MATH PLAY’S EFFECTS ON MATHEMATICS ATTITUDE AND AWARENESS: A RASCH ANALYSIS By Jean Bowen B.Sc., University of Northern British Columbia, 1999 THESIS IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MATHEMATICS UNIVERSITY OF NORTHEN BRITISH COLUMBIA March 2018 © Jean Bowen, 2018 Abstract Where children see mathematics in the world and the attitudes children hold toward mathematics are not thoroughly understood. The Mathematics Attitude Assessment (MAA) and the Difficulty Associating Words with Mathematics (DAWM) are instruments developed to examine mathematics attitude and the propensity of primary students to indicate words as mathematics related. These instruments were used to measure any changes among preintervention, post-intervention and one-month post-intervention of this quasi-experimental design. The intervention was Math Play: a collection of mathematics-based games and activities with emphasis on exploration and problem solving. Rasch analysis and traditional analysis techniques were used to examine the participants, the assessment instruments and any effects. The difference scores were analysed using a 2X2 Two-Factor ANOVA. A statistically significant difference was found for pre-treatment to one-month post-treatment for the DAWM with each participant as an experimental unit. No statistically significant differences were found for the MAA results. ii Table of Contents Abstract ..................................................................................................................................... ii Table of Contents ..................................................................................................................... iii List of Tables ............................................................................................................................. vi List of Figures ......................................................................................................................... viii Acknowledgments ..................................................................................................................... x Dedication ................................................................................................................................ xi Chapter One Introduction and Background ............................................................................. 1 Attitudes Toward Mathematics ........................................................................................... 1 Summary ............................................................................................................................ 11 Statement of Problem ........................................................................................................ 12 The Research Questions ..................................................................................................... 13 Specific Research Questions .............................................................................................. 14 Research Objectives ........................................................................................................... 14 Significance of this Study ................................................................................................... 15 Chapter Two – Rasch Analysis ................................................................................................ 16 Rasch Analysis as a Measurement Instrument .................................................................. 16 What is Rasch Analysis? ..................................................................................................... 16 Chapter Three Methods ......................................................................................................... 29 iii Procedure ........................................................................................................................... 29 Approval Process ................................................................................................................ 29 Group Placement ............................................................................................................... 30 Math Play Facilitators......................................................................................................... 31 Timeline .............................................................................................................................. 32 Assessments ....................................................................................................................... 33 The Assessment Instruments ............................................................................................. 33 Difficulty Associating Words with Math (DAWM) assessment. ..................................... 33 Math Attitude Assessment............................................................................................. 34 Study Design ....................................................................................................................... 36 Intervention Group ............................................................................................................ 37 The Games .......................................................................................................................... 39 Collected Data .................................................................................................................... 41 Ethical Considerations ........................................................................................................ 41 Chapter Four Results .............................................................................................................. 42 Difficulty Associating Words with Mathematics ................................................................ 43 Missing Data Analysis. ........................................................................................................ 60 Traditional Statistical Analysis of the DAWM Results ........................................................ 61 DAWM with Students as the Experimental Unit ................................................................ 64 iv Summary of Analysis of DAWM Assessment Intervention ................................................ 76 DAWM with Schools as a Block and Treatment and Control Separating the Units ........... 77 Math Attitude Assessment (MAA) ..................................................................................... 80 Traditional Statistical Analysis of the MAA Results............................................................ 95 Overall Summary .............................................................................................................. 108 Chapter Five Discussion ....................................................................................................... 110 Conclusion ........................................................................................................................ 111 Delimitations, Limitations and Future Research .............................................................. 114 Implications ...................................................................................................................... 116 Summary .......................................................................................................................... 117 References ............................................................................................................................ 118 Appendix .............................................................................................................................. 123 Appendix A Forms and Consent ....................................................................................... 123 Appendix B Attitude Assessment Instruments ................................................................ 150 Appendix C Games and Instructions ................................................................................ 160 Appendix D Rasch Material .............................................................................................. 200 v List of Tables Table 1 Fit Statistics and Interpretations .............................................................................. 26 Table 2 Reasonable Item Mean-Squares Ranges for Infit and Outfit ................................... 28 Table 3 Excerpt DAWM Participant Measures ...................................................................... 44 Table 4 Item Characteristics for the Difficulty Associating Words with Mathematics DAWM ................................................................................................................................................ 48 Table 5 Anchored Logit Measure, Gender and Group (Treatment or Control) for February, April and May DAWM Excerpt ............................................................................................... 59 Table 6 DAWM Descriptive Statistics by Group and Gender April – February Difference Measure ................................................................................................................................. 63 Table 7 DAWM Descriptive Statistics by Group and Gender May – April Difference Measure ................................................................................................................................................ 63 Table 8 DAWM Descriptive Statistics by Group and Gender May – February Difference Measure ................................................................................................................................. 63 Table 9 DAWM Between-Subject Effects Statistical Significance (p) .................................... 71 Table 10 DAWM Estimated Marginal Means and Standard Error April – February ............. 72 Table 11 DAWM Estimated Marginal Means and Standard Error May – April ..................... 73 Table 12 DAWM Estimated Marginal Means and Standard Error May – February .............. 74 Table 13 The Number of Participants in each Between Subject Factor ............................... 78 Table 14 Excerpt MAA Participant Measures........................................................................ 83 Table 15 Item Difficulty Math Attitude Assessment (MAA) .................................................. 85 Table 16 Items from the MAA Ordered Hardest to Easiest by Logit Measure ..................... 86 vi Table 17 Measure, Gender and Group (Treatment or Control) for February, April and May MAA Excerpt........................................................................................................................... 93 Table 18 MAA Descriptive Statistics by Group and Gender April – February Difference Measure ................................................................................................................................. 96 Table 19 MAA Descriptive Statistics by Group and Gender May – April Difference Measure ................................................................................................................................................ 96 Table 20 MAA Descriptive Statistics by Group and Gender May – February Difference Measure ................................................................................................................................. 97 Table 21 MAA Between-Subject Effects Statistical Significance (p).................................... 103 Table 22 MAA Estimated Marginal Means and Standard Error April – February ............... 104 Table 23 MAA Estimated Marginal Means and Standard Error May – April ...................... 105 Table 24 MAA Estimated Marginal Means and Standard Error May – February ............... 106 vii List of Figures Figure 1. An excerpt from work by Stodolsky, et al., (1991, p. 97) indicting percent of students using the indicated words to describe either mathematics or social studies. ......... 9 Figure 2. A sample of the DAWM assessment. ...................................................................... 34 Figure 3. The statements from the MAA ordered by factor. ................................................. 35 Figure 4. DAWM item and outfit z scores: A graphical display of the fit. .............................. 51 Figure 5. DAWM Persons and items mapped by difficulty. Persons on the left and items on the right. A dot ( ∙ ) represents 1 to 2 people and a number sign (#) represents 3 people. . 54 Figure 6. DAWM item versus average person measure for item endorsement. ................... 56 Figure 7. DAWM frequency distribution and normal curve for April – February data. ......... 65 Figure 8. DAWM frequency distribution and normal curve for May – April data. ................ 66 Figure 9. DAWM frequency distribution and normal curve for May – February data. ......... 67 Figure 10. DAWM box-plot for April – February data test for outliers with difference score scale on the vertical axis. ....................................................................................................... 68 Figure 11. DAWM box-plot for May – April data test for outliers with difference score scale on the vertical axis. ................................................................................................................ 69 Figure 12. DAWM box-plot for May – February data test for outliers with difference score scale on the vertical axis. ....................................................................................................... 69 Figure 13. DAWM estimated marginal means versus group April – February separated by gender. ................................................................................................................................... 73 Figure 14. DAWM estimated marginal means versus group May – April by gender. ........... 74 Figure 15. DAWM estimated marginal means versus group May – February by gender...... 75 viii Figure 16. Treatment and control groups versus blocks of the April – February data .......... 79 Figure 17. MAA items z scores of Outfit. ............................................................................... 88 Figure 18. MAA item z score of outfit item 6 removed. ........................................................ 89 Figure 19. MAA person ability and item difficulty map. ........................................................ 91 Figure 20. MAA item versus person measure for item endorsement. .................................. 92 Figure 21. MAA frequency distribution and normal curve for April – February data. .......... 98 Figure 22. MAA frequency distribution and normal curve for May – April data. .................. 99 Figure 23. MAA frequency distribution and normal curve for May – February data. ......... 100 Figure 24. MAA box-plot for April – February data test for outliers. .................................. 101 Figure 25. MAA box-plot for May – April data test for outliers. .......................................... 101 Figure 26. MAA box-plot for May – February data test for outliers. ................................... 102 Figure 27. MAA estimated marginal means versus group April – February separated by gender with 2XSE error bars. ............................................................................................... 104 Figure 28. MAA estimated marginal means versus group May – April separated by gender with 2XSE error bars. ............................................................................................................ 105 Figure 29. MAA estimated marginal means versus group May – February separated by gender with 2XSE error bars. ............................................................................................... 106 ix Acknowledgments I would like to thank and acknowledge the support of: My family – I would not have been able to straighten this tangled mess of letters, numbers and symbols without your love and encouragement. My committee (in every iteration) – Your guidance and support were un-paralleled and unfaltering. My colleagues – Your patience, understanding and assistance were major contributors to the end-product. My friends – Are you still there? Of course you are! You are always there to lend an ear, a tissue or a critical eye: whichever complimented the situation. School District 57 – Everyone of you was so welcoming and inclusive in your space. This would not of have been possible without you. MATH 190 volunteer facilitators – What can I say? You were all prime figures in allowing this work to happen. Thank you. x Dedication In memory of: Florence Elaine Taylor (nee Coates) Feb 19, 1930 – Dec. 25, 2017 A perfectly cooked breakfast, any kitten or puppy, and a sting of pearls around my neck on my wedding day. and Gracie Brown-Ryburn Leitch (nee Yates) May 15, 1923 – Apr. 10, 2018 “They are in terrible, terrible trouble,” a strong cup of tea in a proper tea cup, and my first Blizzard®. Two amazing, strong women who impacted my life in countless ways. xi Chapter One Introduction and Background Where children see mathematics in the world around them and the attitudes children hold toward mathematics are not thoroughly understood. Attitudes toward mathematics, also referred to as mathematical affect, is an ongoing area of research. Research into attitudes towards mathematics does not all focus on the same aspects of attitude. For example, research into attitudes toward mathematics introduces definitions of attitudes toward mathematics, methods of measuring those attitudes, the results from various attitude assessments and methods used to change those attitudes. Studies in these areas include various age groups. Attitudes Toward Mathematics What is mathematics attitude? The answer to this question is not a simple one. The literature provides a number of definitions of attitudes towards mathematics and extensive information on factors which influence these attitudes. Attitudes towards mathematics is a specific type of mathematical affect or emotion associated with mathematics. Mathematics attitude is often not defined in studies on the subject. Instead the studies focus on factors that influence attitudes. According to Hannula (2002, p.1) mathematics attitude consists of the emotions associated with mathematics. Specifically, Hannula’s emotion-based definition was broken down to, “the emotions aroused in the situation”, “emotions associated with the stimuli”, “expected consequences” and “relating the situation to personal values.” Hannula’s definition is more complex than the earlier definition of Jadav and Quinn 1 (1987) who simply defined liking mathematics as having a positive attitude towards mathematics; a uni-dimensional definition. The study by Jadav and Quinn (1987) was a meta-analysis of articles investigating attitudes toward mathematics. The meta-analysis required that self-concept (or selfimage) not be a factor associated with attitude. This exclusion of self-concept is in contrast to the factors later examined by Marsh and Tapia (2004) and Baumert, Koller, Ludtuke, Marsh and Trautwein (2005) as both studies included self-concept in their factor list. Tapia and Marsh (2004) included self-confidence in the factors they examined which is directly related to self-concept. The study by Baumart et al. (2005) focussed on self concept. The list of factors across studies which may contribute to or define mathematical attitude is extensive and at times contradictory. Uni-dimensional or multifactorial, there is not a widely accepted definition of attitudes toward mathematics. Math attitude assessments instruments. With the variation in the factors being included in the definitions of attitudes towards mathematics, it is not a surprise that there are a number of ways that attitudes toward mathematics have been assessed. Hannula (2002) collected information through dialogues. The dialogues were then examined for statements that described mathematics affect and the researcher provided interpretations on the dialogues. This method was the least common method of information collection. Self-reporting through questionnaires was a far more common method for collecting data. In “A review of instruments created to assess affect in mathematics,” 2 Chamberlin (2010) reviewed a progression of instruments used to assess mathematics affect for secondary school to college age students. The first instrument reviewed was The National Longitudinal Study of Mathematical Abilities (NLSMA). The NLSMA identified attitude as “uni-dimensional” (Chamberlin, 2010). The NLSMA did not define or specify attributes that form attitude. Instead the NLSMA directly asked what the participant’s attitude toward mathematics was. Chamberlin explained that Aiken expanded the attitude assessments to include enjoyment and value of mathematics in the Mathematics Attitude Inventory (MAI) (Chamberlin, 2010). Chamberlin’s review of assessment instruments of attitudes toward mathematics continued with work by Fennema and Sherman from 1976 (Chamberlin, 2010). The next step in the evolution of mathematics affect assessment instruments was Fennema-Sherman Mathematics Attitude Scales (FSMAS) in 1976. The FSMAS instrument assessed mathematics affect based on four components: attitude, self-efficacy, anxiety, and value of mathematics. When Chamberlin did the review in 2010 the FSMAS was still being used. The draw back to the FSMAS was the language used. Chamberlin was concerned that over time the meanings of words change so the test became dated (Chamberlin, 2010). The final instrument reviewed by Chamberlin was the Attitude Towards Mathematics Inventory (ATMI) created by Tapia and Marsh (Tapia & Marsh, 2004). Unlike the NLSMA which examined attitude as a uni-dimensional attribute or the FSMAS which includes attitude in the factors examined when identifying mathematical affect, Tapia and Marsh did not directly include attitude in the factors but instead looked at factors they thought contributed to attitude (Chamberlin, 2010). At the onset of the study, the ATMI 3 assessed attitude defined by six attributes: self-confidence (self-concept), anxiety, value, enjoyment, motivation, and teacher/parent expectations. However, through their analysis of the results they modified the assessment to include self-confidence, value, enjoyment and motivation. (See Appendix B for the complete ATMI) (Tapia & Marsh, 2004). After the review by Chamberlin was published, two of the instruments reviewed were modified: The Fennema-Sherman Mathematics Attitude Scale and the ATMI. The Fennema-Sherman Attitude Scale was re-examined and modified by Doepken, Lawsky, and Padwa (2013) and the language was updated. In 2012, Chapman and Lim reevaluated the ATMI and found such a strong link between motivation and enjoyment that they suggested removing the motivation factor (Chapman & Lim, 2012). Removal of motivation is in contrast to work by Hannula (2006) who focussed exclusively on motivation. It is evident that the creation of an instrument to assess attitudes toward mathematics is a complex and ever developing process. Assessing Attitudes Toward Mathematics in Primary students. All the previously discussed mathematics attitude assessment instruments were designed for and used with students from high school age and up. However, questionnaires are employed with primary students as well. Questionnaires included various factors and were of varying lengths. Tezer and Karasel (2010) used a 10 item questionnaire to assess the attitudes of 230 Grade 2 and 3 students. The students were given the option of a face representing “very happy, happy, neutral and sad” to use to respond to the items. Some example statements include, ‘“What I learn in my math course I use in my daily life” with which 4 idea do you emotionally agree to?’ and ‘”The course I mostly like is math” which facial expression would you reply with for this idea?’ The study reported that a positive attitude toward mathematics was found. Another questionnaire-based mathematics attitude assessment was used by Dowker, Bennett and Smith (2012). They used a 28-question format and focussed on seven areas; “maths in general, written sums, mental sums, easy maths, difficult maths, maths tests, and understanding the teacher”. The participants for this study were 44 grade 3 students and 45 grade 5 students. The participants responded to each item with a self-rating, a degree of liking, level of anxiety and level of unhappiness. An attitude “above neutral” was reported with no gender difference. Another study was a meta-analysis by Quinn and Jadav (1987). They included grade 2 to 6 and examined the results from 1758 students. One of the studies included 11 questions that were not specified in the article nor were any factors specified. A second study in the meta-analysis used the Survey of School Attitudes. The Survey of School Attitudes was designed by T. P. Hogan in 1975 (Jadav & Quinn, 1987). The focus of this study was not to assess attitude as much as it was to look at the relationship between attitude and achievement. The studies that were included in the meta-analysis had four attributes in common: (1) Mathematics and reading evaluative data regarding attitude and achievement had to be collected from testing that occurred at the same time and for the same grade on two or more occasions. 5 (2) The attitude assessment needed to include liking a subject, but no self-concept. (3) The achievement assessment needed to be obtained from specific assessment tests and not teacher grades. (4) The original data had to be available to be re-examined. The component of this study most relevant to attitudes of primary students toward mathematics is point (2) the attitude assessment needed to include liking a subject, but no self-concept. This is counter to some of the pervious attitude assessment instruments (Tapia & Marsh, 2004, Aiken as cited in Chamberlin, 2010). Levine (1972) used a questionnaire with children as young as grade 3. The sample included 144 grade 3, 4, and 6 students. Unlike some of the other studies regarding only attitudes toward mathematics of elementary school students this study was designed to compare attitudes across school subjects. Levine included the following statements: 1) I enjoy studying this subject the most. 2) I do my best work in this subject. 3) I think this subject is the most important subject I study in school 4) My parents are able to help me most in this subject. 5) My parents feel that this should be my best subject. 6) I wish this was my best subject. 7) I feel I need the most help in this subject. 8) I feel my teacher does her (his) best job in teaching this subject. 