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PAGE 1 1 ASSOCIATIONS BETWEEN SELF EFFICACY BELIEFS, SELF REGULATED ELICITING TASKS: AN EXAMINATION OF DIRECT AND INDIRECT EFFECTS By ANU SHARMA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2013 PAGE 2 2 2013 Anu Sharma PAGE 3 3 To my late sister, Shivani Sharda PAGE 4 4 ACKNOWLEDGMENTS The dissertation research and writing process is an ex tensive and tiresome work. I feel blessed to be under the guidance of such a w onderful dissertation committee, Dr. Stephen Pape, Dr. Thomasenia Adams, Dr. David Miller, and Dr. David Therriault who made my journey through research an exciting and fruitful venture. My adviser Dr. Stephen Pape is a phenomenal and inspiring professor. H is thoughtful gui dance and constant encouragement throughout these years have always pushed me to do better in everything I do. This dissertation would not have been possible without his insightful feedback, invaluable advice, and thoughtful comments. He is not only a grea t mentor but also a wonderful person who cares a lot about his students Thank you for standing by me in times of difficulty and showing your confidence in me. I am grateful that his support and help continued even after he took up a position at Johns Hopk ins. Thank you for all the skype meetings. I also want to thank my co chair, Dr. Adams for her generous support. Her suggestions and questions have always prompted me to adopt a different perspective for my work. Thank you for your patience, support, faith, and being there when I needed you the most. Dr. Mil ler has been a great help to me i n terms of constructing appropriate research questions, choosing suitable research design, determining correct research measures as well as analyzing and interpreting data. I have learned a lot from his courses and experience. I am thankful to my external adviser Dr. Therriault for his valuable suggestions, comments, and feedback on my research proposal and dissertation. My special thanks goes to Dr. James Algina for his help with the data ana lysis and interpretation of results. Without his support, I woul d not have been able to PAGE 5 5 understand Structural Equation Modeling. Although he was not on my committee, he spent numerous hours in helping me running data using Mplus software and responding to my questions via emails Thank you for all the help. This resea rch project would not have completed without the support of Dr. Alison, Mr. Bice, Mrs. King, Mrs. Stephenson, and Mrs. Weller as well as the participating students. I am indebted to all the se teachers for letting me into their classrooms to administer surv forms encouraged students to participate in this research project. In addition, I cannot express my gratitude enough to all the participating students for their effort and time. Thanks you for all your support and cooperation in making this a successful project. I would also like to acknowledge a number of friends and colleagues who supported me emotionally when my husband moved to Mississippi and for taking care of my son in my absence Thank you Akshita, Aman, Anit h a, Henna, Maninder, Swapna, and Vinayak for looking after Shivank in times of need. I also thank Sabrina Powell for editing my dissertation and offering suggestions on my writing style. I would like to express my gratitude and appreciation to all my mathematics education colleagues for reading and providing feedback on my chapters as well as listening patiently to me. Thank you Felicia, Jonathan, Julie, Karina, Katherine, Katrina, Maggie, Ricado, Sherri, Tim, Tracy, and Yasemin. I am especially thankful to Sherri for helping me in evaluatin solving results Finally, I express my heartfelt regards to my parents and parents in law for their blessings, supportive words, an d love during this process. I am also thankful to my husband Shekhar for his unwavering love and support through out this journey. A very PAGE 6 6 special thanks goes to my son Shivank for giving me unsolicited hugs and kisses all the time PAGE 7 7 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ .......... 10 LIST OF FIGU RES ................................ ................................ ................................ ........ 11 LIST OF ABBREVIATIONS ................................ ................................ ........................... 12 ABSTRACT ................................ ................................ ................................ ................... 13 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 15 Backgr ound of the Study ................................ ................................ ......................... 18 Mathematical Modeling ................................ ................................ ........................... 21 Self Regulatory Processes and Problem Solving ................................ ................... 24 Statement of the Problem ................................ ................................ ....................... 26 Purp ose of the Study ................................ ................................ .............................. 27 Research Questions ................................ ................................ ............................... 28 Significance of the Study ................................ ................................ ........................ 28 Definition of Terms ................................ ................................ ................................ .. 29 2 LITERATURE REVIEW ................................ ................................ .......................... 32 Mathemat ical Modeling ................................ ................................ ........................... 32 Models ................................ ................................ ................................ .............. 33 Model Eliciting Activities (MEAs) ................................ ................................ ...... 34 Modeling Processes ................................ ................................ ......................... 36 Summary ................................ ................................ ................................ .......... 41 Self Regulatory Processes and Problem Solving ................................ ................... 42 Triadic Reciprocal Interactions ................................ ................................ ......... 43 Cyclical Phases of Self Regulation ................................ ................................ ... 44 Forethought phase ................................ ................................ ..................... 44 Performance phase ................................ ................................ .................... 46 Self reflection phase ................................ ................................ .................. 47 Summary of the Self Regulation Processes ................................ ..................... 47 Self Regulation and Mathematical Problem Solving ................................ ............... 49 Self Efficacy Beliefs and Mathematical Problem Solving ................................ .. 49 Cognitive and Metacognitive Strategies ................................ ........................... 54 Summary ................................ ................................ ................................ .......... 62 3 METHOD ................................ ................................ ................................ ................ 66 PAGE 8 8 Introduction ................................ ................................ ................................ ............. 66 Research Questions ................................ ................................ ............................... 66 Research Hypotheses ................................ ................................ ............................. 66 Pilot Study ................................ ................................ ................................ ............... 68 Participants ................................ ................................ ................................ ....... 68 Measure ................................ ................................ ................................ ........... 69 Procedure ................................ ................................ ................................ ......... 69 Data Analysis ................................ ................................ ................................ ... 70 Research Design ................................ ................................ ................................ .... 72 Determination of Minimum Sample Size ................................ ................................ 72 Method ................................ ................................ ................................ .................... 73 Participants ................................ ................................ ................................ ....... 73 Measures ................................ ................................ ................................ .......... 74 Self efficacy scale ................................ ................................ ...................... 74 Motivated Strategies for the Learning Questionnair e (MSLQ) ................... 75 The modeling test ................................ ................................ ...................... 78 Procedure ................................ ................................ ................................ ............... 81 Data Collection ................................ ................................ ................................ 81 Data Analysis ................................ ................................ ................................ ... 82 Scoring scheme ................................ ................................ ......................... 82 Scoring procedure ................................ ................................ ...................... 84 Descriptive analysis ................................ ................................ ................... 84 Analyses ................................ ................................ ................................ .... 85 Assumptions of the Study ................................ ................................ ....................... 98 4 RESULTS ................................ ................................ ................................ ............. 107 Descriptive Analysis ................................ ................................ .............................. 107 Reliability Estimates ................................ ................................ ....................... 107 Missing Data Analysis ................................ ................................ .................... 107 Descriptive Statistics ................................ ................................ ...................... 108 Multivariate Normality Assumption ................................ ................................ 109 Confirmatory Factor Analysis of the MSLQ Scale ................................ .......... 110 Confirmatory Factor Analysis of the Modeling Self Efficacy Scale ................. 111 Overview of Model Testing ................................ ................................ ................... 112 Research Hypotheses Testing ................................ ................................ .............. 115 5 DISCUSSION ................................ ................................ ................................ ....... 132 Summary of the Findings ................................ ................................ ...................... 132 Reasons for Inconsistent Results and Recommendations for Future Research ... 135 Contributions to the Field ................................ ................................ ...................... 138 Implications ................................ ................................ ................................ ........... 140 Delimitations and Limitations of the Study ................................ ............................ 142 PAGE 9 9 APPENDIX A THE MODELING TEST ................................ ................................ ........................ 147 B SELF EFFICACY SCALE ................................ ................................ ..................... 157 C MOTI VATED STRATEGIES FOR LEARNING QUESTIONNAIRE ....................... 158 D THE MODELING TEST ................................ ................................ ........................ 161 E SCORING RUBRIC FOR MODELING PROBLEMS ................................ ............. 169 LIST OF REFERENCES ................................ ................................ ............................. 171 BIOGRAPH ICAL SKETCH ................................ ................................ .......................... 183 PAGE 10 10 LIST OF TABLES Table page 3 1 Item statistics for the Modeling Self Efficacy scale ................................ ........... 100 3 2 Item Total Correlation Analysis ................................ ................................ ......... 100 3 3 Items for cognitive strategies with three scales ................................ ................ 101 3 4 Items for metacognitive strategies scale ................................ ........................... 102 4 1 Summary of reliability estimates of each scale ................................ ................. 118 4 2 Missing data analysis for the observed indicators of the full model .................. 119 4 3 Missing Value Analysis for each construct ................................ ....................... 121 4 4 Descriptive statistics for the Modeling Self Efficacy scale ................................ 121 4 5 Descriptive statistics for the modeling test ................................ ........................ 122 4 6 Confirmatory Factor Analysis of MSLQ subscal es ................................ ............ 123 4 7 Estimated correlation matrix for the latent vari ables ................................ ......... 124 4 8 Confirmatory Factor Analysis of Modeling Self Eff icacy scale .......................... 124 4 9 Confirmatory Factor Analysis for the full measurement model ......................... 125 4 10 Correlations among latent variables ................................ ................................ 126 4 11 R 2 estimates for each observed and latent dependent variable in the model ... 127 4 12 Model Modification Indices ................................ ................................ ............... 128 4 13 Standardized estimates of the path coefficients in the full structural equation model ................................ ................................ ................................ ................ 129 PAGE 11 11 LIST OF FIGURES Figure page 2 1 Modeling cycles often involves fou r basic steps ................................ ................. 64 2 2 Phases of self re gulation ................................ ................................ .................... 65 3 1 The hypoth esized model ................................ ................................ .................. 103 3 2 The scree plot showing Modeling Self Efficacy scale as one factor model ....... 104 3 3 Problem solving processes involved in mod eling tasks ................................ .... 105 3 4 A basic mediation model ................................ ................................ .................. 106 4 1 The modified measurem ent model ................................ ................................ .. 130 4 2 Standardized path coefficients in the full structural model. ............................... 131 PAGE 12 12 LIST OF ABBREVIATIONS ANOVA Analysis of Variance CCMS Connected Classroom in Promoting Mathematics CCSSM Common Core State Standards for Mathematics CCSSO Council of Chief State School Officers CFA Confirmatory Factor Analysis MAR Missing at Random MCAR Missing Completely at Random MEAs Model Eliciting Activities MI Modification Indices ML Maximum Likelihood MSLQ Motivated Strategies for Learning Questionnaire OECD Organization for Econo mic Cooperation and Development PISA Programme for International Student Assessment RMSEA Root Mean Square Error of Approximation SEM Structural Equation Modeling SPSS Statistical Package for the Social Sciences SRL Self Regulated Learning TIMSS Trends in Mathematics and Science Study TLI Tucker Lewis Index WLSMV Weighted Least Square Means and Variance Adjusted PAGE 13 13 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements f or the Degree of Doctor of Philosophy ASSOCIATIONS BETWEEN SELF EFFICACY BELIEFS, SELF REGULATED ELICITING TASKS: AN EXAMINATION OF DIRECT AND INDIRECT EFFECTS By Anu Sharma August 2013 Chair: Stephen J. Pape Cochair: Thomasenia Lott Adams Major: Curriculum and Instruction Mathematics education currently emphasizes engaging students in mathematical modeling to understand problems of everyday life and society (C ouncil of Chief State School Offic ers ( C C SSO ) 2010; English & Sriraman, 2010; Lesh & Zawojewski, 2007). The Common Core State Standards for mathematics also stress that high school students should develop understanding of algebra, functions, statistics, and geometry in conjunction with mo deling (CCSSO, 2010) A review of mathematical modeling literature indicated a success in solving modeling activities. The present study attempts to fill this gap by examining associat ions between self efficacy beliefs self regulated learning strategies (e.g., cognitive and metacognitive strategy use ) and performance i n modeling tasks. Self efficacy beliefs were measured by developing a new instrument, Modeling Self Efficacy s cale reported use of cognitive and metacognitive strategies were gathered through their responses on the modified version of the Motivated Strategies for Learning Questionnaire (Kaya, 2007). Modeling PAGE 14 14 six modeling problems. These problems were adapted from the PISA 2003 problem solving assessment. The confirmatory factor analysis indicated an acceptable fit of the data with the hypothesi zed measurement model. The structural model tested using Structural Equation Modeling techni ques suggested that perceived modeling self efficacy beliefs ( = .50, p < .001 ) directly and modeling problem s. However, organization strategy use ( p < .05 ) had a p = .08), elaboration ( = .40, p = .41), and metacognitiv e strategies ( = .46, p = .16) on their performance in solving modeling tasks were non significant. efficacy beliefs on modeling task success through their effect on their use of cognitive and metacognitive strateg ies were non significant. The implications for future research along with limitations of this study are discussed. PAGE 15 15 CHAPTER 1 INTRODUCTION One of the core tenets of Common Core State Standards for Mathematics (CCSSM) is to prepare students for the 21 st century global society ( Council of Chief State School Officers (CCSSO), 2010). Towards this end, the Standards for Mathematical Practice speci fy that students should solve real world problems by engaging in modeling activities. Modeling with mathematics is the process of using knowledge and skills from across and within the curriculum to solve problems arising in everyday life, society and work force (CCSSO, 2010). Mathematical modeling is not only an important mathematical practice that teachers should promote through classroom instruction, discussions, and activities but also a conceptual category in high school standards where it is expected t hat students should learn algebra, functions, probability, and statistics in conjunction with modeling (CCSSO, 2010). The authors state that, Modeling links classroom mathematics and statis tics to everyday life, work, and decision making. It is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. Quantities and their relationships in physical, economic, public policy, social and everyday situations can be modeled using mathematica l and statistical methods (CCSSO 2010 p. 72 ) The importance of and need to prepare students for a global society and workforce is further emphasized through statistics that show that U.S students rank significantly below Eu ropean and Asian students on international assessments such as PISA ( Programme for Intern ational Student Assessment ) and TIMSS ( Trends in Mathematics and Science S tudy). PISA 2003 tested the problem solving skills of 15 year old students by examining their and technological society Specifically, it measured the extent to which students across PAGE 16 16 the world can solve real life situations by thinking flexibly and creatively. Sadly, American students ranked 25 th among peers from 38 participating Organization for Economic Cooperation and Development (OECD) countries (Lemke et al., 2004). The U.S. average score on the problem solving scale was also lower than the average OECD score TIMS S 2007, on the other hand, m easured eighth mastery of curriculum based mathematical knowledge and skills. A lthough U.S. students mathematics score was above the TIMSS average score and their performance was better than previous assessment years, only six percent of American students were able to routine problems, and draw and justify s sol ving behaviors, especially in regard to solving problems in real life. One way to do this is by Mathematical Practice puts forth the expectation that students should dev elop expertise in solving real world situations (CCSSO, 2010), the present study is interested in in solving real world modeling tasks, the present study draws upon problem solving literature to explore the degree to which effective problem solving behaviors are associated with Effective problem solving behaviors such as setting appropriate goals, trying alternative solution paths, and perseverance with challenging academic tasks PAGE 17 17 align very closely with the self regulated learning (SR L) behaviors (Pape & Smith, 2002; DeCorte, Verschaffel Se lf regulated students contro l and regulate their thoughts, actions behaviors, and motivation in order to achieve a targeted goal (Schunk & Zimmerman 1994; Zimmerman, 2000). They use effective learning strategies, constantly monitor and assess their progress toward the ir goal, reflect on their thought processes, expend more effort, persist longer, stay motivated on the task and create productive lear ning environments (Schunk & Z immerman, 2008; Zimmerman, 2000). SRL strategies not only enhance academic performance (Dignath, Buetter, & Langfeldt, 2008; Zimmerman, 200 2; Zimmerman & Kitsantas, 2005) but also increase their motivation to learn (P intrich, 1999). Out of numerous behaviors, attitudes, and beliefs exercised by self regulated learners, motivational beliefs such as self efficacy judgments and SRL strategies such as cognitive and metacognitive strategy use engagement and persistence on complex mathematical tasks and their academic performance (De Corte et al. 2000; Hoffman & Spatariu, 2008 ; Pape & Wang, 2003; Puteh & Ibrahim, 2010; Verschaffel et al., 1999). Several studies have reported that self efficacy beliefs are related to and predictive of solving performance ( Greene, Miller, Crowson, Duke, & Alley, 2004; Pajares, 1996; Pajares & Graham, 1999; Pajares & Kranzler, 1995; Pajares & Miller, 1994, Pajares & Valiante, 2001; Pintrich & DeGroot, 1990 ). S solving performance positively impact their engagement, behavior, and cognition during academic activities. Further, s perceived capabilities to use a variety of cognitive and metacognitive strat egies also PAGE 18 18 influence their academic achievement ( Pape & Wang, 2003; Pintrich & De Groot, 1990; Pin trich, Smith, Garcia & McKeachie 1993; Zimmerman & Mar tinez Pons, 1986, 1988, 1990). C ognitive and metacognitive strategies not only help problem solvers in planning, monitoring, evaluating and revising course s of actions, but also encourage them to be more flexible in selecting a solution plan or a strategy. Research also shows that students with high academic self efficacy beliefs are more likely to report using cognitive and metacognitive strate gies and they persist longer to reach their goals ( Bouffard Bouchard, Parent, & Larivee, 1991; Heidari, Izadi, & Ahmadian, 2012 ; N evill, 2008; Pintrinch & DeGroot, 1990 ). The present study built upon and extended e xisting problem solving literature by examining these associations in the context of mathematical modeling and real life problem solving. Specifically, the present study explored associations between motivational beliefs (e.g., self efficacy beliefs), SRL strategies (e.g., cognitive and metacognitive strategies), and modeling outcomes. The next section descri bes the background of the study including the average performance of students on international assessments to illustrate that modeling problems are not only difficult for U.S. students but also challenging for students all over the world. This is followed by a brief overview of mathematical modeling and SRL processes The chapter concludes with a statement of the problem, purpose of the study, research q uestions, and significance of the study. Background of the Study thinkers and effective problem solvers. This is because the kind of mathematical thinking that is needed beyond sch ool has changed significantly with the advent of new communication and collaboration technologies (English, Lesh, & Fennewald, 2008; PAGE 19 19 Lesh, 2000). For example, the actual price of a car is much more than the sticker price. Determining the actual cost of a v ehicle involves interpreting loans, down payment, monthly payments, annual percentage rate, and billing periods. Yet, most classrooms are still not preparing students for life beyond school as they seldom provide students with opportunities to apply what t hey have been learning to understand problems situated in real world cont exts (English et al. 2008). The aver age performance of students from all over the world on international assessments (e.g., PISA, TIMSS) further shows their lack of experience in relation to real life problem solving. PISA 2003 life problem solving skills by measuring the extent to which they can cros s disciplinary situations where the solution path is not immediately obvious and where the content areas or curricular areas that might be applicable are not within a single subject area of oblem solving abilities were measured through three different types of problems including decision making, system analysis and design, and troubleshooting (OECD, 2004). These problems were carefully selected to encompass several problem solving abilities t hat students may need in understanding day to day situations. Some of the problem solving abilities tested by PISA include making appropriate decisions choosing strategically amo ng several alternatives, analyzing situations, describing under lying relation ships, designing systems or diagnosing and rectifying faulty systems In all, 38 countries participated in the PISA 2003 problem solving assessment. solving scale for which the mean score was 500 points. Sadly th e overall problem solving score of only 17 of the 38 PAGE 20 20 participating countries was higher than the OECD average score of 500. T he PISA problem solving scale also d profi ciency levels. Level three represents students with the strongest problem solving skills, and level one denotes students with the weakest problem solving skills. The percentage distribution of 15 year old students on the problem solving scale indicated that 17 percent of the students that part icipated in the PISA problem solving assessment scored below level one, 30 percent at level one, 34 percent at level two, and 18 percent at level three (Lemke et al., 2004). There were only four countries (e.g., Finland, China, Japan, and Korea) that had 3 0 percent or more of their students scoring at level three. In contrast to PISA, TIMSS 2007 measured eighth grade students school based mathematical knowledge and skills (Gonzales et al., 2008 ). achievement was measured by testing t heir subject matter knowledge in the area of number sense algebra, geometry, and data and chance S were assessed in three domains including knowledge of mathematical facts, procedures, and concepts ( knowing ), ability to apply kno wn operations, methods, and strategies ( applying ), and ability to handle unfamiliar situations, complex contexts, and multi step problems ( reasoning ). an average score of 500 (Gon zales et al., 2008). O f the 48 participating countries, the mathematics score of only 12 countries was higher than the TIMSS average score. T here were 18 countries that scored higher than 500 points in the areas of knowing, applying, and reasoning. Further participating performance against international benchmarks of mathematics achievemen t showed that there were only five PAGE 21 21 countries who had a significant percentage of 8 th gr ade students (e.g., 26% to 45%) reaching the advanced level skills includ ing organizing information, making generalizations, solving non routine problems, drawing conclusions, an d justifying solutions (Mullis Martin, & Foy, 2008 ). T he rest of the 43 countries had fewer than 10 percent of their students demonstrating advanced l evel skills. Thus, both PISA and TIMSS assessments point toward the need to improve solving behaviors, especially in regard to solving problems in real life. Learning mathematics with modeling has been cited as one of the possible soluti ons because modeling not only improves transfer but also fosters 21 st century skills of reasoning, critical thinking, and strategic decision making (English, 2011; English & Sriraman, 2010; Lesh & Zawojewski, 2007). Mathematical Modeling According to a mod els and modeling perspective, students understand real world situations by participating in iterative cycles of modeling where they progressively create, test, revise, and refine their mathematical interpretations (Lesh & Harel, 2003). Such interpretations existing knowledge and experiences as well as beliefs and attitudes t hat they bring to the classroom (Eric, 2010) interpretations, and explanati ons of mathematical situations are known as models. Examples of mathematical models include equations, graphs, tables, written symbols, spoken language, diagrams, metaphors, concrete models, or computer based simulations (Lesh, Hoover Hole, Kelly, & Post, 2000; Lesh & Doerr, 2003) Further, mathem