Specialized Rasch Measures Applied at the Forefront of Education (Education in a Competitive and Globalizing World)

EDUCATION IN A COMPETITIVE AND GLOBALIZING WORLD SERIES

SPECIALIZED RASCH MEASURES APPLIED AT THE FOREFRONT OF EDUCATION No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

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EDUCATION IN A COMPETITIVE AND GLOBALIZING WORLD SERIES

SPECIALIZED RASCH MEASURES APPLIED AT THE FOREFRONT OF EDUCATION

RUSSELL F. WAUGH EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2010 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers‟ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Specialized Rasch measures applied at the forefront of education / editor, Russell F. Waugh. p. cm. Includes index. ISBN 978-1-61209-908-8 (eBook) 1. Educational tests and measurements--Cross-cultural studies. 2. Rasch models. I. Waugh, Russell. LB3051.S69 2009 371.26--dc22 2010001176

Published by Nova Science Publishers, Inc. † New York

CONTENTS Preface

xi

Author Biographies Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

xiii

A Rasch Measure of Student Receptivity to Project Work at a Junior College in Singapore Choe Kee Cheng and Russell Waugh

1

Rasch Measures for Sports, Drama and Music Student Self-Views Based on Gardner Intelligences Ahdielah Edries and Russell F. Waugh

23

Teacher Guttman Scales and Teacher Views at an Islamic College Ahdielah Edries and Russell F. Waugh

49

Rasch Measures of Form Constancy of Letters and Numbers, and Letters in Words for Young Children Janet Richmond, Russell F. Waugh and Deslea Konza

65

Rasch Measures of Number Discrimination and Reversal, and Numbers in Calculations for Young Children Janet Richmond, Russell F. Waugh and Deslea Konza

83

Rasch Measures of Self-Discipline and Moderation in Mathematics Education Liu Shiueh Ling and Russell F. Waugh

Chapter 7

Rasch Measures of Dependability and Responsibility Liu Shiueh Ling and Russell F. Waugh

Chapter 8

A Rasch Measure of the Student Entrepreneurial Mindset in Singapore Wong Heng Aik Jason and Russell Waugh

101 131

153

x Chapter 9

Chapter 10

Index

Contents A Rasch Measure Linking Self-Reported Student Attitude and Behavior to Mathematics Radha Devi Unnithan and Russell Waugh

181

A Rasch Measure of University Students‟ Receptivity to Peers with Disabilities Across Two cultures Minoti Biswas and Russell Waugh

205 225

PREFACE This book contains Rasch measurement research papers that were based on part of some of my recent doctoral student theses that I supervised within the Faculty of Education and Arts at Edith Cowan University and the Graduate School of Education at the University of Western Australia (2006 to 2009). My doctoral students worked very hard on their theses investigating the relevant literature, mastering Rasch measurement and the Rasch Unidimensional Measurement Models (RUMM) computer program with its relevant statistics. Most of the doctoral students had families and jobs at the same time as they performed their research and, while all would say that it was a wonderful, rewarding experience during which time they learned a lot, they would also say that it was very hard work writing and re-writing their research output. In this book, they share some of their research output with you. To the best of our knowledge, all the Rasch measures reported here have not been performed by any other researcher in the world, up to the time of the thesis. In making a measure of a variable intended for Rasch analysis, items are designed to be conceptually ordered by difficulty along an increasing continuum from easy to harder for that variable. For the purpose of explanation here, I shall use three items ordered from easy to medium to hard. In designing the items, one keeps in mind that the respondent measures of the variable are conceptualised as being ordered along the continuum from low to high according to certain conditions. The conditions in the three-item example are that respondents with low measures will have a high probability of answering the easy items positively, and a low probability of answering the medium and hard items positively. Respondents with medium measures will have a high probability of answering the easy and medium items positively, and a low probability of answering the hard items positively. Respondents with high measures will have a high probability of answering the easy, medium and hard items positively. These conditions are tested through a Rasch analysis which provides a test of the structure of the variable. Data are collected from respondents on the items and scored dichotomously (0/1), as in, for example, but not limited to, wrong/right, no/yes, none/a lot, disagree/agree, some/often, bad/good, slow/fast, or the items can be scored with three or more responses as, for example, with none (0), some (1), most (2) and always (3). It is better to have an ordered response set and the RUMM computer program will test whether the response categories are being answered consistently and logically for each item. My students and I hope that the Rasch measurement papers in this book will help you in your desire to do some good educational research measurement that improves our knowledge in education for the benefit of young people.

Russell F. Waugh

xii Best wishes Russell Waugh January 2010

In: Specialized Rasch Measures… Editor: Russell F. Waugh, pp. xiii-xiv

ISBN: 978-1-61668-032-9 © 2010 Nova Science Publishers, Inc.

AUTHOR BIOGRAPHIES Minoti BISWAS is a specialist teacher, currently at Narrogin Senior High School in Western Australia. She holds the degrees of BA (Hons.), BEd, MEd, and PhD. Her PhD thesis (2007) was titled University Students‟ Receptivity to Peers with Disabilities. Minoti has extensive experience teaching in India and in Western Australian secondary education. She has a special interest in students with disabilities and her cross-cultural measure of disabilities (India/Western Australia) is important for policy development. CHOE Kee Cheng is a specialist senior teacher in Singapore. He holds the degrees of BA (Hons.), Dip. Ed., MA (English Studies), and EdD. His Doctor of Education thesis (2006) was titled Student Engagement with Project Work in a Junior College in Singapore. Project Work was an initiative of the Singapore Government where all secondary students were required to do Project Work in groups in order to improve their cooperative and creative abilities for the future benefit of their country. Ahdielah EDRIES is Principal of one of three campuses of an Australian Islamic College in Perth, Western Australia and a current Fulbright Scholar. She holds the degrees of BSc, Grad. Dip. of Ed., MEd, and EdD. Many of her students come from war torn countries like Somalia, Ethiopia and Lebanon, and some have not been to school before coming to Australia. Ahdielah works very hard to build up the Australian Islamic College so that its students have the same standard and the same opportunities as other Australian students. Her Doctor of Education thesis (2009) was titled Student and Teacher Identified Attitudes and Needs at an Australian Islamic College. Her research involved investigating the attitudes, interests and needs of the students (but not all of this is reported here). Only some Rasch measures about student self-views based on some Gardner Intelligences are reported in this book. LUI Shiueh Ling is a specialist educator in Singapore. She holds the degrees of Diploma of Teaching, BEd (Hons.), MEd and EdD. Her Doctor of Education thesis (2009) was titled A Caring Thinking Module in Mathematics: Its Impact on Social Attitudes and behavior in Students. Gambling addiction is a problem in many countries, including Singapore, and there is a need to include an understanding of statistics and responsibilities associated with gambling, which was done in the thesis, but not all reported here, in teaching some mathematics. Shiueh Ling has held various senior teaching and administrative positions in

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Russell F. Waugh

Singapore and she recently administered the International Baccalaureate, amongst other things, at a premier high school there. Deslea KONZA is Associate Professor in the Faculty of Education and Arts at Edith Cowan University and a specialist in language education. She holds the degrees of BA, Dip.Ed., Dip. Special Ed., MEd, and PhD. Deslea has had wide experience teaching students of all ages with a range of special needs, including those associated with blindness, profound hearing impairment, intellectual disabilities, physical disabilities and multiple disabilities. She also has wide experience in language development and she has published widely in books and journal articles throughout her university career. Janet RICHMOND has extensive experience in occupational therapy, language and reading education in South Africa, Victoria and Western Australia. She holds the degrees of B. Occ. Therapy, M. Occ. Therapy (Hons), and PhD. Her PhD thesis (2009) was titled: Visual Discrimination of Alphabet Letters and Numbers with Young Children. She continues to help young children improve their reading and language skills mainly through her university work. Rahda Devi UNNITHAN is a specialist Mathematics teacher in Singapore where she has had extensive experience in Mathematics teaching. She holds the degrees of BSc., Post. Grad. Dip. of Ed., MEd and EdD. Her Doctor of Education thesis (2007) was titled Singapore Secondary School Students‟ Conceptions and Misconceptions of Algebraic Equation Solving. Russell F. WAUGH works at two universities in Perth, Western Australia. He is a Professor in the Faculty of Education and Arts at Edith Cowan University and a Senior Research Fellow in the Graduate School of Education at the University of Western Australia, and he supervises doctoral students at both universities. He holds the degrees of BSc, MSc, BEd, MEd, and PhD (UWA). Russell is a former Fulbright Scholar and specializes in Rasch measurement using the Rasch Unidimensional Measurement Models (RUMM) computer program developed by Professors David Andrich, Barry Sheridan and Guanzhong Luo, mainly applied to psychological and educational variables in the human sciences. Russell has published widely through journals and books, nearly all with Rasch measures. Russell can be contacted at [email protected] WONG Heng Aik Jason is a specialist teacher and educator in Singapore. He has taken a special interest in the entrepreneurial mindset of young Singapore students so that they will excel in business later in life for the benefit of themselves and for the benefit of Singapore society. An emphasis on creativity and entrepreneurship in schools was a recent initiative of the Government of Singapore. Wong holds the degrees of BA., Dip. Ed., MEd and EdD. His Doctor of Education thesis (2009) was titled The Entrepreneurial Mindset of Students in Singapore.

In: Specialized Rasch Measures… Editor: Russell F. Waugh, pp. 1-21

ISBN: 978-1-61668-032-9 © 2010 Nova Science Publishers, Inc.

Chapter 1

A RASCH MEASURE OF STUDENT RECEPTIVITY TO PROJECT WORK AT A JUNIOR COLLEGE IN SINGAPORE Choe Kee Cheng1 and Russell Waugh2 1

Singapore Graduate School of Education University of Western Australia 2

ABSTRACT Group Project Work was introduced as a compulsory subject in Singapore to encourage team work, critical thinking and communication. Student Receptivity to Project Work was conceptualised as related to six key aspects (goal management, selfmanagement, learning styles, collaboration, knowledge application, and communication) and a questionnaire composed of 54 stem-items was based on these. Items were conceptualised from easy to hard (nine stem-items for each of the six aspects) and answered in two perspectives (an attitude self-view and a behaviour self-view) making 108 items. Data from 738 students in a junior college were analysed with a Rasch Unidimensional Measurement Model computer program (RUMM2020). Results showed that there was limited support for the model of the variable as 43 items had to be deleted. There was a good fit to the measurement model for the remaining group of 65 items (item-trait chi-square = 594, df=585, p=0.39) and reliability was good (Person Separation Index = 0.95). Items from all six aspects fitted the measurement model. The ideal perspective (this is what I aim for) was easier than the actual behaviour perspective (this is what I actually do), as was conceptualised for the structure of the variable but, for some items, only the ideal perspective, or only the behaviour perspective, fitted the measurement model. Some re-wording of the items, with further Rasch analysis, is needed for further understanding of the variable.

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INTRODUCTION It has become increasingly clear that in today‟s society where human capital is perceived to command premium value in driving the engines of the new knowledge-based economy (Goh, 1997), what is valued in individuals by states are the abilities to be able to think critically, and creatively, work independently, and be adaptable enough to apply one‟s knowledge and learning, innovatively and flexibly (Shanmugaratnam, 2003, 2005; Teo, 2000). In many countries, as such, changes were introduced in the education system to offer students more variety and choice of subjects to develop these qualities (Ball, 1998). Singapore, too, was not unaffected by such changes. The prevailing school of thought since the advent of the 21st century has been that the reorientation of educational emphasis on innovative handling of information technology, team-working, networking, and life-long learning, might lead to a lesser dependence on pen-and-paper type examinations, and to a greater reliance on project work assessment and/or other process skills assessments, which would, ideally, create the type of innovative, adaptable workers for which the Singapore economy was perceived to be in great need. Thus, methods of teaching, and assessment had to emphasise creative problem-solving, and critical pursuits in thinking. Such other generic skills that were deemed to be able to enhance the economic value of the individual included being able to communicate ideas and information, use mathematical ideas and techniques, work and collaborate with others in teams, utilise technology, plan and organise activities; collect, systematise and conduct data analysis, and think critically and creatively to solve problems (Kennedy, 1999). It was in this light that, as conceptualised by the Singapore Ministry of Education, Project Work was introduced as a compulsory academic subject that would infuse these much-desired skills and capabilities in students in Singapore.

WHAT IS PROJECT WORK? Project Work is a curricular programme designed to provide students with opportunities to explore the interconnectedness of subject specific knowledge (Ministry of Education, Singapore, 2004). Above and beyond the fact that the grade obtained for the subject counts for ten per cent of local university admission requirements from 2003 onwards (Nirmala, 1999), it aims to enable students to firstly, apply creative and critical thinking skills; secondly, improve their communication skills (both oral and written); thirdly, foster collaborative learning; and fourthly, develop self-directed inquiry, and life-long learning skills (Ministry of Education, n.d.). Based on the Singapore Ministry of Education‟s guidelines, there are five key features of Project Work: interdisciplinary; involves collaborative learning; requires an oral presentation; focuses on both the process and product; and builds in Just-In-Time Skills instruction (Ministry of Education, 2002). As part of the implementation guidelines, the Singapore Ministry of Education also stipulates that Project Work differs from subject-based projects in that students gather and process information from various sources, and apply, and integrate knowledge, and skills, from different subjects to create new knowledge. Not only that, it has to be carried out during curriculum time so that teachers can work with students more closely. A typical project task, as such, requires students to identify an aspect of society and to evaluate how this particular

A Rasch Measure of Student Receptivity to Project Work at a Junior College…

3

feature can change for the betterment or detriment of the said society. Students have then to come up with measures to enhance or counteract the development of the characteristic of society (Ministry of Education, 2004, 2005). Typically, two project tasks are given to students from which they must select one to do a project on. Each of the tasks revolve around different themes and each task, in turn, is further broken down into a number of bullet points which explicitly instruct students on the type of action they should engage in for different tasks. Changes introduced by the Education Ministry to streamline the structure of Project Work in 2005, and which informs the way Project Work is currently implemented and assessed, have retained the form such project tasks assume.

LITERATURE REVIEW Recent research on the effectiveness of doing project work from students‟ perspectives have reported mixed findings. Several studies have found that students, on the whole, perceive Project Work positively (Chang & Chang, 2004; Chua, 2004; Hays & Vincent, 2004; Lee, 2001; Mueller & Fleming, 2001; Payne, Monk-Turner, Smith & Sumter, 2006; T.L.S. Tan, 2002; Wong, 2001). Such findings can result from the fact that students saw doing project work as a way of acquiring authentic, marketable life skills, such as working with others in a group, data analysis, researching, action planning, and organising, which would be beneficial to their working life. To students, therefore, doing projects was identical to gaining (future) work experience (Bourner, Hughes & Bourner, 2001). Other merits of doing project work included students perceiving a project work environment to be motivating, particularly for groups that could cooperate (Willis et al., 2002), and which provided the opportunity for them to collaborate in teams and think and learn independently (Kucharski, Rust & Ring, 2005; Payne & Monk-Turner, 2006; G.C. I. Tan, 2004). Similarly, students in Singapore, like the students from other parts of the world, perceive Project Work positively for the perceived benefits to be gained from doing the subject: an autonomous learning environment which allows students to explore their own interests (Chin & Kayalvizhi, 2005), have fun (G.C.I. Tan, 2004), collaborative learning opportunities leading to increased peer interaction and friendship (Chin & Kayalvizhi, 2005; Quek & Wong, 2001; G.C.I. Tan, 2004), valuable learning experience (Chin & Kayalvizhi, 2005; O.S. Tan, 2004), and a deeper understanding of the topics under investigation (Chin & Kayalvizhi, 2005; G.C.I. Tan, 2004). However, the body of literature that has reported unfavourable findings have also begun to increase (Cantwell & Andrews, 2002; Phipps, Phipps, Kask, & Higgins, 2001; Zanolli, Boshuizen & De Grave, 2002). These studies challenge the view that students accept, or agree, that the Project Work approach to learning is one that they favoured, and highlight instead the weaknesses of the Project Work instructional and learning approach, especially with regard to issues pertaining to the internal group dynamics of working in a studentoriented learning environment. For example, McPhee (2002) reported that some students in her study felt the demerits of a group project or problem-based learning method keenly, which included group conformity, dysfunctional groups, too much time spent on fruitless discussion, social-loafing, the lack of understanding of content as a result of students not receiving notes and intimidating experience of presenting to peers. Senior students‟

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perceptions of the weaknesses of such a learning method revolve around main issues of group discussions being dominated by individuals, unproductive discussion sessions where no real learning occurs, and an absence of teacher feedback. Nijhuis, Segers and Gijselaers (2005) have also reported that such similar views were echoed by the students in their study who acquired a negative view of time efficacy when they were engaged in a project or problembased learning model. More strikingly, in another study involving 109 students, which used a traditional pre-test and post-test design and occurred over a period of eight weeks, Elen and Clarebout (2001) argued that students did not perceive that more effort put into problembased learning would gain benefits for them. Not only that, their beliefs in the power of technology and problem-based collaboration as aspects of a positive learning environment diminished after the study. The conclusions of these findings seem obvious: students do not have a positive perception of Project Work. In Singapore, while the literature on students‟ perspectives to Project Work has, for the most part, reported positive evaluations, there also appears to be a clear and present theme of student dissatisfaction that has emerged in a number of studies. I.G.C. Tan‟s (2004) study (N=241), for instance, reported that notwithstanding the 652 positive statements that students made of Project Work, another 303 statements related to student preference for the conventional transmission type of teaching, the extent to which doing Project Work was timeconsuming, admission of an inability to learn much vis-à-vis Project Work, and group conflicts. Other researchers‟ studies, such as O.C. Tan (2004), and Chin and Kayalvizhi (2005), made similar findings. While the majority of the 100 undergraduate-participants in O.C. Tan‟s (2004) study had positive learning experiences with the problem-based learning model of Project Work, some participants also acknowledged that: firstly, problem-based learning conflicted with their approach to learning, and learning styles; secondly, there was a lack of overview and direction; thirdly, they lacked mental and emotional readiness for problem-based learning, and felt overwhelmed by numerous issues and self-directed learning; and finally, they were frustrated by a lack of time. Likewise, the 39 Primary Six students in Chin and Kayalvizhi‟s (2005) study reported that the difficulty of coming up with good research designs, and group conflicts, comprised their negative experiences of problem-based learning in Science. These findings offer substantial evidence to suggest that two issues that may commonly plague Project Work is the lack of time and group conflicts. Yet other researchers in Singapore have reported additional workload, and increased stress as weaknesses of Project Work. According to Chang & Chang (2004), students felt that their workload was massive as they had to manage five main academic subjects along with co-curricular activities, community involvement programmes, and enrichment programmes, and now, Project Work – the nature of which required students to work on their projects beyond the time spent on discussions of their ideas for their projects within the classroom during curriculum time. In effect, Project Work vied for the limited amount of time students have to apportion for all their other commitments that, for them, were equally important as Project Work, if not more. Chin and Chia (2004) have argued that one source of stress stemmed from students‟ inability to ask the right questions when working in their project work groups and to have them addressed. These researchers warned that if “students feel they are not making the desired progress, they may become frustrated and unmotivated in pursuing their learning tasks” (Chin & Chia, 2004, p. 724). In summary, although several researchers have asserted that a group-based project or problem-based learning model is one to which students would be receptive, there has also

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been dissenting voices that argue for the contrary. This study was thus an attempt which sought to contribute to the literature on doing project work in relation, specifically, to students in a junior college in Singapore, using a Rasch measurement model as the research base.

AIMS OF STUDY There were two main objectives to this study. Firstly, the study intended to create a questionnaire on Student Receptivity to Project Work based on the following six aspects of Project Work: Goal Management, Self-Management, Learning Styles, Collaboration, Knowledge Application, and Communication. Secondly, the study aim to construct a linear scale of Student Receptivity to Project Work using the RUMM2020 Rasch computer program and thus understand the structure of the variable and its meaning.

METHOD The questionnaire was trialled and revised before being administered to the first year cohort of 913 Science students who took Project Work as a subject in a major junior college in Singapore in 2004. The students were studying for their GCE „A‟ levels before going to university including Physics, Chemistry and Mathematics.

Sample These students were all first year junior college Science students who encountered Project Work for the first time as a subject that was to be compulsorily included in their future application to university entry in Singapore. It was decided that there should be a spread of participants across all the Science classes so that representative student views and perspectives on Project Work from the cohort of first year Science students could be garnered through the writing of diaries, as well as through individual interviews (not reported here).

Administrative and Ethics Approval Prior to the administering of the questionnaire, permission had been obtained from the Principal, as well as the Head of Department for Project Work, for the questionnaire to be administered to the students. Initial approval to conduct the survey was obtained from The University of Western Australia's Ethics Committee. Following this, permission was obtained from the Cluster Superintendent who oversaw the school cluster to which this college belonged, the Principal of the college, as well as the Head of Department for Project Work, to conduct research in the college. Written informed consent was then sought from the participants themselves, and their parents, for their voluntary participation.

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Administering of Questionnaire During the administering of the questionnaire, students read the cover letter which informed them of their right to withdraw from participating in the questionnaire, and signed their consent to participate in this research study, based on the condition that their identities would remain anonymous. The cover and informed consent sheet from the questionnaire were collected before the students began doing the questionnaire proper. Completed questionnaires were subsequently placed into an envelope by the students. Altogether, out of a cohort of first year 913 students, 849 attended on the day, and 738 useable questionnaires, in which all questions were answered, were retained for the Rasch analysis

DESIGN OF THE QUESTIONNAIRE Within each aspect, items were conceptually designed from easy to hard. For example, under Goal Management (Expectations), stem-item 1 was set expectations that I want to achieve in Project Work and this was expected to be relatively easy. Stem-item 2 was do my best to attain the expectations in Project Work that I set for myself and this should be harder because it involves setting attainments compared to expectations. Stem-item 3 was evaluate my performance against the expectations in Project Work that I set for myself and this should be harder still because it involves evaluating which itself is harder and involves more effort than stem-item 2. In a similar way, the items for the other aspects were designed conceptually from easy to hard but they are not described here to avoid repetition. A reader can easily work out the conceptual design by looking at Table 1. All the items were written in a positive sense and answered in two perspectives so it would be evident to students what was being measured, and what they were expected to answer. The two perspectives were what I aim for (to measure what students ideally would like to do and was expected to be easy), and What I actually do (to measure what students do in practice which was expected to be harder). The behaviour was expected to be harder than the attitude because to actually do „something‟ requires effort and includes the attitude to actually do that „something‟. Students were required to answer the items in an ordered response format: none of the time (score 1), some of the time (score 2), most of the time (score 3) and all of the time (score 4), in line with good measurement practice. The conceptual ordering of the item difficulties and the conceptual ordering of the response categories can now be tested by collecting appropriate data and analysing it with a Rasch measurement computer program.

DATA ANALYSIS WITH THE RUMM 2020 PROGRAM The RUMM 2020 program produces tables and graphical output that enables one to check the fit of the data to the measurement model using the Partial Credit Model of Rasch. Firstly, item thresholds and Response Category Curves were inspected so that only those items with ordered thresholds, which indicated that the response categories for the items were answered consistently, and logically, were included in the final linear scale. Then, the itemtrait test-of-fit chi-square was examined to check on the consistency of agreement of the item

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parameters (item difficulties) for all student measures along the scale. After this, individual item fits to the measurement model were examined and those items that did not fit were deleted. Forty-three non-performing items out of 108, as established through the steps above, were subsequently removed from the scale, leaving 65 items whose data fitted the Rasch measurement model and could be used to create a linear scale in which the measures and item difficulties were calibrated on the same scale, including the attitude and behaviour item difficulties.