9) This is my teacher’s favorite subject. (p. 53) 6 Students and their parents were asked to rank English, Mathematics, Science and Social Studies using the nine statements. Results from assessing attitudes toward mathematics. The results from various mathematics attitude assessments are not consistent. Dowker et al. (2012) found that students tend to have a positive attitude toward mathematics. However, Hannula, (2002) reports that attitudes towards mathematics vary but tend to worsen as children progress through school. Arnold, Fisher, Doctoroff, and Dobbs (as cited in Geist, 2010) claim that the attitudes children have form early and are difficult to change. Kogce, Yildiz, Aydin and Altindag (2009) found a correlation between early attitudes and later attitudes of students as they progress from elementary school to secondary school. The role gender has in affecting mathematics attitude varies. No gender difference was found by Kogce et al, 2009, and Ma and Kishor (1997). However, a gender difference was found by Hannula (2002). These findings together could be interpreted as students’ generally positive attitudes toward mathematics are hard to change but if attitudes do change they will decline. This supports an incentive for mathematics researchers and educators to assess attitudes early and take steps to keep attitudes positive. When students like mathematics and how they describe it. Liking of mathematics was found to be based on getting the correct answers and finding the work easy (Stodolsky, Salk, & Glaessner, 1991). They found this was in contrast to the liking of social studies. Social studies was liked if the topic was interesting or the activities were enjoyed. 7 The same study also found that when students were asked to describe mathematics and social studies that the number and nature of the words they chose were very different (Stodolsky et al, 1991). The words and concepts in Figure 1 are a summary of the discussions held with the grade 5 students in the study. The words were not given to them. When describing mathematics and social studies, the responses broke down as in Figure 1. The list for social studies continued with six more categories. What is of note is the difference in the percent of students that agree about the mathematics definition versus that of social studies. Also, the number of categories used to define social studies is greater. Mathematics was described with fewer words that were more specific. Social studies was described with more words that were more general. Games did not appear on the list for mathematics. 8 Words used to define Math Words used to define Social Studies Addition 77% Special place, event, person, period 67% Subtraction 68% History 48% Multiplication 62% About People 30% Numbers 62% Cultures 20% Division 58% Wars 13% Fractions or decimals 30% Projects 12% Measurement 12% Social living 10% Doing Problems 10% Events-dates 10% Word Problems 10% Maps: read & make 8% Geometry 10% Land forms 8% Counting money 7% Reading 7% Telling time 5% Definitions 7% Miscellaneous 13% Countries 7% Figure 1. An excerpt from work by Stodolsky, et al., (1991, p. 97) indicting percent of students using the indicated words to describe either mathematics or social studies. The relationship between mathematics attitude and achievement. The link between attitudes and achievement has been studied extensively but the results from these studies are often contradictory. These contradictions may in part arise because of the different factors used to define attitude and the different instruments used to assess attitude. In a study of seventh graders by Marsh et al, (2005) self-concept, a component of attitude from several assessment instruments, was found to predict achievement. Dowker et al. (2012) also found a link between self-rating and performance in third and fifth grade students. De Lourdes, Monteiro and Peixoto (2012), found that there was not a link between achievement and attitude. This finding did not support work by Anttonen 9 (1969) who found a correlation between attitude and achievement. Moenikia and ZahedBabelan (2009), identified attitude toward mathematics as a cause for achievement in mathematics. The possibility that there is a link between attitude and achievement further supports the need to identify attitude and try to improve it as much as possible. There is a lot of focus on mathematics achievement but studies which identify a relationship between attitude and achievement lend support for the need to focus on attitude and achievement simultaneously. Why Play with Mathematics? How children experience and learn mathematics is diverse. Children will engage in mathematics play during free play sessions (as cited in Capacity Building Series, 2011). Ginsburg goes on to explain that the way a child thinks, “is not limited to the concrete and mechanical; it is often complex and abstract” (Capacity Building Series, 2011, p. 1). MacDonald (2014) states that algorithms are needed to ensure that students are able to solve problems in a timely manner and to allow students to perform mathematics at a higher level. Teaching algorithms is necessary because there are some concepts students need to master that would take too long to solve through discovery if the students found the solutions at all. However, Robinson (2011) has offered the view that we are educating the creativity out of our children. The article by MacDonald referenced work by Mighton, who agreed with the need for discovery and creativity in mathematics (Mighton as cited in MacDonald, 2014). Mathematics education is not limited to the classroom taught topics. 10 “Play expands intelligence, stimulates the imagination, encourages creative problem solving, and helps develop confidence, self-esteem, and a positive attitude toward learning.” This is a quote from Dr. Mustad in an article titled CMEC Statement on Play-Based Learning produced by the Council of Ministers of Education, Canada (2010). Now consider the findings of Geist (2010), who stated that a dislike of mathematics is influenced by high stakes situations and the stress associated with timed tests (especially for female students). Finally add the work of Tapia and Marsh (2004) who found that selfconfidence, enjoyment, motivation and value were four factors which could be used to measure attitudes toward mathematics. These finding could be put together to create a possible instrument to effect attitudes toward mathematics. Summary There is not a concise or consistent definition of attitudes toward mathematics. This contributes to the variety of instruments used when studies address attitudes toward mathematics. Attitudes have been found to be positive but decline as students progress through school. Often work on attitudes toward mathematics had been done with older elementary students to university students. Addressing attitudes toward mathematics in the late primary years may help researchers’ and educators’ understanding of attitudes toward mathematics in the later years. Identifying attitudes toward mathematics and improving attitudes toward mathematics are two separate yet related concepts. Improving attitudes may be done by addressing the aspects of mathematics which are related to why people dislike it and by 11 increasing the exposure to aspects of mathematics through play. The idea of play is supported by the benefits associated with play. Some of the benefits of play directly address factors which have been identified as influences of attitudes toward mathematics. Statement of Problem The focus of this study addresses two components: 1) How can awareness of mathematics in the world around them, mathematics attitude and changes in awareness and attitudes toward mathematics in Primary school students be measured; 2) Where do grade 2 and or 3 students see mathematics in the world around them, what are the attitudes toward mathematics of primary students and does a “Math Play” intervention improve either of these? For the purpose of this study Math Play was created as a collection of mathematics-based games and activities where the emphasis is exploration and problem solving, not correction and criticism. The lack of a consistent definition of attitude toward mathematics and consistently used instruments used to assess those attitudes may contribute to the variety in study results. It is like measuring the quality of a picture without defining quality and without specifying what a picture is. Is it a painting, a polaroid, photo on a smart phone or graffiti on a wall? Is quality a subjective thing or is the interest in framing, lighting, composition, and perspective? A consistent definition and an instrument with known properties that is used repeatedly makes comparison of research findings, changes with time and measuring effects of interventions easier. 12 It is one thing to know what attitudes exist, but if those attitudes are not positive, improving those attitudes is crucial. Introducing mathematics in a way that addresses or removes areas of mathematics that have been identified as being related to negative attitudes toward mathematics may be able to improve attitudes toward mathematics. Mathematics was described using words associated with arithmetic by over half of the students in the study by Slassner et al (1991), and several areas of mathematics were not noted (for example: patterns, probability, graphs, solving equations). The association with arithmetic may be to the detriment of the other areas in day to day life that use and involve mathematics. Trying to expand what students think of math as, and associate math with, may expand their understanding of the subject and its connection to everyday activities they take part in. The Research Questions 1) Do the mathematics instruments Difficulty Associating Words with Mathematics (DAWM), which was created for this study, and Mathematics Attitude Assessment instrument (MAA), which was modified for this study, have the necessary psychometric properties to allow assessment of attitude change in attitudes? 2) If a component of mathematics that does not include timed tests nor being graded, but rather exploration and experience based, (Math Play), is introduced will the overall attitudes toward mathematics improve? Across both genders? And will improvement persist one-month post treatment? 13 Specific Research Questions 1) Is the Difficulty Associating Words with Mathematics assessment (DAWM) an appropriate assessment instrument for the participants? 2) Will Math Play change the propensity of participants’ endorsement of words associated with mathematics? 3) If there is a change in the DAWM, will it persist after Math Play has ended? 4) Is there a gender difference for propensity to associate words with mathematics? 5) Is the Math Attitude Assessment (MAA) an appropriate instrument to assess the attitudes of the participants toward mathematics? 6) Will Math Play change the expressed attitude of the participants toward mathematics? 7) If there is a change in the MAA results, will it persist after Math Play has ended? 8) Is there a difference in the MAA results between the genders? 9) Is there a correlation between the MAA and the DAWM assessment results? Research Objectives There are multiple objectives for this research. First, instruments must be modified/created to assess attitudes toward mathematics and the propensity to identify mathematics in the world that is appropriate for a Primary school student participant group. Second, the psychometric properties of these instruments must be assessed. Third, an intervention must be designed (Math Play - low stress, low risk, high likelihood of success math related games). Fourth, the effectiveness of this intervention must be assessed. 14 Significance of this Study Research significance. This study will contribute to the understanding of attitudes toward mathematics for Primary school students and attitudes toward mathematics measurement instruments for Primary school students as well as the propensity to endorse words and concepts as being mathematics related. Further this study will determine if Math Play has any effect on mathematic affect (the emotions associated with mathematics) and the propensity to endorse words and concepts as mathematics related. Practical significance. This study will contribute a possible technique for improving attitudes toward mathematics as well as increasing awareness of where mathematics is in the world. This study will also provide a possible technique for improving the understanding of what mathematics is through improving the vocabulary of students with relation to mathematics. 15 Chapter Two – Rasch Analysis Rasch Analysis as a Measurement Instrument Studies of mathematics attitude scale development have used a wide variety of methods to investigate the properties of the instruments to be employed. Others who have then used these instruments may or may not adapt the instruments for their target samples. Often a re-evaluation of the properties of the instrument is not done when the instrument is used with a new population. Rasch analysis is an approach to evaluate the instruments. Rasch analysis promises sample free estimates of item difficulty and person ability estimates free of the effects of the sample of items used to measure the person’s characteristics (Bond & Fox, 2012; Linacre, 2012a,b,c,d, n.d.; Lochhead, 2009; Sebok, 2010). What is Rasch Analysis? Not all people are of equal ability and not all questions are of equal difficulty. For example, on a multiplication assessment there could be the following two items: (A) Find the product 2 × 4, and (B) Find the product 246 × 14. The first item (A) would be considered easier than the second item (B). If there were two people in the class, who wrote the test, it would be expected that the mathematics skill of a person who correctly answered only the first questions to be lower than the ability of a person who correctly answered both questions. We can then assess each person’s ability on how many items they got correct; few disagree with this approach – a higher score means more skill. What happens if one takes the difficulty of the question or item into account when assessing ability? If Person M got the first item correct and the second item wrong, and Person N 16 got the first item (the easier item) incorrect and the second item (the harder item) correct: which examinee has a greater ability? Classical Test Theory, observed score practices (Crocker & Algina, 2006) does not address this conundrum – a score of ‘1’ is a score of ‘1’. The issue of the ability of Person N, who scores a difficult item correctly and an easy item incorrectly, is not addressed even if it is noticed. Rasch analysis, developed by Georg Rasch in Denmark (Rasch, 1993), takes both the difficulty of the question and the ability of the respondent into consideration when creating measures for persons and items (Linacre, 2012a). Other statistics (fit statistics) developed in the realization of these Rasch models by Wright, Linacre and others (Huang, 2015; Linacre, 2012a, b, c, d, e) allow for the assessment of aberrant behaviors such as getting easy items incorrect and difficult items correct. Item properties such as difficulty and ease are not confined to dichotomously scored achievement items. Linacre introduces the concept of difficulty and easy using items endorsed by participants indicating their liking of science activities, on a three-point scale. The participants had to indicate “like”, “neutral”, or “dislike”, for items including but not limited to: “watch a rat”, “go to zoo” and “talk w/friends about plants” (Linacre, 2012b, p. 3). The responses were then used to order the items in terms of difficulty. The Rasch analysis of the responses indicated that the order of difficulty from easiest to most difficult, of these sample items, was; “go to zoo”, “talk w/friends about plants”, then “watch a rat” (Linacre, 2012b, p. 5). Other examples Linacre used were: it is easier to “hit a single” than a “home run”, and “division” is more difficult than “addition” (Linacre, 2012b, p. 11). 17 The items included by Linacre in the science example were items a researcher decided to include in the measure of “liking of science” (Linacre, 2012b, p. 3). Andrich explained that psychometric researchers construct variables for use in measurement of the concept of interest, for example how happy someone is (Andrich, 1988). This type of measurement is in contrast to measuring how long a piece of wood is in metres; one metre is always one metre. Rasch analysis is an instrument that creates a measurement system comparable to the metre stick, where an increase of 1 metre is always the same value. The increase of one unit in a Rasch analysis scale is always the same for the set of data being analysed. In general, Rasch models may be thought of as using examinee responses on a set of items to produce estimates of examinee ability and item difficulty that maximize the probability of these combinations of observed responses. Both the examinee ability estimates and the item difficulty estimates are theoretically invariant of the specific situation should the data sufficiently fit the Rasch model. Data that do not fit the Rasch model are considered poor data: similar to outliers in classical measure theory. The idea of the model fitting the data in Rasch Analysis is different from that of traditional statistical analysis where the model is chosen to fit the data. This is explored and discussed at length by Smith and Smith (2004). Smith and Smith (2004) describe the differences in what they refer to as the Traditional Paradigm, where the model fits the data and the Rasch Paradigm where the data fits the model. Chapters 7 and 8 in Introduction to Rasch Measurement are dedicated to this discussion (2004). One of their 18 key points is the ordering of items based on item difficulty should be a direct result of the data not a model. Rasch Analysis includes a variety of techniques – dichotomous and rating scales, and partial credit – only those used in this study will be discussed in further detail. The following symbols will be used in the equations for Rasch measures: 𝑃𝑛𝑖 is the probability of person 𝑛 succeeding on item 𝑖, 𝑛 is the person number 𝑖 is the item number 𝐷𝑖 is the estimated item difficulty 𝐵𝑛 is the estimated person ability. Rasch dichotomous model. The dichotomous Rasch model with person ability 𝐵𝑛 and item difficulty 𝐷𝑖 is given by (Equation 1) (Linacre, 2012a): 𝑙𝑛 ( 𝑃𝑛𝑖 ) = 𝐵𝑛 − 𝐷𝑖 . 1 − 𝑃𝑛𝑖 Linacre (2012c, p. 21) describes this as the “log-odds of a person 𝑛 succeeding on item 𝑖= Ability of person 𝑛 −Difficulty of item 𝑖”. The units used for the log-odds is “logits”. According to Linacre (n.d.) “one logit is the distance along the line of the variable that increased the odds of observing the event specified in the measurement model by a factor of 2.728.., the value of "𝑒”, the base of the “natural” or Napierian logarithms used for the calculation of “log-“ odds.” This process takes responses that are scored zero 19 (failure) or one (success), referred to as dichotomous items, and creates a linear interval scale. Figure 2 shows the probability of success against the difference between person ability and item difficulty. Appendix D expands the explanation of the formation of the graph and the effects of a probability of zero and a probability of 1 for Pni (Linacre, 2012 a, step 109 to 114). Appendix D expands on the development of the person and item measures and shows the sigmodal curve that is asymptotic as the logit difference scores approach infinity and negative infinity. Figure 2. A sample curve for probability of success versus logit difference (Bn-Di) (Linacre, 2012a) There are various computer software programs that may be used for Rasch Analysis: Winsteps, Rumm 2020, Facets, Quest and ConQuest are a few examples (Sick, 2009). The equations described here are those used by Winsteps. Winsteps uses several 20 iterations to estimate Bn and Di. Initially all persons and abilities have the same estimate of zero. These estimations are then modified by the PROX (Normal Approximation) estimation algorithm (Equation 2) (Linacre 2012e): 𝐵𝑛 = 𝜇𝑛 + √1 + 𝜎𝑛2 𝑅𝑛 ln ( ) 2.9 𝑁𝑛 − 𝑅𝑛 where 𝐵𝑛 is the current person ability estimate for person n, 𝜇𝑛 is the mean item difficulty of the items responded to by person n, and 𝜎𝑛 is the standard deviation of the items responded to by person n. 𝑅𝑛 is the raw score for person n and 𝑁𝑛 is the maximum possible score on items responded to by person n. Similarly, item difficulty is estimated as shown in Equation 3 (Linacre, 2012e): 𝐷𝑖 = 𝜇𝑖 + √1 + 𝜎𝑛2 𝑅𝑖 ln ( ) 2.9 𝑁𝑖 − 𝑅𝑖 where 𝐷𝑖 is the current item difficulty estimate for item i, 𝜇𝑖 is the mean ability of the persons who responded to item i, and 𝜎𝑖 is the standard deviation of the person abilities who responded to the item. 𝑅𝑖 is the raw score for item i and 𝑁𝑖 is the maximum possible score on the item. This iteration process is repeated until the change in square sum of the residuals for the overall output for each person and item is less than 0.5 logits. Once this PROX stage is complete the JMLE (Joint Maximum Likelihood Estimation) iterations begin. 21 The JMLE estimation process iterates using Equation 4 (Linacre, 2012e): 𝑦′ = 𝑦 + ∑(𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 − 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑) ∑ 𝑐𝑒𝑙𝑙 𝑣𝑎𝑟𝑖𝑒𝑛𝑐𝑒 where expected value of person A on item 1 is calculated by Equation 5 (Linacre, 2012e): 𝑃{𝐴1 = 1} = 𝑒 (𝑃𝑒𝑟𝑠𝑜𝑛 𝐴 𝑙𝑜𝑔𝑖𝑡 – 𝐼𝑡𝑒𝑚 1 𝑙𝑜𝑔𝑖𝑡) , (1 + 𝑒 (𝑃𝑒𝑟𝑠𝑜𝑛 𝐴 𝑙𝑜𝑔𝑖𝑡 – 𝐼𝑡𝑒𝑚 1 𝑙𝑜𝑔𝑖𝑡) ) and (Equation 6) (Linacre, 2012e): 𝐶𝑒𝑙𝑙 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒(1 − 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒). Equations 4 to 6 are used to calculate both 𝐵𝑛 and 𝐷𝑖 . The JMLE iterations are continued until the residuals are close to zero. Specifically, since Rasch measures are calculated to two decimal places, iterations cease once the residuals are close enough to zero that the reported Rasch measure is not changed. Model Standard Error or Model SE is also reported in a Rasch Analysis. Rasch Analysis reports Model Standard Error for the dichotomous scale, as (Equation 7) (Linacre, 2012e): 𝑆𝐸(𝐵𝑛 , 𝐷𝑖 ) = 1 √∑(𝑃𝑛𝑖 (1 − 𝑃𝑛𝑖 )) . 22 Rasch-Andrich rating scale model. With the introduction of more than two possible responses, the model changes slightly to accommodate. David Andrich (1978, a, b, c) expanded on the Rasch Model to allow the analysis of Likert-type rating scale data. The Rasch-Andrich rating scale model equation (Equation 8) (Linacre, 2012d) is: 𝑃𝑛𝑖𝑗 𝑙𝑛 ( ) = 𝐵𝑛 − 𝐷𝑖 − 𝐹𝑗 . 𝑃𝑛𝑖(𝑗−1) where 𝑗 is a response on a Likert type scale, 𝐹𝑗 is the step calibration or step difficulty and all other variables are as described previously. The rating scale final values are reached after several iterations using similar equations as Equation 2 through 6 (Linacre, 2012e). This model “specifies the probability, 𝑃𝑛𝑖𝑗 , that a person 𝑛 of ability 𝐵𝑛 is observed in category 𝑗 of a rating scale applied to item 𝑖 of difficulty 𝐷𝑖 as opposed to the probability 𝑃𝑛𝑖(𝑗−1) of being observed in category (𝑗 − 1),” (Linacre, 2012a, p 1). Model Standard Error for a Rasch-Andrich rating scale model is calculated using (Equation 9) (Linacre, 2012e): 𝑆. 𝐸. = 1 2 𝑚 √∑𝑛 𝑜𝑟 𝑖(∑𝑚 𝑗=0(𝑗𝑃𝑛𝑖𝑗 − ∑𝑗=0 𝑗𝑃𝑛𝑖𝑗 ) ) Again, the model is linear and the possible responses are modelled into equal difficulty difference intervals. Rasch models and fit. Each Rasch Model is created for the data it is representing. Once the Rasch measures are calculated, the persons and the items are examined for 23 “fit”. “Fit” refers to how well the data fit the model (Linacre, n.d.). If the fit for an item or person is poor, the item or person is referred to as “misbehaving.” Fit is commonly quantified in two ways, mean square and t (or standardized z). The mean square statistic approximates a chi-square distribution. This statistic is divided by its degrees of freedom to produce an expected value of ‘1’ (Bond & Fox, 2012). Fit statistics for Rasch Analysis are part of what make Rasch Analysis so effective. An example that illustrates this effectiveness is the results of the 2002 Olympics. The results from a judge in figure skating were questioned. Linacre (2002) and Looney (as cited in Linacre, 2002) suggests that the patterns, indicated by fit statistics, in the judge’s responses may have been detected had Rasch Analysis been employed. Outfit. Outfit is sensitive to outliers. It is expressed as both an outfit mean square and with an associated t (or standardized z) value. Outfit mean square is calculated as the 2 “average of the standardized residuals, (𝑧𝑛𝑖 )” (Bond & Fox, p 285). This is an unweighted average. As such, “unexpected responses far from a person’s or item’s measure” have more of an effect on the outfit statistic (p 285). The effects of variations in response patterns are described in Appendix D. The outfit mean square equation (Equation 10) is: 2 ∑ 𝑧𝑛𝑖 𝑂𝑢𝑡𝑓𝑖𝑡 = 𝑁 where 𝑧𝑛𝑖 is the standardized residual of each person on each item (Bond & Fox, pp 285286). Specifically, (𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 11) (Linacre, 2012e): 𝑧2 = (𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 −𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑)2 𝜎2 . 24 Infit. Infit is sensitive to patterns in the responses. Infit mean square is a weighted average. As such, there is more of an impact on the infit mean square for “unexpected responses close to person or item’s measure” (Bond & Fox, p 285). The equation for infit mean square (Equation 12) is: 2 ∑ 𝑧𝑛𝑖 𝑊𝑛𝑖 𝐼𝑛𝑓𝑖𝑡 = ∑ 𝑊𝑛𝑖 where 𝑊𝑛𝑖 is the variance for an individual across a given item (Bond & Fox, p 286). Appendix D includes a table that demonstrates the effects of various patterns on the Outfit and Infit mean squares. Both infit and outfit are also described using a t (or a standardized z statistic) (Bond & Fox, 2012). The use of t or z varies, but as the sample size, n, increases (n > 30) the t -distribution approximates the z-distribution. Whether t or z are reported this fit statistic must be examined. In practice, the usual guideline is to examine t or z scores if they exceed the absolute value of 2 (Bond & Fox, p. 286). This holds for both the infit and outfit statistics. For consistency, z will be used. Fit mean square, MS, values are greater than or equal to zero, with an expected value of one. According to the Rasch website (Linacre, n.d.), Mean square values less than 0.5 do not degrade the measure, values from 0.5 to 1.5 are ideal for the measure, 1.5 to 2 do not add to or take away from the measure. Mean square values great then +2 degrade the measure but may be caused by as few as one result (Linacre, 2012e). 25 While Linacre (n.d.) and Bond and Fox (2012) recommend a magnitude of MS approach, others such as, R. M. Smith (Review of Reviews of Bond and Fox, 2002) prefer a statistical probability approach. Standardized fit statistics are based on a z-test with values outside of -2 to +2 (p < .05) are of greatest concern. Items and persons with fit concerns must be examined. The exact values that are of concern vary based on source and sample size (Table 1) (Bond & Fox, 2012 p 240, Linacre, 2012e). Table 1 Fit Statistics and Interpretations Bond and Fox (2012) Linacre (2012e) General Interpretation Mean Square z Mean Square z Response Pattern Variation Interpretation Misfit Type > 1.3 > 2.0 > 1.2 > 2.0 Too haphazard Too much Unpredictable Underfit < 0.75 < -2.0 < 0.8 <-2.0 Too Too little determined Guttman Overfit Table 1 summarizes fit from both Bond and Fox, and Linacre (2012e). The table shows the slight differences in the Mean Square values for underfit and overfit but the z values are the same for both. When z < -2, overfit, the data is too predictable and there may be variables influencing the response that are not being detected. Overfit is also described as redundant, muted or cramped. Guttman refers to a response pattern that does not have a transition area. Most responses, to a dichotomous item, will have a 26 region where the responses vary, for example 1111011010000 the middle four responses are in the transition region. A Guttman response pattern would be 111111000000 (Bond & Fox, 2012, p. 239). When the z score is between -1.9 and 1.9, inclusively, the data is predictable. If z > 2.0 the data are considered underfit, too unpredictable, or noisy. Underfit degrades the quality where as overfit makes the results appear better than they may be (Bond & Fox, 2012). When investigating fit concerns Linacre (Linacre, 2012e) suggests examining outfit (outliers) before infit (inlying patterns) and the size of the concern, as indicated by mean square, before the significance of the concern, as indicated by z. This differs from Bond and Fox (2012) which recommends investigating infit concerns before outfit concerns. Fit guidelines may be affected by both sample size and the stakes around the data collected. As sample size increases, means square values which indicate underfit, decreases (Smith, Schumacker & Bush, 1998). Table 2 (Linacre, 2012e) indicates the changes is the suggested fit mean square statistics for infit and outfit for situations with varying levels of investment. 