atical tasks that require students to understand real life situations through develop ing models are known as model eliciting activities (MEA s ) or modeling tasks PAGE 22 22 (Lesh et al., 2000). Students develop efficient models for these activities by engaging in several modeling cycles ( Mousoulides, Christou, & Sriraman, 2008; Lesh & Doerr, 2003; Lesh & Zawojewski, 2007). E ach modeling cycle consists of four diffe rent modeling processes includi ng (1 ) understanding the modeling task ( description ) (2 ) developing a mathematical model ( manipulation ) (3 ) interpreting the actual situation based on the created model ( prediction ), and (4 ) analyzing and reflecting upon the results ( verification ). Furth ermore, it is important to note that this modeling cycle bears some structural similarity to many of the general problem solving heuristics proposed over the years by researchers such as Polya (1957), Newell and Simon (1972), and Bransford and Stein (1984) devising a plan, carrying out the plan, and looking back. Although the process of describing a modeling task and understanding a problem solving task involve making sense of the task, the cognitive skills required to comprehend modeling tasks are more demanding. Problem solving strategies such as representing the problem separating various parts of a problem, organizing data in the form of a table or making connections between t he known and unknown information may help students in understanding a selecting, quantify 2010, p. 273). for a mathematical problem by looking for a problem having the same or similar PAGE 23 23 unknowns, employing strategies or method s of previously solved problems looking for patterns, making an orderly list, considering special cases, or solving the same problem by making use of smaller numbers On the other hand, during the manipulation phase of the mo deling cycle students generate hypotheses about a given situation by developing a mathematical model that represents relationships between different variables involved in a system. However, it may also include modifying a previously developed model. Durin required to solve a problem as well as make sure that each step is mathematically correct. Model development processes, however, place less emphasis on the precision and accuracy of the solutions and stress the importance of correctly predicting the actual situation based on the model created (e.g., making decisions, designing systems, or diagnosing faulty systems). evelopment cycle encourage students to analyze and look back on their solutions, especially to judge the accuracy of their solutions or models. It is important to note, however, that verification processes involved during model development clearly emphasiz e the need to re understand and re interpret the situation when the created model fails to explain the real world situation. Research also shows that students interpreting modeling activities seldom produce effective models during their engagement with the first cycle of modeling processes (English, 2006; Eric, 2010; Mousoulides, Pittalis, Christou, & Srirama n, 2010 ). The following section describes the cognitive and metacognitive processes involved in solving problems as well as motivational beliefs exhibi ted by effective problem solvers, which align very closely with SRL processes. PAGE 24 24 Se lf Regulatory Processes and Problem Solving S ocial cognitive theory is a useful framework f or understanding SRL behavior s that enhance students solving skills (Zimmer man & Campillo, 2003). The theory describes human functioning in terms of recipro cal interactions between personal v ariables, environmental factors, and behavioral objectives (Bandura, 1986). It presents a view of human agency where people make meaningful and purposeful choices to efficacy beliefs influence their learning behaviors such as choice of problem solving strategies, effort expended, and persistence (Pintrich & De Groot, 1990; Schunk & Mullen, 2012). In turn, about their problem solving capabilities, which motivates them to work harder to produce meaningful solutions. The triadic reciprocal interaction also influences the three cyclical phases of self regulation: forethought, performance, and self reflection. During the f orethought phase problem solvers analyze the requirements of a ta sk, establish achievable goals, and design solution plans by selecting strategies appropriate to achieve these goals (Zimmerman, 2000). These processes are influenced by sever al motivational beliefs such as self efficacy beliefs, outcome expectations, intr insic interest, and goal orientation. Out o f these, self efficacy beliefs involving student lish a particular task have been extensively explored in the context of mathematical problem solving ( Hoffman & Spa tariu, 2008; Pajares, 1996; Pajares & Graham, 1999; Pajares & Miller, 1994 ; Schunk & Pajares, 2009; Schunk & Mullen, 2012). Self efficacious problem solvers set challengin g goals, expend more effort, use effective learning strategies, and persist longer in times of difficulties (Pajares, 2008 ; Schunk & Pajares, 2009). PAGE 25 25 The next phase, p erformance phase comes into play when students are actually engaged in solving mathematics problem s or preparing for a test Effective problem solvers increase their attention and persistence over tasks through self instruction, attention focusing, and task strategies. They also utilize self observation processes, such as self monitoring and self recording, to monitor their pro gress toward the goals as well as to check their understanding of the task (Dabbagh & Kitsantas, 2004) From the problem solving perspective, these are important SRL strategies because they support students in finding errors in their learning and prompt th em to adjust their strategies and procedures in case they are not making adequate progress. Error analyses are closely followed by self evaluation and self reaction processes of the self reflection phase (Cleary & Zimmerman, 2012). Self regulated learners self evaluate their performance against their personal goals and attribute their mistakes to a lack of adequate effort (Zimmerman, 2000; Zimmerman & Campillo, 2003). Effective and ineffective problem solvers have been found to behave a nd react differently in all three phases of self regulation (Clearly & Zimmerman, 2001; Zimmerman & Campillo, 2003). Although there are numerous motivational beliefs and SRL behaviors that support students during problem solving, the present study emphasized the importance of self efficacy beliefs and SRL strategies such as cognitive and metacognitive strategies. Cognitive strategies support students in processing information, such as elaboration, organization, and critical thinking. Elaboration strategies such as paraphrasi ng summarizing, and note taking, facilitate students in developing meaningful representations (or models) for the problem s. The organization strategies such as clustering, outlining and selecting main ideas are helpful in differentiating PAGE 26 26 relevant and irr elevant information These strategies may be useful in solving system analysis and design problems, where students represent relationships among different parts of a system either in the form of a table or a chart. Finally, critical thinking strategies sup port students to make logical decisions and analysis These strategies are considered to b e the most important skills as they help students to think logically, consider alternati ve conceptions of a problem, make effective decisions, reason deductively as well as justify reasoning (Stein, Haynes, Redding, Ennis & Cecil, 2007) Metacognitive strategies including monitoring, controlling, and regulating cognition and learning support students in self evaluating the effectiveness of their models, creating revised models, group decision making, and describing situations using mode ls. These processes are especially important during with the successive modeling cycles of describing, manipulating, predicting and verifying situations using models. The present study examined associations between self efficacy beliefs, self reported use of cognitive strategies (e.g., elaboration, organ ization, and critical thinking) and m etacognitive strategies (e.g., planning, monitoring, and re gulating), and Statement of the Problem Mathematics education currently emphasizes providing students with op portunities to apply mathematical knowledge and skills to u nderstand problems of everyday life and society (CCSSO, 2010; English & Sriraman, 2010; Lesh & Zawojewski, 2007). The CCSSM also emphasize that high school students should develop understanding of algebra, functions, statistics, and geometry within real wo rld contexts. In spite of this, the current mathematics textbooks, teaching practices, and assessment techniques hardly support students in developing understandings and PAGE 27 27 abilities useful for mathematical modeling (Lesh, 2003). The adoption of CCSSM, howeve r, offers some hope regarding preparing students with skills useful for life beyond school. The Standards for Mathematical Practice d escribe the kind of mathematical knowledge and skills to be fostered in classrooms with regard to modeling. As such, they d o not inform teachers about factors that may influence modeling perspective is also of little help because it is still developing. Kaiser, Blomhj, and Sriraman (2006) argue d T he theory of teaching and learning mathematical mode ling is far from being complete. Much more research is needed, especially in order to enhance our understanding on micro levels, meaning teaching and learning problems which occur in particular educ ational settings where students are engaged in examining associations between effective proble m solving behaviors and student success rates on modeling tasks. Purpose of the Study The present study is aimed toward investigating factors that may influence n understanding modeling tasks or real world situations Problem solving literature informs us that self efficacy beliefs (G reene et al., 2004; Pajares & Graham, 1999; Pajares & Krazler, 1995; Pajares & Miller, 1994) and SRL strategy use (Zimmerman & Martinez Pons, 1986, 1988, 190) performance on complex mathematical tasks. Thus, the focus of this research is to examine relationships b etween self efficacy beliefs, M odeling processes including building, describing, testing, revising, manipulating and PAGE 28 28 verifyin g model s align very closely with the three types of problem solving tasks (e.g., decision making, system analysis and design and trouble s hooting tasks) chosen by the PISA 2003 assessment ( Blum, 2011; Mousoulides, 2007). The present study, therefore, exami ned associations between self efficacy beliefs, cognitive and metacognitive decision making, system analysis and design and trouble s hooting tasks. Research Questions The study was guided by three rese arch questions. 1. What are the direct effects of students eff icacy beliefs for modeling t asks on their performance on modeling tasks ? 2. What are the direct effect s of students self reported use of cognitive and metacognitive strategies on thei r performa nce on modeling tasks? 3. What are the indirect effect s of efficacy beliefs for modelin g tasks on their performance on modeling tasks through their effects on their use of cognitive and metacognitive strategies? Significance of the Study The present study was stimulated by the need for research that examines review of the problem solving literature indicated that motivational beliefs, such as self efficacy beliefs, and SRL strategies, such as cognitive and metacognitive strategies, are solving success. Studies that investigated the effects of self confide nce in their ability is positively correlated with their problem solving skills and academic performance ( Pajares, 1996; Pajares & Graham, 1999; Pajares & Kranzler, 1995; Pajares & Miller, 1994, Pajares & Valiante, 2001; Pintrich, 199 9; Pintrich & De Groot PAGE 29 29 positively associated with their learning and problem solving performance (Pape & Wang, 2003; Verschaffel et al., 1999; Zimmerman & Martinez Pons, 1986, 1988, 1990). The pre capabilities as well as use of cognitive and metacognitive strategies are associated with their modeling task success Thus, the results of this study provide researchers and educato tasks. Furthermore, the present study developed a new instrument, the Modeling Self psychometric pro perties of this scale including internal consistency and construct validity were tested using this sample. As such, development of this new scale contributes to the literature related to self efficacy theory and the mathematical modeling field. Definition of Terms COGNITIVE STRATEGIES L processing of information (Pintrich et al., 1993). The present study highlights the importance of three cognitive strategies such as elaboration, organ ization, and critical thinki ng CRITICAL THINKING STRATEGIES These strategies support stude nts in logical decision making especially in consider ing alternati ve conceptions of a problem, making effective decisions, reasoning deductively, and justifying reasoning DECISION MAKING TA SKS These are r eal world problems requiring students to make appropriate decisions by choosing strategically among several alternatives provided under a given set of conditions (OECD, 2004). ELABORATION STRATEGIES Strategies that help students to understand challenging modeling situations by making connection s with their existing mathematical k nowledge and skills (Ormord, 2008). Examples include paraphrasing, summarizing, creating analogies, and explaining ideas to others. PAGE 30 30 MATHEMATICAL MODELING It is the process of using knowledge and skills from across and within the curriculum to solve problems arising in everyday life, society and w orkforce (CCSSO, 2010). METACOGNITIVE STRATEGIES Metacognitive strategies including pl anning, monitoring, controlling, and re gulating cognition and learning support students in self evaluating the effectiveness of their models, creating revised models, group decision making, and describing situations using models. MODEL ELICITING ACTIVITI ES These are real world mathematical situations that are gener ally understood by creating, testing and revising models (Lesh et al., 2000). MODELING OUTCOMES It represents outcomes of engaging students in modeling activities. In this study, modeling ou tcomes include ability to analyze real world problems by drawing effectively on multi disciplinary knowledge, planning, monitoring, and assessing progress, making decisions, troubleshooting faulty systems, or analyzing structures of complex systems. MODEL ING PROCESSES. Modeling processes, such as building, describing, manipulating, predicting, testing, verifying, and revising mathematical interpretations, are the cognitive and metacognitive processes employed by students to produce efficient models. MODE LS Models are conceptual systems that represent how students are thinking, interpreting, and describing modeling tasks (Lesh & Doerr, 2003). Models can be as simple as writing a mathematical equation to depict a relationship between two variables or as co mplicated as creating a spreadsheet to plan an event. ORGANIZATION STRATEGIES Organization strategies, such as clustering, outlining, and selecting main ideas, are helpful in differentiating relevant and irrelevant information. These strategies may be us eful in solving system analysis and design problems, where students are expected to organize information in meaningful ways. REAL LIFE PROBLEMS Real life problems are simulations of the situations and leisure, and in the these problems by applying their personal knowledge and prior experiences. SELF EFFICACY BELIEFS It refer s capabilities to accomplish a particular task (Bandura, 1986). In this study, self accurately solve decision making, system analysis and design, and troubleshooting tasks. PAGE 31 31 S ELF R EGULATED LEARNING STRATEGIES These s trategies refer to actions directed at acquiring information or skill that involves agency, purpose (goals), and instrumentality self Martinez (e.g., elaboration, organization, and critical thinking) and metacognitive (e.g., planning, monitoring, and regulating procedures ) strategies. SELF REGULATION It is the abili ty of learners to control and adapt their cognition, behavior, and emotions in order to achieve a targeted goal (Schunk & Zimmerman, 1994; Zimmerman, 2000). SYSTEM ANALYSIS AND DESIGN TASKS Real world problems that require students to design systems, suc h as diagrams, tables, or flow charts, to represent relationships between variables (OECD, 2004). TROUBLESHOOTING TASKS Real world p roblems that require students to diagno se and repair faulty or underperforming systems (OECD, 2004). PAGE 32 32 CHAPTER 2 LITERATUR E REVIEW As stated in the first chapter, the present study aims to explore relationships examining associations between self efficacy beliefs, cognitive and metacogni tive str ategy use, and success on modeling tasks This chapter provide s a summary of research related to critical processes involved in mathematical modeling and SRL in three major sections. The first section involves an overview of a models and modeling perspective on mathematical learni ng and problem solving The second section is devoted to a discussion of SRL from a social cognitive perspective to explicate the relationship between self regulatory processes and mathematical problem solving. Finally, the last section presents a review o f the research on two aspects of self regulation self efficacy beliefs and students of cognitive and metacognitiv e strategies, and it argues for the positive effect of these constructs on students solving skills. Ma thematical Modeling Mathe matical modeling has been regarded as an effective platform for providing students with experiences that support them in developing mathematical knowledge and skills essential to succeed in life beyond school (English & Sriraman, 2010; Dark, 2003; Galbrait h, Stillman, & Brown, 2010; Lesh & Zawojewski, 2007). During mathematical or predict patterns or regularities associated with complex and dynamically changing (Lesh, 2000, p. 179). They make sense of realistic situations by engaging in the processes of mathematization such as quantifying, organizing, sorting, weighting, and coordinating data. As such, students are provided with many opportunities to PAGE 33 33 exercise ma thematical skills that are needed to understand real world situations. This section provides a brief description of models, model elic iting tasks and modeling processes to illustrate the modeling view of mathematical problem solving. Models According to Hestenes (2010), models represent the structure of a problem solving situation, which include the objects that make up a system as well as the relationships that exist between these objects. Students use models to solve and make predictions about complex p roblem solving situations. Lesh and his colleagues also provided a similar definition and described models as a system that consists of (a) elements; (b) relationships among elements; (c) operations that describe how the elements interact; and (d) patterns or that apply to the relationships and operations. However, not all systems function as models. To be a model, a system must be used to describe another system, to think about it, or to make sense of it, or to explain it, or to make predictions abo ut it ( Lesh et al., 2000, p. 609). In a way, models represent how students are thinking, interpreting, or organizing ideas, and these representations are both internal and external to them (Lesh & Doerr, 2003; Lesh, Doerr, Carmona & Hjalmarson, 2003). In fact external representations such as verbal explanations, mathematical expressions, graphs, diagrams, computer grap hics, or metaphors are The complexity or choice of a mat hematical model is not a matte r of concern provided it fits with a situation; the model can be as large as consistin g of several representations or as small as a simple arithmetic equation or ordinary spoke n language (Lesh & Doerr, 2003). Also, models need to be shareable generalizable and reusable in nature w hich means that models should not only be used to describe a modeling situation for which it PAGE 34 34 is developed but should also be easily adaptable to understand similarly structured situations. Such mode ls are produced when students repeatedly revise and refine their interpretations about the real world situation (Lesh et al., 2000; Lesh & Doerr, 2003; Lesh & Lehrer, 2003). During this process of iterative refinement, students do not create just one model but develop a sequence of models that describe their ways of thinking about a complex modeling situation (Doerr & English, 2003; Larson et al., 2010). One form of such comp lex contexts, which is both model eliciting and thought revealing is model eliciti ng activities or MEAs. Model Eliciting Activities (MEAs) ME As are problem solving activities that are based on real life situations. These activities are carefully designed so that students can use their current mathematical knowledge and understanding to produce powerful, shareable, and re usable models (Lesh, Yoon, & Zawojewski, 2007). Creation of such models involve s identifying, selecting, and collecting releva nt data, describing situations using a variety of representation media, and interpreting the solution repeatedly in the context of a real world situation (Lesh & Doer, 2003). Modeling tasks are also calle d thought revealing activities because models reveal how students are thinking, reasoning explaining, comparing, or hypothesizing about mathemat ical objects, relations, and operations (Lesh et al., 2000) For example, one model eliciting act requires students to develop a procedure for police detectives that help s them to predict the height of a person from the size of a sho e print (Lesh & Doerr, 2003). The students we re told that the procedure developed by them should work for all footprints. Lesh and Doerr reported that a g roup of students estimated the height to be about six times the size of the footprint by using trend estimation technique s rather than setting up a PAGE 35 35 proportion. These students recor ded their height and shoe sizes as well as graphed the measurements by plotting the foot measurements on the x axis and height measurements on the y axis. The line of best fit helped students to determine a relationship between the size of a foot and height of a person. Thus, student concepts a nd relationships. MEAs that provide students with opportunities to develop and test models in order to understand complex real world mathematical problems are designed by taking into account six principles (Lesh et al., 2000). First, solutions to the modeling activities should require students to construct explicit models to describe, explain, and predict about patterns and regularities involved in the situations ( the model construction principle ). Second, the problem solving t asks should be based on authentic situations that students could interpret using their current mathematical knowledge and skills ( the reality principle ). Third, modeling activities should include information that students could use to test and revise their thinking, create alternate solutions, and judge when and how th eir models need to be improved ( the self assessment principle ) Fourth, the context of MEAs should encourage s students to document and record their thinking about problem solving situations, e specially about the givens, goals, and possible solutions as they recursively move through each phase of the modeling cycle ( the construct documentation principle ). Fifth, models (e.g., spreadsheets, graphs, or graphing calculator programs) developed by st udents should be shareable with other people as well as easily modifiable to make sense of situations structurally simila r to the existing task ( the construct shareability and reusability principle ). Finally, model s PAGE 36 36 produced by students should be based on useful metaphors ( the effective prototype principle ). Modeling Processes As stated above, students create models to understa nd and make predictions about modeling situations The process of producing sophisticated models involves the extension of existing knowledge and understanding during which problem solvers repeatedly express, test and modify their i nterpretations about these situations Some important processes employed by effective students (or modelers) limitations and better understanding the problem situation, revising the model and Zawojewski, 2010, p. 239). These modeling processes support students in improving and refining their thinking about the nature of elements involved in the problem, relationships among elements, operations describing how the elemen ts relate to one another, and understanding patterns or regularities in the problem solving si tuation s (Lesh & Lehrer, 2003). modeling processes in terms of modeling cycles. A modeling cycle describes different stages that students have to pass through in order to provid e solutions to real world situations. Lesh and colleagues (Lesh & Doerr, 2003; Lesh & Zawojewski, 2007) proposed a modeling cy cle comprised of four different stages including description, manipulation, prediction, and verification to illustrate ideal model ing behaviors ( see Figure 2 1 ). Description involves understanding the structure of the real world situation by comprehending texts, diagrams, graphs, charts tables, or the context of the situation. It also represents behaviors involved in simplifying com plex situations, such PAGE 37 37 as students making assumptions based on their prior knowledge. Manipulation refers to the act of developing a mathematical model through constructing hypotheses, critically examining mathematical details embedded within the task, and by mathematizing data. Prediction involves interpreting the actual situation based on the created model. Verification involves checking, evaluating, analyzing and reflecting upon the predictions by considering real world constraints as well as communicati ng results. The information gathered through this process supports students in refining and revising their thinking about the mathematical situation, which places them in the next cycle o f the four step modeling process Students typically engage in a seri es of modeling cycles to generate produ ctive interpretations about situations because givens and goal(s) are not clearly defined in modeling situations, and a single modeling cycle is not sufficient to understand a given situation, choose appropriate proce dures, and create effective models (Haines & Crouch, 2010; Lesh & Doerr, 2003). evidence that students go through multiple cycles of modeling processes to develop mathematical m odels that adequately describe complex modeling situations (Amit & Jan, 20120; Doerr, 2007; Doerr & English, 2003; English, 2006; Eric, 2009; Eric, 2010; Mousoulides, Christou, & Sriraman, 2008). Eric (2010) engaged three groups of sixth grade students in a modeling activity: The Floor Covering problem. The activity required students to choose carpet, tiles, or mats as the best covering material for a rectangular floor Students selected the covering material by taking into account the dimensions of each ma terial, cost per unit area of material, cost of loose material for patchwork, and labor expenses involved in cutting the material. They were also provided with actual PAGE 38 38 carpet, tiles, and mats to simulate the situation. Studen written w ork and reflections were used to determine the time spent by students in each phase of the modeling cycle. The timeline diagram across the various modeling stages described the modeling cycle to be iterative in nature. Students in all three groups (2 high ability and 1 mixed ability) chose the best covering material by repeatedly modifying their models, especially by moving cyclically around the four phases of description, manipulation, prediction, and verification as well as by revising their understandin g of models within each of the se four phases. Further, the timeline diagram indicated that the m ixed ability group students produce d more model iterations, and students belonging to the high ability groups revised their models a multiple number of times within a particular modeling phase (e.g., manipu lation). The study also provides personal knowledge and experiences. For example, one of the high ability groups recommended fo r using tiles over cheaper means of using a mat because they considered tiles to be more durable and allergy free compared to the carpets. Doerr and English (2003) also reported that students solve modeling tasks by repeatedly refining their understanding of a modeling situation. They engaged four groups of middle grade students in five mathematically equivalent activities to develop their understanding of the rating system, especially about ranking, sorting, selecting, and weighting data. The five modelin g activities, including Sneakers, Restaurant, Weather, Summer camp, and Crime involved developing a generalizable and reusable rating system often by creating and modifying quantities For example, t he Sneakers problem required students to develop a ratin g system for purchasing a pair of sneakers PAGE 39 39 by brainstorming important factors such as comfort, style, size, cost, brand, quality, and grip. In the Restaurant problem participants were required to determine the most important factors influencing customers to revisit a restaurant based on the survey data S group discussions, and field notes established that they engaged in multiple cyc les of modeling to rank, select, and weight data, especially to describe situations and make effective decisions. Similar to Eric study, students revised and refined their understanding of the real world situation, quantities, and relationships be tween and among quantities at each stage of the modeling cycle. Further, t he successive modeling cycles represent ed a progressive shift in ways of thinking and modeling solutions. For example, a group of students i mproved their ranking system of buying a pair of sneakers from nonmathematical rankings consist ing of personal preferences involved ranking factors by taking into account the nonmathematical