RESULTS The item-trait interaction chi-square with 65 items was 594.2, df=585, p=0.39. This means that there was very good agreement amongst all 738 students, right along the scale, about the difficulties of the 65 items and, hence there was a good overall fit to the measurement model. The Student Separation Index was 0.95 and high. This means that the measures were well-separated in comparison to the measurement errors, as required for good measurement. Based on the Separation Index, the power of the tests-of-fit for this scale was rated as „excellent‟. Of the 65 items that fitted the measurement model, 25 evaluated an ideal aspect (What I aim for), and 40 evaluated an actual aspect (What I actually do). This means that the actual aspect made a stronger contribution to the variable (Student Receptivity to Project Work) than the ideal aspect. Eighteen of the 65 items had ideal and actual aspects that matched each other, and which fitted the measurement model. That is, for nine stem-items, there were nine ideal and nine corresponding actual aspects for those items. Seven ideal and 22 actual aspects of the items made up the remaining 65 items. For Goal Management 15 out 18 items fitted: For Self-Management 8 out of 18 items fitted: For Learning Styles nine out of 18 items fitted: For Collaboration nine out of 18 items fitted; For Knowledge Application 14 out of 18 fitted: For Communication ten out of 18 items fitted. This means that there was only limited support for the conceptualised structure. The item difficulties are given in Table 1. Table 1. Student Receptivity to Project Work (N=738) Item No.

Item wording

GOAL MANAGEMENT Expectations 1-2 Set expectations that I want to achieve in PW. 3-4 Do my best to attain the expectations in PW that I set for myself. 5-6 Evaluate my performance against the expectations in PW that I set for myself Interest 7-8 Read widely to learn more about and find out what is relevant for my topic in PW out of curiosity. 9-10 Display enthusiasm for and commitment to PW. 11-12 Solve problems with which others have difficulty because I am keenly interested. Goal Setting 13-14 Set myself realistic goals in PW.

What I aim for

What I actually do

-0.12 -0.64

0.98 0.40

0.51

No fit

No fit

No fit

0.11 0.38

0.95 1.11

-0.50

0.27

Choe Kee Cheng and Russell Waugh

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Table 1. (Continued). Item No.

Push myself to ensure that I achieve my goals in PW. 17-18 Challenge myself to overcome difficulties to reach my goals in PW. SELF-MANAGEMENT Personal motivation 19-20 Seek to obtain as high a band as possible for PW. 21-22 Like the interaction with peers and social relationships in solving problems for PW. 23-24 Enjoy the intellectual challenge and social learning involved with PW. Time management 25-26 Adhere as closely as possible to my group‟s timeline of work. 27-28 Organise my time to make the optimum use of it for PW. 29-30 Work steadily and consistently for PW, rather than leave everything until the last minute to be done. Tasks 31-32 Decide what tasks should be done at what time for my project.

-0.25

What I actually do 0.59

-0.11

0.83

-0.87 No fit

-0.13 No fit

No fit

No fit

-0.77

No fit

No fit

1.14

-0.26

No fit

-0.43

0.67

33-34

No fit

0.46

No fit

No fit

No fit

No fit

-0.08

No fit

No fit

0.60

No fit

-0.74

No fit

0.18

No fit

No fit

-1.04

-0.04

-1.00

-0.19

No fit

0.18

Item wording

15-16

35-36

Make an effort to achieve these tasks by a set deadline

Challenge myself to use different strategies to accomplish these tasks even when I have difficulties LEARNING STYLES Independent learning 37-38 Rely upon others to guide me on what I need to do for PW. 39-40 Work out for myself exactly what is needed to be done, not just accept what I am told to do for PW. 41-42 Make every effort to seek out necessary information, resources and ways that best enable me to achieve for PW. Learning from others 43-44 Listen to others during group discussions to learn about different points of view in relation to PW. 45-46 Learn actively and deliberately from others who have more knowledge and experience than I have. 47-48 Participate in group discussions to share views and generate new ideas to improve the quality of my project. Learning from the Supervising Tutor (ST) 49-50 Seek and listen to the ST's suggestions on my project. 51-52 Reflect and act upon the ST's suggestions where appropriate for my project. 53-54 Modify or integrate the ST's suggestions with other ideas as best as I can to create new approaches for my project.

What I aim for

A Rasch Measure of Student Receptivity to Project Work at a Junior College… Item No.

Item wording

COLLABORATION Discussion behaviours Spend adequate time discussing how 55-56 improvements to the project can be made. Discuss and implement different strategies to 57-58 overcome difficulties encountered in the project Evaluate the effectiveness of strategies taken to 59-60 improve group progress. Working in teams Pay attention and listen to others during PW 61-62 discussions. Respect others‟ views and show tact in my 63-64 responses as much as I can during PW discussions. Involve others actively, appropriately and 65-66 consistently to the best of my ability during PW discussions Helping each other Volunteer willingly for tasks and responsibilities 67-68 for PW to help others fulfil tasks and meet deadlines. Encourage, motivate and support others to stay 69-70 focused and on-task for PW. Involve others actively to solve difficulties in PW 71-72 effectively as a cohesive group. KNOWLEDGE APPLICATION Initiating research 73-74 Decide on the proper focus of my project. 75-76 Brainstorm and plan how my project can best be completed for PW. 77-78 Seek, summarise and categorise information from a wide range of sources for my project. Managing research 79-80 Examine and analyse information for its usefulness to my project. 81-82 Evaluate the accuracy and credibility of information from different sources. 83-84 Synthesise and relate different information pertinently, cogently and meaningfully to my project discussions, written reports and presentations Applying research 85-86 Recognise the value of the knowledge and skills that I have gained for having done PW. 87-88 Relate the knowledge and skills I gained from PW to other subjects, topics or courses, whenever possible 89-90 Use the knowledge and skills consciously and actively in dealing with other people and situations in life COMMUNICATION Content of presentation 91-92 Link my presentation to my project explicitly. 93-94 Organise my presentation in a coherent fashion. 95-96 Make my presentation captivating and "listenerfriendly".

What I aim for

What I actually do

No fit

0.44

No fit

0.60

0.11

No fit

No fit

-0.88

No fit

No fit

No fit

0.00

-0.47

0.05

No fit

0.34

No fit

0.51

-0.77 No fit

0.15 0.31

-0.72

0.33

No fit

-0.24

-0.95

-0.12

No fit

0.06

0.15

1.06

0.49

No fit

0.46

1.33

No fit No fit No fit

-0.38 -0.70 -0.24

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Table 1. (Continued). Item No.

What I aim for

What I actually do

Presentation matters 97-98 Project my voice and make some eye contact with

No fit

-0.69

99-100

Articulate confidently and fluently on the project.

No fit

-0.45

Express myself with poise, and engage the audience with my personality, appropriate eye contact, tone and relevance of content

No fit

-0.29

Addressing questions 103-104 Provide a relevant answer and elaborate on it for PW.

No fit

0.33

105-106

Offer an insightful response supported with pertinent details for PW.

-1.04

0.59

107-108

Engage the audience with wit, appropriate pacing, and clarity of my response.

-1.20

No fit

Item wording

the audience on PW.

101-102

Table 2. Global Fit Statistics for the Student Receptivity to Project Work Scale Items

Students

Number

65

738

Location mean

0.00

0.86

Standard deviation

0.62

0.81

Fit Statistical mean

0.10

-0.50

Standard deviation

0.99

2.45

Notes on Table 2 1. The item means are constrained to zero by the measurement model. 2. When the data fit the measurement model, the fit statistic approximates a distribution with a mean near zero, and a SD near one (a good fit for the items of this scale, and a not-so-good, but acceptable fit for the student measures).

TARGETING OF THE SCALE Figure 1 shows the item thresholds ordered from easy to hard (at the bottom) and the student measures ordered from low to high (on the top). The thresholds „cover‟ the range of student measures and are thus well targeted at the measures, but some harder items are needed to be added to the scale to „cover‟ the very high student measures.

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Notes 1. The scale is in logits, that is, the log odds of answering the response categories. 2. Student measures (low to high) are placed on the upper side of the scale, and thresholds (easy to hard) are placed on the lower side of the scale. Figure 1. Student Receptivity Measures and Item Thresholds on the same scale.

Figure 2. Student Receptivity by Gender. Note: The RUMM program has the colours wrong: Females are maroon (not red), males are green (not blue).

Choe Kee Cheng and Russell Waugh

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USING THE RESPONSE CATEGORIES LOGICALLY Testing whether the response categories are used logically and consistently is tested by two methods with the RUMM2020 program: (1) threshold ordering and (2) response category curves. Thresholds are points between adjacent categories where the odds are 1:1 of answering in either category. These thresholds should be ordered in line with the ordering of the response categories, if the students use them consistently and logically. Table 3 (below) illustrates that the thresholds for the Goal Management items are ordered in line with the conceptual ordering of the response categories. Threshold data from the other aspects confirmed that the response categories were answered consistently and logically, but are not reported her to avoid repetition. Figure 3 shows a response category curve for item 1 where the probability of answering each of the response categories is given by student measure. For a low measure, there should be a high probability of answering the lowest response category and, as the student measures increase, the probability of answering this low category should decrease and the probability of answering the next category should increase. The process should continue until the highest measures where the probability of answering the highest category should be high and the probability of answering the other categories low. Table 3. Item Thresholds Values for the Goal Management Aspect Item Number

Mean location (Difficulty)

THRESHOLDS 1

2

3

Item 1

-0.12

-1.55

-0.63

1.82

Item 2

0.98

-1.55

1.04

3.46

Item 3

-0.64

-1.86

-1.04

0.97

Item 4

0.40

-2.04

0.35

2.88

Item 5

0.51

-0.65

0.10

2.06

Item 9

0.11

-0.94

-0.32

1.58

Item 10

0.95

-1.07

0.91

3.00

Item 11

0.38

-1.19

0.25

2.06

Item 12

1.11

-1.04

1.33

3.03

Item 13

-0.50

-1.89

-0.81

1.21

Item 14

0.27

-1.57

0.17

2.21

Item 15

-0.25

-1.54

-0.67

1.46

Item 16

0.59

-1.78

0.87

2.68

Item 17

-0.11

-1.75

-0.30

1.72

Item 18

0.83

-1.50

0.91

3.10

Notes on Table 3 1. Mean location (item difficulty) is the mean threshold location. 2. At a threshold, the odds are 1:1 of answering adjacent response categories. In this case, all the thresholds are logically ordered from easy to hard, in line with the ordering of the response categories.

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Figure 3. Item Category Curve for Item 100 (good-fitting item). Note: Threshold 1 is about -2.2 logits; Threshold 2 is about -0.8 logits; and Threshold 3 is about +1.8 logits.

ITEM CHARACTERISTIC CURVE BY GENDER The RUMM program provides characteristic curves for each item and Figure 4 shows such a curve for item 1 by gender. The expected values follow the ogive curve and so the students have answered this item as expected for a good fit to the measurement model and there is no significantly differential item response by gender (chi-square= 2.22, df=1,9, p= 0.14).

Figure 4. Item Characteristic Curve By Gender (Item 1).

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DISCUSSION Meaning of the Scale The 65 items that fitted the measurement model define the variable Student Receptivity to Project Work. The 65 items involve ideal attitudes and behaviour items from six aspects, namely: Goal Management, Self-Management, Learning Styles, Collaboration, Knowledge Application, and Communication. Some items did not have both the attitude and behaviour perspectives fitting the measurement model (only 18 items did so) and from this it might be inferred that some attitudes did not directly influence a corresponding behaviour, contrary to a generally accepted theory (such as proposed by Ajzen, 1989). Where both attitude and behaviour items fitted the measurement model, attitude was easier than behaviour and, in these cases, it can be inferred that attitude did directly affect the corresponding behaviour.

Goal Management: Expectations, Interest, Goal Setting Fifteen out of 18 items for this aspect fitted the measurement model. The five easiest attitude items were (in descending order of easiness): 1. Do my best to attain the expectations in Project Work that I set for myself (ideal expectations); 2. Set myself realistic goals in Project Work (ideal goal-setting); 3. Push myself to ensure that I achieve my goals in Project Work (ideal goal-setting); 4. Set expectations that I want to achieve in Project Work (ideal expectations); 5. Challenge myself to overcome difficulties to reach my goals in Project Work (ideal goal-setting). The five hardest behaviour items for students were (in descending order of difficulty): 1. Solve problems with which others have difficulty because I am keenly interested (actual interest); 2. Set expectations that I want to achieve in Project Work (actual expectation); 3. Display enthusiasm for and commitment to Project Work (actual interest); 4. Challenge myself to overcome difficulties to reach my goals in Project Work (actual goal-setting); and 5. Push myself to ensure that I achieve my goals in Project Work (actual goal-setting).

Self-Management: Personal Motivation, Time Management, Tasks Eight out of 18 items for this aspect fitted the measurement model. The five easiest items are (in descending order of easiness): 1.Seek to obtain as high a band as possible for Project Work (ideal personal motivation); 2.Adhere as closely as possible to my group‟s timeline of work (ideal time management); 3. Decide what tasks should be done at what time for my project (ideal tasks); 4. Work steadily and consistently for Project Work, rather than leave everything until the last minute to be done (ideal time management); and 5.Seek to obtain as high a band as possible for Project Work (actual personal motivation). There are three hard behaviour items (in descending order of difficulty): 1.Organise my time to make the optimum use of it for Project Work (actual tasks); 2. Decide what tasks should be done at what time for my project (actual tasks); and 3. Make an effort to achieve these tasks by a set deadline (actual time management).

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Learning Styles: Independent Learning, Learning from Others, and Learning from the Supervising Tutor Nine out of 18 items for this aspect fitted the measurement model. The five easiest items are (in descending order of easiness): 1. Seek and listen to the supervising tutor‟s suggestions on my project (ideal, learning from the supervising tutor); 2. Reflect and act upon the supervising tutor‟s suggestions where appropriate for my project (ideal, learning from the supervising tutor‟s); 3. Listen to others during group discussions to learn about different points of view in relation to Project Work (actual, learning from others); 4. Reflect and act upon the supervising tutor‟s suggestions where appropriate for my project (actual, learning from the supervising tutor); and 5. Work out for myself exactly what is needed to be done, not just accept what I am told to do for Project Work (ideal, independent learning). There are only three hard items (in descending order of difficulty): 1. Make every effort to seek out necessary information, resources and ways that best enable me to achieve for Project Work (actual, independent learning); 2. Modify or integrate the supervising tutor‟s suggestions with other ideas as best as I can to create new approaches for my project (actual, learning from the supervising tutor‟s); and 3. Learn actively and deliberately from others who have more knowledge and experience than I have (actual, learning from others).

Collaboration: Discussion Behaviours, Working in Teams, and Helping Each Other Nine out of 18 items for this aspect fitted the measurement model. There were two easy items (in descending order of easiness): 1. Pay attention and listen to others during Project Work discussions (actual, working in teams); and 2. Volunteer willingly for tasks and responsibilities for Project Work to help others fulfil tasks and meet deadlines (ideal, helping each other). The five hardest items (in descending order of difficulty) were: 1. Discuss and implement different strategies to overcome difficulties encountered in the project (actual, discussion behaviours); 2. Involve others actively to solve difficulties in Project Work effectively as a cohesive group (actual, helping each other); 3. Spend adequate time discussing how improvements to the project can be made (actual, discussion behaviours); 4. Encourage, motivate and support others to stay focused and on-task for Project Work (actual, helping each other); and 5. Evaluate the effectiveness of strategies taken to improve group progress (ideal, discussion behaviours).

Knowledge Application: Initiating Research, Managing Research, and Applying Research Fourteen out of 18 items for this aspect fitted the measurement model. The five easiest items are (in descending order of easiness): 1. Evaluate the accuracy and credibility of information from different sources (ideal, managing research); 2. Decide on the proper focus of my project (ideal, initiating research); 3. Seek, summarise and categorise information from

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Choe Kee Cheng and Russell Waugh

a wide range of sources for my project (ideal, initiating research); 4. Examine and analyse information for its usefulness to my project (actual, managing research); and 5. Evaluate the accuracy and credibility of information from different sources (actual, managing research). The five hardest items (in descending order of difficulty) are: 1. Use the knowledge and skills consciously and actively in dealing with other people and situations in life (actual, applying research); 2. Recognise the value of the knowledge and skills that I have gained for having done Project Work (actual, applying research); 3. Relate the knowledge and skills I gained from Project Work to other subjects, topics or courses, whenever possible (ideal, applying research); 4. Use the knowledge and skills consciously and actively in dealing with other people and situations in life (ideal, applying research); and 5. Seek, summarise and categorise information from a wide range of sources for my project (actual, initiating research).

Communication: Content of Presentation, Presentation Matters, and Addressing Questions Ten out of 18 items for this aspect fitted the measurement model. The five easiest items are (in descending order of easiness): 1. Engage the audience with wit, appropriate pacing, and clarity of my response (ideal, addressing questions); 2. Offer an insightful response supported with pertinent details for PW (ideal, addressing questions); 3. Organise my presentation in a coherent fashion (actual, content of presentation); 4. Project my voice and make some eye contact with the audience on Project Work (actual, presentation matters); and 5. Articulate confidently and fluently on the project (actual, presentation matters). There are two very hard items (in descending order of difficulty): 1. Offer an insightful response supported with pertinent details for Project Work (actual, addressing questions); 2. Provide a relevant answer and elaborate on it for Project Work (actual, addressing questions).

WHY DON’T 43 ITEMS FIT THE MEASUREMENT MODEL? The RUMM program does not explain why 43 items did not fit the measurement model: it just provides statistics to show that 43 items did not fit and these items were discarded. Three items (65, 99 and 101) were deleted because students did not use the response categories consistently and it is difficult to understand why. Maybe the items were too complicated and need to be simplified but they were made complex because some hard items were needed for good targeting. Most items did not fit the measurement model because students could not agree on their difficulty. What seemed to be happening was that a particular group such as those with medium measures might be split with say 50% agreeing that it was a medium difficulty item and the other 50% saying that it was a hard item (or sometimes an easy item). This seemed to occur because of difficulties in particular Project Work groups as not all of them ran smoothly. Sometimes, students with different measures (either low or high) could not agree on the difficulty of particular items. For example, stem-item 3 in the actual perspective did not fit because, apparently, some students with high measures did not evaluate their

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performance in their Project Work group (and thus found this item hard) and some with high measures did evaluate their performance and found that it was only of medium difficulty. A different example is stem-item 11 where there was such variability in the cohesion within the Project Work groups (some students considered that they did most of the work and some others did very little, some were friendly and others were not) that friction occurred making problems with relationships. Where this occurred, there were clearly problems for students agreeing with the difficulties of the items. When one looks through the items that did not fit the model and conducts interviews with the students afterwards, it is possible in many cases to see why the disagreements about difficulties occurred.

CONCLUSION This paper has described the creation of a reliable, unidimensional scale of Student Receptivity to Project Work based on six aspects and 65 items, using the Partial Credit Model of Rasch and the RUMM 2020 computer program. There was good individual-item fit, good global-item fit, good Separation Index, and good targeting of the items of this scale. Of the 65 items that fitted, and comprised the linear scale, there are 25 self-reported ideal (attitude), and 40 self-reported actual (behaviour), items. The results show the structure of the variable and that it involves both attitude and behaviour for six aspects: (1) goal management, (2) selfmanagement, (3) learning styles, (4) collaboration, (5) knowledge application, (6) and communication. Of the six aspects, the two easiest aspects are Communication, and Learning Styles, while the two hardest are Goal Management, and Knowledge Application. Females have significantly higher measures than males on the scale Receptivity to Project Work (t=6.90, df=736, p=0.0000). The findings above clearly point to a need for more understanding of the various ways in which Project Work has impacted on students, and on their perspectives of doing Project Work.

REFERENCES Ball, S. (1998). Big policies/small world: An introduction to international perspectives in education policy. Comparative Education, 34 (2), 119-130. Bourner, J., Hughes, M., & Bourner, T. (2001). First-year undergraduate experiences of group project work. Assessment and Evaluation in Higher Education, 26 (1), 19-39. Chang, T.T., & Chang, A. (2004). Assessing Project Work: Teachers and students‟ perspectives. In A. Khoo, M.A.Heng, L. Lim, & R.P. Ang (Eds.), Innovation and Diversity in Education (pp. 64-79). Singapore: McGraw-Hill. Chin, C., & Chia, L.G. (2004). Problem-based learning: using students‟ questions to drive knowledge construction. Science Education, 88, 707-727. Chin, C., & Kayalvizhi, G. (2005). What do pupils think of open science investigations? A study of Singaporean primary 6 pupils. Educational Research, 47 (1), 107-126. Chua, J.J. (2004). Evaluating the effects of Project Work in learning in a primary school. Unpublished master‟s thesis, National Institute of Education, Nanyang Technological University, Singapore.

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Goh C.T. (1997). Shaping our future: Thinking schools, learning nation. Speech by Prime Minister Goh Chok Tong at the Opening of the 7th International Conference on Thinking at the Suntec City Convention Centre Ballroom. Retrieved Mar 1, 2006, from http://www.moe.gov.sg/speeches/1997/020697_print.htm Hays, J.R., & Vincent, J.P. (2004). Students‟ evaluation of problem-based learning in graduate Psychology course. Teaching of Psychology, 31 (2), 124-126. Lee, L.C. (2001). Evaluating critical thinking pedagogy to support primary school Project Work using an action research approach. Unpublished master‟s thesis, National Institute of Education, Nanyang Technological University, Singapore. Kennedy, K.J. (1999). Constructing the School Curriculum for the Global Society. Innovating schools, rapporteur. Donald Hirsch. Paris: Organisation for Economic Co-operation and Development. Ministry of Education, Singapore (n.d.) Retrieved Jan 1, 2006, from http://www.moe.gov.sg/projectwork/ Ministry of Education, Singapore. (2002). Project Work Student Handbook. Singapore: Ministry of Education. Mueller, A., & Fleming, T. (2001). Cooperative learning: Listening to how children work at school. The Journal of Educational Research, 94 (5), 259-265. Nirmala, M. (1999, July 14). New criteria for university entry in 2003. The Straits Times, p.1. Payne, B.K., Monk-Turner, E., Smith, D., & Sumter, M. (2006). Improving group work: Voices of students. Education, 126 (3), 441-448. Shanmugaratnam, T. (2003). The next phase in education: Innovation and enterprise. Speech by Mr Tharman Shanmugaratnam, Acting Minister for Education, at the Ministry of Education Work Plan Seminar 2003 at Ngee Ann Polytechnic. Retrieved Mar 11, 2006, from http://www.moe.gov.sg/speeches/2003/sp20031002_print.htm Shanmugaratnam, T. (2005). Achieving quality: Bottom up initiative, top down support. Speech by Mr Tharman Shanmugaratnam, Minister for Education, at the Ministry of Education Work Plan Seminar 2005 at Ngee Ann Polytechnic Convention Centre. Retrieved Mar 11, 2006, from http://www.moe.gov.sg/speeches/2005/ sp20050922_ print.htm Teo C.H. (2000). Ability-driven education – Putting the system in place. Speech by RAdm Teo Chee Hean, Minister for Education and Second Minister for Defence at the Work Plan Seminar 2000 at Nanyang Polytechnic Auditorium. Retrieved Mar 11, 2006, from http://www.moe.gov.sg/speeches/2000/sp23092000_print.htm Tan, T.L.S. (2002). Using Project Work as a motivating factor in lower secondary Mathematics. Unpublished master‟s thesis, National Institute of Education, Nanyang Technological University, Singapore. Wong, H.M. (2001). Project work in primary schools. Unpublished master‟s thesis, National Institute of Education, Nanyang Technological University, Singapore.

