Supporting Secondary Students' Perseverance for Solving

SUPPORTING SECONDARY STUDENTS’ PERSEVERANCE FOR SOLVING

CHALLENGING MATHEMATICS TASKS

Joseph DiNapoli

A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Education

Summer 2018

SUPPORTING SECONDARY STUDENTS’ PERSEVERANCE FOR SOLVING

CHALLENGING MATHEMATICS TASKS

Joseph DiNapoli

Approved: ______Chrystalla Mouza, Ed.D. Interim Director of the School of Education

Approved: ______Carol Vukelich, Ph.D. Dean of the College of Education and Human Development

Approved: ______Douglas J. Doren, Ph.D. Interim Vice Provost for the Office of Graduate and Professional Education

I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.

Signed: ______Amanda Jansen, Ph.D. Professor in charge of dissertation

Signed: ______Alfinio Flores, Ph.D. Member of dissertation committee

Signed: ______James Hiebert, Ph.D. Member of dissertation committee

Signed: ______James A. Middleton, Ph.D. Member of dissertation committee

ACKNOWLEDGEMENTS

Realizing my dream of earning a Ph.D. in Mathematics Education would not have been possible without the support of two special groups of people: the wonderful community at the University of Delaware and my loving family. First and foremost, I would like to express my appreciation for my advisor, Dr. Amanda Jansen. Thank you for being the best possible mentor I could ever have imagined. Your constant encouragement and meaningful feedback helped shape my development in so many productive ways.

Also, your tireless work ethic and unparalleled ambition have set such a strong example for the professor I hope to become. I want you to know that I regard the privileged decision to work under your tutelage to be the best decision of my professional life. Your positive influence will continue to guide me throughout my career in education.

To my committee members, Drs. Jim Hiebert, Alfinio Flores, and Jim Middleton: thank you for your thoughtful engagement with my ideas throughout the dissertation process. Each of you provided a unique expertise from which I learned so much. Thank you for your time and energy in making me a better researcher.

To the mathematics education faculty members at the University of Delaware: thank you for being such passionate and enthusiastic scholars. It is abundantly clear to me why the doctoral program at UD is one of the best in the country. You are all wonderful at your jobs.

Thank you to the mathematics education doctoral students with which I have had the pleasure of working over my four years at UD. The sense of community amongst grad students is a prominent reason why this program is so terrific. I want to single out and thank Emily Miller and Alison Marzocchi for their mentorship and kindness; I owe so much of my success to you both. I also want to especially thank Tony Mixell and

Siobahn Suppa for their empathy and friendship as we navigated the stages of the Ph.D. journey together.

To my parents and sister: thank you for your unwavering support along every step of the way these past four years. It was your initial encouragement that gave me the confidence to pursue my Ph.D. dream, even if it meant leaving a teaching job I enjoyed.

Your genuine interest in my journey has been so motivating to me. I would not be here without you.

Last, but, most assuredly, not least, I would like to recognize my wife, Kendi, to whom this dissertation is dedicated. This achievement would not have been possible without your love and sacrifice. Everything I have accomplished and everything I will accomplish is because of your support. Thank you, Kendi.

TABLE OF CONTENTS

LIST OF TABLES ...... ix LIST OF FIGURES ...... x ABSTRACT ...... xii

Chapter

1 DISTINGUISHING BETWEEN GRIT, PERSISTENCE, AND PERSEVERANCE FOR LEARNING MATHEMATICS WITH UNDERSTANDING ...... 1

Grit ...... 5

Measures/Operationalization ...... 7 Associations between Grit and Learning ...... 9 Critiques of Grit Relative to Learning Mathematics with Understanding ...... 11

Persistence...... 15

Measures/Operationalization ...... 17 Associations between Persistence and Learning ...... 20 Critiques of Persistence Relative to Learning Mathematics with Understanding ...... 24

Perseverance ...... 28

Measures/Operationalization ...... 31 Associations between Perseverance and Learning ...... 37 Critiques of Perseverance Relative to Learning Mathematics with Understanding ...... 43

Toward a Distinction between Constructs ...... 46 Conclusion ...... 55

2 SUPPORTING SECONDARY STUDENTS’ PERSEVERANCE FOR SOLVING CHALLENGING MATHEMATICS TASKS ...... 56

Conceptual Framework: Perseverance in Problem-solving ...... 58 Supporting Student Perseverance with Mathematical Tasks ...... 62

Scaffolding Perseverance via Mathematical Task Structure and Teacher Questioning ...... 64 Considering Embedded Conceptualization Scaffolds for Perseverance Support ...... 70

Next Steps in Perseverance Research ...... 75

Methods...... 76

Participants and Context ...... 77 Data Collection ...... 79

Think-aloud Interviews ...... 80

Mathematical Tasks ...... 80 Scaffold Conditions ...... 82

Video-reflection Interviews ...... 85 Exit Interviews ...... 86

Data Analysis ...... 87

Three-Phase Perseverance Framework ...... 89 Coding Decisions ...... 92 Reliability Procedures ...... 95

Results ...... 95

Result 1: Higher Quality Perseverance on Scaffolded Tasks than on Non- Scaffolded Tasks, Especially during Additional Attempt...... 96

James’s Perspective: A “Life-Preserver” amidst “Chaos” ...... 101

James’s Experience with Cross Totals ...... 102 James’s Experience with Triangular Frameworks ...... 110 James’s Overall Experience ...... 114

Supporting Points of View: Tom and Marcia ...... 115

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Result 2: Perseverance Improved in Quality over Time ...... 117

Sandra’s Perspective: “Getting Better at Sticking with It” ...... 122

Sandra’s Experience with Cross Totals ...... 123 Sandra’s Experience with Sidewalk Stones ...... 127 Sandra’s Experience with Skeleton Tower ...... 133 Sandra’s Overall Experience...... 143

Supporting Points of View: Dennis and Laurie ...... 145

Summary of Results ...... 148

Discussion ...... 149

Limitations and Future Research ...... 154

Conclusions and Implications ...... 156

REFERENCES ...... 159

Appendix

A PRETEST ITEMS ...... 178 B PARTICIPANTS’ TIME SPENT ON EACH TASK ...... 181 C THINK-ALOUD INTERVIEW PROTOCOL ...... 182 D MATHEMATICAL TASKS...... 183 E VIDEO-REFLECTION INTERVIEW PROTOCOL ...... 193 F EXIT INTERVIEW PROTOCOL ...... 194 G PROBLEM-SOLVING HEURISTICS ...... 195 H SOLUTIONS TO MATHEMATICAL TASKS ...... 198 I INSTITUTIONAL REVIEW BOARD APPROVAL LETTERS ...... 200

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LIST OF TABLES

Table 1 Grit Scale (Grit-O) Items ...... 7

Table 2 Short Grit Scale (Grit-S) Items ...... 9

Table 3 Persistence Scale Items ...... 18

Table 4 Three-Phase Perseverance Framework ...... 32

Table 5 Distinctions between Grit, Persistence, and Perseverance ...... 47

Table 6 Mathematical Task Details ...... 84

Table 7 Three-Phase Perseverance Framework ...... 90

Table 8 Three-Phase Perseverance Framework Scores ...... 97

Table 9 Perseverance Frequencies in Additional Attempt Phase ...... 98

Table 10 James’s Three-Phase Perseverance Framework scores for Cross Totals and Triangular Frameworks ...... 101

Table 11 Summary of Regression Analyses for Participants’ Three-Phase Perseverance Framework Scores ...... 119

Table 12 Sandra’s Three-Phase Perseverance Framework scores for Cross Totals, Sidewalk Stones, and Skeleton Tower ...... 123

LIST OF FIGURES Figure 1 Sidewalk Stones Task ...... 16

Figure 2 Anghileri’s (2006) Scaffolding Framework ...... 72

Figure 3 Scaffolding Perseverance in Problem-solving...... 73

Figure 4 James’s Initial Conceptualization for Cross Totals ...... 103

Figure 5 James’s Guessing-and-checking during Initial Attempt on Cross Totals .....104

Figure 6 James’s Observations of the Given Example during Additional Attempt on Cross Totals ...... 107

Figure 7 James’s Own Examples during Additional Attempt on Cross Totals ...... 108

Figure 8 James’s Guessing-and-checking during Initial Attempt on Triangular Frameworks...... 112

Figure 9 James’s Rules for Triangular Frameworks ...... 112

Figure 10 Mean Three-Phase Perseverance Scores over Time ...... 118

Figure 11 Sandra’s Initial Conceptualization for Cross Totals ...... 124

Figure 12 Sandra’s Own Example during Initial Attempt on Cross Totals ...... 125

Figure 13 Sandra’s Rules for Cross Totals ...... 126

Figure 14 Sandra’s Initial Conceptualization for Sidewalk Stones ...... 127

Figure 15 Sandra’s Table of Values during Initial Attempt on Sidewalk Stones ...... 128

Figure 16 Sandra’s New Diagram during Additional Attempt on Sidewalk Stones .....131

Figure 17 Sandra’s Amended Table of Values during Additional Attempt on Sidewalk Stones ...... 131

Figure 18 Sandra’s Rules for Sidewalk Stones ...... 131

Figure 19 Sandra’s Initial Conceptualization for Skeleton Tower ...... 134

Figure 20 Sandra’s Counting of Cubes during Initial Attempt toward First Objective on Skeleton Tower ...... 134

Figure 21 Sandra’s Diagram during Additional Attempt toward First Objective on Skeleton Tower ...... 138

Figure 22 Sandra’s First Rule for Skeleton Tower ...... 138

Figure 23 Sandra’s Table during Additional Attempt toward Second Objective on Skeleton Tower ...... 140

Figure 24 Sandra’s Second Rule for Skeleton Tower ...... 142

Figure 25 Scaffolding Perseverance in Problem-solving: A Cyclical Representation...... 152

ABSTRACT

Perseverance is a key process through which mathematics can be learned with understanding. However, withstanding such uncertainty can be difficult for students to endure and necessitates support. In this study, I investigated ways in which embedded scaffolds encouraged 10 ninth-grade students’ perseverance for solving a series of analogous challenging mathematics tasks. I designed a Three-Phase Perseverance

Framework to capture the student perspective of how they persevered, both before and after reaching a perceived impasse. I conducted think-aloud interviews, video-reflection interviews, and an exit interview with each student as he or she engaged with one task per week for five weeks. Three tasks were embedded with conceptualization scaffolds prompting students to record their initial conceptual thinking prior to engagement; two tasks had no scaffolds. Results showed that students persevered significantly more on scaffolded tasks than on non-scaffolded tasks, with the most notable difference occurring after students encountered an impasse. Also, the quality of students’ perseverance improved over time, more so when working on scaffolded tasks than on non-scaffolded tasks. Students attributed much of their perseverance success to their preliminary conceptualization work prompted by the scaffolds. The findings suggest these scaffolds supported perseverance in problem solving in a cyclical manner, as students were encouraged to revisit their initial conceptual thinking upon impasse and re-initiate and re- sustain their productive struggle by exploring a different set of mathematical ideas.

xii

Furthermore, the data show malleability of perseverance, suggesting students can improve their perseverance in problem solving over time through carefully designed deliberate practice.

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Chapter 1

DISTINGUISHING BETWEEN GRIT, PERSISTENCE, AND PERSEVERANCE

FOR LEARNING MATHEMATICS WITH UNDERSTANDING

Learning mathematics with understanding focuses on conceptual understanding, that is, the connections learners make among mathematical facts, procedures, and ideas

(Hiebert & Carpenter, 1992). In contrast to procedural skills, such as executing rehearsed mnemonic devices or memorized algorithms to solve problems, conceptual understanding is developed as mental connections become richer and more widespread (Hiebert, 2013).

In the context of problem-solving, a key process that supports the development of conceptual understanding is productive struggle. The purpose of this paper is to carefully examine constructs that are commonly associated with such struggle, namely grit, persistence, and perseverance, and to review and distinguish the ways in which these constructs offer insights into learning mathematics with understanding.

The idea of struggle has long been recognized as key to learning mathematics with understanding (Dewey, 1910; Festinger, 1957; Pólya, 1971), and Hiebert and colleagues (1996) conceptualize productive struggle as grappling with key mathematical ideas that are within reach, but not yet well formed. It is in this process of productively struggling to overcome mathematical obstacles that learners begin to make their own mathematical meanings and build their conceptual understanding (Bjork & Bjork, 2011;

Hiebert & Grouws, 2007; Kapur, 2014; Warshauer, 2014). Yet, developing conceptual

1 understanding by incorporating productive struggle can be a challenge for both teachers and students. Teachers must provide their students with opportunities to wrestle with challenging mathematics, and students must be willing to do the wrestling amid frustration or difficulties.

Wrestling with mathematical ideas while managing frustration, however, can be grueling for students. Some students may prefer to avoid engaging in struggling to solve challenging tasks. As Schoenfeld (1988) posits, students may come to believe that “if you can’t solve a problem in a short amount of time, you should simply give up” (p. 159).

This mindset may limit students’ opportunities to learn and explains the recent emphasis on developing productive student dispositions toward mathematics concept learning in guiding texts like the National Research Council’s (NRC) Adding it Up (2001), the

Common Core State Standards for Mathematics (CCSSM) (2010), and the National

Council for Teachers of Mathematics’ (NCTM) Principles to Actions (2014). The NRC defines a productive disposition as a “habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy”

(Kilpatrick, Swafford, & Findell, 2001, p. 116).

The focus on developing a productive disposition toward struggling with mathematical ideas and connections represents a nationwide mathematics education investment in supporting students to stay engaged with difficult mathematics content. A key feature of such engagement is the continuation of effort, despite challenges and setbacks, to encourage deeper mathematics learning. This commitment is evident in the

2 first Standard for Mathematical Practice in the CCSSM (2010), “[students should] make sense of problems and persevere in solving them” (p. 6). As such, dispositional factors like grit, persistence, and perseverance have entered into the discourse about mathematics learning, both among researchers and practitioners. These terms are being used, sometimes synonymously, to refer to the ideas of sticking-with-it or never-giving-up in the context of students working with challenging mathematics.

In addition to informing teaching practice, researchers have focused empirical studies on investigating grit, persistence, and perseverance in the context of mathematics learning. Research on learning mathematics with understanding should be seeking to uncover the nuanced ways learners use both cognitive and non-cognitive skills to make meaning of important mathematical ideas (Cai et al., 2017). Research emphasizing dispositional constructs like grit, persistence, and perseverance provides insight into students’ engagement with mathematics learning, and these factors contain both cognitive and non-cognitive components. In the context of working on a challenging mathematical task, when learners enact a productive disposition, they are aware of obstacles that they encounter when solving problems, develop ideas about their options for navigating these obstacles, and also exercise a willingness or desire to keep trying despite inevitable setbacks. Hence, research incorporating the perspectives of grit, persistence, and perseverance are particularly important because they address the resilience and self- regulation that is necessary to withstand the frustration and uncertainty of productive struggle and to continue move forward toward meaning-making. Therefore, with such

3 dispositional factors as potential sources of support for productive struggle, and with productive struggle as a key process by which learning mathematics is supported, it is valuable to investigate constructs like grit, persistence, and perseverance to understand students’ experiences while learning mathematics.

A fine-grained analysis of scholarship focused on these dispositional constructs unveils ample distinctions between grit, persistence, and perseverance. These constructs’ respective definitions, operationalizations, usage in research, and potential for uncovering insights into how students learn mathematics have distinct differences. These distinctions are important to make in order to help researchers select and define the particular construct best suited to investigate their particular phenomenon of interest. Accordingly, this paper elaborates upon the ways in which grit, persistence, and perseverance have been defined and operationalized in research on mathematics learning.

Specifically, in the context of research on mathematics learning, the questions that guided my analysis of the constructs were:

1. How are grit, persistence, and perseverance defined? 2. How are grit, persistence, and perseverance operationalized? 3. How are grit, persistence, and perseverance associated, or not, with learning mathematics? 4. What are the similarities and differences between grit, persistence, and perseverance?

In the following sections, I will depict each construct separately before comparing and contrasting them. I will first review how grit, persistence, and perseverance are all uniquely defined, the ways that the constructs are each typically measured, and then

4 present findings associated with learning (with an emphasis on learning mathematics, if applicable), as well as critique of how each construct fits into the growing call for research on learning mathematics with understanding. Then, I will consider grit, persistence, and perseverance in relation to one another, making explicit the differences between each dispositional construct. I will show how each construct affords different insights into mathematics learning at various grain-sizes. I will end by advocating for researchers to purposefully select and define constructs to reflect their phenomenon of interest, and argue that examining students’ perseverance with challenging mathematics, compared to studying grit or persistence, better informs the research community about how students navigate obstacles to learn mathematics with understanding.

Grit

To achieve goals of learning mathematics for understanding, the trait of grit epitomizes hard work and a steadfast will to never give up. First introduced by

Duckworth (2006), grit is “[continuance of effort]1 and passion for specific, high goals, sustained over years” (p. 73). Grit was originally introduced and studied to help explain differences in achievement across all domains between people of similar intelligence and resources. The concept of grit was hypothesized by Duckworth, Peterson, Matthews, and

Kelly (2007) to be just as important as talent when explaining why individuals were

1 Duckworth et al. (2007) use perseverance here, but to avoid confusion between constructs, I will use continuance of effort, a phrase Duckworth and colleagues suggest is synonymous with their perspective of perseverance.

5 successful in achieving their long-term goals. Framing grit as an individual’s “sustained commitment to their ambitions” (p. 1088), Duckworth and colleagues (2007) believed such relentless hard work, over time, could propel someone into the upper echelons of achievement, perhaps before a more-gifted but less-gritty peer.

A key feature of this non-cognitive construct is the nature of the long-term goal toward which an individual is striving. For example, grit can help explain a student achieving goals like earning a college degree in mathematics or scoring highly on an

Advanced Placement calculus exam. Such achievements require years of preparation, dedication, and strenuous hard work, maintaining both interest and effort despite distractions, failure, adversity, and plateaus in progress (Duckworth et al., 2007).

Duckworth and colleagues (2007) allude to grit metaphorically via a long-distance runner: “The gritty individual approaches achievement as a marathon; his or her advantage is stamina. Whereas disappointment or boredom signals to others that it is time to change trajectory and cut losses, the gritty individual stays the course” (p. 1088). Most runners would agree, finishing a 26.2 mile race is indeed a long-term goal. Moreover, numerous challenges emerge during the hours spent running – physical pain, mental anguish, and other competitors who appear faster and stronger. It takes grit to keep running despite these challenges. It is no different for the mathematics student with a passion to learn. Learning mathematics with understanding is a marathon that requires stamina. There will always be setbacks and challenges, such as making mistakes,

6 struggling to understand a concept, and peers who appear to be smarter and progressing faster, but the gritty individual stays the course.

