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Development and Calibration of a Primary School Mathematics Diagnostic Test Based on Item Response Theory

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Development and Calibration of a Primary School Mathematics Diagnostic Test Based on Item Response Theory.


IRT method has the capacity to produce item and person parameters that are, respectively, examinee and test sample independent.

It was, therefore, the purpose of this study to develop and calibrate a Primary School Mathematics Diagnostic Test (PRISMADAT (Forms A and B)) based on IRT for use on primary four pupils. The design of the study was instrumentation.

Seven research questions were posed and 10 hypotheses were formulated to guide the study. A sample of 4,844, selected through multistage cluster sampling plan from a population of 57,954 primary four pupils in Nasarawa State, during 2011/2012 academic session, took part in the study.

The PRISMADAT (Forms A and B with 65 items on each form (totalling 130 items)) which was the main instrument in the study, was constructed based on the use of Primary School Mathematics Skills Deficiency Identification Test Blue Prints (MSDITBP’s).

The MSDITBP’s were prepared based on the contents of the first four years (primaries 1-4) of the 9-year Basic Education Mathematics curriculum. Pilot study report showed that the PRISMADAT was unidimensional and had Cronbach Alpha reliability of 0.94.

Data were collected from six local government areas (LGA’s) in Nasarawa State. Findings of the study showed that PRISMADAT Forms A and B are unidimensional and very reliable. Item difficulty parameter values range from -0.29 to 4.95 and from -0.97 to 3.21 for PRISMADAT Forms A and B respectively.

Their discrimination parameters range from 0.19 to 4.70 and from 0.25 to 5.88 respectively. With regard to the guessing parameters, PRISMADAT Forms A and B, ranged from 0.00 to 0.38 and from 0.00 to 0.29 respectively.

There was no sufficient evidence to show that a significant difference exists between the means of the parameters of items on PRISMADAT Form A and the means of corresponding parameters of items on PRISMADAT Form B.

The conclusion was that a Primary School Mathematics Diagnostic Test (PRISMADAT Forms A and B) has been developed and calibrated based on the item response theory (IRT).


Title Page i
Certification ii
Approval Page iii
Dedication iv
Acknowledgements v
Table of Contents viii
List of Figures xi
List of Tables xii
Abstract xiii


Background of the Study 1
Statement of the Problem 12
Purpose of the Study 14
Significance of the Study 15
Scope of the Study 18
Research Questions 19
Hypotheses 19


Conceptual Framework 23
Diagnostic Tests and Diagnostic Testing 23
Test Construction Procedures Based on CTT and IRT 25
Mathematics in the 9-Year Basic Education Curriculum 33
Theoretical Framework 35
Classical Test Theory (CTT) 35
Item Response Theory 36
IRT Models 37
IRT Assumptions 46
Empirical Studies 54
Summary of Literature Review 85


Design of the Study 88
Area of the Study 88
Population of the Study 89
Sample and Sampling Technique 90
Instrument for Data Collection 91
Validation of Instrument 94
Pilot study Report 95
Training of Research Assistants 97
Method of Data Collection 98
Method of Data Analysis 99


Results 101
Summary of the Findings 122


Discussion of the Findings 124
Conclusion 131
Implications of the Research Findings 131
Recommendations 133
Limitations of the Study 134
Suggestions for Further Research 134
Summary of the Study 135


1.1 Background of the Study

Mathematics is among the core subjects on both the primary and secondary school curricula. It is an essential tool for success in the study of science and technology therefore its teaching and evaluation are emphasized in primary and secondary schools.

Teachers are required to give continuous assessment tests to measure pupils’ progress and achievement in all subjects. The tests which are basically teacher made do not usually have known psychometric properties (validity, reliability, etc) therefore may not be valid for assessment  of learners.

Moreover, there is “the fact that skill required in test construction is not always developed by teachers who may be highly proficient in other phases of the teaching-learning process” (Skinner, 1984, p. 686).

Most teachers lack competence in test construction (Emenogu, 2003). Gage and Alto (1974) assert that most teachers lack the ability to construct diagnostic tests.

Most of the test instruments constructed by teachers are achievement tests. Not all teachers determine the psychometric properties of the test instruments they construct and use in assessing the pupils they teach.


Adedoyin, O.O., Nenty, H.J. and Chilisa, B. (2008). Investigating the Invariance of Item Difficulty Parameter estimates based on CTT and IRT. Educational Research and Review. 3(2), 083-093. Available online at Accessed on Thursday 15/04/2010.

Adisa, A. (2011). Nasarawa State – “Home of Solid Mineral”. Available online at: Accessed on Monday 24/02/2014

Ahmann, J.S. and Glock, M.D. (1981). Evaluating Student Progress. Boston: Allyn and Bacon, Inc.

Alexopoulos, D.S. (2010 ). Classical Test Theory. In Neil j. Salkind and Kristin Rasmussen, Encyclopedia of Measurement and Statistics. Available Online at http://srmo.Sagepub.Com/view/encydopedia-of-measurement-and-statistics/n82.xml Accessed on Monday 03/01/2011

Ali, A. (2006). Conducting Research in Education and the Social Sciences. Enugu: TIAN Ventures.

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