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APPENDIX A Goal Management Item Location Goal Management Item 1 -0.12 Item 2 0.98 Item 3 -0.64 Item 4 0.40 Item 5 0.51 Item 9 0.11 Item 10 0.95 Item 11 0.38 Item 12 1.11 Item 13 -0.50 Item 14 0.27 Item 15 -0.25 Item 16 0.59 Item 17 -0.11 Item 18 0.83

SE

Residual

df

ChiSq

Prob

0.06 0.06 0.06 0.06 0.05 0.05 0.05 0.05 0.06 0.06 0.05 0.06 0.06 0.05 0.06

-0.48 -0.77 -1.06 -1.81 1.13 0.68 -1.10 0.04 0.08 0.16 0.50 -0.47 -1.19 -0.61 -1.10

723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66

3.81 9.69 6.77 21.97 9.05 5.60 11.83 3.95 14.03 2.92 5.01 12.98 15.38 9.53 16.95

0.92 0.38 0.66 0.01 0.43 0.78 0.22 0.91 0.12 0.97 0.83 0.16 0.08 0.39 0.04

Notes: 1. Location refers to the difficulty of the item on the linear scale. 2. SE refers to standard error, that is, the degree of the uncertainty in a value; in this case, the standard error for each of the items is low (0.05 to 0.06 logits). 3. Residual represents the difference between the expected value on an item, calculated according to the Rasch measurement model, and its actual value. 4. df (degrees of freedom) refers to the number of scores in a distribution that are free to change without changing the mean of the distribution. 5. ChiSq means chi-square. 6. Prob means probability, and refers to the levels of certainty to which an item fits the measurement model, based on its chi-square. 7. All the numbers are given to two decimal places because the errors are to two decimal places.

Goal Management has three stem-items ordered conceptually under Expectations, three stem-items conceptually ordered under Interest and three conceptually ordered under Goal Setting. The first result to note is that three of the 18 items did not fit the measurement model.

Choe Kee Cheng and Russell Waugh

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APPENDIX B Fit of Items based on the Self Management aspect of Student Receptivity to Project Work Item Location Self Management Item 19 -0.87 Item 20 -0.13 Item 25 -0.77 Item 28 1.14 Item 29 -0.26 Item 31 -0.43 Item 32 0.67 Item 34 0.46

SE

Residual

df

ChiSq

Prob

0.06 0.05 0.06 0.05 0.05 0.06 0.05 0.05

0.05 -1.43 -0.91 0.91 2.43 -0.19 -0.83 0.02

723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66

10.22 10.12 11.64 15.10 12.84 9.76 6.68 12.16

0.33 0.34 0.23 0.09 0.17 0.37 0.67 0.20

APPENDIX C Fit of Items Based on the Learning Styles Aspect of Student Receptivity to Project Work Item Location Learning Styles Item 39 -0.08 Item 42 0.60 Item 44 -0.74 Item 46 -0.18 Item 49 -1.04 Item 50 -0.04 Item 51 -0.99 Item 52 -0.19 Item 54 0.18

SE

Residual

df

ChiSq

Prob

0.05 0.06 0.06 0.05 0.06 0.05 0.06 0.05 0.05

1.37 0.29 1.47 0.60 -0.31 2.40 -0.61 2.12 -0.02

723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66

10.25 4.62 14.25 5.54 8.21 11.71 6.26 10.84 8.74

0.33 0.87 0.11 0.78 0.51 0.23 0.71 0.29 0.46

APPENDIX D Fit of Items based on the Collaboration aspect of Student Receptivity to Project Work Item Collaboration Item 56 Item 58 Item 59 Item 62 Item 66 Item 67 Item 68 Item 70 Item 72

Location

SE

Residual

df

ChiSq

Prob

0.44 0.60 0.11 -0.88 0.00 -0.47 0.05 0.34 0.51

0.06 0.06 0.05 0.06 0.06 0.06 0.06 0.05 0.05

-0.05 -0.29 0.68 0.40 0.68 -0.47 -0.77 -1.06 -0.55

723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66

9.66 6.34 9.53 7.12 11.21 2.92 8.28 11.00 4.90

0.38 0.71 0.39 0.62 0.26 0.97 0.51 0.28 0.84

A Rasch Measure of Student Receptivity to Project Work at a Junior College…

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APPENDIX E Fit of Items based on the Knowledge Application aspect of Student Receptivity to Project Work Item Location Knowledge Application Item 73 -0.77 Item 74 0.15 Item 76 0.31 Item 77 -0.72 Item 78 0.33 Item 80 -0.24 Item 81 -0.95 Item 82 -0.12 Item 84 0.06 Item 85 0.15 Item 86 1.06 Item 87 0.49 Item 89 0.46 Item 90 1.33

SE

Residual

df

ChiSq

Prob

0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.05 0.05 0.05 0.05 0.05

-1.22 -0.48 0.23 -1.06 0.62 0.64 -1.27 0.03 -0.18 0.88 0.27 2.64 2.18 1.24

723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66

8.44 8.87 4.14 10.82 14.35 4.40 12.60 6.70 3.50 10.72 7.95 15.93 10.71 9.74

0.49 0.45 0.90 0.29 0.11 0.88 0.18 0.67 0.94 0.30 0.54 0.07 0.30 0.37

APPENDIX F Fit of Items based on the Communication aspect of Student Receptivity to Project Work Item Communication Item 92 Item 94 Item 96 Item 98 Item 100 Item 102 Item 104 Item 105 Item 106 Item 107

Location

SE

Residual

df

ChiSq

Prob

-0.38 -0.70 -0.24 -0.69 -0.45 -0.29 0.33 -1.04 0.59 -1.20

0.06 0.06 0.06 0.06 0.06 0.06 0.05 0.06 0.05 0.07

-0.02 -1.12 0.35 0.48 0.38 0.56 0.94 0.03 0.79 -0.25

723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66 723.66

6.85 14.40 8.91 4.00 2.81 4.91 4.96 12.71 5.60 10.80

0.65 0.11 0.45 0.91 0.97 0.84 0.84 0.18 0.78 0.29

In: Specialized Rasch Measures… Editor: Russell F. Waugh, pp. 23-47

ISBN: 978-1-61668-032-9 © 2010 Nova Science Publishers, Inc.

Chapter 2

RASCH MEASURES FOR SPORTS, DRAMA AND MUSIC STUDENT SELF-VIEWS BASED ON GARDNER INTELLIGENCES Ahdielah Edries1 and Russell F. Waugh2 1

2

Australian Islamic College Faculty of Education and Arts; Edith Cowan University Mount Lawley; Western Australia

ABSTRACT A co-educational Independent Australian Islamic College has three campuses which cater for migrant students from war-torn countries and others with culturally and linguistically, diverse backgrounds. This paper is part of a larger study to identify the strengths and interests of Islamic students, across eight of Gardner‟s intelligence domains, as perceived by the students, so that the College could better meet the needs of these students. This study is important for the Islamic College because it is hoped that the study will lead to the provision of opportunities for students to increase their confidence, self-esteem and motivation, and to achieve better in academic and non-academic areas. Student self-views were based on three aspects: (1) Things I really like; (2) Things I enjoy; and (3) Things I prefer, with items answered in two perspectives What I would like to do and What I actually do. This paper reports a Rasch analysis of student self-views based on three Gardner Intelligences: Sports, Drama and Music (N=321). All 12 items fitted the measurement model for Sports Self-Views, 9 out of 12 items for Drama Self-Views and all 12 items for Music. For all items, students found it easier to say what they would like to do than to actually do it. The item-trait interaction chi-squares are respectively: x2 =69.56, df=48, p=0.02; x2 =43.39, df=36, p=0.41and x2 = 52.85, df = 48, p= 0.29 showing no significant interaction between student measures and item difficulties along the scale, thus supporting uni-dimensional scales. The Person Separation Indices are respectively 0.88, 0.89 and 0.88 with standard errors of about 0.10 logits showing acceptable separation of measures compared to errors, and improvements could be made by adding more items to all measures.

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BACKGROUND FOR CHAPTERS TWO AND THREE Three campuses of an Australian Islamic College cooperate and work together by following the same policies to provide a holistic education for all the students by integrating Islamic values and information technology in all the subject matter. The first campus opened its doors in February, 1986. This campus was established with two teachers and 50 students and it delivered an academic education based on a framework of Islamic ethos and values. Its students have now been incorporated into three main campuses. An addition was constructed in 1990 with its Technological Centre being developed in 1994. Its total enrolment in 2009 is 550 students. The second campus was established in 1996 and a new double-storey building on the school ground was opened in March 2003 to accommodate the increasing number of enrolments. This college caters for students between Kindergarten and Year 10 (5 to 15 years old), total enrolment in 2009 is 708 students. The third campus is the most recent addition and this was purchased in 2000 to cater for the increasing number of Islamic high school students. This College offers Kindergarten through to Year 12 and has been successful in producing graduates who have entered tertiary education in recent years (N=120). This year (2009), the three campuses of the Australian Islamic College have approximately 2300 students, over 200 teachers and supporting staff (Magar, 2008 p. 13). Small class sizes in the Colleges enable students and teachers to interact more efficiently and productively then they could with larger classes. Class sizes range from around 25-30 students in Lower Primary with a Teacher and Teacher Aide per class (Kindy – Year 1) to 20-25 students in Middle Primary and in High School (Edries, 2008). The philosophy, Islamic Values and Academic Excellence for Your Children’s Success in this Life and the Hereafter, sums up how the Islamic Colleges govern themselves and educate their students. All classroom curricula are based on the Western Australian Curriculum Framework (Curriculum Framework, 2008), outcome-based teaching and learning where teachers plan, conduct lessons and assess their students through outcomes. Portfolios, fortnightly assessments, formal testing (English literacy and numeracy), National English Testing (National Assessment Program Literacy and Numeracy), Interschool Competitions (University of New South Wales) and progress maps are used to record students‟ academic performance throughout their education at the Colleges. Islamic values are integrated in all the different subject areas, thus allowing students to learn classroom concepts and relate them to life and their faith. The students are encouraged to excel in secular and non-secular subjects, integrate with other communities and strive towards a goal in life. The Islamic Colleges encourage academic excellence in all areas, as well as good behaviour and conduct through a reward system amongst the students where positive points may be accumulated to be exchanged for a prize at the end of a week (Edries, 2008). On the other hand, a negative-point system also exists to discourage anti-social behaviour and to monitor the students‟ conduct through parent-teacher conferences and shared concerns. The school conducts several inter-school visits with neighbouring government and private schools, where students meet to exchange ideas about religion and form friendships outside their Islamic communities. Other types of visits include celebratory occasions such as Harmony Day, and Inter-Faith Sports for Peace Carnival (Edries, 2008). In the primary Islamic colleges, students attend English, Mathematics, Science, Society &

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Environment lessons. In addition to these core subjects, the students have to attend the LOTE (Arabic), Islamic Studies, Sports, Health, Art and Computing classes (Deria, 2006a). Each college day includes six, 45-50 minute periods, a recess and lunch break, prayers after lunch and morning and afternoon assemblies. Most of the teachers are trained in Australia and they are encouraged to utilise the best and latest pedagogy in their classrooms, share resources and promote the best teaching methods at the weekly meetings after school (Deria, 2006a). In the secondary Islamic colleges, qualified teachers are employed to give the students the best education possible, especially in Senior School (Deria, 2006b). The boys and girls are separated into single-sex classes and are taught by male and females teachers respectively. The aim is to discourage any distraction in class and allow more comfortable interactions between staff and students than might be possible with mixed boys and girls classes. Secondary college subjects taught at Middle school level are English, Mathematics, Science, Society & Social Environment, Computing, Art, Health, Sports and Islamic Studies (Deria, 2006b). In the senior years (Years 11 & 12), the subjects vary a little depending on student choices and numbers; examples of Tertiary Entrance Ranked subjects include LOTE-Arabic, English, Senior English, English as a Second Language (ESL), English Literature, Geography, History, Political and Legal Studies, Calculus, Geometry & Trigonometry, Physics, Chemistry, Biology, and Human Biology (Deria, 2006c). Each College runs its own education support system where the educational and emotional needs of the students are catered for by a school psychologist through consultations and testing. Education support classes are provided for the weaker students from each year group and, while gifted and talented classes are offered for the advance students of each year group, resources are limited and the Colleges would like to do more in this regard (Edries, 2008). Behaviour management consultations (such as the National Safe School‟s Friendly Policy, 2008) are provided where needed. Although all the students share the same faith of Islam, they originate from various cultural groups. A small percentage (20%) of students‟ are Australian-born (first generation Australians) and the next largest group (30%) of students was born in Iraq and Somalia. The remaining students (50%) come from all the continents of the world such as Africa, America, Asia, Europe and the Pacific Islands. Due to this diversity of cultures, the students speak a variety of languages and dialects, as well as conforming to different customs in areas of community life. The wide variety of cultural groups in the Australian Islamic College has positive aspects and limitations. Such a diverse group of students in one College makes the teaching of diversity, multiculturalism and tolerance easier because the students can learn from each other and they bring their own „voice‟ to the classroom. Celebrations of multiculturalism brings a deeper meaning when the students come dressed in their cultural clothing, bring home-cooked food of their nations and share experiences from their country of origin. However, such a diverse cultural mix also has its problems because of difficulties in communication, low English literacy acquisition and cultural conflicts. College Parent-Teacher conferences require translators and newsletters sent home have to be translated in other languages such as Bosnian, Somali, Arabic and Afghan (Edries, 2009). The Islamic Colleges also spend a lot of funds purchasing and maintaining literacy and numeracy programs that are IT-based for every classroom, such as the Accelerated Reader and Accelerated Mathematics Programs (Edries, 2008). These programs encourage users to take some charge of their learning and become independent learners to a greater extent. The Accelerated Reader Program is linked to the

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Ahdielah Edries and Russell F. Waugh

Colleges‟ library catalogues and students borrow a book every week to read and then sit for a computer quiz that assesses their reading level and directs them to another reading level book. The Accelerated Mathematics Program is a collection of Mathematics topics (categorized by year groups) that the computer produces, such as exercises, tests and revision questions for students to complete and that are later marked through the computer program (Edries, 2008). Every student has a different Mathematics level and can progress at his or her own pace, thus making their learning less stressful and more positive. These programs are conducted as extension exercises to the students‟ usual study load. The Australian Islamic Colleges have New Arrivals classes that are an essential starting point for many of its migrant students. In these classes, specialist teachers help to integrate the students into the formal education system and provide the basic English literacy skills to cope with their age-level classes in six months to a year‟s time. However, this time duration isn‟t sufficient for most of these students to achieve at the mainstream standard. They often find it very difficult to cope at the standard of the mainstream, not because they are academically incapable, but because they simply haven‟t mastered the English language yet (Edries, 2009; personal observation as Principal). Many of the classroom teachers have to adapt their pedagogy to suit the multiple student needs in their classrooms. In the Islamic College mainstream classrooms, the assumption that all the students who have come from overseas, or government schools in Perth, have basic English literacy and numeracy skills is challenged when these students still fall into the lower percentile of academic performance compared with their same year level peers across Western Australia. In addition to the usual pressure to produce excellent results nationally (Tertiary Entrance Ranking table), the College also has to contend with national testing such as the National Assessment Program in Literacy and Numeracy (previously known as Western Australian Literacy and Numeracy Assessment) each year (Edries, 2008). The Islamic Colleges have taken this challenge to improve English Literacy and Numeracy standards by adapting their hiring policy to encourage very motivated teachers, and by retaining excellent staff through meritocracy-based salary reviews (Magar, 2008a). Handling such a dependent and disadvantaged group of students (and their families) is also draining on the resources of the Australian Islamic College as there are insufficient funds and staffing to implement all that is considered necessary. Computers are essential tools for the IT-based programs and the growing number of students will require more access to up-todate computers and computer programs. The computer programs are expensive and have to be maintained by support staff, thus adding further staffing costs. To prepare the students for national testing and general school work, the Islamic Colleges also run after-school, weekend and holiday classes that see committed teachers volunteer their time and effort to help the students achieve a better understanding of their school work. All these mean that the Colleges run almost all day, every day, and this accrues high energy costs (Edries, 2009; personal observation as Principal). Cultural conflicts and misunderstandings are unavoidable in the Islamic Colleges, especially when new students and families arrive in Australia for the first time. These people bring their own conceptions about education and social interactions, and handling confrontations is an acquired skill at the Colleges. The administration, Heads of schools and teachers have to deal with these interactions sensitively to avoid escalation in conflict (Edries, 2009; personal observation as Principal). Education Support services are essential in this area but often there is a lack or unavailability of resources, including interpreters and mediators.

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The Islamic Colleges often have to rely on external services offered by independent organizations, or small government funded associations, to assist with helping these new families adapt to new life in Australia and therefore ease their children into the Australian school system. Families who come from war-torn countries, or who have lived in refugee camps for several years, tend to arrive with children who have emotional and social difficulties at school thus affecting their already limited schooling and literacy acquisition (Edries, 2009; Haig & Oliver, 2007). Generally, the Intensive English classes are small in number (N=15) and the students are given quality time, thus affording them the opportunity to make some progress. However, some students still experience great difficulty in their learning, and continue to struggle throughout the mainstream classes (Edries, 2009). This is why interventions involving programs based on multiple intelligence theory could be important to the Islamic College students. If student abilities across Gardner‟s intelligences can be ascertained, then programs can be tailored to give each student some success in at least one subject, involving at least one of the intelligences.

RATIONALE FOR THE STUDY There are few significant studies (Haig & Oliver, 2007) which have dealt with the attitudes and needs of students from cultural and linguistically diverse (CALD) backgrounds in Australia (especially migrants and refugees from war-torn countries); and no recent studies in Australia could be found that have investigated how Australian schools could effectively improve and enhance the learning of these migrant students. Since the majority of the students within the Islamic Colleges are born overseas (particularly from war-torn countries), or live in families with parents born overseas, the students‟ English literacy and numeracy limitations have to be addressed by any Islamic College, as little help comes from the homes. In particular, research has never been undertaken in the Western Australian Islamic Colleges to investigate the intelligences, strengths and interests of the students, and the teacher perceived needs of their students. Cook-Sather (2002, p.3) stated that there are certain pedagogical merits in understanding and utilising student perspectives in schools. Choe (2006) in his recent study in Singapore asserts that hearing and listening to student voices is important in education because of the various ways it can improve educational practices, reinform existing conversations about educational reform, and point to the discussions and reform efforts yet to be undertaken. Students‟ who feel engaged with what they are learning, feel empowered because their views are valued, or teachers who, in seeing the world from the students‟ point of view, adopt more effective instructional approaches to teach their students, but these statements arose from research with advantaged students, not migrant students with poor English and numeracy skills. Secondly, there is a need to address the issues that students from Culturally and Linguistically Diverse Backgrounds have to contend with in a traditional classroom. On a daily basis, the administration and staff at the Islamic Colleges grapple with issues such as helping students deal with their past traumatic experiences and their new way of life in Australia; the consequences of severely disrupted prior schooling; a lack of social understanding and learning strategies to process content; and poor literacy in their first

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Ahdielah Edries and Russell F. Waugh

language (which is needed to support the acquisition of a second language); meeting classroom demands for literacy and communication; obtaining appropriate learning resources; and a lack of parental support due to lack of education. According to Haig and Oliver (2007) these deficiencies are compounded by the complexity and specificity of cognitive academic knowledge used in schools. Cognitive development which has taken place over many years in the classroom is clearly one of the key elements which students from Non-English speaking backgrounds are missing when they have had interrupted (or no prior) schooling (Miller, Mitchell, & Brown, 2005). This makes it difficult for staff at the Australian Islamic College to target learning at the right level, as students in their classes have variable skills and gaps in their learning. It is hoped that the current study will provide some information to develop and implement effective school and educational practices to cater for its students at the Islamic colleges in Perth. Student learning, motivation and engagement in a variety of academic and non-academic areas are important for success at school (Bragg, 2005). Whilst there are numerous factors that impact on the wellbeing, learning and teaching of students, it is not in the scope of the present study to investigate all of these. The present study will only focus on the need to determine student interests in relation to some Gardner intelligence areas and their perceived interests and strengths. Gardner views "intelligence" as a biological and psychological potential that is capable of being realized to a greater or lesser extent in everyone, depending on one's experience, education, social environment, and other factors (Gardner, 1993a). The majority of students attending the Islamic Colleges in Perth overcome obstacles such as adjusting to their host country, facing confronting rules and issues in school (something to which a large proportion of these students have not been exposed to in their own countries); and, coupled with compliance to learning from a traditional curriculum (which often does not take into account their prior background and past experiences) makes school difficult for them. When students‟ views, strengths and interests are known, action can be taken to address difficulties that students might be facing and further develop their strengths. Consequently, this study attempts to use students‟ and teachers‟ perspectives on what can be done to enhance student learning, and how their needs can be effectively met within the school environment. Unidimensional, linear scales for self concepts relating to Interpersonal, Intrapersonal, English, Mathematics, Art, Sport, Music, and Drama domains were created using a Rasch Measurement Model computer program, RUMM 2020 (Andrich, Sheridan & Luo, 2005). This paper reports on the RUMM 2020 output, that is, the statistics showing a good fit to the measurement model for the data relating to the Interpersonal and Intrapersonal domains. These domains were chosen to present first because they are important to students who come from war-torn countries like Lebanon, Somalia, Iraq and Ethiopia. While the academic subject self concepts and sports self-concepts are important too, it was considered, more important to be able to make valid inferences about student Interpersonal and Intrapersonal self-concepts first. Rasch measures for student self-views based on other Gardner Intelligences were made as part of a larger study, but not reported here.

