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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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A Rasch Analysis of Math Testsusing M.In.E.R.Va. Platform
Grazia Messineo - Salvatore Vassallo
ICETC2014
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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General frameThe Web Platform
The problem
Students must achieve some automatism in eachdiscipline to succeedMany students have difficulties in Mathematics(see Ocse-Pisa reports)Traditional ways (courses) to help students aredifficult to implement, also because of their costs
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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General frameThe Web Platform
Goals
To help students to recover their deficiencies in adisciplineTo provide teachers a tool to assign common orpersonalized tasks to studentsTo provide teachers a tool to automatically verifystudents’ achievements
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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General frameThe Web Platform
The structure: contents
Theoretical schemes to strengthen or createknowledgeGuided exercises to illustrate the correct way forsolving the problemsConsolidation and self-assessment exercisesPerformance assessment exercises
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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General frameThe Web Platform
The structure: feedback
Assignment of a scoreAutomatic answers correctionHint to get a solutionReferences to theoretical contentSolution (complete or in partial form)
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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General frameThe Web Platform
How it works
The software consists of some LATEX files: Thepackage minerva, the master files, and the files ofthe exercisesEach file of exercises contains one or morevariants of a basic exerciseThe output is a pdf file with interactive contentmanaged by Javascript
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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General frameThe Web Platform
Advantages
Possibility of being used offlineWide range of exercises:
Multiple choice testsOpen questions (with powerful automaticcomputation)Multiple answers questions (with different marksfor any question)
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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General frameThe Web Platform
Advantages
Use without installing any additional applicationExcellent display of mathematical textIndependence from operating systemAbility to use the same tool in different situationsand disciplines
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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General frameThe Web Platform
The generation of a test
A variant is randomly chosenVariants are randomly scrambledAnswers are shuffledValues are assigned to random parametersA test is produced (PDF file)
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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General frameThe Web Platform
The url: http://progettominerva.unicatt.it
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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General frameThe Web Platform
Students’ interface - 1
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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General frameThe Web Platform
Students’ interface - 2
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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General frameThe Web Platform
Teachers’ interface - 1
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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General frameThe Web Platform
Results
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Secondary Schools
Trial has been going on since school year 2011-12Results before and after the use of the tool
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
University
Trial started in February 2014Students can perform exercises and studytheoretical material onlineA group gives a part of the final exam online:figure shows positive marks.
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Rasch model
Mathematical model to compute the probability𝑝𝑖𝑗 that the subject 𝑖 with ability 𝜃𝑖 answerscorrectly to question 𝑗 with difficulty 𝛽𝑗Rasch measure based on the difference 𝜃𝑖 − 𝛽𝑗Data collected in a raw score matrix: 𝑛 rows(subjects) and 𝑚 columns (items)Conversion of values of measures of ability anddifficulty into a common scale (logit)
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
The analysis performed in 2014
Data collected from a test of MathematicsTwo sets of students: 20 of Catholic Universityand 16 of a Secondary SchoolQuestionnaire with 16 multiple choice questions,8 on derivatives and 8 on domains
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Summary Statistics
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Summary Statistics interpretation
For students: separation index lower than 2 andreliability index lower than 0.8
→ more items maybe needed for a better separation of ability levels(actually we can divide the sample into 3 levels)For items: separation index lower 3 and reliabilityindex lower than 0.9 → the students’ sample isnot large enough to confirm the item difficultyscale.
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Summary Statistics interpretation
For students: separation index lower than 2 andreliability index lower than 0.8→ more items maybe needed for a better separation of ability levels(actually we can divide the sample into 3 levels)
For items: separation index lower 3 and reliabilityindex lower than 0.9 → the students’ sample isnot large enough to confirm the item difficultyscale.
Messineo - Vassallo Rasch Analysis in Minerva Platform 20/37
The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Summary Statistics interpretation
For students: separation index lower than 2 andreliability index lower than 0.8→ more items maybe needed for a better separation of ability levels(actually we can divide the sample into 3 levels)For items: separation index lower 3 and reliabilityindex lower than 0.9
→ the students’ sample isnot large enough to confirm the item difficultyscale.
Messineo - Vassallo Rasch Analysis in Minerva Platform 20/37
The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
Other Tables
Initial dataIndividual AnalysisDifferential Analysis
Summary Statistics interpretation
For students: separation index lower than 2 andreliability index lower than 0.8→ more items maybe needed for a better separation of ability levels(actually we can divide the sample into 3 levels)For items: separation index lower 3 and reliabilityindex lower than 0.9 → the students’ sample isnot large enough to confirm the item difficultyscale.
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Person-Item Distribution Map
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Person-Item Distribution Map
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Person-Item Distribution Map
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Person-Item Distribution Map
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Person-Item Distribution Map interpretation
students’ ability is on the left, from highest (top)to lowest (bottom);items are on the right, ordered by difficulty, frommost difficult to easiest;both are measured on the same scale;average of difficulty of items is set equal to 0;students’ mean value is 0.12 → students’performance is a little above the expectedperformance.
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Guttman Scalogram
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Guttman Scalogram
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Guttman Scalogram
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Guttman Scalogram
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Guttman Scalogram interpretation
𝐴 → unexpected correct answer@ → unexpected wrong answerhelps identify “lucky guesses” and other patternsof response
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Item Category/Option/Distractor Frequencies
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Scatterplot
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
Scatterplot interpretation
Comparison of performance in the two groupsoutliers are the items outside the two bands thatrepresent the limits of the confidence interval(95%)DOM20 is significatively more difficult forUniversity studentsDER15 is significatively more difficult forSecondary School students
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
THANK YOU
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Initial dataIndividual AnalysisDifferential Analysis
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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DIF Plot for students
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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DIF Plot - interpretation
Plot shows the average difference for each groupbetween the observed and expected responseDER15, DER03, DOM20 show significativedifferences in the two groups
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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Scatterplot Female-Male
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The M.In.E.R.Va ProjectStatistical Analysis and Rasch Model
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DIF Plot for items
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