27 Table 2 Reasonable Item Mean-Squares Ranges for Infit and Outfit Type of Test Range Multiple Choice (High stakes) 0.8 – 1.2 Multiple Choice (average stakes) 0.7 – 1.3 Rating Scale (Survey) 0.6 – 1.4 Clinical observation 0.5 – 1.7 Judged (agreement encouraged) 0.4 – 1.2 To avoid any confusion the guideline being used can be stated. Fit statistics indicate where problems may occur and are examined for both the participants and the items. Any participant or item outside the fit guidelines, decided on, should be examined. Together, with the use of Rasch analysis techniques the result is a linear measure that fit the data and takes ability of participants together with item difficulty to create measures that are invariant across the population. Granger (2010, p. 7) summarized it by saying, “Rasch analysis provides an internally valid measure that, when developed from an appropriate sample, is independent of the particular sample to which it is applied, meaning that the findings for the sample extrapolate to its population.” The idea of independence of the sample is further discussed in Smith and Smith (Smith & Smith, 2004). These properties of Rasch analysis make it an excellent choice for both major goals of this study. 28 Chapter Three Methods Procedure Between December 2014 and June 2015, 12 grade 2 and/or grade 3 classrooms participated in a Math Play intervention designed to increase awareness of mathematics in the world and improve attitude toward mathematics. The sample came from grade 2 and/or grade 3 classes identified with the help of Cindy Heitman, District Principal, Curriculum and Instruction, School District 57. Split classes were included because of the limiting factor if they were excluded – due to school sizes, several schools do not have classes with only grade 2 or grade 3 students – therefore requiring classes to be exclusively grade 2 or grade 3 would reduce the number of classes that could participate. Approval Process Tentative approval for the Math Play study was granted by Ms. Cindy Heitman, District Principal, Curriculum and Instruction, School District 57. Final approval from School District 57 was conditional on UNBC Research Ethics Board (REB) approval and the supervising committee approval. These approvals were all granted: forms are located in Appendix A. Approval to work in each school was obtained from the school principals. Once participating schools were confirmed, the individual teachers were contacted for approval. Finally, the parents of the children and the children in the class were contacted to seek consent. Information letters and consent forms were provided to all parties involved. Copies of sample documents are located in Appendix A. Participation in the assessments was voluntary and could be withdrawn at any time by any party. The children were also given the opportunity to withdraw any time during the assessment 29 component. For the classes assigned to the treatment group, the entire class participated in the Math Play sessions. No data were collected from the children during these sessions. Group Placement Between February and the end of May half of the classes were involved in Math Play intervention and assessments. For the same time frame, the other classes, the control group, only participated in attitude assessments. To ensure all participants, the elementary students, and their teachers and aids, were given access to any benefits associated with Math Play, Math Play sessions took place in control group classes after the final attitude assessments were complete – in late May and June. Placement into treatment or control group was not completely random. It was based on which group the teacher requested (when the requests could be accommodated) and matching the class. For example, when matching classes, if there were two grade 3 classes in a single school, one was placed in the control group and the other was in the treatment group. If there was only one grade 3 class from a specific school the class was paired with a grade 3 class from another school, one in treatment and one in the control group. The only class that was not paired that way was one grade 3/4 split class where the grade 3 students took part in the study. This class was paired with a grade 3 class. There was one class that consisted of two individual classes merged into one with two teachers working in the classroom. This merged class was part of the treatment group. 30 Math Play Facilitators Students from the University of Northern British Columbia’s MATH 190 (Math for Elementary Educators) class facilitated the Math Play sessions in the grade 2/3 classrooms. To avoid confusion, for the purpose of this study facilitator refers to MATH 190 student volunteers and teacher refers to the grade 2/3 classroom teacher. There were two facilitators per class, with one exception. The class that was two classes merged into one had four facilitators due to the doubled class size. Participation as facilitators in the Math Play study was part of the MATH 190 course work but an alternative was provided if MATH 190 students did not want to or were not able to participate. All the students from MATH 190 who wanted to take part were able to, with one exception. This student’s availability did not match with any of the times that the classroom teachers had provided. The total number of volunteer facilitators was 14. Before facilitators entered the classrooms, they obtained criminal record checks. After obtaining criminal record checks and prior to the intervention, the facilitators visited their treatment group class for one hour a week for two weeks. This was to mitigate a possible Hawthorn effect and familiarize the facilitators and the classroom teachers with each other (Oswald, Sherratt & Smith, 2014). The researcher was present to observe all Math Play sessions and the pre-intervention visits. For the intervention, facilitator visits lasted for one hour, and occurred once per week for a total of five weeks (5 hours). If a regular visit was scheduled for a holiday or non-instructional day the Math Play session was rescheduled. Time was allotted for training of and feedback from the facilitators. 31 Initially, facilitators were going to be placed in the classes based on the results of a mathematics attitude assessment, which the facilitators took before the study began (Tapia & Marsh, 2004) but due to scheduling restrictions that was not feasible. Instead, the placement depended on matching the times the classroom teachers had available and the time the facilitators were available. Facilitators were responsible for transportation to and from the Math Play sessions. They were also responsible for reviewing and understanding the instructions provided on session facilitation (Appendix C). Short overview meetings to review instructions and finalize the plan for each session were held immediately prior to each Math Play session. Feedback was gathered after each Math Play session. This allowed for immediate problem resolutions if any arose. Timeline Recruitment of classes began December 2014. The timeline was restricted by the semester dates of Winter 2015. Recruitment of facilitators was not able to begin until the Winter 2015 semester began. Normally School District 57’s Spring Break and UNBC’s Reading Break do not coincide but winter of 2015 the two breaks coincided: February 16 to March 1, 2015. • December to January 2014: recruit classes to participant in the study – 12 classrooms in total 32 • January 2015: Facilitator recruitment, and facilitators had criminal record checks, did self-attitude assessments using the ATMI and trained for intervention, classroom teachers handed out and collected consent forms, • February 2 – 15, 2015: Attitude assessment in the classes where Math Play will occur and facilitators perform classroom support for an hour a week • February 16 – March 1 2015: Spring Break for School District 57/Reading Break for UNBC • March 2 – April 2 2015: Facilitators lead Math Play sessions in the class for one hour a week • April 7 – 10 2015: Attitude Assessment in the class all classes • End of May: Attitude Assessment in all classes • June 2015: Math Play visits in control group classes Assessments The classroom teacher administered the attitude assessments. This was done as indicated in the timeline. The exact day the assessments took place in each class was decided by the teacher, based on their schedule, as long as it was in the indicated weeks. The Assessment Instruments Difficulty Associating Words with Math (DAWM) assessment. Difficulty Associating Words with Mathematics, (DAWM), was designed to examine the propensity of students to indicate they associate words and/or concepts with mathematics. The DAWM assessment was inspired by the ideas in work by Stodolsky, et. al. (1991) who 33 complied a list of words students used while discussing mathematics. The DAWM was then created: a list of words and/or concepts that may be identified as mathematics related (Figure 2). The DAWM was designed to assesses a student’s propensity to indicate words or concepts as being related to mathematics. Figure 2. A sample of the DAWM assessment. Math Attitude Assessment. Attitudes were measured using an assessment instrument modelled after the ATMI by Martha Tapia and George E. Marsh II (2004) (Appendix B). After conferring with Dr. Cindy Hardy and Ms. Cindy Heitman, School District 57 District Principal, Curriculum and Instruction, it was decided that 10 to 15 questions would be better suited to the age group than the 40 questions in the original ATMI. Quinn and Jadav (1987) used as assessment with 11 questions on a group of similar 34 ages. Dowker et al. (2012) used a total of 28 questions. To meet the requirements set out by School District 57, together with the lengths of previous assessments, the MAA assessment length was set at 12 statements. Consistent with the work by Tapia and Marsh (2004), the DAWM contained statements based on the following factors: selfconfidence (SC), value (V), enjoyment (E) and motivation (M). Due to the similarity of the target age group, the assessment language was similar in complexity to that used by Levine, (1972) as the study by Levine was designed for grade 3 students. Math Attitude Assessment (MAA) 1. SC Math is easy. 2. SC I know I can get math questions right. 3. SC I find math hard. 4. V Math is useful. 5. V I can think of ways to use math. 6. V Math is useless. 7. E Math makes me feel happy. 8. E I am sad when I have to do math. 9. E Math makes me scared. 10. M I want to do math next year. 11. M When I grow up I want a job that uses math. 12. M Next year I want to stay away from math. Figure 3. The statements from the MAA ordered by factor. There were three response categories: Yes (agree), Sometimes (neutral), or No (disagree). The students were told to circle the response that best described how they felt 35 (Appendix C). The assessment statements were read to the students by their classroom teachers. Basic background information was collected: age, grade, and gender. Each students’ name was assigned a random number for identification. This was done to ensure that the students’ results are kept confidential. Study Design This was a quasi-experimental pre-post treatment design. Lack of randomness characterizing it as a quasi-experimental design (Hurlburt, 2006). Schools and teachers were self selecting. Classes had to be kept intact, so participants were not randomly assigned to the treatment or control group. This meant that if there were variables that caused differences in the groups they may affect the results. The control group should have matured at a similar rate while being exposed to comparable curricular content helping to reduce the non-random placement of the participants (Creswell, 2014; Hurlburt, 2006; Pagano, 1998). Each classroom teachers knew that their class was part of the treatment group and this may have affected topics discussed in class or ways that they approached topics. The parents knew if their child was part of the treatment group and this too may have affected home behaviour. The participants in the treatment group may have known they were part of the treatment group. Instructions were given that requested that the connection between Math Play and the assessments not be made for the students, but how consistently these instructions were followed can not be confirmed. Participant 36 awareness could contribute to the possibility of Hawthorn effect. To mitigate the effects of the Hawthorn effect pre-visits took place. A study by Oswald et al (2014) suggested six steps to help reduce the Hawthorn effect and found having the participants in the study have a good relationship with the researchers and have the participants feel comfortable were the most important steps in Hawthorn effect mitigation. The pre-visits occurred to allow relationships between the facilitators and observer with the student participants to develop. Intervention Group Classroom visits pre-Math Play. During the two weeks from February 2-15 the facilitators were in the classroom to provide support. They were there for one hour a week for two weeks. The amount of time allotted to these class visits was dictated by the restrictions on the timeline. The teacher instructed the facilitators on their role during the visits. Most pre-intervention visits consisted of the facilitators listening to a lesson then circulating to support students on their work for that lesson. In one class visit the facilitators took part in an outdoor school play session. Math Play sessions. Before each Math Play session written instructions that included “Before you start you will need,” “The goal of this game,” “How to play,” “What to do to teach the game,” “Ways to modify the game,” “Ways to correct without it feeling like criticism”, “Avoid statements like…” and “Ask,” were distributed to the facilitators. Facilitators were provided with instructional videos (Appendix C for a complete list of games and instructions). 37 During the Math Play sessions, from March 2, 2015 through April 2, 2015, teachers were asked to be present to supervise the classroom (student behaviour), but did not participate directly in the Math Play sessions, with a few exceptions. If the class was not understanding the instructions or if a facilitator was absent the teacher supported the facilitator(s). The classroom teachers were given specific instructions on the language to be used (positive) and the instructions for the activities. All supplies for Math Play, other than pencils, were brought in by the facilitators. Classroom teachers were instructed to not bring any of the games used during Math Play sessions into the class while the research was taking place. The teachers were asked to teach math as they normally would. Each Math Play session involved three games or activities with one exception. The day the facilitators taught the participants “Magic Number” they had an extra game. These games and activities were a combination of paper and pencil games and storebought games. Each session involved at least one of each type – paper and pencil and store bought. Games were only used once in the five-week intervention. The Math Play sessions started with the entire class working on a game together facilitated by the volunteers. The class was then given the opportunity to try the game on their own with the facilitators circulating to help. Group work was encouraged. After the initial activity, the class was split in two groups, with each facilitator presenting and supporting a new game. The facilitators then switched groups. This method resulted in approximately 20 minutes per game. 38 The Games The games were selected to not directly overlap with grade 2 and 3 curriculum. This was done to avoid curriculum-based games being presented in one class but not in others. Most of the games were selected for their focus on logic and problem-solving strategies that must be employed to complete the game. A complete list of the games used and the instructions for the games can be found in Appendix C. During the Math Play sessions the students were encouraged to find the mathematics in each game. The facilitator tried to help the students identify them. Some facilitators had more success with this than others did. The facilitators were instructed to only use positive language. Success on a game was not based on getting the “right answer”, instead the focus was on attempting to complete the game, making progress toward completion, and/or figuring out what success would look like for that game. Phrases like, “Your solution does not look quite like what I got, can you tell me what you did?” or “Your first step looks great, can you tell me how you did that and use that same strategy to complete the next step?” were encouraged. Facilitators were instructed to avoid phrases like, “That is wrong,” or “You need to correct that.” This approach was used to try to avoid the feeling of getting the wrong answer which was identified as a being related to a negative attitude toward mathematics (Stodolsky, et al, 1991). Focus was placed on taking part and engaging with the activities not getting the right answer. 39 Paper and pencil games. Paper and pencil games varied but often were based on grid style games. Some students chose to work independently, and others worked in groups. A focus was allowing the students to work in a way they were comfortable with. They were not forced into working groups. An example of game a paper and pencil game used follows. Several of the games came from a collection compiled by Susan Milner (Milner, n.d.). Hidato. Hidato is a grid game. The grids were made of squares connected to each other. The goal is to place a natural number in each square in the grid so that consecutive numbers are able to be connected by a horizontal, diagonal or vertical chain. Some of the squares were already filled. This game was presented on a large version made for demonstration purposes. The facilitators explained the game and worked through several examples with the students and then the students were given paper versions of the game and encouraged to work through them. A variety of difficulty levels were available to accommodate the difference in abilities in the classes. For students who were not able to place the numbers in the correct places they were encouraged to figure out which numbers were missing and place those numbers in the grid. Boxed games. A variety of store bought games were brought in. Like the paper and pencil games, these games were worked on in groups and independently. An example of a boxed or store bought games follows. A full list of games and instruction is in Appendix C. 40 Q-bitz. Q-bitz is a pattern matching game. A card that shows a pattern is selected. There are 16, six-sided dice. Each die has a pattern on each side. The 16 dice were then oriented in a four by four layout to match the pattern on a card. The variety in the difficulty level of the cards was appropriate for the group and enable pattern matching success for most of the students. If students made their own patterns, instead of trying to match the pattern cards, they were encouraged to describe the pattern they made. Collected Data The information in the assessments was coded and entered in a secure file to ensure privacy. Information collected regarding the students was handled by the classroom teacher and researcher. When the data were entered it was separated from any identifiers of the participants except a number which was recorded on the original assessments and a separate file. Ethical Considerations Research Ethics Board approval was obtained. The original application was resubmitted to accommodate for an increase in the number of classes involved in the study. Ethics is always of concern. To minimize the concern of shared confidential information the volunteers did not at any point handle or see the personal information of the participants. The teacher handled the consent forms and the consent forms were then secured and given to the researcher. The facilitators signed confidentiality forms to ensure they were aware of and would respect confidentiality concerns. 41 Chapter Four Results Two measures were used to examine the attitude and effect of Math Play with 163 grades 2 and 3 students in schools in Prince George in School District 57: The Difficulty Associating Words with Mathematics (DAWM) and the Math Attitude Assessment (MAA). The data were analysed with Rasch Analysis using Winsteps (Version 3.92.1) followed by the Rasch measures being used in traditional statistical analysis techniques employing SPSS (Version 24). The data were also sorted, organized and graphed using Excel 2016. The analysis was done on the DAWM data followed by the MAA data. The analyses took place in six stages. First, the participants’ responses were analysed using Rasch Analysis. Second, Rasch Analysis techniques were employed to examine the items in the assessments. Third, commonly used Rasch Analysis techniques were used to examine scale. Fourth, each participants’ measure was determined for each of the three assessments, using anchored item difficulty. Fifth, the adequacy of the data for use in traditional statistical analysis was explored using SPSS. Finally, data was analysed using traditional statistics methods in SPSS (Field, 2011). In the first four stages of analyses Rasch analysis techniques were employed for data preparation, then traditional statistical analysis techniques were used in the investigation of the treatment effect related to the research questions of this thesis. 42 Difficulty Associating Words with Mathematics Rasch Analysis DAWM. Rasch Analysis of the DAWM assessment took place in four stages. First, participants’ responses were analysed, to determine if any participant’s results warranted being removed from the study. Second, items were analysed, in the DAWM assessment to determine if all items should be retained. Third, the scale in the assessment instrument was examined to determine if it was appropriate for the participants. Fourth, anchored values were determined for further analysis. Rasch Analysis of participant results DAWM February treatment and control. The responses from all 163 participants were analyzed in Winsteps to examine the participants for misbehaving responses. Displayed in Table 3 is a selection of high, low and midrange endorsement rates for the 19 words or concepts, referred to as items, used in the DAWM. Each row corresponds to one student. Gaps indicate where entries are not reported due to space considerations in this table. The “Score” in Table 3 indicates the actual number of words or items the participant indicated as being related to mathematics. Each row represents the results from a single participant. The mean score was 6.3 with a standard deviation of 3.3. The second column reports “% Endorse”. It is the percent of the 19 items which the participant indicated as being associated with mathematics. The scores are converted to Logit Measures (Equation 2, 4, 5 & 6). The “Logit Measure” column represents the students’ propensity to associate words with mathematics. Figure 5 also shows individual participant’s logit measures on a map know as a “Wright map”. The larger the number, 43 the higher the participants’ propensity was to endorse words as being associated with mathematics. For example, a participant with a logit measure of -11.92 (in the last row of Table 3) is least likely to endorse words as being associated with mathematics (they selected two of the 19 possible choices) where as a participant who endorsed all 19 words had a logit measure of -4.60. Table 3 Excerpt DAWM Participant Measures % LOGIT SCORE ENDORSE MEASURE 19 100.0 -4.60 19 100.0 -4.60 19 100.0 -4.60 18 94.7 -4.64 15 80.0 -4.83 15 80.0 -4.83 14 73.7 -5.02 MODEL INFIT INFIT OUTFIT OUTFIT SE 0.21 0.21 0.21 0.24 0.39 0.39 0.52 MS 0.25 0.25 0.25 0.22 0.38 0.26 0.60 z -0.6 -0.6 -0.6 -0.4 -0.1 -0.3 0.1 MS 0.53 0.53 0.53 0.48 0.62 0.41 0.55 z -0.6 -0.6 -0.6 -0.7 -0.2 -0.5 0.1 CORR .00 .00 .00 .25 .42 .50 .50 9 9 8 47.4 47.4 42.1 -6.75 -6.75 -7.14 0.62 0.62 0.66 1.51 1.46 0.75 1.6 1.5 -0.7 1.59 1.25 0.43 0.8 0.6 -0.2 .50 .54 .75 4 21.1 -9.51 1.03 3.09 2.0 3.11 1.5 .48 3 3 3 2 15.8 15.8 15.8 10.5 -10.68 -10.68 -10.68 -11.92 1.10 1.10 1.10 1.08 0.22 0.22 0.22 0.91 -1.4 -1.4 -1.4 -0.1 0.06 0.06 0.06 0.17 -0.8 -0.8 -0.8 -0.5 .86 .86 .86 .68 44 The “Model SE”, or Model Standard Error is a measure of the “best case” error (Equations 7 & 9). The general trend is that as the score decreases, and therefore the Logit Measure decreases, the Model SE increases. The grey row in Table 3 is an example of a participant with infit MS of 1.51, which does not add or impede the results, and an infit z of 1.6 which is between -2 and +2, and indicates the data were predictable. There are two values associated with infit and outfit statistics, the mean square, “MS” and standardized fit statistic, “z”. The fit statistic is a mean square value made of a ratio of an approximate Chi-square statistic divided by its degrees of freedom (Equations 10 & 12). The expected value is ‘1’ but values range from zero to infinity. Infit is sensitive to patterns and outfit is sensitive to outliers. The Mean Square indicates the magnitude of the randomness and the z is an indicator of the probability of fit of the model. Mean Square values were examined and given more weight when identifying areas of concern than the z statistic. Giving the Mean Square values more attention is suggested by Linacre (2012b). An entry with a mean square greater than ‘2’ is considered to possibly degrade the measure but this value can be caused by as few as one observation. At a p <.05, Mean Square values between 1.5 and 2.0 do not impede or help the measure. Mean square values between 0.5 and 1.5 contribute the most to the measure. Values less than 0.5 do not degrade the measure but are considered an indication of a lack of new information and may cause high reliability and separation coefficients. The z statistic is affected by large sample size, which we have, and large misfit values for z may not be as much of a concern as they would be if there was a small sample size. Mean Square concerns are addressed before z concerns. 