Measures/Operationalization

The operationalization of grit considers both the long-term consistency of interests and long-term continuance of effort. Because the grit construct is meant to describe a personality trait, Duckworth and colleagues (2007) developed a 27-item Likert survey asking participants to discern the degree to which a particular statement described how they tend to pursue long-term goals. The research team generated one scale to assess continuance of effort that “tapped the ability to sustain effort in the face of adversity” (p.

1090). The team generated another scale to assess consistency of interests, that is, to reveal the motivation behind such sustained effort. Motivators such as “subjective interest, [being] afraid of change, compliant with the expectations of others, or unaware of alternative options” (p. 1090) were included. After several rounds of correlation analysis, internal reliability analysis, efficiency analysis, and exploratory factor analysis,

Duckworth et al., (2007) retained 12 of the 27 original items to construct the 2-factor Grit

Scale (Grit-O) (see Table 1 for items).

Table 1

Grit Scale (Grit-O) Items (Duckworth et al., 2007)

Factors and Grit Scale Items Consistency of Interests Continuance of Effort I often set a goal but later choose to I have achieved a goal that took years of pursue a different one.* work.

New ideas and new projects sometimes I have overcome setbacks to conquer an distract me from previous ones.* important challenge. I become interested in new pursuits every I finish whatever I begin. few months.* My interests change from year to year.* Setbacks don’t discourage me. I have been obsessed with a certain idea I am a hard worker. or project for a short time but later lost interest.* I have difficulty maintaining my focus on I am diligent. projects that take more than a few months to complete.* Participants rated items on a 5-point scale from 1 = not at all like me to 5 = very much like me. *Item was reverse scored.

The Grit Scale (Grit-O) theorized grit as a compound trait comprising stamina in dimensions of interest and effort, but had room for improvement regarding predictive validity and efficiency (Duckworth & Quinn, 2009). Across six studies, Duckworth and

Quinn (2009) employed item analysis, confirmatory factor analysis, validity analysis, and stability analysis to validate a more efficient measure of grit, the Short Grit Scale (Grit-

S). Grit-S retained “the 2-factor structure of the original Grit Scale (Grit-O) with four fewer items and improved psychometric properties” (Duckworth & Quinn, 2009, p. 166)

(see Table 2 for the retained items in Grit-S). Used to measure grit for pursuing long-term goals across fields and domains, including goals of mathematical understanding and meaning-making, Grit-S is considered the most valid self-reporting measure of its kind today.

Table 2

Short Grit Scale (Grit-S) Items (Duckworth & Quinn, 2009)

Factors and Grit Scale Items Consistency of Interests Continuance of Effort I often set a goal but later choose to I finish whatever I begin. pursue a different one.* New ideas and new projects sometimes Setbacks don’t discourage me. distract me from previous ones.* I have been obsessed with a certain idea I am a hard worker. or project for a short time but later lost interest.* I have difficulty maintaining my focus on I am diligent. projects that take more than a few months to complete.* Participants rated items on a 5-point scale from 1 = not at all like me to 5 = very much like me. *Item was reverse scored.

Associations between Grit and Learning

Research on grit has suggested several positive outcomes relative to long-term student learning and achievement. During initial research validating the construct, grit predicted educational attainment and retention in scholarly programs, grade-point average

(GPA), and competitive academic achievement for subjects of different ages and contexts, including adult volunteers (ages 25-65+), West Point cadets (ages 18-22), and

Scripps National Spelling Bee contestants (ages 7-15) (Duckworth, 2006; Duckworth et al., 2007). Among large samples of adults, grit predicted higher education levels and traits supportive of achievement motivation, including conscientiousness, agreeableness, extraversion, and openness to experience (Duckworth et al., 2007; Duckworth & Eskreis-

Winkler, 2013). Such findings suggest that the willingness and stamina to stick-with

9 long-term goals over time makes possible the achievement of higher educational degrees and accolades.

Duckworth and colleagues (2007) also found that undergraduate students’ GPAs were positively associated with grit scores. However, students with high GPAs and high grit scores tended to have lower SAT scores than their peers with lower evidence of grit.

These correlational data suggest that perhaps grit can supplement ability to support academic achievement. In other words, among relatively intelligent individuals, those who may be less bright than their peers can compensate by working harder with more determination (Moutafi, Furnham, & Paltiel, 2005). These contentions were additionally supported by data that showed the importance of grit in extracurricular accomplishment.

Duckworth et al. (2007) found that grittier children competing in the Scripps National

Spelling Bee more diligently prepared and performed better than their less gritty peers.

Upon validation of the grit scale (Duckworth et al., 2007; Duckworth & Quinn,

2009), research has continued to show grit as a determinant of student achievement

(Duckworth, Kirby, Tsukayama, Berstein, & Ericsson, 2011; Duckworth, Tsukayama, &

May, 2010; Duckworth, Quinn, & Tsukayama, 2012; Eskreis-Winkler, Shulman, Beal, &

Duckworth, 2014; West et al., 2015; Yeager et al., 2014), teacher retention and effectiveness (Duckworth, Quinn, & Seligman, 2009; Robertson-Kraft & Duckworth,

2014), and general life successes (Duckworth & Gross, 2014; Von Culin, Tsukayama, &

Duckworth, 2014). Working with a large sample of 8th grade public school students, West and colleagues (2015) determined a positive, significant relationship between grit and

10 mathematics test-score gains. These findings suggest non-cognitive skills like grit may play a key role in academic performance in mathematics.

Teachers may also benefit from being gritty. As a result of two longitudinal studies of novice teachers in low-income districts, including teachers of mathematics at primary and secondary levels, Robertson-Kraft and Duckworth (2014) found:

Teachers in their first and second year in the classroom who had demonstrated higher levels of grit [via Grit-S] in their pursuits prior to entering teaching were more likely to remain in the classroom for the school year and, among those who stayed, to make academic gains with their students. (p. 17)

Coupled with similar results demonstrating how teachers reporting higher grit celebrate greater academic gains from their students (Duckworth et al., 2009), these findings suggest that students’ long-term understandings may be affected by their teachers’ grit in addition to their own.

In the context of pursuing long-term goals, measures of grit have generally correlated with measures of achievement. Grit has predicted various educational successes from the perspective of both the student and the teacher. In all, the research base on grit supports Duckworth’s (2006) original hypothesis that grit is a powerful forecaster of achievements that necessitate continuous effort and decided interest over extended periods of time.

Critiques of Grit Relative to Learning Mathematics with Understanding

Although grit has been predictive of long-term achievements, it is met with some criticism for lacking insights for the processes of learning mathematics with

11 understanding. Much of the research base on grit suggests that gritty students work hard and feel determined, but there is little evidence of specific connections to understanding mathematics over time. Few grit studies have focused on relationships with mathematics at all, and those that have only considered broad-strokes achievements, such as SAT scores (see Duckworth et al., 2007), standardized state tests (see West et al., 2015), or passing one’s mathematics courses to help graduate high school (see Eskreis-Winkler et al., 2014). Prior evidence that grit supports a person’s capability with achieving high scores on a mathematics exam or passing a mathematics class simply demonstrates that a gritty individual has achieved these things, but fail to explain how grit helps individuals actually learn mathematics with understanding, primarily by making their own connections between important mathematical ideas (Hiebert, 2003).

Other related criticisms of grit research have concentrated on the outcomes in structured versus unstructured learning environments (Farrington et al., 2012; Grohman,

2014; Gutman & Schoon, 2013; Headden & McKay, 2015; Howard, 2015; Ivcevic &

Bracket, 2014; Sparks, 2014). In their synthesis of how new research can boost student engagement, Headden and McKay (2015) set forth the general critique that grit may help explain how individuals achieve in proceduralized settings focused on routine, but the relationships between grit and creative work are far less apparent. As part of a study on the relationships between undergraduates’ non-cognitive traits and creative problem- solving, Grohman (2014) found that grit did not predict creative achievement or ingenuity. Similarly, Ivcevic and Bracket (2014) found that grit was unrelated to several

12 measures of high schoolers’ success, and posited that grit did not help explain how students regulate their behavior during emotion-laden situations, such as sticking-with-it during particularly challenging moments of academic work. These findings suggest that grit may forecast success in predictable situations with solution pathways or procedures to follow, but grit may not explain the intricacies of navigating uncertain situations requiring exploration, such as making progress on a challenging mathematical task for which a solution path is not known.

Duckworth et al.’s (2007) conceptualization of grit clearly recognizes the salience of “the amount of energy one invests in a particular task at a given moment in time …

[and] the consistency of one’s long-term goals and the stamina with which one pursues those goals” (p. 1098. Items in Grit-O and Grit-S, such as “I am diligent”, “Setbacks don’t discourage me”, and “I often set a goal but later choose to pursue a different one” appear to measure what they are intended to measure, but offer little insight into what and how someone may have learned in a given moment in time.

Also unclear is the nature of obstacles obstructing a long-term achievement. Not knowing the ways in which an individual was challenged during his or her time spent pursuing a goal clouds the issue of whether or not achieving such a goal implies grit, by default. In this way, measuring a person’s high score on the grit scale may just be measuring achievement of a long-term goal in general, not necessarily a struggle over years despite setbacks and frustrations at key moments in time. This is not to say that achieving a summative goal cannot be difficult: certainly a consistency of interest and

13 effort over a long period of time is useful for success toward almost any accomplishment.

But without knowing the details of the ways in which an individual was challenged, it is difficult to know what a high grit score reveals about a person other than that he or she achieved a long-term goal.

Although grit has been shown to coincide with several types of summative achievements and is often celebrated as a key predictor for successful learning in general, the construct has been criticized for lacking insights into the process by which rich mathematical meanings are developed. Grit does not explain the details of key moments in time when an individual was challenged and pushed to the brink of giving up, nor does it explain any creative insights that may have been developed. Learning mathematics with understanding requires many moments of approaching and grappling with unfamiliar challenges to create meaning (Hiebert & Grouws, 2007), yet the key mechanisms of how a gritty individual navigates such obstacles in the moment, propelling him or her forward toward success in the long term, are unclear. It is in those details that help explain how an individual becomes gritty, and how he or she incrementally learns mathematics with understanding. Therefore, to better learn about those key moments, researchers should consider engagement and experiences toward more short-term goals, within those trying moments when individuals encounter specific obstacles. The next section will detail a related construct, persistence, which shifts the analytic focus to such in-the-moment engagement.

Persistence

During challenging moments along the pathway toward mathematical meaning- making, the construct of persistence portrays when a person refuses to quit when faced with setbacks. Peterson and Seligman (2004) define persistence as “voluntary continuation of a goal-directed action in spite of obstacles, difficulties, or discouragement” (p. 229). The concept of persistence describes steadfast, in-the-moment effort in the face of challenge – refusing to quit by continuing one’s particular effort before the task is accomplished. Scholars posit persistence does not alone guarantee accomplishment, but accomplishment is often unattainable without it (Jacobs, Prentice-

Dunn, & Rogers, 1984; Janoff-Bulman & Brickman, 1982; Meyer, 1987; Multon, Brown,

& Lent, 1991).

Contrary to grit, the construct of persistence includes the nature of the short-term goal toward which an individual is working (Peterson & Seligman, 2004). Although the conceptualization of persistence primarily concerns short-term goals, consistent persistence on short-term goals can accumulate and develop into long-term achievements

(Constantin, Holman, & Hojbota, 2011). In this way, the construct of persistence can be considered through the perspective and long and short-term endeavors, with a general emphasis on the short-term.

Trying to solve a challenging mathematical task is an example of a short-term goal in which an individual could be engaged. Consider the task Sidewalk Stones in

Figure 1. To persist in achieving the goal of Sidewalk Stones, an individual must (a) be

15 met with an obstacle of some kind, and (b) endure and attempt to overcome such an obstacle. In these ways, persistence embodies self-regulation through tenacity. With the goal of solving the problem in mind, regulating one’s own behavior underlies the dichotomous battle between continuing engagement and giving up (Carver & Scheier,

2001).

Figure 1. Sidewalk Stones Task

Assuming the individual did not immediately know how to solve Sidewalk Stones from the outset, he or she may begin working on the task by looking for patterns between the different examples of square stones. Using this problem-solving heuristic, the individual may be counting the different shaded stones and noting common characteristics, variations, or differences amongst Pattern #1, Pattern #2, and Pattern #3.

If the individual has not yet generalized the relationship between the different sidewalk stones after a concerted first effort, the essence of persistence is embodied in the ways he or she responds to these struggles – by staying the course and continuing effort using his or her chosen plan of attack. The individual simply starts over, spending effort looking again for similarities and differences between the different patterns of squares, hoping to come upon a generalizable pattern that had eluded him or her previously. This steadfast insistence and dedication to his or her problem-solving strategy describes an individual persisting in the moment toward a short-term goal of developing mathematical understanding.

Measures/Operationalization

Persistence measures have taken the form of self-reports of a tendency to continue effort as well as observations of such engaged behavior. A number of questionnaires include a persistence scale or subscale to help discern an individual’s propensity for sustaining involvement and expending effort while facing obstacles (Cloninger,

Przybeck, & Svrakic, 1991; Cloninger, Pryzbeck, Svrakic, & Wetzel, 1994; Constantin,

Holman, & Hojbota, 2011; Lufi & Cohen, 1989; McCown & Johnson, 1989; Tangney,

Baumeister, & Boone, 2004; Twenge, Tice, & Harter, 2001; Wallack, Goodale, Wijting,

& Smith, 1971). In an effort to synthesize past measures of persistence, Constantin,

Holman, and Hojbota (2011) developed and validated a three-dimensional, 13-item model dividing the construct based on the proximity of desired ends: short-term pursuits, long-term pursuits, and recurrence of unattained pursuits (see Table 3 for items). Items

17 addressing short-term pursuits aim to apprehend an individual’s “ability to remain focused on the goals at hand and to prolong effort in the face of boredom, fatigue or stress” (p. 102). Because short-term pursuits can amass and be directed toward more distal goals, item addressing long-term pursuits aim to assess one’s “ability to remain committed to resource-consuming, higher-order goals that require prolonged investment, despite failures or short-term hedonic costs” (p. 103). Lastly, items addressing the recurrence of unattained pursuits aim to assess “the tendency to continue the pursuit of past, unaccomplished goals” (p. 104), in part to help discern an individual’s willingness to spend long-term effort in response to short-term failure.

Table 3

Persistence Scale Items (Constantin et al., 2011)

Dimensions and Persistence Scale Items Short-term Pursuits I have a good capacity to focus on daily tasks. Once I decide to do something, I am like a bulldog: I don’t give up until I reach the goal. I continue a difficult task even when the others have already given up on it. The more difficult a task is, the more determined I am to finish it. Long-term Pursuits I remain motivated even in activities that spread on several months. Long term purposes motivate me to surmount day to day difficulties. I purposefully pursue the achievement of the projects that I believe in. I keep on investing time and effort in ideas and projects that require years of work and patience. Recurrence of Unattained Pursuits I often come up with new ideas on an older problem or project. From time to time I imagine ways to use opportunities that I have given up. Even though it doesn’t matter anymore, I keep thinking of personal aims that I had to give up. I often find myself thinking about older initiatives that I had abandoned.

It’s hard for me to detach from an important project that I had given up in favor of others. Participants rated items on a 5-point scale from 1 = in a very low degree to 5 = in a very high degree.

Perhaps the most common measures of persistence involve observations of engaged behaviors toward short-term goals. Framed in the experimental tradition of assessing endurance to discomfort and self-control (Howells, 1933), persistence has often been operationalized as the amount of time spent on difficult tasks (e.g., Bandura &

Schunk, 1981; Caroll, 1963, 1985; DiPuala & Campbell, 2002; Fisher, 1996; Helmke &

Schrader, 1996; Hyland et al., 1988; Meyer, 1987; Shrauger & Sorman, 1977; Starnes &

Zinser, 1983; Tafarodi & Vu, 1997; Weiss & Sherman, 1973; Volet, 1997). Outside of strict time calculations, persistence on challenging tasks has been dichotomously measured by asking the individual if he or she would like to stop or continue working after a certain amount of time (e.g., Ainley, Hidi, & Berndorff, 2002; Ainley, Hillman, &

Hidi, 2002; Tulis & Fulmer, 2013). Additionally, persistence has been operationalized without any time component at all. Instead, the number of subtasks attempted or completed within a task (e.g., Bandura & Schunk, 1981; Multon, Brown, & Lent, 1991), or the number of rounds exploring a task has served as a persistence assessment (e.g.,

Niemivirta & Tapola, 2007; Vollmeyer & Rheinberg, 2000). In all, each of these measures of persistence emphasizes whether or not an individual stays engaged with a task, not the ways in which a person is engaging.

Associations between Persistence and Learning

Relatively little research has focused on the primary purpose of gaining insight into persistence, yet numerous studies have found associations between persistence and short-term learning (Shechtman et al., 2013). Often measured as a result of a different variable in which researchers were more interested, persistence coexists with several outcomes facilitative of in-the-moment learning. For instance, there is much evidence of a symbiotic relationship between persistence and self-efficacy in the context of students engaging in learning activities of various structures (Bandura, 1986; Bandura & Schunk,

1981; Carver, Blaney, & Scheier, 1979; Duval, Duval, & Mulilis, 1992; Jacobs, Prentice-

Dunn, & Rogers, 1984; Janoff-Bulman & Brickman, 1982; Meyer, 1987; Multon, Brown,

& Lent, 1991; Niemivirta & Tapola, 2007; Weiss & Sherman, 1973). Bandura and

Schunk (1981) explored the effects of encouraged goal structures in an elementary-age self-directed mathematics learning environment. They found that children persisted more and perceived greater self-efficacy on more complex subtraction tasks when adhering to short-term, proximal goals compared to more long-term, distal goals or no goals at all.

Specifically, children were asked to work independently across seven sessions on a 42- page series of subtraction tasks of increasing difficulty. The researchers encouraged children in the proximal goal group to complete about six pages of tasks per session.

Children in the distal group were encouraged to complete all 42 pages by the end of the seventh session, and children in the control group were not given any goal suggestions.

Under proximal goals, children attempted more subtraction tasks, spent more time on

20 more challenging subtraction tasks, and judged themselves as more competent than all other groups. The authors suggest, at least in part, that students found it easier to persist toward manageable investments of time and energy toward short-term goals. Such persistence, then, often resulted in making mathematical progress, and thus, greater perceptions of self-efficacy.