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GARDNER INTELLIGENCES Definitions of Gardner’s Intelligence Domains The Intelligence Domains are based on Gardner‟s Multiple Intelligence Theory (1993, 1999, 2004) and the definitions below are taken from his work. Linguistic Intelligence refers to the capacity to use words effectively, whether orally or in writing. This includes the ability to manipulate the structure and syntax of language, the sounds of language, the meanings of language, and the practical uses of language, such as for explaining, remembering, and persuading. This intelligence is evident in children who demonstrate strength in the language arts: speaking, writing, reading, listening. They learn best by saying, hearing and seeing words and they are good at memorizing names, places, dates and trivia. These students have always been successful in traditional classrooms because their intelligence lends itself to traditional teaching. Logical/Mathematical Intelligence refers to the capacity to use numbers effectively and to reason well. This includes awareness of logical patterns and relationships, functions, and cause and effect. This intelligence is evident in children who display an aptitude for numbers, reasoning and problem solving. They like to participate in experiments, figure things out, ask questions, explore and discover patterns and relationships. They learn best by categorizing, classifying, and working with abstract patterns. These children typically do well in traditional classrooms where teaching is logically sequenced and students are asked to conform. Visual/spatial Intelligence refers to the ability to perceive the visual and spatial world accurately, including sensitivity to colour, line, shape, form, space, and the relationships between them. Includes the capacity to visualize, make graphic representations, and orient oneself in spatial surroundings. This intelligence is evident in children who learn best visually and organise things spatially. They like to see what you are talking about in order to understand. They enjoy charts, graphs, maps, tables, illustrations, art, imagining things, puzzles, costumes (basically anything that is eye catching). Students learn best by visualizing, dreaming, working with colours and pictures. Bodily/Kinesthetic Intelligence refers to the ability to use one's whole body to express ideas and feelings, and the ability to fashion or transform with one's hands. This includes skills such as coordination, balance, dexterity, strength, flexibility, speed, and other physical skills. This intelligence is evident in children who experience learning best through activity: games, movement, hands-on tasks, building. They learn best by touching, moving, interacting with space and processing knowledge through bodily sensations. These children are often labelled „overly active‟ in traditional classrooms where they are told to sit and be still. Musical Intelligence refers to the ability to perceive, distinguish between, and express oneself in musical forms. It includes sensitivities to rhythm, pitch or melody, timbre, and tone colour. It can apply to either an intuitive grasp of music, or an analytic or technical understanding of it, or both. This intelligence is evident in children who learn well through songs. They like to sing, hum tunes, listen to music, patterns, rhythm, play an instrument and respond to musical expression. These children are good at picking up sounds, remembering melodies, noticing pitches and rhythms and keeping time. It is easy to overlook children with this intelligence in traditional education.

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Interpersonal Intelligence refers to the capacity to perceive and distinguish differences in the moods, intentions, motivations, and feelings of others. It includes sensitivity to facial expressions, gestures, and body language. This intelligence also includes the ability to respond to these cues effectively, to work well with others, and to lead. This intelligence is evident in children who are noticeably people orientated and outgoing, and do their learning cooperatively. They are good at understanding people, leading others, organizing, communicating, manipulating and mediating conflicts. They learn best sharing, helping others and asking for help. These children may have typically been identified as „talkative‟ or „too concerned about being social‟ in a traditional setting. Intrapersonal Intelligence refers to the capacity for self-knowledge and understanding, and the ability to act on the basis of that knowledge. It includes having an accurate picture of one's own strengths and limitations, inner moods, intentions, feelings, motivations, needs, and desires, and a capacity for self-discipline and self-esteem. This intelligence is evident in children who are especially in touch with their own feelings, values, and ideas. They may tend to be more reserved, like to work alone and reflect on problems, but they are actually quite intuitive about what they learn and how it relates to themselves. They are good at being independent, and learn best by being given time to think. Naturalist Intelligence refers to the capacity of children to love the outdoors, to be with animals, and to go on field trips. They are good at categorizing, organizing a living area, planning a trip, preservation and conservation. Learns best by studying natural phenomenon in a natural setting learning about how things work. These children love to pick up on subtle differences in meanings. The traditional classroom has not been accommodating these children. Existentialist Intelligence refers to the capacity of children to learn in the context of where humankind stands in the „big picture‟ of existence. They ask, “Why are we here?” and “What is our role in the world?” This intelligence is seen in the discipline of philosophy. Gardner (1999) suggested that the nine intelligences very rarely operate independently. Rather, the intelligences are used concurrently and typically complement each other as individuals develop skills or solve problems. Viewed in this way human intelligence is not restricted to only the more narrow linguistic and mathematical abilities measured by the common standardized tests in which high scores traditionally described students in school as being „smart‟. There are research studies that confirm that by addressing students‟ culture, language, and social status with appreciation, inclusion, and sensitivity increases students‟ academic successes (Grant & Tate, 1995; Jimenez, 1997). This chapter reports on the RUMM 2020 output and data analysis for the data relating to Sports, Drama and Music. This analysis features important inferences that can validly be made from the reliable Rasch-created linear measures, describes the RUMM 2020 output for item difficulties (easiest and hardest), and the weakest 25 student measures for Sports, Drama and Music Self-Concepts. Data from the targeting graphs by gender enable valid inferences to be made by identifying the easy and hard items and the weak students who may require intervention. A summary of main findings is listed at the end of this chapter. Suggested improvements and implications for helping and improving students are not discussed here in order that the actual results can be separated from the discussion and implications which are presented in the last thesis chapter.

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INITIAL RASCH ANALYSIS The RUMM 2020 computer program (Andrich, Sheridan & Luo, 2005) was used to analyse the 12 items for each of the Self-Concepts for Sports, Drama and Music, separately, based on the Partial Credit Model of Rasch (Masters, 1997). While the 12 items for the Sports Self-Concept and Music Self-Concept fitted the measurement model, only nine items fitted the measurement model for Drama Self-Concept. For Drama Self-Concept, items 5, 6 and 10 were deleted and the RUMM analysis repeated. For the Music Self-Concept, the response categories for items 9 and 11 were collapsed from three to two because the two higher response categories were not answered consistently and logically, and the RUMM analysis repeated.

Final Rasch Analysis The results of the final Rasch analyses are now presented for the Self-Concepts of Sports (12 items), Drama (9 items) and Music (12 items).

Dimensionality The item-trait chi-square values for the Self-Concepts of Sports, Drama and Music are, respectively , x2 = 69.56, df=48, p=0.02; x2 = 43.39, df=36, p=0.41; and x2 = 52.85, df=48, p=0.29. These mean that there is good agreement (less so for Sports Self-Concept) amongst the students about the difficulties of the items along scales and that there is a good fit of data to the measurement model involving a unidimensional trait (less so for Sports Self-Concept) Reliability The standard errors of measurement were about 0.1 logits for Sports, Drama and Music Self-Concepts, and the Person Separation Indices are 0.88, 0.89 and 0.88 respectively, indicating excellent separation of measures in comparison to the errors. These indicate that the measures are well separated in comparison to the errors. Table 1. Reliability Indices for Sports, Drama and Music Self-Concept Sports Drama Music

Standard Error of Measurement 0.10 0.10 0.10

Person Separation Index 0.88 0.89 0.88

Power of test-offit Excellent Excellent Excellent

Item Fit to the Measurement Model The RUMM 2020 program calculates individual item fits to the measurement model and these are given in Tables 2, 3 and 4 respectively for the Sport, Drama and Music Self-Concepts. The tables are accompanied by some explanatory notes relating to individual item fits to the measurement model. From the analysis it was established that for Sport self-concept, 12 out of

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12 items fitted the measurement model with a probability greater than p=0.01 and, the power of test-of-fit is excellent, based on a Person Separation Index of 0.88. For the Drama SelfConcept, all nine items fitted the measurement model with a probability of greater than p=0.05 and, the power of the test-of-fit were excellent, based on a Person Separation Index of 0.89. For Music Self-Concept, all 12 items fitted the measurement model with a probability greater than p=0.04 and, again the power of the test-of-fit were excellent, based on a Person Separation Index of 0.88. Table 2. Fit of items to Rasch Measurement Model (Sport Self-Concept) Item 1 2 3 4 5 6 7 8 9 10 11 12

Location -0.59 -1.21 -0.11 -0.71 1.31 0.99 0.02 -0.41 0.78 0.16 0.19 -0.32

SE 0.11 0.13 0.10 0.11 0.09 0.09 0.10 0.10 0.09 0.09 0.09 0.10

Residual -1.36 -1.02 -1.81 -0.86 1.54 2.35 -1.15 -0.82 -0.17 -0.21 0.38 0.90

df 241.00 241.00 241.00 241.00 241.00 241.00 241.00 241.00 241.00 241.00 241.00 241.00

Chi-Square 14.51 1.24 8.92 27.2 3.50 13.00 5.21 10.05 3.91 4.04 0.92 1.55

Probability 0.01 0.87 0.06 0.61 0.48 0.01 0.27 0.04 0.42 0.40 0.92 0.82

Table 3. Fit of items to Rasch Measurement Model (Drama Self-Concept) Item 1 2 3 4 5 6 7 8 9 10 11 12

Location +0.80 -0.48 +0.39 -0.75 N/F N/F +0.30 -0.69 +0.49 N/F -0.57 -0.38

SE 0.10 0.09 0.10 0.09 N/F N/F 0.09 0.09 0.09 N/F 0.09 0.09

Residual -1.46 -0.44 0.07 -0.85 N/F N/F -0.67 0.35 0.64 N/F 0.11 -0.17

df 252.85 252.85 251.06 251.06 N/F N/F 252.85 252.85 251.95 N/F 252.85 252.85

Chi-Square 5.70 2.51 4.58 4.14 N/F N/F 3.33 3.42 5.78 N/F 1.42 1.97

Probability 0.22 0.64 0.33 0.39 N/F N/F 0.49 0.22 0.10 N/F 0.84 0.74

Explanatory Notes on Table 2, 3 & 4 1. Location refers to the difficulty of the item on the linear scale. 2. SE refers to standard error, that is, the degree of the uncertainty in a value: in this case, the standard error for each of the items is reasonable, ranging from 0.09 to 0.14 logits (for table 1) and 0.08 to 0.1 logits (for table 2) respectively. 3. Residual represents the difference between the expected value on an item, calculated according to the Rasch measurement model, and its actual value. 4. df (degrees of freedom) refers to the number of scores in a distribution that are free to change without changing the mean of the distribution. 5. Chi-square is the statistic used to determine fit to the measurement model.

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6. Probability, refers to the levels of certainty to which an item fits the measurement model, based on its chi-square. 7. N/F means no fit (hence, the item was deleted) 8. All values are given to two decimal places because the errors are to two decimal places. Table 4. Fit of items to Rasch Measurement Model (Music Self-Concept) Item

Location

SE

Residual

df

Chi-Square

Probability

1

-0.67

0.09

0.10

268.63

1.73

0.79

2

-1.05

0.10

-1.72

268.63

9.86

0.04

3

0.60

0.09

0.70

267.72

4.59

0.33

4

-0.54

0.09

1.37

267.72

3.56

0.47

5

-0.22

0.09

0.25

268.63

3.13

0.54

6

-0.59

0.09

-0.54

268.63

5.91

0.21

7

1.05

0.10

-0.37

267.72

1.15

0.89

8

-0.08

0.10

-1.52

268.63

5.64

0.23

9

0.91

0.14

-0.21

268.63

3.28

0.51

10

-0.11

0.09

0.90

267.72

9.53

0.05

11

0.58

0.13

0.39

268.63

1.89

0.76

12

0.12

0.09

0.10

267.72

2.58

0.63

Item-Person Fit Interactions The fit residual data for both items and students (Self-Views in Sports) has a mean near zero and a standard deviation near one showing that the data fit the measurement model satisfactorily and this means that there is a good consistency of student-item response patterns (see Table 5). The fit residual data for Drama Self-Views (see Table 6) and for Music SelfViews (see Table 7) similarly show good consistency of student-item response patterns. Table 5 Global Item and Person Fit to the Measurement Model for Sports Self-Views Item Locations

Item Residuals

Student Locations

Student Residuals

Mean

0.00

-0.19

1.27

-0.22

SD

0.73

1.26

1.29

1.01

Table 6. Global Item and Person Fit to the Measurement Model for Drama Self-Views Item Locations

Item Residuals

Student Locations

Student Residuals

Mean

0.00

-0.23

0.24

-0.22

SD

0.83

0.68

1.53

1.03

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Table 7. Global Item and Person Fit to the Measurement Model for Music Self-Views Item Locations

Item Residuals

Student Locations

Student Residuals

Mean

0.00

-0.05

-0.19

-0.17

SD

0.67

0.91

1.36

1.04

Explanatory Notes on Tables 5, 6, & 7 1. Item location is item difficulty in logits 2. Person location is person measure in logits 3. SD is standard deviation 4. The mean item difficulty is constrained to zero by the RUMM 2020 program 5. Fit residuals are the difference between the actual values and the expected values calculated according to the measurement model (standardised the data fit the measurement model (a good fit for these data). They have a mean near zero and an SD near 1 when the data fit the measurement model (as is the case for these data). 6. All values are given to two decimal places because the errors are to two decimal places.

TARGETING

Notes on Figures 1, 2 and 3 1. The scale is in logits, that is, the log odds of answering the response categories. 2. Person measures (low to high) are given on the upper side in logits. 3. Item thresholds (easy to hard) are given on the lower side in logits. Figure 1. Person measure/Item Threshold Graph for Sport Self-Views.

It can be seen from the distribution of items in Figure 1 that the item difficulties mostly cover the middle range (-1.6 to +1.4 logits) of Sports Self-View measures. This indicates that some easier and harder items need to be added in any revision of the scale to cover students

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with the lowest measures (-2.73 logits) and students with the highest (+3.41 logits) measures. Eight students (of which seven were female) had a very low Sports Self-View, their raw scores ranging from 0 (extremely low) to a score of 7 for the lower end of the scale. Similarly, fifty-six students (nine females compared with their (47) male counterparts) had an extremely high self-view of Sport with a raw score of 24. Hence, targeting could be improved by adding some easy and hard items appropriate for students with the lowest and the highest measures and this would improve the measure.

Note: The Drama Self-View measures are on the upper side from low to high and the item thresholds are on the lower side of the graph from easy to hard. Figure 2. Person measure/Item Threshold Graph for Drama Self-Views.

Figure 2 shows that two-hundred and twenty-six students (102 males and 124 females) out of a total of three-hundred and twenty-one students were able to answer the items positively relating to Drama Self-Views (that is, the item difficulties mostly covered the middle range (-1.8 logits to +1.4 logits) of Drama Self-View measures. Thirty-two students (twenty-six males and six females) were in the lower range (-1.8 logits to -3.22 logits) of the scale with scores ranging from 0 (extremely low) to a score of 2. In contrast, sixty-three students (twenty males compared with their (43) female counterparts) were in the higher range (+1.4 logits to +3.24 logits) of the scale with scores ranging from 12 to 19 (extreme high), indicating that some easier and harder items need to be added in any revision of the scale to cover the students with these measures.

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Figure 3. Person measure/Item Threshold Graph for Music Self-Views.

Figure 3 shows the item thresholds for Music Self-Views range from easy (about -1.4 logits) to reasonably hard (about +1.4 logits); and the student measures calibrated on the same scale from low (about -3.25 logits) to high (about 3.21 logits). Forty-seven students (thirtythree males and fourteen females) had a very low Self-Views of Music, their raw scores ranging from 0 (extremely low) to a score of 3 for the lower end of the scale. Similarly, thirtyfour students (seven males compared with their (27) female counterparts) had an extremely high Self-View of Music with a raw score of 22. These measures indicate that some harder items need to be added to better target the attitudes and behaviours of students with high measures (that is, students who obtained scores from 14 to 22).

INFERENCES FROM THE LINEAR SCALES Since there was a good or reasonable fit to the measurement model for all three Self-Views, valid inferences can now be made about item difficulties and student measures on Sports, Drama and Music Self-Views. As stated previously, there is a prediction that the attitude items should be easier than the behaviour items and, where both fit the model, attitude items are easier than their corresponding behaviour items. Where attitudes are easier than their corresponding behaviors, it can be inferred that attitudes influence corresponding behavior. Tables 8, 9, and 10 below show the Item Difficulties for Student Sport Self-View, Drama Self-View, and Music Self-View respectively.

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Table 8. Item Difficulties for Student Sport Self-View Item

1 2 3 4 5 6

SPORT SELF-CONCEPT Things that I really like Playing sports Playing competitive games and sports Things I enjoy Watching sporting programs on TV like Fox Sport Being involved with sports and games Things I prefer Playing in sports teams where others watch me Playing ball games where I have to react fast

What I actually do

What I‟d like to do

-0.59 -0.11

-1.21 -0.71

+1.31 +0.02

+0.90 -0.41

+0.78 +0.19

+0.15 -0.32

Table 9. Item Difficulties for Student Drama Self-Views Item

DRAMA SELF-CONCEPT Things that I really like Being involved in drama classes Acting out different scenes and characters Things I enjoy Watching movie stars act out scenes and characters in movies Being involved with acting and plays Things I prefer Acting in plays where others watch my presentation Acting in musicals, comedies and plays

1 2 3 4 5 6

What I actually do

What I‟d like to do

+0.81 +0.39

-0.47 -0.75

N/F

N/F

+0.30

-0.69

+0.49 +0.50

N/F -0.57

Table 10. Item Difficulties for Student Music Self-Views Item

1 2 3 4 5 6

MUSIC SELF-CONCEPT Things that I really Singing and listening to music Playing a musical instrument Things I enjoy Watching musical shows on TV such as Idol and Video Hits. Being involved with musical productions Things I prefer Playing a musical instrument in a band or group Writing and reading music

What I actually do

What I‟d like to do

-0.67 +0.60

-1.05 -0.53

-0.22

-0.59

+1.05

-0.08

+0.91 +0.58

-0.12 +0.12

Explanatory Notes for table 8, 9, and 10 N/F means no fit (hence, the item was deleted). All values are given to two decimal places because the errors are to two decimal places.

ITEM DIFFICULTIES For the Sport Self-View attitudes (What they would like to do), students found it very easy to say that they would like to play sports (difficulty -1.21 logits), and they found it moderately easy to say that they would like to play competitive games and sports (difficulty -0.71 logits).

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Students found it moderately hard to have an atittude that they would prefer to play in sports teams where other people watch them (difficulty +0.15 logits), and very hard to have an attitude that they like to enjoy watching sporting programs like „Fox Sport‟ on TV (difficulty +0.90 logits). In contrast, for actual behavior (What they actually do), students found it moderately easy to actually enjoy playing competitive games and sports (difficulty -0.11 logits). They found it moderately easier to actually play sports without competition (difficulty -0.59 logits). Students found it hard to actually play in sports teams where they are being watched by others (difficulty +0.78 logits), and very hard to say that they actually enjoyed watching sporting programs like „Fox Sport‟ (difficulty +1.31 logits). For the Drama Self-View attitudes (What they would like to do), students found it very easy to say that they would like to hold an attitude to like acting out different scenes and characters (0.75 logits). They found it moderately easy to have an attitude that they would like to enjoy being involved with acting and plays (difficulty -0.48 logits), and moderately easy to have an attitude that they prefer to act in musicals, comedies and in plays (difficulty -0.57 logits). In contrast, for actual behavior (What they actually do), students found it moderately hard to actually enjoy being involved in acting and plays (difficulty +0.30 logits), and moderately hard to actually enjoy acting out different scenes and characters (difficulty +0.39 logits). Students found it moderately hard to actually enjoy acting in musicals, comedies and plays (difficulty +0.50 logits), and very hard to be involved in Drama classes (difficulty +0.80 logits). For Music Self-View attitudes (What they would like to do), students found it very easy to say that they would really like to sing and listen to music (difficulty -1.05 logits), moderately easy to say that they would like to enjoy watching musical shows on TV like, „Idol‟ and „Video Hits‟ (difficulty -0.59 logits), and moderately easy to say that they would like to enjoy being involved with musical productions (difficulty -0.08 logits). They found it moderately hard to say that they would like to prefer to read and write music (difficulty +0.12 logits). In contrast, for actual behavior (What they actually do), they found it easy to actually like singing and listening to music (difficulty -0.67 logits) , and moderately easy to actually watch musical shows on TV like, „Idol‟ and „Video Hits‟ (difficulty -0.22 logits). Students found it hard to actually write and read music (difficulty +0.58 logits), hard to actually play a musical instrument (difficulty +0.60 logits), and very hard to actually play a musical instrument in a band or group (difficutly +0.91 logits) and very hard to actually be involved with musical productions (difficulty +1.05 logits).

LOWEST STUDENT MEASURES OF SELF-VIEWS The lowest 25 student measures were identified for Sports, Drama, and Music Self-Views (see Table 11, 12 and 13 respectively). These are the students who could be given extra support with these self-concepts to encourage them to take part in school activities better, to gain selfconfidence and to improve their knowledge and abilities. The 25 number cut-off for the lowest measures is somewhat arbitrary, but the highest measure in each of these tables represents answers of „some of the time‟, which represents a low self-view in Sports, Drama and Music respectively. A discussion of what could be done to help these students is given in the last chapter but, in the present chapter, the weakest students will be identified.

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Comments on the 25 students who have been identified as having a low Sport Self-Views (Table 11), a low Drama Self-View (Table 12), and a low Music Self-View (Table 13) are now given. The majority of students in the bottom twenty-five were females (20) compared to their male (5) counterparts. The raw scores indicate that these females have a very low Sport Self-Concept and might require urgent intervention. The above results are indicative that males have different perceptions of Sport than females, and this needs to be reviewed. Four students (2 males and 2 females) have the highest measure of 10, which represents answers of „some of the time‟ to only 10 of the 24 items, representing a low self-view in Sport. Table 11. Students with Lowest 25 Measures for Sport Self-View ID 237 45 145 153 14 243 228 233 241 91 165 108 2 49 46 40 231 62 3 242 52 55 251 299 11

Raw Score 1 3 3 4 5 6 6 7 8 8 8 8 8 8 9 9 9 9 9 9 9 10 10 10 10

Student Measure -2.73 -1.81 -1.81 -1.51 -1.26 -1.04 -1.04 -0.84 -0.66 -0.66 -0.66 -0.66 -0.66 -0.66 -0.48 -0.48 -0.48 -0.48 -0.48 -0.48 -0.48 -0.31 -0.31 -0.31 -0.31

SE 0.85 0.59 0.59 0.53 0.49 0.47 0.47 0.45 0.43 0.43 0.43 0.43 0.43 0.43 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42

Residual -0.29 -0.02 0.76 1.18 -1.57 0.84 -0.35 -0.35 1.16 0.78 -0.34 -0.11 -1.62 0.56 0.27 -0.96 -0.82 -0.04 1.09 0.34 0.72 1.87 0.87 0.05 -1.63

Table 12. Students with Lowest 25 Measures for Drama Self-View ID 37 230 188 215 1 181 285 196 226 123 302 180

Raw Score 0 0 0 0 0 0 0 0 0 1 1 1

Student Measure -3.22 -3.22 -3.22 -3.22 -3.22 -3.22 -3.22 -3.22 -3.22 -2.38 -2.38 -2.38

SE 1.23 1.23 1.23 1.23 1.23 1.23 1.23 1.23 1.23 0.88 0.88 0.88

Residual -0.90 -0.90 0.41

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Table 12. (Continued). ID 257 267 179 197 175 172 250 318 279 272 299 222 221

Raw Score 1 1 1 1 1 1 1 1 1 1 2 2 2

Student Measure -2.38 -2.38 -2.38 -2.38 -2.38 -2.38 -2.38 -2.38 -2.38 -2.38 -1.81 -1.81 -1.81

SE 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.69 0.69 0.69

Residual -0.90 -0.90 -0.90 -0.34 -0.58 -0.90 -0.90 -0.90 0.25 -0.46 0.24 0.20 -0.50

Table 13. Students with Lowest 25 Measures for Music Self-View ID 287 174 175 171 267 268 271 180 112 250 285 17 189 305 155 191 193 54 49 46 226 32 197 179 279

Raw Score 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1

Student Measure -3.25 -3.25 -3.25 -3.25 -3.25 -3.25 -3.25 -3.25 -3.25 -3.25 -3.25 -3.25 -3.25 -3.25 -3.25 -3.25 -3.25 -3.25 -2.51 -2.51 -2.51 -2.51 -2.51 -2.51 -2.51

SE 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1.15 0.82 0.82 0.82 0.82 0.82 0.82 0.82

Residual -0.69 0.64 -0.29 -1.15 -1.15 -0.72 -0.29

Nine students (7 males and 2 females) had an extreme low raw score of 0 (-3.22 logits), and will require urgent extra support in Drama. Thirteen students (11 males and 2 females) had a score of 1, and will also require extra support. The remaining two students (lowest 25 measures) had a raw score of 2. These students answered, „some of the time‟ to the items, and will also benefit from extra support and attention in Drama. More males had a low Music Self-View than females (that is, 18 males compared with 7 females), indicating that intervention may be required as well as extra support in Music for

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these males. Raw scores for these students were quite low ranging from 0 to 1 (-2.51 to -3.25 logits).