45 Any participants whose MS or z fell outside of the indicated ranges were further examined to determine if it was advisable to remove their data from the study. No participants were removed because even though their values did fall outside of the ideal ranges the responses were recorded properly and there was no indication that they were not a reflection of the participant’s propensity to indicate a word as associated with mathematics. The last column indicates the correlation between the participant results and the ability measure, referred to as a point-measure correlation. The participants who endorsed all 19 words had a correlation of zero indicating no relationship between the participant’s ability and the measure. Except for the three participants who endorsed all 19 words, all participants showed a correlation greater than zero. Of the original 163 participants, 135 wrote all three assessments (February, April and May) an attrition rate of 17.1%. Of the 135 participants in the DAWM data, 66 selfidentified as male and 69 self-identified as female. The mean age of the 163 students was 7.89 with a Standard deviation of 0.63. The minimum age was 7 and the maximum age was 10. Of the 163 who completed the original assessment, only 4 students did not fill in an age. With the exception of the participants who did not write all three assessments, all participants were retained. Rasch Analysis of the items for the DAWM. Difficulty Associating Words with Mathematics (DAWM) for the entire group in February was used to assess the DAWM as an instrument and to create an anchor for item values for future analysis. The items 46 (words) showed an item reliability of .98 – indicating enough items of varying difficulty were in the assessment. The Item Real Separation was 7.51. Item Real Separation can be interpreted as the 19 items can be separated into 7 statistically distinct endorsement levels. The collection of words showed a range of difficulties of endorsement from -7.71 to 3.22 (Table 4). Individual item summary statistics are given in Table 4. Person Real Separation of 1.68 and Person Reliability of.74 were identified. The Person Real Separation of 1.68 suggests that the assessment may not be quite sensitive enough to separate people of high and low abilities (any score <2) but is close to the 2 guideline. The assessment may benefit from more items. The assessment is sensitive enough to identify treatment. Person Reliability is analogous to traditional statistics “test” reliability often reported using Cronbach Alpha. Person Reliability is close to the .8 guideline for low stakes standardized exams and exceeds the .7 recommended for classroom exams (Wells & Wollack, 2003). This Person Reliability indicates a good internal consistency. The logit measures of Item 7 and 12, “maps,” and “working on maps,” respectively is an extra indication of the participant’s consistency in their responses. Table 4 shows a range from easiest to associate mathematics with (most likely word to be associated with mathematics), “adding” with a logit measure of -7.71 (at the bottom of Table 4) to the most difficult word(s) for the students to associate with mathematics (least likely word(s) to be associated with mathematics), “walking a dog” had a logit measure of 3.22, at the top of Table 4. The standard error ranges between 0.2 and 1.06. The mean of the item logit measures is zero, as is standard practice for Rasch Analysis. 47 Frequency of word selections for the original 163 participants ranged from a maximum endorsement of 162 for “adding”, 159 for “subtracting”, and 158 participants selecting “counting”, down to 11 selecting “singing” and “playing tag” and a minimum of 10 selecting “walking a dog” (Table 2). Table 4 Item Characteristics for the Difficulty Associating Words with Mathematics DAWM MODEL Item 4 5 14 13 18 9 8 3 19 10 7 12 17 16 15 2 11 6 1 Word(s) walking a dog singing playing tag watching a movie painting cleaning your room cooking doing a maze playing board games puzzles maps working on maps studying people playing cards reading writing counting subtracting adding INFIT OUTFIT SCORE % EDDORSE LOGIT MEASURE SE MS z MS z CORR. 10 11 11 6.1 6.7 6.7 3.22 3.05 3.05 0.43 0.40 0.40 1.10 0.78 1.11 0.4 -0.6 0.5 0.45 0.47 0.96 -0.8 -0.8 0.2 .49 .54 .44 12 13 7.4 8.0 2.89 2.75 0.38 0.37 0.10 1.02 0.1 0.2 0.59 0.72 -0.6 -0.3 .49 .48 16 31 9.8 19.0 2.40 1.25 0.33 0.24 0.80 0.86 -0.8 -1.0 0.39 0.66 -1.3 -0.8 .56 .56 32 19.6 1.20 0.24 0.84 -1.1 0.73 -0.6 .56 38 39 45 23.3 23.9 27.6 0.88 0.83 0.55 0.22 0.22 0.21 0.92 0.91 0.95 -0.7 -0.8 -0.4 0.93 0.77 0.69 -0.1 -0.6 -1.0 .53 .55 .55 49 30.8 0.38 0.21 0.93 -0.6 0.69 -1.1 .56 58 35.6 0.02 0.20 1.25 2.6 1.12 0.5 .44 61 79 112 158 159 162 37.4 48.5 68.7 96.9 97.5 99.4 -0.10 -0.76 -2.04 -5.82 -6.04 -7.71 0.20 0.19 0.21 0.54 0.56 1.06 1.13 0.90 1.25 0.93 1.53 0.56 1.5 -1.2 2.2 0.0 1.2 -0.3 1.04 0.83 2.28 1.82 3.22 0.02 0.2 -0.7 3.9 1.1 1.9 -2.6 .48 .57 .38 .19 .04 .24 48 From Table 4, infit ranges from 0.56 to 1.53 with z scores from -1.2 to 2.6. Outfits range from 0.02 to 3.22 with z scores from -2.6 to 3.9. Indicating only four items of possible concern: • Item 17, “studying people”, infit z = 2.6 • Item 2, “writing”, infit z = 2.2, outfit MS = 2.28 and z = 3.9 • Item 6, “subtracting”, outfit MS = 3.22 • Item 1, “adding”, outfit z = -2.6. For the items with fit concerns: • Item 17, “Studying people”, from, Table 4, was kept because infit and outfit MS are considered before z and the infit and outfit MS values were within the 0 to 2 range. • Item 2, “Writing”, showed higher values for outfit MS (2.28) and z score (3.9) indicating random response or outliers (an indication of a few students not selecting writing even though most did) but was kept in the assessment (Table 4). Infit MS is considered before z and infit MS is less than +2. • Item 6, “Subtracting”, showed a high outfit value of 3.22 with z = 1.9. outfit of 3.22 indicates a few random response or outliers (an indication of a few students not selecting subtracting even though most did with relative ease) which did not justify removal (Table 4). 49 • Item 1, “Adding”, does not add information to the test (infit of 0.56 and outfit of 0.02) but was kept because it does not take away from the information of the test. The lack of variation is expected. Low fit scores are a result of the ease of recognition the item (Table 4). The data were examined with the guideline of an MS > 1.3 is considered underfit, unpredictable, noisy, or having unmodeled noise and an MS < 0.7 is considered overfit, (Bond & Fox, 2012) too predictable, redundant, muted, or cramped. Underfit degrades the quality where as overfit makes the results appear better than they may be. From Table 4, Items 1, 2, 11, and 6 demonstrate underfit and so are unpredictable. Items 4, 5, 13, 18, 9, 8, 3, 7, 12 and 1 show overfit so are behaving predictably and therefore redundantly. These Items were retained. Overfit and underfit items do not need to be removed from the assessment but need to be examined. The last column in Table 4 indicated the Pearson-Point Correlation of the item with the assessment. Correlations ranged from .04 to .57. The words easily endorsed by the participants (counting, subtracting and adding) had the lowest correlations (.19, .04 and .24 respectively). Sixteen of the nineteen items in the DAWM assessment were indicated to have a moderate positive correlation (.38 to .57). No items showed a negative correlation. There is only one item with a very low correlation (r = .04), Item 6. Having one item with a very low correlation is not a concern. The common mathematics related word(s), “Counting”, “Subtracting” and “Adding” were included in the assessment so the young participants had words they would more commonly associate with mathematics. As can be seen by the scores in Table 50 3 there was a drop in endorsement of words shown above those three words, indicating that several students may have only endorsed those three words. The values from Table 4 can be represented as the bubbles shown in Figure 4. The bubble plot is a combination of measures, errors and fit. The items that where endorsed most easily and often and had the most fit issues (Items 2, 11, 6 and 1) can be seen standing out from the majority of the other items. The large impact of one participant not endorsing Item 1 is show in the large bubble on the far left bottom of Figure 4. DAWM Item Difficulty versus Outfit z Less Measures More 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -4 -2 Overfit 0 Outfit z 2 4 6 Underfit Figure 4. DAWM item and outfit z scores: A graphical display of the fit. Items with -2 < zMS < +2, on the horizontal axis, are generally of no concern. The radius of the bubble is determined by relative Standard Error. The placement along the vertical axis is determined by the Logit Measure. The zMS outfit determines the left to right positioning. Item 2 is underfit. It is unpredictable but is retained because writing is 51 an important component of mathematics. Item 1 when its error in taken into consideration, may still be within the –2 to +2 guideline values. All Items were to be retained in the DAWM assessment. There were a few items with outfit concerns but infit was considered before outfit and infit MS was less than +2 for all for the items. Item 6, “subtracting”, has a low correlation, was highly endorsed and kept because it was one of the common mathematics related words. The DAWM assessment item logit measures will be used as anchors in future analyses. Rasch Analysis of the dichotomous scale for the DAWM. Figures 5 and 6 are common displays used in Rasch analysis to examine an assessment instrument. Figure 5 is a person and item map, referred to as a ‘Wright’ map. The numbers on the far left (-12 to 3) indicate the logit measure. On the left of the centre line, the participants are mapped. The number sign (#) represents 3 participants and the dot ( ∙ ) represents one or two participants. On the right side of the centre line, the WA followed by a number, represent each item. For example, “WA4”, at the top of the map, represents Item 4, ”walking a dog”. Along the centre dashed line are “S”, “T”, and, “M”. The “S” represents one standard deviation. The “T” represents two standard deviations and the “M” represents the mean of the data on the corresponding side. The separation index indicated seven statistically distinct groups. Figure 5 suggests four to five groups that may have differed in difficulty in a practical way. The ease of indicating “adding”, “counting” and “subtracting” as being related to mathematics can be seen by their positions on the bottom right. The position of most of the participants below the easiest items suggests the DAWM difficulty level was not consistent with the 52 abilities of the participants. This suggests the difficulty of the items exceeded the ability of most of the participants. It would be analogous to giving a grade two class a mathematics achievement test designed to assess the ability of a grade 3 class. The scale is not ideal for these participants at the time of the assessment. 53 Figure 5. DAWM Persons and items mapped by difficulty. Persons on the left and items on the right. A dot ( ∙ ) represents 1 to 2 people and a number sign (#) represents 3 people. 1 1 While there are more advanced graphics available, for simplicity and consistency (over decades of use) the Rasch displays are used for Figures 5, 6, 19 and 20. 54 Figure 6 is a scale assessment figure used to analyse the dichotomous scale. It shows the average person logit measure for the endorsement of each item as a “1” and the average logit measure of the persons who did not endorse the word as a “0”. For example, the average logit measure of a person not choosing Item 1, “adding”, was about -7 and the average logit measure of a person choosing Item 1 was about -0.7. The “m” indicated missing data. In a dichotomous scale like this the “0” is expected to be to the left of the “1” indicating the participants who do not endorse the item have a lower average propensity or lower ability to endorse than the participants who do endorse the item. All but one of the items, Item 6, “subtracting”, show at least one logit difference between the average of the participants endorsing the item and the ones not endorsing the item. Figure 6 shows how a participant’s average ability to endorse Item 4, “walking a dog” is higher than the ability of the average participant’s score who endorsed Item 1 “adding”. For items like 1, 6 and 11, which were endorsed by almost every participant, the zero estimate is unstable because the average ability of the participants who did not endorse the item is calculated using few participants (in the case of “adding” all but one participant endorsed it). 55 Figure 6. DAWM item versus average person measure for item endorsement. As a result of the item analysis, all items were retained in further analyses and all item logit measures were determined for anchoring. All participants were also retained for further analyses. Anchoring is a common technique in Rasch Analysis. It is defined as, “measures obtained from one analysis (or construct theory) imposed on another to place it in the same frame of reference,” (Linacre, 2012d). All items other than Item 1, “adding” and Item 11, “counting” were anchored using the item logit measures in Table 4 calculated 56 using the February data from the 163 participants. Items 1 and 11 were not anchored because of their easy of endorsement and their fit concerns. With anchoring the number of items anchored should be at least 5 and the fit should be good and the items should have a range of logit measures (Linacre, 2012e). Though Item 6 and 2 did not have ideal fit, they were anchored to ensure a range of anchoring items across the logit measures. Anchored Logit Measures for February, April and May DAWM. With anchors set, person participation confirmed, and item retention confirmed the logit measures were determined for the participants in February, April and May. An excerpt of the measures is found in Table 5. Of the original 163, the 135 who wrote all three assessments were kept for the final analysis. Analysis of the groups. The 135 participants were from 12 classes in School District 57. Classes were grade 2, grade 2/3 split and grade 3 (one was a grade 3/4 split from which only grade 3 students participated). Participants were not able to be randomly assigned to the treatment or control group because of the restrictions on the class that they were in and the need to match classes. Neither could the participants be pre-tested to assign them to groups. Both the treatment group and the control group wrote the DAWM assessment and the MAA in February (pre-treatment), April (posttreatment) and May (a month after treatment ended). The items for DAWM and MAA were the same on successive sittings but the items were presented in a different order for each assessment. Note, one class wrote the February assessment after our first visit (not a Math Play visit but the first visit). 57 Table 5 is an excerpt from the complete list of logit measures for the control, “C”, and treatment, “T”, for all three assessments with anchoring. The table was ordered based on the logit measure in February (lowest to highest). Blank lines show where data were not presented due to space considerations. The first column contains the student ID. The second column indicates the self identified gender of the participants. The logit measure in Table 5 indicate a propensity to endorse word(s) as being associated with mathematics. For example, the participant who received a 6.03 in February associated all 19 word(s) with mathematics and the participant who received a -6.19 only endorsed 1 word as being associated with mathematics. Therefore, in this scale the higher the measure the higher the propensity of the participant to endorse word/concepts as mathematics related. In Table 5, three rows have been highlighted. The first highlighted row shows a male from the control group with a logit measure of -6.19 in February and increased to 1.63 in April and decreased to 0.11 in May. This participant had a difficult time associating regular words and concepts with mathematics in February but in April, when presented with the same words, endorsed more words as being related to mathematics. However, by May, his propensity had decreased. The second highlighted row is the results from a female in the treatment group. She showed a decrease from the February logit measure of -0.82 to -2.36 in April and then another decrease to -4.14 in May. The third participant highlighted is another male but from the treatment group. Unlike the other highlighted participants, this participant showed an increase in his logit measure each time he wrote 58 an assessment. He had a higher propensity to endorse the words as mathematics related on each successive assessment. The changes observed for the participants varied. Table 5 Anchored Logit Measure, Gender and Group (Treatment or Control) for February, April and May DAWM Excerpt Person ID Gender 57518 83816 80911 M M M 21512 26214 59311 23213 23813 48220 M F M F M M -2.36 -2.36 -2.36 -2.36 -2.36 -2.28 -1.45 -2.36 -2.36 -6.31 0.11 -0.82 -0.32 -0.32 -0.32 0.5 1.62 -3.56 T T T T T C 7520 39216 87816 4810 M M F F -1.43 -1.43 -1.43 -0.82 -0.82 3.66 0.93 -2.36 0.11 2.00 2.40 -4.14 C C C T 46210 815 66621 41410 45918 52013 76414 F F M M F M F -0.32 -0.32 -0.32 -0.32 0.11 0.11 0.11 0.89 1.63 2.01 1.63 -1.43 0.11 -0.82 1.25 1.62 2.00 2.00 -0.32 0.11 0.11 T T C T C T T 45115 73421 81317 57814 M M F M 2.82 6.03 6.03 6.03 2.40 0.88 2.40 0.51 2.00 -0.81 3.30 3.30 T C C T Feb Apr May Group Logit Logit Logit Measure Measure Measure -6.19 1.63 0.11 C -5.12 0.88 0.11 C -4.09 -6.31 -4.14 T 59 This concludes the Rasch Analyses portion of the results for the DAWM. First, the participants were analysed. All participants who wrote all three assessments were retained for further analysis. Second, the items were analysed, and all items were retained in the DAWM assessment. Third, anchor values were determined for future analysis. Finally, initial Logit Measures were determined for all participants, using anchoring, for the February, April and May assessments. Missing Data Analysis. Though Rasch Analysis accommodates for missing data for a participant, sources indicate participants must be paired for ANOVA (Laerd, n.d.; Pagano, 1998; Plonsky, 1997) (the next stage of analysis). Of the original 163 participants 135 were present for all three assessments. There was no reason to suspect the missing data was anything other than random, therefore two options were explored for how to accommodate for missing data: omitting missing data or imputing data based on the regression equations. The benefit of imputing data was an increase in sample size. The disadvantage was “overestimates [in] model fit and correlation estimates” and “weaken[ed] variance”, (Humphries, n.d.). The benefit of deleting missing data was ease and comparisons of the actual data. The disadvantage of deleting data was loss of data and therefore reduction of power and if the missing data was not random, a bias may be introduced (Humphries, n.d.). The regression line created using the February control group, 𝑥𝑖 , and the April control group, 𝑦̂𝑖 , was used to predicted values of missing data in those groups. The regression equation for the control group was 𝑦̂𝑖 = .3998𝑥𝑖 + .1764. Once the imputed data were present a t-test was done to compare the February mean measure with and without data imputed. 60 There was not a statistically significant difference between the February control group with and without the imputed data (p = .875). The same process was repeated for the April control group data and again no statistically significant difference was found (p = .964). As a result, all missing data were omitted. Data from participants who did not write all three assessments were omitted from the traditional statistical analysis portion of the analysis. Traditional Statistical Analysis of the DAWM Results Adequacy for Traditional Statistical Analyses of the DAWM results. The logit measures mentioned in Table 5 were used in SPSS for further analyses. Various analyses were attempted but were not used because they did not satisfy the necessary assumptions on data behavior. Two-Way Repeated Measure ANOVA also called a TwoFactor Repeated Measure ANOVA would have been best suited to the situation based on three repeated measures of both a treatment and control group and so was the preferred method of analysis. This technique requires sphericity. A data set passes the test for sphericity if the population variance in all possible combinations is equal (van den Berg, 2017). The data for this study resulted in a significance of p = .003 on the test for sphericity using Mauchly’s W and therefore the assumption of sphericity was violated. Next, ANCOVA was considered but the lack of homogeneity of regression lines for the February, April and May data (homogeneity of variance) disallowed its use. Two different approaches were used to analyse the DAWM data: three 2X2 TwoFactor (treatment by gender) ANOVAs on the difference scores (April – February, May – April and May – February) with each student as an experimental unit and a three way 61 ANOVA (treatment, gender and block) on the difference scores with each class as an experimental unit and each school as a block. The adequacy of the data for use in both ANOVA analyses is explored further in this chapter. The choice of method of analysis was guided by Tabachnick and Fiddell (2007). Descriptive statistics for the DAWM difference scores. The final count for the participants was 50 in the control group and 85 in the treatment group. The difference in size is due, in part, to one treatment group class consisting of two classes combined. Tables 6, 7 and 8 contain the descriptive statistics for April – February, May – April, and May – February. Most of the difference Rasch Measures are positive indicating a higher Rasch Measure on the second assessment. An example of a negative difference is in Table 6: Female Control Group (-0.36), indicating a mean Rasch Measure higher in February than in April. The mean difference in the Rasch Measures for the May – April Control Group Females was zero (Table 7) and close to zero for the Female Treatment group. The statistical significant of the various mean difference Rasch Measures are examined further using ANOVA, later in this chapter. Tables 6, 7 and 8 contain the descriptive statistics for the DAWM data. In Tables 6, 7 and 8 n represents the sample size. Determining if the differences in these descriptive data are of a statistically significant level are explored later. 62 Table 6 DAWM Descriptive Statistics by Group and Gender April – February Difference Measure Group Control Gender Female Male Total Treatment Female Male Total Mean -0.36 0.84 0.24 0.90 0.27 0.60 SD 1.49 3.09 2.47 1.79 2.06 1.94 n 25 25 50 44 41 85 Table 7 DAWM Descriptive Statistics by Group and Gender May – April Difference Measure Group Control Gender Female Male Total Treatment Female Male Total Mean 0.00 -0.45 -0.23 -0.05 0.52 0.23 SD 1.12 1.79 1.49 1.75 1.86 1.82 n 25 25 50 44 41 85 Table 8 DAWM Descriptive Statistics by Group and Gender May – February Difference Measure Group Control Gender Female Male Total Treatment Female Male Total Mean -0.36 0.39 0.01 0.85 0.79 0.82 SD 1.71 2.93 2.40 1.83 2.29 2.05 n 25 25 50 44 41 85 63 DAWM with Students as the Experimental Unit Checking assumptions for analysis. The difference scores were examined to determine if it was advisable to use ANOVA. The assumptions were: the dependent variable has to be continuous, the independent variables must consist of independent categories, observations must be independent, homogeneity of variance for each combination of groups, data must be approximately normal, and no significant outliers (Laerd, n.d.). Normal distribution, homogeneity of variance and independent samples were the only assumption that needed to be met according to Field (2009) and only Normality and homogeneity of variance was required by Pagano (1998). Some sources replace the requirement for homogeneity of variance with homogeneity of error in variance (Kutner et al, 2004). Levene’s test is suggested for testing variance (Laerd, n.d.) but Levene’s test tests for equality of error variance so satisfies both assumption. The Rasch measure for each student is a continuous scale, satisfying the first assumption. The independent variables Gender and Treatment or Control group are independent, meeting the second assumption. No participants were in both the Treatment and Control group, meeting the assumption of observations being independent. Levene’s test for equality of error Variances results indicated it was reasonable to assume equal variance (April – February p = .056, May – April p = .130, May – February p = .141). The other assumptions were also examined. The April – February difference score data do not show normality (p = .001, Shapiro-Wilk’s) indicating that the assumption of normality is not statistically evident. The May – April data also do not show normality (p = .000, Shapiro- Wilk’s). The May – 64 February data do appear normal (p = .093, Shapiro-Wilks’). ANOVA is robust to nonnormal data and the histograms suggest that though the data is not normal, the data does somewhat follow the normal curves. (Figure 7, 8 and 9). Due to the sample size, n, being large (n > 50) it is advised to look beyond the test statistics (Field 2009; Laerd, n.d.). As can be seen in Figure 7 (April – February histogram) there are a few data points at the low end that may be contributing to the non-normal results but as discussed previously there is no valid reason to reject the participants’ data. Figure 7. DAWM frequency distribution and normal curve for April – February data. In Figure 7 the high peak of entries just above zero and the few entries below -5.00 can be observed not following the normal curve. 65 Figure 8. DAWM frequency distribution and normal curve for May – April data. Figure 8 also shows a peak near zero and a few participants near the ends of the normal curve but outside of those entries does have a general normal shape. Figure 9 shows the May – February data which does have an approximately normal distribution (p = .093). 