rankings of all the groups and aggregating the number of times f actor s occurred most frequently at the top two positions However, when factors could not be ranked using frequency ranking strategy, students used pairwise comparisons to compare the relative order of the factors. Th ese findings further prove when students are engaged in the successiv e sequences of the modeling cycle, they not only create efficient models but also develop a deeper understanding of the constraints and limitations associated with th eir models at each stage of the model developmen t (Zawojewski, 2010). MEAs are cognitively demanding tasks (Blum, 2011; English, 2010; Gai lbraith, 2011), PAGE 40 40 and s tudents need to work harder, persist longer, show greater interest in learning, and expend a lot of effort and time to produce possible solutions. Researchers who tracked students typically spent two 40 minute class sessions to understand and descr ibe a single modeling task (Doerr & English, 2003; Eric, 2009; Eric, 2011). Within the field of academic motivation, t hese behaviors are found to be associated with self efficacy omplish a particular task (Schunk & Pajares, 2009). Also, there is considerable evidence in the problem solving literature that higher sense of self efficacy beliefs significantly affect solving achievement, self regulation, amo unt of effort, and persistence on complex mathematical tasks (Chen, 2002; Nicolidau & Philippou, 2004; Pajares & Graham, 1996; Pajares & Krazler, 1995; Pajares & Miller, 1994). Given this literature th e present study hypothesized that self efficacy belief s would have positive effects T he modeling problems used in the Doerr and English (2003) study emphasized the use of cognitive and metacognitive strategies. As mentioned earlier, students in this study developed rating systems for five different real world situations by ranking quantities, identifying relationships between and among quantities, and selecting appropriate operations and representations. Such notions of ranking, sorting, selecting, organizing, transforming, and weighting data may require the use of cognitive strategies. Specifically, elaboration strategies may be useful in summarizing data sets or information provided in the problem, explaining ideas to others, noting important points, and negotiation of con jectures and clarification of explanations. Organization PAGE 41 41 strategies may be useful in transforming and organizing information in meaningful ways, modifying quantities, and ranking multiple factors. Describing, explaining, and predicting actual situations wi th the help of rating systems (or models) may require the use of critical thinking strategies. Metacognitive strategies may play an a ctive role when students refine their models during multiple cycles of interpretations, descriptions, conjectures, explanat ions, and justifications. Additional studies that examine the role of cognitive and metacognitive strategies in mathematical modeling will be discussed in the next section. Summary Researchers (e.g., Blum, 2011 ; English & Sriraman, 2010; Lesh & Zawojewski, 2007) as well as current ma thematics standards (e.g., CCSS M) argue for engaging students in mathematical modeling to provide them with opportunities to use and apply mathematical knowledge and skills in solving real world situations Modeling tasks are co mplex problem solving activities that are solved by iteratively creating, testing, and representations of their ideas that describe how they are thinking, organizing, and interpreting information provided in the modeling tasks ( Lesh & Doerr, 2003 ). Students use m odeling processes to understand and solve these real world problems ( Lesh at al., 2000 ). Modeling processes incl ude explaining the problem, describing the problem, building a mathematical model, connecting the model with the real world situation, predicting real world problems, and verifying the solution within the context of the real world situation. A brief review of research focused on designing and tracking modeling activities informs us that self efficacy beliefs and SRL strategy use modeling tasks. Self efficacy beliefs may influence rmance, persistence, PAGE 42 42 and efforts in solving complex modeling tasks. The self reported use of SRL strategies, such as cognitive and metacognitive strategies, may support learners in interpreting real world situations and developing efficient models. In orde r to test these hypotheses, the present study examined associations bet ween self efficacy beliefs, cogniti ve and The next section describes the importance of self regulatory processes tha t play a major role in students being active agents of their own learning process. Self R egulatory Processes and Problem Solving In this section, self regulation is discussed from a social cognitive perspective, which may provide a useful framework to und e rstand the proactive and independent working style of students engaged in mathematical modeling. Modeling as a Standard for Mathematical Practice requires students to apply their mathematical knowledge and skills to solve problems arising in a real world e nvironment. Self regulation theory may help us to understand how students engaged in complex modeling activities control and regulate their behaviors during iterative modeling cycles Social cognitive theory defines self regulated students vely, motivationally (Zimmerman, 1989, p. 329). Such students regulate their motivation and behavior s by establishing realistic and attainable learning goals, monitoring and assessing pro gress towards these goals, and setting revised goals and actions (Zimmerman, 1989, 2000). In general, social cognitive th eory views self regulation as (1 ) an interaction among personal, behavioral, an d environmental factors, and (2 ) comprised of cognitive and metacognitive processes as well as self motivational beliefs. PAGE 43 43 Triadic Reciprocal Interactions A central tenet of social cognitive theory is that human functioning involves reciprocal interactions among personal, behavioral, and environmental factors (Bandura, 1997; Zimmerman, 2000). During these triadic interactions, problem solvers not only control t heir behaviors and environments but also are influenced by them. Personal variables include covert processes (e.g., cognitive and metacognitive processes), beliefs (e.g., self efficacy beliefs), and affective factors (e.g., perceptions of satisfaction or dissatisfaction) that students use to acquire knowledge and skills (Zimmerman, 1989). Behavioral factors involve making changes in behavior to im prove l earning, overcoming anxiety, and reducing perceptions of low self efficacy (Zimmerman, 1989, 2000). Examples of critical behaviors are keeping track of problem solving strategi es through journal writing self evaluating performances, making appropri ate choices, increasing effort or persistence toward the task, and verbalizing thoughts. Environmental factors comprise the social and physical environment of the problem solver such as the nature of the task posed, statements communicated or feedback pro vided by the environment including teachers The triadic model assumes that personal, behavior al, and environmental variables are distinct from each other but constantly influence each other in a reciprocal fashion. For example, the environment influences b ehavior when a teacher shares a modeling task with the students and directs their attention (behavior) t oward it. Behavior affects the environment when students do not understand the complex mathematical task and the teacher (environment) supports studen ts to understand the task through scaffolding Students s such as the use of cognitive and metacognitive strategies raise their self efficacy beliefs for solving tasks which further influence PAGE 44 44 person al factors such as increased persistence as well as effort expended in interpreting modeling tasks The social cognitive theory presents a view of personal agency through which students exert control over their thoughts, feelings and actions (Schunk & Pajares, 2005). Researchers have found that effecti ve and ineffective problem solvers regulate and control aspects of personal, behavioral and immediate learning environment s differently (Clearly & Zimmerman, 2001). Cyclical Phases of Self Regulation The social cognitive theory of self regulation segment s behaviors aimed toward accomplishing a task into three phases including forethought, performance and self reflection that are associated with specific cognitive and metacognitive strategies (Zimmerman, 2002; Zimmerman & Campillo, 2003) (s ee Figure 2 2). Forethought phase The forethought phase processes involve planning and preparation efforts before engaging in a task Self regulated students proactively engage in goal setting and strategic planning by analyzing the problem solving tas k, setting realist ic goals, and activat ing problem solving strategies (Zimmerman, 2002). Carefully selected methods and strategies enhance students and motoric execution (Zimmerman & Campillo, 2003). Self regulatory processes are influenced by several self motivational efficacy beliefs), their beliefs about the future benefits of engaging in a task ( outcome expectations ) their natural interest in a topic ( intrinsic interest ) and their general reasons for engaging in a task ( goal orientation ) The present study focused particularly on self efficacy beliefs, which refer to es to accomplish a particular task PAGE 45 45 (Bandura, 1997). In academic settings, particularly sc hool mathematics, self efficacy refers to students ematics probl ems, perform mathematics related tasks, or engage in mathema tical activities (Pajares, 1996). Self efficacy b eliefs are task and situation specific judgments, which are reported in relation to a goal (Pajares, 2008). For example, students may hold high self efficacy beliefs for solving routine math ematics problems that require procedural knowledge of basic math ematics rules and low efficacy beliefs for solving real world problems that require conditional knowledge of when to use a particular math ematical rule or strategy. This is from several factors such as the nature of the tasks, amount of effort required, skills needed, and environmental factors. These beliefs reflect students dgments of performing a task in future rather than their actual performance level. In actual repor ting, students may underestimate or overestimate their judgments about their own competence (Pajares & Miller, 1994; Pajares & Kranzler, 1996; Schunk & Pajares, 2009). Poor calibration mainly occurs because students fail to understand the complexity involv ed in the task and cognitive demands posed by it (Schunk & Pajares, 2009) Academic self efficacy plays an important role during the three phases of self regulation (Schink & Ertmer, 2000; Zimmerman, 2002a). D uring the forethought phase self efficacy bel iefs influence goal setting as well as the strategies selected to accomplish a task (Schunk, 2000). Students with high self efficacy beliefs set proximal and challenging goals, whereas those with low efficacy tend to stay away from difficult tasks (Schunk, 1983a; Zimmerman, Bandura & Martinez Pons, 1992). Additionally, self PAGE 46 46 monitor comprehension, evaluate goal progress, and create supportive learning ullen, 2012, p. 222). Thus, motivational beliefs impact the extent to which students engage in self control and self observation processes of the performance phase. Performance phase Performance phase p rocesses come into play when students are actually inv olved in doing a task, such as solving a mathematics problem or preparing for a test Self regulated students utilize several self control processes, such as self instruction, attention focusing, and task strategies, to increase their focus, attention, and persistence towards the task (Zimmerman & Campillo, 2003). Specifically, they control their cognition by employing cognitive strategies such as organization, elaboration, and critical thinking. They manage their behaviors and actions by using self instruc tion strategies such as overt or covert self verbalization. Self observing or monitoring processes closely follow the self control processes. Self monitoring involves tracking eir methods and strategies (Cleary & Zimmerman, 2012). These processes inform students about their progress toward their goals and motivate them to adjust their strategies and behaviors as necessary. Further, the extent to which students use these processe s is governed by the beliefs they hold about their own capabilities. Self efficacious students during their engagement with tasks use produc tive problem solving strategies and are more likely to monitor their performances as well as assess their progress t oward goals ( Schunk & Mullen, 2012 ). They work harder, persist longer, and persevere in difficult times (Pintri ch, Roeser, & DeGroot, 1994; Zimmerman & Martinez Pons, 1990 ). PAGE 47 47 Self reflection phase The self reflection phase is contingent upon cognitive and behavioral monitoring and tracking of the problem solving steps E ffective students improve their learning strategies and problem solving performance b y engaging in processes such as self judgment and self reaction Specifically, during this phase students evaluate their progress by comparing their current performances with their previous achievements and by tracking the ways in which they have improved (Zimmerman, 2000; Zimmerman & Campillo, 2003). Self evaluative judgments are closely linke d to the attributions students make for their successes and failures. These attribution judgments play a crucial role during the self reflection phase, as students who attribute the cause of their failures to low ability react negatively and refrain from e ngaging in the same task with greater effort. On the contrary, students who attribute their mistakes to poor problem solving strategies believe that they can correct their mistakes by improving their strategies. Further it is important to note that studen ts who are confident in their abilities attribute their poor performance to the lack of effort or strategy use. Additionally, they are more likely to revise their strategies and goals. Summary of the Self Regulation Processes The social cognit ive view of self regulated learning postulates that human functioning occurs as a result of the reciprocal interaction between personal, behavioral, and environment factors (Zimmerman, 2000). During this interaction, each of the three factors not only infl uences the other two factors but also is affected by them. The theory also hypothesizes that self regulatory processes and motivational beliefs are enacted within three cy clical phases of forethought, performance and self reflection (Zimmerman & Campillo, 2003) During the forethought phase, self efficacious problem PAGE 48 48 solvers analyze the task and plan strategically to accomplish self set goals. Performance phase progress toward their goals. Students who believe in their academi c abilities use effective learning strategies for learning the material as well as adjust their strategies and methods by engaging in self observation and self monitoring procedures Self reflection processes involve reactions to their learning o utcomes, especially how they perf ormed against a self set goal. Inconsistencies between the established goals and actual performance guide problem solvers to modify and revise their g o als and strategies, which places them into the next SRL cycle (Clearly & Zimmerman, 2012) Furthermore, these behaviors are structurally similar to the modeling behaviors exhibited by students who are engaged in the successive modeling cycles. During these modeling cycles, students iteratively test, revise, and modify their m athematical interpretations until they develop a model that adequately describes a modeling situation. Similar to self regulated learners, students engaged in c omplex modeling activities are typically required to analyze tasks, select appropriate mathemati cal concepts and operations to mathematize realistic situations, create and keep track of their models, and refine their interpretations iteratively when the created model(s) fails to predict the actual situation (Amit & Jan, 2010; English, 2006; Eric, 201 0; Mousoulides et al., 2010). Because self regulation involves proactive processes and beliefs to acquire self set goals (Zimmerman, 1989), SRL behaviors and motivational beliefs may eal world modeling problems. T he present study hypothesized that these behaviors and beliefs argues for PAGE 49 49 these relationships through an examination of associations between self efficacy beliefs, self regulatory strategies and problem solving (or modeling) achievement Self Regulation and Mathematical Problem Solving In this section, self efficacy beliefs and SRL strategies will be explained. The first subsection describes research studie s that explored correlations between self efficacy problem solving performance. Additionally, studies will be discussed that established correlations between self reported use of cognitive and metacogni tive strategies. The second sub section includes reported use of cognitive and metacognitive strategies and their performance on academic tasks Self Efficacy Beliefs and Mathematical Problem Solving As stated earlier, t he social cognitive theory highlights that human functioning occurs as a result of reciprocal interactions among personal variable s behavioral factors, and environmental influences. Further, the theory states that proac tive self reflecting and self regulated learners have the capacity to take control of their thoughts, feelings, and actions. According to Bandura (1997), self regulated learners display this sense of personal agency because of the beliefs they hold about themselves and their capabilities. The present study is interested in understanding the influence of self efficacy perceptions real world or modeling tasks. Research shows that self efficacy beliefs corr elate posit academic achievement and problem solving success ( Chen, 2003; Greene, Miller, Crowson, Duke, & Alley, 2004; Nicolidau & Philippou, 2004; Pajares, 1996; Pajares & Graham, 1999; Pajares & Kranzler, 1995; Pajares & Miller, 1994, Pajares & Valiante, 2001; Pintrich & DeGroot, 1990). For example, Pajares and Miller (1994) measured PAGE 50 50 mathematics confidence in 350 college undergraduate students Participants also reported their math ematics self concept, math ematics anxiety, perceived usefulness o f mathematics, prior experience (number of years/semesters in mathematics), and math ematics performance. math ematics performance was measured through a test composed of items from the National Longitudinal Study of Mathematics Abilities (Pajares & Miller, 1994) Correlations between all independent variables ( e.g., self confidence, self concept, math ematics anxiety, perceived usefulness, and prior experience) and math ematics performance were found to be significant. However, students beliefs about their capabilities to solve problems were found to be the most predictive of their actual ability. In fact, self efficacy had stronger direct effects on performance than any other variable. The previous study was replicated and extended by examinin g the role of self efficacy b eliefs in a high school setting and by taking into account students mental ability (Pajares & Kranzler, 1995). Two hundred and seventy three high school students reported their mathematics self efficacy and math ematics anxiety. performance was measured through a problem solving test consisting of 18 items focu sing on arithmetic, algebra, and geometry The correlations between math ematics self efficacy beliefs, math ematics ability, anxiety, prior experience, and math ematics performance were found to be significant. Moreover Pajares and Kranzler found stronger d irect effects of students efficacy beliefs on math ematical problem solving performance even after controlling for general mental ability. Pajares further reported PAGE 51 51 that self efficacy beliefs are helpful in predicting the problem solving performance of not only college and high school students but also middle grade students (Pajares & Graham, 1999). S elf efficacy beliefs are more strongly related solving achievement when other factors are studied within a statistical model (Nicolidau & Philippou, 2004). In this study, 238 fifth grade students reported their self efficacy beliefs and attitudes toward mathematics. solving achievement was measured in terms of their succ ess on a test consisting of 10 word problems and 10 r outine problems. In agreement with research findings discussed above, researchers found that self efficacy beliefs had solving performance ( = .55, p < .001) than their attitudes toward mathematics ( = .37, p < .001). Self efficacy beliefs not only influence problem solving achievement directly but strategies (Bouf fard Bou chard et al. 1991; Heidari et al. 2012 ; Pintrinch & DeGroot, 1990 ; Zimmerman & Bandura, 1994). In a study involving 173 seventh grade students Pintrich and De Groot (1990) examined the correlation s between orientation ( e.g., self ef ficacy beliefs ), self regulated learning strategy use ( e.g., cognitive, metacognitive and effort management strategies), and academic performance in science and English classrooms. The Motivated Strategies for Learning Questionnaire (MSLQ) consisti ng of 56 items was used to collect data related to self efficacy beliefs and SRL strategies. Academic achievement data were collected through igher academic self PAGE 52 52 efficacy was associated with studen performance. This implies that students who report high levels of academic self efficacy would also report using various cognitive and self regulative metacognitive strategies, and they were more likely to persi st under difficult learning activities. Self efficacy a cademic performance on seatwork, academic essays, and exams when cognitive variables were included in the statistical analyses Based on these fin dings self efficacy beliefs Zimmerman and Bandura (1984) also reported that self efficacy for writing influence well as indirectly through personal goal setting. Approximately 95 college undergraduates reported on two questionnaires that measure their self efficacy for writing as well as the extent to which they regulate their writing activities (e.g., planning, or ganizing, and revising compositions). Self efficacy and goal setting together accounted for 35% of the variance in academic achievement. These findings were supported in other subject areas (e.g., English) as well ( Greene et al., 2004) A total of 220 hi gh school students responded to a self report questionnaire measuring their self beliefs use of cognitive strategies, mastery goals, performance goals, and academic achievem ent in English classes S elf efficacy and use of meaningful strategies had the str ongest direct effect on students achievem ent among other variables such as mastery and performance goal orientations. Further, self meaningful strategies as well as indirectly affect students through their use of cognitive strategies. Supporting these findings, Bouffard Bouchard PAGE 53 53 et al. (1991) also reported a positive association between self efficacy beliefs and school junior and 44 high school senior students in nine comprehension tasks involving replacement of irrelevant words efficacy beliefs were correlated with several dependent variables such as the number of times they checked the ti me (monitoring of time), their persistence over a task, self evaluation, and their performance on the test. Students who believe in their English reading comprehension skills were found to display greater performance monitoring, task persistence as well as performed better on the comprehension tasks than students with low self efficacy beliefs. Nevill (2008) also found significant correlations between reading self efficacy beliefs and regulation of cognition. A convenience sample of 84 elementary students r eported their self efficacy beliefs on a reading scale called The Reader Self Perception s cale Behavior Rating Inventory of Executive Function achievement was measured on an oral reading fluency test. Nevill reported that students who we re self efficacious about their reading abilities we re more likely to regulate their thought pr ocesses than students who he ld low self efficacy beliefs in reading Similar findings were reported by another study involving 50 high school junior Iranian students majoring in English translation ( Heidari et al. 2012 ). Students responded to a self efficacy belief questionnaire and vocabulary learning strategy questionnaire. High self efficacy beliefs significantly correlated with more diverse use of vocabulary learning strategies. Based on these results, researchers concluded that PAGE 54 54 students who believed they were capable of reading would use vocabulary strategies more frequently a nd effectively. Taking into account the positive correlations established between students perceived self efficacy beliefs and academic performance, the present study extend ed past research by e xploring the influence of self efficacy beliefs performance in solving modeling tasks The indirect effects of self efficacy beliefs on were also investigated. The next section describes associations between SRL stra tegies (e.g., cognitive and metacognitive strategy use) and mathematical problem solving success. Cognitive and Metacognitive Strategies solve academic tasks (Garcia & Pintrich, 1994). Zimmerman and Martinez Pons (1986) actions directed at acquiring information or skill that involve agency, purpose (goals), and instrumentality s elf perceptions b y a learner Although self regulated students util ize various strategies, the present study focused on the use of cogniti ve and metacognitive strategies. Cognitive strategies are learning organization and critical thi nking (Pape & Wang, 2003; Pintrich & De Groot, 1990; Pintrich et al., 1993; Zimmerman & Mar tinez Pons, 1986, 1988, 1990). Elaborative strategies are higher order learning str ategies that support students' acquisition of information by in tegrating new mater ial with existing knowledge (Ormord, 2008; Schunk & Zimmerman, 1998). Some elaborative strategies used by effective problem solvers include paraphrasing or summarizing material, creating analogies, productive note PAGE 55 55 taking, explaining the material to others, and asking or answering questions to clarify understanding or improving comprehension (Kitsantas & Dabbagh, 2010). These strategies may be useful to understand modeling activities, where students are required to make sense of real world situations using t heir current mathematical knowledge and skills. Similar to elaborative strategies, organizational strategies also help students in building connections between different ideas as well as to arrange the material meaningfully (Ormord, 2008). Organizational strategies such as selecting and outlining important ideas or topics, developing concept maps, or representing concepts graphically support students in distinguishing relevant from irrelevant material and in placing similar ideas together. These strategi es may be helpful in mathematizing realistic modeling tasks, which include sorting, quantifying, organizing, and selecting large data sets. In addition to cognitive strategies, metacognitive strategies may influence students solving performance (Pape & Wang, 2003; Pintrich & De Groot, 1990; Pintrich et al., 1993; Zimmerman & Martinez Pons, 1986, 1988, 1990). Metacognitive strategies are typically comprise d of three different types of processes including planning goals, monitoring acti ons, and regulating strategies or methods. Planning strategies assist students in analyzing the task as well as setting appropriate goals, especially by activating existing knowledge and experiences. These strategies may help students in organi zing their t houghts and selecting concepts and strategies useful in understanding modeling tasks PAGE 56 56 Monitoring strategies focus students keep track of their strategies and actions. Students may monitor their work by tracking t heir attention while working on a task or using questions to check their understanding of the task These strategies may help students in evaluating their thinking about a modeling situation or finding limitations in their models Regulating strategies are closely connected to the monito ring strategies, as they involve students evaluative judgments made using monitoring strategies. These strategies may help students in improving their models because they encourage them to modify their stra tegies or methods of inquiry by acquiring more information or reviewing initial models Similar to academic self efficacy, learning strategies also influence the cyclical phases of self regulation. Specifically, strategic students analyze the task, develo p learning plans to achieve their academic goals, choose task appropriate strategies, and organize, monitor and regulate their thought processes throughout the three phases. Several researchers have reported that cognitive and metacognitive strategy use i s problems solving and academic performance (Pape & Wang, 2003; Pintrich, 1989; Pintrich & De Groot, 1990; Verschaffel et al., 1999; Zimmerman & Martinez Pons, 1986, 1988, 1990). For example, Pintrich and De Gr oot (1990) reported that seventh cognitive (e.g., elaboration, organization and critical thinking) and self regulative metacognitive strategies were significant predictors of their academic performance in science and English classes. Zimmerman and Martinez Pons (1986) also reported similar results in a study with 40 high school students. They explored the learning PAGE 57 57 strategies used by high and low achieving students during learning. Participants reported use of learning strate gies in relation to six problem situations were categorized into 14 self regulated behaviors, such as self evaluation, organization and transformation, goal setting and planning, keeping records, and monitoring. Researchers found that high and low achievi ng groups differed significantly in their strategy use, frequency of using each strategy, and consistency of using a particular strategy. Specifically, high achieving students reported greater use