Gender Differences The graphical data for gender is displayed in Figure 4 for Sports Self-Concept, Figure 5 for Drama Self-Concept and Figure 8.6 for Music Self-Concept respectively. Males had a higher mean Sports Self-Concept than females and this is statistically significantly higher (t=7.8, df=319, p-0.000). For Music Self-Concept, females had a mean higher Self-Concept than males and this is statistically significantly higher (t=6.7, df=319, p=0.000). For Drama Self-Concept, females had a higher mean Self-Concept than males and this is statistically significantly higher (t=4.25, df=319, p=0.000).

Figure 4. Target Graph by Gender for Sport Self-Views. Note: There is a colour error in the RUMM program. Purple corresponds to red (female) and green corresponds to blue (male). Self-View measures are on the upper side and item difficulties are on the lower side in standard logit units.

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Figure 5. Target Graph by Gender for Drama Self-Views. Note: There is a colour error in the RUMM program. Purple corresponds to red (female) and green corresponds to blue (male). Self-View measures are on the upper side and item difficulties are on the lower side in standard logit units.

Note: There is a colour error in the RUMM program. Purple corresponds to red (female) and green corresponds to blue (male). Self-View measures are on the upper side and item difficulties are on the lower side in standard logit units. Figure 6. Target Graph by Gender for Music Self-View.

Further conclusions can be made from the graph for Sports Self-Views (Figure 4). From about -1.00 to -2.73 logits, there are 2.1% of students (1 male and 6 females), who have extremely low levels of Sport Self-Views. From -1 to 0 logits, 11.5% of students (9 males and

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28 females) have low levels of Sport Self-Views. From 0 to +1.0 logits, there are 30.8 % of students (34 males and 65 females) who have a medium level of Sport Self-Views. From+1.0 to +2.0 logits, there are 28% of students (37 males and 53 females) who have moderately high levels of Sport Self-Views. From +2.0 to +3.41 logits, there are 27.7% of students (68 males and 21 females) who have very high levels of Sport Self-Views. From Figure 5, further conclusions can be made about Drama Self-Views. There were 25.2 % of students (60 males compared with 21 females) who had low levels of self-view (1.00 to -3.25 logits). The majority of students (50.8%), comprising of 72 males and 91 females had a self-view in the range –1.00 to +1.00 logits. The remaining 25.5 % (25 males and 57 females) had a high self-view in the +1.0 to + 3.2 logits range, with 10 males and 19 females having a very high score of 3.2 logits. From Figure 6, further conclusions can be made for Music Self-Views. There were 21.2 % of students (60 males compared with 21 females) who had low levels of self-view (-1.00 to -3.25 logits). The majority of students (59.8%), comprising of 88 males and 104 females had a self-concept in the range –1.00 to +1.00 logits. The remaining 19.3 % (12 males and 50 females) had a high self-view in the +1.0 to + 3.3 logits range.

SUMMARY OF FINDINGS The reliability of the scale data relating to Sport Self-Views (12 items), Drama SelfViews (10 items), and Music Self-Views (12 items) was shown by: 1. Good global and person item fit to the measurement model, and good individual fit to the measurement model. 2. The standard errors of measurement are about 0.1 logits for Sports, Drama and Music Self-Views, and the Person Separation Indices are 0.88, 0.89 and 0.88 respectively, indicating excellent separation of measures in comparison to the errors. 3. The item-trait interaction for the 12 items of Sports Self-Views, the 9 items of Drama Self-Views, and the 12 items of Music Self-Views were satisfactory, indicating a satisfactory overall fit to the measurement model for each variable and the measurement of a uni-dimensional trait. 4. Reasonable targeting of the items against the person measures, although some easier and harder items need to be added for any future use of the three scales. Since the scale data were shown to be reliable, the following valid inferences were drawn from the scales. 1. All attitude relationships (What I would like to do) were easier than their corresponding actual behaviors (What I actually do). 2. The easiest attitude item (What I would like to do) for Sports Self-View was playing sports (and very easy, at -1.21 logits). The hardest attitude item (What I would like to do) for Sports Self-Views was watching sporting programs like „Fox Sport‟ on TV (and hard at +0.90 logits). The easiest behaviour item (What I actually do) for Sports Self-Views was playing sports (moderately easy at -0.59 logits). The hardest behavior

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3.

4.

5. 6. 7.

item (What I actually do) for Sport Self-Views was watching sporting programs like „Fox Sport‟ on TV (and very hard at +1.31 logits). The easiest attitude item (What I would like to do) for Drama Self-Views was acting out different scenes and characters (and very easy, at -0.75 logits). The hardest attitude item (What I would like to do) for Drama Self-Views was being involved in Drama classes (and easy at -0.48 logits). In contrast, the easiest behavior item (What I actually do) for Drama Self-Views was being involved in acting and plays (moderately hard at +0.30 logits). The hardest behavior item (What I actually do) for Drama Self-Views was being involved in Drama classes (and very hard at +1.39 logits). The easiest attitude item (What I would like to do) for Music Self-Views was singing and listening to music (and very easy, at -1.05 logits). The hardest attitude item (What I would like to do) for Music Self-View was reading and writing music (and moderately hard at +0.12 logits). In contrast, the easiest behavior item (What I actually do) for Music Self-View was singing and listening to music (and easy at 0.67 logits). The hardest behavior item (What I actually do) for Music Self-View was being involved with musical productions (and very hard at +1.05 logits). Males have a statistically significantly higher Sports Self-Views than females. The weakest male and female students have been identified for possible intervention for improving their enjoyment and achievement in sports. Females have a statistically significantly higher Drama Self-Views than males. The weakest male and female students have been identified for possible intervention for improving their enjoyment and achievement in Drama. Females have a statistically significantly higher Music Self-Views than males. The weakest male and female students have been identified for possible intervention for improving their enjoyment and achievement in Music.

REFERENCES Allerup, P. (1997). Rasch meassurement theory. In J. P. Keeves (Ed.), Educational Research, Methodology, and Measurement: An International Handbook (2nd ed., pp. 863-874). Cambridge University Press, UK: Elsevier Science Ltd Publishers. Andrich. (1988a). Rasch models for measurement. Paper presented at the Sage university on quantitative applications in the social sciences, series number 07/068, Newbury Park, CA: Sage Publications Andrich, D. (1988b). A general form of Rasch's Extended Logistic Model for partial credit scoring. Applied Measurement in Education, 1(4), 363-378. Andrich, Sheridan, B., & Luo, G. (2005). RUMM: A windows-based item analysis program employing Rasch unidimensional measurement models. Perth:WA: RUMM Laboratory. Armstrong, T. (1994). Multiple Intelligences in the classroom. Alexandria, VA: Association for Supervision and Curriculum Development. Armstrong, T. (2000). Multiple intelligences in the classroom. (2nd ed.). Alexandria, VA: Association for Supervision and Curriculum Development.

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Bragg, J. (2005). The Effects of Problem-Based Learning on Student Engagement and Motivation. In L. P. McCoy (Ed.), Studies in Teaching 2005 Research Digest. WinstonSalem, NC: Wake Forrest University. Campbell, B., Campbell, L., & Dickinson, D. (1992). Teaching and Learning Through Multiple Intelligences. Australia: Hawker Brownlow Education. Choe, K. C. (2006). Student Engagement with Project Work in a Junior College in Singapore. Unpublished Doctor of Education thesis, Graduate School of Education, University of Western Australia. Christodoulou, J. A. (2009). Applying multiple intelligence: how it matters for schools today, 25 years after its introduction by Howard Gardner. School Administrator, 66(2), 22-25. Cook-Sather, A. (2002). Authorising Students' Perspectives: Toward trust, dialogue, and change in education. Educational Researcher, 31(4), 3-14. Curriculum Framework. (2008). Retrieved 25 March, 2008, from http://www.curriculum. wa.edu.au/pages/framework00.htm Deria, A. (2006a). Australian Islamic College Primary Student Handbook. Perth, WA: Australian Islamic College Deria, A. (2006b). Australian Islamic College Middle School Student Handbook. Perth, WA: Australian Islamic College Deria, A. (2006c). Australian Islamic College Senior School Student Handbook. Perth, WA: Australian Islamic College Edries, A. (2008). Australian Islamic College (Dianella) Annual Report. Perth, WA: The Australian Islamic College. Edries, A. (2009). Personal observation as Principal. Unpublished memo, Australian Islamic College, Perth, WA Furnham, A., Clark, K., & Bailey, K. (1999). Sex differences in estimates of multiple intelligences. European Journal of Personality, 13, 247-259. Furnham, A., & Ward, C. (2001). Sex differences, test experience and self-estimation of multiple intelligence. New Zealand Journal of Psychology, 30, 52-60. Furnham, A., Wytykowska, A., & Petrides, K. V. (2005). Estimates of multiple intelligences: A study of Poland. European Pyschologist, 10, 51-59. Gardner, H. (1983). Frames of Mind: The Theory of Multiple Intelligences (Second ed.). London: Fontana Press. Gardner, H. (1993a). Frames of Mind: The Theory of Multiple Intelligences (2 ed.). London: Fontana Press. Gardner, H. (1999). Multiple Intelligence Theory. Australian Journal Of Education, 43(1), 289. Gardner, H. (2004). Audiences for the Theory of Multiple Intelligences. Teachers College Record, 106(1), 212-220. Gardner, H., Goleman, D., & Csikszentmihalyi, M. (1998). Optimising Intelligences: Thinking, Emotion & Creativity. On Video [Video]: National Professional Resources, Inc. Goodnough, K. (2001). Multiple intelligence theory: a framework for personalising science curricula. School Science and Mathematics, 101(4), 180-193. Grant, C. A., & Tate, W. E. (1995). Multicultural education through the lens of the multicultural education research literature. In J. A. Banks & C. A. M. Banks (Eds.),

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Handbook of Research on Multicultural Education. New York: MacMillan, Harvard University Library. Haig, Y., & Oliver, R. (2007). Waiting in Line: African Refugee Students in Western Australian Schools. Bunbury: WA. Hickey, M. G. (2004). "Can I pick more than one Project?" Case Studies of Five Teachers Who Used Multiple Intelligence-Based Instructional Planning. Teachers College Record, 106(1), 77-86. Jimenez, R. T. (1997). The strategic reading abilities and potential of five low-literacy Latino readers in middle school. Reading Research Quarterly, 32(3), 363-383. Johnson, M. (2007). An Extended Lietarure Review: The Effect of Multiple Intelligences on Elementary Student Performance. Unpublished Master of Science in Education, Dominican University of California, San Rafael, CA. Kornhaber, M. L., Fierros, E., & Veenema, S. (2004a). Multiple Inteligences: Best ideas from theory and practice. Needham Heights: MA: Allyn & Bacon. Leitao, N. C. (2008). Teacher-Student Relationships in Primary Schools in Perth. Unpublished Doctor of Education thesis, Edith Cowan University, Perth. Loori, A. A. (2005). Multiple intelligences: A comparison study between the preferences of males and females. Social Behaviour and Personality, 33(1), 77-88. Luo, G. (2007). The relationship between the Rating Scale and the Partial Credit Models, and the implication for disordered thresholds of Rasch models for polytomous items. In E. V. Smith & R. M. Smith (Eds.), Rasch measurement: Advanced and specialized applications (pp. 181-201). Maple Grove, MN: JAM Press. Magar, A. (2008a). Australian Islamic College Annual School Report. Perth, WA: Australian Islamic College Magar, A. (2008b). Teacher Induction Booklet. Perth, WA: Australian Islamic College Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrica, 47, 149-174. Masters, G. N. (1988). The analysis of partial credit scoring. Applied Measurement in Education, 1(4), 279-297. Masters, G. N. (1997). Partial Credit Model. In J. P. Keeves (Ed.), Educational Research, Methodology and Measurement: An International handbook (2nd ed., pp. 857-863). Cambridge, UK: Cambridge University Press. Michell, J. (1990). An introduction to the logic of psychological measurement. Hillsdale, NJ: Lawrence Erlbaum Associates. Michell, J. (1997). Quantitative science and the definition of psychology. British Journal of Psychology, 88, 355-383. Michell, J. (1999). Measurement in psychology: A critical history of a methodoological concept. Cambridge, UK: Cambrige University Press. Miller, J., Mitchell, J., & Brown, J. (2005). African Refugees with interrupted schooling in the highschool mainstream:Dilemmas for teachers. Prospect, 20(2), 19-33. Moran, S., Kornhaber, M., & Gardner, H. (2007). Multiple Intelligences: building active learners. Teacher, 177, 26-30. Nolen, J. L. (2003). Multiple Intelligences in the Classroom. Education (Chula Vista, Calif.), 124(1), 115-119. Park, J., & Niyozor, S. (2008). Madrasa education in South Asia and Southeast Asia: Current issues and debates. Asia Pacific Journal of Education, 28(4), 323-351.

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Rasch, G. (1960/1980). Probabilistic Models for Intelligence and Attainment Tests. Chicago: IL: MESA Press. Shah, S. (2008). Leading multi-ethnic schools: Adjustments in concepts and practices for engaging with diversity. British Journal of Sociology of Education, 29(5), 523-536. Waterhouse, L. (2006). Multiple Intelligences, the Mozart Effect, and Emotional Intelligence: A critical review. Educational Psychologist, 41(4), 207-225. Waugh, R. F. (2003a). Measuring Attitudes and Behaviours to Studying and Learning for University Students: A Rasch Measurement Model Analysis. Journal of Applied Measurement, 4(2), 164-180. Waugh, R. F. (2003b). On the Forefront of Educational Psychology. New York: Nova Science Publishers. Waugh, R. F. (2005b). Frontiers in Educational Psychology. New York: Nova Science Publishers. Waugh, R. F. (2006). Rasch Measurement. In N. J. Salkind (Ed.), Encyclopedia of Measurement and Statistics (Vol. 3, pp. 820-825). Thousand Oaks, CA: Sage Publications. Whitaker, D. (2002). Multiple intelligences and after-school environments. Nashville: TN: School-Age NOTES. White, J. (1998). Do Howard Gardner's multiple intelligences add up? London: Institute of Education, University of London. Wikipedia. (2009). Theory of multiple intelligences. Retrieved 23 April, 2009, from http://www.en.wikipedia.org/wiki/Theory_of_multiple_intelligences Wright, B. D. (1999). Fundamental Measurement for Psychology. In S. E. Embretson & S. L. Hershberger (Eds.), The New Rules of Measurement: What every psychologist and educator should know (pp. 65-104). Mahwah, NJ: Lawrence Erlbaum Associates.

In: Specialized Rasch Measures… Editor: Russell F. Waugh, pp. 49-64

ISBN: 978-1-61668-032-9 © 2010 Nova Science Publishers, Inc.

Chapter 3

TEACHER GUTTMAN SCALES AND TEACHER VIEWS AT AN ISLAMIC COLLEGE Ahdielah Edries1 and Russell F. Waugh2 1

2

Australian Islamic College Faculty of Education and Arts; Edith Cowan University Mount Lawley; Western Australia.

ABSTRACT A co-educational Independent Australian Islamic College has three campuses which cater for migrant students from war-torn countries and others with culturally and linguistically, diverse backgrounds. This paper is part of a larger study to identify teacher views about the needs of students and the College. This study is important for the Islamic Colleges because it is hoped that the study will lead to improvements to the College and opportunities for Islamic students to have opportunities to achieve better in academic and non-academic areas. Three Guttman scales measured teacher perceptions (N=32) of: (1) Priority Activities Providing Links to the Western Culture; (2) General Types of Resources Needed; and (3) School Needs for Professional Areas. Teachers views were requested and they were analysed to produce eight propositions.

INTRODUCTION The background to this chapter is given in Chapter Two. This chapter explains the analysis of the main needs of students and the College, as perceived by their teachers. Data were collected from three Guttman Scales: (1) three items relating to Priority Activities Providing Links to the Western Culture (see Table 9.1); (2) three items relating to Measuring General Types of Resources Needed to meet the students needs (see Table 9.2); and (3) three items relating to School Needs for Professional Areas (see Table 9.3). Three reliable, but non-linear Guttman Scales (1944, 1950) were created, where the items were arranged in order of difficulty from easy to hard, and the total raw score on these three items arranged from low to high respectively. The open-ended questions answered

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by the teachers that could lead to improvements to be made within the College were analysed by creating some propositions that could be supported by many of the teachers. These were made by abstracting similar comments and then combining them into suitable propositions. A summary of findings is listed at the end of this chapter. While a Rasch measurement model can be used to create a linear unidimensional scale, as was done for the students‟ self-concepts in the previous chapter, large samples (N=250+) are needed to do so. When samples are low, as in the present teacher sample (N=32), the next best measurement model to use is a Guttman Scale which is undimensional but non-linear. This means that, while the scale measures one trait, equal differences between total raw scores on the scale do not represent equal amounts of the trait. Guttman scales are difficult to construct because they require a special structure of ordered items. In the three Guttman scales created for the present study, the items are conceptually ordered from easy (item 1) to hard (item 3). Teachers who answer the hardest item 3 positively are logically expected to answer items two and one positively. Teachers who answer the second item positively (but not the hardest item) are expected to answer item 1 positively. It is often difficult to create this ordered structure of items and, in practice, there are often deviations from the ideal pattern. Guttman (1944, 1950) suggested that up to a 10% error rate could be tolerated so that valid inferences might still be made from the scale.

FIRST GUTTMAN SCALE The first Guttman Scale (see Table 1) measured Priority Activities Providing Links to the Western Culture (N=32). This includes adding Singing, Music, Drama, Cultural Activities, Sports and Games to provide a holistic curriculum and a College environment that will enable the students to integrate into the Australian community more easily. While there was a reasonable fit to the Guttman pattern, this pattern was not ideal. The responses of teachers 1001, 1003 and 1027 did not fit the ideal pattern, but 87 out 96 (90.1%) of the responses did fit the Guttman pattern and this is within the ten percent error limits set by Guttman. Thus, it can be claimed that the scale is unidimensional and reliable so that valid inferences can be drawn from it. It was conceptualised that adding Singing, Music or Drama would be the easiest item to include in the curriculum to improve links to the western culture, but this was still expected to be judged as hard by the teachers. This is because of some restrictions pertaining to music under Islamic teachings and also due to some possible parental discontent given to teachers in some cases by some parents. From Table 1, this item was indeed the easiest (but it is still hard) as only 18 out of 32 teachers supported it in their priority statements, but 12 out of the 32 did not state it as a priority (with some strongly against it), indicating that it is a hard item and its implementation would probably be divisive in the College. It was expected that item 2, adding Cultural Activities, would be harder than item 1 because not all our students are fully integrated into the community (due to their past experiences in other Islamic countries and their limited experience with Australian culture which is quite different from their own) and because they and their families want to „hold onto their Islamic identity and culture‟ to a large extent.

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It was conceptualised that item 3, adding Sports and Games, would be the hardest item because girls have different perceptions about sport than boys in the Islamic culture. This is at least partly due to girls being required to adhere to the Islamic female dress code (that is, wearing the veil, long sleeve shirt and long trousers) and perceived parental disapproval with regard to certain games and sports that would be deemed inappropriate, especially for girls. Table 1 shows that item 3 is harder than item 2 which, in turn, is harder than item 1. Item 3 is very hard as only seven out of 32 teachers (21.8%) stated it as a priority for the College. It should be noted that 12 teachers scored zero on this Guttman Scale. That is, these 12 teachers (37.5% of teaching staff) did not state that any of these three additions listed in this scale were a priority for the College and student improvement. Some of the additions they suggested include extra support in each of the core subjects to improve language comprehension skills to enable students to understand the concrete meaning behind the words, open-ended investigations to be included into programs, providing hands on activities to develop students critical perception and abstract thinking, identify and understand students emotional intelligence (that is, encourage students to be aware of how they feel when they find a piece of work difficult), closer monitoring of individual education plans for students who are gifted and who require educational support, and to develop student intrapersonal skills in the classroom. Table 1. Priority Activities Providing Links to the Western Culture (non-linear scale) Teacher

1031 1032 1014 1022 1007 1004 1013 1027• 1015 1029 1000 1024 1003• 1011 1025 1026 1028 1030 1009 1001• 1006 1023 1021 1020 1019 1018 1017 1016

Add Singing, Music, or drama (item 1, easiest) Y Y Y Y Y Y Y Y Y Y Y Y N Y Y Y Y Y Y N N N N N N N N N

Add Cultural Activities (Item 2, harder) Y Y Y Y Y Y Y N Y Y Y Y Y N N N N N N Y N N N N N N N N

Add Sports & Games (Item 3, hardest) Y Y Y Y Y Y Y Y N N N N N N N N N N N N N N N N N N N N

Total Score (non-linear) 3 3 3 3 3 3 3 2 2 2 2 2 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0

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Table 1. (Continued). Teacher

1012 1010 1002 1008

Add Singing, Music, or drama (item 1, easiest) N N N N

Add Cultural Activities (Item 2, harder) N N N N

Add Sports & Games (Item 3, hardest) N N N N

Total Score (non-linear) 0 0 0 0

Explanatory Note for Table 1 1. All responses were converted into a raw score (that is, Yes = 1 and No = 0) 2. A dot next to the teacher number (1001, 1003 and 1027) indicates that the teacher‟s pattern of responses do not conform to the Guttman pattern. 3. Knowing only a teacher‟s raw total score on the Guttman scale predicts the exact pattern of responses, if there is a good fit to the measurement model.