Additionally, Niemivirta and Tapola (2007) found evidence of a dynamic relationship between persistence and self-efficacy. Working with 100 ninth-grade students, participants in small groups worked on a task intended to induce complex mathematical problem-solving. While working, student self-efficacy was assessed three times. Niemivirta and Topola (2007) found an increasing overall trend of self-efficacy, suggesting students’ self-efficacy became stronger as they continued to engage with the task. They concluded that subsequent changes in self-efficacy appeared to be a function of the students’ unfolding task-specific experiences and overall persistence.

Much research also points to ties between persistence and students’ affective experiences relative to work on challenging tasks (Ainley, Hidi, & Berndorff, 2002;

Ainley, Hillman, & Hidi, 2002; D'Mello & Graesser, 2012; D'Mello, Lehman, & Person,

2010; Eccles & Wigfield, 2002; Efklides & Petkaki, 2005; Elliot & Dweck, 2005;

Graesser & D'Mello, 2012; Hembree, 1988; Linnenbrink & Pintrich, 2004; McLeod &

Adams, 1989; Pekrun, Goetz, Daniels, Stupnisky, & Perry, 2010; Pekrun & Linnenbrink-

Garcia, 2012; Tulis & Ainley, 2011; Tulis & Fulmer, 2013; Zeidner, 2007). For instance,

Tulis and Fulmer (2013) found that persistence was related to both positive-activating

21 and negative-activating emotions. Working with 141 seventh-grade students, participants were asked to individually engage with a challenging mathematics task. Participants described the current state of their general affect and emotions three times during engagement: at the outset, after 1 minute and 30 seconds, and after 3 minutes. Also, after

5 minutes had passed, participants were provided with the option to stop or continue working on the task, and if students wished to continue they were told they could stop at any point thereafter. Tulis and Fulmer (2013) found the decision to continue working to be positively related to both enjoyment, a positive-activating emotion, and anxiety, a negative-activating emotion. The authors concluded that “activating emotions, regardless of whether these emotions are positive or negative, can support students’ decision to persist” (p. 43).

Another example of research examining the relationship between persistence and affect includes Mukhoiyaroh, Atmoko, & Hanurawan’s (2017) investigation of inquiry- based learning and students’ learning persistence. Measuring persistence with Constantin et al.’s (2011) validated scale, Mukhoiyaroh et al. (2017) found a significant effect of inquiry-based pedagogy in an eighth-grade class integrating science, technology, engineering, and mathematics (STEM) on student self-reports of persistence and emotional commitments to academic learning compared to a more traditional teaching style in an analogous class. The authors concluded that students’ learning within a STEM environment conducive for autonomy can influence persistence and emotional regulation

22 during engagement with learning tasks. This is likely because such freedoms encourage genuine exploration and authentic interest in the inquiry process.

Although they rarely have been directly studied, the benefits of persistence are widely recognized relative to short-term, and ultimately accumulating to long-term, learning. Studies involving persistence generally connect steadfast continuations of effort to more positive learning outcomes. Peterson and Seligman (2004) synthesize three chief conclusions applicable to learning mathematics with understanding within research including persistence as a measure. First, persistence may enhance an individual’s emotional enjoyment of subsequent success. Enduring the difficulty of continuing effort in the face of learning challenging mathematics might make the eventual achievement more satisfying, thereby developing a level of interest and pleasure in productive struggle. Second, persistence may improve one’s skills and resourcefulness. Overcoming new mathematical obstacles inevitably requires developing new approaches and new ways to solve problems. In this way, persisters learn new ways to persist, which may be applicable to future meaning-making scenarios. Thus, persistence tends to produce further persistence. Lastly, and arguably most importantly, persistence is necessary to accomplish almost all difficult learning goals. The very nature of learning mathematics with understanding requires persistence because the process is filled with obstacles with which to wrestle. Peterson and Seligman (2004) posit:

Persistence increases one's chances of attaining difficult goals. Relatively few major undertakings are marked by a steady stream of progress and positive feedback. Setbacks and problems are typically encountered, and

these can be discouraging, but if the person gives up, he or she will not reach those goals. Persistence is thus often necessary if success is to be achieved. (p. 238)

Quitting at the first sign of struggle essentially assures failure, while willingness to continue one’s effort and withstand the discomfort of struggle prolongs the opportunity to learn. Therefore, in a domain where struggle is inevitable, such as learning mathematics with understanding, persistence is a valuable tool.

Research incorporating persistence has yielded several general findings supportive of educational gains. Rarely studied directly, persistence has been shown to correlate with in-the-moment factors like self-efficacy, enjoyment, and anxiety – all of which have been shown to facilitate short-term mathematics achievements in certain contexts. Because learning mathematics requires wrestling with difficult ideas that are not immediately apparent (Hiebert & Grouws, 2007), the research involving persistence implies a consistent relationship between staying engaged despite challenges and making mathematical progress.

Critiques of Persistence Relative to Learning Mathematics with Understanding

Persistence is generally celebrated as a key component of learning, yet studying persistence to better understand the meaning-making process in mathematics still has its shortcomings. Most notably, research incorporating persistence has yielded relatively few insights about how persisters are spending their time while persisting. This is a function of the method by which the construct is measured. Peterson and Seligman (2004) warn that “simply measuring how long someone works at a task does not adequately capture

24 the essence of [persistence] because continuing to perform something that is fun or rewarding does not require one to endure and overcome setbacks” (pp. 229-230). In this way, counting the minutes spent working on a challenging mathematical task reveals very little about the obstacles encountered during engagement, if he or she encountered any obstacles at all, or how an individual navigated such obstacles to persist toward a solution. Yet, time-on-task remains as arguably the most widely-used measure of persistence.

Persistence, by definition, encapsulates one’s effort “in spite of obstacles, difficulties, or discouragement” (Peterson & Seligman, 2004, p. 229), but, similar to grit, one is left guessing about the degree to which an individual was challenged or discouraged. Although tasks in research involving persistence were designed to warrant difficulty, most analyses are not at a fine enough grain-size to reveal the nature of challenges nor the individuals’ specific learning gains while overcoming such assumed challenges. In this way, once again similar to grit, an individual achieving his or her short-term goal of solving a theoretically challenging mathematics problem may mean, by default, that he or she persisted. This is despite the fact it is unclear the degree to which the individual was challenged and the specifics of the mathematical meanings developed through the student’s time-on-task.

Other critiques of a time-on-task operationalization of persistence concern the different rates and purposes by which students work. In their study of motivation and learning in a complex system, Vollmeyer, Rollett, & Rheinberg (1998) found that

25 counting the minutes spent working on a task was an unreliable predictor of achievement on that task. Undergraduate participants were tasked with solving a mathematical system in the context of biochemistry, relating three input and three output variables.

Researchers found the time spent exploring different aspects of the system to be very inconsistent and did not correlate with the performance measures of the task. For example, spending more time exploring a relationship between certain input and output variables did not lead to more knowledge about the system. Some participants performed highly after relatively short amounts of time, while others worked much longer and achieved less. Vollmeyer, Rollett, & Rheinberg (1998) concluded that time on task was not a reliable measure of persistence in this context because learners could have different reasons to spend more or less time on the task (e.g., a learner who wants to discover the system’s structure right now, as opposed to a learner who does not give a full effort and starts daydreaming or pseudo-engaging).

To remove time from the measurement method, Vollmeyer and Rheinberg (2000) used the number of rounds of exploration of the complex system as a measure of persistence in a related follow-up study. Researching the effect of persistence as a mediator of motivation on problem-solving, Vollmeyer and Rheinberg (2000) found that undergraduate participants who persisted through more rounds exploring the task actually performed worse than less persistent individuals. The authors speculate prior knowledge differences as a possible explanation to these findings. Still, such results echo the overall wonderment about how persisters are spending their time while persisting.

The construct of persistence captures steadfast, in-the-moment effort toward a challenging task. Any effort, let alone consistent sustained effort, is certainly useful when striving to make connections between student engagement and student achievement.

Assuming an individual is giving his or her best effort, the more time spent and the more willing one is to engage with a challenge, the more opportunities one has to learn from that challenge. However, incorporating a persistence lens when analyzing student engagement during challenges omits key features of learning mathematics with understanding. Specifically, it is unclear how persisting learners overcome setbacks in the moment to develop the meanings behind the mathematics, that is, to uncover relationships between the mathematical ideas present (Hiebert, 2003). Also, observing whether or not an individual gives up on a task does not discern if he or she encountered an obstacle. In this manner, the mechanisms within wrestling with an obstacle are unexplained by persistence analysis. Simply refusing to give up does not guarantee successful learning. Counting the time spent on a task or assessing the willingness to sustain effort on a task does not unpack the actual engagement with unfamiliar challenges that is so crucial for meaning-making in mathematics. For researchers to better understand how students learn mathematics with understanding, a more fine-grained analysis of how students spend their time engaging is necessary. Perseverance, a related dispositional construct that more closely considers the learner’s perspective around challenge and struggle, is unpacked next.

Perseverance

As inevitable obstacles are approached during the process of learning mathematics with understanding, perseverance describes in-the-moment tenacity towards accomplishing a goal while also accounting for the malleability to alter a strategy when necessary. Different from the individual factors of grit and persistence, perseverance has been conceptualized both from the individual and collective perceptions. From the individual perspective, DiNapoli (2018) defines the construct of perseverance as

“initiating and sustaining, and re-initiating and re-sustaining, in-the-moment productive struggle in the face of one or more obstacles, setbacks, or discouragements” (p. 4). In contrast, Sengupta-Irving and Agarwal (2017) define perseverance as a collective enterprise, as “peers engaging together in productive struggle with effort over time” (p.

116). The authors use the term “collective” (Sengupta-Irving & Argarwal, 2017, p. 115) to account for both (a) the aggregation of peers’ individual productive struggle and (b) the group’s efforts to productively struggle in a collaborative sense.

In the context of working on a challenging mathematical task, both definitions reject the idea that whenever children solve a problem they have, by default, persevered.

Instead, perseverance requires an effortful response to the initial obstacle of a non-routine task, and altered efforts in response to any setbacks or difficulties that may arise in the work that follows. It is making explicit the nature of the challenges at hand that separates perseverance, in part, from persistence and grit. Individuals or groups that try to persevere past a mathematical obstacle are not guaranteed success, but are equipped with

28 more resources to approach the learning opportunity than those giving up on the mathematical task.

Similar to persistence and contrary to grit, the concept of perseverance concerns working toward a short-term goal with the recognition that a consistent tendency to persevere, over time, can accumulate and develop into long-term pursuits. During mathematical problem-solving, it is the emphasis on the flexibility to change a strategy upon setbacks, or “amending one’s plan of attack” (Middleton, Tallman, Hatfield, &

Davis, 2015, p. 5), that distinguishes perseverance from persistence. In this way, perseverance has been discussed as “productive persistence, defined as tenacity plus effective strategy [use]” (Dolle, Gomes, Russel, & Byrk, 2013, p. 451), because an individual or group may need to employ different plans to make progress on a problem if their first plan was unsuccessful. At the core of perseverance is the intersection of self- regulatory and metacognitive skills: regulating oneself to not only continue engagement in the face of challenge, but to notice progress or the lack thereof and decide to change course when necessary (Carver & Scheier, 2001; Flavell, 1976; Lepper, 1988;

Schoenfeld, 1992; Schoenfeld & Sloane, 2016; Zimmerman, 1990).

Gresalfi and Barnes (2015) suggest how mathematical or consequential justification can inform a decision to continue to persevere and change course. Achieving mathematical awareness through effort, that is, using made mathematical connections

(i.e., “this plan is not working because the slope of my function is incorrect”) to justify taking a different approach demonstrates the productive relationship of perseverance and

29 mathematical meaning-making. In contrast, achieving consequential awareness through effort, that is, using consequences (i.e., “this plan is not working because I’m not getting the right answer”) to justify a decision to change course showcases a less mathematically productive outcome.

Again, consider solving the Sidewalk Stones task (see Figure 1) as an example of a short-term goal in which an individual or group could be interested. To persevere in achieving the task goal, one must individually or collectively (a) be met with an initial obstacle of some kind, (b) endure and make a diligent initial attempt to overcome such an obstacle, and (c) if the prior attempt is unsuccessful, recognize this setback and endure and change solution strategies to make a diligent additional attempt(s) to overcome such an obstacle. Assuming Sidewalk Stones is a non-routine task for an individual or group, not immediately knowing how to proceed to discern the block rules constitutes the first obstacle toward which they can persevere. To persevere, an individual or group must make an effort to begin working toward the task goal, perhaps by counting the different shaded stones and noting similarities and differences amongst the provided Patterns. If the problem is not solved via this initial attempt, an individual or group must recognize this difficulty and alter their plan to continue their engagement, perhaps by drawing a larger sidewalk pattern, Pattern #4, to help them discern a relationship between the different blocks and the size of the Pattern. This process of initiating, sustaining, and altering effort toward a short-term task goal can continue indefinitely, ideally until

30 mathematical progress is made or the problem is solved, and describes perseverance from both an individual and group perspective.

Measures/Operationalization

Perseverance measures are a fine-grained account of an individual or group’s engagement with a challenging task, particularly focused on identifying when obstacles surfaced and how such obstacles were navigated (Bass & Ball, 2015; DiNapoli, 2017,

2018; Morales, Jr. & DiNapoli, under review; Sengupta-Irving & Agarwal, 2017;

Warshauer, 2014). In the context of an individual student working on a non-routine mathematical task, DiNapoli (2018) operationalized perseverance using a three-phase engagement framework: the Entrance Phase, the Initial Attempt Phase, and the

Additional Attempt Phase (see Table 4). The structure of the framework was developed using an inductive approach, emerging from the following data sources: observation, in- the-moment interview as the student thought-aloud, and video reflection interview as the student reviewed clips of his or her own engagement. The framework was designed to reflect perspectives of concept (CCSSI, 2010; Dolle et al., 2013; Middleton et al., 2015), problem-solving actions (NCTM, 1989, 2000; Pólya, 1971; Schoenfeld, 1979, 1985,

1988, 1989; Schoenfeld & Herrman, 1982; Schoenfeld & Sloane, 2016; Silver, 2013;

Wilson, Fernandez, & Hadaway, 1993), self-regulation (Baumeister & Vohs, 2004;

Carver & Scheier, 2001; Flavell, 1976; Schoenfeld, 1992; Winne, 2010; Zimmerman &

Schunk, 2011), and making and recognizing mathematical progress (Gresalfi & Barnes,

2015). To substantially capture the student point of view during engagement with tasks,

31 coding decisions, or whether or not certain engagement constituted evidence of perseverance, largely depended on student cues from in-the-moment think-alouds and video reflection interviews. For instance, evidence of mathematical progress consisted of the student himself or herself affirming, describing, and/or defending his or her point of view that he or she better understood the mathematical context embedded in a task after his or her effort(s).

Table 4

Three-Phase Perseverance Framework (DiNapoli, 2018)

Entrance Phase Component Evidence of Appropriate Task Clarity Objectives were understood Initial Obstacle Solution pathway not immediately apparent Initial Attempt Phase Component Evidence of Perseverance Initial Effort Engaged with task Sustained Effort Used problem-solving heuristics to explore task Outcome of Effort Made mathematical progress Additional Attempt Phase (after perceived impasse) Component Evidence of Perseverance Initial Effort Engaged with task Sustained Effort Used problem-solving heuristics to explore task Outcome of Effort Made mathematical progress Note: If a task had multiple objectives, work toward each objective would require perseverance analysis While initially reading the task, the Entrance Phase captured whether the student understood the entirety of what the task was asking (clarity component) and if the student immediately knew how to solve the problem (initial obstacle component). Essentially, this phase established the appropriateness of a task for perseverance analysis with a

32 particular individual. An ideal outcome in the Entrance Phase was a student affirming the task made sense and was within his or her perceived realm of capability, yet he or she did not immediately know a procedure by which to solve it. In this case, the task was deemed as appropriate for perseverance measurement and participants entered into the next phase of engagement.

The Initial Attempt Phase examined whether and how the student initiated and sustained his or her effort, and the outcome of such effort as he or she worked toward solving all parts of the problem. Since the student did not immediately know a solution pathway, evidence of perseverance included deciding to engage with the task at all (initial effort component). If the student decided to pursue solving the problem, evidence of perseverance included using a problem-solving strategy to diligently explore the uncertain nature of the mathematical situation (sustained effort component). In this context, a problem-solving strategy was defined as using one or more heuristics, in conjunction, to work toward a solution. As a result of these diligent efforts, evidence of perseverance included making mathematical progress toward better understanding the mathematical relationships or solving the problem outright (outcome of effort component).

In the event the student did not solve the problem after making an initial attempt, the Additional Attempt Phase aimed to measure if and how the student amended his or her original problem-solving plan, and the outcome of such efforts as he or she worked to overcome any setbacks and toward solving all parts of the problem. To enter this phase

(marking the end of an initial attempt and the beginning of a second attempt), the student must first be stuck, that is, he or she perceived to have exhausted all options immediately stemming from his or her first plan of attack and are unsure of how to keeping moving forward toward finding a solution. At this point, evidence of perseverance included deciding to re-engage with the task using a different strategy – one that was not used during the initial attempt (initial effort component). Assuming the student decided to pursue solving the problem with a new plan of attack, additional evidence of perseverance included diligently exploring the mathematical situation with the new strategy (sustained effort component), and making new progress toward better understanding the mathematics involved in the task, or solving the problem outright

(outcome of effort component). Depending on the number of times a student amends his or her original problem-solving plan, or the logistical constraints of scheduling time for students to work, a student could continue working in the Additional Attempt Phase, making third attempts, fourth attempts, and so on.

In sum, DiNapoli’s (2018) three-phase engagement framework measures a student’s in-the-moment perseverance by first determining in the Entrance Phase if the task with which the student is working necessitates perseverance. The components within the Initial and Additional Attempt Phases then detail the evidence of perseverance at specific moments during problem-solving with a challenging mathematical task. It is important to note that DiNapoli’s (2018) perseverance framework prizes how students spend their time – spending effort to grapple with important mathematical ideas and

34 making mathematical progress – not time on task or necessarily getting correct answers.

This focus encourages a mathematics-as-activity perspective, in which doing mathematics constitutes the process of thinking mathematically, while simultaneously wrestling with inherent obstacles toward the goal of better understanding a mathematical structure embedded in a task or situation (Schoenfeld, 1994, 2016).