of all SRL strategies such as organizing, transforming, mai ntain in g records, and monitoring. In a study with middle grade students, Pape and Wang (2003) examined sixth and seventh academic achievement, problem solving behaviors, and prob lem solving success. On a self report Strategy Questionnaire adapted from Zimmerman and Martinez study, students reported the strategies they used during reading and mathematical problem solving situations, frequency of using these strategies and confidence in using the se strategies solving behavior and success in problem solving were examined by engaging them in videotaped think aloud sessions. The high and low achievement group students did not differ signific antly in relation to the number of strategies used, frequency of using each strategy, and confidence ratings. However, high achieving students used more sophisticated strategies than low achieving students with respect to mathematical problem solving situa tions. For example, they solved mathematics problems by understanding the context of the problem as well as by transforming information into meaningful representations. PAGE 58 58 studie such as sorting, quantifying, categorizing, dimensionalizing, and weighting data, in simplifying situations and building models (Doerr & English, 2003; English, 2006; Eric, 2009, Mousoulides, Pitt alis, Christou, & Sriraman, 2010). English (2006) examined sixth mathematization processes by understanding how they modeled a situation involving creation of a consumer guide for deciding the best snack chip. During the study, students wer e engaged in whole class discussion s consumers, various consumer items, criteria that consumers might consider in purchasing an item Students sort and organize data by identifying and ranking important factors related to the snack (e.g., chip size, cost, freshness, moistness, crunchiness, guarantee, and quality), assigning and negotiating ratings, quantifying qualitative data such as taste, raising sample issues, and rev ising strategies repeatedly to prepare a consumer guide. Such actions and behaviors may require the use of cognitive strategies. Specifically, students may need to use elaboration strategies to select relevant material, organization strategies to arrange i nformation meaningfully, and critical thinking strategies to critically interpret quantitative data. Further, decisions involving revision s of models may involve critical and logical decision making. Mousoulides et al. (2010) showed that older students (e. g., 8 th grade) are more likely to employ superior mathematization processes and produce more efficient models in comparison to younger students (e.g., 6 th grade). Researchers in this study compared and contrasted the modeling and mathematization processes of sixth ( n = 19) and PAGE 59 59 eighth grade ( n required them to choose three part time and three full time vendors based on the number of hours worked and money collected by nine vendors. Studen ts in both the grade s engaged in several mathematization processes to organize and explore data as well as to rank and select employees. However, eighth grade students presented more refined and sophisticated models as they considered all the relevant vari ables and possible relationships to identify patterns and relations. Further, 6 th grade students did not verify their models in the real world context whereas 8 th graders interpret ed their models several number of times within the context of the situation to select employees. These students also indulged in metacognitive activity such as reflecting and revising models based on the suggestions provided by others (e.g., teacher or teammates) Further, the time spent (four 40 minute sessions) by students in b oth the groups to find a solution for the problem indicated that modeling problems require a lot of time and effort. Furthermore, the iterative process of describing, testing, and revising models requires the use of metacognitive skills such as planning, monitoring, and revising importance of metacognitive knowledge and str ategies during mathematical modeling can be found in studies conducted by Kramarski and his colleagues investigating the influence of metacognitive instruction in understanding real world problems (e.g., Mevarech & Kramarski, 1997; Kramarski, Mevarech, & A rami, 2002; Kramarski, 2004 ) In an attempt to support students in understanding authentic mathematical tasks, PAGE 60 60 Kramarski et al. (2002) engaged seventh grade ( N = 91) students in two different types of learning environment s, cooperative learning with metacognitive instruction and cooperative learning with no metacognitive instruction. In the cooperative metacognitive condition, students were provided experience in using metacognitive questions to discuss an d solve standard math ematics problems. Metacognitive questions focused on the cogn itive processes of comprehending the problem (e.g., what is the problem all about?) constructing connection s between new and previously solved problems (e.g. what are the similarities or differences between the current and already solved problem?), using appropriate strategies (e.g., what are the strategies/tactics/principles appropriate for solving the problem and why?), and reflecting (e.g., what did I do wrong here?). S tudents in the cooperative learning group discussed the problems in a group without undergoing any training on using metacognitive questions. Each student shared his/her solution process with the whole group and discussed the problem collectively to provide a common solution. T he effect of the co operative metacognitive learning environment was evaluated by testing students on authentic and standard mathematical tasks prior to and following the intervention T he authentic tasks used by Kramarski and his colleagues align very closely with the decision making tasks used in the PISA 2003 assessment. For examp le, the posttest problem involved ordering pizzas for a party after considering a variety of information such as the price of the pizza, size of the pizza and the number of toppings. T hese t asks required students to make a decision after taking into account a host of different factors as well as considering several constraints. The standard mathematical PAGE 61 61 test involved 41 multiple choice questions focusing on whole numbers, fractions, decimals and percent age s. Students in the cooperative metacognition group scored better on authentic and standard tasks and they provided better justifications than the students in the cooperative group. Further, b oth high and low achievers in the cooperative meta cognitive group outperformed their peers in the cooperative group. In a follow up study, Kramarski (2004) examined the effect of cooperative metacognitive environment to construct and interpret graphs Eighth grade students were eith er placed into cooperative learning environment s or cooperative metacognitive learning environment s In each environment, students learned several graphical concepts including understanding of slope, intersection point and rate of change as well as variou s methods of interpreting graphs M etacognitive instruction significantly enhances students to construct and interpret graphs. The students trained in metacognitive instruction were also found to possess less alternative conceptions about graph in terpretation In contrast to these experimental studies, Magiera and Zawojewski (2011) investigated the influence of metacognitive knowledge using an exploratory approach They claimed that small group mathematical modeling provide s contexts for activatin g metacognitive aspects of thinking because during these discussions students interpret diverse perspectives of group members, explain and justify their own reasoning, and seek mathematical consensus. Magiera and Zawojewski examined the metacognitive behaviors of three ninth grade students within a coll aborative learning environment where they collectively solved five modeling problems. After the problem solving was complete students watched the video records of each modeli ng session to explain and PAGE 62 62 justify their thought processes about understanding and solving modeling problems into metacognitive awareness, evaluation, and regulation. The f requency of occurrence of each behavior was also noted. Students predominantly engaged in metacognitive evaluation and regulation followed by awareness of their thought processes. Based on this literature, the present study hypothesized the positive effect of self reported use of Summary In this secti on, three components of SRL: S elf efficacy beliefs, cognitive strategies, and metacogn itive strategies were described along with their inf luence on students solving performance and achievement Students who believe in their competence set challenging goals, select productive strategies, persist longer and expend more effort toward academic tasks (Schunk & Pajares, 2008) E fficacious students are also more likely to correctly solve problem solving tasks (Pajares & Miller, 1995) C ognitive and metacognitive strategies, which support students in organizing though t processes as well as in planning, monitoring and regulating problem solv ing behaviors were also described Several studies have found self reported use of cognitive and metacognitive strategies to be correlated solving achievement (Pape & Wang, 2003; Pintrich & De Groot, 1990; Zimmerman & Martinez Pons, 1986, 1988, 1990) T he use of metacognitive questions has been found to be useful i n solving authentic and real world mathematical tasks ( Mevarech & Kramarski, 1997; Kramarski et al. 2002; Kramarski, 2004 ) Further, academic self efficacy facilitate s the use of cognitive and metacognitive strategies (Bouffard Bouchard et al., 1991; Pintrich & DeGroot, 1990) Based on the review of literature, this study examine d PAGE 63 63 associations between self efficacy beliefs, cognitive and metacognitive strategy use solving three different types of modeling problems including decision making, system analysis and design and troubleshooting PAGE 64 64 Figure 2 1 Modeling cycles often involves four basic steps including descript ion, manipulation, prediction and verificati on (Lesh & Doerr, 2003, p. 17) PAGE 65 65 Figure 2 2 Phases of self r egulation (Zimmerman, 2002, p.13) PAGE 66 66 CHAPTER 3 METHOD Introduction The goal of this study was to gain un derstanding of the relationships between SRL and mathematical modeling by examining how self efficac y beliefs, cognitive strategies, and metacognitive strategies we re associated with success rates on solving modeling tasks. Based on the purpose o f the present study, three key research questions were investigated. Research Questions 1. What are the direct effects of students efficacy beliefs for modeli ng tasks on their performance on modeling tasks? 2. What are the direct effects of students self reported use of cognitive and metacognitive strategies on their performance on modeling tasks? 3. What a re the indirect effects of students efficacy beliefs for modeling tasks on their performance on modeling tasks through their effects on their use of cognitive and metacognitive strategies? Research Hypotheses Past research studies have reported a positive correlation between self efficacy beliefs and problem solving achievement (Hoffman & Spatariu, 2008; Pajares, 1996; Pajares & Graham, 1999; Pajares & Kranzler, 1995; Pajares & Miller, 1994, Pintrich & DeGroot, 1990). These studies provide evidence that beliefs about their competence are a significant predictor of their problem solving success even after controlling for mental ability (Pajare s & Kranzler, 1995) and mathematics anxiety (Pajares & Graham, 1999) Efficacy judgments positively influence students engagement and persistence with complex tasks as well as amount of cognitive effort exerted during problem sol ving activities (Schunk & Mullen 2012; Schunk & Pajares, PAGE 67 67 2009). Self regulated learning strategies such as cognitive and metacognitive strategies academic achievement and problem solving skills ( Pape & Wang, 2003; Pintrich & De Groot, 1990; Pintrich et al., 1993; Zimmerman & Bandura, 199 4; Zimmerman, & Martinez Pons, 1986, 1988, 1990). Cognitive strategies such as elaboration, organization, and critical thinking influence processing of information, which further helps them to bette r understand the problem and create superior solutions. Pintrich and colleagues ( e.g., Pintric h, 1989; Pintrich et al., 1991) found that students who report using more cognitive and metacognitive strategies solve more problem s olving tasks correctly and receive higher grades Further, they reported that students who believe in their abilities are mor e cognitively engaged and display greater use of cognitive and metacognitive strategies in solving mathematics problems (Pintrich & De Groot, 1 990). Similar results were reported by Bouffa rd Bouchard et al. (1991), who found that high school students with high self efficacy beliefs for academic tasks displayed greater monitoring on academic performance and persisted longer than students with low s elf efficacy beliefs. Given the literature in the field, a statistical model was developed that self efficacy beliefs, cognitive and metacognitive strategy use to their performance on a modeling test ( see Figur e 3 1). B ased on the literature three hypotheses were proposed for this study. First, it wa s hypothesized that students efficacy beliefs for the modeling tasks would have a positive direct influence on their ability to correctly solve problems on th e modeling test. Second, it wa s hypothesized that students reported use of cognitive and metacognitive strategies wou l d directly influence their performance on the modeling PAGE 68 68 test Third, it wa efficacy beliefs for mod eling tasks wou l d have a positive indirect influence on their performance on modeling tasks through the positive effect on their use of cognitive and metacognitive strategies. Pilot Study In order to answer these research questions, two major steps were ta ken related to data collection procedures. First, problems from the PISA problem solving assessment were revised to contextualize them within partic surroundings Second, a Modeling Self Efficacy scale was developed to collect data about p conducted to test the psychometric proper ties of this scale including item analysis, internal consistency, content validity, and construct validity. Participants One hundred and fifty one 10 th grade students were selected through convenience sampling from three different locations. Ou t of these, 91 students between the ages of 15 and 18 were engaged from a local research developmental school 46 rising tenth graders between the ages of 14 and 15 were engaged from a summer science camp hosted b between the ages of 14 and 18 were involved from a summer camp program organized at a local community college. The total sample co nsisted of 17 fourteen year old, 37 fifteen year old, 34 sixteen year old, 37 seventeen year old, and 26 eighteen year old students. The mean age of all the participants was 16.18 years ( SD = 1.28). The sample included about 60% females ( n = 90) and 40% males ( n identified ethnicity included 53.6% White ( n = 81), 18.5% African American ( n = 28), PAGE 69 69 16.6% Hispanic ( n = 25), 7.3% Asian ( n = 11), 0.7% Native Hawaiian ( n = 1), and 3.3% others ( n = 5). Measure The Modeling Self Efficacy surve y consisted of nine modeling problems (see Appendix A). Each modeling problem was followed by four self efficacy questions determining the information, and correctly solvi ng the modeling problem Following Bandura (2006) recommendation, s tudents rated their confidence on a 100 point scale ranging from 0 ( not at all sure ) to 100 ( very sure ) (see Appendix B ) Procedure Students were invited to participate in the study by p roviding them information about the purpose of the study, tasks involved, the benefits and risks involved in joining this study, and the confidentiality of their responses. Students who returned signed consent forms were issued alphanumeric codes to mainta in anonymity. Before administering the questionnaire, the researcher highlight ed the importance of reporting accurate efficacy judgments and request ed students to provide their honest opinions. Participants completed the self efficacy survey in approximately 25 minutes. It is important to note that participants did not solve any of the modeling problems. They reported their judgments for understanding and solving modeling problems after reading them. Additionally, five students from the research developmental school with different ability levels were e ngaged in think aloud interviews to a scertain that students understoo d these problems and could solve them. These students were selected based on the recommendations of the classroom teachers. PAGE 70 70 Data Analysis Descriptive analysis of the scale (see Table 3 reported high levels of self confidence in understanding and solving the Cinema Outing ( M = 87.78, SD = 13.84), Library System ( M = 87.72, SD = 17.28), and amp ( M = 886.48, SD = 12.80) problems. They reported almost similar confidence ratings for the Hospital ( M = 85.0, SD = 15.92), Holiday ( M = 84.18, SD = 17.30), and Energy Needs ( M = 83.40, SD = 18.85) problems. Students appeared to be least confident in s olving the Course Design ( M = 81.92, SD = 16.40), Irrigation ( M = 77.86, SD = 18.34), and Freezer ( M = 77.85, SD = 20.29) problems. Internal consistency estimates ensure that across items when a single form of a test is administered (Kline, 2005) It is important to have high internal consistency among the items within a scale so that all the items are efficacy beliefs for modeling tasks. The full scale was found to have Cronbac alpha coefficient equal to .89. Additionally, item total correlation analyses were performed to ensure that all the items on the scale were homogeneous. Kline (2000) suggested .30 as an acceptable corrected item total correlation for the inclusion of an i tem. Item total correlations (see Table 3 2) for the scale, ranged from .48 to .77 suggesting that all the items were adequately measuring a single underlying construct. Factor analysis was performed to est ablish con struct validity that also means identi fying any underlying association between the items on the scale. Principal Component Analysis with varimax rotation indicated a single factor model because the first factor accounted f or 54.5 modeling ta sks in comparison to the second f actor that only accounted for 9.3 % of the PAGE 71 71 total variance. The scree plot showed that the second (9.3%) and third (8.5%) factors were similar in magnitude (see Figure 3 2). Further, inspection of the component matrix table s howed that all items load strongly on the single underlying construct (all factor loadings were higher than .67). Content validity measures whether the wording and format of the questions on a scale are consistent with the construct of interest. The items on the self efficacy scale were reviewed and verified by experts in the field including researchers familiar with the psychological construct and people with measurement expertise. Further, five students from a research developmental school with differe nt ability levels were engaged in think aloud interviews during which they were encouraged to verbalize the steps taken by them in solving modeling tasks. The major purpose of the think aloud interview was to ascertain whether the modeling tasks were suffi ciently aloud interviews revealed that problems on the modeling test were not very challenging for tenth grade students between 16 to 18 years of age. Accordingly, four students fr om lower grade levels, specifically 8 th and 9 th grade were selected to solve the modeling problems. These students were interviewed individually and were encouraged to share their thought processes about the strategies they used in understanding and solvi ng these problems. Since the modeling problems were found to be sufficiently challenging for eighth and ninth grade students, it was decided to recruit eighth and ninth grade students between 13 to 15 years of age for the dissertation study. PAGE 72 72 Research De sign The p resent study measure d the degre e of association between self efficacy beliefs, self reported use of cognitive and metacognitive strategies, and performance on the modeling test. Thus, this study followed a correlation al research design to determine wh ether and to what degree the variables involved in the study were related to one an other (Clark & Creswell, 2010). I t is important to note that correlational research is not causal in nature. As such, no attempts were made to establish ca use effect relationships among the variables. Data were collected in the form of self report questionnaire s and responses on a modeling test. S tructural equation modeling (SEM) techniques were used to explain relationships among the variables und er investigation (Byrne, 2012; Kline, 2005/2011). SEM helps to estimate and test direct and indirect effects of the latent variables involved in a system through a series of regression equations. Latent variables, such as self efficac y beliefs, cognitive s trategies, and met acognitive strategies are constructs that can neither be observed nor measured directly. Rather, t hey are indirectly measured by using observed variable s that reflect different characteristic s of the desired construct. For example, in th is study students meta cognitive strategies was measured indirectly through their ratings on nine items measuring their ability to plan, monitor, and regulate goals or problem solving strategies. The structural relationships among the variables can be represented in the form of a statistical model (see Figure 3 1) Determination of Minimum Sample Size Determining appropriate sample sizes for research studies is crucial to detect ing statistically signifi cant relation s if they exist. Meeting the crite ria for the minimum sample size decreases the probability of committing a Type II error (failing to detect relations PAGE 73 73 among the variables when they do exist) or increases the power of a study. With regard to SEM and confirmatory factor analysis, there is li ttle consensus in the research community concerning minimum sample size requirements (Kline, 2011; Mundfrom, Shaw & Ke, 2005). Various methods have been suggested in this regard such as a minimum sample size approach of 200 participants, es timating sample sizes by using the N:q rule, where N is the number of participants and q is the number of parameters included in the statistical model, or through conducting power analysis (Jackson, 2003; Kline, 1998, 2005, 2011; Marsh, Balla, & McDonald, 1988; Mundfrom et al., 2005). In determining sample sizes through the N:q rule, it is unclear how many participants (e.g., 20, 10 or 5) should be selected for each statistical variable (Kline, 2005/2011; Jackson, 2001/2003). Kline (2005/2011) suggested that sample sizes greater than 200 are large enough for statistical model testing as well as to obtain a desired level of statistical recruit ed 225 eighth and ninth graders from a local research developmental sch ool university. Method Participants A total of 325 eighth and ninth grade students in 13 classrooms were invited to participate in this study. Out of these, 236 (72.6%) students returned the signed parent consent and stude nt assent forms. Eleven students were absent on the day the questionnaires were administered. Thus, 225 eighth ( n = 88, 39.11%) and ninth grade ( n = 137, 60.8%) students participated in the study. The average age of the participants was 14.22 with a stand ard deviation of 0.85. The number of female students ( n = 122, 54.2%) was slightly higher than the number of male students ( n = 103, 45.8%). PAGE 74 74 Participants reported their ethnicity as White ( n = 111, 49.3%), African American ( n = 46, 20.44%), Hispanic ( n = 3 3, 14.6%), Asian ( n = 12, 5.33%), and Native Hawaiian ( n = 1, 0.44%). The remaining students reflected their ethnic background as either a combination of these categories ( n n = 3, 1.33%). Measures T hree instruments were used to measure the desired constructs including: (1 ) a se lf efficacy scale developed ments on the modeling tasks, (2 ) a modified version of the MSLQ as developed for Connected Classroom in Promoting Mathematics (CCMS) project an d use of cognitive and me tacognitive strategies, and (3 ) a modeling test adapted from PISA 2003 problem solving items to measure students modeling outcomes. Self efficacy scale The self efficacy scale assess ed stud problems. Students provided judgments of their perceived capability to correctly solve modeling problem s after reading each problem on the test. Specifically, they respond ed to four questions that f efficacy judgments including, 1. How sure are you that you can understand this mathematical problem? 2. How sure are you that you can determine a strategy to solve this problem? 3. How sure are you that you can determine the information required to solve this problem? 4. How sure are you that you can solve this mathematical problem correctly? Students record ed the strength of their efficacy beliefs on a 100 point scale, divided into 10 unit inter vals ranging from 0 ( not at all sure ) to 100 ( very sure ) (see Appendix B ). Psychometric properties of the Modeling Self Efficacy scale tested during the pilot study sug gest that the scale has high internal consistency ( = .89). PAGE 75 75 Motivated Strategies for the Learning Questionnaire (MSLQ) The MSLQ is a self motivational orientation and their use of learning strategie s in studying material for a college course (Pintrich et al., 1991; Pintrich et al., 1993). The motivation and learning strategies section together represent 15 subscales (or constructs) with a total of 81 items The present study used four sub scales from the learning strategies section elaboration, organization, critical thinking, and metacognitive self regulation. Students reported their cognitive and metacognitive strategy use o n a seven point Likert scale from Higher scores indicate greater levels of the constructs being measured or greater reported strategy use. The MSLQ is a widely used questionnaire t hat has been validated by a variety of empirical studies. Pintrich and his colleagues (1991) claim that the MSLQ scales can be used collective ly as well as independently. Since the original MSLQ instrument was developed for colleg e students, the present s tudy use d a version of the questionnaire that was modified for the CCMS project and used by Kaya (2007) Modifications in this questionnaire were made to meet the cognitive level s of middle grade students Also, some items were reworded to reflect motivati onal beliefs and use of learning strategies in reference to mathematics. The modified questionnaire included 67 items related to student motivation, cognitive and metacognitive strategy use, and management strategies (Kaya, 2007). Because the present study focused items related to cognitive strategy use and 9 items concerned with metac ognitive strategy use were included in the self report questionnaire (see Appendix C ). Kaya PAGE 76 76 (2007) adminis tered the modified MSLQ to 1,626 Algebra I students to test the in ternal consistency of the scale and the reliability estimates indicate d high item total correlations. The e lab oration (6 items, = .78 ), organization (4 items, = .73), critical thinking ( 5 items, = .76), and metacognitive self regulation ( 9 items, = .83) Further, three problematic items on the original metacognitive self regulation sub scale of the MSLQ were deleted (Kaya, 2007). Cogn itive s trategies. The cognitive strategies section includes 15 items across three sub scales including elaboration, organization, and critical thinking (Kaya, 2007) (s ee Table 3 3 ). The elaboration sub learning st rategies such as paraphrasing, summarizing, and note taking. These strategies support learners to process information more deeply through translating new information into their own words and creating mental models of a problem by associat ing the informatio n given in a problem to their existing knowledge (Ormrod, 2008 ). S use of elaborative strategies was measured through six items (e.g., I try to relate ideas in this subject to those in other courses whenever possible ). Elaboration strategies may be helpful in solving all three types of problems chosen for the PISA problem solving assessment. Relating the problem solving situations to what students already know may help them to solve decision making, system analysis and design, and troubleshooting pr oblems. The o rganization subscale consisting of four items measure d the extent to which students use learning strategies such as clustering, outlining, and selecting main ideas to differentiate relevant and irrelevant information (e.g., I make simple charts, diagrams, PAGE 77 77 or tables to help me organize course material). These strategies may be useful in solving decision making tasks that require students to identify the relevant alternatives and constraints. Organization strategies may also be helpful in solving system analysis and design problems, where students represent rela tionships among different parts of a system in the form of a table or a chart. Finally, the critical thinking sub scale includes five items that measure d the degree to which learners apply their prior knowledge and skills to think ing logically about new si tuations (e.g., I treat the course material as a starting point and try to develop my own ideas about it ). Critical thinking skills are considered to be the most important skills for solving modeling tasks as these strategies support learners to think anal ytically consider alternative conceptions of a problem, make effective decisions, reason deductively as well as justify their reasoning Metacognitive s trategies. T he metacognitive self regulation sub scale consisting of nine items measured the extent to w hich students (1 ) plan their goals or activities (e.g., Before I study new material thoroughly, I often skim it to see how it is organized), (2 ) monitor their actions to enhance attention and to self evaluate their progress (e.g., I ask myself questions to make sure I understand the