SECOND GUTTMAN SCALE The second Guttman Scale (see Table 2) measured General Types of Resources Needed (N=29). The responses of four teachers (numbers 1005, 1011, 1029 and 1032 did not fit the ideal Guttman pattern but 83 out of 87 responses (95%) did fit the Guttman pattern, well within the 10% error limit set by Guttman (1950). Hence, it can be claimed that the scale is sufficiently unidimensional and reliable to enable valid inferences to be drawn from it. It was conceptualised that adding IT Learning Centres, Library resources and a Naturalistic Centre would be an easy item for teachers to list on the needs of the College. This is because the College already implements some of these variables to some extent, except for the naturalistic centre which needs some further investigation. A Naturalistic Centre might include developing an outdoor classroom, collecting objects from nature, initiating projects from the food chain and water cycle, researching environmental issues, researching local and global environmental concerns, categorising species of animals and plants, outdoor activities such as camping, hiking, or climbing, setting up sensory skill activities, and showing DVD‟s about nature, Science or animals. From table 9.2, this item was the easiest as 21 out of 29 teachers (72.4%) supported it in their needs statements. It was noted that 8 out of 29 teachers (27.6%) did not make this a priority need. It was conceptualised that item 2, adding Resources for Sports and Physical Activities (Ovals) would be hard. This was because of limited support for this in the Islamic culture (referred to previously) and because of the lack of physical space in the current school grounds. From Table 2, item 2 is harder than item 1 as only 12 out of 29 teachers (41.4%) supported it in their needs statements and 17 out of 29 teachers (58.6%) did not include it their needs statements. It was conceptualised that item 3, adding Extra-Curricular Activities, would be the hardest for a number of reasons. About 75% of the students are taken home by bus at 3.50 pm daily and most parents would not be expected to allow their children to participate in extracurricular activities after normal school hours despite the desirability of these activities being

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supervised at the College. From Table 2, item 3 was harder than item 2 and very hard. Only 7 out of 29 teachers (24%) stated that extra-curricular activities were needed. It should be noted that 4 out of 29 teachers (13.79%) did not list any of these three resources in this scale as a priority for the College and the students. Some of the additions they suggested include developing student interpersonal and intrapersonal skills in the classroom and encouraging students to do more outside activities, exposing students to opportunities to demonstrate their knowledge, teacher punctuality, teacher motivating students, hard but fair discipline, presenting interesting lessons, introducing varied teaching and learning techniques to pique student interests to enhance their learning. These teachers also felt that there should be more parental support and involvement with their children‟s learning, as well as the need for the College to address the educational needs of students who were not performing well academically. Table 2. Measuring General Types of Resources Needed (non-linear scale) Teacher

Add IT Learning centres, Resources, Naturalist, (Item 1, easiest)

1021 1000 1004 1007 1013 1031 1024 1019 1020 1002 1022 1005• 1009 1010 1011• 1012 1014 1015 1016 1017 1023 1026 1028 1029• 1032• 1006 1025 1018 1030

Y Y Y Y Y Y Y Y Y Y Y N Y Y N Y Y Y Y Y Y Y Y N N N N N N

Add Resources – Ovals Physical Activities (Item 2, harder) Y Y Y Y Y Y Y Y Y N N Y N N Y N N N N N N N N Y N N N N N

Add Extra Curricular Activities (Item 3, hardest) Y Y Y Y Y Y N N N N N N N N N N N N N N N N N N Y N N N N

Total Score (non-linear)

3 3 3 3 3 3 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0

Explanatory Note for Table 2 1. All responses were converted into a raw score (that is, Yes = 1 and No = 0)

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Ahdielah Edries and Russell F. Waugh 2. A dot next to the teacher number (1005, 1011, 1029 and 1032) indicates that the teacher‟s pattern of responses do not conform to the Guttman pattern. 3. Knowing only a teacher‟s raw total score on the Guttman scale predicts the exact pattern of responses, if there is a good fit to the measurement model.

THIRD GUTTMAN SCALE The third Guttman Scale measured School Needs in Professional Areas (N=32). This includes incorporation of Gardner‟s intelligence domains into the curriculum, adding pastoral care and educational support, adding a differential curriculum to cater for individual student needs. The responses of four teachers (numbers 1018, 1002, and 1023), did not fit the ideal Guttman pattern but 90 out of 96 (93.7%) of responses did fit the pattern, well within the 10% error limits set by Guttman (1950). Hence, it can be claimed that the scale is unidimensional and reliable so that valid inferences can be made from it. It was conceptualised that item 1, incorporating Gardner‟s Intelligence Domains into the curriculum would be an easy item for teachers to list on what could be done to meet the needs of their students. This was expected due to the dedication and commitment of present staff to enhancing the learning of their students, to helping students achieve their potential across many subject areas, and by seeking „other‟ ways, consistent with Islamic culture, of helping their students learn and achieve. From Table 3, this item was the easiest as 16 out of 32 teachers (50%) supported this in their statements, but it was still hard. It was noted that 12 out of 32 teachers (37.5%) did not mention this in their statements. So there is only moderate support for implementing item 1 into the school curriculum to improve the academic and social attitudes and achievements of the students. It was conceptualised that item 2, adding Pastoral Care and Educational Support, would be harder than item 1. This means that whilst staff realise the importance of extra educational support for disadvantaged students and the need for greater pastoral care within the College, they perceive a greater need for academic and professional support for all their students. Although the College has policies in place in relation to this aspect, it was expected that most teachers probably believe that more has to done in the academic areas of learning before resources are placed into pastoral care and educational support. It was expected too that staff would want more professional support staff such as a school nurse, a school psychologist and a school dentist. For these reasons, it was expected that adding Pastoral Care and Support, while needed would be of great value to the students at the College, it would have a lower position of need than academic support and professional support, as judged by the teachers. From Table 3, item 2 is harder than item 1 as only 12 out of 32 teachers (37.5%) supported it in their needs statements and 20 out of 32 teachers (62.5%) did not support it in their needs statements. So there is not strong support to implement item 2 aspects at the College. It was conceptualised that item 3, Providing Opportunities for Students and having a Differential Curriculum (that is, a holistic curriculum that comprises of academic and nonacademic subjects) would be the hardest because the College has many practices in place to meet the needs of all its students. However, due to limited finances, limited resources, lack of physical space, lack of support from parents, students prior experiences (and trauma), students academic ability (or lack thereof), it is not always possible to accomplish these goals.

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From Table 3, item 3 is harder than item 2 and very hard. Only 6 out of 32 teachers (18.8%) stated it as a priority for the College. Hence, there is not strong support for implementing item 3 at the College. It should be noted that 12 out of 32 teachers (37.5%) did not list any of these aspects from Scale 3 as a Professional Need for the students and the College. They suggested things such as providing more resources, professional development for ESL in the mainstream, encourage parents to learn along with their own children, improving remedial teaching for literacy and numeracy, improving thematic approaches to learning, providing opportunities for groups with similar intelligences to meet on a regular basis and focus on their strengths to boost their learning, helping staff to develop whole school policies to cover all eights areas of learning, adopting a more practical approach to teaching - not just pencil and paper, but restructuring lessons to incorporate learning through investigation rather than content based. Table 3. School Needs in Professional Areas (non-linear scale) Teacher

1016 1017 1020 1032 1014 1018• 1019 1022 1028 1005 1000 1003 1002• 1004 1006 1007 1008 1009 1010 1023• 1021 1024 1025 1026 1029 1031 1027 1001 1015 1030 1011 1012

Incorporating Gardner‟s Intelligence Domains in Academic Work (item 1, easiest) Y Y Y Y N N Y Y Y Y Y Y N Y Y Y Y Y Y N N N N N N N N N N N N N

Adding Pastoral Care and Support

(Item 2, harder) Y Y Y Y Y Y Y Y Y Y N N Y N N N N N N Y N N N N N N N N N N N N

Adding a Differential Curriculum to Cater for Individual Needs (Item 3, hardest) Y Y Y Y Y Y N N N N N N N N N N N N N N N N N N N N N N N N N N

Total Score (non-linear)

3 3 3 3 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

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Explanatory Note for Table 3 1. All responses were converted into a raw score (that is, Yes = 1 and No = 0) 2. A dot next to the teacher number (1002, 1018, and 1023) indicates that 3. the teacher‟s pattern of responses do not conform to the Guttman pattern. 4. Knowing only a teacher‟s raw total score on the Guttman scale predicts the exact pattern of responses, if there is a good fit to the measurement model.

PROPOSITIONS FROM THE TEACHER OPEN-ENDED QUESTIONS Teachers‟ responses to the open-ended questions were read and similar comments were placed in similar groups under appropriate headings. Thereafter, similar comments were reread to check that they had been placed in the correct group together under the correct headings. The comments were checked again and the following eight propositions were created from the teachers‟ comments now under common, appropriate headings. Some partquotations supporting the eight propositions are given below with a number referring to the particular teacher (not identified by name for ethical reasons). The comments and propositions were meant to add further information in relation to issues measured in the Guttman scales.

PROPOSITION 1 While it may seem reasonable to introduce singing, music or drama into the College as part of a way to forge some links between the Islamic students and the other Australian students, it is a potentially divisive issue. If singing, music or drama were to be introduced at the College, it would have to be done in a sensitive way, probably in very small stages after extensive consultation with staff and parents. Adding singing, music or drama to the curriculum did not have universal support with the teachers as they are aware that music is not well supported in the Islamic culture, hence, they do not see this addition as a priority for the College. Remarks from staff who support adding of singing, music or drama are evident in the following comments: The two main domains that I believe are not being met within my classroom are the musical domain and the naturalist domain. While they have been met partially, they can certainly be improved on. (1028) Music and associated learning areas e.g. drama and dance. Improve Interpersonal relations particularly with other cultural groups and experiences. Be more flexible in areas of music and literature provided to students and adopt the multicultural Australian way of life. (1031) Put music and dance in the curriculum. Singing is great for health and mood. Help students to become independent thinkers that are able to solve their own problems. (1022) We do not use and practice music as part of our curriculum. Offer music in our curriculum and or make music an option for students and parents to select. (1011)

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PROPOSITION 2 The majority of staff agreed that the College already had some practices in place to cater for the non-academic needs of their students. The practices they referred to include sport, art, gardening, excursions to recreational places, the supportive pastoral care within the College, community service participation, and inter-faith school visits. However, some teachers seem to be suggesting that more could be done to meet the non-academic needs of the students in the area of activities, games and sports to provide better links with the western culture. Others, for example, suggest helping students to become independent learners and thinkers, teaching students about life skills and issues, and introducing extra-curricular activities to enrich the students‟ lives. If extra activities, games and sports were to be introduced at the College, this would have to done in a sensitive way, probably in small stages after extensive consultation with staff and parents. Adding games and sports did not have universal support amongst the teachers and many did not see it as a priority despite games and sports being a strong priority in Australian culture. The non-academic needs that the majority of staff deem necessary to meet the needs of their students are outlined in the following comments: Incorporate more practical hands on learning areas e.g. vet subjects and not focus on pure academic areas (sic). Be more flexible in areas of music and literature provided to students and adopt the multicultural Australian way of life. Be more emphasis on structured sport in school (sic). Incorporate and integrate slowly into the learning areas. Engage students in more puzzles and strategy games where student use logic, problem solving and critical thinking. (1031) I believe that we as a school do our best to meet the needs of the students. However areas which we could improve on are the following; more computers that are faster, bigger, library with a variety of books for students to read and use for projects, cleaner canteen with a variety of healthy foods, more sports equipment and a oval so children can play freely and get some exercise. (1020) Closer monitoring of individual education plans for students who are gifted and remedial. - More technology used inside and outside of the classroom. - Provide a more rounded education. Emphasize importance of the 4 secular areas but provide more opportunities for learning in the areas of health, sport, art, incursions and excursions. More community involvement with those from a non-Muslim background. (1012) More group work - allows those with high interpersonal skills to shine, continued use of pictorial representation and drama to explain concepts and vocabulary. Greater use of games in learning followed up by written work to allow for differing skill/intelligence. Provide a more open-learning environment with a smaller emphasis on strict use of text books and a cross curricular approach to planning. (1008)

PROPOSITION 3 Whilst some teachers acknowledged the importance of pastoral care and educational support for students within the College, they perceived a greater need to enhance the academic and professional support for their students. The needs to which the teachers are referring include improving academic areas of learning for all students (that is, enhancing the learning of their students, to helping students achieve their potential across many subject areas, and by seeking „other‟ ways, consistent with Islamic culture, of helping their students

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Ahdielah Edries and Russell F. Waugh

learn and achieve) and to provide more professional support staff such as a school nurse, a school psychologist and a school dentist. These teachers were suggesting things such as professional development for teachers to enable them to develop whole school policies to cover all eight areas of learning, more focus on remedial teaching in literacy and numeracy, provision of opportunities for groups with similar intelligences to meet on a regular basis and focus on their strengths to boost their learning, adopting a more practical approach to teaching and learning. It was acknowledged that implementation in the College of the suggestions would require extra resources, specialist teachers, ovals, adding learning centres, more co-operative learning in classes, enhancing IT within the College, reviewing the teaching and learning methodologies of staff and students respectively. However, whilst all these are interesting suggestions, the College would not be able to accommodate some of these suggestions due to financial difficulties as a result of most of the parents paying very little or no fees. Teachers provided the following comments: The existing learning programme has to be modified. We tend to teach from the syllabus only which excludes real life situations. We should get more professionals to visit students with special needs at school to assess their educational needs and assist teachers in developing programmes. We should promote differential learning, offer more subjects. (1010) Provide more opportunities for logical/mathematical students to compete in interschool events-designing, creating solutions for scientific/mathematical problems through some fun activities interactive, free from tests). Provide after-school specialists for various student interests, such as, drama media school's newsletter created by talented students, art, other languages, playing instruments, sports and photography. (1017) We need to cater more for the spatial/body/kinesthetic intelligence domain student. Our linguistic/logical centered approach caters for a select few. We need to extend our topics across all learning areas. (1013) Closer monitoring of individual education plans for students who are gifted and remedial. More technology used inside and outside of the classroom. Provide a more rounded education. Emphasize importance of the four secular areas but provide more opportunities for learning in the areas of health, sport, art, incursions and excursions. More community involvement with those from a non-Muslim background. (1012) Fundraise to better facilitate and resource catering for a variety of intelligences….. (1000) I feel the school can provide the necessary needs by buying equipment for the above discussed. Funds should be raised to cater for all this. (1016) Provide better resources for the areas of creative and practical arts…. (1014) Provide more resources, PD in ESL in the mainstream (1021) …. Pastoral care for students who have school and home life issues e.g. school chaplain in public schools…. (1012)

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PROPOSITION 4 The staff was divided with regard to whether extra-curricular activities should be added to the current academic curriculum. While there was some support for this, there was some comment against introducing any extra-curricula activities, due to extenuating factors such as, the length of the school day, student transport, parental disapproval and students‟ unwillingness to participate in certain activities, limited access to resources and finances to run these activities. In contrast with this, some teachers suggested adding academic-type activities such as introducing practical hands-on activities, puzzles and games to encourage logical thinking, activities that tap into students individual strengths and interests, more sophisticated IT, better resources to encourage creativity, activities to promote the inclusiveness of the eight intelligences to enhance the current academic curriculum. The following quotes by staff highlight their relevant perceptions. Provide better resources for the areas of creative and practical arts. Modify the timetable to ensure more time is provides for physical activity. Vastly enhance the computer network throughout the college and provide programs that improve the learning environment of the students. (1014) I feel the need for specialist teachers in the areas of art, drama and sport rather than P.E (sic). More funds needs to be placed in the areas mentioned previously e.g. Bodily/kinesthetic (sic), musical and naturalist intelligence. (1015) Provide more opportunities for logical/mathematical students to compete in interschool events-designing, creating solutions for scientific/mathematical problems through some fun activities interactive, free from tests). Provide after-school specialists for various interests drama media school's newsletter created by talented students, art, other languages, playing instruments, sports and photography. (1017) In subjects such as math/science in order to improve understanding and develop skills I can apply in my lesson strategies such as; Body/kinesthetic - Science; students to enact kinetic theory of matter, structure of the action, planets and space - model. More (sic) skills re-inforcement in the science laboratory. Math Algebra - enact shapes, estimate distance. (1006) I believe that we as a school do our best to meet the needs of the students. However areas which we could improve on are the following; more computers that are faster, bigger, library with a variety of books for students to read and use for project, cleaner canteen with a variety of healthy foods, more sports equipment and a oval so children can play freely and get some exercise. (1020)

PROPOSITION 5 Adding IT Learning Centres or library resources to enhance the learning and cater for the needs of students was very strongly supported by 72% of the staff. It was stated by some teachers that the College was already effectively catering for some students‟ needs related to these areas, however, they felt that the College could do more with better resources, such as, for example, better IT facilities and Learning Centres, more „hands-on‟ activities to draw on

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student prior learning experiences and to make their learning more practical. It was also suggested that students should be exposed to opportunities to demonstrate their knowledge, and to pique their interests. These suggestions are described by the following quotes. Provide better resources for the areas of creative and practical arts. Modify the timetable to ensure more time is provided for physical activity. Vastly enhance the computer network throughout the College and provide programs that improve the learning environment of the students. (1014) Extra support in English; - More focus on problem solving;- Open ended investigations to be included into programs;- improve facilities and IT support. (1006) I believe that we need to encompass much more hands on learning in order to cater for those children who struggle with written work. To do this we also need space to set up learning centers (sic). Maybe if we do get the new mosque/building, the old mosque can be put to good use. By making use of thematic learning we can look at creating a whole school curriculum that builds on previous learning. (1008) Students are lacking in critical perception, abstract thinking and inability to operate as an independent learner mainly because they do not have hands on experiences. Most of the time (sic) its teacher- student traditional teaching and learning. (1010) I believe that we as a school do our best to meet the needs of the students. However areas which we could improve on are the following; more computers that are faster, bigger, library with a variety of books for students to read and use for project, cleaner canteen with a variety of healthy foods, more sports equipment and a oval so children can play freely and get some exercise. (1023) Provide more resources, PD in ESL in the mainstream, encourage parents to learn along with their own children. (1024)

PROPOSITION 6 Some teachers suggested the introduction of a Naturalist Centre (which could be very meaningful for the students especially taking their backgrounds and prior learning experiences into consideration). A Naturalist Centre could be good for the students because it could tap into their strengths (without the need to show their lack of academic ability). It can also enhance their undiscovered strengths, and provide students the opportunity to develop naturally by being involved with practical „hands-on‟ learning in a non-intrusive environment. A Naturalist Centre could involve developing an outdoor classroom, collecting animals, plants and rocks from nature, growing plants and vegetables, caring for animals, researching local and global environmental issues and concerns, and categorising species of animals and plants. Linked to this could be outdoor activities such as camping and hiking, setting up sensory skill activities, and showing DVD‟s about nature, Science and animals. Naturalist: We could have a little garden patch with chicken grow vegetables (sic) to cater for children that display naturalist intelligence. (1017)

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I feel that the students‟ spatial awareness is lacking and bodily/ kinesthetic experiences. Naturalist activities such as cooking and recycling needs must be integrated in school life. (1024) ….. Naturalist domain - more emphasis in this area particularly forming or joining youth groups, scouts, brownies and girl guides. (1031) Do more naturalistic hands on activities such as caring for animals, worms and composting. (1005) Plan more tactile activities, more building, creating and designing of physical materials and projects (sic). Hands on activities (sic). Space is an issue but the year 7 garden is excellent it would be great to get the kids involved in planting and tending plants. I would love to do cooking with my kids.... (1025) …. Naturalist domain - more emphasis in this area particularly forming or joining youth groups, scouts, brownies and girl guides. (1034)

PROPOSITION 7 Incorporating Gardner‟s Intelligence Domains into the curriculum was strongly supported by teachers on what could be done to meet the needs of their students. This support was primarily due to staff wanting to explore other avenues to enhance the learning of their students, to helping students achieve their potential across many subject areas, consistent with Islamic culture, of helping their students learn and achieve. Staff suggested improving the academic and social attitudes and achievements of their students by introducing activities linked to the eight intelligences, adopting thematic approaches to make students‟ learning more meaningful, incorporating varied learning techniques to enhance students interest to attain their full potential, providing opportunities for deep critical thinking, investigative open-ended tasks, and to plan more inclusive projects for students to stimulate their learning. These ideas are described below: I can use the ideas of the seven intelligences to improve the abilities of my students through using more open-ended tasks and through allowing students to compete a task using a variety of methods which is best suited to their intelligence. (1028) I believe that in my class I already use most of the 7 intelligences with my students. However there are a few of the intelligences that I would like to put into practice if I was given the opportunity in order to improve our students learning. I would like the opportunity to teach my students a musical instrument. It would be useful to take students on camp to learn more about the environment and learn how to live /survive in the bush. I would also like to provide my students with the opportunity to act in plays to boost their confidence especially students who are very shy. I would like to give my students the opportunity to take part in building models, more hands on activities to improve themselves and boost their confidence. (1020) Plan more tactile activities, more building, creating and designing of physical materials and projects (sic). More hands on activities. Space is an issue but the year 7 garden is excellent it would be great to get the kids involved in planting and tending to

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Ahdielah Edries and Russell F. Waugh plants. I would love to do cooking with my kids. Drama and public speaking could be greater valued and respected, this helps confidence and improves the presentation. (1022) Program in a holistic way (that is, using a thematic approach). Divide children according to their strengths (within a year level) across say 3 intelligences then provide opportunities for the groups to come together on a regular basis and focus on their strengths to boost learning. (1021) In subjects such as math/science in order to improve understanding and develop skills I can apply in my lesson strategies such as; Body/kinesthetic - Science; students to enact kinetic theory of matter, structure of the action, planets and space - model. Skills reinforcement in using science lab. Math Algebra; enact shapes, estimate distance. Interpersonal skills and linguistic. (1006)

PROPOSITION 8 While it may seem natural that the College should provide reasonable physical space for the students there was little support by the teachers to add an oval as a necessary resource. This was primarily due to the lack of physical space within the College (already having some inadequate sport and play areas), limited finances to upgrade, need for better resources, and the perception by staff, parents and the community that there were „other student needs‟ that should be addressed with more priority, such as developing student interpersonal and intrapersonal skills in the classroom, exposing students to opportunities to demonstrate their knowledge, introducing varied teaching and learning techniques to pique student interests and enhance their learning, greater parental support and involvement with their children‟s learning, improving literacy and numeracy, and addressing the educational needs of students who were not performing well academically, that could be of more benefit to the students. Remedial teaching for literacy and numeracy with a strict timetable. More resources to help slow learners. More(sic) extra curricular activities to increase focusing and to remain on task. (1030) Intrapersonal - for the Islamic student, deep thinking is encouraged and could be a great advantage to successful learning and also to reflect conscientiously in achieving their learning goals across the curriculum framework. Students need to develop their faculty of 'deep thinking'. Different ways of learning gives a better chance of understanding. Students are given the choice as to which of the seven intelligences they wish to undertake. Students think about what they can improve. (1019) Playground - An oval to run and play on - Specialist teachers; Art to teach subject as it should be taught. (1007) Teach students about the reality of working and their future. Real issues about what they will face in the future - taxes, salaries and job options so that they have a better idea of what they want to do and work towards a goal. Motivation and confidence to study. I have spoken to many students and they have all expressed their interest in this area. Also, the power of reflection, letting children keep a diary to record reflections to better themselves and learn from their mistakes. (1023)

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I believe that there is a significant number of students suffering from lack of parenting. We need to work with the parents to maximize student learning. … Students‟ emotional issues (particularly in upper primary/high school) need to be addressed. (1018)

SUMMARY OF FINDINGS Data were collected from thirty-two teachers to ascertain their views about the College meeting the needs of its students. The data involved two parts; (1) three Guttman Scales and (2) answers to open-ended questions from which eight propositions were created. In part 1, three Guttman Scales were created for three items measuring „Priority Activities Providing Links to the Western Culture‟, three items measuring General Types of Resources needed to meet the students‟ needs, and three items measuring School Needs in Professional Areas. The items were conceptually arranged in order of difficulty from easy to hard, and the total raw score on its three items arranged from low to high measured the three variables in a unidimensional, but non-linear scale. All three scales had acceptable teacher item-response patterns. In part 2, the eight propositions were created by collating similar comments under appropriate headings and then creating propositions that reflected the teacher comments. Following this, the original teacher data were re-read to check that the data supported the eight propositions. As a summary, nine main inferences can be drawn from the data. 1. Adding singing, music or drama to the curriculum was supported by 59% of teachers in their priority statements. However, 41% teachers did not see it as a priority, as there could be some adverse implications stemming from this that would be divisive in the College. 2. Adding Cultural Activities to the curriculum was considered to be hard, as some of the teachers expressed concern that there may be some cultural barriers that would need to be addressed first before implementing any activities. 3. Only 24% of teachers stated that adding sports and games to the curriculum was a priority for the College. This was due to some gender issues that would need to addressed with sensitivity and consultation, particularly with regard to the female students. 4. Adding IT Learning Centres, Library Resources and a Naturalist Centre was well supported as the College already implements (at least partially) most of these variables, but the Naturalist domain needs to be further investigated. 5. Due to lack of physical space within the College grounds, and a „limited‟ sports program currently being offered within the College, only 41.4% supported Adding Resources for Sports and Physical Activities (Ovals) as resources that are needed within the College to meet the needs of the students. 6. There was a unanimous agreement amongst staff that adding Extra-Curricular Activities, whilst necessary, would not be able to be incorporated into the school curriculum as most parents would not agree to allow their children to participate due to the College‟s long hours and most students needing the school transport after school.