In the context of a group of students collectively working on a challenging mathematical task in a classroom setting, Sengupta-Irving and Agarwal (2017) studied a group’s perseverance by focusing on interactions that may have evidenced productive struggle. The goal of their analysis was to better “demarcate the boundaries of perseverance as collective enterprise” (p. 124). Such perseverance was evidenced by students modeling or representing ideas to their peers, considering others’ explanations, or monitoring their own problem-solving while wrestling with important mathematical ideas that were just beyond reach. The research team video recorded and observed groups of four students working collaboratively on a challenging mathematical task. They systematically chronicled when a struggle emerged, what led to the struggle, and details of the talk and activity that preceded and followed each episode of struggle. As such, the discursive, in-the-moment indicators of potential productive struggle in this collaborative context were:

(1) Conflict in declared solution or strategy; (2) Declaration of uncertainty about solution or strategy; (3) Declaration of inelegant or inefficient strategy; (4) Clarification of task expectations or features; (5) Seeking expert support [from teacher] (p. 122)

In addition to identifying indicators of potential productive struggle, Sengupta-

Irving and Agarwal’s (2017) perseverance framework specified five outcomes of collective engagement immediately after an opportunity for productive struggle emerged:

(1) Problem-solving but not persevering because of agreement on incorrect answer; (2) Problem-solving but not persevering because solution is immediately obvious; (3) Perseverance in problem-solving but not as collective enterprise; (4) Perseverance in problem-solving as collective enterprise (brief); (5) Perseverance in problem-solving as collective enterprise (extended) (p. 124)

In this way, the authors identified relationships between particular indicators and subsequent perseverance to detail perseverance as collective enterprise in this context.

Viewing perseverance as a collective construct “capitalizes on [the] opportunity to build stronger and more supportive classroom communities – communities that will prove themselves unbreakable in the midst of great struggle” (Sengupta-Irving & Agarwal,

2017, p. 134). Thus, this perspective encourages the design of learning environments that bring students together to productively struggle with unformulated mathematical ideas that collective effort will bring within reach. It remains to be seen the specific ways in which developing such collective perseverance with challenging mathematics, with students working collaboratively, may influence the development of individual perseverance, with the same students working individually (Bass & Ball, 2015).

Associations between Perseverance and Learning

Studies specific to perseverance endorsing conceptualizations that focus on in- the-moment engagement around obstacles, although relatively few, have made deep connections to students learning mathematics with understanding. By studying how students navigate the challenging moments during mathematical work when a formulated idea is just out of reach, research on perseverance reveals the dynamics of how children overcome setbacks to make their own connections between important mathematical ideas

(Bass & Ball, 2015; DiNapoli, 2017, 2018; Morales, Jr. & DiNapoli, under review;

Sengupta-Irving & Agarwal, 2017; Warshauer, 2014). Further, such research has bared great insights into how teachers can support students in their meaning-making process.

From the individual learning perspective, DiNapoli (2017) found that students who reported holding generally positive motivations may not know what it means to persevere to accomplish their goal of understanding mathematics. In this study, DiNapoli

(2017) explored the relationship between 11 ninth-grade students’ self-reported achievement goals and their subsequent perseverance on a mathematical task with which they were unfamiliar yet interested. Via a problem-solving interview, the researcher discerned the goal(s) toward which each student worked and measured perseverant actions using the three-phase framework in Table 4. Results showed that students may have the best intentions to persevere and develop their own mathematical understandings in the face of a challenge, but hold a distorted view of what counts as productive effort.

In this study, students shared how satisfied they felt from their effort, even though they only worked for mere seconds on a challenging problem. The same students shared how their teachers consistently encouraged approaching challenges and not giving up, suggesting students may be interpreting this message in a binary way: you can either try or not try, there is no in-between. Students may exert minimum effort to check-the-box of perseverance and feel proud of their work, but not make real progress. This inquiry into student perseverance suggests that teachers should be more explicit about what counts as perseverance and help children realize the difference between minimal and meaningful effort in the learning process.

Studying student perseverance over a series of several challenging mathematical tasks, DiNapoli (2018) found that ninth-graders persevered more on tasks embedded with prompts to promote conceptual thinking, and demonstrated an improvement in quality of perseverance over time, compared to their work on similar tasks without such scaffolds.

Using problem-solving interview and video retrospection interview methodologies,

DiNapoli (2018) worked in-depth with 10 students as they worked and thought-aloud on a challenging task. Then, the research facilitated a discussion with each individual participant about how he or she experienced the specific moments when perseverance was necessary while together watching a video of his or her engagement immediately thereafter. Over a five-week period, each participant worked on five analogous mathematical tasks for which he or she had demonstrated prerequisite knowledge. Each student worked on one task per week in a random order. Three of the tasks were

38 embedded with scaffolds designed to activate prior knowledge and/or support planning a solution pathway, both methods of conceptualizing a mathematical situation while maintaining cognitive demand (Anghileri, 2006); two of the tasks had no scaffolds.

Using the three-phase perseverance framework (see Table 4), results showed that these ninth-grade students’ perseverance was supported by these scaffolds, but the type of scaffold did not seem to matter. The most prevalent difference in demonstrated perseverance between tasks with and without scaffolds occurred in the Additional

Attempt Phase. After making an initial attempt at solving a task and reporting they were unsure how to continue, the quality of students’ perseverance in their additional attempt was significantly better while working on scaffolded tasks compared to non-scaffolded tasks. After engaging with the scaffolds, students were more often amending their original plan of attack, deeply exploring the task with a new set of exploratory heuristics, and making additional mathematical progress as a result of their additional efforts. On the non-scaffolded tasks, students were much more likely to give up earlier and opt out of making an additional attempt altogether. From the participant point of view, several students cited their initial conceptualization work as the primary reason for their higher- quality perseverance on the scaffolded tasks. By initially making explicit the important mathematical ideas and the connections between them, participants found such work helpful after making a mistake, even when they were at their most frustrated, because they could more easily reconsider their original mathematical conceptualization, make some adjustments in their thinking, and try something new. Without such

39 conceptualizations written down, many participants cited disorganization and frustration as reasons for giving up after encountering a setback.

Additionally, participants’ perseverance improved in quality over time, more so across their work on scaffolded tasks than on non-scaffolded tasks. While reflecting on their experiences, participants reported they believed they were getting better at making progress on these challenging tasks. Many cited the phenomenon of momentum, or a drive to finish what they had started, as responsible for their improvement. Others cited more cognitive gains, reporting they found it easier to design plans of attack, both for their first and additional attempts, the more opportunities they had to practice their perseverant engagement. Since participants had a significantly easier time initiating first and additional efforts on tasks with scaffolds, this research suggests that giving students a formal space to think and record their ideas can encourage subsequent perseverance in ways that will facilitate making meaningful mathematical connections.

From the collective learning perspective, Sengupta-Irving and Agarwal (2017) aimed to describe what collective perseverance looked like, and found that the most perseverant groups of students were working in a collaborative zone of proximal development. That is, these students grappling with the mathematics at similar levels of struggle, while also able to awaken possible mathematical connections in one another through collaboration. The researchers observed 16 fifth-grade students engage with six challenging tasks, working in groups of four, over the course of six days. By identifying in-the-moment indicators of potential productive struggle and examining the

40 perseverance (or lack thereof) demonstrated by a group immediately afterwards,

Sengupta-Irving and Agarwal (2017) found that peer dynamics both constrained and advanced children in persevering.

During work on one task, a group of two boys and two girls had trouble working together. The boys were overcritical of the girls’ initial solution strategy because they thought it was inefficient. This critique seemed to rupture the good will necessary for preserving together. The children decided to continue working in smaller groups of two, sacrificing the opportunity to learn from more diverse perspectives in problem-solving to gain a deeper understanding of the mathematics. The researchers determined that only the girls in this group were engaged in real productive struggle, because the boys seemed to never encounter a true obstacle. Thus, at the level of the entire group, the students failed to evidence perseverance as collective enterprise.

During work on another task, a different group of two boys and two girls demonstrated a productive collaboration. The children worked at times in smaller groups of two and then together in one group of four to persevere past several obstacles during problem-solving. The group of girls initiated effort with a visual representation strategy, while the group of boys began with a numeric strategy. Several times during their work, the group convened as a collective to share how they were thinking about solving the problem from their own points of view. It is through these collective efforts that each group member advanced their understanding of the mathematical situation and was able to progress with their work in meaningful ways. Ultimately, the group of boys and the

41 group of girls both productively struggled and solved the task in different ways. This group demonstrated perseverance as collective enterprise in their willingness to share their struggles and learn from one another.

In sum, Sengupta-Irving and Agarwal (2017) stressed the importance of teachers creating learning environments with norms that sustain collective perseverance. Such norms include shared student accountability, student-driven inquiry, and, most importantly, grouping students by a collaborative zone of proximal development. This research points to substantial learning gains to be had through collective perseverance, but only if each member of the collective has similar opportunities for productive struggle.

Although rather uncommon, research on perseverance has made progress describing challenging moments for learners during work with mathematical tasks.

Unlike grit and persistence, constructs that have primarily been studied as a related outcome with achievement, perseverance has been studied directly to reveal the ways in which students initiate, sustain, re-initiate, and re-sustain productive struggle within the learning process. As a result of such work, perseverance has operationalized from the individual and collective perspective, and has been implemented in empirical work to uncover resources – such as conceptual thinking scaffolds and collaborative discourse – supportive of productive struggle with challenging mathematics.

Critiques of Perseverance Relative to Learning Mathematics with Understanding

There is general consensus that perseverance is vital for learning mathematics because struggling with important mathematical ideas while recovering from setbacks is fundamental for the sense making process (Hiebert & Grouws, 2007). Yet, critiques exist regarding the explicit relationship between student perseverance and conceptual development. While studies on perseverance (e.g., DiNapoli, 2018; Morales, Jr. &

DiNapoli, 2018; Sengupta-Irving & Agarwal, 2017) uncover resources learners can leverage to further propel their perseverance with a mathematical challenge, the specific learning gains are often unclear. In these studies, the main focus of research is the perseverance itself, not necessarily the resulting mathematical gains. Although an important factor of operationalizing perseverance in the moment is making mathematical progress, the details of such progress are not well defined. For instance, DiNapoli (2018) accounts for making mathematical progress primarily from the student point of view, that is, if the student believes he or she better understands the mathematical situation as a result of his or her engagement, then he or she made mathematical progress. The student perspective is certainly valued in research on perseverance, but more emphasis on specifically how students are constructing and clarifying the relationships between mathematical ideas is warranted in future research.

Also, the way perseverance has been studied has been critiqued. For instance,

Sengupta-Irving and Agarwal (2017) contend that perseverance should be investigated from a collaborative, not individual, point of view. They argue that framing perseverance

43 as a collective enterprise offers a vantage point from which to consider how children collaborate around challenging mathematics. This assertion derives from the well-known relationship between group dynamics and mathematical learning and problem-solving, namely that peer collaboration can encourage making connections between important mathematical ideas (Boaler, 2008; Esmonde & Langer-Osuna, 2013; Goldin, 2000;

Schoenfeld, 1992; Schoenfeld & Sloane, 2016). In practice, encouraging collective perseverance can help build more equitable and effective learning communities (Boaler

& Staples, 2008; Morales, Jr. & DiNapoli, under review; Sherin, 2002; Silver & Smith,

1996), help teachers create and sustain opportunities for groups of children to persevere

(Sengupta-Irving, 2014), and build understanding of how the context might constrain its emergence (Gresalfi, 2009). The authors insist that “the problem with an individual perspective on perseverance is that it presumes the child is the cause of success/failure

(i.e., she lacks perseverance) without considering what role the context plays in supporting or constraining her efforts” (p. 117).

The trouble with Sengupta-Irving and Agarwal’s (2017) critique of the individual perspective is that it is rooted in research on persistence, not perseverance. Sengupta-

Irving and Agarwal (2017) reference works from the mid-twentieth century (e.g.,

Altshuler & Kassinove, 1975; Briggs & Johnson, 1942; Feather, 1961, 1966; Ryans,

1938a, 1938b; Schofield, 1943) to paint an “empirical history of the construct” (p. 117) and to describe “how perseverance has typically been studied” (p. 117). However, these studies all focused on the notion of persistence, not perseverance, by operationalizing

44 effort through time on task. Studies like these over-simplify the idea of spending time to grapple with a challenge because their measures do not consider how that time was spent.

Researching perseverance from the individual perspective does not discount what can be learned from the collective perspective, but instead tries to better understand what states of engagement are enacted by person when wrestling with a mathematical challenge. Such states are not inert traits of these individuals, but a collection of in-the- moment mindsets, motives, and actions potentially living within all persons that can be brought to the surface in certain contexts (Goldin, Epstein, Schorr, & Warner, 2011). This point of view is rooted in the fundamentals of mathematics learning, as explained in

Adding It Up, in which the National Research Council posits that mathematically proficient children develop a productive disposition and a respect for perseverance:

Productive disposition refers to the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics. If students are to develop conceptual understanding, procedural fluency, strategic competence, and adaptive reasoning abilities, they must believe that mathematics is understandable, not arbitrary; that, with diligent effort, it can be learned and used; and that they are capable of figuring it out. Developing a productive disposition requires frequent opportunities to make sense of mathematics, to recognize the benefits of perseverance, and to experience the rewards of sense making in mathematics. (Kilpatrick, Swafford, & Findell, 2001, p. 131)

The individual perspective should always be considered in context, however. The individual’s prior experiences, interest level, achievement goals, self-efficacy, and other motivators all play a role in one’s willingness to persevere in a given situation (Middleton

45 et al., 2015). The structure of a mathematical task and the level of challenge are also important contextual variables to recognize in this work (DiNapoli, 2016).

In all, both perspectives, the individual and collective, offer fruitful lenses through which to study perseverance. The individual point of view reveals the inner-mechanisms of how a student can manage to struggle to make sense of mathematics, even after a setback or discouragement. The collective point of view reveals the ways in which a community of learners can approach, sustain, and make progress upon a mathematical challenge, even after a conflict in a solution or strategy. This outlook is echoed in

NCTM’s Principles to Action (2014), asserting that mathematics instruction should consider perseverance from both standpoints. They affirm “effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships” (p. 48). Extending this notion to include making multiple attempts to grapple with mathematical ideas and relationships, even after a mistake or setback, encapsulates the importance of teachers providing students opportunities to persevere.

Toward a Distinction between Constructs

Grit, persistence, and perseverance are three dispositional factors that embody the principle of continuing effort in the face of a challenge. The process of learning mathematics for understanding is full of challenges because one must wrestle with ideas that are not immediately apparent to help make sense of them. From an analytical

46 perspective, each construct reveals something different about a student’s engagement with mathematics. On the surface, grit, persistence, and perseverance all describe the idea of sticking with it, a “stamina” (Duckworth et al., 2007, p. 1098) of sorts while navigating obstacles during pursuits of learning mathematics. Yet, each construct explains the details of such stamina at varying grain-sizes. Therefore, it is important to make explicit the distinctions between grit, persistence, and perseverance and how they can be used in the context of research on learning mathematics for understanding. The primary differences between the constructs are the focused timeframe and the details revealed about how one spends his or her time while engaged with challenging mathematics (see Table 5).

Table 5

Distinctions between Grit, Persistence, and Perseverance

Construct Definition Primary Timescale Grain-size of Measure Analysis Grit The [continuance Questionnaire: Long-term, Determines if of effort] and Grit-S scale – a tendency students’ self- passion for Consistency of over time reports of focused specific, high Interests and hard work and goals, sustained [Continuation] summative over years of Effort mathematical (Duckworth, (factors) achievements are 2006, p. 73). (Duckworth & associated. Quinn, 2009). Persistence The voluntary Observed Short-term, Quantifies for continuation of a engagement: in the how long a goal-directed the amount of moment student engaged action in spite of time spent with a obstacles, working on a mathematical difficulties, or difficult task. challenge,

discouragement suggesting more (Peterson & opportunities to Seligman, 2004, learn emerge p. 229). from more time spent on task. Perseverance Initiating and Observed Short-term, Describes student sustaining, and engagement: in the engagement with re-initiating and Three-Phase moment a mathematical re-sustaining, in- Perseverance challenge, the the-moment Framework – ways in which productive Initial Effort, effort was struggle in the Sustained initiated and face of one or Effort, sustained before more obstacles, Outcome of and after setbacks, or Effort setbacks, and the discouragements. (components) apparent (DiNapoli, 2018, (DiNapoli, mathematical p. 4). 2018). progress made.

Grit, persistence, and perseverance differ in terms of the timeframe concerning their nature. Grit describes stamina across challenges toward a long-term goal, while persistence and perseverance depict stamina within effort toward a short-term goal. For persistence and perseverance, such short-term goals are often solving challenging mathematical tasks. Students must persist or persevere despite setbacks to begin to develop an understanding about the mathematics embedded in a task; all of this stamina occurs in-the-moment while grappling with obstacles in the problem. For this reason, persistence2 and perseverance are framed not as a character trait, but as a state of

2 At times, persistence has been framed as a measure via questionnaire of one’s tendency to not give up on a task (e.g., Constantin et al., 2011), which implies a trait of one’s character. However, persistence has most often been measured via time-on-task in the moment, which implies a state of engagement, not a characteristic. Therefore, here we consider persistence as a time-on-task measure.

48 engagement in those trying moments. Grit, on the other hand, implies a consistency of effort toward a summative achievement, such as earning a mathematics degree. This measure concerns the tendency of a person to continue to work toward a long-term goal, thus grit is framed as a trait of one’s character. These types of achievement are presumably a result of the accumulation of in-the-moment persistence or perseverance with formative achievements. Both achievements, solving a challenging mathematical task or earning a mathematics degree, imply effort was spent toward developing one’s mathematical understanding, but the dispositional factors at play are operating within different timeframes.