material I have been s tudying in this class), and (3 ) regulate their cognitive strategies and goals (e.g., When studying for Items included on the metacogni tive self regulation scale are provided in Table 3 4 Prominent researchers in the field of mathematical modeling reported that metacognitive strategies such as planning, monitoring and regulating strategies support students in self evaluating the ef fecti veness of their models, creating revised models, and describing PAGE 78 78 situations using models ( Blum, 2011; Magiera & Zawojewski, 2011; L esh, Lester & Hjalmarson, 2003) The modeling test The third instrument used in this study was a test comp osed of six real wo rld situations modeling success competence (see App endix D ). The modeling test was developed by adapting problems from the PISA 2003 problem solving assessment (OECD, 2004). These problems were selected because researchers in the field of mathematical modeling often regard PISA problems as complex modeling tasks ( Blum, 2011; Carriera, Amado, & Lecoq, 2011; Maa 2011; Mousoulides, 2007 ; Mousoulides, Christou, & Sriraman, 2008 ). The PISA 2003 problems ha ve been empirically examined and validated with students from 41 countries and the overall reliability of the problem solving scale from which these items were adapted was very high ( = .87 ) The modeling performance throu gh three different types of tasks: decision making, system analysis and design, and troubleshooting. It included six modeling problems, with t wo problems for each type of task. The problem solving (or modeling) processes involved in solving decision makin g, system analysis and design, and troubleshooting tasks include understanding the problem, characterizing the problem, representing the problem, solving the problem, reflecting on the solution, and communicating the solution (see Figure 3 3). Understand in g the problem is very similar to the description process of the modeling cycle. It involves making sense of the context and information given in the problem (e.g., text, diagrams, formulas or the tabular data) by utilizing prior knowledge and experiences. Characterizing includes identifying relevant variables involved in the PAGE 79 79 problem and hypotheses; and retrieving, organizing, considering, and critically evaluating contextual OECD, 2004, p. 27). Further, learners establish relationships between the variables by representing the situation in tabular, graphical, symbolic, or verbal forms. In order to successfully solve these problems, students need to make predictions about real world problems. This includes making a ppropriate decision s in the case of decision making tasks, analyzing or designing system s in the case of system analysis and design tasks, and diagnosing faulty system s in the case of troubleshooting tasks Verification involves evaluating results within the context of the real world situation (OECD, 2004). Finally, communication involves selecting eff ective methods of communication to report solutions such as choosing appropriate forms of media and represent ations. The decision making tasks measure d the extent to which students could make appropriate decisions by choosing strategically among several alternatives provided under a given set of conditions The decision making skills were tested through Cinema O uting and Energy Needs (see Appendix D ) problems These problems involve d a variety of information, and students were required to understand and provide solutions to these problems by identifying the constraints given in the situation, translating the info rmation into meaningful representations, and making a decision after systematically considering all the alternatives and constraints (OECD, 2004). For example, the Energy Needs problem required students to select suitable food for a person after calculatin g his/her required daily energy needs. In order to calculate the energy needs of a person, students need ed to integrate two or more pieces of information such as age, gender, PAGE 80 80 activity level, and occupation of a person. to make accurate de cisions is largely affected by the number of factors pr esent in a problem, especially in separating the relevant from the irrelevant information. The system analysis and design tasks required students to identify complex relationships among the variables o r to design a system by satisfying all the conditions given in a problem. The two system analysis and design problems included in the test were and Course Design (see Appendix D ). Similar to decision making tasks, system analysis and design problems i nvolved a variety of information and students need ed to sort through the information in order to depict relationships among the variables. But unlike decision making problems, all the alternatives were not given and the constraints were not obvious. For ex ample, the problem involved assignment of children and teachers into different dormitories by matching the capacity of eac h dorm with the number and gender of the people It required thorough understanding of the context of the situation, list of adults and children and knowledge of the dormitory rules. As such, these problems require d students to think logically and critica lly about all the variables as well as constantly monitor, reflect, and adjust their actions. The third type of task, troubleshooting problems, required students to diagnose, rectify, and improve a faulty or underperforming system. The modeling test inclu ded two trouble shooting problems, Irrigation and Freezer (see Appendix D ). In order to solve these problems, students need ed to understand the main components of a system as Additionally, the y were required to understand how different components of a system interact with each other PAGE 81 81 causally (OECD, 2004). Based on this understanding, students we re required to di agnose a malfunction of a system and propose an appropriate solution. Students could communicate their recommendations by either drawing a diagram or writing a problem solution report. For example, the Freezer problem was a trouble shooting item, where students need ed to diagnose the probable cause of a malfunctioning freezer based on sev eral variables such as the knowledge of the manual, the functioning of the warning light, the state of the temperature control, and external indications about the freezer motor. It is important to note that all six problems on the modeling test were either open ended having more than one correct solution or in multiple choice format Procedure Data Collection The present study was conducted during the fall of 2012 at a developmental research school. Students were recruited following the approval of procedures from the University of Florida Institutional Review Board All the students complete d the MSLQ questionnaire a s well as solv e d modeling problems after rating their confidence for solving these problems. Both the questionnaires and modeling test were administered during regular class periods in two sittings. The MSLQ survey took approximately 10 15 minutes to complete. Before the administration of the questionnaire, students were instructed to respond to the items with reference to their mathematics classroom. Students took approximately 15 minutes in rating their confidence in solving modeling t asks Finally, they solve d problems on the modeling test in approximately 30 minutes. During the modeling test administration, students were encouraged to not only solve but also to provide justifications for their responses. The present study, however, did not PAGE 82 82 take into account s test. Data Analysis Scoring scheme the scoring system used in the PISA 2003 problem so lving assessment (see Ap pendix E ). Students could earn a maximum of 2 points for some problems (e.g., Cinema ), while other problems (e.g., Irrigation, Freezer ) were worth a maximum of 1 point. As such, they could earn a max imum of 10 points on the modeling test. The Cinema Outing problem required students to identify movies that three friends could watch together upon analyz ing the duration and show times for each movie. Students received a maximum score of 2 for correctly c choice questions, and a partial score of 1 for answering all but one of the questions correctly. They received zero points for incorrectly answering more than two multiple choice questions. The Energy Needs pr oblem required students to suggest a suitable food for a person that aligns with his or her energy needs. To receive full credit, students needed to show all the calculations including the total energy of the fixed price menu, sum of the fixed price menu a energy intake for the day, and difference between conclusion. Students received partial credit in two ways, either by showing all the calculations correctly bu t providing a wrong conclusion, or making a minor error in one of the calculations steps leading to a wrong conclusion. Students, however, did not get any credit for simply calculating total energy of the fixed price meal. PAGE 83 83 The was an open ended problem involving assign ment of adults and students to different dormitories based on the dormitory rules. A full credit response involved allocating people to eight dormitories after ensuring the total number of girls (e.g., 26), boys (e.g., 20), an d adults (e.g., 4 female s and 4 males) were equal to the required number. Further, the total number of people in each dormitory should not exceed the number of beds, adults and children in each dormitory should be of the same gender, and there should be at least one adult sleeping in each dormitory. Students receive d a partial credit of 1 point if they violate d at most two of the recommended six conditions. They did not receive any credit for violating more than two conditions. The Course Design problem required students to sequence 12 college courses over a three year period F ull credit involved listing subjects by satisfying the two recommended conditions. Students received partial credit of 1 point i f they list ed all the subjects in proper order excep t mechanics and e conomics. However, they received no credit for completing the whole table correctly but failing to put e lectronics courses (e.g., Electronics (I) and Electronics (II)) into the table The Irrigation problem required students to decide whet her the water would Students earn ed a full credit of 1 point on correctly answering all the three multiple choice questions. Unlike other problems, they could not ear n any partial credit for this problem. The Freezer is a multiple choice problem that required students to detect whether the warning light of a malfunctioning home freezer was working properly when the temperature was controlled at different positions In order to receive full credit, st udents needed to answer all three questions PAGE 84 84 correctly. Similar to the Irrigation problem, no partial credit was given if any of the three responses were incorrect. Scoring procedure The researcher and a fellow mathematics e ducation graduate student scored between the two raters to ensure the consistency of the implementation of the scoring rubric. It es tend to make exactly the same Tinsley & Weiss, 2000, p. 99). Interrater agreement s kappa and values higher than .80 are generally considered to be acceptable (Tinsley & Weiss, 2000). Th e present study found high interrater agreement between the two raters ( = .96). Descriptive analysis Descriptive analysis was conducted by reporting reliability estimates, patterns of missingness, and descriptive statistics for each construct. The relia bility estimate s for elaboration, organization, critical thinking, metacognitive self regulation subscales as well as self efficacy and modeling test were determined by calculating Cronb alpha. Coefficient alpha measures internal consistency of a scale that refers to the (Kline, 2005, p.59). In the social sciences, acceptable reliability estimates range from .70 t o .80 (Kline, 2005). Missing values analysis procedure including pattern of missing data was performed using the SPSS statistical software Specifically, univariate descriptive statistics including non missing values, means, standard deviations, and number a nd percent of missing values were computed. PAGE 85 85 Descriptive statistic s including mean scores fo r each subscale in the MSLQ a nd mean self efficacy scores for each modeling problem were calculated. By following the recommendations made in the MSLQ manual, mea n scores for the elaboration, organization, critical thinking, and metacognitive self regulation subscales were s across all the items within that subscale. Further, partici efficacy scores for th e six mod eling problems were calculated by efficacy questions (i.e., confidence in understanding the problem, determining information, determining strategies, and solving the problem). Further construc t validity for the MSLQ and Modeling Self Efficacy scale was examined through Confirmatory Factor Analysis (CFA). In CFA, relationships between the observed indicators and underlying latent constructs known as factor loadings, are specified a priori based on the review of the literature (Byrne, 2012) The hypothesized model is tested by examining goodness of fit indices such as, Chi square test statistics c omparative fit index (CFI), Tucker Lewis index (TLI) and root mean square erro r of approximation (R MSEA). In the present study, the CFA of the MSLQ was conducted by allowing elaboration, organization, critical thinking, and metacognitive self regulation items to load freely on their corresponding latent factors. The CFA of the Modeling Self Efficacy sca efficacy scores for each of the six modeling problems on the overall modeling self efficacy latent variable. Analyses Data were analyzed using SEM techniques. The statistical calculations such as estimating fit indices, errors and model parameters were performed using the Mplus version 7 program. The hypothesized statistical model was tested using weighted least PAGE 86 86 square means and variance adjusted (WLSMV) estimator. This estimator was selected beca use it produces accurate parameter estimates and unbiased standard errors with varying sample sizes ( N = 100 to 1,000) and models in which observed variables are measured on widely varying scales (Brown, 2006; Muthn & Muthn, 1998 2012). It also provides superior measurement model fit and more precise factor loadings with categorical data. SEM is a useful methodology to study relations among observed and unobserved (i.e., latent) variables in both experimental and non experimental settings (Byrne, 2009/20 12; Hoyle, 1995). Such relations, however, are not causal in nature (Kline, 2005). These relations can be represented in the form of a series of structural equations as well as depicted pictorially in the form of a struct ural model (Byrne, 2012). The hypot hesized structural model that guide d the present study is presented in Figure 3 1 self reported use of elaboration, organization, and critical thinking strategies. Thus students mean ratings for the elaboration, organization, and critical thinking subscales were loaded on the cognitive strategy latent variable. S use of metacognitive strategies was measured indirectly through their ratings on nine items measuring their se lf reported use of planning, monitoring, and regulating strategies. It is important to note that mean ratings for each metacognitive item were loaded on the overall meta cognitive latent variable. efficacy beliefs for modeling tasks were indirectly measured in terms of their confidence in solving decision making, system analysis and design, and troubleshooting tasks. Self efficacy beliefs for decision making system analysis and PAGE 87 87 the Cinema Outing and Energy Needs, Ch Camp and Course Design and Irrigation and Freezer problems re s pectively. Modeling outcomes were in directly making, system analysis and design, and troubleshooting tasks. Modeling success rates for decision making, system analysis and design, and troubleshooting tasks were calculated by averaging st scores in Irrigation and Freezer problems, respectively. SEM determines the extent to which the hypothesized model fits with a set of data obtained from a given sample The general structural equation model consists of two sub models: a measurement model and a structural model (Byrnes, 2012; Hoyle, 1995 ; Kline, 2005 ). The measurement model determines how well the latent variables are described by the observed variables. This model is analogous to confirmatory factor analysis because it indicates how each observed measure (e.g., items or subscales on the questionnaire) loads on a particular factor ( i.e. latent variable ) The second com ponent is the structural model, which defin es relations among the unobserved latent variables (s) In the present study, the struct ural model describes relationships between modeling self efficacy beliefs (exogenous latent variable) and cognitive strategies, metacognitive strategies, and modeling tasks (endogenous latent variables). Benefits of SEM. There are several benefits of usin g SEM over multivariate procedures such as Analysis of Variance (ANOVA) and multiple regressions (Byrne, PAGE 88 88 2012). First, SEM involves a confirmatory approach to the data analysis because relations among the variables are specified a priori based on the revi ew of the literature Second, SEM provides explicit estimates for the measurement errors, which are not assessed correctly using traditional multivariate procedures. Measurement errors are associated with observed variables, and accounting for such errors results in accurate estimation of the structural relations between the observed and latent variables. Third, SEM allows researchers to test several hypotheses and make inferences based on both latent and observed variables. Five basic s teps of SEM. SEM in volves five basic steps including model specification, model identification, model estimation, model testing, and model modification (Hoyle, 1995). Model specification involves proposing a model by reviewing relevant theory and liter ature (e.g., Figure 3 1 ). Specifically, it includes establishing observed variables that can appropriately measure the latent variables as well as defining relations between observed and latent variables. and A model is said to be identified if it ) there must be at least as many observations as free model parameters (df M ) every uno Structural models may be under identified, just identified, or over identified. If the number of free parameters exceeds the number of observations, a model is said to be u nder identified and cannot be estimated. A just identified model fits the data perfectly, as it involves only one possible set of values for the parameters. In general, over PAGE 89 89 identified models, in which the number of observations is more than the number of independent parameters, are preferred as they facilitate statistical model testing. The m odel estimation process yields parameter values such that the (i.e., residual) between the sample covariance matrix and the population covariance matrix implied by the model during this stage initial values are plugged in for all the parameters and then the model is estimated iteratively using an estimator, such as WLSMV, until the discrepancy between sample and population covariance matrix is minimum. This is also known as model convergence. The model fit test is one of the most crucial steps of SEM since it assesses the extent to which the observed data fit the proposed statistical model. SEM allows researc hers to test theoretical propositions by determining the goodness of fit between the hypothesized statistical model and the data collected from the population of interest. Byrne (2012) describes the model fitting procedure in SEM as follows: Data = Model + Residual variables, the observed and latent variables, and if possible relations between laten t variables as well. This model is generally hypothesized after the review of extant literature. data. As such, goodness of fit is the variance in the data that is not exp lained by the proposed model. PAGE 90 90 The model fi t is generally evaluated based on two broad criteria: (1 ) the g oodness of fit statistic and (2 ) the individual parameter estimate (Byrnes, 2012). Byrnes further recommends that the model should be examined based on several criteria. Some prominent goodness of fit statistics that indicate overall fit of the model include chi square test of model fit ( 2 ), the root mean square error of approximation (RMSEA), the c omparative fit index (CFI), and the Tucker Lewis inde x (TLI) The chi square index evaluates the discrepancy between the population covariance matrix and the sample covariance matrix This means as the chi square values increases, the fit of the model becomes worse. The null hypothesis in a chi square goodne ss of fit test states that the hypothesized model fits the data. In other words, factor loadings, factor variances and covariances, and residual variance for the model under study are valid (Byrne, 2012). M plus typically calculates this statistic as ( N ) F m in where N represents the sample size and F min is the minimum fit function (Byrne, 2012) The probability value associated with 2 determines the fitness between the hypothesized model and the model obtained from the sample population. It represents the likelihood that the chi square test statistic is greater than the 2 value when the null hypothesis is true. Thus higher p values ( p > .05) indicate closer fitness between the two types of models. It is important to note, however, that c hi large sample sizes frequently results in rejection of the hypothesized model. The effect of large sample sizes can be reduced by dividing the chi square index by the degrees of freedom (Kline, 2005). Higher correlations among obs erved variables also increase the probability of rejecting the null hypothesis (Miles & Shevlin, 2007). This occurs because higher correlations among the variables give greater power to the tested model causing PAGE 91 91 an increase in chi square index. It is there fore, recommended to check the results obtained from chi square with other f it indices such as CFI, TLI, or RMSEA. Both CFI and TLI are incremental indices of fit in SEM, which measure the relative improvement in fit of the hypothesized model in compariso n to the baseline model (Byrne, 2012). The baseline model is also called the null or independence model that assumes zero covarianc e among the observed variables (Kline, 2005 ). The values of CFI lie between .0 and 1.0, with va lues greater than .95 indicati ng that the population matrix fits closely with the hypothesized model (Byrne, 2012) TLI is called the nonnormed index since values lie beyond the norm al range of .0 to 1.0. Similar to CFI, TLI values close to .95 indicate that the hypothesized model is a good fitting model. chosen parameter estimates would fit the populat ion covarianc as cited in Byrne, 2012). It assumes that the model does not fit the sample data perfectly Unlike chi square, it is not sensitive to large sample sizes. RMSEA values less than .05 are considered a good fit, values in the range of .06 and .08 are considered a moderate fit, and val ues greater than .10 indicate poor fit. The goodness of fit statistics evaluate model fitness by concentrating on the model as a whole. On the other hand, the individual parameter estimates focus on 77). Specifical ly, parameter estimates assess the degree to which statistical estimates are consisten t with the proposed model, such as correct sign and si ze Values of the estimated parameters that fall beyond the required range, such as correlations greater PAGE 92 92 than 1.00 or negative variances, represent incorrect estimates. The estimated standard errors with extremely large and small values also indicate poor m odel fit. Model modification generally occurs when the original mode l does not fit the data as indicated by the goodness of fit indices. It involves adding or removing statistical paths as suggested by the residuals and modification indices (MI) obtained from running the original model (Hoyle, 1995 ). Byrne further specifie d that statistical paths in the proposed model should not be modified solely on the basis of modification indices, the suggested paths should also be theoretically appropriate. Assumptions of SEM. There are two basic assumptions of structural equation mode ling: independence assumption and multivariate normality assumption. The Y from X for one case is The independence assumption requires indepe ndent observations obtained through random sampling. This assumption is usually violated in social and behavioral sciences because most often participants are nested within schools or classrooms or they are not selected through random sampling. Nonrandom sampling does not provide accurate estimates of variances and covariances associated with the latent constructs (McDonald & Ho, 2002). In the present study, the independence assumption was violated because students belong ed to different classroom s within a school and they we re selected through nonrandom sampling. Fabrigar, Wegener, MacCallum, and Strahan (1999) suggested that in the case of convenience sampling, researchers should refrain from selecting participants who are relatively homogeneous with respect to the factors of interest. In the present study, the impact of this assumption violation was reduced to some extent because the PAGE 93 93 selected students belonged to varying socioeconomic status and different cultural backgrounds. Multivariate norma lity means that observations are drawn from a continuous and multivariate normal population (Kline, 2005) The violation of this assumption results in substantial overestimation of goodness of fit statistics (e.g., 2 CFI, TLI, RMSEA) and underestimation of standard error estimates. Although the measurement model in the present study was tested using the WLSMV estimator, which produces accurate parameter estimates under non normality, the multivariate normality assumption was tested by computing univariate skewness and kurtosis values for each variable. Both skew and kurtosis describe the distribution of observed data around the mean. Skewness indicates whether the observed scores are above (negative skew) or below the mean (positive skew) (Kline, 2005). On the other hand, kurtosis values suggest whether the multivariate distribution of the observed variables has high peak and heavier tails (positive kurtosis) or the curve is flat with light tails (negative kurtosis). Further, it is important to note that sk ew values influence tests of means, whereas kurtosis values impact tests of variance and covariance (DeCarlo, 1997 as cited in Byrne, 2012). Considering the fact that SEM is based on analysis of covariance structure, parameter estimates and standard errors tend to be more influenced by abnormal kurtosis values in comparison to skew values. Kline (2005) reported that skewness greater than 3.0 generally suggests a serious problem. Kurtosis values greater than 10.0 might be interpreted as a sign of a problem w hile the values greater than 20.0 may point to a serious problem. These reports were used as a point of reference for the examination of the multivariate normality of the current data. PAGE 94 94 Handling Missing Data. Missing data causes pr oblems for researchers us ing SEM techniques. Choosing the most appropriate method for handling missing data is of utmost importan ce because applying inappropriate methods may lead to bias in standard errors and test statistics (Allison, 2003). According to Widaman (2006), the caus e of missing data can be due to item nonresponse, scale nonresponse, or dropout of the participants during the course of a study. Item nonresponse may occur when a participant does not respond to a particular item because of temporary lack of attention, in ability to comprehend a situation, or personal issues. Scale nonresponse occurs when a participant fails to respond to all the items pertinent to a particular construct (e.g., if a participant does not respo nd to all six items of the elaboration scale). Further, missing data are classified either as missing completely at random (MCAR) or missing at random (MAR) (Kline, 2005; Widaman, 2006). MCAR means that the missingness is completely random and is not predictable from either the observed variables or la tent variables in the study. The missing data were tested for MCAR assumes that data are missing completely at random. Hence, p values less than .05 significance level indic ate s data are not missing completely at random. MAR means that the missingness is unpredictable from the latent variables as well as the observed variable for which it is a missing data indicator. However, it is predictable from other observed variables. Some common techniques to handle missing data include listwise deletion, pairwise deletion, mean imputation, and Maximum Likelihood (ML) estimation (Enders & Bandalos, 2001). Listwise deletion methods remove the complete record of a participant PAGE 95 95 with any mi ssing values. Although this method is very easy to implement, it results in the los s of valuable data leading to small sample size that further decreases power and accuracy, and biased parameter estimates when the data are not MCAR (Arbuckle, 1996; Wothke, 2000). Nevertheless, it is recommended to use listwise deletion if the percentage of observations that contain missing values is reasonably low (less than 5%) (Bentler, 2005; Hair, Black, Babin, Anderson, & Tatham, 2006 ). The second method to handle miss ing data involves pairwise deletion. In this method, cases are excluded only when the case has missing data for a variable that is part of the data analysis. Similar to listwise deletion, it is easy to apply and results in less loss of data but has several disadvantages. First, the pairwise deleted correlation matrix may not be positive, which means certain mathematical operations with the matrix will be difficult to carry out. Second, it results in biased parameter estimates when the data are not MCAR. Thi rd, pairwise deletion raises the tendency to reject the statistical model (Enders, 2001). Fourth, it produces standard error estimates that may not be consistent with true standard errors. This problem arises because it