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Ahdielah Edries and Russell F. Waugh 7. There was strong support to incorporate Gardner‟s Intelligence Domains into the curriculum due to the dedication and commitment of the staff to enhance and optimise the learning of their students. 8. Although the College has Policies in place, as well as excellent staff members who strive to meet the needs and well-being of all their students, most staff believe that more has to be done to meet the needs of the students. 9. Most staff acknowledged that they strive to meet the needs of all their students, however, due to limited finances, limited resources, lack of physical space, lack of support from parents, students prior experiences (and trauma), students academic ability (or lack thereof), it is not always possible to accomplish goals that they would like to for their students.

In: Specialized Rasch Measures… Editor: Russell F. Waugh, pp. 65-81

ISBN: 978-1-61668-032-9 © 2010 Nova Science Publishers, Inc.

Chapter 4

RASCH MEASURES OF FORM CONSTANCY OF LETTERS AND NUMBERS, AND LETTERS IN WORDS FOR YOUNG CHILDREN Janet Richmond, Russell F. Waugh and Deslea Konza Faculty of Education and Arts, Edith Cowan University, Perth, Western Australia

ABSTRACT English and number literacy are very important topics and the Australian Government runs numeracy and literacy tests, administered through the State Education Departments, for all Year 3 (8 years old), Year 5 (10 years old) and Year 7 (12 years old) students. Results of these tests are reported to schools and parents with a view to ensuring that all children meet certain literacy standards and that children who are „falling behind‟ are detected early so that remedial work can be given. Rasch measures were then created with the RUMM2020 computer program for visual discrimination regarding Form Constancy of Letters and Numbers (FCL&N) and Letters in Words (LinW). The student sample was N=324 pre-primary and primary students in Perth, Western Australia, aged 49 years old. Data on 24 items for FCL&N and 75 items for LinW, where each item was scored in one of two categories (wrong scored zero and correct scored one), were Rasch analysed to create two linear scales. Six of the initial 24 items for FCL&N were deleted due to item misfit statistics, leaving 18 items. Forty-one of the original 75 items for LinW were deleted due to item misfit, leaving 34 items. The final data for FCL&N and LinW were used to create two highly reliable, linear, uni-dimensional scales (Student Separation Indices of 0.94 and 0.97 and Cronbach Alphas of 0.94 and 0.97, respectively) where the items are ordered from easy to hard and the student measures from low to high on the same scale. The two scales showed no statistically significant interaction of student measures on item difficulties along the scale, meaning that there was good agreement about the item difficulties along each scale, and each scale was unidimensional. The item-trait chi-squares are respectively, χ² = 69.69, df=54, p=0.07, and χ² = 117.59, df=102, p=0.14. The fit residual statistics for each of the two scales was reasonable and the targeting was reasonable.

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INTRODUCTION This report presents a Rasch analysis with the RUMM 2020 computer program (Andrich, Sheridan & Luo, 2005) in which two linear, unidimensional scales were created: (1) Form Constancy of Letters and Numbers, and (2) Letters in Words. These two scales relate to visual perceptual concepts of „form constancy‟ and „figure ground‟. This report describes the measurement results in terms of Rasch measurement fit statistics including global item and person fit to the measurement model, dimensionality, person separation indices, distribution of item-person interactions, and discrimination. Some discussion is included of the non-fitting items, as well as good fitting items, and the person-item threshold distribution (targeting). This is followed by mean Rasch measures by group and final items for the Form Constancy and Figure Ground Scales discussion. Finally, inferences drawn from the linear Rasch measurement data analysis and the summary of the results are presented.

INITIAL RASCH ANALYSIS An initial Rasch analysis was performed on the original items for Form Constancy of Letters and Numbers (24 items) and Letters in Words (41 items) where each item was scored in one of two categories (incorrect answer scored zero and correct answer scored one). Six of the initial 24 items of Form Constancy of Letters and Numbers were deleted due to item misfit statistics. The remaining 18 items were found to have an excellent fit to the measurement model for the 324 persons included in this study. For Letters in Words, seven of the initial 41 items were deleted due to item misfit statistics. The remaining 34 items displayed an excellent fit to the measurement model. The Rasch analysis with the RUMM program does not indicate how to alter an item in order to make it fit the measurement model. In order to include, in a future measure, the deleted items which were initially considered conceptually valid, would need to be changed and re-tested.

FINAL RASCH ANALYSIS RESULTS The following material shows the results for the final Rasch analysis for the three scales: (1) Form Constancy of Letters and Numbers (18 items), and (2) the Figure Ground Scales of Letters in Words (34 items).

Summary of Fit Statistics The RUMM2020 program estimates an item-person interaction which establishes the overall fit statistics that determine whether the item estimations contribute meaningfully to the measurement of one construct. This calculation thus examines the consistency with which students responses agree with the calculated difficulty of each item on the scale. The standardised fit residual statistics (see Table 1) have a distribution with a mean near zero and

Rasch Measures of Form Constancy of Letters and Numbers…

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a standard deviation near one when the data fit the measurement model, as is the case with these three measures. This means too that there is a good pattern of person and item responses consistent with a Rasch measurement model.

Dimensionality For Form Constancy of Letters and Numbers, there was an item-trait interaction chisquare of 69.69 with df=0.94 and a probability of 0.07. This means that the scale is constructed with acceptable, but not ideal, agreement amongst the students about the linear progressive difficulty of the items. The item-trait interaction chi-square for Letters in Words was 117.59 with df=0.97 and a probability of 0.14, showing a similar acceptable agreement amongst the students about the linear progressive difficulty of the items along the scale. This means that the students agree as to which items are easy, which are of medium difficulty and which are hardest. Table 1. Global Item and Student Fit Residual Statistics (N=324) ITEMS Location Fit Residual Form Constancy of Letters and Numbers (I=18) Mean 0.00 -0.45 Standard Dev. 0.65 0.89 Letters in Words (I=34) Mean 0.00 -0.68 Standard Dev. 0.82 1.11 Numbers in Calculations (I=15) Mean 0.00 -0.35 Standard Dev. 0.59 0.04

PERSONS Location

Fit Residual

+1.97 2.06

-0.20 0.75

+2.00 2.56

-0.43 1.25

+1.29 2.11

-0.08 0.84

Comment on Table 1: Fit residuals have a mean near zero and a standard deviation near one when the data fit the measurement model (as is the case here). This reflects good consistency of item and student scoring patterns.

Person Separation Index The Person Separation Index is an estimate of the true score variance among the students and the estimated observed score variance using the estimates of their ability measures and the standard error of these measures (Andrich & van Schoubroeck, 1989). For Form Constancy of Letters and Numbers and Letters in Words, the Person Separation Indices are 0.94, and 0.97 respectfully. For a good measure, it is desirable that this index should be 0.9 or greater, as it is an indicator that the student measures are separated by more than their standard errors. Based on this index, the Form Constancy of Letters and Numbers and Letters in Words scales demonstrate very good separation of measures in comparison to the errors of measurement.

Janet Richmond, Russell F. Waugh and Deslea Konza

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Individual Item Fit Items are ordered by calibrated values to evaluate their fit to the measurement model. The location of each item on the scale is the item difficulty in standard units, called logits (log odds of answering successfully). All the items in Form Constancy of Letters and Numbers fit the measurement model with probabilities greater than p=0.03 (see Table 2). The residuals shown in Table 2 represent the difference between the observed responses and the expected responses calculated from the Rasch measurement parameters. Standardised residuals should fall within the range of -2 and +2. Table 2 shows that all items for Form Constancy of Letters and Numbers have acceptable residuals except for item 14. For Figure Ground Letters in Words, all the items fit the measurement model with probabilities greater than p=0.06 (see Table 3), but a few of the residuals are a little outside what might be considered good limits. Table 2. Individual Item Fit Statistics for Form Constancy of Letters and Numbers Item

Location

SE

Residual

DegFree

ChiSq

DegFree

Prob

18

-0.93

0.26

0.49

143.56

9.11

3

0.03

1

-0.72

0.25

-0.61

143.56

1.35

3

0.72

23

-0.70

0.25

+0.08

143.56

4.50

3

0.21

21

-0.63

0.24

-0.36

143.56

1.31

3

0.73

19

-0.50

0.24

+0.86

143.56

4.53

3

0.21

20

-0.40

0.23

+0.74

143.56

6.23

3

0.10

5

-0.33

0.23

-0.08

143.56

0.77

3

0.86

2

-0.33

0.23

-0.67

143.56

1.97

3

0.58

3

-0.26

0.23

-0.49

143.56

5.09

3

0.17

8

-0.12

0.23

-1.01

143.56

8.32

3

0.04

14

+0.10

0.22

-2.41

143.56

6.24

3

0.10

13

+0.28

0.21

-0.71

143.56

7.18

3

0.07

16

+0.29

0.21

-1.65

143.56

3.81

3

0.28

11

+0.51

0.21

+0.22

143.56

1.42

3

0.70

17

+0.70

0.20

-1.46

143.56

2.13

3

0.55

9

+0.71

0.20

+0.26

143.56

0.98

3

0.81

7

+0.93

0.20

-1.58

143.56

3.43

3

0.33

-0.03

143.56

1.32

3

0.70

4

+1.39

0.19

Notes on Table 2 and 3: 1. Location refers to the difficulty of the item on the linear scale. 2. SE means Standard Error, and refers to the degree of uncertainty in a value. 3. Residual represents the difference between the expected value of an item, calculated according to the Rash measurement model and the actual value. 4. DegFree stands for degrees of freedom, and refers to the number of scores in a distribution that are free to change without changing the mean distribution. 5. ChSq stands for Chi-square 6. Prob relates to the probability based on the Chi-square and refers to the levels of certainty to which an item fits the measurement model.

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Rasch Measures of Form Constancy of Letters and Numbers… Table 3. Individual Item Fit Statistics for Figure Ground Letters in Words Item

Location

SE

Residual

DegFree

ChiSq

DegFree

Prob

11 13 26 9 3 15 23 6 27 12 19 5 20 7 8 25 10 16 18 29 17 14 30 22 24 31 41 33 28 36 40 35 38 34

-1.16 -1.13 -1.13 -1.07 -0.98 -0.83 -0.81 -0.78 -0.71 -0.68 -0.65 -0.61 -0.55 -0.53 -0.41 -0.38 -0.22 -0.02 +0.17 +0.30 +0.36 +0.43 +0.68 +0.69 +0.70 +0.84 +0.86 +0.88 +0.96 +0.99 +1.08 +1.17 +1.19 +1.36

0.27 0.27 0.27 0.27 0.26 0.25 0.25 0.25 0.24 0.24 0.24 0.24 0.23 0.23 0.23 0.23 0.22 0.21 0.20 0.20 0.20 0.20 0.19 0.19 0.19 0.19 0.19 0.19 0.18 0.18 0.18 0.18 0.18 0.18

-0.68 -0.57 -0.59 -1.18 -0.23 -1.11 -1.96 +0.69 +0.31 -1.52 -2.12 -1.67 -1.71 -0.58 -1.42 -0.88 -0.69 -0.49 -1.56 -0.57 0.55 -0.59 -0.22 +1.98 -0.33 +0.75 -1.03 -2.31 +0.65 -2.40 +0.09 -0.53 -2.13 +1.97

176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65 176.65

3.20 4.06 5.23 2.09 1.17 3.78 3.00 2.43 2.85 3.29 4.92 2.24 4.24 3.18 3.13 2.65 6.37 6.68 5.05 0.82 3.00 0.57 4.64 7.48 2.33 4.87 2.19 3.25 0.72 6.82 0.86 3.65 3.34 3.52

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

0.36 0.25 0.16 0.55 0.76 0.29 0.39 0.49 0.41 0.35 0.18 0.52 0.24 0.37 0.37 0.45 0.09 0.08 0.17 0.84 0.39 0.90 0.20 0.06 0.51 0.18 0.53 0.36 0.87 0.08 0.84 0.30 0.34 0.32

Targeting The RUMM2020 program produces a student-measure item-difficulty or targeting graph on which the student measures are placed on the same scale as the item difficulties in standard units called logits. For Form Constancy of Letters and Numbers (see Figure 1), this targeting graph shows that the student measures cover a range of about -3.5 to +3.5 logits and the item difficulties cover a range of about -1.0 to +1.4 logits. From the graph it can be seen that many students (about 245) were able to answer the items correctly, while about 30 students were unable to answer any of these items correctly. This indicates that the targeting of the items needs to be improved in any future use of the scale by adding in some easier and more difficult items to „cover‟ the students with the lowest and highest measures.

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Note: Student measures are on the upper side in logits. Item difficulties are on the lower side of the same scale in logits. Many students (about 245) answered the items correctly. Figure 1. Targeting Graph for Form Constancy of Letters and Numbers.

Note: Student measures are on the upper side in logits. Item difficulties are on the lower side of the same scale in logits. Many students (about 175) answered the items correctly. Figure 2. Targeting Graph for Figure Ground Letters in Words.

For Figure Ground Letters in Words (see Figure 2), the targeting graph shows that the student measures cover a range of about -4.4 to +4.3 logits and the item difficulties cover a range of about -1.2 to +1.4 logits. From the graph it can be seen that many students (about 205) were able to answer the items correctly, while about 45 students were unable to answer these items. This indicates that the targeting of the items needs to be improved in any future use of the scale by adding in some easier and more difficult items to „cover‟ the students with the lower and higher measures.

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Discrimination Item Characteristic Curves examine the relationship between the expected response and the mean group student measures. These curves display how well the item discriminates between groups of persons. An example of one item characteristic curve for each of the three constructs will be presented. Figure 3 shows the Item Characteristic Curve for Item 1 Form Constancy of Letters and Numbers. This curve shows that the item discriminates well for students with different measures. The Item Characteristic Curves for all the other items were checked and found to be satisfactory (but are not reported here to avoid unnecessary repetition).

Figure 3. Item Characteristic Curve: Item 1 – Form Constancy of Letters and Numbers.

Figure 4. Item Characteristic Curve: Item 16 –Figure Ground of Letters in Words.

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Figures 4 shows the Item Characteristic Curves for Item 16 of Figure Ground of Letters in Words. This item discriminates well for students with different measures. The Item Characteristic Curves for all the other items in both measures were checked and found to be satisfactory (but are not reported here to avoid unnecessary repetition).

Consistency of Use of Scoring Categories The RUMM2020 program produces graphs of the scoring categories for each item. The Scoring Category Curves show the relationship between the probability of scoring in each category (zero for incorrect answer and one for correct answer) on each item. Figure 5 is the Scoring Category Curve for item 1 of Form Constancy of Letters and Numbers. This figure shows that the scoring was done logically and consistently. When students have low measures on item 1, then they have a high probability of obtaining a zero score (answer incorrect) and, when they have a high measure, they have a high probability of scoring 1 (answer correct). The Scoring Category Curves for all the other items were checked and they were satisfactory too. The Scoring Category Curves for all the items of the other variable, Figure Ground Letters in Words, were checked and they were also found to be satisfactory, but they are not presented here to avoid repetition.

Figure 5. Scoring Category Curve: Item1 – Form Constancy of Letters and Numbers.

CHARACTERISTICS OF THE SAMPLE (FCLN, FGLIW) The measures for Form Constancy of Letters and Numbers (FCLN) were displayed in a graphical format separated by gender (Figure 6), type of school (Figure 7), age (Figure 8), grade (Figure 9) and whether intervention had been received (Figure 10). Females have a higher mean measure than males for Form Constancy of Letters and Numbers but this is not statistically, significantly different (t=0.76, df=321, p=0.25). Public school students have a higher mean measure than private school students for Form Constancy of Letters and

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Numbers but this is not statistically, significantly different (t=0.93, df=321, p=0.18). As would be expected, the mean measures generally increased by age from four years of age (lowest) to nine years of age (highest) and this was statistically, significantly different (t=7.9, df=65, p=0.000). Again, as expected, the mean measures generally increased by grade from Pre-primary (lowest) to Year 3 (highest) and this was statistically, significantly different (t=12.0, df=126, p=0.000). The mean measures for intervention/ no intervention was not statistically, significantly different (t=0.88, df=321, p=0.20).

Note: There is a colour error in the RUMM program. Purple represents the females (not red) and green represents the males (not blue). Figure 6. Target Graph by Gender for Form Constancy of Letters and Numbers.

Note: There is a colour error in the RUMM program. Purple represents other schools (not red) and green represents the public schools (not blue). Figure 7. Target Graph by Type of School for Form Constancy of Letters and Numbers.

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Figure 8. Target Graph by Age for Form Constancy of Letters and Numbers.

Note: There is a colour error in the RUMM program. Pre-primary is represented by green (not blue), Year 1 is represented by purple (not red), Year 2 is represented by pink (not green), and Year 3 is represented by maroon (not purple). Figure 9. Target Graph by School Year for Form Constancy of Letters and Numbers.

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Figure 10. Target Graph by Intervention for Visual Form Constancy of Letters and Numbers Note: There is a colour error in the RUMM program. Green represents no intervention and purple intervention.

The graphical data for Figure Ground Letters in Words were checked in the RUMM computer program but is not produced here to avoid repetition but the graphs are similar to those produced for Form Constancy of Letters and Numbers. Females had a higher mean measure than males for Figure Ground Letters in Words but this is not statistically, significantly different (t=1.90, df=321, p=0.025). Public school students had a higher mean measure than private school students for Figure Ground Letters in Words and this is statistically, significantly different (t=3.6, df=321, p=0.000) in favour of the public schools. As would be expected, the mean measures generally increased by age from four years old (lowest) to ten years old or older (highest) and this was statistically, significantly different (t=8.10, df=66, p=0.000). Again, as expected, the mean measures generally increased by grade from Pre-primary (lowest) to Year 3 (highest) and this was statistically, significantly different (t=21.2, df=127, p=0.000). While the mean measure for no intervention was higher than for intervention, this was not statistically, significantly different (t=0.71, df=321, p=0.25).

FINAL ITEMS FOR THE FORM CONSTANCY AND FIGURE GROUND SCALES The final 18 items and their difficulties are presented, in order from easiest to hardest, in Table 4 for Form Constancy of Letters and Numbers. The students found it easy to identify the reversed item for the letter „a‟ and for numbers. They found it moderately easy to identify the reversed letters that are not often reversed in the font used in this scale (e.g. e, b, c), moderately difficult to identify letters that could be reversed or letters that had a body and a tail (e.g. s, q, y) and most difficult to identify the reversed letters that are commonly written in a reversed orientation by young students (e.g. j, g, d).

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In the Figure Ground Letters in Words (see Table 5 for the 34 item difficulties ordered from easy to hard), students found it easy to identify words as correct when they did not contain a reversed letter, such as: the, ran, that, know, and moderately easy to identify words as correct or incorrect when they had a mixture of long and short letters, for example ,

,

. Longer words containing a reversed letter were moderately difficult for students

to identify as correct or incorrect, for example , , ; while the most difficult words to identify as correct or incorrect were those with reversed orientation of g and u (e.g.

,

,

).

Table 4. Difficulties for 18 Final Items in Form Constancy of Letters and Numbers

Note: Items are ordered from easiest (item 18, -0.93 logits) to hardest (item 4, +1.39 logits).

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Table 5. Difficulties for 34 Final Items in Figure Ground Letters in Words Scale

Note: Items are ordered from easiest (item 11, -1.16 logits) to hardest (item 34, +1.36 logits).

COMMENTS ON THE NON-FITTING ITEMS DELETED FROM THE THREE SCALES Six items were deleted from the Form Constancy of Letters and Numbers Scale due to poor fit to the Rasch measurement model. Usually the main reason for non-fit is poor agreement in regard to the item difficulty. For example, half of the medium ability students may say an item is easy and half say that it is hard, thus it does not fit the measurement model and is deleted. The six items deleted in Form Constancy of Letters and Numbers Scale were: f, k, p, t, 6, and 9. The students may have disagreed on these letters due to the font used in this assessment, however it was noted that many students chose the upper case letter or the same letter as the reversed letter in a number of these situations as well as the same number or the number that had been made smaller. It is also of particular interest that most of the letters and

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numbers deleted due to disagreement were the letters and numbers that students often tend to reverse. In Figure Ground Letters in Words, seven of the original 41 words were deleted due to non-fit to the Rasch measurement model. The deleted letters were the words one (with reversed e), come, ate (with reversed t), think, fast (with reversed t), together (with reversed h) and never (with reversed n). It is noticeable that five of the words with poor fit had reversed letters; however there is no noticeable pattern of the similarity of letter or position of the reversed letter in the words. The font used in the assessment may have been a contributing factor to the students‟ interpretation of these words; however this does not present as an obvious influencing factor.

INFERENCES FROM THE MEASURES OF THE THREE LINEAR RASCH SCALES Linear scales were created showing good fit to the measurement model for the Form Constancy of Letters and Numbers, Figure Ground Letters in Words and Figure Ground Numbers in Calculations. Valid inferences can now be made about the student measures for form constancy and figure ground perception from these three linear scales. The bottom 49 student measures for Form Constancy of Letters and Numbers have been taken because these students all scored 6/18 or less, meaning that they were the students who were unable to identify the letters (other than a) and were only able to achieve some of the items that contained numbers. Twenty-two students had a score of zero with a location of -3.45, a standard error of 1.24. These student measures are presented in Table 7. The students who scored zero in Form Constancy of Letters and Numbers were unable to answer any of the items correctly, suggesting that they either misunderstood the instruction or are unable to identify when numbers or letters are reversed when the letters and numbers are presented in a variety of fonts. Students who scored 6 had difficulty identifying the reversed letters, but were more capable when identifying reversed numbers in different fonts. Students scoring poorly in Form Constancy of Letters and Numbers have difficulty identifying when letters and numbers of differing fonts are reversed and may need extra assistance to improve this skill. The bottom 53 student measures for Figure Ground Letters in Words have been taken because these students scored less than 17 out of 34, meaning that they were unable to identify more than half of the items as having or not having a reversed letter within the word. These student measures are presented in Table 8. Students, who scored 7, were only able to correctly identify items where no reversed letters occurred in the word. The students scoring 17 correct answers were able to identify words containing no reversals and the easiest four words containing a reversed letter. The four easiest items containing a reversed letter consisted of three words where a letter with a body as well as a head (long letter) and one short letter with only a body. These student measures identify students who may require assistance to improve their skill in identifying when a letter is reversed within a word. They may also be the students who reverse their letters in reading, spelling and or writing.