While pursuing a goal related to learning mathematics with understanding, grit, persistence, and perseverance all reveal something different about how one maintains his or her stamina. Grit and persistence describe the generic action of staying the course, while perseverance digs deeper into the specific actions one takes to continue effort despite obstacles and disappointments. Questionnaires like the Grit-S Scale (Duckworth

& Quinn, 2009) aim to discern whether or not an individual is tenacious in his or her work to achieve his or her goal in spite of challenges. Similarly, time-on-task measures of persistence imply the level of tenacity exerted in the moment. Thus, studying mathematics learning via grit and persistence affords information about if an individual decided to stay the course in a particular timeframe, but not how. Although affirmative responses to Grit-S (Duckworth & Quinn, 2009), for example, suggest an individual is consistent, over time, with his or her efforts in the face of challenge, such reports do not

49 detail how those efforts were maintained. It is apparent the gritty individual leveraged his or her stamina to reach the finish line (i.e., earned a math degree), but it is not clear how exactly he or she overcame the many obstacles to get there. Likewise, the more time spent working on a challenging task, for instance, demonstrates a stamina and an in-the- moment commitment to not give up. Yet, it is the details of how this time was spent that is missing. It is clear this individual persisted in his or her efforts to stay engaged with the task, but it is unclear what exactly he or she did to overcome obstacles and make mathematical progress despite setbacks, if he or she did at all.

Perseverance, on the other hand, affords more insights into how effort was spent while maintaining stamina during challenging moments with mathematics. For instance, the Three-Phase Perseverance Framework (DiNapoli, 2018) first accounts for whether the task necessitated perseverance, and if so, secondly captures the student’s problem-solving strategies used to continue engagement, overcome obstacles and make progress, and, most importantly, thirdly make an additional attempt to stay engaged with the task if an initial attempt was unsuccessful. Because learning mathematics with understanding can involve multiple obstacles, it is likely one must tenaciously and flexibly try several different strategies to make progress toward solving a problem.

Apart, these self-regulatory facets of tenacity and flexibility can be limited in the context of learning mathematics. A student can demonstrate tenacity with a persistent stubbornness to never give up, but make little progress toward his or her goal if the plan is flawed. Conversely, one can have multiple productive ideas for how to solve a

50 challenging task for which a solution pathway is unknown, but without thorough and diligent exploration of those strategies he or she will never know which strategy works well and which strategy does not. In this case, making mathematical progress is unlikely.

These two facets necessitate and complement one another to help describe in-the-moment perseverance. Thus, in addition to determining if a student was engaged with a mathematical task that was indeed challenging, measures of perseverance seek to depict the quality of such engagement by considering the specific actions students take when solving challenging problems, flexibility to amend one’s plan, and overall productivity of effort in the context of learning mathematics.

Therefore, studies including grit, persistence, or perseverance tell us something different about students learning mathematics with understanding. Learning mathematics with understanding requires ample opportunities for students to productively struggle with challenging content to discover and describe relationships between mathematical ideas at play. Research on learning mathematics with understanding is constantly looking for ways different learners navigate obstacles successfully and how they continue to propel themselves forward, despite struggles and setbacks. Although studies of grit and persistence have been tied to mathematical achievements in general – such as gritty students scoring highly on the SATs (see Duckworth et al., 2007) or persistent students celebrating increased perceptions of confidence with mathematical problem-solving (see

Niemivirta & Topola, 2007) – little evidence exists that using these constructs as an analytical tool reveals much about the process of meaning-making in mathematics. Self-

51 reports of grit depict a general willingness to maintain focus and effort toward a goal, but nothing is revealed about what a student does during uncertain moments when obstacles present themselves. Observed persistence depicts continued engagement with a task, but nothing is revealed about the quality of that engagement. Researchers simply know a student reported as gritty or were observed to be on-task, yet they do not know the details of how the presumed hard work was done. Learning to be gritty or persistent certainly would have its academic advantages, but the lack of details provided by these constructs make it difficult to extract ideas for how to help develop grit or persistence in students.

Instead, social-psychological interventions (e.g., teaching growth-mindset and malleable- intelligence theory (see Blackwell, Trzesniewski, and Dweck, 2007)) are often tasked with trying to develop such non-cognitive factors in students. Without specific empirical data for how students became gritty or managed to persist with a mathematical challenge, these interventions can be ill-received and ineffective because they offer no concrete strategies for students to practice to help generate long-lasting effects (Yeager & Walton,

2011).

Perseverance, conversely, has been operationalized in ways to provide such empirical data as students traverse in-the-moment mathematical obstacles. Thus, perseverance can be operationalized through an analytic framework that details the ways in which students productively struggle and incrementally make connections between important mathematical ideas. Studies of perseverance can inform researchers of specific conditions that encourage perseverance in specific ways, such as the finding that

52 prompting students to make a list of potentially relevant mathematical ideas prior to working on a mathematical task can encourage them to amend their original problem- solving plan (if it was unsuccessful) and try a different heuristic to make progress (see

DiNapoli, 2018). Further, discussing key decision-making moments with participants themselves can offer a valuable perspective from which to interpret findings, such as learning that students felt more invested in a task after making a list of mathematical ideas. As a result, it was easier for them to explore uncertain terrain as a result of such an investment (see DiNapoli, 2018). These types of results are helpful for researchers in this context because they reveal exactly what students are doing while wrestling with mathematical ideas and offer practical suggestions about how one might overcome in- the-moment mathematical obstacles.

It is important to note that studying perseverance is not as simple as providing students a mathematical task and expecting them to productively struggle. No person is always perseverant. Perseverance is not a character trait; it is a state of engagement idiosyncratic in nature (Sengupta-Irving & Agarwal, 2017). Perseverance with a challenging task is influenced by a variety of environmental factors, including interest level (Hidi & Renninger, 2006; Ryan & Deci, 2000), goals for engagement (DiNapoli,

2017; Elliot, 2005), pre-requisite knowledge (Schoenfeld & Herrmann, 1982; Silver,

2013), self-efficacy (Bandura, 1993; Linnenbrink & Pintrich, 2003), uncertainty affect

(Pekrun, Goetz, & Titz., 2002; Wang et al., 2015), and the structure of the task itself

(DiNapoli, 2016, 2018; Lotan, 2003; Stein, Grover, & Henningsen, 1996). Thus,

53 examining student perseverance requires careful planning and consideration of the situational variables at play. In fact, if even one variable is incompatible with a student in a particular situation, a reasonable response to a challenge task may be to not persevere

(see Star, 2015). Every student perseveres, or does not persevere, in different ways for different reasons, so research on perseverance should describe explicitly the conditions under which any study was conducted, and what was gleaned about learning mathematics in that specific context.

Researchers should carefully select and define constructs to reflect their phenomenon of interest. As it stands, grit, persistence, and perseverance are often used interchangeably, which leads to a misconception of what can be learned through the lens of each construct. For example, in studies of grit, Duckworth and colleagues consistently refer to “perseverance”, but in a long-term, less nuanced sense than argued in this paper.

Christensen & Knezek (2014) discuss measures of “perseverance” in their study on motivation and study habits, yet use grit questionnaires to do so. Marshall (2017) aimed to depict “perseverance” in an algebra class, yet largely relied on time-on-task calculations, or persistence, to describe the extent of student effort. In the context of learning mathematics, studies of grit and persistence are equipped to draw conclusions far different, and far less detailed, than studies of perseverance. These constructs deserve distinction to clarify what an investigation is studying and how it is relevant to understanding mathematics and mathematics education in general.

Conclusion

Grit, persistence, and perseverance are distinct, dispositional factors that afford different insights into how students learn mathematics with understanding. Although each construct refers generally to not giving up in the face of a challenge, perseverance digs far more deeply into how students spend their time navigating mathematical obstacles than does grit or persistence. For the purposes of research on meaning-making in mathematics, studying how students persevere with challenging problems can reveal the ways in which they overcome setbacks and make their own mathematical connections in the moment – revelations that can pragmatically inform teachers for how to set up mathematical activities and encourage students to necessitate and nurture such practices.

This is not to say that research incorporating grit and persistence is not useful in mathematics education. Grit research can establish associations between accomplishments and consistency of interest and effort, which can posit the construct as a predictor of general mathematical achievements in the long term. Research incorporating persistence can efficiently describe the time a student spent engaging with a challenge, and thus, the opportunity he or she had to learn from that challenge. Yet, in the context of learning more about how students learn mathematics with understanding, examining students’ perseverance with challenging tasks has the most potential to bare the most fruitful and fine-grained insights.

Chapter 2

SUPPORTING SECONDARY STUDENTS’ PERSEVERANCE FOR SOLVING

CHALLENGING MATHEMATICS TASKS

The process of developing new understandings involves students persevering past obstacles with the appropriate support (Dewey, 1910; Festinger, 1957; Pólya, 1971). To make sense of concepts, Dewey (1910) suggested to prize the “perplexity, confusion, or doubt” (p. 12) that accompanies solving challenging tasks. It is through wrestling at moments of challenge and withstanding uncertainty that one can better develop deep content knowledge. In the context of problem-solving with challenging mathematics, this paper aims to investigate the ways in which student perseverance can be supported, both during moments of significant uncertainty and over extended periods of time.

While learning mathematics, students can develop their understandings through productive struggle as they grapple with mathematical ideas that are within reach but not yet well formed (Hiebert, 2003; Hiebert et al., 1996). It is through this productive struggle with challenging mathematics that students can make more explicit connections between ideas, thus beginning to learn mathematics with understanding (Kapur, 2010,

2011; Hiebert, 2013; Hiebert & Grouws, 2007; Warshauer, 2014). Additionally, reconciling moments of impasse at times of significant uncertainty or discouragement is a critical component of mathematics learning. Struggling to approach, reach, and make continued progress despite a perceived impasse puts forth cognitive demands upon the

56 learner that are conducive for development of conceptual ideas (Collins, Brown, &

Newman, 1988; VanLehn et al., 2003; Zaslavsky, 2005). Building from these ideas, perseverance can be viewed as initiating and sustaining, and re-initiating and re- sustaining, in-the-moment productive struggle in the face of one or more obstacles, setbacks, or discouragements.

However, withstanding struggles can be difficult for students to endure in the moment. After all, the very nature of what it means to persevere implies tolerating some level of discomfort. Accordingly, recent mathematics reform movements have set explicit expectations for encouraging and improving students’ perseverance in problem-solving

(CCSSM, 2010; Kilpatrick, Swafford, & Findell, 2001; NCTM, 2014). To support students, mathematics teacher educators have sought to uncover and describe classroom scaffolding practices that can support student perseverance with challenging mathematics, including structuring a task to activate pre-requisite knowledge (Bass &

Ball, 2015) and encouraging conceptual thinking through teacher questioning (Freeburn

& Arbaugh, 2017). Although these scaffolding practices appear to have potential to support students to initially persevere as they attempt to solve a challenging task, questions remain about how to continue to encourage student perseverance at further moments of perceived impasse. Also, there is more to understand about whether and how these scaffolds help improve students’ perseverance over time. Therefore, the purpose of this study is to investigate the ways in which scaffolding mathematical tasks can support students’ perseverance in-the-moment while problem-solving, during an initial attempt at

57 solving and after an encountered impasse, and how such scaffolds help perseverance improve over time. In the following sections, I will define the construct of perseverance and review the literature on scaffolding practices that can support students’ perseverance in problem-solving.

Conceptual Framework: Perseverance in Problem-solving

In the context of problem-solving, I define perseverance as initiating and sustaining, and re-initiating and re-sustaining, in-the-moment productive struggle in the face of one or more obstacles, setbacks, or discouragements. My definition is similar to

Middleton, Hatfield, Tallman, and Davis’ (2015) and Dolle, Gomes, Russel, and Bryk’s

(2013) definitions of perseverance3 because it describes the extent of one’s flexible effort towards a problem-solving goal. My definition differs, though, in the way it specifies diligent and tenacious effort as productively struggling to begin and continue engagement amidst uncertainty and upon unveiled challenges. Thus, my definition of perseverance leverages the meaning of productive struggle to capture how one initiates and sustains effort toward a goal that is not immediately apparent, as well as how one re-initiates and re-sustains effort toward the same goal if and when a substantial obstacle reveals itself as a result of one’s engagement. The moment this substantial obstacle is encountered is

3 Middleton et al. (2015) define perseverance as “the continuance of effort, carried out in a thorough and diligent manner, towards some perceived goal while overcoming difficulties, obstacles, or discouragement along the way by amending one’s plan of attack” (p. 4-5). Dolle et al. (2013) define perseverance as “tenacity plus effective strategy” (p. 451).

58 defined here as an impasse, or when a person is unsure how to continue his or her engagement and admittedly “gets stuck” (VanLehn et al., 2003, p. 220). When trying to solve a problem for which a solution pathway is not known, it is engaging in productive struggle at the outset and also after a perceived impasse that helps describe perseverance and the productive process by which mathematical understandings can be developed.

When researchers investigate perseverance, they seek to locate and unpack the details of the in-the-moment productive struggles that support the development of mathematical concepts. Colloquially, people usually think about perseverance as working hard toward accomplishing a goal and never giving up. In education, words like grit and persistence are often interpreted as synonymous with perseverance, yet these three constructs describe different phenomena that enable distinguished operationalizations in a research context (see DiNapoli, 2018).

Grit is defined by Duckworth and colleagues (2007) as “perseverance and passion for specific, high goals, sustained over years” (p. 73) and is measured via the Grit-S self- report questionnaire (Duckworth & Quinn, 2009). This measure of the continuation of passionate work and interest level over years despite failures and progress plateaus is framed as a trait of one’s character (for instance, see Robertson-Kraft & Duckworth,

2014). This is different from a perseverance perspective, which is contextual – that in- the-moment productive struggles are always being influenced by environmental motivations rather than being fixed as a capacity or ability (Bass & Ball, 2015; Middleton et al., 2015). Relative to building mathematical meanings through productive struggle,

59 research on grit does not adequately consider the specific moments of struggle and how they are navigated during problem-solving, and thus, is far disconnected from the goals of perseverance research.

Persistence, as defined by Peterson and Seligman (2004), is the “voluntary continuation of a goal-directed action in spite of obstacles, difficulties, or discouragement” (p. 229). This describes a steadfast, in-the-moment effort while pursuing a more short-term goal. These authors explain this effort as refusing to quit a task before it has been accomplished, which can result in tenaciously sticking to an initial strategy even if it may not be working. The primary measure of persistence has been calculating one’s time on task (for instance, see Tulis & Fulmer, 2013), implying the goal of such research to be documenting if one has stayed engaged during problem-solving.

The staying-the-course perspective of persistence research diverges from that of perseverance because it reveals little about how time was spent during problem-solving, nor any insights about the productive struggling that may or may not have occurred.

Perseverance is a distinct perspective in the way it prizes in-the-moment engagement in the face of challenging mathematics. As inevitable obstacles are approached during the process of learning mathematics with understanding, perseverance describes the specific ways one navigates a challenge, while also accounting for the malleability to alter a strategy when necessary. Middleton et al. (2015) identify

“amending one’s plan of attack” (p. 5) at moments of perceived discouragement as a key component of perseverance because such in-the-moment problem-solving flexibility

60 affords more opportunities for productive struggle, rather than succumbing to frustration and disengaging form the learning opportunity altogether. Thus, at the core of perseverance is the intersection of self-regulatory and metacognitive skills: regulating oneself to not only continue engagement in the face of challenge, but to notice progress or the lack thereof and decide to change course when necessary (Carver & Scheier, 2001;

Flavell, 1976; Lepper, 1988; Schoenfeld, 1992; Schoenfeld & Sloane, 2016; Winne &

Azevedo, 2014; Zimmerman, 1990). Further, seeking evidence of mathematical progress is crucial for describing perseverance. Working through the struggles that epitomize perseverance is meant to be productive, that is, a coveted result of perseverant engagement is improved understanding. Gresalfi and Barnes (2015) suggest student perceptions of mathematical meaning-making can play a large role in informing a decision to continue to persevere upon a mistake or setback during problem-solving.

Achieving mathematical awareness through effort, that is, using personal mathematical connections to justify taking a different approach upon a perceived impasse demonstrates the productive relationship between perseverance and noticing mathematical learning gains.

In the context of working on a challenging mathematical task, the conceptualization of perseverance considered here rejects the idea that whenever children solve a problem they have, by default, persevered. Instead, perseverance requires an effortful response to the initial obstacle of a non-routine task, and altered efforts in response to any setbacks or difficulties that may arise in the work that follows. It is

61 making explicit the nature of the challenges at hand that separates perseverance, in part, from persistence and grit. Trying to persevere past a mathematical obstacle does not guarantee success, but does equip one with more resources to approach the learning opportunity than those choosing not to persevere.

It is important to remember that the nature of perseverance is situative, that is, conditions in the local context affect whether a student may persevere or not with a challenging mathematical task. Situational variables like interest in the task at hand (Hidi

& Renninger, 2006; Middleton & Jansen, 2011), the alignment between prior knowledge and the challenge of the task (Middleton et al., 2015), the problem-solving goals driving engagement with the task (DiNapoli, 2017; Elliot, 2005), task-related self-efficacy

(Linnenbrink & Pintrich, 2003), and emotional responses to engaging with a challenging task (Goldin, 2002; Pekrun, Goetz, & Titz, 2002; Wang et al., 2015) all play a role in encouraging or constraining one’s decision to persevere despite an obstacle in a given moment. Factors like these act within a person, yet the expression of such factors are a result of engagement with the task itself. Thus, to encourage perseverance for all students, the affordances of the mathematical task at hand – like how it can be scaffolded to meet students’ needs – warrants careful consideration.

Supporting Student Perseverance with Mathematical Tasks

Despite widespread educational support around the notion of struggling with challenging mathematics, both students and teachers can be reluctant to engage in and offer opportunities for perseverance (DiNapoli & Marzocchi, 2017). While trying to

62 solve a mathematical task requiring perseverance, students may aim to escape the discomfort of perseverance altogether and disengage their effort once a substantial obstacle has been encountered. Explanations of such avoidances of perseverance have been hypothesized across the literature, including student beliefs about mathematics being quick and routine (Schoenfeld, 1988; Schoenfeld & Sloane, 2016), overwhelming student discomfort when trying to re-engage effort after an impasse (Kapur, 2010), and teachers over-scaffolding instruction and reducing the need for perseverance, at least in part due to relieving their own discomfort observing their students uncomfortably struggling with the mathematics at hand (Stein, Grover, & Henningsen, 1996).

Therefore, since perseverance can still often be avoided or resisted, and because learning through problem-solving is a vital component of learning mathematics (English

& Gainsburg, 2016; Pólya, 2014), it is not surprising that the expectations for scaffolding student perseverance in problem-solving have been made more explicit in the United

States. Guiding texts like the National Research Council’s (NRC) Adding it Up

(Kilpatrick et al., 2001), the National Council for Teachers of Mathematics’ (NCTM)

Principles to Actions (2014), and the Common Core State Standards for Mathematics

(CCSSM) (2010) all address ideas solidifying perseverance and the necessary support systems as chief components of learning mathematics with understanding. The NRC stresses support for students developing “a belief in diligence and one’s own efficacy” over time (Kilpatrick et al., 2001, p. 116) as they encounter mathematical problems that necessitate perseverance. NCTM (2014) advises for teaching practices that encourage and

63 nurture perseverance over time, namely by “consistently provid[ing] students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships” (p. 48). Finally, the

CCSSM (2010) specifically reference perseverance in the first Standard of Mathematical

Practice, emphasizing the need for supporting students to “make sense of problems and persevere in solving them” (p. 6). Consequently, recent literature has sought to make explicit the most prevalent scaffolding practices that can support and help develop student perseverance with challenging mathematics, namely designing tasks with a particular structure and teacher questioning techniques.