uses different sample sizes for esti mating different parameters. Another way of handling missing data is mean imputation, which involves substituting missing values with the mean score of that observed variable. Imputation allows researchers to include subjects with missing values in data an alysis, but it distorts the shape of the distribution of the data as well as relationships between variables. It also results in reduced variance and underestimated standard e rrors. In general, this method is not an appropriate method to handle missing dat a. PAGE 96 96 In comparison to these methods, maximum likelihood (ML) estimation has been regarded as one of the most promising methods to handle missing values in SEM because it can handle missing data under the MAR assumption (Byrne, 2012; Kline, 2005). Unlike mean imputation and listwise deletion, ML neither fills in the missing values nor discards the data. Rather, it uses all the available data to produce parameter Enders, 2010) Specifically, it identifies population parameter values and by using a log likelihood function generates sample estimates that best fit the data. Further, Byrne (2012 ) indicated several benefits of using this method. First, in comparison to listwise and pairwise estimates, ML estimation provides more reliable and efficient solutions under MCAR assumption Second, ML offers reliable estimates even when the data values are missing under MAR conditions. Third, ML estimation does not cause any problems with t he covariance matrices that occur during the case of pairwise deletion. In order to use the ML estimation method to handle missing data, several conditions must be satisfied including existence of a valid model, large sample size, multivariate normal distr ibution for observed variables, and use of a continuous scale fo r the observed variables (Byrne 2001). The most challenging assumption in the present study would be the treatment of ordinal scale variables (e.g., Lik ert scale) as continuous. Byrne (2001) suggested that the violation of this assumption can be handled if the observed variables have multivariate normal distribution and include four or more categories. As discussed before, unless extreme values for skewness and kurtosis are detected, ML method s provide reliable estimation. PAGE 97 97 Multicollinearity. Multicollinearity is yet another serious issue influencing SEM analyses. It occurs when there exist high inter correlations among the latent variables causing the dependent variable to load on more than on e factor. It is problematic because it produces singular covariance matrices and makes some mathematical calculations difficult to carry out (Kline, 2005). In the present study, multicollinearity between the latent variables was reported using the correlat ion matrix with correlations greater than .90 indicating multicollinearity. Direct and Indirect Effects. Di rect effects between the exogeneous (e.g., self efficacy beliefs) and endogeneous variables (e.g., cognitive strategies, metacognitive relationship between two variables are also known as mediators. Similar to direct effects, indirect effects of the vari ables are also interpreted as path coefficients. Indirect effects are generally estimated as the product of two path coefficients. For example, the basic mediation model as shown in Figure 3 4 consists of three variables, the independent or the exogenous v ariable X the dependent or the endogenous variable Y and the mediator M X in a model predicting M from X, Y from M and X respectively (see Figure 3 the direct effect of X on Y the indirect effect of X on Y through M The total effect is equal to the sum of the direct effect of X on Y and the indirect effect through the mediating variable M Additionally, it PAGE 98 98 i s important to note that an independent variable ( X ) can indirectly influence a dependent variable ( Y ) through a mediating variable ( M ) even if X and Y are not correlated (Mathieu & Taylor, 2006). Although there are several methods available (e.g., Sobel t est, bootstrapping, and the empirical M test) to test the statistical significance of the mediating variables, researchers (e.g., MacKinnon, Lockwood, & Williams, 2004; Preacher & Hayes, 2004; Shrout & Brogler, 2002) now advocate using bootstrapping proced ures for several reasons. First, bootstrapping is already implemented in some SEM software such as M plus. Second, unlike the Sobel test, it does not assume normality of the sampling distribution of the indirect effect (Preacher, Rucker, & Hayes, 2007). Thi rd, unlike M test, it can be used to test the effect of the mediating variables in complex path models (William & Mackinnon, 2008). Fourth, it provides better estimates in small to moderate samples. In bootstrapping method, samples are drawn with replaceme nt from the population of interest. Then, the indirect effect is estimated for each resampled data set. This process is repeated for k (e.g., 1000) number of times, which on completion provides k estimates of the indirect effect. The distribution of these k estimates serves as the empirical approximation of the sampling distribution of the indirect effects. Bootstrapped standard errors and confidence intervals for the indirect effects are calculated using this distribution. In Mplus bootstrapped standard e rrors and confidence intervals for the indirect effects can be requested by specifying the number of bootstrap draws to be used in the computation. Assumptions of the Study The study holds three assumptions. First, students engaged in this study wou ld make accurate self efficacy judgments for modeling tasks. Second, students would PAGE 99 99 express their true feelings and provide honest reports about their use of cognitive and efficacy judgments and their responses on the self report questionnaire would not be affected by any social or peer pressure. Third, students would expend a lot of effort in solving modeling tasks. PAGE 100 100 Table 3 1. Item statistics for the Modeling Self Efficacy scale N Mean Standard Deviation SE1 150 87.78 13.84 SE2 150 83.40 18.85 SE3 150 84.18 17.30 SE4 150 86.48 12.80 SE5 150 81.91 16.40 SE6 150 87.71 17.27 SE7 150 77.87 18.34 SE8 150 77.85 20.29 SE9 150 85.05 15.92 Note. SE1 = Cinema Outing SE2 = Energy Needs SE3 = Holiday SE4 = SE5 = Course Design SE6 = Library System SE7 = Irrigation SE8 = Freezer SE9 = Hospital Table 3 2. Item Total Correlation Analysis Scale Mean if Item Deleted Scale Variance if Item Deleted Corrected Item Total Correlation Squared Multiple Correlation Alpha if Item Deleted SE1 664.47 9923.79 .773 .691 .868 SE2 668.85 9722.87 .583 .410 .881 SE3 668.06 10262.51 .480 .279 .889 SE4 665.76 10362.06 .660 .497 .876 SE5 670.33 9732.35 .693 .540 .871 SE6 664.53 9783.92 .633 .533 .876 SE7 674.38 9533.13 .633 .478 .874 SE8 674.40 9219.43 .671 .510 .874 SE9 667.20 9759.22 .709 .560 .870 Note. SE1 = Cinema Outing SE2 = Energy Needs SE3 = Holiday SE4 = SE5 = Course Design SE6 = Library System SE7 = Irrigation SE8 = Freezer SE9 = Hospital PAGE 101 101 Table 3 3 Items for cognitive strategies with three scales Scales Elaboration 1. When reading (your mathematics textbook) for this class, I try to relate the material to what I already know. 2. I try to understand the material in this class by making connections between the 3. I try to apply ideas from course readings (your math ematics textbook) in other class activities such as lecture and discussion. 4. When I study for this class, I pull together information from different sources, such as lectures, readings, and discussions. 5. I try to relate ideas in this subject to those in other courses whenever possible. 6. When I study for this course, I write brief summaries of the main ideas from the readings (your mathematics textbook) and my class notes. Organization 1. When I study the readings (your mathematics textbook) for t his course, I outline the material to help me organize my thoughts. 2. When I study for this course, I go through the readings (your mathematics textbook) and my class notes and try to find the most important ideas. 3. I make simple charts, diagrams, or tables to help me organize course material. 4. When I study for this course, I go over my class notes and make an outline of important concepts. Critical Thinking 1. I often find myself questioning things I hear or read in this course to decide if I fin d them convincing. 2. When a theory, interpretation, or conclusion is presented in class or in the readings, I try to decide if there is good supporting evidence. 3. I treat the course material as a starting point and try to develop my own ideas about it. 4. I try to play around with ideas of my own related to what I am learning in this course. 5. Whenever I read or hear an assertion or conclusion in this class, I think about possible alternatives. PAGE 102 102 Ta ble 3 4 Items for metacognitive strategies scale Metacognitive Self regulation 1. When I become confused about something I'm reading for this class, I go back and try to it out. 2. If course readings are difficult to understand, I change the way I read the material. 3. Before I study new course material thoroughly, I often skim it to see how it is organized. 4. I ask myself questions to make sure I understand the material I have been studying in this class. 5. I try to change the way I study in order to fit the course requirements and the way m y teacher presents the material. 6. I try to think through a topic and decide what I am supposed to learn from it rather than just reading it over when studying for this course. 7. When studying for this course I try to determine which concepts I don't u nderstand well. 8. When I study for this class, I set goals for myself in order to direct my activities in each study period. 9. If I get confused taking notes in class, I make sure I sort it out afterwards. PAGE 103 103 Figure 3 1 The hypothesized model depicting relationships between self efficacy beliefs, cognit ive and metacognitive strategy use and perfo rmance on model eliciting tasks PAGE 104 104 Figure 3 2. The scree plot showing Modeling Self Efficacy scale as one factor model PAGE 105 105 Decision making Syste m ana l ysi s an d desig n T rou b le shooting Goals Choosing among alte r n a ti v es under constraints Identifying the r el a tionships bet w een pa r ts of a system and/or designing a system to exp r ess the r el a tionships bet w een pa r ts Diagnosing and co r r ecting a f aulty or unde r perfo r ming system or me c hanism P r ocesses i n v ol v ed Unde r standing a situ a tion whe r e the r e a r e s e v eral alte r n a ti v es and constraints and a speci ed task Unde r standing the info r m a tion th a t c haracte r ises a g i v en system and the r equi r ements associ a ted with a speci ed task Unde r standing the main fe a tu r es of a system or me c hanism and its malfunctionin g and the demands of a speci c task Identifying r el ev ant constraints Identifying r el ev ant pa r ts of the system Identifying causal l y r el a ted v a r ia b les Rep r esenting the possi b le alte r n a ti v es Rep r esenting the r el a tionships among pa r ts of the system Rep r esenting the functioning of the system Making a decision among alte r n a ti v es Ana l ysin g o r designin g a syste m th a t c a p t u r e s th e r e l a t i o n s h i p s bet w ee n p a r t s Diagnosing the malfunctioning of the system and/or p r oposing a solution Che c king and ev alu a ting the decision Che c king and ev alu a ting the ana l ysis or the design of the system Che c king and ev alu a ting the diagnosis/solution Com m unic a ting or justifying the decision Com m unic a ting the ana l ysis or justifying the p r oposed design Com m unic a ting or justifying the diagnosis and the solution P ossi b le sou r ces of compl e xity Number of constraints Number of inte r r el a ted v a r ia b les and n a tu r e of r el a tionships Number of inte r r el a ted pa r ts of the system or me c hanism and the w a ys in whi c h these pa r ts interact Numbe r an d typ e o f r e p r e s e n t a t i o n s use d ( v e rb a l picto r ial nu m e r i c a l ) Numbe r an d typ e o f r e p r e s e n t a t i o n s use d ( v e rb a l picto r ial nu m e r i c a l ) Number and type of r ep r esent a tions used ( v erbal, picto r ial, nume r ical) Figure 3 3. Problem solving (modeling) processes involved in three different types of problem solving (modeling) tasks (OECD, 2004, p. 29) PAGE 106 106 Figure 3 4. A basic mediation model with X as an independent variable, Y as a dependent variable, and M as an intervening variable PAGE 107 107 CHAPTER 4 RESULTS This chapter describes results of descriptive analysi s including internal consistency of each construct, patterns of missingness, assumptions of str uctural equation modeling, and Confirmatory Factor A nalysis (CFA) of the MSLQ and Modeling Self Efficacy scale s It also includes results of CFA of the full measurement model and r esults obtained from testing the structural model Descriptive Analysi s Reliability Estimates Results of the reliability estimate for each scale, provided in Table 4 1, indicated that co efficient alpha ranged from .60 to .89. T he reliability estimate of the Modeling Self E fficacy scale was very high w ith coefficient alpha equal to .89. As minimum acceptable level of reliability in social sciences varies from .70 to .80, the low reliability estimate s of the organization subscale ( = .61) and the modeling test ( = .60 ) cause d concerns with regard to the results of this study. Missing Data Analysi s Missing data or observations with missing values were examined by performing missing value analysis. The percentage of missing val ues for each observed variable wa s not more than 1% (see Table 4 2) and remained under 3% for each latent construct (see Table 4 3). Sinc e the missingness in each case wa s under 5%, no missing pattern analysis was conducted. Furthe r, in the present study data were analyzed using the WLSMV estimator, which e stimates models by using all the available data except for those cases with missing data on the exogenous observed variables (Muthn & Muthn, 2002). As evident from the Table 4 2 there were no missing values for the self efficacy PAGE 108 108 variables, which were the exogenous variables in the present study WLSMV utilized all the available 225 cases without either imputing data or eliminating cases based on the assumption that missingness was completely at random (MCAR). Descriptive Statistics sco res for each subscale in the MSLQ were calculated by averaging the items o n a subscale as recommended in the MSLQ manual (Pintrich et al., 1991). Descriptive analysis of t he MSLQ subscales (see Table 4 3 ) indicated that means of the elaboration ( M = 3.99, SD = 1.14) and organization subscales ( M = 3.92, SD = 1.24) were higher than the critical thinking subscale ( M = 3.62, SD = 1.24). The overall mean score for the metacognitive self regulation scale was 4.25 ( SD = 1.05). efficacy scores for each of the six modeling problems efficacy questions. For example, the mean self efficacy score for the Cinema Outing problem was calc understanding the problem, determining a strategy, determining the information, and correctly solving a problem. Table 4 4 shows the descriptive analysis of the Modeling Self Efficacy scale. C onsistent with the pilot study results, e ighth and ninth grade students reported higher levels of self confidence in solving the Cinema Outing ( M = 82.02, SD = 17.44) and Camp ( M = 81.00, SD = 16.80) problems than the Energy Needs problem ( M = 76.0, SD = 20.34). Students a ppeared to be least confident in solving the Course Design ( M = 72.93, SD = 21.73), Irrigation ( M = 71.74, SD = 21.61), and Freezer ( M = 72.1 6, SD = 21.55) problem s PAGE 109 109 S tudents engaged in this study also solved six modeling problems. Some of the problems required a definite solution including yes or no whereas some problems required students to provide explanations of their solutions. The minimum and maximum scores receive d by students on each problem vary between 0 and 2. Table 4 5 shows that st udents earned the highest average score for the Cinema Outing ( M = 0 .85, SD = 0 .62 ) and ( M = 0 .78, SD = 0 .72 ) problems. These scores were consistent with their high self efficacy beliefs reported for these tasks. With regard to the Course Design and Freezer problems, there were some inconsistencies between the levels of confidence reported by students and their scores on these problems Students reported similar l evels of confidence for both the problems, but the mean score for the Course D esign ( M = 0 .77, SD = 0 .88 ) was much higher than the Freezer problem ( M = 0 .37, SD = 0 .49 ). The mean for the Energy Needs and Irrigation problems were .74 ( SD = 0 .86 ) and .43 ( SD = 0 .50 ) respectively. Multivariate Normality Assumption Multivariate normal ity assumption implies that each individual variable is normally distributed and combinations of such variables are distributed as multivariate normal (Kline, 2005/2011). The departure from normality causes the chi square test statistic to be larger than e xpected and standard errors to be smaller than they should be. Although parameter estimates, standard errors, and test statistics in the present study were computed using a robust WLSMV estimator that produces accurate results under both normal and non nor mal distributions (Byrne, 2012; Muthn & Muthn (2002), it is always a good idea to check the distribution of each variable for univariate normality. The skew and kurtosis values as prese nted in Tables 4 3, 4 4 and 4 5 for each variable were within reasonable ranges. Skewness values for each variable were not PAGE 110 110 0 .530) and kurtosis values stayed ption of multivariate normality was satisfied. Confirmatory Factor Analysi s of the MSLQ Scale In an attempt to gather evidence of construct validity for the MSLQ sca le, individual parameters, such as factor l oadings and factor correlations, were examined through CFA procedures. In general, the values for factor loadings should be moderately high to establish significant relationships between observed indicators and their corresponding latent variables. On the other hand, values for factor correlations shou ld be minimal to discriminate latent variables from one another. The item level factor analysis of the MSLQ scale provided an acceptable fit to the data in terms of Chi square, CFI, TLI, and RMSEA fit indices ( 2 [246 df, N = 225] = 442.55 p < .001 CFI = .92, TLI = .91, RMSEA = .06 with 90% CI [.05, .06]). The standardized parameter estimates for the factor loadings, provided in Table 4 6 indicated that all observed indicators had standardized factor loadings on their common factors greater than .30. Fur ther, all indicators (i.e. items on each subscale) in the model had statistically significant stand ardized factor loadings ( p < .001 ) confirmin g that observed indicators for each construct were correlated. Construct corr elations, presented in Table 4 7 i ndicated high correlation between latent factors suggesting low discriminant validity between them. Specifically, statistically significant correlations were found between metacognitive self regulation scale and indicators of the cognitive strategi es such as elaboration ( r = .876 p < .0 01), critical thinking ( r = .730 p < 001), and organization ( r = .716 p < .001) subscales. PAGE 111 111 Such high correlations were also found in Kaya (2007) study that used the same version of the MSLQ. These correlations were expected because one of the important aspects of metacognitive strategies is to enable learners to control, monitor, and regulate their cognitive processes which involve the use of elaboration, organization, and critical thinking strate gies (Pintrich, 2002; Pintrich et al., 1993). The high correlations between metacognitive self regulation and elaboration, organization, and critical thinking subscales suggest ed potential multicollinearity problems between metacognitive self regulation an d cognitive strategies scale Confirmatory Factor Analysi s of the Modeling Self Efficacy Scale Before model testing, CFA of the Modeling Self Efficacy scale was conducted to investigate whether the factor loading pattern established during the pilot study fits the data from a new sample. The CFA f or the Modeling Self Efficacy scale was conducted n ratings for the six modeling problems calculated across four self efficacy questions, on the overall modeling self efficacy latent variable For Cinema Outing problem calculated over four sel f efficacy items including understanding the problem, determining a strategy, determining information, and solving the problem. The goodness of fit indices such as chi square, RMSEA, CFI, and TLI indicated that the model fits the data well ( 2 [9 df, N = 2 25] = 13.48, p = .14, C FI = .99, TLI = .99 RMSEA = .05 (with 90% CI lower bound = .00 and upper bound = .10 )). P arameter estimates shown in Table 4 8 indicated that each observed variable had statistically significant ( p < .001) standardized factor loading on the overall Modeling Self Efficacy scale. PAGE 112 112 Overview of Model Testing Model testing in the current study was performed by using the two step modeling approach including verifying the measurement model and testing the full Stru ctural Equation Model (Kline, 2005/2011). During the first step, confirmatory factor ana l ysis of the measurement model is performed to determine relationships between the observed indicators and the continuous latent variables. On obtaining an acceptable m easurement model, the second part of the two step modeling procedure i s performed to test relationships among latent variables. As such, under the full SEM model both the mea s urement and structural models a re tested. Results of a confirmatory factor analy sis indicated an acceptable fit of the data with the hypo thesized measurement model The goodness of fit test results were 2 [129 df, N = 225] = 250.60, p < .001, CFI = .92, TLI = .90 and RMSEA = .06 with 90% CI [.05, .07]. Further, a ll observed indicato rs in the model had statistically significant ( p < .001 ) standardized factor loadings on their corresponding laten t factors A s expected a large correlation ( r = .96 p < .001 ) was found between cognitive and metacognitive factors indic ating multicollinearity problems In order to improve the model fit and to reduce correlations between the cognitive and metacognitive factors, modification indices for the measurement model were reviewed to check for adding cross loadings between metacognitive factor indicators (e.g., nine metacognitive items) and cognitive factor indicators (e.g., elaboration, organization, and critical thinking). However, no cross loadings were found between these two highly correlated factors. In such cases, Grewal, Corte, an d Baumgartner (2004) suggest re specifying the statistical model. PAGE 113 113 Accordingly, the latent cognitive factor in the original model was replaced by three latent factors including elaboration, organization, and critical thinking. The modified model (see Figure 4 1 ) provided a much better fit to the data with regard to CFI (.94), TLI ( .94 ), and RMSEA (.05 with 90% CI lower bound = .04 and upper b ound = .06 ) fit indices. However, Chi square statistics ( 2 [390 df, N = 225] = 595.97, p < .001 ) suggested a large di screpancy between the sample covariance matrix and the restricted population covariance matrix. As depicted in Figure 4 2 the model consisted of six correlated latent factors including self efficacy for modeling tasks, metacognitive, critical thinking, or ganization, elaboration strategies, and modeling tas ks. Standardized factor loadings, factor correlations, and R 2 estimates are present ed in Table 4 9, 4 10, and 4 11 The evaluation of t he factor loadings in Table 4 9 indicated that all the observed indi cators had standardized factor loadings greater than .30 on their common factors, which suggest ed that they adequately represent their underlying latent variables. The ratio of each parameter estimate to its corresponding standard error was greater than 1. 96, indicating that all the estimates were st atistically significant. All the stan d ard errors were in good order since t hey were n either very large nor too small. Excessi vely large standard errors make the test statistic for the relate d parameter s difficult to compute and standard errors approaching zero result in undefin ed test statistic (Bentler as cited in Byrne, 2012 ). On reviewing factor corr elations presented in Table 4 10 it was found that some of the bivariate correlations among latent fa ctors were in the expected directions while efficacy beliefs for modeling tasks were found ical PAGE 114 114 thinking ( r = .383, p < .001 ) an d elaboration ( r = .320, p < .001 ) as well as metacognitive strategies ( r = .330, p < .001 ). Also, self efficacy beliefs had a significant moderate n modeling tasks ( r = .542, p < .001 ). However, correlations betwee n modeling self efficacy beliefs and organization stra tegies were not significant at the .05 level ( r = .009, p = .910). Similar inappropriate relationships were observed among other variables as well. Specifically, insignificant correlations were found be r = .095, p = .288) and their success rate in the area of mathematical modeling. Critical thinking strategies ( r = .029, p = .769) and elaboration strategies ( r = .058, p = .517) also did not correl ate ormance on the modeling test. Further, a significant negative correlation was found between organization strategies and modeling task success ( r p = .005). As found in the CFA for the MSLQ high correlations existed between metacognitive strategies and critical thinking ( r = .731, p < .001 ), organization ( r = .714, p < .001 ) and elaboration ( r = .876, p < .0 01 ) strategies. Kline (2005 ) specified that correlations higher than .85 lea d to mu lticollinearity problems. Thus, high correlations between elaboration and metacognitive self regulation factors caused concerns for multicollinearity. Therefore, Modification Index (MI) va lues, as presented in Table 4 12 were reviewed to identify whether observed indicators of elaboration, critical thinking, and metacognitive factors were cross loading on more than one factor. For the purposes of this study, parameters having MI values greater than or equal to 10.00 were reported. On inspecting these param eters, it was found that most of the MI values were very small and not worthy of inclusion in a subsequent model. For example, a MI PAGE 115 115 value of 11.39 suggested that if ORG, which was designed to measure organization strategies, were to load additionally onto metaco gnitive factor, the overall model Chi s quare value would decrease by 11.39. Moreover, such modifications (e.g., combining organization and metacognitive subscales) did not make sense theoretically because organization, elaboration, critical thinking, and metacognitive self regulations are distinct construct s (Pintrich et al., 1991). As such, no further actions were taken. R 2 estimates reported in Table 4 11 represent the proportion of variance in each observed variable that can be explained by the lat ent construct to which it is linked. All R 2 estimates were found to be reasonable as well as statistically significant except two n troubleshooting tasks, that had R 2 values equal to .18 and .18 respectively. Research Hypotheses Testing The full structural model was estimated by specifying all the structural regression paths including effects of self s elf reported use of elaboration, organization, critical thinking, and metacognitive strategies; effects of self efficacy beliefs and use of elaboration, organization, critical thinking, and metacognitive s; indirect effects of self efficacy beliefs on modeling tasks success via use of elaboration, organization, critical thinking, and metacognitive strategies. Unfortunately, estimation of the full structural model did not converge when the default starting values supplied by Mplus were used. As a result, the produce a model statement, which includes final estimates as the starting values (Muthn & Muthn, 1998 2012). On using thes e starting values, the model estimation PAGE 116 116 terminated normally. The full structural model fit adequately to the given da ta with regard to CFI, TLI and RMSEA fit indices ( 2 [390 df, N = 225] = 595.97, p < .00 1 CFI = .95, TLI = .94 RMSEA = .05 (with 90% CI lower bound = .04 and upper bound = .0 6)). As presented in Figure 4 2 and Table 4 13 eighth and ninth grade self efficacy beliefs for modeling tasks showed significant positive direct effects on critical thinkin g ( = .38, p < .001 ), elaboration ( = .3 2, p < .001 ), and metacog nitive strategies ( = .33, p < .001 understanding mathematical modeling tasks also report ed the use of these strategies. Surprisingly, results indicated that perceived modeling self efficacy ( = .009, p = .910) credible evidence was found to support an association between modeling self efficacy and organization st rategy use. Perceived mode ling self efficacy ( = .50, p < 001 ) directly positively predicted In other words, students who reported greater self efficacy for solving modeling tasks were more l ikely to correctly solve the modeling problems. C ontrary to the hypothesized relationship organiza tion strategy use ( p = .004 ) had a significant negative direct effect on performance o n the modeling test. This means that students who reported using more organization strategies tended to get lowe r scores on the modeling test. p = .08), elaboration ( = .40, p = .41), and metacognitive strategies ( = .46, p = .16) on their performance in solving modeling tasks were non significant. Therefore, the data did not provide any PAGE 117 117 evidence of the direct effects of critical thinking, elaboration, and metacognitive strategies on their performance in modeling abili ty test. efficacy for modeling tasks directly predicted their reported use of critical thinking, elaboration, and metacognitive strategies but the direct effects of cognitive and metacognitive strategies on mo deling task success were non efficacy for modeling on their performance in solving modeling tasks through its effect on their use of critical thinking ( p = .10), organization ( = .006, p = .91), elaboration ( = .128, p = .41), and metacognition