66 Figure 9. DAWM frequency distribution and normal curve for May – February data. Further examination of normality was warranted. Though the Shapiro-Wilk’s tests suggest the data is not normal (except May – February) the Kurtosis values together with skewness indicate the April – February data do not have an undue level of lack of normality (except April – May). According to Tabachnick and Fiddell (2007) when determining a level of skewness, examine how far from mean the centre of the distribution is and Kurtosis values other than zero are an indicator of peaks that are not ideal. More specifically, if skewness is below -0.5 or above 0.5 and Kurtosis is below -2 or above +2 concerns for normality are justified (Brown, 2016). April – February Kurtosis value was 1.731 (SE = .414). The Kurtosis value for May – April is 2.534 (SE = .414) which does support lack of normality. May – February Kurtosis value is 1.049 (SE = .414) supporting the normality found previously. Skewness was also minimal for two data sets (April – February .127± .209 and May – February -.030 ± .209). May – April skew of .870 ± 67 .209 was a bit high. Together these results suggest the level of lack of normality is not at a level of concern (except for May – April). In a data set of 135, approximately seven participants would be expected to be outliers (∝ = .05). Figures 10, 11 and 12 show box-plots of the difference scores. Figure 10. DAWM box-plot for April – February data test for outliers with difference score scale on the vertical axis. Figure 10 shows 13 outliers. This is the largest number of outliers on any of the difference scores. This is more than the expected seven outliers on a group this size. 68 Figure 11. DAWM box-plot for May – April data test for outliers with difference score scale on the vertical axis. There were a total of six outliers reported for the May – April data (Figure 11), which is less than the seven expected. Figure 12. DAWM box-plot for May – February data test for outliers with difference score scale on the vertical axis. 69 There were a total of 8 outliers reported in the May – February data (Figure 12), only one more than the expected seven. The assumptions for use of ANOVA, (continuous variable, matched pairs, no significant outliers, normality and equal variance (Laerd, n.d.) were met with the following exceptions: the May – April data lacks normality, and April – February data had 13 outliers. ANOVA is however robust to a lack of normality (Field, 2009). One source indicated that the ANOVA results for data like April – February must be interpreted with caution due to the number of outliers (Laerd, n.d.). However, another source replaces the assumption of no outliers with an assumption of equal sample size (Plonsky, 1997). Yet another source has equality of variance and equal sample size as the only two assumptions for ANOVA (Pagano, 1998, p 418). Only results from participants who wrote all three assessments (February, April and May) were used, therefore sample sizes were equal. Also, quality of variance is met by all three sets of difference scores. Therefore, a 2X2 Two-Factor ANOVA was used to analyse the data (April – February, May – April and May – February). Traditional Statistical Analysis of the treatment on DAWM assessment. Three 2X2 Two-Factor (group and gender) ANOVA tests on the April – February, May – April and May – February difference scores were preformed. The significance results for the Tests of Between-Subject Effects are reported in Table 9. For each of the three difference scores (April – February, May – April and May – February) logit measures, the between subject effect was examined. Table 9 shows the p values for the between subject effects. A significant interaction was found between 70 Group (treatment and control) and Gender (p = .017) in April - February. The interaction is illustrated in Figure 13 (graphically shown as crossing lines). No other statistically significant interactions were reported (last row Table 7). The power for the April – February data was .22. The power for the May – April and the May – February data were higher at .72 and .79 respectively. Table 9 DAWM Between-Subject Effects Statistical Significance (p) Source Gender Treatment Gender and treatment April – Feb p .563 .829 .017 May – April p .838 .127 .092 May – Feb p .378 .040 .307 A statistically significant difference was found between the treatments group and the control group in the May – February data (p = .040, Cohen’s d = 0.337) (Table 7) with a small to medium treatment effect size (Partial ETA of .032, Cohen’s d = 0.337). There was not a significant difference between gender or treatment groups for the May – April data (p =.838, p = .127 respectively) with a minimal treatment effect size (Partial ETA .018, Cohen’s d = 0.30). Due to the interaction in the April – February data (p= .017) and the possible interaction in May – April (p = .092), the most interpretable data set is May – February. A trivial effect size (Cohen’s d = 0.15) was determined for the April – February data and a small effect size (Cohen’s d = 0.30) was determined for the May – April data. The means and standard errors for each of the difference scores are shown in Tables 10, 11, and 12. 71 Table 10 DAWM Estimated Marginal Means and Standard Error April – February Group Gender Control Female Male Female Male Treatment Estimated Marginal Mean -.356 .840 .901 .271 2xSE .848 .848 .638 .622 As opposed to the earlier means reported, the means in Tables 10, 11, and 12 are Estimated Standard Means, which are the means adjusted for the other variables present. Table 10 shows the April – February data. Twice the Standard Error is also reported in Table 10. In Figures 13, 14, and 15, the data from Tables 10, 11 and 12 are represented graphically. The interaction, characterized by the crossing of the lines in a graph, is observed in the April – February data and the possible interaction in the May – April data are visible as the crossing lines. E. Wagenmakers, A. Krypotos, A. H. Criss, and G. Iverson (2012) explore interactions, characterised by the crossing of lines, and indicate that they are nonremovable and caution the interpretation of any main effect when a interaction is present. 72 Estimated Marginal Means Versus Group April – February Separated by Gender DAWM Logit Measeure Estimated Marginal Means 2 1.5 1 0.5 0 -0.5 -1 -1.5 Control Treatment female male Figure 13. DAWM estimated marginal means versus group April – February separated by gender. Figure 13 shows the results from Table 10 graphically. Logit measure is on the vertical axis. The left column shows the control group separated by gender. The right column shows the treatment group separated by gender. Table 11 DAWM Estimated Marginal Means and Standard Error May – April Group Gender Control Female Male Female Male Treatment Estimated Marginal Mean -.003 -.454 -.051 .523 2xSE .678 .687 .512 .528 Table 11 shows the May – April data. The data are represented graphically in Figure 14. 73 Estimated Marginal Means Versus Group May – April Separated by Gender DAWM Logit Measeure Estimated Marginal Means 1.5 1 0.5 0 -0.5 -1 -1.5 Control Treatment Female Male Figure 14. DAWM estimated marginal means versus group May – April by gender. Figure 14 is a visual representation of Table 11. Table 12 DAWM Estimated Marginal Means and Standard Error May – February Group Gender Control Female Male Female Male Treatment Estimated Marginal Mean -.359 .386 .850 .794 2xSE .876 .876 .660 .684 Table 12 shows a range of Estimates Marginal Means between -0.359 and 0.850 with twice Standard Error between .660 and .876. Figure 15 shows this information graphically. The power for the May – February data was .79. 74 Estimated Marginal Means Versus Group May – February Separated by Gender DAWM Logit Measeure Estimated Marginal Means 2 1.5 1 0.5 0 -0.5 -1 -1.5 Control Treatment female male Figure 15. DAWM estimated marginal means versus group May – February by gender. The final trend of increase in the estimated marginal means can be seen in Figure 15. There is no statistically significant difference between male and female participants in the control and treatment groups, however when the means of the female and male participants are taken into consideration together there is a statistically significant difference between the control and treatment groups with a small to moderate effect size (p = .040, Cohen’s d = 0.337). The effect size for the female participants was large (Cohen’s d = 1.380). The male participants showed a medium effect size (Cohen’s d = .466). 75 Summary of Analysis of DAWM Assessment Intervention All participants’ scores were retained for the analyses if they wrote all three assessments (February, April and May). Though some participants did not perfectly fit the Rasch model, their results were retained for analysis. There was enough variety in the item difficulty (item reliability of .98). Rasch Analyses indicated the 19 items, though difficult for the participants, had 7 statistically different levels of difficulty (Item Separation 7.51), that the items varied enough in difficulty, the assessment had good internal consistency and an appropriate reliability for a low stakes test (Person Reliability .74) (Wells & Wollack, 2003). The assessment may not have been sensitive enough to reliably determine individual participant’s scores (Person Separation 1.68). All items were also retained for the analyses. The average logit measure of participants who did not endorse a word as being associated with mathematics (0) was lower than the average ability of participants who did endorse a word as being associated with mathematics (1) indicating the scale was appropriate. Anchors were used for all but Item 1 and 11. Overall, the DAWM instrument is reasonably suitable for the task of assessing the treatment effect in this study. Summary of DAWM traditional statistical analysis with students as the experiment unit. The difference scores of the logit measures for each assessment (April – February, May – April, May – February) for the DAWM met the assumptions necessary to preform a 2X2 Two Factor (treatment, gender) ANOVA. A statistically significant interaction was indicated for the April – May data (p = .017). The April – February data had a power of .22 well below the .8 guideline. The only statistically significant difference 76 was identified for the May – February data for treatment (p = .040) with a small to medium effect size (Partial ETA 0.032, Cohen’s d = 0.337). The power for the May – February data was .79. Close to the .8 guideline. For the May – February data the effect size for the female participants was larger than that of the male participants (Cohen’s d = 1.38 and d = .466 respectively). This indicated a difference between control and treatment from before the intervention to a month after the intervention ended. DAWM with Schools as a Block and Treatment and Control Separating the Units The experimental units. The study took place in five schools. In three of the schools there was one class in the control group and one class in the treatment group. In one school there were two treatment group classes and one control group class. In another school there was one treatment group class and two control group classes. To facilitate comparing blocks (schools) if a school had more than one class in either the treatment or control group the classes were combined into one. The result was five blocks each with a treatment group and a control group (each of these groups as an experimental unit). Table 13 contains the summary of the number of participants separated by treatment and control, gender, and block. 77 Descriptive data for each group in each block is not being included. Table 13 The Number of Participants in each Between Subject Factor Treatment 50 Control 85 Male 66 Female 69 Block 1 24 Block 2 50 Block 3 27 Block 4 22 Block 5 12 Checking assumptions for analysis. The difference scores were examined at length with each student as an experimental unit. Due to the robustness of ANOVA to non-normal data, normality of errors was explored. The variance was investigated using Levene’s test and at a significance level of ∝ = 0.5, equal variance was reported for May – April (p = .290) and May – February (p = .406). With a significance level of ∝ = .01 for April – February (p = .010). All other assumptions were met as discussed previously. 78 Summary of traditional statistical analysis of DAWM with Blocks. Three difference scores were analysed in a Three Factor (Treatment, Gender and Block) ANOVA (April – February, May – April and May – February), resulting in the model: 𝑦𝑖𝑗𝑘 = 𝜇 + 𝜏𝑖 + 𝛾𝑗 + 𝛽𝑘 + 𝜏𝑖 ∗ 𝛾𝑗 + 𝜏𝑖 ∗ 𝛽𝑘 + 𝛾𝑗 ∗ 𝛽𝑘 + 𝜏𝑖 ∗ 𝛾𝑗 ∗ 𝛽𝑘 + 𝜀𝑖𝑗𝑘 , where 𝜏𝑖 represents the treatment, 𝛾𝑗 represents the gender and 𝛽𝑘 represents the block effects. For the April – February data, statistically significant main effect was identified for Block (p = .025), however there was also a statistically significant interaction between Block and Treatment (p = .001). Figure 16 indicating that there was an interaction between the intervention and the schools which indicated different schools responded differently to the treatment. Treatment and Control Versus Blocks April - February Estimated Means 8 6 4 2 0 1 2 3 -2 4 5 Block t c Figure 16. Treatment and control groups versus blocks of the April – February data 79 The May – April Three - Way ANOVA did not indicate any statistically significant effects or interactions. The May – February data was also analysed. As with the April – February data, there was a statistically significant Block effect (p = .004) and an interaction between Block and Treatment (p = .001). For the Treatment, Gender and Block a statistically significant interaction was identified (p = .016). There were no Blocks reported by Tukey HSD as being statistically significantly different. Math Attitude Assessment (MAA) Each time the participants wrote this assessment (MAA) the order the statements were presented in was different. There were a total of four versions of the MAA. During data entry an error was found in one version of the MAA. One version (Version 2) of the MAA had Statement 11 (I know I can get math questions right,) twice and Statement 1, (Math is easy,) was not present (Appendix B). Rasch Analysis can be done with missing values. A quasi random process was used to select one of the responses for Statement 11 and Statement 1 was left as missing data. Of the original 163 participants, 44 received Version 2, the affected MAA. The analysis for the MAA data was conducted in the same order as the DAWM. The participants, items and scale were examined, then anchor values set using Rasch Analysis techniques then the difference scores were used in traditional statistical analysis. Rasch Analyses MAA. Rasch Analysis of the MAA results took place in four stages. First, the participants were analysed to determine if any participants warranted removal 80 from the study. Second, the items in the MAA were assessed to determine if all items should be retained. Third, the scale in the assessment was examined to determine if it was appropriate for the participants. Forth, anchored values were determined for further analysis. Rasch Analysis of participant’s results MAA February treatment and control. All participants data for February were analysed together in Winsteps to identify misbehaving data. Table 14 shows an excerpt from the full person measure results. All participants data were examined if they had a high infit or outfit value, but no responses warranted removal. The MAA included 12 statements with, “No”, “Sometimes”, and “Yes” as the responses. “No” was scored as “0”, “Sometimes” as “1” and “Yes” as “2”. There were both positively and negatively phrased statements. For example, Statement 6 was, “Math is useful”, and statement 2 was, “Math is useless”. Negatively worded items (statements) were reverse coded. If a participant responded “Yes” to “Math is useless,” an original score of “2” was been assigned to the response but then switched to a “0” for further calculations. Table 14 includes 9 columns of information from the MAA. The “Count” column refers to the number of statements to which the participants responded. Unclear responses were entered as missing data. The “Score” is the result the sum based on the responses, after re-framing scores based on negative language. For example, 12 responses of “Yes” would result in a score of 24. The Logit Measure is affected by both the count and the score. The grey cells in Table 14 illustrate that if the scores are equal, for example 11, the participant with the lower count received the higher Logit Measure 81 (more positive attitude toward mathematics). If the counts are equal, for example 12, the participant with the higher score receives the higher Logit Measure (more positive attitude toward mathematics). The Model Standard Error column in Table 14 is the Model Standard Error as described previously (Equation 9). Generally, the higher the SE the higher the logit measure and visa versa. There are exceptions to this trend. 82 Table 14 Excerpt MAA Participant Measures COUNT SCORE 12 24 12 24 12 24 12 23 LOGIT MODEL INFIT MEASURE SE MS 4.48 1.84 1 4.48 1.84 1 4.48 1.84 1 3.24 1.03 1.01 z 0 0 0 0.35 OUTFIT MS 1 1 1 0.73 z 0 0 0 0.22 CORR 0 0 0 0.22 12 11 12 21 20 20 2.02 2.46 1.67 0.63 0.75 0.56 0.96 0.66 2.35 0.13 -0.31 2.13 0.67 1.09 1.97 -0.27 0.39 1.44 0.35 0.38 0.11 12 11 11 19 19 19 1.38 1.91 1.91 0.52 0.64 0.64 0.68 0.78 0.53 -0.66 -0.22 -0.79 0.62 0.46 0.48 -0.66 -0.68 -0.64 0.71 0.81 0.67 12 12 17 17 0.9 0.9 0.46 0.46 1.29 0.66 0.82 -0.9 1.36 0.59 0.9 -1 0.05 0.65 12 11 16 16 0.7 1.04 0.44 0.49 0.46 1.32 -1.84 0.85 0.58 1.96 -1.14 1.75 0.68 0.34 11 10 11 12 11 11 0.27 0.2 -0.08 0.43 0.46 0.43 0.69 1.22 1.8 -1.01 0.72 2.16 0.71 1.24 1.72 -0.84 0.7 1.8 0.52 0.1 0.27 11 10 11 12 12 10 11 8 7 6 5 4 2 1 -0.63 -0.63 -1.06 -1.38 -1.65 -2.19 -3.11 0.45 0.48 0.48 0.5 0.55 0.75 1.02 1.22 0.67 0.99 0.42 0.71 1.16 1.15 0.71 -0.89 0.09 -1.59 -0.5 0.45 0.48 1.09 0.62 1.29 0.43 0.63 1.52 7.65 0.34 -0.85 0.7 -1.15 -0.47 0.79 2.6 0.7 0.75 0.31 0.64 0.39 -0.16 -0.75 83 Fit statistics in Table 14 vary, as they did with the DAWM data. If a participant responded positively to all 12 items (statements) their infit and outfit were 1 with a z = 0 which indicates that the person is helpful to the measure. One student (the bottom row in Table 14) has an extreme outfit value of 7.65 with z = 2.6. Any participants whose fit statistics were outside the ideals, were reviewed. The equations used for the RaschAndrich rating scale are in Chapter 2, Equations 8, 9, 10, 11 and 12. All participants’ individual scores were positively correlated to the group scores, except for the two participants in the bottom two rows. For the three participants who scored 24, correlation was zero, expected when there is no variation in score. Though not all participants fit the model ideally, all participants who wrote all three MAA assessments (February, April and May) were retained for traditional statistical analysis, to be consistent with the DAWM assessment analysis. Of the 163 students who participated, 135 responded to all three assessments (February, April and May). A few of these were different from the respondents in the DAWM data. This can be observed in the gender breakdown. Of the 135 participants 67 self identify as female and 68 self identified as male. This resulted in the same attrition rate as the DAWM, 17.1%. Rasch Analysis of Items for February MAA. Rasch Analysis of the Math Attitude Assessment results for the entire group was done to assess the MAA as an instrument. The items showed a Rasch person reliability of .67 – indicating a need for more items in the assessment. Recall, person reliability is test reliability which is analogous to reporting Cronbach Alpha, so the test reliability is close to the .7 guideline for a low-stakes test (Wells & Wollack, 2003). A Person separation of 1.43 was reported – indicating that while 84 the assessment may be suitable for detecting group differences, it is not sensitive enough for scoring individuals. The items in the MAA showed an item reliability of .96. This value indicates a large enough sample size with enough variety is item difficulty. The item separation was 4.80. This value indicates the items can be separated into 4 statistically significantly different levels of difficultly. Table 15 contains the Logit Measure, Fit information and Correlation for the 12 items in the MAA. The item number, in the first column, corresponds to the statements listed in Table 16. The “Count” column indicates the number of responses to the statement. The “Score” column indicates the sum of the response values (sum of zeroes, ones and twos). Generally, as the “Score” increases the logit measure for that item decreases. Table 15 Item Difficulty Math Attitude Assessment (MAA) Item No. 12 4 7 1 11 3 8 5 9 6 2 10 Count 158 157 153 114 158 152 155 159 150 156 154 151 Score 111 191 187 152 219 215 227 241 232 248 261 264 Logit Model Measure SE 1.80 0.13 0.56 0.13 0.54 0.13 0.25 0.15 0.12 0.13 0.03 0.13 -0.10 0.14 -0.27 0.14 -0.34 0.15 -0.48 0.15 -0.93 0.17 -1.19 0.18 Infit MS 1.16 0.76 0.79 0.76 1.17 0.86 0.96 0.92 1.03 1.55 1.17 1.10 Infit z 1.5 -2.7 -2.3 -2.1 1.6 -1.3 -0.3 -0.6 0.3 3.8 1.1 0.6 Outfit MS 1.60 0.75 0.85 0.88 1.39 0.72 1.01 0.70 0.88 1.89 1.16 1.64 Outfit z 3.5 -2.2 -1.2 -0.8 2.5 -2 0.1 -1.9 -0.6 3.9 0.7 2.1 Corr 0.58 0.64 0.53 0.50 0.34 0.67 0.58 0.64 0.49 0.31 0.44 0.41 85 The statement order shown in Table 16 is based on the order indicated in Table 15. They are ordered from the item found most difficult to endorse at the top, to the easiest to endorse at the bottom. The “N” beside several of the items in Table 16 indicate the items whose scores were reversed due to negative language. Table 16 Items from the MAA Ordered Hardest to Easiest by Logit Measure Item 12 4 7 1 11 3 8 5 9 6 2 10 Statement When I grow up I want a job that uses math. Math makes me feel happy. I find math hard. N Math is easy. I know I can get math questions right. I want to do math next year. I am sad when I do math. N Next year I want to stay away from math. N I can think of ways to use math. Math is useful. Math is useless. N Math makes me scared. N The “Logit Measure” column in Table 15 indicates the difficulty level of the item to which the participants responded positively. The higher scored items have a lower logit measure as the measure is ‘difficulty’ not ‘easiness’. For example, it was harder for a participant to indicate they want a job that uses mathematics when they grow up than it was to indicate mathematics does not scare them. The item logit measures ranged from 1.19 to 1.8 and their Model Standard Errors between .13 to .18. The errors increased as the logit measures decreased. Items 6 and Item 2 received similar Logit Measures. The similar measures for items 2 and 6 is further indication of reliability in participant 86 responses. This is not surprising due to the similarities in the items. The difference in the Logit Measures for Items 4 and 10 may be an indication that the participants did not interpret the opposite of “scared” as “happy”. Infit ranged from 0.76 to 1.55 with z scores from -2.7 to 3.8. Outfit ranged from 0.7 to 1.89 with z scores from -2.5 to 3.9. All MS are less then +2, however some of the items have z values outside of the -2 to +2 ideal range. MS was given priority over z values when determining item retention as suggested by Bond and Fox (2012). The z = 3.8 and 3.9 for Item 6 and z = 3.5 for Item 12 are greater than 3 and of particular concern. Exploration of the removal of Item 6 is discussed further later in this chapter. The last column is correlation. Of the 12 statements, all items showed a correlation greater than +.3. This indicates that all of the 12 statements showed a moderate positive correlation between the items and the assessment. The largest correlation of .67 was identified for Item 3, “I want to do math next year.” Both Item 12 and Item 5, “When I grow up I want a job that uses math,” and “Next year I want to stay away from math,” respectively had a correlation of .64. The three items that were most positively correlated with the assessment were the items that were interested in Motivation. The lowest correlations were identified for Item 6, “Math is useless,” correlation .31, Item 10, “Math is useful,” correlation .34, and Item 11, “I know I can get math questions right,” also with a correlation of .34. Items 6 and 10 assessed Value and Item 11 assessed Self-Confidence. The results from Table 16 are presented in a more visual representation in the bubble plot in Figure 17. The Logit Measure is on vertical axis. Overfit and underfit are 87 indicated in the horizontal axis. Items 6 and 12 are of concern, showing underfit (unpredictable responses). MAA Item Difficulty versus Outfit z Measures More 3 2 1 0 Less -1 -2 -4 -2 0 2 4 6 Figure 17. MAA items z scores of Outfit. In Figure 17 the similarity of the Model Standard Error is reflected in the similar radii for all items. Items with zOF <-2 or zOF > +2 are generally of concern. Removal of item was not advised due to the item reliability score indicating that more items would be beneficial. However, item assessment was performed with item 6 removed; Figure 18 shows how Items 11 and 12 have an increased z in the new item assessment. The outfit z 88 score for Item 11 went from 2.5 to 3.0 and the outfit z score for item 12 went from 3.5 to 3.7 when item 6 is removed. MAA Item Difficulty versus Outfit z Without Item 6 Measures More 3 2 1 0 Less -1 -2 -4 -2 0 2 4 6 Figure 18. MAA item z score of outfit item 6 removed. Removal of Item 6 did not improve the assessment. The results from Figure 18 helped confirm that all items should be retained. If removing an item of high z score on outfit improved the others or even kept the z scores for the other items the same there may have been a reason to remove an item but there is no evidence that removal would be a benefit. Each of the 12 statements belong to the categories Self-Confidence, Value, Motivation or Enjoyment. When attempts were made to separate the 12 statements into 89 their respective categories for further analysis, there were too few statements in each category to allow for meaningful results. All items were retained for the MAA. The fit statistics did not indicate the need for removal and more items were indicated as being beneficial. This was further evidence to support full item retention. The logit measure for each statement was used as an anchor for further Rasch Analysis. Rasch Analysis of Scale for MAA. Comparison of the abilities of the participants to the difficulty of the assessment was done using Figure 19, the “Wright” map (Bond & Fox, 2012). The “X” indicates a single participant on the left and “MA#” indicates the item. For example, “MA1” is Item 1, “Math is easy.” Figure 19 shows the item difficulties are contained within the person abilities. The item separability of 4.80 indicating four statistically significantly different levels of difficulty and these are suggested in the map. Several participants’ abilities are higher than the most difficult item (Item 12). The mean participant Logit Measure is higher than the mean Item Logit Measure. Together these results suggested the assessment was relatively easy for the participants. 