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Rasch Measures of Form Constancy of Letters and Numbers… Table 7. Lowest 49 Student Measures for Form Constancy of Letters and Numbers ID 151 199 167 166 165 164 324 162 203 153 163 150 27 21 19 4 3 108 37 156 323 161 200 64 110

Raw score 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

Location

SE

Residual

ID

-3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -3.45 -2.62 -2.62 -2.04

1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 1.24 0.89 0.89 0.71

-0.93 0.16 -0.58

80 5 119 18 84 111 223 76 78 23 268 49 66 46 16 224 319 234 65 205 83 22 297 317

Raw score 3 3 3 4 4 4 4 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6

Location

SE

Residual

-1.62 -1.62 -1.62 -1.28 -1.28 -1.28 -1.28 -0.99 -0.99 -0.99 -0.99 -0.99 -0.99 -0.99 -0.99 -0.99 -0.73 -0.73 -0.73 -0.73 0.73 -0.73 -0.73 -0.73

0.62 0.62 0.62 0.57 0.57 0.57 0.57 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.54 0.52 0.52 0.52 0.52 0.52 0.52 0.52 0.52

-0.59 -0.43 0.12 0.30 1.06 -0.95 -0.11 -1.09 -1.09 -0.62 -1.09 -1.09 0.56 -1.09 -0.99 -0.85 -1.28 -0.60 1.18 1.33 0.35 -0.95 0.51 0.85

Table 8. Lowest 53 Student Measures Figure Ground Letters in Words ID 324 65 66 80 82 83 150 64 156 79 164 166 167 276 199 203 205 151 23 18 81 12

Raw score 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Location

SE

Residual

ID

-4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20 -4.20

1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22

-

24 25 26 27 2 84 67 202 237 162 3 57 20 4 8 209 110 78 74 62 206 208

Raw score 0 0 0 0 0 6 7 8 8 9 9 9 12 13 13 13 14 14 14 14 15 15

Location

SE

Residual

-4.20 -4.20 -4.20 -4.20 -4.20 -1.69 -1.49 -1.31 -1.31 -1.15 -1.15 -1.15 -0.69 -0.55 -0.55 -0.55 -0.41 -0.41 -0.41 -0.41 -0.27 -0.27

1.22 1.22 1.22 1.22 .122 0.46 0.44 0.42 0.42 0.41 0.41 0.41 0.38 0.38 0.38 0.38 0.37 0.37 0.37 0.37 0.37 0.37

0.38 0.04 -1.60 1.29 -1.10 -1.01 1.99 -0.89 -1.45 -1.45 0.13 1.66 -2.72* 0.24 -2.16 0.19 -2.54*

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Table 8. (Continued). ID

Raw score

Location

SE

Residual

ID

Raw score

Location

SE

Residual

323

0

-4.20

1.22

-

114

16

-0.14

0.37

-3.30*

5 16

0 0

-4.20 -4.20

1.22 1.22

-

111 317

16 17

-0.14 -0.01

0.37 0.37

-1.20 -1.05

22 37

0 0

-4.20 -4.20

1.22 1.22

-

200

17

-0.01

0.37

4.15*

Notes on Table 8:*: Fit residual value exceeds limit set for test of fit.

SUMMARY OF FINDINGS Linear scales were created for Form Constancy of Letters and Numbers and Figure Ground Letters in Words using the RUMM2020 Program (Andrich, Sheridan, & Luo, 2005). The reliability of the two scales was shown by: 1. Global item fit as well as person item fit to the measurement model; 2. Good Person Separation Indices indicating that the person measures were reasonably well, or acceptably well, separated in relation to the errors; 3. Good item-trait interaction chi-squares indicating the measurement of a unidimensional trait; 4. Targeting of items against the person measures was reasonable, but indicates the need for easy and more difficult items in the scales for future use. Valid inferences may be drawn from the scales as the scale data were shown to be reliable. Inferences are that it is easiest for students to identify reversed numbers in a variety of fonts rather than the reversed letters and that the most difficult letters for students to identify as reversed when presented among a variety of fonts were long letters as in z, j, g, and d. For Form Constancy of Letters and Numbers, girls scored more highly than boys, but this was not statistically significant. There was no statistical significant difference between private and public schools, although public schools scored a higher mean average. Furthermore, there was as expected, a statistically significant difference in the performance of students as their age and grade increased, with younger students in lower grades scoring significantly lower than the older students in the higher grades. Students with the lowest scores were those who had most difficulty identifying reversed letters and numbers among a selection of letters and numbers presented in a variety of fonts. For Figure Ground Letters in Words the girls scored a higher mean average than boys, but this was not statistically significant. Public schools scored a statistically significant higher mean value than private schools. The younger students in the lower grades scored a lower mean value than the older students in the higher grades and this was statistically significant as would be expected. Students with the lowest scores had difficulty identifying words that contained a reversed letter as opposed to words that did not have a reversed letter embedded in the word.

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REFERENCES Andrich, D., Sheridan, B.E., & Luo , G. (2005). Rasch Unidimensional Measurement Models (RUMM2020): A windows-based item analysis program employing Rasch models. Perth, WA: RUMM Laboratory Andrich, D. & van Schoubroeck, L. (1989). The General Health Questionnaire: A psychometric analysis using latent trait theory. Psychological Medicine, 19, 469-485 Rasch, G. (1960/1980). Probabilistic models for some intelligence and attainment tests (expanded edition). Chicago, IL: MESA Press (original work published in 1960).

In: Specialized Rasch Measures… Editor: Russell F. Waugh, pp. 83-99

ISBN: 978-1-61668-032-9 © 2010 Nova Science Publishers, Inc.

Chapter 5

RASCH MEASURES OF NUMBER DISCRIMINATION AND REVERSAL, AND NUMBERS IN CALCULATIONS FOR YOUNG CHILDREN Janet Richmond, Russell F. Waugh and Deslea Konza Faculty of Education and Arts, Edith Cowan University, Perth, Western Australia.

ABSTRACT Number literacy is a very important topic and the Australian Government runs numeracy and literacy tests, administered through the State Education Departments, for all Year 3 (8 years old), Year 5 (10 years old) and Year 7 (12 years old) students. Results of these tests are reported to schools and parents with a view to ensuring that all children meet certain numeracy standards and that children who are „falling behind‟ are detected early so that remedial work can be given. Rasch measures were created with the RUMM2020 computer program for Visual Discrimination of Numbers (VDN) and Figure Ground Numbers in Calculations (FGNC). The student sample was N=324 pre-primary and primary students in Perth, Western Australia, aged 4-9 years old. Data on 20 items for VDN and 28 items for FGNC, where each item was scored in one of two categories (wrong scored zero and correct scored one), were Rasch analysed to create two linear scales. Six of the initial 20 items for VDN were deleted due to item misfit statistics, leaving 14 items. Thirteen of the initial 28 items for FGNC were deleted due to item misfit statistics, leaving 15 items. The final data for VDN and FGNC were used to create two highly reliable, linear, uni-dimensional scales (Student Separation Indices of 0.75 and 0.95 respectively) where the items are ordered from easy to hard and the student measures from low to high on the same scale. The two scales showed no statistically significant interaction of student measures on item difficulties along the scale, meaning that there was good agreement about the item difficulties along each scale, and each scale was unidimensional. The item-trait chi-squares are respectively, χ² = 68.34, df=0.92, p=0.12, and χ² = 58.83, df=60, p=0.52. The fit residual statistics for each of the two scales was reasonable and the targeting was reasonable.

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INTRODUCTION: LITERATURE REVIEW A National Literacy and Numeracy Plan was instituted in Australia to improve literacy and numeracy standards in the Australia. To identify students at risk, the National Assessment Program Literacy and Numeracy (NAPLAN) (Department of Education and Training, 2008) has been instituted to assess children in Year Three, Five and Seven, however this does not fit with early identification of students as stated by the Australian Council for Educational Research. In addition, the final report of literacy and numeracy review in Western Australia found that there was a need for pre-primary diagnostic assessment of numeracy skills to identify the students at risk (Department of Education and Training, 2007), while the Western Australian Government has developed a plan to improve the numeracy outcomes of students in Western Australia (Government of Western Australia, 2007). These policies and plans require relevant, linear, user friendly assessments to identify students at risk so that the plans to improve numeracy skills at the earliest opportunity can be implemented. Children having difficulty with the mechanics of mathematics (dyscalculia) are slow to grasp the relative size of figures, to learn tables, to remember the sequencing of digits, and to understand the meaning of mathematical signs or master fractions (Green & Chee, 1997). To manage mathematics as an academic subject, children need to use visual imagery in order to display planning, problem solving, and organisation, as well as have a good working memory (Green & Chee, 1997; Loikith, 1997). This link between symbolic language and mathematics was also identified by Johnson and Myklebust (1978), who found that the practical function in mathematics was to express quantitative and spatial relationships and the theoretical function in mathematics was to facilitate thinking. In addition, Lucas and Lowenberg (1996) separated mathematical concepts into two major aspects: (1) recognition and manipulation of numbers, and (2) acquisition and application of the language of mathematics, which in turn makes problem solving possible. To carry out mathematical computations, children must have an understanding or grasp of basic perceptions of shape, space, symbols, copying and numeracy (Chinn, 2002; Miles, Chinn, & Peer, 2000; Schneck, 1996). Furthermore, the manipulation of numbers in mathematics also requires good visual perceptual skills such as visual discrimination, directionality, sequencing, organisation of work (spatial), correct alignment of columns for calculation (placement of number values), figure ground and memory (Chinn, 2002). For example, many rows of calculations on a worksheet could be disorganising for the child with figure-ground problems. Spatial perceptual skills are required in geometry and visual memory is required when multiple steps are required in a sum (Schneck, 1996). A number of authors agree that to solve mathematical problems, understand geometric relationships and use graphs, children require recognition skills, the ability to discriminate and the ability to compare objects, form and space (including inversions, rotations and distortions) (Chinn, 2002; Fisher, Murray, & Bundy, 1991; Hung, Fisher, & Cremak, 1987; Levine, 1991; Schneck, 1996). Siegel (1999) described dyscalculia as “a crippling ailment that prevents one from learning math” (p. 305), while others (Fisher et al., 1991; Lucas & Lowenberg, 1996) found that difficulties with language may affect mathematical skills in the area of problem solving where problems are written in words rather than numbers. It has also been found that some learners had specific learning difficulties in mathematics where they could manipulate

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numbers orally and mentally, but were unable to record the responses as mathematical manipulations were primarily conducted in the right cerebral hemisphere of the brain while writing was primarily conducted in the left cerebral hemisphere (Fisher et al., 1991; Lucas & Lowenberg, 1996). The right hemisphere has an important role in understanding and applying mathematical concepts. Fisher et al. (1991) suggested that this deduction was based on associations between visual-spatial abilities and the understanding of mathematical concepts. The visual-spatial abilities can be determined in picture completion and copying tasks which are important predictors of arithmetic (mathematical) achievement (Belka & Williams, 1979; Sorter & Kulp, 2003). Thus, it appears that mathematical ability is affected by visual perceptual skills. These visual perceptual skills include, but are not limited to, visual memory, visual sequential memory, visual perception and specifically visual spatial ability (Belka & Williams, 1979; Chinn, 2002; Fisher et al., 1991; Green & Chee, 1997; Hung et al., 1987; Levine, 1991; Miles et al., 2000; Schneck, 1996; Simpson, 1987).

RASCH REPORT This report presents a Rasch analysis with the RUMM 2020 computer program (Andrich, Sheridan & Luo, 2005) in which two linear, unidimensional scales were created: (1) Visual Discrimination of Numbers and (2) Figure Ground Numbers in Calculations. These two scales relate to visual perceptual concepts of „visual discrimination‟ and „figure ground‟. This report describes the measurement results in terms of Rasch measurement fit statistics including global item and person fit to the measurement model, dimensionality, person separation indices, distribution of item-person interactions, and discrimination. Some discussion is included of the non-fitting items, as well as good fitting items, and the personitem threshold distribution (targeting). This is followed by mean Rasch measures by group and final items for the Visual Discrimination and Figure Ground Scales discussion. Finally, inferences drawn from the linear Rasch measurement data analysis and the summary of the results are presented.

INITIAL RASCH ANALYSIS An initial Rasch analysis was performed on the original items for Visual Discrimination of Numbers (20 items) and Figure Ground Numbers in Calculations (28 items) where each item was scored in one of two categories (incorrect answer scored zero and correct answer scored one). Six of the initial 20 items of Visual Discrimination of Numbers were deleted due to item misfit statistics. The remaining 14 items were found to have a good fit to the measurement model for the 324 persons included in this study. For Figure Ground Numbers in Calculations, 13 of the initial 28 items were removed because of item misfit statistics with the remaining 15 items were found to have an excellent fit to the measurement model. The Rasch analysis with the RUMM program does not indicate how to alter an item in order to make it fit the measurement model. In order to include, in a future measure, the deleted items

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which were initially considered conceptually valid, these would need to be changed and retested.

FINAL RASCH ANALYSIS RESULTS The following material shows the results for the final Rasch analysis for the two scales: (1) Visual Discrimination of Numbers (14 items), and (2) the Figure Ground Scale of Numbers in Calculations (15 items).

Summary of Fit Statistics The RUMM2020 program estimates an item-person interaction which establishes the overall fit statistics that determine whether the item estimations contribute meaningfully to the measurement of one construct. This calculation thus examines the consistency with which students responses agree with the calculated difficulty of each item on the scale. The standardised fit residual statistics (see Table 1) have a distribution with a mean near zero and a standard deviation near one when the data fit the measurement model, as is the case with these three measures. This means too that there is a good pattern of person and item responses consistent with a Rasch measurement model.

Dimensionality For Visual Discrimination of Numbers, there was an item-trait interaction chi-square of 68.34 with df = 0.92 and a probability of 0.12. This means that the scale is constructed with acceptable agreement amongst the students about the linear progressive difficulty of the items. For Numbers in Calculations, the item-trait interaction chi-square was 58.83 with df=0.93 and a probability of 0.52 respectively, showing very good agreement amongst the students about the item difficulties along the scale. This means that the students agree as to which items are easy, which are of medium difficulty and which are hardest. Table 1. Global Item and Student Fit Residual Statistics (N=324) ITEMS Location Visual Discrimination of Numbers (I=14) -0.42 -0.42 0.92 0.92 Numbers in Calculations (I=15) Mean 0.00 Standard Dev. 0.59

Fit Residual

PERSONS Location

Fit Residual

-0.45 0.89

+1.97 2.06

-0.20 0.75

-0.35 0.04

+1.29 2.11

-0.08 0.84

Comment on Table 1: Fit residuals have a mean near zero and a standard deviation near one when the data fit the measurement model (as is the case here). This reflects good consistency of item and student scoring patterns.

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Person Separation Index The Person Separation Index is an estimate of the true score variance among the students and the estimated observed score variance using the estimates of their ability measures and the standard error of these measures (Andrich & van Schoubroeck, 1989). For Visual Discrimination of Numbers and Numbers in Calculations, the Person Separation Indices are 0.75 and 0.95 respectfully. For a good measure, it is desirable that this index should be 0.9 or greater, as it is an indicator that the student measures are separated by more than their standard errors. Based on this index, the Visual Discrimination of Numbers demonstrates an acceptable separation, while Figure Ground Numbers in Calculations scale demonstrates very good separation of measures in comparison to the errors of measurement.

Individual Item Fit Items are ordered by calibrated values to evaluate their fit to the measurement model. The location of each item on the scale is the item difficulty in standard units, called logits (log odds of answering successfully). All the items in Visual Discrimination of Numbers fit the measurement model with probabilities greater than p=0.05 (see Table 2). The residuals shown in Table 2 represent the difference between the observed responses and the expected responses calculated from the Rasch measurement parameters. Standardised residuals should fall within the range of -2 and +2. Table 2 shows that all items for Visual Discrimination of Numbers have acceptable residuals. Table 2. Individual Item Fit Statistics for Visual Discrimination Numbers Item 6 17 2 4 7 3 19 10 9 18 5 13 11 20

Location -1.79 -1.34 -1.08 -0.81 -0.79 -0.78 -0.38 -0.33 0.69 0.93 0.96 1.43 1.53 1.78

Notes on Table 2:

7. 8. 9.

SE 0.33 0.28 0.25 0.23 0.23 0.23 0.21 0.20 0.16 0.16 0.16 0.15 0.15 0.15

Residual -0.98 -0.43 0.68 -1.65 -1.46 -1.25 -0.19 -0.89 -0.07 0.33 -0.75 -0.38 -1.30 1.80

DegFree 191.29 191.29 191.29 191.29 191.29 191.29 191.29 191.29 191.29 191.29 191.29 191.29 191.29 191.29

ChiSq 1.92 3.26 8.63 3.98 6.20 5.83 5.60 2.89 1.98 1.22 6.18 2.75 9.58 8.32

DegFree 4 4 4 4 4 4 4 4 4 4 4 4 4 4

Prob 0.75 0.52 0.07 0.41 0.18 0.21 0.23 0.58 0.74 0.87 0.19 0.60 0.05 0.08

Location refers to the difficulty of the item on the linear scale. SE means Standard Error, and refers to the degree of uncertainty in a value. Residual represents the difference between the expected value of an item, calculated according to the Rash measurement model and the actual value. 10. DegFree stands for degrees of freedom, and refers to the number of scores in a distribution that are free to change without changing the mean distribution. 11. ChSq stands for Chi-square

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12. Prob relates to the probability based on the Chi-square and refers to the levels of certainty to which an item fits the measurement model.

For Figure Ground Numbers in Calculations, all the items fit the measurement model with probabilities greater than p=0.08 (see Table 3) and residuals are very satisfactory. Table 3. Individual Item Fit Statistics for Figure Ground Numbers in Calculations Item

Location

SE

Residual

DegFree

ChiSq

DegFree

Prob

13

-0.94

0.20

-0.46

159.60

2.45

4

0.65

12

-0.88

0.20

-0.44

159.60

3.26

4

0.52

7

-0.64

0.20

+0.61

159.60

2.00

4

0.74

14

-0.57

0.19

-1.19

159.60

4.76

4

0.31

8

-0.32

0.19

+0.73

159.60

2.19

4

0.70

11

-0.17

0.19

-0.27

159.60

1.93

4

0.75

10

-0.05

0.19

+1.22

159.60

2.22

4

0.70

9

+0.03

0.18

+1.12

159.60

4.67

4

0.32

20

+0.22

0.18

-1.33

159.60

3.99

4

0.41

21

+0.22

0.18

-1.52

159.60

5.38

4

0.25

16

+0.38

0.18

-0.29

159.60

4.55

4

0.34

15

+0.40

0.18

+0.79

159.60

3.36

4

0.50

25

+0.43

0.18

-1.52

159.60

8.21

4

0.08

27

+0.81

0.18

-0.79

159.60

2.90

4

0.58

-1.97

159.60

7.00

4

0.14

24

+1.07

0.18

Notes on Table 3:

1. 2. 3. 4. 5. 6.

Location refers to the difficulty of the item on the linear scale. SE means Standard Error, and refers to the degree of uncertainty in a value. Residual represents the difference between the expected value of an item, calculated according to the Rash measurement model and the actual value. DegFree stands for degrees of freedom, and refers to the number of scores in a distribution that are free to change without changing the mean distribution. ChSq stands for Chi-square Prob relates to the probability based on the Chi-square and refers to the levels of certainty to which an item fits the measurement model.

Targeting The RUMM2020 program produces a student-measure item-difficulty or targeting graph on which the student measures are placed on the same scale as the item difficulties in standard units called logits. For Visual Discrimination of Numbers (see Figure 1), the targeting graph shows that the student measures cover a range of about -1.2 to +3.8 logits and the item difficulties cover a range of about -1.8 to +1.8 logits. From the graph it can be seen that many students (about 215) were able to answer the items correctly and the targeting of the items needs to be improved in any future use of the scale by adding in some harder items to „cover‟ the students with the higher measures.

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Note: Student measures are on the upper side in logits. Item difficulties are on the lower side of the same scale in logits. Many students (about 215) answered the items correctly. Figure 1. Targeting for Visual Discrimination Numbers.

For Figure Ground Numbers in Calculations (see Figure 2), the targeting graph shows that the student measures cover a range of about -3.4 to +3.3 logits and the item difficulties cover a range of about -1.0 to +1.2 logits. From the graph it can be seen that many students (about 195) were able to answer the items correctly, while about 42 were unable to answer any items correctly, thus the targeting of the items needs to be improved in any future use of the scale by adding in some easier and more difficult items to „cover‟ the students with the lower and higher measures.

Note: Student measures are on the upper side in logits. Item difficulties are on the lower side of the same scale in logits. Many students (about 215) answered the items correctly. Figure 2. Targeting for Figure Ground Numbers in Calculations.

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Discrimination Item Characteristic Curves examine the relationship between the expected response and the mean group student measures. These curves display how well the item discriminates between groups of persons. An example of one item characteristic curve for each of the two constructs will be presented. Figure 3 shows the Item Characteristic Curve for Item 13 Visual Discrimination of Numbers. This curve shows that the item discriminates well for students with different measures. The Item Characteristic Curves for all the other items were checked and found to be satisfactory (but are not reported here to avoid unnecessary repetition).

Figure 3. Item Characteristic Curve: Item 13 – Visual Discrimination Numbers.

Figure 4. Item Characteristic Curve: Item 4 – Figure Ground Numbers in Calculations.

Figure 4 shows the Item Characteristic Curves for Item 4 of Figure Ground Numbers in Calculations. These items discriminate well for students with different measures. The Item

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Characteristic Curves for all the other items in the measure were checked and found to be satisfactory (but are not reported here to avoid unnecessary repetition).

Consistency of Use of Scoring Categories The RUMM2020 program produces graphs of the scoring categories for each item. The Scoring Category Curves show the relationship between the probability of scoring in each category (zero for incorrect answer and one for correct answer) on each item. Figure 5 is the Scoring Category Curve for item 2 of Visual Discrimination of Numbers. This figure shows that the scoring was done logically and consistently. When students have low measures on item 2, then they have a high probability of obtaining a zero score (answer incorrect) and, when they have a high measure, they have a high probability of scoring 1 (answer correct). The Scoring Category Curves for all the other items were checked and they were satisfactory too. The Scoring Category Curves for all the items of the other variable, Figure Ground Numbers in Calculations, were checked and they were also found to be satisfactory, but they are not presented here to avoid repetition.

Figure 5. Scoring Category Curve: Item2 – Visual Discrimination of Numbers.

CHARACTERISTICS OF THE SAMPLE (VDN AND FGNC) The measures for Visual Discrimination of Numbers (VDN) were displayed in a graphical format separated by gender (Figure 6), type of school (Figure 7), age (Figure 8), grade (Figure 9) and whether intervention had been received (Figure 10). Females have a higher mean measure than males for Visual Discrimination of Numbers but this is not statistically, significantly different (t=1.78, df=320, p=0.04). Public school students have a higher mean measure than private school students for Visual Discrimination of Numbers and this is not statistically, significantly different (t=1.39, df=320, p=0.03). As would be expected, the mean measures generally increased by age from four years old (lowest) to ten years old or

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older (highest) and this was statistically, significantly different (t=8.79, df=65, p