strategies ( = .15, p = .18) were non significant. PAGE 118 118 Table 4 1. Summary of reliability estimates of each scale Scales Number of Items Self efficacy .89 6 Elaboration .73 6 Organization .61 4 Critical thinking .76 5 Metacognitive self regulation .78 9 Modeling test .60 6 Note. N = 225 PAGE 119 119 Table 4 2. Missing data analysis for the observed indicators of the full model N M S.D. Missing Count Percent MSLQ elab1 225 4.73 1.77 0 0 .0 elab2 225 3.79 1.83 0 0.0 elab3 225 5.03 1.71 0 0.0 elab4 225 2.13 1.56 0 0.0 elab5 225 4.38 1.76 0 0.0 elab6 224 3.90 1.87 1 0 .4 org1 225 2.96 1.61 0 0.0 org2 225 5.05 1.80 0 0.0 org3 225 3.10 1.88 0 0.0 org4 225 4.59 1.99 0 0.0 ct1 225 3.89 1.72 0 0.0 ct2 225 5.05 1.80 0 0.0 ct3 225 3.52 1.65 0 0.0 ct 4 225 3.82 1.84 0 0.0 ct5 225 3.38 1.77 0 0.0 mcsr1 224 5.29 1.50 1 0 .4 mcsr2 224 3.17 1.65 1 0 .4 mcsr3 225 3.61 2.03 0 0.0 mcsr4 225 4.27 1.84 0 0.0 mcsr5 225 3.64 1.64 0 0.0 mcsr6 225 4.14 1.60 0 0.0 mcsr7 225 5.45 1.58 0 0.0 mcsr8 225 4.13 1.94 0 0.0 mcsr9 224 3.90 1.87 1 0 .4 Self Efficacy Scale q1se1 225 83.78 18.52 0 0.0 q1se2 225 79.47 20.17 0 0.0 q1se3 225 83.64 18.15 0 0.0 q1se4 225 81.20 19.84 0 0.0 q2se1 225 77.78 21.05 0 0.0 q2se2 225 74.84 21.15 0 0.0 q2se3 225 77.07 21.98 0 0.0 q2se4 225 74.58 22.73 0 0.0 q3se1 225 82.58 17.56 0 0.0 q3se2 225 79.51 18.40 0 0.0 q3se3 225 82.62 17.02 0 0.0 q3se4 225 79.29 19.19 0 0.0 q4se1 225 75.24 21.85 0 0.0 q4se2 225 71.47 22.36 0 0.0 q4se3 225 74.53 22.51 0 0.0 q4se4 225 70.49 24.26 0 0.0 q5se1 225 72.62 22.61 0 0.0 q5se2 225 71.07 21.62 0 0.0 q5se3 225 72.89 22.56 0 0.0 PAGE 120 120 Table 4 2. Continued N M S.D. Missing Count Percent Self Efficacy Scale q5se4 225 70.40 24.13 0 0 .0 q6se1 225 73.33 20.98 0 0.0 q6se2 225 71.07 22.09 0 0.0 q6se3 225 73.91 22.37 0 0.0 q6se4 225 70.36 24.30 0 0.0 Modeling Tasks mod1 219 0 .85 0 .62 6 2.7 mod2 219 0 .74 0 .86 6 2.7 mod3 219 0 .78 0 .72 6 2.7 mod4 219 0 .77 0 .87 6 2.7 mod5 219 0 .43 0 .49 6 2.7 mod6 219 0 .37 0 .48 6 2.7 Note. MCSR = Metacognitive Self Regulation, ELAB = Elaboration, ORG = Organization, CT = Critical Thinking mod1 = Cinema Design, mod5 = Irrigation, mod6 = Freezer problems PAGE 121 121 Table 4 3. Missing Value Analysis for each construct N M S.D. Missing Skewness Kurtosis Count Percent Elaboration 225 3.99 1.14 0 0 .0 0 .102 0 .303 Organization 225 3.92 1.24 0 0 .0 0 .139 0 .341 Critical Thinking 225 3.62 1.24 0 0 .0 0 .297 0 .269 MCSR 225 4.25 1.05 0 0 .0 0 .385 0 .100 Self Efficacy 225 75.98 18.2 4 0 0 .0 0.884 0.721 Modeling Tasks 219 3.93 2.41 6 2.7 0.229 0.954 Note. MCSR = Metacognitive Self Regulation Table 4 4. Descriptive statistics for the Modeling Self Efficacy scale N M S.D. Skewness Kurtosis SE1 225 82.02 17.44 1.632 3.295 SE2 225 76.06 20.34 1.012 0 .609 SE3 225 81.00 16.80 1.214 1.627 SE4 225 72.93 21.73 0 .898 0 .284 SE5 225 71.74 21.61 0 .672 0 .354 SE6 225 72.16 21.55 0 .882 0 .382 Note. SE1 = Self efficacy for Cinema Outing, SE2 = Self efficacy for Energy Needs, SE3 = Self efficacy efficacy for Course Design, SE5 = Self efficacy for Irrigation, SE6 = Self efficacy for Freezer problems PAGE 122 122 Table 4 5 Desc riptive statistics for the modeling test Problems N Min Max M S.D. Skewness Kurtosis Cinema Outing 219 0.00 2.00 0 .85 0 .621 0 .111 0 .474 Energy Needs 219 0.00 2.00 0 .74 0 .862 0 .530 1.452 Camp 219 0.00 2.00 0 .78 0 .723 0 .367 1.025 Course Design 219 0.00 2.00 0 .77 0 .876 0 .473 1.535 Irrigation 219 0.00 1.00 0 .43 0 .496 0 .288 1.935 Freezer 219 0.00 1.00 0 .37 0 .485 0 .522 1.743 Note. N = 225 ( Six students did not complete the modeling test) PAGE 123 123 Table 4 6 Confirmatory Factor Analysis of MSLQ subscales with WLSMV parameter estimate Parameter Standardized Factor Loading Standard Error Est./S.E. Two tailed p value MCSR 1 .56 .05 11.55 .000 MCSR2 .54 .05 11.15 .000 MCSR3 .53 .05 10.46 .000 MCSR4 .67 .04 15.67 .000 MCSR5 .51 .05 9.53 .000 MCSR6 .62 .05 13.12 .000 MCSR7 .52 .06 9.50 .000 MCSR8 .55 .05 11.17 .000 MCSR9 .48 .06 8.73 .000 CT1 .47 .06 7.58 .000 CT2 .71 .04 18.40 .000 CT3 .69 .04 17.95 .000 CT4 .59 .05 11.98 .000 CT5 .76 .04 19.19 .000 ORG1 .63 .06 10.94 .000 ORG2 .50 .06 8.58 .000 ORG3 .49 .07 7.19 .000 ORG4 .69 .06 12.18 .000 ELAB1 .44 .05 8.19 .000 ELAB2 .62 .04 14.03 .000 ELAB3 .61 .04 14.11 .000 ELAB4 .56 .06 10.14 .000 ELAB5 .67 .04 17.49 .000 ELAB6 .70 .04 20.26 .000 Note. MCSR = Metacognitive Self Regulation, ELAB = Elaboration, ORG = Organization, CT = Critical Thinking PAGE 124 124 Table 4 7 Estimated correlation matrix for the latent variables MCSR Critical Thinking Organization Elaboration MCSR 1.000 Critical Thinking .730 1.000 Organization .716 .402 1.000 Elaboration .876 .862 .632 1.000 Key: p< .05 MCSR = Metacognitive Self Regulation Table 4 8 Confirmatory Factor Analysis of Modeling Self Efficacy scale with WLSMV estimator Parameter Standardized Factor Loading Standard Error Est./S.E. Two tailed p value SE1 .72 .04 19.47 .000 SE2 .82 .03 30.58 .000 SE3 .75 .03 21.98 .000 SE4 .75 .03 22.54 .000 SE5 .77 .03 24.18 .000 SE6 .79 .03 26.48 .000 Note. SE1 = Self efficacy for Cinema Outing, SE2 = Self efficacy for Energy Needs, SE3 = Self efficacy efficacy for Course Design, SE5 = Self efficacy for Irrigation, SE6 = Self efficacy for Freezer problems PAGE 125 125 Table 4 9 Confirmatory Factor Analysis for the full measurement model Parameter Standardized Factor Loading Standard Error Est./S.E. Two Tailed P value Modeling Self Efficacy SE DM .89 .02 45.18 .000 SE SAD .85 .02 38.55 .000 SE T .84 .02 39.98 .000 MSLQ Subscales MCSR1 .57 .05 11.88 .000 MCSR2 .53 .05 10.88 .000 MCSR3 .52 .05 9.8 5 .000 MCSR4 .68 .04 15.58 .000 MCSR5 .49 .05 9.19 .000 MCSR6 .62 .05 13.02 .000 MCSR7 .53 .05 9.79 .000 MCSR8 .55 .05 10.76 .000 MCSR9 .49 .05 8.97 .000 CT1 .45 .06 7.30 .000 CT2 .71 .04 18.27 .000 CT3 .69 .04 18.01 .000 CT4 .61 .05 12 .29 .000 CT5 .77 .04 19.68 .000 ORG1 .63 .06 11.18 .000 ORG2 .50 .06 8.55 .000 ORG3 .49 .07 7.30 .000 ORG4 .67 .06 12.44 .000 ELAB1 .43 .05 7.96 .000 ELAB2 .62 .04 14.05 .000 ELAB3 .62 .04 14.06 .000 ELAB4 .55 .05 9.74 .000 ELAB5 .68 .03 18.00 .000 ELAB6 .70 .03 19.66 .000 Modeling Tasks Decision making task s .56 .07 7.59 .000 System analysis tasks .70 .08 8.56 .000 Troubleshooting tasks .42 .08 4.96 .000 Note. SE DM = Self Efficacy for Decision making tasks, SE SAD = Self Efficacy for System Analysis and Design tasks, and SE T = Self Efficacy for Troubleshooting tasks. MCSR = Metacognitive Self Regulation, ELAB = Elaboration, ORG = Organization, CT = Critical Thinking PAGE 126 126 Table 4 10 Correlations among latent variables Parameter Standardized Correlations Standard Error Est./S.E. Two Tailed P value MCSR SE .33 .07 4.82 .000 CT SE .38 .07 5.85 .000 CT MCSR .73 .04 18.14 .000 ORG SE .01 .08 0 .11 .910 ORG MCSR .71 .05 14.39 .000 ORG CT .40 .07 5.74 .000 ELAB SE .32 .07 4.60 .000 ELAB MCSR .87 .03 29.01 .000 ELAB CT .86 .04 24.21 .000 ELAB ORG .63 .06 10.34 .000 Modeling Tasks SE .54 .08 7.17 .000 Modeling Tasks MCSR .10 .09 1.06 .288 Modeling Tasks CT .03 .10 0 .29 .769 Modeling Tasks ORG .28 .10 2 .005 Modeling Tasks ELAB .06 .09 0 .65 .517 Note. MCSR = Metacognitive Self Regulation, ELAB = Elaboration, ORG = Organization, CT = Critical Thinking, SE = Self Efficacy PAGE 127 127 Table 4 11 R 2 estimates for each observed and latent dependent variable in the model Observed Variable Estimate Standard Error Est./S.E. Two Tailed P value Self Efficacy (decision making) .79 .03 22.59 .000 Self Efficacy (system analysis) .73 .04 19.28 .000 Self Efficacy (troubleshooting) .71 .04 19.99 .000 MCSR1 .33 .06 5.94 .000 MCSR2 .28 .05 5.44 .000 MCSR3 .27 .05 4.93 .000 MCSR4 .46 .06 7.79 .000 MCSR5 .24 .05 4.60 .000 MCSR6 .39 .06 6.51 .000 MCSR7 .28 .06 4.90 .000 MCSR8 .30 .05 5.38 .000 MCSR9 .24 .05 4.48 .000 CT1 .21 .06 3.65 .000 CT2 .50 .06 9.13 .000 CT3 .48 .05 9.00 .000 CT4 .37 .06 6.15 .000 CT5 .59 .06 9.84 .000 ORG1 .40 .07 5.59 .000 ORG2 .25 .06 4.28 .000 ORG3 .24 .07 3.65 .000 ORG4 .45 .07 6.22 .000 ELAB1 .18 .05 3.98 .000 ELAB2 .39 .06 7.03 .000 ELAB3 .38 .05 7.03 .000 ELAB4 .30 .06 4.87 .000 ELAB5 .4 6 .05 8.99 .000 ELAB6 .49 .05 9.83 .000 Decision making tasks .31 .08 3.80 .000 System analysis tasks .50 .12 4.28 .000 Troubleshooting tasks .18 .07 2.48 .013 Note. MCSR = Metacognitive Self Regulation, ELAB = Elaboration, ORG = Organization, CT = Critical Thinking PAGE 128 128 Table 4 12 Model Modification Indices Parameters M.I. E.P.C. CT BY ELAB1 12.09 ORG BY MCSR1 11.39 ORG BY ELAB1 21.43 0.63 ORG BY ELAB2 16.66 ORG BY ELAB4 17.34 0.63 Note. MCSR = Metacognitive Self Regulation, ELAB = Elaboration, ORG = Organization, CT = Critical Thinking, M.I. = Modification Index, E.P.C. = Expected Parameter Change PAGE 129 129 Table 4 13 Standardized estimates of the path coefficients in the full structural equation model Parameter Standardized Estimate Standard Error Est./S.E. Two Tailed p value Self Efficacy ON Critical Thinking .38 .07 5.42 .000 Self Efficacy ON Organization .01 .08 0 .11 .910 Self Efficacy ON Elaboration .32 07 4.60 000 Self Efficacy ON Metacognitive .33 .07 4.82 .000 Critical Thinking ON Modeling tasks .59 .35 1.70 .088 Organization ON Modeling Tasks .62 .21 2.90 .004 Elaboration ON Modeling Tasks .40 .49 0 .81 .417 Metacognitive ON Modeling Tasks .46 .33 1.40 .161 Self Efficacy ON Modeling tasks .50 .10 4.78 .000 Note. Statistically significant paths are in boldface. PAGE 130 130 Figure 4 1 The modified measurement model depicting relationships between modeling self efficacy beliefs, use of elaboration, organization, critical thinking, metacognitive strategies, and modeling task success. Note. SE for DM = Self Efficacy for Decision making tasks, SE for SAD = Self Efficacy for System Analysis Design tasks SE for TS = Self Efficacy for Troubleshooting tasks, MCSR = Metacognitive Self Regulation, ELAB = Elaboration, ORG = Organization, CT = Critical Thinking PAGE 131 131 Figure 4 2 Standardized path coefficients in the full structural model. Note. p < .05 PAGE 132 132 CHAPTER 5 DISCUSSION Summary of the Findings The primary purpose of this study was to examine associations between self efficacy beliefs, self regulated learning behaviors, and Towards this end, three researc h hypotheses were tested. First, students efficacy beliefs for the modeling tasks w ere hypothesized to have a positive direct influence on their ability to correctly solve problems on the modeling test. Second, students r eported use of cognitive and metacognitive strategies was hypothesized to directly influence their performance on the modeling test Third, efficacy beliefs for modeling tasks were hypothesized to have a posit ive indirect influence on their performance on the modeling test through the positive effect on their use of cognitive and metacognitive strategies. This investigation was guided by prior research indicating & Kranzler, 1995; Nicolidau & Philippou, 2004) as well as their self reported use of cognitive and metacognitive strategies (e.g., Pape & Wang, 2003; Pintrich & DeGroot, 1990; Zimmerman & Martinez Pons, 1986, 1988, 1990) p roblem solving and mathematics achievement. The present study, however, is different from these studies in a few respects. First, the present study examined the influence of self efficacy beliefs and SRL strategy x real world problems (i.e. modeling tasks). Second, research studies such as Bouffard Bouchard et al. (1991), Mousoulides and Philippou (2005), Pintrich and DeGroot (199 0), and Kaya (2007) studied the impact o f SRL strategy use by in dicating elaboration, critical thinking, and organization strategies PAGE 133 133 as observed indicators for the cognitive strategy latent variable. Owing to high multicollinearity found between cognitive and metacognitive strategy scales, the present study used items on the MSLQ to define elaboration, critical thinking, and organization latent variables in the modified measurement model rather than including them as observed indicators for defining the cognitive latent variable. There is strong evidence that ompetence are related to as well as predictive of their problem solving achievement ( Chen, 2003; Greene et al., 2004; Nicolidau & Philippou, 2004; Pajares & Graham, 1999 ; Pajares & Kranzler, 1995; Pajares & Miller, 1994 ; Pajares & Valiante, 2001; Pintrich & DeGroot, 1990 ) Consistent with the problem solving literature, the findings of the present study indicated that self efficacy beliefs are associated with tasks. That is, students who reported higher levels of confidence for understanding modeling tasks were more successful in solving these tasks. Further, research has shown that students who believe in their competence are more likely to employ sophisticated cognitive and metacognitive strategies to understand and solve acad emic or problem solving tasks (Bouffard Bouchard et al., 1991; Greene et al., 2004; Pintrich & DeGroot, 1990; Zimmerman & Bandura, 1984; Zimmerman & Martinez Pons, 1990). Similarly, the findings of the present investigation indicated that self efficacy bel iefs reported use of cognitive and metacognitive strategies. Specifically, students who perceived themselves capable of understanding and solving modeling tasks also tended to report using elaboration, crit ical thinking, and metacognitive strategies as they engage in mathematical activities. PAGE 134 134 The present study, however, did not find significant association between self efficacy reported use of organization strategies. Further, sign ificant associations have been identified in the literature between SRL strategy use and student problem solving performance (Pape & Wang, 2003; Pintrich & DeGroot, 1990; Zimmerman & Martinez Pons, 1986, 1988, 1990). These studies have established that stu dents who are successful in solving problem solving tasks tend to report using more sophisticated learning strategies. These results were not confirmed in the present study because no significant associations were found aboration, critical thinking, and metacognitive strategies and their success on the modeling tasks The use of organization strategies, however, was negatively associated with performance on the modeling test. That is, students who reported highe r use of organization strategies received lower scores on the modeling test. This result was also found in previous studies (e.g., Mous oulides & Philippou, 2005; Kaya 2007) although they examined the direct effect of thematics achievement. The negative association between the self the modeling tasks might have occurred because of t he low reliability estimate ( = .61) o f the organization scale indicat ing that items on the scale might not be consistently measuring the required construct. With regard to the third objective of this study, th e findings of the present study contradict earlier assertions made by Bouffard Bouchard et al. (1991), Heidari et al ( 2012 ), Pintrinch and DeGroot ( 1990 ), and Zimmerman and Bandura (1994) Specifically, the findings did not provide evidence for the indirect effects of self efficacy PAGE 135 135 elaboration, organization, critical thinking, and metacognitive strategy use. These results were expected because no significant associations were found between the mediating variables (i.e., elaboration, organization, critical thinking, and metacognitive strategies) and the dependent variable (i.e the modeling task success) (Zhao, Lynch, & Chen, 2009). Further, the current study attempted to provide a valid and reliable instrument to of the Modeling Self Efficacy scale evaluated during the pilot and main studies indicated that the items efficacy beliefs for understanding and solving modeling tasks ( = .89). The construct validity of the scale esta blished using confirmatory factor analysis revealed that the items had significantly high factor loadings, ranging from .72 to .82, on the overall modeling self efficacy latent variable. These findings suggest that the Modeling Self Efficacy scale is a dep endable instrument modeling tasks. Reasons for Inconsistent Results and Recommendations for Future Research In this section, possible reasons for finding results inco nsistent with the past literature will be explored and based on that recommendations for future research projects will be offered One of the possible explanations would be that the MSLQ gnitive and metacognitive strategies in relation to real world problem solving. T he MSLQ is a retrospective measure requiring students to self report their use of SRL strategies PAGE 136 136 based on recollections of past experiences (Zimmerman, 2008). Further self re ports such as the MSLQ regulatory behaviors (Cleary as cited in National Research Council, 2011 ). Cleary stated that self reports the characteristics of self regulated learning but they do so in a National Research Council 2011 p. 88 ). He indicated two potential problems of using self are validity i ssues involved in using self report questionnaires that do not measure context specific SRL behaviors. T self reporting SRL behaviors varies across tasks as well as subject areas ( Zimmerman & Martinez Pons, 1986, 1988, 1990 ). Second, self reports are often incongruent with the strategies actually employed by students in doing specific academic tasks (Winnie & Jamieson Noel, 2002). This mismatch between the strategies reported and actually used again indicates that students use different learning strategies for different tasks. In contrast to using self report measures, future research studies should consider involved in solving model ing tasks, personal diaries in which students record their thoughts and problem solving strategies toward solving modeling problems and think aloud engaging in modeling tasks ( Cleary as cited in National Research Council, 2011; Zimmerman, 2008) Although th ese measures are very time consuming, they provide a Another suggestion would be modifying items on the MSLQ scale to more c losely align with the strategies used by students when engaged in modeling activities. For example, one of the elaboration PAGE 137 137 items used in this study was: When I study for this class, I pull together information from different sources such as lectures, readi ngs, and discussions we have in class A revised elaboration item more applicable w ithin the modeling context might be: I solve math problems in everyday life by applying math learned in school (e.g., through lectures, readings, math text book and discussi ons) An organization item used in this study was: When I study the readings (your mathematics textbook) for this course, I outline the material to help me organize my thoughts This item could be revised as: When I read math problems that are not immediat ely resolvable I outline the material to help me organize my thoughts Second it is likely that the eighth and ninth grade students who participated in this study might not have enough experience solving real world PISA problems. The low reliability est imate ( = .60) of the modeling test further indicates that the test was not is well documented in the mathematical modeling literature that PISA problems are valid to test Christou, & Sriraman, 2008). Perhaps, pilot testing PISA problems for item difficulty and item discrimination might have resulted in developing a modeling test that more reliably measures in solving modeling tasks. Third the rubric used by PISA 2003 problem solving assessment (see Appendix E). This rubric was selected because with the same scoring system from where these problems were obtained. However, the PAGE 138 1 38 for correctly solving many o f the sub questions or performing many of the mathematical steps. As such, it resulted in restricting the variance of scores. For example, the Cinema Outing problem required students to answer all six multiple choice questions correctly in order to receive full credit (i.e. 2 points). S tudents did not earn partial credit even if they answered four out of six multiple choice questions correctly. Similarly, in the Irrigation and Freezer problems students received full credit for answering all three multiple choice questions correctly. They did not earn partial points for correctly answering one or two of the required three sub questions. This narrow scoring rubric restricted the range of scores and may not be appropriate for s on the modeling test. Perhaps a more robust and comprehensive rubric that provides students partial credit for correctly answering even sub questions would have been more suitable. Thus, future investigation might include developing a more comprehensive scoring rubric for the modeling test. Contributions to the Field The primary objective of this study was to examine relationships between self efficacy beliefs for solving complex modeling tasks, self reported use of cognitive and metacognitive strategies, and t he direct and indirect effects of these variables on world modeling tasks. The present study contributed to research in mathematics education in several ways. A significant contribution of this study to the mathematics edu cation literature was the creation of a statistical model connecting self outcomes. This model responded to a need in mat hematical modeling research by investigating factors that might influence The fit indices for the measurement model suggested an adequate fit for the data and PAGE 139 139 structural model indicated positive association between self efficacy beliefs and Further, c onsidering self efficacy beliefs have never world modeling problems, the significant relationship established between these constructs contributes significantly to both academic self efficacy and mathematical modeling literature. Another significant contribution of this study is the develop ment of a reliable and valid efficacy beliefs for correctly solving real world modeling tasks. In the field of educational psycho logy, self efficacy beliefs have been achievement, and self regulation (Schunk & Mullen, 2010; Schunk & Pajares, 2008). Further, there is a growing body of literature sugges ting the need to engage students in mathematical modeling for instilling 21 st century workforce skills (English & Sriraman, 2010; Kaiser, Blum, Ferri, & Stillman, 2011; Lesh & Doerr, 2003). In contrast to word problems usually found in school mathematics solutions to modeling problems situated in real world contexts are not readily available ( Lesh, Yoon, & Zawojewski, 2007; Verschaffel, van Dooren, Greer, & Mukhopadhyay, 2010). To correctly solve modeling problems, students need to understand the context of the situation select or acquire appropria te mathematical concepts, procedures and problem solving strategies for describing the situation and interpret ing the solution (Blum, 2011 ; Verschaffel et al., 2010 ). As a result, it would not be appropriate to efficacy for modeling tasks by merely asking their confidence in solving these problems, which is the typical way of measuring self efficacy beliefs for solving mathematical tasks. Bandura (2006) also argued that behavior is better predicted PAGE 140 140 self efficacy beliefs for processes or actions needed to exhibit a particular behavior (e.g., modeling task success). The d evelopment of a Modeling Self Efficacy scale not only fulfilled this need but also contribute d to the growing litera ture of self efficacy theory and mathematical modeling field The data also provided evidence for the reliability and construct validity of the scale suggesting its use for future research purposes. Implications The present study found high correlations between the metacognitive self regulation scale and the observed indicators for the cognitive strategy scale such as elaboration ( r = .876, p < .001), critical thinking ( r = .730, p < .001), and organization ( r = 716, p < .001) subscales. The high multicollinearity found between the cogni tive and metacognitive strategies indicated that the two scales might be measuring a similar construct. Therefore, one of the major theoretical implications of this study is that t he measurement of cognitive and metacognitive constructs might not be easy for researchers. Artzt and Armour Thomas (1992) also indicated that although cognitive distinct controlling, monitoring, and regulating cognitive processes, and cognitive activities such as the use of elaboration, organization, and critical thinking strategies may implic itly in volve the use of metacognitive actions. As a result, it is difficult to categorize a particular problem solving behavior as purely cognitive or purely metacognitive. For these reason s Artzt and Arthur Thomas advocate for observing students during s mall group problem solving. The small group problem solving not only provides natural settings for activating cognitive and metacognitive strategies but also offers researchers PAGE 141 141 with opportunities to differentiate problem solving behaviors into cognitive and metacognitive activities own actions The present study also has some practical implications for the educators. The study provided evidence that self efficacy is an important factor impacting performance in solving modeling tasks. influences the amount of effort and time they expend (Schunk & Pajares 2008). Therefore, teachers should support students in raising their self efficacy beliefs for solving complex modeling problems. The self e fficacy literature especially that stems from offers several suggestions for raising self efficacy beliefs for solving modeling tasks As peer relationships be come increasingly important in adolescence (Schunk & Meece, 2006), teachers may provide students with vicarious learning experiences to raise their self efficacy beliefs. Specifically, creating opportunities to observe peers with similar or higher ability levels struggle and eventually succeed when engaged in cognitively demanding modeling problems may motivate students to exert significant effort, time, and energy towards understanding and solving modeling problems. Instructional practices such as providin g students with effective feedback and engaging them in self efficacy beliefs (Schunk & Mullen, 2012) Teacher feedback intended to encourage and make students aware of their capabilities supports them in believing themselves capable of solving complex modeling tasks. This is because students doubt their own competencies and hearing positive performance related statements from teachers or their peers provide them with information about how well they are lea rning and performing on these tasks. PAGE 142 142 In addition to performance feedback, teachers should provide students with attribution feedback encouraging them to attribute their success to effort and failure to lack of effort. This would motivate students with low abilities to work harder and persist longer on academic tasks. Further, teachers should educate students to self reflect, self monitor, and self evaluate their solution processes (Schunk & Ertmer, 2000; Schunk & Pajares, 2008). Such metacognitive processes own learning progress, which further motivate s them to persist at ta sks and more cognitively engage in them. Furthermore, creating positive and supportive learning environments such as encouraging students to partici pate in classroom discussions, explaining their thought processes, and focusing on the process rather than the correct efficacy beliefs. Delimitations and Limitations of the Study This study is delimited in seve ral ways. First, SRL processes in the present study are limited to self efficacy beliefs and cognitive and metacognitive strategy use. According to the model of self regulation of learning proposed by Zimmerman and Campillo (2003), effective problem solver s engage in several self regulatory processes such as goal setting, strategic planning, self control, self observation, self judgment, and self reaction processes, and they exhibit a variety of motivational beliefs such as self efficacy, outcome expectation, task value, and goal orientation. Although all these variables are important, inclusion of too many variables in the statistical model would have been difficult to study and manage Additionally, considering too many variables reduces the efficiency of a statistical model as it results in over fitting of a model with the sample data (Kline, 2005). The definition of cognitive strategy use was also delimited to self reported use of el aboration, organization, and critical thinking strategies. The use of PAGE 143 143 rehearsal strategies such as naming, reciting, or repeating material for learning was deliberately excluded as these strategies were not identified in prior literature as effective in he lping students understand complex modeling problems. Second, the definition of mathematical modeling taken up in this study is somewhat limited. According to Julie (2002), there are two approaches to the teaching of mathematical modeling: modeling as vehi cle and modeling as content. The modeling as vehicle approach uses mathematical modeling activities as a p latform for teaching curriculum based mathematical knowledge and skills. The primary purpose of this a particular content area by using real world contexts. The modeling as content approach, which is also the focus of the present study, involves the process of solving problems arising in other discipline areas or in real world environment s by making use of curriculum based mathematics. This approach was appropriate for the present investigation as it was interested in examining in solving modeling prob lems. T he Standar ds for Mathematical Practice also utilize modeling as content approach to exemplify the modeling expectations. Specifically, modeling practice requires students to apply mathematical concepts to understand problems situated in real world contexts. Further, the present study focused on the extent to which students can utilize school based knowledge and skills to solve real world problems that students might find in their personal life, work, and leisure. Thus, this definition of mathematical modeling may be limited in promoting the essential 21 st century skills and abilities. PAGE 144 144 reported use of cognitive and metacognitive strategies was measured through the MSLQ questionnaire. B y adopting the modeling perspective put forth by the Standards for Ma thematical Practice (CCSSO, 2010) the focus of this study was to examine the extent to which students use and apply learning strategies acquired in school s to solve problems situated in real world