Scaffolding Perseverance via Mathematical Task Structure and Teacher

Questioning

The structure of a mathematical task or activity can encourage perseverance, including features like multiple entry points and non-obvious solution paths (Bass & Ball,

2015; DiNapoli, 2016, 2017; Fitzpatrick & Dominguez, 2017; Horn, 2005; Kapur, 2010,

2011; Kapur & Bielaczyc, 2012; Madden & Gonzales, 2017; Middleton et al., 2015;

Sengupta-Irving & Agarwal, 2017; Sorto, McCabe, Warshauer, & Warshauer, 2009; Star,

2015; Stein & Lane, 1996; Stein & Smith, 1999; Taylor, 2015; Warshauer, 2014).

Perseverance is only necessary when a student encounters an obstacle while problem- solving – that is, when the solution pathway for a task is not immediately known or apparent. Thus, tasks that may require perseverance should be complex enough such that students must overcome challenges to make their own mathematical connections (Kapur

& Bielaczyc, 2012). However, in addition to such mathematical hurdles, the task should invite initial engagement via perceived familiarity (Bass & Ball, 2015). Without such familiarity, individuals may judge this task as too challenging and outside of his or her perceived capabilities. Ill-structured problems, though, seemingly grow in complexity as they unfold, and students find themselves engaging with a task that may not be so familiar after all, providing an opportunity for perseverance. Such tasks have been described as having a low-floor (relatively familiar prerequisite knowledge, multiple entry points) and a high-ceiling (non-obvious solution paths that may showcase previously veiled mathematical connections) (DiNapoli, 2016).

In a task meant to warrant perseverance, establishing a low-floor means that students can enter into the task using their prerequisite mathematical knowledge. If a student has previously demonstrated a basic understanding of a particular mathematical topic, then such a task can leverage that prerequisite knowledge to encourage initial engagement. For example, consider the mathematical topic of generalizing patterns: a topic believed to be foundational for all mathematics learning (CCSSM, 2010; Lee,

1996), yet is a common trouble spot for many students studying algebra (Lee, 1996;

Mason, 1996; Zazkis & Liljedahl, 2002). If a student has shown an understanding in recognizing and generalizing a pattern of cells multiplying over time, say with a simple equation like 푦 = 2푛, then a follow up task meant to encourage perseverance may introduce a similar context of two variables increasing, but the equation that generalizes

65 the situation may be more complex, like 푦 = 8푛2 + 4푛 + 1. The low-floor structure of this follow up task is the similar context of a relationship between two quantities.

Tasks with high-ceilings often require critical thinking and reasoning skills, creativity, and innovation rather than memorization and procedural fluency. High-ceiling tasks require higher-level cognitive demands and push the students beyond their initial prior knowledge. Stein and Lane (1996) describe lower- and higher-level cognitive demand. Tasks requiring lower-level cognitive demands focus on memorization or following procedures without making mathematical connections. Tasks requiring higher- level cognitive demands allow for students to follow broad procedures that have “close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts” (Stein & Lane, 1996, p. 58). Or, higher-level cognitive demand tasks may involve no procedures at all, and instead “require complex and nonalgorithmic thinking [and] require students to explore and understand the nature of mathematical concepts, processes, or relationships (Stein & Lane, 1996, p. 58). Thus, the higher-level demands of following procedures while making mathematical connections or doing mathematics without any required procedural component help establish a task’s high-ceiling.

Opportunities to engage with tasks scaffolded with low-floor/high-ceiling features offer students first-hand experience as to why these problem-solving skills are valued in mathematics and can be called upon to help achieve mathematical goals. Low-floor/high- ceiling problems situated in a relatable context offer even more opportunities for

66 perseverance, as students can analyze complex issues related to their everyday lives (Star,

2015; Taylor, 2015). Bass and Ball (2015) used a task scaffolded with low-floor/high- ceiling components to expose mathematics learners to the process of a proof of impossibility. This task had no solution and addressed a common misconception that all math problems have quick and obvious pathways toward solutions, usually brought on by repeated exposure to rote tasks (Schoenfeld, 1988). By grappling with an unsolvable problem, participants in this study had to persevere past multiple obstacles as the task unfolded, broadening their perspective of what mathematics looks like.

In addition to using tasks scaffolded with low-floor/high-ceiling components, teachers can scaffold student perseverance with mathematics through questioning techniques (Bass & Ball, 2015; Bieda & Huhn, 2017; Freeburn & Arbaugh, 2017;

Housen, 2017; Kress, 2017; Sengupta-Irving & Agarwal, 2017; Sorto, McCabe,

Warshauer, & Warshauer, 2009; Warshauer, 2014). At moments when students are stuck,

Freeburn and Arbaugh (2017) suggest using assessing questions to elicit student thinking

(e.g., How are you thinking about this?), advancing questions to extend student thinking

(e.g., How do you think your ideas can help us solve this problem?), and initiating new ideas for students to explore using judicious telling (e.g., What would happen if you tried this idea?). Sengupta-Irving and Agarwal (2017) used similar inquiries to encourage collective meta-cognition amongst groups of students collaborating. Such non-leading teacher questioning techniques provides access to in-the-moment student perspectives

67 and leverages such perspectives to encourage students to keep wrestling with the ways in which they are building their own mathematical understandings.

Kress’ (2017) discussion of teachers introducing essential questions for problem- solving aims to assist students in relying less on others’ procedures and more on one’s own thinking to persevere with a challenging mathematics task. The questions include:

1. What do you notice? 2. What additional information or clarification would be helpful? 3. What can you do or figure out? 4. How do you know that your work and/or answer are accurate? 5. Is there another way you could approach this problem? 6. What else can you say about the problem, and what else would you like to know? (p. 193)

Recounting her own teaching experiences, Kress (2017) reported the success of such guiding questions in helping students become more comfortable and independent problem solvers. With the essential questions solidified as part of the classroom culture, students asked less for step-by-step strategies to solve a problem and instead were often able and willing to persevere to solve unfamiliar problems. Sorto et al. (2009) and

Warshauer (2014) both share similar findings, arguing for responding to student struggles with teacher questions that maintain the cognitive demand of the situation. In these two studies, students working with challenging mathematical tasks were better able to organize their ideas and initially persevere when teachers scaffolded their moments of struggle with affordance questions, or teacher responses that limited teacher intervention while encouraging students to build on their ideas and be their own source of progress.

Although scaffolding practices like low-floor/high-ceiling task designs and non- leading teacher questioning techniques are meant to support student perseverance, the situative nature of perseverance makes it difficult to construct learning environments appropriate for all students’ needs. For instance, empirical studies of perseverance, although there are few, show that students need further support to continue to persevere with a low-floor/high-ceiling task; the design of the task helped students initially engage with the challenge, but students often gave up as a result of their first attempt at solving when they reached an impasse (DiNapoli, 2017; Kapur, 2010, 2011; Warshauer, 2014).

Further, these studies largely focused on students’ perseverance at single moments, and did not consider the ways in which scaffolds may have helped students develop and improve their perseverance practice over time.

Additionally, even though teachers asking assessing and affordance questions helped students stay engaged in productive struggle, teachers could not attend to all students’ needs, due to the logistical constraints of a busy classroom (Warshauer, 2014).

This left some students without opportunities to respond to teacher questioning and the benefits for perseverance that come with it. For the students that did receive ample attention, teachers still often over-scaffolded low-floor/high-ceiling tasks by revealing to students a key problem-solving step (Warshauer, 2014). Scaffolding practices like this – including partitioning tasks into steps, providing routine guidance, or modeling a solution to an analogous problem – reduce the need for student perseverance at all because the cognitive demand of the task has been lowered (Collins & Kapur, 2010). Different

69 scaffold methods provide different opportunities for students to persevere, and over- simplifying an in-the-moment obstacle lessens such opportunities. Therefore, to maintain the integrity of a task’s perseverance requirements, a scaffold must “[simplify] the task so that it could be solved while maintaining task complexity” (Stein, Grover, & Henningsen,

1996, p. 467).

Considering Embedded Conceptualization Scaffolds for Perseverance Support

Anghileri (2006) describes three levels of scaffolds that can be applied to tasks to help students problem solve (see Figure 2). Not all support levels encourage perseverance, however. The first level of scaffolding concerns making environmental provisions, which in this context involves restructuring the task into procedural parts and/or providing direct instruction for each part. The second level of scaffolding concerns explaining, reviewing, and restructuring the task. Within the task itself, these scaffolds may involve providing access to a worked-out analogous problem from which students can infer strategies applicable to the task at hand. Or, such scaffolds might simplify the task so that understanding can be built in progressive steps towards the larger problem.

Though these scaffolding strategies can assist students in having success in problem- solving, they also lessen the cognitive demand of the task by offering procedures to follow and/or reducing the need for student exploration in the task to discern important mathematical relationships.

The third level of Anghileri’s (2006) scaffolding framework, however, goes beyond replicating taught routines and solving isolated problems, and addresses

70 developing conceptual thinking. In this context, task scaffolds can provide opportunities for students to conceptualize the situation by making connections from their prior knowledge to the task at hand (e.g., Before you start, what mathematical ideas do you think might be important for solving this problem? Write down all of your ideas in detail.), and/or by developing and mapping out their own strategies for problem-solving

(e.g., Before you start, what steps do you think you might take to solve this problem?

Write down all of your ideas in detail.). Such conceptualization scaffolds provide a structure for thinking and acting that can organize a problem-solving plan coming entirely from a student’s own ideas (Tharpe & Gallimore, 1988).

Prompts depicting Anghileri’s (2006) third-level scaffolds are not unlike the teacher questioning techniques shown to encourage perseverance in the moment (e.g.,

Freeburn & Arbaugh, 2017; Kress, 2017). Also, these scaffolds are often embedded into the task itself. This can help quell the classroom logistical concern of teachers attending to every student, and it can allow more student opportunity to think independently and identify any relationships between mathematical ideas (Reiser & Tabak, 2014). Since the students themselves are the source of these prior knowledge connections and mapped out plans, these types of scaffolds maintain task complexity and do not lessen the perseverance requirements of the task.

Figure 2. Anghileri’s (2006) Scaffolding Framework

Relative to problem-solving, Anghileri’s (2006) conceptualization scaffolds align with Pólya’s (1971) general problem-solving procedures through which problem solvers approach a task. Pólya’s four-stage model includes (1) understanding the problem and then (2) devising a plan, next (3) carrying out the plan, and lastly (4) reflecting on the effectiveness of the plan and/or correctness of the solution. While stages (3) and (4) help describe the initiation and sustaining of productive struggle necessary for perseverance, it is stages (1) and (2) that can support those efforts by encouraging students to conceptualize the mathematical situation at hand (see Figure 3). Task scaffolds that provide opportunities to make connections from prior knowledge to the task at hand may help students understand the problem and extract important information that activates what they already know about relevant mathematical topics. Also, task scaffolds that address developing and mapping out problem-solving strategies ahead of time may help

72 students devise a plan that might help them start working on the problem and continue to work as they explore each stage of their plan.

Figure 3. Scaffolding Perseverance in Problem-solving

Scaffolding that prompts conceptualizing of a mathematical situation encourages initial student engagement on low-floor/high-ceiling tasks (Holton & Clarke, 2006;

Hmelo-Silver & Barrows, 2006; Ozsoy & Ataman, 2017; Reiser, 2004; Schoenfeld,

1992; van de Pol, Volman, & Beishuizen, 2010). When students have opportunities to make connections to what they already know and record all of their ideas and plans prior to the actual execution of problem-solving strategies, they are better suited to initiate and sustain their productive struggle with the task. The frustration associated with getting started on an unfamiliar mathematical task can be a key reason for students choosing not to persevere (DiNapoli, 2017). However, attending explicitly to known conceptions can serve as a springboard into more easily entering a problem; students can start to think about how to incorporate such ideas to make progress (Cheng et al., 2017). Further, a daunting experience for students is not knowing if the strategy they are initially

73 considering will actually help them solve the problem. In other words, individuals can be reluctant to persevere and invest the effort needed to explore whether a strategy is helping if he or she is not certain of success (DiNapoli, 2017). Yet, applying one’s conceptual thinking to devising and executing a problem-solving plan ahead of time can help students stay engaged with a challenging task because they are essentially “self- scaffolding” (Holton & Clarke, 2006, p. 142) by following their own step-by-step instructions to explore the situation. Following in this theoretical framework for supporting perseverance in problem-solving, innovative applications like CueThink

(2018) have entered the mathematics education space as a resource to help students explicitly attend to conceptual thinking while problem-solving (see http://www.cuethink.com/howitworks/).

Despite ample research on how conceptualization scaffolds encourage initial engagement with challenging mathematical tasks, it is less clear how these scaffold methods affect student perseverance upon reaching a perceived impasse – a moment that could be leveraged for key conceptual learning gains (e.g., Baturo, Cooper, Doyle, &

Grant., 2007; Buchbinder & Zaslavsky, 2008). If scaffolds are to support student perseverance during problem-solving, they must not only support initiating and sustaining productive struggle at the outset, but also support re-initiating and re- sustaining productive struggle after a student encounters a significant setback. Therefore, in the context of working on low-floor/high ceiling mathematical tasks, perseverance research needs to consider next the ways in which conceptualization scaffolds encourage,

74 if they do at all, student perseverance after reaching a perceived impasse during problem- solving.

Next Steps in Perseverance Research

Taken together, the literature associated with supporting student perseverance has made great strides to clearly demonstrate how certain scaffolding practices can help encourage student perseverance for solving challenging mathematics tasks. Building from these ideas, the next steps for perseverance research includes investigating how student perseverance can be supported via scaffolds at moments of impasse. The notion of perseverance includes initially engaging in productive struggle with an uncertain mathematical task, but just as important is the continuation of productive struggle after encountering a significant setback during problem-solving. Another important next step is to examine how such scaffolding might help nurture student perseverance and help improve it over time. Developing perseverance practices in students is an important goal in mathematics education (CCSSM, 2010; Kilpatrick et al., 2001; NCTM, 2014) and implies a need for research to consider how scaffolding can help change the ways in which students persevere for the better. As such, the purpose of this study is to investigate the effects of embedded conceptualization scaffolds on student perseverance with a series of low-floor/high-ceiling problem-solving tasks. This will extend the literature to clarify the ways in which students draw upon their initial conceptualizations to overcome perceived impasses in the moment and continue to persevere, as well as to detail how such efforts work to improve students’ perseverance in specific ways over

75 time. In doing so, I will introduce a student-centered perseverance framework to measure and describe in-the-moment perseverance. Also, I will employ methods that prioritize students’ in-the-moment voices to glean exactly how they felt supported in their perseverance with challenging mathematics tasks, especially during moments of discouragement, frustration, and impasse.

Methods

In this study, I investigate the relationship between an individual student’s perseverance and his or her initial engagement with a scaffolding prompt which encouraged conceptualizing of the mathematical situation. Participants worked independently on a set of five low-floor/high-ceiling mathematical tasks, some of which were embedded with conceptualization scaffolds while some others were not. All tasks required similar prerequisite knowledge and prompted students to generalize a mathematical situation. Each participant engaged with one task per week. Informed by participants’ own in-the-moment actions and their reflections on such actions, student perseverance on each task was described and analyzed based on an operationalization of the construct aligned with the conceptual framework. Therefore, this study aimed to address two research questions:

RQ1: What is the impact on students’ perseverance of scaffolds designed to prompt students to attend to their conceptualization of the task situation? RQ2: Were there any differences in changes in students’ perseverance over time, depending upon whether or not they engaged in scaffolded or non-scaffolded tasks? If so, what were these differences?

Participants and Context

The participants for this study were 10 ninth-grade students from one suburban- area high school algebra class. This school was located in the Mid-Atlantic region of the

United States and had demonstrated recent mathematics assessment scores near the median of the state. I chose to study ninth-graders’ perseverance because the transition from middle school to high school is often accompanied by increasingly negative attitudes and motivations toward mathematics learning (Fredericks & Eccles, 2002).

Additionally, algebra is often considered a gateway course to higher-level mathematics

(Cavanaugh et al., 2005). Therefore, understanding students’ perseverance and how to better support it in this context is particularly important for mathematics learning.

The selection of participants for this study was not random, but instead participants were chosen from a larger pool of interested students with similar mathematics experiences and similar demonstrations of prerequisite knowledge (see more below). Each interested student had been in the current school district for at least the last three school years, was from the same high school, was taking the same algebra course, and was in the same class section with the same teacher. The curriculum that guided this course was Glencoe/McGraw-Hill’s Algebra 1 (Holliday et al., 2012). This particular section of algebra was taught by the mathematics department chair, a teacher with 16 years of middle and high school teaching experience.

In general, the pedagogical approach for this class by the classroom teacher was direct instruction. Most lessons consisted of a warm-up, lecture, and classwork. Lessons

77 often began by engaging students with a warm-up activity, a task of a recently taught procedural skill. Next, the teacher usually delivered a lecture on the topic at hand, essentially introducing problems to solve and rehearsing, as a class, procedures by which to solve them. Most lectures contained built-in exercises for students to solve independently. At times, the teacher would ask students to talk with one another about their problem-solving ideas, but this was rare. Lastly, classes usually ended with time for students to work independently or in groups on a classwork assignment. This assignment usually consisted of rehearsing procedures they had just learned, and was often assigned for homework if the students did not finish it in class.

To help choose participants, each interested student was given a mathematics pretest (see Appendix A) containing items that assessed understanding of prerequisite knowledge. This knowledge was necessary to initially engage with each of the low- floor/high-ceiling mathematical tasks with which participants would be engaging in this study (see Table 6). Such knowledge included investigating a mathematical relationship given a set of rules and describing a mathematical relationship in a visual pattern. From the group of 21 interested students, I chose the 10 participants that best demonstrated understanding of these mathematical topics. There were six female participants and four male participants. All participants were between the ages of 14 and 15 years old.

Participants’ races were not disclosed.