90 Figure 19. MAA person ability and item difficulty map. 91 Figure 20 indicates the average person measure for the endorsement of each statement or item and examines scale. Figure 20. MAA item versus person measure for item endorsement. In Figure 20 the average person measure for response to each item is shown. The zero, one, two order that is expected for a scale is observed for all but one item. “MA6”, “Math is useful.” is the only item that does not follow the ideal of “0” then “1” then “2” as the response order. Indicating the response of “Sometimes” “Math is useful” is easier than a “Yes” or “No” response. Note the “m” indicates missing data. 92 Anchored Logit Measures for February, April and May MAA. With person participation confirmed, item retention confirmed and therefore anchors set (using the February data), the Logit Measure for each participant was determined for each of the three assessments (February, April and May). Samples of the values are given in Table 17. Blank lines indicate where data were not reproduced in the table due to space. Table 17 Measure, Gender and Group (Treatment or Control) for February, April and May MAA Excerpt Person ID Gender 91918 79510 FEB APR MAY Group Logit Logit Logit Measure Measure Measure M -2.23 -0.24 1.41 C F -1.63 -1.01 -1.16 T 78618 66621 M M 0.16 0.17 0.77 -0.19 0.49 -0.02 C C 80220 38412 68421 29511 44214 85414 321 M M F F F F M 0.55 0.59 0.67 1.36 1.36 1.36 1.41 -0.02 -0.02 0.33 1.8 1.49 1.49 0.15 -0.6 -0.05 -0.02 1.75 1 2.2 1 C T C T T T C 77011 45918 M F 1.64 1.7 2.17 2.43 1.51 1.82 T C 36913 29610 26214 M M F 2.36 2.43 2.46 1.8 3.44 1.18 1.82 2.08 4.75 T T T 7520 23421 M F 4.46 4.54 3.29 4.53 1.76 4.75 C C 93 The first column in Table 17 indicates the student ID number and the second column indicates self identified gender. Table 17 was ordered based on the logit measure for February from lowest to highest. The logit measure here is an indication of a propensity to indicate a positive attitude toward mathematics as indicated by responses to the MAA. The participant with a logit measure of 4.54 in February responded in a more positive way to the assessment than the participant with a logit measure of -2.23 and would be said to have a more positive attitude toward mathematics. Referenced values are highlighted in Table 17. The -2.23 Logit Measure is the most negative response and 4.54 is the most positive response found in February. Overall the maximum Logit Measure was 4.75 and the minimum measure was -4.62. Both the maximum and minimum occurred in May. The change in Logit Measures was not consistent. Some participants’ Logit Measures increased in each MAA. An example of this is participant 85414. She started at 1.36 then increased to 1.49 and ended at 2.20. Others decreased, like participant 68421 (0.67, 0.33 and -0.02 respectively). And others were inconsistent in their change, showing increases and decreases, like participant 29610. A value of one was assigned to neutral responses. A value of zero was assigned to negative responses and a value of two was assigned to positive responses. The higher logit measure indicates the participants propensity to indicate a more positive attitude toward mathematics. Participants with a score of 12 out of 12 responses received a Logit Measure of approximately zero, -0.02. Participants with a score of 12 out of fewer than 94 12 responses received a positive Logit Measure. If the score divided by the number of responses was greater than one, a positive Logit Measure resulted. Therefore, a negative Logit Measure would indicate a negative attitude and a positive Logit Measure would indicate a positive attitude. The mean anchored Logit Measure for February was 1.085. This indicates an overall positive attitude toward mathematics using the MAA, which is supported by a mean score of 15.8 (SD = 4.5) on the February data for all 163 participants. The average Logit measures range from .0905 for February control group to 1.501 for the May treatment group. Overall, the mean attitudes were slightly positive for February, April and May. This concludes the Rasch Analysis portion of the results of the MAA. Each participant now has a Logit Measure indicating their attitude toward mathematics as measured by the MAA. An overall slightly positive attitude was found for the participants. Next the Rasch Logit Measures were used in a traditional statistical analysis approach to determine if any differences are present between the control and treatment group while considering gender, with both the students as experimental units and then with schools as blocks. Traditional Statistical Analysis of the MAA Results Justification of the Traditional Statistical Analysis method used for MAA results. The logit measures from Table 13 were used in SPSS for analyses of treatment effect. To be consistent with the analyses methods with the two assessments (DAWM and MAA) difference scores were tested for use in a 2X2 Two-Factor (group and gender) ANOVA with students as the experimental units, as was a Three-way (Treatment, Gender, Block) 95 ANOVA . The data from participants who wrote all three assessments was used. The final count was 67 participants who self identified as females and 68 who self identified as males. The control group had 51 participants and the treatment group had 84 participants. Descriptive Statistics of the MAA Difference Scores. Table 18, 19, and 20 contains the descriptive statistics for the difference in Rasch Measures for the MAA data: (April – February, May – April, and May – February. The n represents the size of each sample. Determining if the differences indicated between treatment and gender are of a statistically significant level is investigated later in this chapter. As with the DAWM results there are both positive and negative differences. Table 18 MAA Descriptive Statistics by Group and Gender April – February Difference Measure Group Control Gender Female Male Total Treatment Female Male Total Mean 0.01 0.18 0.10 -0.02 0.05 0.02 SD 0.70 1.01 .88 1.31 1.14 1.23 n 24 27 51 43 41 84 Table 19 MAA Descriptive Statistics by Group and Gender May – April Difference Measure Group Control Gender Female Male Total Treatment Female Male Total Mean 0.30 -0.04 0.12 0.27 0.31 0.29 SD 1.18 0.81 1.01 1.07 1.54 1.32 n 24 27 51 43 41 84 96 Table 20 MAA Descriptive Statistics by Group and Gender May – February Difference Measure Group Control Gender Female Male Total Treatment Female Male Total Mean 0.31 0.14 0.22 0.25 0.36 0.31 SD .82 1.10 0.97 1.16 1.61 1.40 n 24 27 51 43 41 84 Checking Assumptions for analysis with students as the experimental unit. As with the DAWN assessment, the MAA results are continuous dependent variable with matched pairs. The difference scores were examined for equality of variance, normality and outliers to determine if the other assumptions for use of ANOVA were met. Due to the nature of the assessment, an ∝ = .01 was appropriate. With an alpha of .01 all three sets of difference scores showed homogeneity of variance. The April – February had a p of.026 on the Levene’s Test. The April – February data had a p of .019. The May – April data showed equality of variance with a p of .319. Homogeneity of variance for the 2X2 Two-Factor ANOVA was adequately met. Normality was tested using the Shapiro-Wilk’s test and May – April and May – February both report p = .000 so are not normal. April – February is normal (p = .301). Due to the sample size, n, being large (n > 30) it is advised to look beyond the test statistics. The histograms for the data further the investigation into normality. In further exploring normality Figures 21, 22, and 23, show both the frequency distributions and the normal curves. The difference between the normal April – February data compared to the May – February and the April – February data is illustrated. 97 Figure 21. MAA frequency distribution and normal curve for April – February data. Figure 21 supports the Shapiro-Wilk’s p = .301 to indicate normality. There is evidence of outliers, but normality is evident in the shape of the frequency distribution graph and its relationship with the normal curve. Figure 22 shows the frequency distribution for May – April and the normal curve. Outliers are evident. There is a concentration of responses near the mean of zero. There is a loose approximation of the normal curve by the frequency distribution. 98 Figure 22. MAA frequency distribution and normal curve for May – April data. Figure 23 of the frequency distribution of the May- February data though not normal (Shapiro- Wilk’s p = .000) does show a pattern close to a normal curve. The outliers close to -6 are clear and a drop in frequency just above the mean of zero are clearly visible. 99 Figure 23. MAA frequency distribution and normal curve for May – February data. Skew and Kurtosis were examined to further explore normality. May – February skew was -.484 with Kurtosis of 2.922. May – April skew was 1.014 and Kurtosis was 5.259. Skew standard error was .209 and Kurtosis standard error was .414. With the guideline of ± 2 for Kurtosis, neither the May - April data nor the May – February data can claim normality. ANOVA, however, is robust to non-normality so will still result in meaningfully interpretable results. The outliers suggested in the frequency distributions (Figure 21, 22, and 23) can be more readily observed in the Box-Plots below (Figures 24, 25, and 26). For a sample size of 135, seven outliers would be expected. The April – February data shows 3 outliers (Figure 24). The May – April data shows 7 outliers (Figure 25). The May – February data shows 5 outliers (Figure 26). 100 Figure 24. MAA box-plot for April – February data test for outliers. Figure 24 shows 3 outliers. It also shows equal whiskers, minimal skewing and is about centered at zero. A mean of zero is expected for Rasch logit measures. Figure 25. MAA box-plot for May – April data test for outliers. 101 There were a total of 7 outliers reported for the May – April data seen in Figure 25. Again, there is a zero centre, minimal skewing and equal whiskers. Figure 26. MAA box-plot for May – February data test for outliers. There were a total of 5 outliers reported in the May – February data reported in Figure 26. Though this data shows an outlier between -4 and -6, the whiskers are equal, the centre is zero and there is minimal skewing. Summary of Assumptions for data to be use in 2X2 Two-Factor ANOVA. To use a 2X2 Two-Factor (Treatment and Gender) ANOVA assumptions must be met. The assumption of a continuous variable, matched pairs and no significant outliers was met for all three difference scores (April – February, May – April and May – February). The assumption of homogeneity of variance (HOV) was also met (∝ = .01). Normality is reasonable to assume in April – February (p = .301). There were minor indications of 102 deviation from normality. However, histograms and boxplots suggest these variations are of minor importance and ANOVA is robust to non-normal data. The assumptions to use a 2X2 Two Factor ANOVA were met to a sufficient level that the output from the 2X2 TwoFactor ANOVA will be able to be interpreted in a meaningful way. Traditional Statistical Analysis of the treatment on MAA with students as the experimental unit. Three 2X2 Two-Factor (Treatment and Gender) ANOVA tests were preformed on April – February, May – April and May – February difference scores were preformed. The significance results for the tests of Between-Subject Effects are reported in Table 21. No statistically significant differences were reported between the groups (all p > .30, ∝ = .05). Table 21 MAA Between-Subject Effects Statistical Significance (p) Source Gender Treatment Gender and treatment April – Feb p .543 .704 .800 May – April p .479 .469 .381 May – Feb P .885 .718 .535 The lack of a statistically significant difference is supported and represented visually in Figures 27, 28, and 29. Tables 22, 23, and 24 summarize the values used to generate the plots in the corresponding figures. 103 Table 22 MAA Estimated Marginal Means and Standard Error April – February Group Control Treatment Gender Estimated Marginal Mean Female 0.008 Male -0.018 Female 0.179 Male 0.053 2xSE 0.456 0.428 0.340 0.348 The means vary from -0.018 to 0.179 with standard errors from 0.170 to 0.228. Figure 27 is a graphical representation of the information in Table 22. Logit Measeure Estimated Marginal Means Estimated Marginal Means Versus Group April Feb Separated by Gender MAA 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 Control Treatment female male Figure 27. MAA estimated marginal means versus group April – February separated by gender with 2XSE error bars. 104 Table 23 MAA Estimated Marginal Means and Standard Error May – April Group Gender Control Female Male Female Male Treatment Estimated Marginal Mean 0.304 0.271 -0.039 0.308 2xSE 0.496 0.468 0.370 0.380 The graphical representation of the data from Table 23 can be seen in Figure 28. Estimated Marginal Means Versus Group May April Separated by Gender MAA Logit Measeure Estimated Marginal Means 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 Control Treatment female male Figure 28. MAA estimated marginal means versus group May – April separated by gender with 2XSE error bars. The MAA estimated marginal means are reported in Table 24 for May – February. The means vary from 0.140 to 0.360 and the errors vary from 0.192 to 0.257. 105 Table 24 MAA Estimated Marginal Means and Standard Error May – February Group Gender Control Female Male Female Male Treatment Estimated Marginal Mean 0.312 0.253 0.140 0.360 2XSE 0.514 0.484 0.384 0.394 A more graphical representation of this data is shown in Figure 29. Logit Measeure Estimated Marginal Means Estimated Marginal Means Versus Group May Feb Separated by Gender MAA 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 Control Treatment female male Figure 29. MAA estimated marginal means versus group May – February separated by gender with 2XSE error bars. The power for all difference scores were low: April – February was .03, May – April was .28 and May – February was .12. The effect size was small for all difference scores as well: April – February Cohen’s d = .091, May – April Cohen’s d = .168 and May – February Cohen’s d = .093. 106 Summary of analysis of MAA intervention with students as the experimental unit. All participants’ results were retained for analyses if they wrote all three assessments (February, April and May). The average Logit Measures were slightly positive for all groups on each assessment, with a mean score supporting a positive overall attitude. Though some participants did not fit the Rasch model, no justification was found for removals. Sample size was determined to be large enough (item reliability .96). The Person Reliability (test reliability) of .67 indicated the MAA was only able to separate participants in to two ability levels so more items would have been beneficial. Item Separation of 4.80 indicated the items could be groups into 4 levels of difficulty. The person Separation of 1.43 indicated the MAA may not be sensitive enough to separate participants indicating a positive attitude toward mathematics on the MAA from a negative attitude. The MAA was also indicated as being too easy for the participants. The Math Attitude Assessment results when reported as difference scores of Rasch Logit measures met the assumptions necessary to obtain meaningful results from difference scores analysed using the 2X2 Two-Factor (Treatment and Gender) ANOVA. No statistically significant differences were reported in the 2X2 Two-Factor ANOVA. The power for each data set was low (April – February .03, May – April .28 and May – February .12). Summary of MAA Interventions with Blocks. A Three Factor (Treatment, Gender and Block) ANOVA was conducted on the same five Block grouping as the DAWM data for all three difference scores. Assumptions for ANVOA were explored previously except for variance due to the change in experimental units. The April – February and May – April 107 data showed equality or error of variance using Levene’s test (p = .057 and p = .154 respectively). The May – February data was not indicated as having equality of error in variance (p =.000) indicating that any significant results must be interpreted conservatively. No significant differences were indicated for treatment, gender, or block. No interactions were indicated and no differences between individual blocks were indicated (all 𝑝 > .05). Overall Summary In summary, for both the DAWM and the MAA the participants, items and scale were analysed using Rasch analysis. All participants and all items were retained for both assessments. DAWM assessment items difficulty did not match the participants abilities’. The items were more difficult than was well suited to the participants and the DAWM could have benefitted from more items but overall was an effective assessment instrument. The MAA difficulty level was appropriate for the students, if a bit easy, but could have benefitted from more items. No correlation was found between the February results for the DAWM and the MAA when comparing the individual students (r = .044). Difference scores of Logit Measures were used in traditional statistical analysis to test for a treatment effect. Two different approaches were used: each student as an experimental unit and each school as a block with Treatment and Control within each block as the experimental units. When analysing the data with each student as an experimental unit, assumptions for use of ANOVA were met to a satisfactory level of both assessments (DAWM ∝ =.05, MAA ∝ =.01). For the DAWM difference score data, a statistically significant interaction 108 was identified between group and gender for April – February data (p = .017). For the DAWM data, a statistically significant difference was identified between the treatment and control group for the May – February data (p = .040, Cohen’s d = 0.337, power = .79). When considering the MAA difference score data, no statistically significant interactions or differences were identified and the power for all data sets was low (.03 to .28). Cohen’s d values for April – February, May – April and May – February were all trivial. When analysing the data in blocks, the assumption of equal variance (or equal error in variance) was satisfied for the May – April and May – February data (p =.290 and p =.406, respectively) but not for the April – February data (p = .000). The lack of equal variance indicates that any significant findings for the April – February results must be interpreted conservatively. For the April – February data a significant effect was found for block (p = .025) and a significant interaction found identified for block and treatment (p = .001). A significant difference was identified for block 4 and 5 (p =.035). No statistically significant differences or interactions were identified for the May – April data. The May – February data indicated a statistically significant block effect (p = .004), as well as statistically significant interactions for treatment and block (p = .001) and treatment, gender and block (p =.016). 109 Chapter Five Discussion Assessing and improving attitudes toward mathematics is a common area of concern for educators, researchers and parents. Assessing where mathematics is recognized in the world or what students recognize as being related to mathematics may help us understand what impacts mathematics attitude. This study was designed to assess and improve both the student’s propensity to indicate words and/or concepts as being related to mathematics and attitudes toward mathematics, of late primary, early elementary school students. The assessment was comprised of two parts: the Difficulty Associating Words with Mathematics (DAWM) and the Math Attitude Assessment (MAA). DAWM assessment assessed children’s propensity to associate real world situations with mathematics. MAA was a modified version of the ATMI (Tapia & Marsh, 2004). The MAA assessed children’s’ attitudes toward mathematics using 12 statements focused on four areas, Self-Confidence, Motivation, Value and Enjoyment. The resulting three items per area was insufficient to examine the separate factors, so only one global attitude score was obtained for each participant. The assessments were done three times: pre-intervention, post-intervention, and one-month post intervention. The intervention, Math Play, is defined as play involving math related toys, games and activities with the purpose of exploration, discovery and problem solving. Math Play does not involve testing or criticisms. Analysis was done to determine if Math Play effects existed and whether or not there were differences across gender. 110 Conclusion Math Play was able to increase propensity of primary students to indicate words and/or concepts as being mathematics related from pre-intervention to one-month post intervention (when each participant was an experimental unit) suggesting further investigation into the effectiveness of Math Play is warranted. However, Math Play was not able to change attitudes toward mathematics. Is the Difficulty Associating Words with Mathematics assessment (DAWM) an appropriate assessment instrument for the participants? The difficulty associating words with mathematics (DAWM) assessment was found to be more difficult than would be ideal for grade 2 and 3 students for assessing the propensity to endorse words as being associated with mathematics (Figure 5). However, while this instrument had enough adequately preforming items to detect group differences it may not have been sensitive enough to separate high performing participants from low performing participants. The DAWM would benefit from more items and also had items that did not add to the assessment quality such as the word “adding”. The common mathematics related words (adding, subtracting and counting) were included in the assessment for face validity purposes. Due to the age of the participants, if it may have been confusing if there were not any words in the assessment that were commonly related to mathematics. Will Math Play change the propensity of participants endorsement of words associated with mathematics? A statistically significant interaction between treatment and gender from pre-intervention to post intervention was identified (p =.017) (Tables 9), 111 indicating that the male and female students responded differently to the intervention. The lack of a difference, immediately following intervention, was consistent for both methods of analysis If there is a difference in propensity, will it persist after Math Play has ended? When each participant was an experimental unit, no difference in propensity was found, pre-intervention to post-intervention. However, there was an increase in the long-term effect (pre-treatment to one-month post treatment) (Table 9). Math Play was found to have a small to medium effect Partial ETA = .032, and Cohen’s d = .337) on the grade 2 and 3 students’ propensity to indicate words and/or concepts as being related to mathematics. The power of the effect was .79, indicating a 79% probability of detecting an effect in the May – February data if there was one there to detect. ( If there was only a change immediately after intervention that did not persist that would indicate the need to persist with Math Play. Is there a gender difference for propensity to associate words with mathematics? No statistically significant differences were identified between the participants who self identify as male versus those who self identify as female (Table 9 and Figures 13, 14, and 15) for either experimental unit. The lack of a gender difference is consistent with some work done on mathematics affect and inconsistent with others. Altindag et al (2009), de Lourdes Meta et al. (2012) and Moenikia and Zahed-Babelan (2009) all found no gender difference, whereas, Watt (2004) did. 112 Is the math attitude assessment (MAA) an appropriate instrument to assess the attitudes of the participants toward mathematics? The MAA was not as good a measurement instrument as the DAWM. It matched the ability level of the participants, but it may not have been sensitive enough to separate students with positive attitudes from those with negative attitudes. The MAA was sensitive enough to detect group effect. The assessment also did not have enough items to be able to separate the statements based on factors of attitude that were being targeted. The MAA instrument would need improvement for future use but was judged adequate for assessing treatment efficacy and gender differences. Will Math Play change the attitude of the participants toward mathematics? Math Play did not have a measurable effect on the attitudes of grade 2 and 3 students toward mathematics. Though no difference was reported it was clear from observation that the participants were having a positive experience. Some participants asked when Math Play sessions would happen again and smiled and laughed when they were engaged in the activities. Some participants asked where to get the games that were introduced or if the same games could be brought back the next week. Changes were observed from the first Math Play sessions to the next. When the Math Play sessions first started some participants appeared reluctant to take part but by the end of the first day everyone in every class appeared to be taking part enthusiastically. Every child in every class participated. Many of the teachers commented 113 on the level of engagement and the positive feedback they were getting from the participants. Studies indicate that attitudes toward mathematics form early and are difficult to change (as cited in Geist, 2010). The observations may be an indication of state versus trait (Heffner, 2017). The session’s engagement level and responses may indicate that the participants were in a state of enjoyment when they were engaged in Math Play. Their trait may have been a more negative view on mathematics. With evidence suggests changing attitudes toward mathematics is difficult (as cited in Geist, 2010), it is reasonable to assume it would take more than five, one-hour sessions to affect attitude at meaningful level and at a level associated with changing a trait. Is there a difference in the change of MAA results between the genders? No gender differences were found for the MAA data. Delimitations, Limitations and Future Research Choices and circumstances caused limitations to this quasi-experimental study (Creswell, 2014; Hurlburt, 2006; Pagano, 1998). The volunteers who were facilitating the Math Play had limited time available. An increase in treatment time would be advisable in future research. Random assignment was not possible for the schools, teachers, students nor facilitators. The teachers were randomly assigned to the treatment or control group (with one exception). Increased randomization could increase the power of the results of the study (Creswell, 2014; Hurlburt, 2006; Pagano, 1998). 114 Having trained facilitators with a pre-measured attitude facilitate the interventions would be advisable in future research. This may control for attitude differences. The age of the participants and the suggestion by the school district limited the length of the MAA. As a result of the limited length, there were not enough items per category to allow meaningful analysis for the separate factors involved in the MAA: selfconfidence, value, enjoyment and motivation. Identifying the factors that contribute to the attitude of Primary students and then creating an assessment instrument with at least five items that address those factors would be advisable in future research. The teachers’ attitudes were not assessed, nor that of the parents or siblings of the participants. Some studies suggest teacher attitude and parent attitude can impact the students’ attitudes (Gunderson, 2012). Further studies examining the impact of teacher, parent, sibling and overall classroom attitudes toward mathematics would be advisable. Another recommendation for future investigation is expanding the sample to include more schools. This would allow more thorough investigation with schools as blocks and the classes in the schools as experimental units. This would be advisable because results at a classroom level, as opposed to the individual level, would better motive teachers to use Math Play as a resource for improvements in awareness of mathematics in the world and attitudes toward mathematics. 