contexts. Therefore, the MSLQ scale which measured studen based mathematical tasks, was considere d appropriate. Fourth, the present study engaged eighth and ninth grade students between 13 and 15 years of age. The PISA problems, however were specif ically designed for tenth grade students between 15 to 16 y ears of age (OECD, 2004) This decision was made because think aloud interviews conducted during the pilot study indicated that PISA problems were not challenging for tenth grade studen ts aged 15 t o 18 years of age. This may have been the case b ecause the study was conducted in a research developmental school where students are regularly engaged in inno vative educational projects. Further, there is evidence when students are engaged in think aloud i nterviews, they are more likely to provide correct responses to real world challenging tasks (Selter, 1994, 2001). This is because interview questions suc h as what exactly are you doing or why are you doing it, prompt students to reflect on their problem procedures and solutions. As a result, students may be more likely to provide correct responses. The study has some limitations that need to be acknowledged. First, a limitation of any correlational research study is that correlations between two or more variables cannot be interpreted in terms of causal relationships. For example, the present study PAGE 145 145 suggests that there is a relationship between self performance on the modeling tasks, but the findings do not indicate a causal relationship between increased self efficacy beliefs and correct modeling solution s Second, data were collected using self report questionnaires. Although survey methods are helpful in collecting large amounts of data in a relatively short period of time, there is an underlying assumption that participants provide honest responses to survey questions. The tendency of some participants to provide socially desirable responses might have i ntroduced bias into the results Third, the present study found low reliability estimate for the organization subscale ( = .61) although prior studies (e.g., Kaya, 2007) reported this subscale to have good internal consistency ( = .72). The low reliability index might indicate that the organization strategy subscale is not a reliable measure but in the present study it might be an issue of sample size Fur ther, the modeling test included problems adapted from the PISA 2003 problem solving assessment. These problems were situated within the real life contexts as wel l as embedded within the subject areas of mathematics, science, and reading (OECD, 2004). The individual differences in reading, cognitive ability, their familiarity with the context of the problem, socioeconomic status, gender, and their prior mathematics achievement. The achievement. Summary The main objective of this study was to examine the influence of self performance in solving modeling tasks. The findings of the present study provide PAGE 146 146 elf efficacy beliefs are significantly associated with modeling task success. The study, however, did not provide evidence for the direct influence of SRL strategy use on correctly solving modeling tasks. Further, the structural model did not provide evide nce for the indirect influence of self efficacy beliefs mediated by SRL strategy use on modeling task success. Future researchers might consider modify ing the PISA scoring rubric to capture the range of mathematical skills display ed by students. This may r esult in increasing the reliabili ty of the modeling test. T hey might modify the MSLQ scale involving revision of items with respect to real life problem solving. Finally they should consider giving these problems to tenth grade students between 15 to 16 y ears of age. PAGE 147 147 APPENDIX A THE MODELING TEST ______________________________________________________________________ (a) DECISION MAKING TASKS 1. CINEMA OUTING James, a 15 year old, wants to organize a cinema outing with two of his friends, who are of the same age, during the one week Spring Break. The break begins on Saturday, March 24 th and ends on Sunday, April 1 st James asks his friends for suitable dates and times for the outing. He received the following information. Mike: ome on Monday and Wednesday afternoons for music practice Richard: s to movies suitable for his age and does not walk home. They will fetch the boys home at any time up to 10 p.m. James checks the movie times for the Spring Break. He finds the following information. Regal Cinema 3702 West University Avenue, Gainesville FL 32607 Advance Booking Number: (352) 373 4277 Bargain Day Tuesdays: All films $3 Films showing from Friday March 23 rd for two weeks: Children in the Net 1hr and 53 min 2:00 PM (Mon Fri only) 9:35 PM (Sat/Sun only) Suitable only for persons of 12 years and over Pokamin 1 hr and 45 min 1:40 PM (Daily) 4:35 PM (Daily) Parental Guidance. General viewing, but some scenes may be unsuitable for young children Monsters from the Deep 2 hrs and 44 min Enigma 2 hrs and 24 min PAGE 148 148 7:55 PM (Fri/Sat only) Suitable only for persons of 18 years and over 3:00 PM (Mon Fri only) 6:00 PM (Sat/Sun only) Suitable for persons of 12 years and over Carnivore 2 hrs and 28 min 6:30 PM (Daily) Suitable only for persons of 18 years and over King of the Wild 1 hr and 3 minutes 6:30 PM (Mon Fri only) 6:50 PM (Sat/Sun only) Suitable for persons of all ages Question 1: CINEMA OUTING Taking into account the information James found on the movies, and the information he got from his friends, which of the six movies should James and the boys consider watching? Movie Should the three boys consider watching the movie? Children in the Net Yes/No Monsters from the Deep Yes/No Carnivore Yes/No Pokamin Yes/No Enigma Yes/No King of the Wild Yes/No 2. ENERGY NEEDS This problem is about selecting suitable food to meet the energy needs of a person in Florida. The following table shows the recommended energy needs in kilojoules (KJ) for different people. PAGE 149 149 DAILY ENERGY NEEDS RECOMMENDED FOR ADULTS MEN WOMEN Age (years) Activity Level Energy Needed (KJ) Energy Needed (KJ) From 18 to 29 Light Moderate Heavy 10660 11080 14420 8360 8780 9820 From 30 to 59 Light Moderate Heavy 10450 12120 14210 8570 8990 9790 60 and above Light Moderate Heavy 8780 10240 11910 7500 7940 8780 ACTIVITY LEVEL ACCORDING TO OCCUPATION Light Moderate Heavy Indoor sales person Teacher Construction worker Office worker Outdoor salesperson Laborer Housewife Nurse Sportsperson Samantha Gibbs is a 19 friends invite her out for dinner at a restaurant. Here is the menu. MENU energy per serving (KJ) Soups: Tomato Soup 355 Cream of Mushroom Soup 585 Main Courses: Mexican Chicken 960 Caribbean Ginger Chicken 795 Pork and Sage Kebabs 920 Salads: Potato Salad 750 Spinach, Apricot and Hazelnut Salad 335 Couscous Salad 480 Desserts: Apple and Rasberry Crumble 1380 Ginger Cheesecake 1005 Carrot Cake 565 PAGE 150 150 Milk Shakes: Chocolate 1590 Vanilla 1470 The restaurant also has a special fixed price menu. Fixed Price Menu (50 dollars) Tomato Soup Caribbean Ginger Carrot Cake QUESTION 2: ENERGY NEEDS Samantha keeps a records of what she eats each day. Before dinner on that day her total intake of energy had been 7520 kJ. Samantha does not want her total energy intake to go below or above her recommended daily amount by more than 500 kJ. Decide whether kJ of her recommended energy needs. Show you work. 3. HOLIDAY This problem is about planning the best route for a holiday. s 1 and 2 show a map of the area and the distance between towns. 1: Map of roads between towns PAGE 151 151 QUESTION 3: HOLIDAY Calculate the shortest distance by road between Nuben and Kado. Distance: ________________ miles. ______________________________________________________________________ (b) SYSTEM ANALYSIS AND DESIGN TASKS The Florida Gator Community Service is organizing a five six children (26 girls and 20 boys) have sig ned up for the camp, and 8 adults (4 men and 4 women) have volunteered to attend and organize the camp. PAGE 152 152 Dormitory Allocation. Fill the table to allocate the 46 children and 8 adults to dormitories, keeping to all the rules Name # of Boys # of girls Name(s) of adult(s) Red Blue Green Purple Orange Yellow White 5. COURSE DESIGN A technical college offers the following 12 subjects for a 3 year course, where the length of each subject is one year. PAGE 153 153 QUESTION 5: COURSE DESIGN Each student will take 4 subjects per year, thus completing 12 subjects in 3 years. A student can only take a subject at a higher level if the student has completed the lower level(s) of the same subject in a previous year. For example, you can only take Business Studies Level 3 after completing Business Studies Levels 1 and 2. In addition, Electronics Level 1 can only be taken after completing Mechanics Level 1, and Electronics Level 2 can only be taken after completing Mechanics Level 2. Decide which subjects should be offered for which year, by completing the following table. Write the subject codes in the table. Subject 1 Subject 2 Subject 3 Subject 4 Year1 Year 2 Year 3 6. LIBRARY SYSTEM The John Hobson High School library has a simple system for lending books: for staff members the loan period is 28 days, and for students the loan period is 7 days. The following is a decision tree diagram showing this simple system: PAGE 154 154 The Green wood High School library has a similar, but more complicated, lending system: For books (not including journals) that are not on the reserved list, the loan period is 28 days for staff and 14 days for students. For journals that are not on the reserved list, the loan period is 7 days for everyone. Persons with any overdue items are not allowed to borrow anything. QUESTION 6: LIBRARY SYSTEM You are a student at Greenwood High School, and you do not have any overdue items from the library. You want to borrow a book that is not on the reserved list. How long can you borrow the book for? Answer: ______________ days ______________________________________________________________________ ( c) TROUBLESHOOTING TASKS 7. IRRIGATION Below is a diagram of a system of irrigation channels for watering sections of crops. The gates A to H can be opened and closed to let the water go where it is needed. When a gate is closed no water can pass through it. This is a problem about finding a gate, which is stuck closed, preventing water from flowing through the system of channels. Michael notices that the water is not always going where it is supposed to. He thinks that one of the gates is stuck closed, so that when it is switched to open, it does not open. QUESTION 7: IRRIGATION Michael used the following gate settings to test the gates. Table 1: Gate Settings A B C D E F G H Open Closed Open Open Closed Open Closed Open PAGE 155 155 Michael finds that, when the gates have the Table 1 settings, no water flows through, Decide for each problem case below whether the water will flow through all the way. e, and justify your response. Problem Case Will water flow through all the way? Gate A is stuck closed. All other gates are working properly as set in Table 1. YES / NO Gate D is stuck closed. All other gates are working properly as set in Table 1. YES / NO Gate F is stuck closed. All other gates are working properly as set in Table 1. YES / NO 8. FREEZER Jane bought a new cabinet type freezer. The manual gave the following instructions: Connect the appliance to the power and switch the appliance on. o You will hear the motor running now. o A red warning light (LED) on the display will light up. Turn the temperature control to the desired position. Position 2 is normal. Position Temperature 1 5F 2 0.399F 3 5.80F 4 13F 5 25.6F The red warning light will stay on until the freezer temperature is low enough. This will take 1 3 hours, depending on the temperature you set. Load the freezer with food after four hours. Jane followed these instructions, but she set the temperature control to position 4. After 4 hours, she loaded the freezer with food. After 8 hours, the red warning light was still on, although the motor was running and it felt cold in the freezer. QUESTION 8: FREEZER Jane wondered whether the warning light was functioning pr operly. Which of the following actions and observations would suggest that the light was working properly? PAGE 156 156 Action and Observation Does the observation suggest that the warning light was working properly? She put the control to position 5 and the red light went off. Yes / No She put the control to position 1 and the red light went off. Yes / No She put the control to position 1 and the red light stayed on. Yes / No 9. HOSPITAL The cardiology department at a local hospital employs 5 doctors. Every doctor can work from Monday to Friday and examine 10 patients per day. In a whole year (365 days, 52 weeks) a cardiologist can have 25 days for holiday s and 26 days off for attending seminars and the weekends. QUESTION 9: Can the 5 cardiologists deal with the 12000 patients that are expected to arrive at the hospital during the following year? If not, what do you suggest that the hospital can do? Explain your answer. PAGE 157 157 APPENDIX B SELF EFFICACY SCALE Th efficacy related to each problem on the modeling ability test. Students will read each problem and respond to following questions on a scale ranging from 0 to 100. 1. How sure are you that you can u nderstand this mathematical problem? 0 10 20 30 40 50 60 70 80 90 100 Not at all Sure Moderately Sure Very Sure 2. How sure are you that you can determine a strategy to solve this problem? 0 10 20 30 40 50 60 70 80 90 100 Not al all sure Moderately Sure Very Sure 3. How sure are you that you can determine the information required to solve this problem? 0 10 20 30 40 50 60 70 80 90 100 Not at all Sure Moderately Sure Very Sure 4. How sure are you that you can solve this mathematical problem correctly? 0 10 20 30 40 50 60 70 80 90 100 Not at all Sure Moderately sure Very Sure PAGE 158 158 APPENDIX C MOTIVATED STRATEGIES FOR LEARNING QUESTIONNAIRE ______________________________________________________________________ _____________ Participant Number: _____________ __________ Month of birth: _____________ Year of birth: __________ Grade in school: _____________ Gender: Male Female Ethnicity: American Indian Asian Black or African American H ispanic or Latino/a Native Hawaiian or Pacific Islander White, non Hispanic Other (please specify) The following questions ask about your learning strategies and study skills for YOUR mathematics class. When the questions ask you about the re adings for the class think about reading the textbook that you have for your mathematics class or other materials your teacher might give you to read or study from. Again, there are no right or wrong answers. Answers the questions about how you study in this class as accurately as possible. Use the same scale to answer the remaining questions. If you think the statement is very true of you, fill in the circle next to 7 ; if a statement is not at all true of you, fill in the circle next to 1. If the statement is more or less true of you, find the number between 1 and 7 that best describes you. Not at all true O 1 O 2 O 3 O 4 O 5 O 6 O 7 Very true of me PAGE 159 159 Motivated Strategies for Learning Questionnaire Not at all true Very true of me 1. When I study the readings (your mathematics textbook) for this course, I outline the material to help me organize my thoughts. O 1 O 2 O 3 O 4 O 5 O 6 O 7 2. I often find myself questioning things I hear or read in this course to decide if I find them convincing. O 1 O 2 O 3 O 4 O 5 O 6 O 7 3. When I become confused about go back and try to it out. O 1 O 2 O 3 O 4 O 5 O 6 O 7 4. When I study for this course, I go through the readings (your mathematics textbook) and my class notes and try to find the most important ideas. O 1 O 2 O 3 O 4 O 5 O 6 O 7 5. If course readings (your mathematics textbook) are difficult to understand, I change the way I read the material. O 1 O 2 O 3 O 4 O 5 O 6 O 7 6. When a theory, interpretation, or conclusion is presented in class or in the readings (your mathematics textbook), I try to decide if there is good supporting evidence. O 1 O 2 O 3 O 4 O 5 O 6 O 7 7. I make simple charts, diagrams, or tables to help me organize course material. O 1 O 2 O 3 O 4 O 5 O 6 O 7 8. I treat the course material as a starting point and try to develop my own ideas about it. O 1 O 2 O 3 O 4 O 5 O 6 O 7 9. When I study for this class, I pull together information from different sources, such as lectures, readings (your mathematics textbook), and discussions we have in class. O 1 O 2 O 3 O 4 O 5 O 6 O 7 10. Before I study new course material thoroughly, I often skim it to see how it is organized. O 1 O 2 O 3 O 4 O 5 O 6 O 7 11. I ask myself questions to make sure I understand the material I have been studying in this class. O 1 O 2 O 3 O 4 O 5 O 6 O 7 12. I try to change the way I study in order O 1 O 2 O 3 O 4 O 5 O 6 O 7 PAGE 160 160 to fit the course requirements and the way my teacher presents the material. 13. I try to think through a topic and decide what I am supposed to learn from it rather than just reading it over when studying for this course. O 1 O 2 O 3 O 4 O 5 O 6 O 7 14. I try to relate ideas in this subject to those in other courses whenever possible. O 1 O 2 O 3 O 4 O 5 O 6 O 7 15. When I study for this course, I go over my class notes and make an outline of important concepts. O 1 O 2 O 3 O 4 O 5 O 6 O 7 16. When reading (your mathematics textbook) for this class, I try to relate the material to what I already know. O 1 O 2 O 3 O 4 O 5 O 6 O 7 17. I try to play around with ideas of my own related to what I am learning in this course. O 1 O 2 O 3 O 4 O 5 O 6 O 7 18. When I study for this course, I write brief summaries of the main ideas from the readings (your mathematics textbook) and my class notes. O 1 O 2 O 3 O 4 O 5 O 6 O 7 19. I try to understand the material in this class by making connections between the readings (your mathematics textbook) and the concepts from my O 1 O 2 O 3 O 4 O 5 O 6 O 7 20. Whenever I read or hear an assertion or conclusion in this class, I think about possible alternatives. O 1 O 2 O 3 O 4 O 5 O 6 O 7 21. When studying for this course I try to determine which concepts I don't understand well. O 1 O 2 O 3 O 4 O 5 O 6 O 7 22. When I study for this class, I set goals for myself in order to direct my activities in each study period. O 1 O 2 O 3 O 4 O 5 O 6 O 7 23. If I get confused taking notes in class, I make sure I sort it out afterwards. O 1 O 2 O 3 O 4 O 5 O 6 O 7 24. I try to apply ideas from course readings (your mathematics textbook) in other class activities such as lecture and discussion. O 1 O 2 O 3 O 4 O 5 O 6 O 7 PAGE 161 161 APPENDIX D THE MODELING TEST ______________________________________________________________________ (a) DECISION MAKING TASKS 1. CINEMA OUTING James, a 15 year old, wants to organize a cinema outing with two of his friends, who are of the same age, during the one week Spring Break. The break begins on Saturday, March 24 th and ends on Sunday, April 1 st James asks his friends for suitable dates and times for the outing. He received the following information. Mike: y and Wednesday afternoons for music practice Richard: suitable for his age and does not walk home. They will fetch the boys home at any time up to 10 p.m. James checks the movie times for the Spring Break. He finds the following information. Regal Cinema 3702 West University Avenue, Gainesville FL 32607 Ad vance Booking Number: (352) 373 4277 Bargain Day Tuesdays: All films $3 Films showing from Friday March 23 rd for two weeks: Children in the Net 1hr and 53 min 2:00 PM (Mon Fri only) 9:35 PM (Sat/Sun only) Suitable only for persons of 12 years and over Pokamin 1 hr and 45 min 1:40 PM (Daily) 4:35 PM (Daily) Parental Guidance. General viewing, but some scenes may be unsuitable for young children Monsters from the Deep 2 hrs and 44 min Enigma 2 hrs and 24 min PAGE 162 162 7:55 PM (Fri/Sat only) Suitable only for persons of 18 years and over 3:00 PM (Mon Fri only) 6:00 PM (Sat/Sun only) Suitable for persons of 12 years and over Carnivore 2 hrs and 28 min 6:30 PM (Daily) Suitable only for persons of 18 years and over King of the Wild 1 hr and 3 minutes 6:30 PM (Mon Fri only) 6:50 PM (Sat/Sun only) Suitable for persons of all ages Question 1: CINEMA OUTING Taking into account the information James found on the movies, and the information he got from his friends, which of the six movies should James and the boys consider watching? Movie Should the three boys consider watching the movie? Children in the Net Yes/No Monsters from the Deep Yes/No Carnivore Yes/No Pokamin Yes/No Enigma Yes/No King of the Wild Yes/No 2. ENERGY NEEDS This problem is about selecting suitable food to meet the energy needs of a person in Florida. The following table shows the recommended energy needs in kilojoules (KJ) for different people. PAGE 163 163 DAILY ENERGY NEEDS RECOMMENDED FOR ADULTS MEN WOMEN Age (years) Activity Level Energy Needed (KJ) Energy Needed (KJ) From 18 to 29 Light Moderate Heavy 10660 11080 14420 8360 8780 9820 From 30 to 59 Light Moderate Heavy 10450 12120 14210 8570 8990 9790 60 and above Light Moderate Heavy 8780 10240 11910 7500 7940 8780 ACTIVITY LEVEL ACCORDING TO OCCUPATION Light Moderate Heavy Indoor sales person Teacher Construction worker Office worker Outdoor salesperson Laborer Housewife Nurse Sportsperson Samantha Gibbs is a 19 friends invite her out for dinner at a restaurant. Here is the menu. MENU energy per serving (KJ) Soups: Tomato Soup 355 Cream of Mushroom Soup 585 Main Courses: Mexican Chicken 960 Caribbean Ginger Chicken 795 Pork and Sage Kebabs 920 Salads: Potato Salad 750 Spinach, Apricot and Hazelnut Salad 335 Couscous Salad 480 Desserts: Apple and Rasberry Crumble 1380 Ginger Cheesecake 1005 Carrot Cake 565 PAGE 164 164 Milk Shakes: Chocolate 1590 Vanilla 1470 The restaurant also has a special fixed price menu. Fixed Price Menu (50 dollars) Tomato Soup Caribbean Ginger Carrot Cake QUESTION 2: ENERGY NEEDS Samantha keeps a records of what she eats each day. Before dinner on that day her total intake of energy had been 7520 kJ. Samantha does not want her total energy intake to go below or above her recommended daily amount by more than 500 kJ. Decide whether kJ of her recommended energy needs. Show you work. ____________________________________________________________________ (b) SYSTEM ANALYSIS AND DESIGN TASKS 3 The Florida Gator Community Service is organizing a five six children (26 girls and 20 boys) have signed up for the camp, and 8 adults (4 men and 4 women) have volunteered to attend and organize the camp. PAGE 165 165 QUESTIO N 3 Dormitory Allocation. Fill the table to allocate the 46 children and 8 adults to dormitories, keeping to all the rules Name # of Boys # of girls Name(s) of adult(s) Red Blue Green Purple Orange Yellow White 4 COURSE DESIGN A technical college offers the following 12 subjects for a 3 year course, where the length of each subject is one year. PAGE 166 166 QUESTION 4 : COURSE DESIGN Each student will take 4 subjects per year, thus completing 12 subjects in 3 years. A student can only take a subject at a higher level if the student has completed the lower level(s) of the same subject in a previous year. For example, you can only take Business Studies Level 3 after completing Business Studies Levels 1 and 2. In addi tion, Electronics Level 1 can only be taken after completing Mechanics Level 1, and Electronics Level 2 can only be taken after completing Mechanics Level 2. Decide which subjects should be offered for which year, by completing the following table. Write the subject codes in the table. Subject 1 Subject 2 Subject 3 Subject 4 Year1 Year 2 Year 3 PAGE 167 167 ______________________________________________________________________ (c) TROUBLESHOOTING TASKS 5 IRRIGATION Below is a diagram of a system of irrigation channels for watering sections of crops. The gates A to H can be opened and closed to let the water go where it is needed. When a gate is closed no water can pass through it. This is a problem about finding a g ate, which is stuck closed, preventing water from flowing through the system of channels. Michael notices that the water is not always going where it is supposed to. He thinks that one of the gates is stuck closed, so that when it is switched to open, i t does not open. QUESTION 5 : IRRIGATION Michael used the following gate settings to test the gates. Table 1: Gate Settings A B C D E F G H Open Closed Open Open Closed Open Closed Open Michael finds that, when the gates have the Table 1 settings, no water flows through, Decide for each problem case below whether the water will flow through all the way. justify your response. Problem Case W ill water flow through all the way? Gate A is stuck closed. All other gates are working properly as set in Table 1. YES / NO Gate D is stuck closed. All other gates are working properly as set in Table 1. YES / NO Gate F is stuck closed. All other gates are working properly as set in Table 1. YES / NO PAGE 168 168 6 FREEZER Jane bought a new cabinet type freezer. The manual gave the following instructions: Connect the appliance to the power and switch the appliance on. o You will hear the motor running now. o A red warning light (LED) on the display will light up. Turn the temperature control to the desired position. Position 2 is normal. Position Temperature 1 5F 2 0.399F 3 5.80F 4 13F 5 25.6F The red warning light will stay on until the freezer temperature is low enough. This will take 1 3 hours, depending on the temperature you set. Load the freezer with food after four hours. Jane followed these instructions, but she set the temperature control to position 4. After 4 hours, she loaded the fre ezer with food. After 8 hours, the red warning light was still on, although the motor was running and it felt cold in the freezer. QUESTION 6 : FREEZER Jane wondered whether the warning light was functioning properly. Which of the following actions and ob servations would suggest that the light was working properly? Action and Observation Does the observation suggest that the warning light was working properly? She put the control to position 5 and the red light went off. Yes / No She put the control to position 1 and the red light went off. Yes / No She put the control to position 1 and the red light stayed on. Yes / No PAGE 169 169 APPENDIX E SCORING RUBRIC FOR MODELING PROBLEMS 1. CINEMA OUTING SCORING Full Credit (Score 2) Partial Credit (Score 1) No Credit (Score 0) If the answers are in the order: Yes, No, No, No, Yes and Yes One incorrect answer Other responses 2. ENERGY NEEDS SCORING Full Credit (Score 2) Partial Credit (Score 1) No Credit (Score 0) Food from the fixed price menu does not contain enough energy for Samantha to keep within 500 KJ of her energy needs. The following steps are necessary: (i) Calculation of the total energy of the fixed price menu: 355+795+565=1715 (ii) Recognition that S recommended energy need is 9820 KJ. (iii) Calculating 7520+1715=9235 and showing that Samantha would be more than 500 KJ below her recommended energy need. (iv) Conclusion that the fixed price menu does not contain enough energy. Correct me thod, but a minor error or omission in one of the calculation steps leading to a correct or incorrect, but consistent, conclusion. 1715+7520=9235, this is within 500 of Or Correct calculations, but conclusion Other responses, including No, Samantha should not order from the fixed price menu 1715 is above 500 KJ, so Samantha should not have this Or Correct reasoning in words but no s shown. That is partial credit needs to have some supporting s. The fixed price menu does not have enough KJ, so Samantha should not have it. Full Credit (Score 2) Partial Credit (Score 1) No Credit (Score 0) Six conditions to be satisfied Total girls = 26 Total boys = 20 One or two conditions (as mentioned in the first column) violated. Violating the same condition more than once will Other responses. PAGE 170 170 Total adults = four female and four male Total (children and adults) per dormitory is within the limit for each dormitory. People in each dormitory are of the same gender. At least one adult must sleep in each dormitory to which children have been allocat ed. be counted as ONE violation only. Forgetting to count the adults in the tally of the number of people in each dormitory. The number of g irls and the number of boys are interchanged (no. of girls = 20, no. of boys = 26), but everything else is correct. (Note that this counts as two conditions violated) The correct number of adults in each dormitory is given, but not their names or gender. (Note that this violates both condition 3 and condition 5). 5. 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PAGE 183 183 BIOGRAPHICAL SKETCH Anu Sharma graduated from Punjab University, Chandigarh (India) in the year 1997 with a Bachelor of Science in m athematics physics, and c hemistry In 1998, s he completed her Bachelor of Education from the same university. After graduation, she taught elem entary level mathematics and science for 10 year s at Kundan Vidya Mandir (KVM) School in Ludh iana, India. Anu also served as the coordinator for academics and co curricular activities, supporting other faculty in curricular, instructional, and assessment p lanning as well as organizing and managing various co curricular activities. While teaching at KVM School, she also earned a degree in Master of Mathematics from Himachal Pradesh Unive rsity, Shimla, India in 2006. Shortly after coming to the United States in 2008, Anu enrolled at the University of Florida as a graduate student to begin her PhD in Curriculum and Instruction with emphasis in mathematics education. As part of her doctoral program, she also earned a cognate in Educational Psychology. She passed her qualifying exams in December 2010 and her dissertation proposal was approved in September 2011. She received her PhD from the University of Florida in the summer of 2013 with an aim to continue exploring the possibilities of integrating mode ling activities and self regulated learn ing within regular classrooms. After graduating, she joined the Centre for Educational Testing and Evaluation at the University of Kansas as a post doctoral researcher. 