The participants for this study were chosen to have some similarities regarding their past learning opportunities in mathematics. They have also demonstrated a

78 familiarity with particular mathematical ideas such that their experience with a similar mathematical situation might invite initial engagement. To help establish a rapport with potential participants, I attended their math class for several weeks prior to data collection and interacted with students by co-teaching one lesson with their teacher and helping students with their daily classwork assignments. All names reported in this paper are pseudonyms.

Data Collection

Both research questions guiding this study address the ways in which students persevered, or did not, while working on a series of challenging mathematical tasks. Each participant was observed engaging with five low-floor/high-ceiling tasks, one per week.

Because this study prioritized the student point of view of the perseverance experience, I incorporated several opportunities for participants to make explicit their in-the-moment perspectives during problem-solving. For each task and each participant, I conducted think-aloud interviews while he or she worked and video-reflection interviews immediately after he or she finished working. Additionally, once a participant had engaged with all five tasks (and thus all five think-aloud interviews and video-reflection interviews), I conducted an exit interview to give each participant an opportunity to comment on his or her overall experience working on the five low-floor/high-ceiling tasks. Considering all data sources, I conducted 11 interviews with each participant, or

110 interviews in total for this study.

Think-aloud interviews. Guided by Ericsson and Simon’s (1993) methods for collecting verbal reports of in-the-moment experiences, I conducted five think-aloud interviews (one per task) with each participant in order to provide opportunities to persevere past mathematical obstacles. In each think-aloud interview, the participant worked on a low-floor/high-ceiling task while thinking out loud. These interviews lasted anywhere from 5-45 minutes, depending on for how long a participant stayed engaged with a task (see Appendix B for participants’ time spent on each task). I did not help the participants with their problem-solving. Instead, I only reminded them to think aloud and prompted them to clarify what they were thinking or doing, when necessary. Two particularly important moments of clarification occurred near the beginning of each think-aloud interview. After the participant had read the task but before he or she started working, I asked the participant to clarify (in his or her own words) what he or she would be trying to accomplish while working on this task, and if he or she knew exactly how to accomplish it from the start. These questions helped ensure if the participant had the opportunity to persevere with the task at hand. The ideal conditions for perseverance consisted of a student understanding the objectives of the task yet not knowing exactly how to achieve such objectives. These interviews were video recorded, transcribed, and I collected his or her written work for later analysis. See Appendix C for the full think- aloud interview protocol.

Mathematical tasks. The mathematical tasks with which each participant engaged across the think-aloud interviews were from the Mathematics Assessment Project, part of

80 the Mathematics Assessment Resource Service (MARS). The specific five tasks (Cross

Totals, Sidewalk Stones, Skeleton Tower, Table Tiling, and Triangular Frameworks) (see

Appendix D) align with the same topics included on the pretest to insure that each participant had demonstrated an understanding of some of the prerequisite knowledge needed to solve these problems (see Table 6). The Cross Totals and Triangular

Frameworks tasks provide a set of rules for a particular situation and asked students to investigate and describe a mathematical relationship. The Table Tiling, Sidewalk Stones, and Skeleton Tower tasks provide a visual pattern and asked students to investigate and describe a mathematical relationship.

All think-aloud interview tasks involved generalizing mathematical relationships and are considered analogous because they are all “expert tasks, [which are] rich, less structured tasks requiring strategic problem-solving skills as well as content knowledge”

(MARS, 2015). Moreover, “expert tasks aim to cover the full range of [Common Core

State Standards for Mathematical] Practice” (MARS, 2015), including “students should make sense of problems and persevere in solving them” (CCSSM, 2010, p. 6). Thus, each of these tasks had a similar low-floor/high-ceiling structure that can support perseverance by motivating initial engagement via familiar content, but ultimately complex enough to beget uncertainty and necessitate struggle. I also solicited six independent mathematics education experts to rate the difficulty of these tasks; they reported no perceived differences. Additionally, each participant in this study was given an opportunity to

81 reflect on task difficulty after his or her participation and reported no perceived differences (see section on exit interviews).

To help insure that each participant worked toward similar objectives for each task, I adapted tasks (when necessary) to make the problem-solving goals specific and clear (e.g., “Your goal while working on this task is to ______.”). Each task had two specific objectives for the participant to accomplish. Though there are several goals likely driving the problem-solving actions of a student (see Elliot & McGregor, 2001, for their theory about competence and valence influencing student goal setting), embedding a clear goal in the task itself made it more likely that all participants perceived an explicit goal toward which they could work – an important condition for perseverance (Middleton et al., 2015).

Scaffold conditions. For the conditions necessary to answer RQ1, I embedded four different scaffolds directly into the mathematical tasks (see Table 6). These conditions were based on Anghileri’s (2006) conceptualization scaffolds, which encourage conceptual thinking yet preserve cognitive demand. The assignment of particular tasks into particular scaffold conditions was random to avoid any potential bias.

The Activating Prior Knowledge (APK) scaffold was assigned to Cross Totals.

This is a conceptualization scaffold that read: Before you start, what mathematical ideas do you think might be important for solving this problem? Write down all of your ideas in detail. This scaffold appeared after the statement of the task goal and prompted

82 participants to brainstorm about the specific concepts they think they might need to solve the problem, before they started working on the problem.

The Solution Pathway (SP) scaffold was assigned to Sidewalk Stones. This is another conceptualization scaffold that read: Before you start, what steps do you think you might take to solve this problem? Write down all of your ideas in detail. Like the

APK scaffold, this prompt appeared after the statement of the task goal and prompted participants to brainstorm about the specific problem-solving steps they might need to execute to solve the problem, before they started working on the problem.

The Activating Prior Knowledge (APK) and Solution Pathway (SP) scaffold was assigned to Skeleton Tower. Both of these scaffolds appeared after the statement of task goal and prompted participants to conceptualize the situation in two ways, before they started working on the problem.

The remaining tasks, Table Tiling and Triangular Frameworks, were assigned the non-scaffold condition. That is, they did not contain an embedded prompt asking participants to explicitly conceptualize the situation in some way. Instead, Table Tiling and Triangular Frameworks contained a trivial prompt, appearing after the statement of the task goal, to help mirror the task structure of the other scaffolded tasks. This prompt read: Before you start, do you know what you are trying to accomplish while working on this task? Circle yes or no. This prompt was considered trivial because a version of this question was asked in the think-aloud interview for all tasks.

The order in which each participant worked on each task was mostly random to avoid bias, with the exception of Triangular Frameworks always being the fifth and final task, without a scaffold. The original research design only included four conditioned mathematical tasks: Cross Totals (APK), Sidewalk Stones (SP), Skeleton Tower (APK &

SP), and Table Tiling (non-scaffolded). Including Triangular Frameworks in the pool of mathematical tasks was a post-hoc decision because it became apparent during data collection that participants were engaging with the scaffolds in scaffolded tasks in the same ways, despite the different types of scaffolds. Participants did not perceive4 any differences in what the APK, SP, and APK & SP scaffolds were asking them to do, and essentially treated all scaffold prompts as an APK prompt by recording their conceptual brainstorming before starting any scaffolded task. From a scaffolding perspective, this meant that three tasks were conditioned with a conceptualization scaffold (Cross Totals,

Sidewalk Stones, and Skeleton Tower) and only one was not (Table Tiling). Therefore, to increase the pool of non-scaffolded tasks for better comparison purposes, I administered

Triangular Frameworks (non-scaffolded) to each participant in his or her final think- aloud interview.

Table 6

Mathematical Task Details

Problem-solving Scaffold Pretest Primary Mathematical Topic(s) Interview Task Condition Item

4 Almost all participants (9 out of 10) mentioned during their think-aloud interviews that the scaffold prompts were ostensibly equivalent.

Describing a relationship given a set of Cross Totals APK 2 rules Sidewalk Stones SP Describing a relationship in a visual pattern 2 APK & Skeleton Tower Describing a relationship in a visual pattern 1 SP Table Tiling None Describing a relationship in a visual pattern 1 Triangular Describing a relationship given a set of None 1 Frameworks rules Note: APK = Activating Prior Knowledge; SP = Solution Pathway

Video-reflection interviews. Informed by Schön’s (1983) concept of reflective practice, Ericsson and Simon’s (1993) stimulated recall protocol, and Webel’s (2013) use of video clips to inspire authentic recollection, I conducted five video-reflection interviews (one per task) with each participant. These interviews lasted approximately

30-60 minutes. The purpose of the video-reflection interviews was to provide opportunities for participants to reveal and explain any in-the-moment cognitive and emotional activity they experienced during their time working on a mathematical task.

Video-reflection interviews were conducted immediately after each think-aloud interview, the ideal time according to Ericsson and Simon (1993) because “much information is still in [the participant’s short term memory] and can be directly reported or used as retrieval cues” (p. 19). In each video-reflection interview, the participant and I watched the video recording of his or her think-aloud interview conducted just moments prior. I asked participants to best reconstruct their thoughts and feelings at particular moments during their engagement. The moments in which I was most interested were those that define perseverance in the moment: when participants appeared to initiate productive struggle, sustain productive struggle, realize their mathematical progress,

85 encounter an impasse, and if they appeared to re-initiate and re-sustain their productive struggle to realize additional progress. A participant was considered to have reached a perceived impasse if he or she affirmed they were substantially stuck and unsure how to continue (VenLehn et al., 2003). Thus, I asked non-leading questions throughout the interview, including “what were you doing/thinking/feeling in this moment?” as we watched the video recording together. See Appendix E for the full video-reflection interview protocol. These questions gave the participant a chance to explain if and how he or she was persevering, or were not persevering, during his or her work with a challenging task. These interviews were audio recorded and transcribed for later analysis.

The video-reflection interviews proved to be a fruitful outlet by which participants shared explicit details about their in-the-moment perseverance from their point of view – details they did not share during their think-aloud interviews. Although these reflections include a hindsight bias, Schön (1983) argued that a reflection-in-action and reflection-on-action perspective can be most beneficial to learn about meaningful moments during deep engagement because participants often have difficulty articulating their full experience as it unfolds. Thus, including an opportunity for participants to make a present rationalization of such moments via video-reflection helps complement and clarify any insights gleaned from a prior think-aloud interview and supports a more quality participant introspection.

Exit interviews. Lastly, after conducting all think-aloud and video-reflection interviews, I conducted an exit interview with each participant. These interviews lasted

86 approximately 20-30 minutes. The purpose of the exit interview was to allow for participants to make any last comments or reflections about their entire experience in this study. To help motivate conversation, I made available to them their written work on each of the five mathematical tasks. Each exit interview began with me asking “Is there anything you’d like to share about your overall experience working with these five problems?”, which then unfolded into a conversation directed by whatever the participant chose to share. Additionally, to discern from the student point of view about variability of challenge, I asked participants to comment on the difficulty of each task. It is important to note that almost all participants (9 out of 10) affirmed that the five tasks were comparably difficult (e.g., “all of them were hard, like the same”, “these problems were all kinda similar to me”; “I really didn’t know what to do for any of them”). The other participant (1 out of 10) perceived Skeleton Tower (a scaffolded task) and Table Tiling (a non-scaffolded task) as the two most difficult tasks, in no particular order. I also used the exit interview as a way of member checking. I shared with participants my observations of what they revealed during their think-aloud and video-reflection interviews and asked them if they would change anything about my general findings to more closely align with their experiences. See Appendix F for the full exit interview protocol. These interviews were audio recorded and transcribed for later analysis.

Data Analysis

The goal of my data analysis was to describe and examine the ways in which participants persevered during their think-aloud interviews, and their reasons for why

87 they persevered, between scaffolded and non-scaffolded task conditions and also over the course of five weeks. To measure and operationalize perseverance, I developed the

Three-Phase Perseverance Framework (see Table 7). First, the framework considered if the task at hand warranted perseverance for a participant (the Entrance Phase). Next, it considered the ways in which a participant initiated and sustained productive struggle

(the Initial Attempt Phase). Lastly, the framework considered the ways in which participants re-initiated and re-sustained productive struggle if they reached an impasse as a result of their first attempt (the Additional Attempt Phase). Mathematical productivity was determined based on the extent to which the participant perceived himself or herself as better understanding the mathematical situation as a result of his or her efforts. A participant need not correctly solve the task in its entirety to demonstrate maximum perseverance as measured by the framework. The Three-Phase Perseverance

Framework captures perseverance itself, not an overall achievement. Therefore, a participant could demonstrate ample perseverance while working with a task through building incremental understanding via effort, yet not completely solve the task.

To inform my measurement decisions about how each participant persevered with each task, I viewed the video footage of each think-aloud interview, studied the accompanying written work, and listened to the accompanying video-reflection interview. I also read each think-aloud and video-reflection transcript simultaneously. To ascertain, from the participant point of view, why participants were persevering and why it might be changing over time, I read transcripts from the think-aloud, video-reflection,

88 and exit interviews. All transcripts were coded using the open-coding process to reveal and establish themes about how and why participants persevered or did not persevere

(Strauss & Corbin, 1990).

Three-Phase Perseverance Framework. The structure of the Three-Phase

Perseverance Framework (see Table 7) was developed using an inductive approach, emerging from the data sources of this study. This perseverance framework was designed to reflect perspectives of concept (CCSSI, 2010; Dolle, Gomez, Russell, & Bryk, 2013;

Middleton et al., 2015), problem-solving actions (NCTM, 1989, 2000; Pólya, 1971;

Schoenfeld, 1979, 1985, 1988, 1989; Schoenfeld & Herrmann, 1982; Schoenfeld &

Sloane, 2016; Silver, 2013; Wilson, Fernandez, & Hadaway, 1993), self-regulation

(Baumeister & Vohs, 2004; Carver & Scheier, 2001; Flavell, 1976; Schoenfeld, 1992;

Winne, 2010; Zimmerman & Schunk, 2011), and making and recognizing mathematical progress (Gresalfi & Barnes, 2015). To substantially capture the student point of view during engagement with tasks, coding decisions – or whether or not certain engagement constituted evidence of perseverance – largely depended on student cues from each think- aloud and video-reflection interview. For instance, evidence of mathematical progress consisted of the student affirming, describing, and defending his or her point of view that he or she better understood the mathematical context embedded in a task after his or her effort(s). Specific examples of data analysis decisions are included in the presentation of the results.

Table 7

Three-Phase Perseverance Framework

While initially reading the task, the Entrance Phase captured whether the participant understood the entirety of what the task was asking (clarity component) and if the participant immediately knew how to solve the problem (initial obstacle component).

Essentially, this phase established the appropriateness of a task for perseverance analysis with a particular individual. An ideal outcome in the Entrance Phase, from the researcher perspective, was a participant affirming the task made sense and was within his or her perceived realm of capability, yet he or she did not immediately know a procedure by which to solve it. In this case, the task was deemed as appropriate for perseverance measurement and participants entered into the next phase of engagement. It is important to note that for all think-aloud interviews (50 out of 50), participants affirmed both

90 components in the Entrance Phase of the Three-Phase Perseverance Framework: they understood the objectives of the task, but did not immediately know ways in which to achieve them.

The Initial Attempt Phase examined whether and how a participant initiated and sustained his or her effort, and the outcome of such effort as he or she worked toward solving all parts of the problem. Since the student did not immediately know a solution pathway, evidence of perseverance included deciding to engage with the task at all (initial effort component). If the student decided to pursue solving the problem, evidence of perseverance included using a problem-solving strategy to diligently explore the uncertain nature of the mathematical situation (sustained effort component). In this context, a problem-solving strategy was defined as using one or more heuristics, in conjunction, to work toward a solution. See Appendix G for a list of heuristics suggested by NCTM (1989, 2000). As a result of these diligent efforts, evidence of perseverance included making mathematical progress toward better understanding the mathematical relationships or solving the problem outright (outcome of effort component). See

Appendix H for task solutions.

In the event a participant did not solve the problem after making a first attempt, the Additional Attempt Phase aimed to capture if and how the participant amended his or her original problem-solving plan, and the outcome of such efforts as he or she worked to overcome any setbacks and toward solving all parts of the problem. To enter this phase

(marking the end of a first attempt and the beginning of a second attempt), the participant

91 must first reach a perceived impasse. Evidence of a perceived impasse was when a participant voiced during his or her think-aloud, or clarified during his or her video- reflection interview, that he or she was substantially stuck and unsure how to continue with the task (VenLehn et al., 2003). This implied the participant had exhausted all of his or her perceived options immediately stemming from carrying out a first plan and were unsure of how to keeping moving forward toward finding a solution (Warshauer, 2014).

At this point, evidence of perseverance included deciding to re-engage with the task using a different strategy – one that was not used during the first attempt (initial effort component). Assuming the student decided to pursue solving the problem with a new plan of attack, additional evidence of perseverance included diligently exploring the mathematical situation with the new strategy (sustained effort component), and making new progress toward better understanding the mathematics involved in the task, or solving the problem outright (outcome of effort component). Due to the logistical constraints of scheduling participants’ interviews for this study, I only analyzed one additional attempt at solving the problem in the Additional Attempt Phase.

Hypothetically, a participant could continue his or her effort in the Additional Attempt

Phase – making a second, third, or fourth additional attempt, and so on – but such efforts were not considered here.

Coding decisions. I used a point-based analysis of the Three-Phase Perseverance

Framework to help inform deeper qualitative investigation of the specific ways in which participants were persevering on scaffolded and non-scaffolded tasks. Each participants’

92 experiences with each task were analyzed using the framework, and each component in the Initial Attempt and Additional Attempt Phases were coded as 1 or 0, as affirming evidence of the component or otherwise, respectively. Thus, participant problem-solving experiences that demonstrated evidence for more components from the Initial and

Additional Attempt Phases were considered examples of higher quality perseverance compared to those experiences that demonstrated evidence for less components.

Coding decisions were based on explicit quotes from the participants’ think-aloud and video-reflection interviews. For instance, participants earned a 1 for an Outcome of

Effort component only if they affirmed perceiving mathematical progress. Such decisions were not based on my own observations or perceptions. More examples of coding decisions are reported together with rich descriptions of participants’ engagement in the

Results section. Since each task had two objectives and six components per objective, there were 12 framework components to consider, per participant, per task. Thus, Three-

Phase Perseverance scores ranged from 12 to 0, depicting optimal to minimal demonstrated perseverance in this context, respectively5.

Once Three-Phase Perseverance scores were determined for each participant’s experiences with all five tasks (three scaffolded, two non-scaffolded), I ran statistical

5 In the event a participant solved a task objective during their Initial Attempt Phase, and thus did not require an Additional Attempt Phase, they earned the maximum score for that objective. This was to depict their optimal outcome as a result of their productive effort on that objective. Across the 50 think-aloud interviews, and thus 100 objectives toward which participants worked, this happened five times.