115 Implications Research implications. Further to the suggestions previously mentioned, research needs to be undertaken to determine what can be done to create a concise definition of attitudes toward mathematics. Various studies assess attitudes in various ways with various instruments (Chamberlin, 2010; Chapman & Lim, 2012; Hannula, 2002; Tapia & Marsh, 2004;). It is hard to be consistent in interpretation of and the study of something that is not consistently defined. It also may be necessary to develop different assessment instruments targeted to different age groups. How incidents affect attitudes toward mathematics needs to be determined. When discussing attitudes, Hannula (2002) referred to a particular incident contributing to the formation of a negative attitude toward mathematics. Are attitudes toward mathematics formed over long-term influences and then changed by an event? How static are attitudes toward mathematics? The propensity to associate words with mathematics can be influenced and increased in as little as five weeks with only an hour a week. Would more exposure to Math Play or Math Play like activities improve attitudes? Practical implications. One teacher whose class was in the study said that Math Play had changed the way she thought of mathematics. The results suggest support for her statement with the increase in propensity to indicate words as being mathematics related (p =.040). Several of the teachers purchased some of the games and activities that were brought in (after the study was complete) and invited us back. These were the responses of the people who work with the students everyday implying the differences 116 they say from Math Play were impactful and beneficial. An increase in the understanding that mathematics is all around us may have other benefits. Summary Math Play was not shown to have an effect on attitudes toward mathematics held by primary students. An attitude is a trait (Heffner, 2017) held by an individual, and hard to change (Geist, 2010). An increase in intervention duration together with a refinement of the Mathematics Attitude Assessment is recommended for future studies. 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Journal for Research in Mathematics Education. 3(1), 51-58. Linacre, M. (2012a). Winsteps Tutorial 1, Retrieved from http://www.winsteps.com/a/winsteps-tutorial-1.pdf Linacre, M. (2012b). Winsteps Tutorial 2, Retrieved from http://www.winsteps.com/a/winsteps-tutorial-2.pdf Linacre, M. (2012c). Winsteps Tutorial 3, Retrieved from http://www.winsteps.com/a/winsteps-tutorial-3.pdf Linacre, M. (2012d). Winsteps Tutorial 4, Retrieved from http://www.winsteps.com/a/winsteps-tutorial-4.pdf Linacre, M. (2012e). Retrieved from http://www.winsteps.com Linacre, M. (Retrieved on April 25, 2018). Institute for objective measurement. Institute for Objective Measurement. Retrieved from www.rasch.org Linacre, M. (2002). Judging Debacle in Pairs Figure Skating. Retrieved from https://www.rasch.org/rmt/rmt154a.htm Linacre, M. (2002b) Review of Reviews of Bond & Fox. Rasch Measurement, Transactions of the Rasch Measurement SIG American Educational Research Association. 16(2). Retrieved from https://www.rasch.org/rmt/rmt162.pdf Lochhead, L., (2009). Assessment of perceived functional capacity: Using Rasch Analysis to evaluate the measurement properties of four perceived pain & disability scales (master’s thesis). University of British Columbia. Ma, X., & Kishor, N. (1997) Assessing the relationship between attitude toward mathematics and achievement in mathematics: A meta-analysis. Journal for Research in Mathematics Education. 28(1). 26-47. 120 MacDonald, M. (2014). Professors debate the best way to teach math. University Affairs. Retrieved from http://www.universityaffairs.ca/features/feature-article/how-toteach-math/ Marsh, H.W., Trautwein, U., Ludtke, O., Koller, O., & Baumert, J. (2005). Academic selfconcept, interest, grades, and standardized test scores: reciprocal effects models of causal ordering. Child Development, 76(2). 397-416. Milner, S. (n.d.). Susan’s Math Games. Retrieved on April 25, 2018, Retrieved April 25, 2018 from http://susansmathgamesca.ipage.com/puzzles-games/ Moenikia, M., & Zahed-Babelan, A. (2010). A study of simple and multiple relations between mathematics attitude, academic motivation, and intelligence quotient with mathematics achievement. Procedia Social and Behavioral Sciences. 2. 1537-1542. Oswald, D., Sherratt, S., & Smith, S. (2014). Handling the Hawthorn effect: the challenges surrounding a participant observations, Review of Social Studies, 1(1). Pagano, R. R. (1998) Understanding statistics in the behavioral sciences (5th ed.). Pacific Grove: CA, Brooks/Cole Publishing Company. Plonsky, M. (1997), Analysis of variance – two way. Retrieved from https://www4.uwsp.edu/psych/stat/index.htm Quinn, B., & Jadav, A. D. (1987). Causal relationship between attitude and achievement for elementary grade mathematics and reading. The Journal of Educational Research. 80(6), 366-372. Rasch, G. (1993). Probabilistic models for some intelligence and attainment tests. MESA Press Robinson, K. (2006). Do schools kill creativity? [Video file]. Retrieved from https://www.ted.com/talks/ken_robinson_says_schools_kill_creativity Sebok, S., (2011). “Pick Me, Pick Me, I Want to Be a Counsellor”: Assessment of a MedCounselling application selection process using Rasch analysis and generalizability theory (master’s thesis). University of Victoria Sick, J. (2009). Rasch analysis software programs. Shiken: JALT Testing & Evaluation SIG Newsletter. 13, 13-16. Smith, R. M., Schumacker, R. E., & Bush. MJ. (1998). Using item mean squares to evaluate fit to the Rasch model. Journal of Outcome Measurement, 2, 66-78. 121 Smith, E., & Smith, R. (2004). Introduction to rasch measurement. Maple Grove, Minnesota: JAM Press Statistical sampling and regression: t-distribution, Retrieved from http://ci.columbia.edu/ci/premba_test/c0331/s7/s7_4.html Stodolsky, S., Salk, S., & Glaessner, B. (1991). Students views about learning math and social studies. American Educational Research Journal, 28, 89-116. Tabachnic, Fidell. (2007). Using multivariate statistics (6th ed.). Pearson. Tapia, M., & Marsh II, G.E. (2004). An instrument to measure mathematics attitudes. Academic Exchange Quarterly, 8. Tezer, M., & Karasel, N. (2010). Attitudes of primary school 2nd and 3rd grade students towards mathematics course. Procedia Social and Behavioral Sciences, 2(2), 58055812. van den Berg, R. G., (2017). SPSS Repeated Measures ANOVA Tutorial. Retrieved from https://www.spss-tutorials.com/spss-repeated-measures-anova/#repeatedmeasures-anova-assumptions Wagenmakers, E., Krypotos, A., Criss, A. H., and Iverson, G (2012) On the interpretation of removable interactions: A survey of the field 33 years after Loftus, Memory and Cognition, 40(20), 145-160 Watt, H. (2004). Development of adolescents’ self-perceptions, values, and task perceptions across gender and domain in 7th- through 11th-grade Australian students. Society for Research in Child Development, 75(5). 1556-1574. Wells, C.S., & Wollack, J.A. (2003). https://testing.wisc.edu/Reliability.pdf 122 Appendix Appendix A Forms and Consent • UNBC REB approval • Consent forms Principals Teachers, Participants • Confidentiality form 123 124 125 Oct. 19, 2014 3333 University Way Prince George BC V2N 4Z9 Dear Principal, My name is Jean Bowen. I am working on a Master’s in Mathematics at UNBC on math education. I am seeking permission to conduct research in your school during the 20142015 school year. The focus of the study is attitudes towards mathematics in the primary grades. This project has been reviewed by Cindy Heitman, District Principal of Curriculum and Instruction, my supervising committee and the University of Northern British Columbia’s Research Ethics Board (REB). Research Objective: 1) To assess attitudes toward mathematics of Grade 2 and/or 3 students. 2) To attempt to improve attitudes through the introduction of Math Play. For the purpose of this study Math Play is defined as play involving math related toys, games and activities with the purpose of exploration, discovery and problem solving. Math Play does not involve testing or criticisms. 126 Proposed Plan: I would like to enlist the help of eight (or 12) classes of Grade 2 and/or 3 students. In four (or six) of these classes attitude assessment will occur. In the other four classes’ attitude assessment as well as Math Play sessions will take place. Parents will receive information letters, be invited to information sessions and be invited to email me with any questions. In classes where Math Play takes place the entire class will take place in the play sessions. Consent forms will be required to be signed by parents before their child(ren) may take part in the attitude assessment portion of the study. In classes where Math Play takes place, attitudes would be assessed four times throughout the year: October (or later due to the strike), end of January, beginning of April and May/June. In the other four classes assessment will happen in October (or later) and June. Attitude assessments will be administered by the classroom teacher and would take approximately 15 minutes each time. A volunteer student from the UNBC MATH 190 (Mathematics of Elementary Teachers) would attend one hour a week for the first two weeks as in class support. After Spring Break the intervention would begin. The Intervention: For an hour a week from March 2, to April 2 the MATH 190 student would return to lead sessions in Math Play. Math Play is defined as play involving math related toys, games and 127 activities with the purpose of exploration, discovery and problem solving. Math Play does not involve testing or criticisms. Ethical Considerations For the Participants: Participation is voluntary for the attitude assessment portion. As such you or any participant may withdraw from the study at anytime. Any participant who withdraws will have all information that has been collected destroyed unless it has already been reported. Due to the two fold nature of the study (attitude assessment and Math Play) participation in the Math Play sessions without attitude assessment information being collected is easy to facilitate. This is done to make the process easier for the classroom teacher should a parent decide that they do not want attitude information collected on their child(ren). The entire class will participate in the Math Play session. You or the classroom teacher may stop participation in the study at any time. A child may also choose to stop participation in the attitude assessment portion of the study at any time. Privacy and Confidentiality: There will be no identifiers for the school, teachers, or students in the report. Only grouped data will be reported. Students will be assigned number identifiers. The only copy of the document that connects the student and their numbers will be kept under lock and key at UNBC. My supervisor and I will be the only people with access to the 128 information. Information will be retained for one year after the final paper has been approved and then will be shredded. Potential Benefits: • • • • Learning through play is supported in literature. Several of the activities will involve logic puzzles. Development of logic skills is academically beneficial. This study may improve our understanding of primary students’ attitudes toward mathematics and some of the factors that influence such attitudes. This study may give us a new way to improve primary students’ attitudes towards mathematics. Potential Risks: • Student participants will receive five hours less instruction time with their classroom teacher. This is not a significant amount of time in comparison to the total instruction time that takes place in a year. Student participants will not miss out on instruction time as the entire class will be participating. Sharing Results: • The final report will be provided to the principals, the teachers, School District 57 and the parents of the children in the class. Thank you for considering participating in this research project. I would appreciate the opportunity to work with you and your staff. If you or any of your staff have any questions please feel free to contact me or my supervisor, Dr. Jennifer Hyndman. I contacted via email at bowenj@unbc.ca. Dr. Jennifer Hyndman can be contacted at Jennifer.hyndman@unbc.ca. Concerns or complaints should be directed to the REB (phone: 250-960-6735: email reb@unbc.ca). 129 Sincerely, Jean Bowen I, _______________________________________ principal of ______________________________ give permission to Jean Bowen, Masters student at University of Northern British Columbia, to include the above mentioned school in the described research project. ________________________________ ___________________ Signature Date 130 Teacher Information My name is Jean Bowen. I am working on a Master’s in Mathematics at UNBC on math education. I am seeking permission to conduct research in your school during the 20142015 school year. The focus of the study is attitudes towards mathematics in the primary grades. This project has been reviewed by Cindy Heitman, District Principal of Curriculum and Instruction, my supervising committee and the University of Northern British Columbia’s Research Ethics Board (REB). Research Objective: 1) To assess attitudes toward mathematics of Grade 2 and/or 3 students. 2) To attempt to improve attitudes through the introduction of Math Play. For the purpose of this study Math Play is defined as play involving math related toys, games and activities with the purpose of exploration, discovery and problem solving. Math Play does not involve testing or criticisms. Proposed Plan: Your class would be one of 8(or 12)classes of Grade 2 and/or 3 students that will be part of the study. The study consists of two parts. Part one is assessing attitudes of students toward mathematics. Part two is introducing Math Play. Only half of the classes will take part in the Math Play. 131 Part One Attitudes would be assessed four times throughout the year: October, end of January, beginning of April and May/June. This is a questionnaire comprised two sections: section 1 is circling words that the students associate with mathematics, and section two is 12 statements where the student circles yes, maybe or no. As the classroom teacher you would be asked to administer the questionnaire to avoid the added disturbance of having extra people in the classroom. The assessment would need to be read to the students. The assessment would take approximately 15 minutes to administer each time. Parents will receive information letters, be invited to information sessions and be invited to email me with any questions. Consent forms will be required to be signed by parents before their child(ren) may take part in the study. You would be asked to hand out the permission forms and collect them to be held until I collect them. (This paragraph was moved.) Part Two A volunteer student from the UNBC MATH 190 (Mathematics of Elementary Teachers) would attend one hour a week for the first two weeks as in class support from February 2, 2015 to February 13, 2015. After Spring Break the intervention would begin. The Intervention: 132 For an hour a week from March 2, to April 2 the MATH 190 student would return to lead sessions in Math Play. Four of the classes will participate in both Part 1 and Part 2. The other four classes will participate in only Part 1- attitude assessments. I would like to invite your class to participate in both Part 1 and 2 of the study. Ethical Considerations For the Participants: Participation is voluntary. As such you or any participant may withdraw from the study at anytime. Any participant who withdraws will have all information that has been collected destroyed unless it has already been reported. Due to the two fold nature of the study (attitude assessment and Math Play) participation in the Math Play sessions without attitude assessment information being collected is easy to facilitate. This is done to make the process easier for the classroom teacher should a parent decide that they do not want attitude information collected on their child(ren). Privacy and Confidentiality: There will be no identifiers for the school, teachers, or students in the report. Students will be assigned number identifiers. The only copy of the document that connects the student and their numbers will be kept under lock and key at UNBC. My supervisor and I will be the only people with access to the information. Information collected will be kept for a year after the final paper has been completed, it will then be shredded 133 Potential Benefits: • • • • Learning through play is supported by several studies. Several of the activities will involve logic puzzles. Development of logic skills is academically beneficial. This study may improve our understanding of primary students’ attitudes toward mathematics and some of the factors that influence such attitudes. This study may give us a new way to improve primary students’ attitudes towards mathematics. Potential Risks: • Student participants will receive five hours less instruction time with their classroom teacher. This is not a significant amount of time in comparison to the total instruction time that takes place in a year. Student participants will not miss out on instruction time as the entire class will be participating (except the students who do not have permission to do so). Sharing Results: • The final report will be provided to the principals, the teachers and the school district. Thank you for considering participating in this research project. I would appreciate the opportunity to work with you and your staff. If you or any of your staff have any questions please feel free to contact me or my supervisor, Dr. Jennifer Hyndman. I contacted via email at bowenj@unbc.ca. Dr. Jennifer Hyndman can be contacted at Jennifer.hyndman@unbc.ca. Concerns or complaints should be directed to the REB (phone: 250-960-6735, email: reb2unbc.ca). 134 Sincerely, Jean Bowen If you agree to have your class be part of this study please fill out and return the portion below to me. I, _______________________________________ of ______________________________ give permission to Jean Bowen, Master’s Student at University of Northern British Columbia, to include my class in the described research project. This class will take part in both attitude assessment (Part One) and Math Play (Part Two). ________________________________ ___________________ Signature Date 135 Dear Parents, My name is Jean Bowen. I am working on a Master’s in Mathematics at UNBC on math education. I am seeking permission to conduct research in your school during the 20142015 school year. The focus of the study is attitudes towards mathematics in the primary grades. Students from your child’s class will be participating in this study. Research Objective: 1) To assess attitudes toward mathematics of Grade 2 and/or 3 students. 2) To attempt to improve attitudes through the introduction of Math Play. For the purpose of this study Math Play is defined as play involving math related toys, games and activities with the purpose of exploration, discovery and problem solving. Math Play does not involve testing or criticisms. The Plan: The study consists of two parts. Part one is assessing attitudes of students toward mathematics. Part two is introducing Math Play. Part One Attitudes would be assessed four times throughout the year: October, end of January, beginning of April and May/June. This is a questionnaire comprised two sections: section 1 is circling words that the students associate with mathematics, and section two is 12 statements where the student circles yes, maybe or no. As the classroom teacher you 136 would be asked to administer the questionnaire to avoid the added disturbance of having extra people in the classroom. The assessment would need to be read to the students. The assessment would take approximately 15 minutes to administer each time. Part Two A volunteer student from the UNBC MATH 190 (Mathematics of Elementary Teachers) will attend one hour a week from February 2, 2015 to February 13, 2015 as classroom support. For an hour a week from March 2, to April 2 the MATH 190 student will return to lead sessions in Math Play. Your child’s class has been selected to participate in Part 1 of the study. 137 Ethical Considerations This study has been reviewed by your child’s principal, your child’s teacher, the school district, my supervisory committee and the University of Northern British Columbia Research Ethics Board (REB). Concerns or complaints should be taken to the UNBC REB (phone: 250-960-6735, email: reb2unbc.ca). For the Participants: Participation in the attitude assessment is voluntary. As such, participation may withdrawn from the study at anytime. Any participant who withdraws will have all information that has been collected destroyed unless it has already been reported. Privacy and Confidentiality: There will be no identifiers for the school, teachers, or students in the report. Students will be assigned number identifiers. The only copy of the document that connects the student and their numbers will be kept under lock and key at UNBC. My supervisor and I will be the only people with access to the information. Information collected will be kept for one year after the final paper has been approved and then will be shredded. Potential Benefits: • • Learning through play is supported not only in literature but in SD57. Several of the activities will involve logic puzzles. Development of logic skills is academically beneficial. 138 • • This study may improve our understanding of primary students’ attitudes toward mathematics and some of the factors that influence such attitudes. This study may give us a new way to improve primary students’ attitudes towards mathematics. Potential Risks: • • [This paragraph will be omitted as it does not apply to the classes where Math Play is not taking place. Student participants will receive five hours less instruction time with their classroom teacher. This is not a significant amount of time in comparison to the total instruction time that takes place in a year. Student participants will not miss out on instruction time as the entire class will be participating (except the students who do not have permission to do so).] There are no notable risk for the students who will be taking part in the assessments only. Sharing Results: • The final report will be provided to the principals, the teachers and the school district. Thank you for considering participating in this research project. I would appreciate the opportunity to work with you and your staff. If you or any of your staff have any questions please feel free to contact me or my supervisor, Dr. Jennifer Hyndman. I contacted via email at bowenj@unbc.ca. Dr. Jennifer Hyndman can be contacted at Jennifer.hyndman@unbc.ca. Sincerely, Jean Bowen 139 Dear Parents, My name is Jean Bowen. I am working on a Master’s in Mathematics at UNBC on math education. I am seeking permission to conduct research in your school during the 20142015 school year. The focus of the study is attitudes towards mathematics in the primary grades. Your child’s class will be participating in this study. Research Objective: 1) To assess attitudes toward mathematics of Grade 2 and/or 3 students. 2) To attempt to improve attitudes through the introduction of Math Play. For the purpose of this study Math Play is defined as play involving math related toys, games and activities with the purpose of exploration, discovery and problem solving. Math Play does not involve testing or criticisms. The Plan: The study consists of two parts. Part one is assessing attitudes of students toward mathematics. Part two is introducing Math Play. Part One Attitudes would be assessed three times throughout the year: February, beginning of April and May/June. This is a questionnaire comprised two sections: section 1 is circling words that the students associate with mathematics, and section two is 12 statements 140 where the student circles yes, maybe or no. The assessment would take approximately 15 minutes to administer each time. Part Two Volunteer students from the UNBC MATH 190 (Mathematics of Elementary Teachers) will attend one hour a week from February 2, 2015 to February 13, 2015 as classroom support. For an hour a week from March 2, to April 2 the MATH 190 student will return to lead sessions in Math Play. The entire class will participate in the Math Play sessions. No information will be collected from your child(ren) during these sessions. I may be present as an observer during these times. Your child’s class will be participating in both parts of the study. To preserve the integrity of the study please do not discuss the connection between the two parts of the – the attitude assessment and the Math Play- with your child until after the study is complete. Ethical Considerations This study has been reviewed by your child’s principal, your child’s teacher, the school district, my supervisory committee and the University of Northern British Columbia Research Ethics Board (REB). Concerns or complaints should be taken to the UNBC REB (phone: 250-960-6735, email: reb2unbc.ca). 141 For the Participants: Participation is voluntary. As such participation may withdrawn from the study at anytime. Any participant who withdraws will have all information that has been collected destroyed unless it has already been reported. Privacy and Confidentiality: There will be no identifiers for the school, teachers, or students in the report. Students will be assigned number identifiers. The only copy of the document that connects the student and their numbers will be kept under lock and key at UNBC. My supervisor and I will be the only people with access to the information. Information collected will be kept for one year after the final paper has been approved and then will be shredded. Potential Benefits: • • • • Learning through play is supported not only in literature but in SD57. Several of the activities will involve logic puzzles. Development of logic skills is academically beneficial. This study may improve our understanding of primary students’ attitudes toward mathematics and some of the factors that influence such attitudes. This study may give us a new way to improve primary students’ attitudes towards mathematics. Potential Risks: • Student participants will receive five hours less instruction time with their classroom teacher. This is not a significant amount of time in comparison to the total instruction time that takes place in a year. Student participants will not miss out on instruction time as the entire class will be participating (except the students who do not have permission to do so). 142 Sharing Results: • The final report will be provided to the principals, the teachers and the school district. Thank you for considering participating in this research project. I would appreciate the opportunity to work with you and your staff. If you or any of your staff have any questions please feel free to contact me or my supervisor, Dr. Jennifer Hyndman. I contacted via email at bowenj@unbc.ca. Dr. Jennifer Hyndman can be contacted at Jennifer.hyndman@unbc.ca. Sincerely, Jean Bowen 143 Child Information To be shared with the students. Researcher: Jean Bowen, Masters Student in Mathematics, UNBC Your class has been selected to participate in an attitude assessment portion of a study. What the study is about: Grade 2 and/or 3 students in Prince George will be participating in a research study. Please check each box below to indicate that you and your child have read and understood each statement. If you both agree that the student may participate in the study, please complete and sign the form. Privacy:  The identity of your child will be kept private.  