Appendix 1. First Questionnaire
Zedland’s postal charges are based on the weight of the
goods you want to mail, as shown in the table below (approxi-
mated to the nearest gram).
Which one of the following graphs is the best representation
of the postal charges in Zedland?
(The horizontal axis shows the weight in grams and the ver-
tical axis the charge in zeds).
A. Graph A
B. Graph B
C. Graph C
D. Graph D
The chart below shows the changes in speed of a racing car
on the second lap of a circuit 3 kilometers long.
Where is the slower speed during the second round?
A. At the starting point
B. At about 0.8 km
C. At about 1.3 km
D. In the middle of the round
Appendix 2. Second Questionnaire
A student has obtained a grade of 7/10 in a Math test.
You want to express this grade in fifteenths using the criteria
suggested in the figure.
If p is the grade in fifteenths which corresponds to 7/10, then:
A. 11 < p < 11.1
B. 11.1 < p < 11.2
C. 11.2 < p < 11.3
D. 11.3 < p < 11.4
Consider the angles α e β as in the figure which of the fol-
lowing relationships is correct?
A. tan β < cos α
B. sin β < cos α
C. cos β > cos α
D. tan β > tan α
A number sequence 012
,, ,xxx is defined as:
for any i ≥ 2.
Then x6 is equal to
Shown in the following graph is the cost of a phone call in
function of time.
What is the cost in euro of a 20 minute phone call?
C. A. PENSAVALLE, G. SOLINAS
Open Access 773