No wherein the final report will specific schools, teachers or students be identified.  All information with ways to identify specific children and schools will be kept locked at UNBC. Participation:  Participation in the study is voluntary.  If you give permission now, but wish to withdraw your child from the study at any time you may do so. Any information already collected will be destroyed and withdrawn from the study as long as the request is before the report is written.  The principal or teacher can withdraw the school or class from the study. Questions: 144 If you have any questions or concerns please contact me, Jean Bowen, at bowenj@unbc.ca, or my supervisor, Dr. Jennifer Hyndman at Jennifer.hyndman@unbc.ca. To give permission for your child to participate in the study please fill in the second page of this form and have your child return it to their teacher by ___________________________. 145 Attitudes Toward Mathematics and Math Play Please fill in this form and have your child return it in the sealed envelope to their classroom teacher. Parent/Guardian I, __________________________________________________ give permission for my child, (print full name) _______________________________________________ to participate the attitude assessment. (print full name) ____________________________________ __________________________ ___________________ Signature of Parent/Guardian Relationship to child Date Student I, __________________________________________________,wish to participate the attitude 146 (print full name) assessment. ____________________________________ Signature of Student __________________________ Date 147 The Effects of Math Play on the Attitudes of Primary Students Towards Mathematics Oath of Confidentiality The project named above is being undertaken by Jean Bowen at the University of Northern British Columbia. The objectives of this work are to: 1. Assess attitudes of Grade 2/3 students. 2. Introduce Math Play. For the purpose of this paper, Math Play is defined as play involving math related toys, games and activities with the purpose of exploration, discovery and problem solving. Math Play does not involve testing or criticisms. The results of this project may be used to increase our understanding of what influences the attitudes of primary students toward mathematics. As an assistant working on this project I, (print name) agree to: 1. Treat as confidential all information learned through my interactions with the students participating in this project. I will accomplish this by not discussing or sharing confidential information learned in the context of this project in any form or format with anyone other than Jean Bowen, the classroom teacher, and the project supervisor, Dr. Jennifer Hyndman. 2. I will not be handling an personal information from the primary students. I further understand and agree that this oath of confidentiality will continue in force indefinitely, even after I cease being an assistant on this project. Name of Assistant: (print): _________________________ (sign): __________________________ Date: ____________________ Researcher’s Name: : (print): _______________________ (sign): ________________________ Date: ____________________ If you have any questions or concerns about this project, please contact: 148 Jean Bowen, bowenj@unbc.ca This study has been reviewed by the Research Ethics Board at the University of Northern British Columbia. For questions regarding participants rights and ethical conduct of research, contact the Office of Research and Graduate Programs at (250) 960-6735. 149 Appendix B Attitude Assessment Instruments • DAWM • MAA • ATMI 150 151 152 153 154 155 156 157 158 159 Appendix C Games and Instructions • • • • • • • • • • • • • • • Towers Tri-Hex Tri-hexaflexagon Tantrix Rectangles Blink Magic Numbers Magic Squares Neighbours Qwirkle Set Suspend Telaga Buruk Q-Bitz Hidato 160 Towers Before you start you will need. • • A floor version of towers. There are 16 towers: 4 of each height, 1 unit, 2 units, 3 units and 4 units tall. brains The goal of this game is to place a tower of each height in each row and column based on the clues. REMEMBER ENJOYMENT AND SUCCESS ARE KEY! Help them achieve it. How to play? • • • Place a tower of each height in each column and row based on the clues. The numbers on the outside of the grid tell you how many towers you can see in front of you from that spot. If there is a number given in the grid, a tower of that height is placed in that spot. What to do to teach the game. • • • Place the towers in a stair like fashion. 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 Have the children come to the side by the 1’s and imagine they are something small right in front of one of the rows of numbers, looking toward the 4 tower. Ask them how many towers they can see? 4 Have the children move to facing the 4 towers. 1 161 • • • • • • • Show them the board and tell them that the numbers on the outside tell you how many towers you can see from that spot. Tell them that each row and each column must have one of each height. Assign each child a tower. That is the only tower that student may move. Have them work together to place the towers. Do this for at least two complete games. When a group of students is ready to tackle the game on their own, give them a bag with the smaller towers to work with and they can try it on their own. If a student is able to they can try it without the towers and just do it on paper. What math is hidden? • • Problem solving Trying to switch to doing a three dimensional problem with only paper and pencil. Ways to modify the game. • • • If Ways to correct without it feeling like criticism. Avoid statements like, “That is wrong.” Ask: o “Is that tower going to work there?” o “Do you see a tower that is missing from that row/column?” 162 163 Tri-Hex Before you start you will need. • • A tri-hex sheet Coloured disc The goal of this game varies. See sheet. REMEMBER ENJOYMENT AND SUCCESS ARE KEY! Help them achieve it. How to play? • • • • There are three versions. The instructions are on the sheet. Separate the children into pairs. Give each pair a sheet and the discs. What to do to teach the game. • Have the children gather with you and teach them the first two versions. What math is hidden? • Problem solving Ways to modify the game. • • • Should not need to be modified. Can just take turns placing discs. Can move the discs around following a path. Ways to correct without it feeling like criticism. • • Avoid statements like, “That is wrong.” Ask: o “Where can you place that disc?” o “How can you block the other player?” 164 165 Tri-hexaflexagon Before you start you will need crayons, felts or pencil crayons. The goal of this activity is to see the magic. REMEMBER ENJOYMENT AND SUCCESS ARE KEY! How to play and teach the game. • • • • • • Show the children the template. Ask how many sides does a piece of paper have? 2 Show them a folded trihexaflexagon. Ask them how many sides it has? 2 Give them each a trihexaflexagon and get them to go colour both sides. It is best if each side is different: different pattern or different colours. When they are done show them how to fold them. o “See how one side is pink and one side is green. If I fold it up like this, you can only see pink, if I unfold it what should I see?” o Now show them how to fold their own. Get them to repeat the folds until they get their original colours back. How many folds does it take? Can you fold it the other way? o How many sides does it really have What math is hidden? • • • • • Hexagon – a 6 edges shape made of straight lines This hexagon has all its edges the same length. Would it work it the edges were different lengths? NO Equilateral triangles – triangles with all edges equal in length 6 triangles in the hexagon Ways to modify this activity. • • You may have to do a lot of help with getting the folds to work at first. Just colouring it and seeing the triangles. Ways to correct without it feeling like criticism. 166 • • • You should not need to do much here. Avoid statements like, “That is wrong.” Ask leading questions. o “Can we try folding it a different way?” o “I like the colour you are using but could we use a different one for the other side?” – if a child really wants both sides to be the same colour that is fine too. 167 168 Tantrix Before you start you will need. • Trantrix sets. Half the class split into four groups. The goal of this game is to complete as many loops as you can. REMEMBER ENJOYMENT AND SUCCESS ARE KEY! Help them achieve it. How to play? • • • • • Place all the tiles face down, so you can see the numbers on the back. Take the first three tiles. Notice how the highest number you have is yellow. This means when you flip those three tiles over you will make a yellow loop. When you have done that with three tiles add the fourth. The colour of the loop you are trying to make is the colour of the highest number you are using to make the loop. On the tenth tile you can make a yellow loop, a red loop and a blue loop but not all at the same time. Any lines that touch that are not part of the loop must also be the same colour. What to do to teach the game. • • • • • • • Get the kids to sit with you on the floor. Put one set of tantrix out and show them the numbers. Pick the first three tiles.]show them the highest number being yellow. Flip the tiles and make a yellow loop. Explain the rest of the rules. If need be make the loop with the fourth tile. Have them make suggestions on where to place the tiles. Split them into groups. (It usually will work better if you split them that if you let them do it themselves.) What math is hidden? • • Problem solving Spatial awareness 169 Ways to modify the game. • If making the other pathways one colour each is causing problems they can focus on just the main colour. Ways to correct without it feeling like criticism. • • Avoid statements like, “That is wrong.” Ask: o “Is the whole path the same colour?” o “Is the loop complete?” o “Maybe that piece would work better in a different place?” 170 Rectangles Before you start you will need. • Pencil and eraser The goal of this game is to complete as many rectangle grids as you can. REMEMBER ENJOYMENT AND SUCCESS ARE KEY! Help them achieve it. How to play? • Place each number in a rectangle that is made up of as many individual squares as the number indicates. o Remember that a square is a rectangle. What to do to teach the game. • • • • • • Put up a rectangle example grid. Explain the rules Walk them through as few examples Ask them where to place the rectangles As with hidato, let them make mistakes so they can see how to identify them and fix them As you are explaining the game point out the math (it got too busy the first week to discuss the math with them while they were playing.) What math is hidden? • • • • Squares are rectangles Prime numbers have to be done in a line (2, 3, 5, 7) composite numbers can be made many ways (12 can be 1x12, 2x6, 3x4) Multiplication logic Ways to modify the game. 171 • • If a student needs extra help walk them through it or have them work with someone who is getting it more easily If need be they can just draw rectangles that work for most of the numbers that is great too. Ways to correct without it feeling like criticism. • • Avoid statements like, “That is wrong.” Ask leading questions. o “That rectangle seems to be in the way? Can we move one of those rectangles?” o “Starting with the larger numbers may help?” 172 173 Blink Before you start you will need. • Deck of Blink cards. The goal of this game is get rid of all your cards as fast as you can. REMEMBER ENJOYMENT AND SUCCESS ARE KEY! Help them achieve it. How to play? • • • • • • • Split the deck(s) evenly between all the people. 6 per group works well. Lay three cards in the center. Each person puts three cards in their hand. As they play a card they replace it with a card from their stack. A card can be played if it matches colour, number or shape. When you say go they do this as fast as they can. Make sure you lose. Help anyone that is falling too far behind. What to do to teach the game. • • • • • Get the kids to sit with you on the floor. Show them the different colours, shapes and numbers. Explain the rules. Have everyone place a card or two from their hand to get the hang of it first if needed. Smile as they laugh and play. What math is hidden? • • • Matching Patterns Numbers 174 Ways to modify the game. • • • I have played this with very young kids and it has worked. If the group is having a hard time take turns instead of going super fast. They can help each other place cards. Ways to correct without it feeling like criticism. • • Avoid statements like, “That is wrong.” At most they will play a wrong card. Make sure they understand the rules and if need be slow the game down by turn taking. Instructions from Boardgames.about.com 175 Magic Numbers Before you start you will need. • • Magic numbers grid sheets Pencil and scrap paper may be needed by some The goal of this game is to learn a magic trick and practice adding. (shhh don’t focus on the addition practice part!!) REMEMBER ENJOYMENT AND SUCCESS ARE KEY! Help them achieve it. How to play? • • • • Have the other person pick any number that appears in the magic numbers rectangles. They are NOT to tell you the number. They place a disc beside each rectangle that has the number in it. You tell them their magic number. To find the number you add the top left hand number in each rectangle that they placed a disc beside. What to do to teach the game. • • • • • • • • • You will pick one student (or two) and find their magic number. Then you will teach them how the trick works. The student(s) will them go do the trick with a class mate and teach it to them. Repeat. You can go around and teach it to more students or supervise the kids as they do it. This will need to be group dependant. If you are in a grade 2 class you may need to restrict the students to picking numbers less than 20. If a rectangle with the number in it is missed then the trick will not work. If addition is not perfect then the trick won’t work. They may need help with the addition. You can pair students up. 176 What math is hidden? • • Magic Adding Ways to modify the game. • • • Restrict the numbers to lower numbers, not all 60. Help them with the addition. Have them work together to find the sums. Ways to correct without it feeling like criticism. • • Avoid statements like, “That is wrong.” Ask: o “Are you sure you placed a disc by every rectangle that has your number in it?” o “You may want to double check your addition?” 177 178 Magic Squares Before you start you will need. • • • 4 of each coloured disks Game board Partner The goal of this game is to place one piece of each colour in each row, each column and each main diagonal. REMEMBER ENJOYMENT AND SUCCESS ARE KEY! Help them achieve it. How to play? • • Place one disk of each colour in each row and each column. Do the 3X3 then move on to the 4X4. What to do to teach the game. • • • Show them the board, use the 3X3 one. Remind them that a row is horizontal and a column is vertical. Place a different colour in each of the top spots and let them do the rest on their own at their places with a partner. What math is hidden? • • Problem solving Strategies Ways to modify the game. • • Each row can be the same colour. Each column can be the same colour. Ways to correct without it feeling like criticism. 179 • • Avoid statements like, “That is wrong.” Ask: o “Do you see that colour in that row already?” o “Would that colour be better in a different spot?” 180 181 Neighbours Before you start you will need. • • Each child needs a neighbours sheet, pencil and ereaser. You need the large white neighbours paper and white board markers. The goal of this game is to place the numbers 1-4 (1-5) in the grid following the arrows to indicate which are neighbours. REMEMBER ENJOYMENT AND SUCCESS ARE KEY! Help them achieve it. How to play? • • • Any squares that have arrows between them are numbers that are neighbours. Any squares without arrows between them may not be neighbouring numbers. 1 ↔ 2 ↔ 3 ↔ 4 so 1 is neighbours with only 2, 2 is neighbours with 1 and 3, ect. Only one of each number 1, 2, 3, 4, (5) may appear in each row and column. What to do to teach the game. • • • • • • • • Get the kids to sit with you on the floor. Draw 1 ↔ 2 ↔ 3 ↔ 4 and talk through who is neighbours with whom. Explain what the goal of the game is. Have them help you with the first one. As always let them make mistakes and help them identify where the mistake is and help walk them through fixing it but let them try to fix it on their own first. Ask leading questions. If some of them clearly get how to do it, they may go try on their own (and with others sitting near them). One of you can circulate around the kids working on their own while the other continues with the group. If there are some who need more help keep them with the other volunteer and you can continue to work through more examples with them. As they understand they can move to working on their own. Some may need to stay with one of you the whole time this game is played. If so you may wish to trade roles part way through. Keep the 1 ↔ 2 ↔ 3 ↔ 4 with a note that 1 and 4 are not neighbours somewhere where everyone will be able to see it. 182 What math is hidden? • • Problem solving. 4X4 =16, 5X5=25 Ways to modify the game. • Some may only be able to place neighbouring numbers in the squares without worrying about one of each number in each row and each column. Ways to correct without it feeling like criticism. • • Avoid statements like, “That is wrong.” Ask: o “Is that a neighbour?” o “Does that number already appear in that row/column?” o “Did you see the hint they gave us by telling us that that square has to be a ____________?” o “What number that is a neighbour is missing from this row/column?” 183 184 Qwirkle Before you start you will need. • A smile The goal of this game is to use up as many tiles as they can as a group. REMEMBER ENJOYMENT AND SUCCESS ARE KEY! Help them achieve it. How to play? • • • • • • • • • Have a group of six (or so half of your group for each bag) work from each bag. Each student takes 7 tiles. You put out a tile. Pick a person to go first and then work around the circle. They take turns adding tiles to the collection. For each tile they place they take another from the bag. This continue until time is up or the bag is empty. A row is built from tiles of the same colour or the same shape. There are 6 colours and 6 shapes. On your turn you can add as many tiles to a single row as you can as long as they follow the pattern. After a row has either all the colours or all the shapes it is full and can not be built on any more. You can build off of other rows as long as any tile you touch you follow the rule of that row. What to do to teach the game. • • • Have the students sit around you. Take some tiles out of the bag. Tell them the basic rules and them have them help you place several tiles until most, if not all, of the students understand the rules. 185 • When everyone understands separate them into two groups and get them set up. Do not separate them until they have all the instructions they need because once you separate then it will be harder to get them to listen. What math is hidden? • • • Grids Patterns matching Ways to modify the game. • • If a student is not able to play the game the way it is designed they can be given a collection of tiles to place on their own to make their own game. If matching the tiles in all directions is too much just make them match in one direction. Ways to correct without it feeling like criticism. • • Avoid statements like, “That is wrong.” Ask leading questions. o “Is that the same shape or colour?” o “Do you see one of the shape already in that row?” o “Can you see a different place that may be better for that tile?” 186 Set Before you start you will need. • A Set deck of cards. The goal of this game is find out how to always win. REMEMBER ENJOYMENT AND SUCCESS ARE KEY! Help them achieve it. How to play? • • • • • • • • Do not tell the children the rules. Tell them that a set is a group of three cards that follows a set of rules. Start taking sets. Get them to try to find sets. If a set they try is not a Set tell them the card that is an issue and see if they can tell you what card would make a set. The group can help with this. A set is a collection of three cards that is either the same or different across each trait. There is colour (red, green or purple), number (1, 2 or 3), shading (empty, striped or solid) and shape (diamond, squiggle or oval). The video may really help for this one. What to do to teach the game. • As above. What math is hidden? • • • • Problem solving Strategies Patterns Research Ways to modify the game. 187 • Play it as a group. Ways to correct without it feeling like criticism. • • Avoid statements like, “That is wrong.” Ask: o “What would finish this set?” o “Use the words Same and Different. Can we always say them?” 188 Suspend Before you start you will need. • Half the class broken into 3 groups The goal of this game is to place all of the bars. REMEMBER ENJOYMENT AND SUCCESS ARE KEY! Help them achieve it. How to play? • • • • • Place all the bars with coloured ends in a pile. Place one orange ended bar on the suspend structure. Take turns placing a bar on the structure. If a piece falls add it to the pile people can take from. Try to place all the bars. What to do to teach the game. • Tell them the rules. What math is hidden? • • • Problem solving Strategies Balancing weights like you balance an equation. Ways to modify the game. • You should not need to but if they modify it that is fine. Ways to correct without it feeling like criticism. • • Avoid statements like, “That is wrong.” Could say: 189 o “If it is too heavy on one side what can we do to balance it out?” o “Is that piece going to be heavy enough/too heavy?” 190 Telaga Buruk Before you start you will need. • • Game sheet 2 play pieces The goal of this game is to block the other player. REMEMBER ENJOYMENT AND SUCCESS ARE KEY! Help them achieve it. How to play? • • • Place your pieces as show. Take turns moving to the open space but you must stay on the lines. You may not jump over another player or share a space. What to do to teach the game. • • • • Show them the board set up. Show them a couple example moves. Put them into pairs. Have them play a few times. Switch partners and try again. What math is hidden? • • Problem solving Strategies 191 Ways to modify the game. • There is not very much you can do to modify this game. It should not be needed. Ways to correct without it feeling like criticism. • • Avoid statements like, “That is wrong.” You may only need to remind what the instructions are for this game. Yes there is an end to the game. 192 193 Q-Bitz Before you start you will need. • A smile The goal of this game is to have the class complete as many of the pattern cards as possible. REMEMBER ENJOYMENT AND SUCCESS ARE KEY! Help them achieve it. How to play? • • • • With about half the class (up to 16 children) have them work in pairs. Each pair gets one set of cubes and a base. Each pair gets a pattern card. Once the pattern card is complete they bring it to you, you check the pattern and if it is correct the pair gets a new card. What to do to teach the game. • • • • Have to group join you on the floor. Bring out a pattern card and a set of cubes and a base. Explain that the colour on the cube is equivalent to the black on the card. Go through an example of completing the pattern on the card with as much input from the group as possible. What math is hidden? • • • • • Logic Problem solving Pattern recognition Rotation of shapes 194 Ways to modify the game. • • • If a student is not able to follow the pattern you can complete the first row for them. If a student is not able to follow the pattern and makes their own pattern you can write the pattern that the student created. (If all a student is capable of is stacking the blocks or making them into a train that can be their pattern.) Ways to correct without it feeling like criticism. • • Avoid statements like, “That is wrong.” Ask leading questions. o “Would it help if we rotate that block?” o “Is that pattern exactly the same as the one on the card?” o “If you place the cubes directly on the pattern card would that help? Do they match now?” o “It may help to take all the cubes off the base and start fresh.” o “Can you see where the card and your cubes are different?” 195 Hidato Before you start you will need! • • Pencil Eraser The goal of the game is to place all the numbers from 1-9 (or higher for bigger games). REMEMBER ENJOYMENT AND SUCCESS ARE KEY! Help them achieve it. How to Play? • • • • Numbers must make a chain that links side to side (horizontally), up and down (vertically) or diagonally. The starting number is bold and the last number is bold. If there is a square that has an X in it you cannot place a number in it and you cannot pass through it. Sometimes working backwards helps, especially for the bigger puzzles! 1 9 3 4 6 7 196 What to do to teach the game. • • • • • • • Start by doing a few with them until they understand the game. Talk them through the reasons for placing the numbers where you do if need be but see if they can place the numbers. Have them go to their places with the first puzzle. Have them work together to find the path (unless the teacher indicates that the student it to work alone.) If need be place students that are having success with others that are struggling. (Check with the teacher before making any suggestions like this.) Exchange solved puzzles for new ones. If a student is finding it very easy and wants a challenge, skip up to a harder version. What math is hidden? • • • Ordering numbers Writing the numbers properly Grid - multiplying 3 X 3 means 9 squares 4X 4 means 16 squares ect • • What do horizontal and vertical mean? Ask them!! Ways to modify the game. • Write all the numbers in from 1 – 9 (or higher). Ways to Correct without it feeling like criticism. • • Avoid statements like, “That is wrong.” Ask leading questions that help them get to the answer but so they feel like they have done it themselves. o “Are you sure that number goes there?” o “Can you think of another number that would work better there?” 197 o “I think that number needs to be written a different way, what do you think?” o “If you put that number there will it get you to the next number?” o “Your path looks blocked by that number can you move it to a different place?” 198 199 Appendix D Rasch Material • • Rasch logit scale explanation Fit Statistics Infit and Outfit 200 Rasch Logits Scale Explanation. 201 202 Examples of Outfit and Infit Mean Square values for various response patterns. From https://www.rasch.org/rmt/rmt82a.htm 203 204