93 tests to help reveal some trends of the perseverance data critical to answering the research questions. To help address RQ1, I conducted matched-paired t-tests to compare participants’ perseverance scores between their work on scaffolded and non-scaffolded tasks. To help address RQ2, I conducted a regression analysis of participants’ perseverance scores on scaffolded tasks over time, and on non-scaffolded tasks over time.

The outcomes of these tests informed how I proceeded to more deeply analyze the data.

For example, because significant differences in perseverance scores were apparent between scaffold conditions in the Additional Attempt Phase alone (see Table 8 in

Results), I sought through my qualitative analysis of all data sources to uncover exactly why participants were persevering differently in the Additional Attempt Phase (after encountering an impasse) on scaffolded tasks versus non-scaffolded tasks. This coding process consisted of an initial examination of each interview to allow themes to emerge relative to a phenomenon of interest (e.g., perseverance in the Additional Attempt Phase), and was followed by a more detailed identification of influential factors, voiced by participants, that helped unpack an overarching theme. For instance, one overarching theme that emerged from coding the think-aloud, video-reflection, and exit interviews was that participants’ additional attempts benefited from their initial conceptualization work prompted by scaffolded tasks. Participants voiced specific reasons explaining that general theme, including improvements in their mathematical thinking and feeling more organized in their problem-solving. More examples of emergent themes and influential factors are described together in the Results section.

Reliability procedures. I enlisted help from two coders, one mathematics education faculty member and one mathematics education graduate student, to independently code approximately 14% of the data set (15 out of the 110 total interviews). The purpose of this coding help was to determine how consistent my codes were with their coding of demonstrated perseverance (via the Three-Phase Perseverance

Framework) and the themes and factors that help explain why participants persevered.

Each coder analyzed five randomly selected participant’s experiences with a mathematical task. Thus, they each independently coded a think-aloud interview, the associated video-reflection interview, and the exit interview for five different participants. Between the three of us, our inter-rater reliability was 93%. We discussed and remedied all coding disagreements, and I refined my perseverance framework and description of themes and influential factors as a result of these discussions.

Results

I will report two primary results that address each of the two research questions:

RQ1: What is the impact on students’ perseverance of scaffolds designed to prompt students to attend to their conceptualization of the task situation?

RQ2: Were there any differences in changes in students’ perseverance over time, depending upon whether or not they engaged in scaffolded or non-scaffolded tasks? If so, what were these differences?

Relative to RQ1, participants in the study demonstrated higher quality perseverance on scaffolded tasks compared to when they worked on non-scaffolded tasks. This finding was especially prevalent in the Additional Attempt Phase of participants’ engagement.

Relative to RQ2, participants’ perseverance changed over time by improving in quality while working on scaffolded and non-scaffolded tasks, but the quality of perseverance was improving more strongly in the context of working on scaffolded tasks than with non-scaffolded tasks. Further, participants were able to notice and articulate specific ways in which their perseverance was improving across their work on scaffolded tasks, and not on non-scaffolded tasks. In the sections that follow, I will first elaborate upon each result at the group level through both the quantitative and qualitative findings. Then, to illustrate each finding from the student perspective, I will detail one representative participant’s experiences more deeply, followed by several participants’ points of view that substantiate those experiences.

Result 1: Higher Quality Perseverance on Scaffolded Tasks than on Non-Scaffolded

Tasks, Especially during Additional Attempt

Participants demonstrated higher quality perseverance when working on scaffolded tasks (S) compared to on non-scaffolded tasks (NS), in general, as evidenced by significantly greater mean Three-Phase Perseverance Framework scores (MS = 8.73,

SDS = 2.07, MNS = 6.00, SDNS = 2.60; t(9) = 6.816, p < .001) (see Table 8), as well as by the reported nature of their engagement while problem-solving. Differences in participants’ perseverance in the Additional Attempt Phase of the Three-Phase

Perseverance Framework drove this general finding. Participants demonstrated higher quality perseverance after encountering a perceived impasse while working on scaffolded tasks (S) compared to on non-scaffolded tasks (NS). This was evident by significantly

96 greater mean Additional Attempt Phase perseverance scores (MS = 3.67, SDS = 2.50, MN

S= 1.60, SDNS = 2.58; t(9) = 4.083, p = .003) (see Table 8), as well as by the reported nature of their engagement while problem-solving.

Table 8

Three-Phase Perseverance Framework Scores

Total Perseverance Scores (Maximum of 12 Points) Task Type Mean n Scaffold 8.73 30 (3 tasks per 10 participants) Non-Scaffold 6.00 20 (2 tasks per 10 participants) Difference 2.73*** Additional Attempt Perseverance Scores (Maximum of 6 Points) Task Type Mean n Scaffold 3.67 30 (3 tasks per 10 participants) Non-Scaffold 1.60 20 (2 tasks per 10 participants) Difference 2.07** ** p < .01, *** p < .001

The most prevalent difference between participants’ perseverance on scaffolded and non-scaffolded tasks was whether and how they re-initiated and re-sustained a productive additional attempt at solving the problem. This means that while working on scaffolded tasks, participants often continued to productively struggle toward a solution after reaching a perceived impasse. This was not often the case after reaching an impasse while working on non-scaffolded tasks. When considering the specific Additional

Attempt Phase components of re-initiating, re-sustaining, and outcome of effort, participants working on scaffolded tasks demonstrated more aspects of perseverance compared to their work on non-scaffolded tasks (see Table 9). These frequency differences further suggest a general effect of the conceptualization scaffolds on

97 participants’ perseverance upon impasse. Participants in this study were 45% more likely to re-initiate their effort, 35% more likely to re-sustain their effort, and 37% more likely to be productive in their new efforts after encountering an impasse while working on scaffolded tasks compared to non-scaffolded tasks. Additionally, although the goal of the

Three-Phase Perseverance Framework was to capture perseverance itself, not necessarily the achievement of solving a problem outright, participants’ work in the Additional

Attempt Phase on scaffolded tasks did more often resulted in a complete solution than their related work on non-scaffolded tasks.

Table 9

Perseverance Frequencies in Additional Attempt Phase

Component On Scaffolded Tasks On Non-Scaffolded Tasks Re-Initiated Effort 80% (24 out of 30) 35% (7 out of 20) Re-Sustained Effort 70% (21 out of 30) 35% (7 out of 20) Productive Outcome of 57% (17 out of 30) 20% (4 out of 20) Effort Solved Outright 37% (11 out of 30) 10% (2 out of 20)

While talking about their engagement with the five problem-solving tasks in the study, all participants (10 out of 10) reported a general positive effect of their preliminary conceptualizing work prompted by the scaffolds embedded in the scaffolded tasks on their mathematical engagement. Related, none of the participants (0 out of 10) engaged in any noticeable preliminary conceptualizing work on tasks that did not prompt it, i.e., the non-scaffolded tasks. In other words, across the 20 think-aloud interviews I conducted with participants working on non-scaffolded tasks, never was there a time that a

98 participant explicitly wrote down or spoke about any of his or her ideas regarding an initial conceptualization of the mathematical situation before he or she initiated an effort toward a solution, if he or she decided to initiate an effort. Most of the participants (8 out of 10) explicitly mentioned in their interviews that the reason they did not engage in such preliminary work on non-scaffolded tasks was because the task did not specifically ask them to do so, even though they recognized such work was helpful to them. Therefore, in this way, participants made clear one reason they persevered more on scaffolded tasks was because they engaged in the process of initially conceptualizing the situation, as prompted by the scaffolds.

Several more specific themes emerged from the analysis of all interviews that help explain why participants found it easier to persevere on scaffolded tasks compared to on non-scaffolded tasks, especially during an additional attempt at solving after reaching an impasse. Almost all participants (9 out of 10) shared that their preliminary conceptualization work was particularly helpful after reaching an obstacle, specifically because they could re-visit their initial mathematical ideas about the situation during the moment when they most wanted to give up on the task at hand. In that moment, when they had already made a mistake and were not sure how to continue to persevere, these participants reported it being easier to think mathematically about the situation compared to during their work on non-scaffolded tasks when they had not recorded such thinking.

As a result of such mathematical thinking in those trying moments, many participants (7 out of 10) shared how they considered a different mathematical idea(s) to explore to

99 continue to persevere. By thinking about different connections between mathematical ideas they had recorded in their initial conceptualization of the situation, these participants reported it less difficult to re-initiate and re-sustain their effort exploring their ideas, and often deepened their mathematical understanding of the situation as a result of such additional efforts.

Almost all participants (9 out of 10) also reported that their work on scaffolded tasks felt more organized and structured than their work on non-scaffolded tasks. As a result, several participants (6 out of 10) mentioned a feeling of momentum during their engagement with scaffolded tasks. These participants shared that they felt more invested in the task because the scaffold prompt helped them initially engage somehow with the mathematical ideas, and once they delved into the problem it was easier to keep progressing. Conversely, most participants (8 out of 10) specifically reported feeling lost and disorganized after reaching an impasse during their work on non-scaffolded tasks, and often cited such affect as responsible for deciding not to continue to persevere.

The next two sections illustrate representative participants’ experiences working on scaffolded and non-scaffolded tasks. I will first unpack and compare James’s experiences as he engaged with a scaffolded task, Cross Totals, and a non-scaffolded task, Triangular Frameworks. James’s demonstrated perseverance with these two tasks is representative of the above findings because he leveraged his conceptualization work on

Cross Totals to continue to persevere past a setback, compared to his work on Triangular

Frameworks in which he chooses to give up when reaching a discouraging obstacle. After

100 recounting James’s demonstrated perseverance and point of view, I will share the perspectives of Tom and Marcia, both of which align with James’s experiences in their exit interviews as they reflected upon their overall problem-solving experiences.

James’s Perspective: A “Life-Preserver” amidst “Chaos.” Describing and comparing James’s engagement with Cross Totals and Triangular Frameworks (see

Appendix D for tasks) is illustrative of how perseverance was better supported in scaffolded tasks compared to non-scaffolded tasks, especially by helping participants make a quality additional attempt at solving. For James, Cross Totals was the fourth overall task upon which he worked and the third and final scaffolded task. Triangular

Frameworks was his fifth and final task, the second of two non-scaffolded tasks. James earned a Three-Phase Perseverance Framework score of 12 for his work on Cross Totals and 6 for his work on Triangular Frameworks (see Table 10), with the most notable difference being his lack of an additional attempt on Triangular Frameworks. While reflecting on his work on Cross Totals, James specifically referenced the conceptualization work prompted by the scaffold, explaining the benefit of revisiting such work during his most frustrated moments while problem-solving.

Table 10

James’s Three-Phase Perseverance Framework scores for Cross Totals and Triangular Frameworks

Triangular Cross Totals (S, INITIAL ATTEMPT PHASE Frameworks (NS, 4th) 5th) Evidence of Perseverance Obj. 1 Obj. 2 Obj. 1 Obj. 2

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Initial Effort Engaged with the task     Sustained Used problem-solving Effort heuristics to explore the task     Outcome of Made mathematical progress Effort     Triangular Cross Totals (S, ADDITIONAL ATTEMPT PHASE Frameworks (NS, 4th) 5th) Evidence of Perseverance Obj. 1 Obj. 2 Obj. 1 Obj. 2 Initial Effort Engaged with the task     Sustained Used problem-solving Effort heuristics to explore the task     Outcome of Made mathematical progress Effort     Total Perseverance Score 12 6 Notes: S = Scaffolded Task; NS = Non-Scaffolded Task; Obj. = Objective; 4th = Fourth task overall; 5th = Fifth task overall

James’s experience with Cross Totals. While working on Cross Totals, James began his think-aloud by affirming he understood the objectives of this task. He said,

“I’m gonna try to find a way to say the possible [cross totals] and the impossible [cross totals].” Before starting, he also affirmed that he has never seen this problem before and said, “I have no idea what to do.” This was evidence in the Entrance Phase of the Three-

Phase Perseverance Framework that the task was appropriate for James from a perseverance analysis perspective.

Next, James began his work with the scaffold prompt. He was thinking about apparent mathematical ideas present in the problem that might be helpful to him. First, he started to explore the parameters of possible and impossible cross totals by finding the sums of integers 1 through 5 and 5 through 9. Then he noticed that the integer in the

102 middle of the cross total would be included in both crosses, the horizontal and vertical.

Last, James thought that based on the relationships between these ideas, that most of the cross total solutions would be sums between 20 and 30. While he was engaging with the conceptualization scaffold, James was recording his observations and ideas (see Figure

4). During his Cross Totals video-reflection interview, James further reflected back upon what he did during his work with the scaffold prompt:

I: Can you describe you experiences here as you worked on [the scaffold prompt]? James: Well I was just writing down stuff I noticed to be true, or probably true. And there was no wrong answer here in the beginning so I was just trying to narrow it down. I: What do you mean narrow it down? James: Like for any new problem there are so many things you can focus on. So it gave me lots less to work with so I could better understand what I was going to do. I felt pretty confident at this point because I had some ideas about it.

Figure 4. James’s initial conceptualization for Cross Totals

After recording his initial ideas, James started his first attempt at solving the problem. During his think-aloud he first shared that he will initiate his effort by trying to

“find some possible ones first, and those might connect to impossible ones.” In this way,

James is thinking about his objectives regarding possible and impossible cross totals

103 simultaneously. He chose guess and check as his problem-solving heuristic and began to sustain his effort by inserting the integers along two lines, with one integer in the middle of both lines, and finding their sums (see Figure 5). These two lines represented the horizontal and vertical lines that could make up a cross. James was not having success in finding examples of possible cross totals, as he had hoped. After guessing and checking for two different possible cross totals, he said during his think-aloud:

James: That doesn’t work…that doesn’t work either…It’s not the same on each of them. I thought it would work for any numbers really. This is harder than I thought. I: What are you thinking now? James: I’m thinking I’m stuck. This is harder than I thought.

Figure 5. James’s guessing-and-checking during initial attempt on Cross Totals

James affirmed he perceived an impasse (“I’m stuck”) as a result of his first attempt at solving the problem. While reviewing the moments leading up to his obstacle during the video-reflection interview, James explained he was “getting very frustrated” because he anticipated this problem was going to be simple. He shared “it was stressful here because I never learned this and there was not any way I would know how to do it. I definitely panicked.” James explained his thought process for choosing guess and check as his heuristic. He thought he could just “pick any numbers and they would work”, so his plan was to guess and check for possible cross totals and determine the rules for both

104 possible and impossible solutions based on any examples he might have found. James also revealed during his video-reflection interview that his first attempt at solving helped him learn more about both objectives within the mathematical situation. He said, “I definitely saw here that you couldn’t just throw numbers in, you had to think about big numbers and small numbers to get possible ones to work and to figure out why some won’t ever work.”

In all, James’s first attempt at solving Cross Totals demonstrated perseverance in all components of the Initial Attempt Phase of the Three-Phase Perseverance Framework.

He initiated his effort towards both objectives, finding general rules that describe possible and impossible cross totals, by stating a plan to explore both ideas simultaneously. He sustained his effort toward both objectives by guessing and checking with different integers, thinking about the underlying mathematical ideas that might make the possible ones possible and the impossible ones impossible. Lastly, he perceived he was making mathematical progress toward better understanding the mathematical relationships involved with constructing possible and impossible cross totals.

After James revealed he was “stuck” during his think-aloud, he paused and then admitted “I don’t know what to do now.” He started looking around at his work on the papers in front of him and eventually pointed to his list of mathematical ideas under the scaffold prompt and said, “Well I haven’t used this yet really. Something is probably special about the middle number.” During his video-reflection interview, he explained his point of view during these moments:

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I: Can you describe your experience here as you pointed to the middle number idea? James: Well I had kind of forgotten that I wrote all this stuff down. So I saw again that the middle number was very important. It was kind of like my life-preserver. I: What do you mean life-preserver? James: Like it kept me from giving up. I really wanted to stop because I thought the problem would be easy, you know? But then I saw my earlier idea that I had forgotten and it kind of pushed me on. It made the problem easy again…not like easy to figure out, but like easy to try on…because I had another idea to try. It was something I could do, and I thought it would be important.

James’s idea to focus on the role of the cross totals’ middle number to make an additional attempt at solving was from his initial conceptualization of the situation, which occurred through his engagement with the scaffold. During his think-aloud, he shared his plan of re-initiating his effort by focusing on the role of the middle number for possible and impossible cross totals. He began re-sustaining his effort by changing his point of view, a different problem-solving heuristic, and examined the provided example. He noticed that “this one has 9 in the middle to get a 27 and I can see how they are balancing the big and small numbers... and maybe there is something about the evens and odds.”

James began to write down his observations around the provided example (see Figure 6) and stated “I need to think about how all this might work.” In his video-reflection interview, James shared how these were the moments when he started to build momentum toward solving the problem:

I: Can you describe your experience here as you studied the given example? James: Well now I was starting to figure things out. Like I paid attention to the 9 in the middle, and then it really started to make sense that

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the 9 was going to be added twice because it’s in the horizontal and vertical. And then it’s like I just got on a roll. I started to see other stuff, like the big and small number placement, and evens and odds. I just wanted to try my own example to see if I could get it to work.

Figure 6. James’s observations of the given example during additional attempt on Cross Totals

James continued his efforts during his think-aloud by logically reasoning, another problem-solving heuristic, about “balancing the numbers in the cross” of his own examples (see Figure 7). He began filling his cross with a 1 in the middle because he wanted to “try an odd number here.” Then, he continued to insert integers into the different lines of the cross, considering their position carefully. He explained, “If I put the

9 in [the horizontal] line, then I will probably need to put the 8 in [the vertical] line to balance them out.” James went on to follow this logic of considering the magnitude of each number to fill his cross completely and found that he created a cross total of 23.

Recognizing that the given example and his example both produced cross totals with odd integers in the middle, he created a second example with 4 in the middle “see what happens with evens.” He began inserting integers into the cross, following his balancing

107 logic, and eventually said, “I don’t think it’s going to work…you can’t fit these other numbers in to get [the horizontal and vertical lines] to be the same.”

Figure 7. James’s own examples during additional attempt on Cross Totals

James finished his work during his think-aloud by creating one additional example of a possible cross total (with 5 as the middle integer), and one additional example of an impossible cross total (with 6 as the middle integer). He then began to write down his rules to meet the objectives of the task. James’s primary rules were about the nature of the middle integer in the cross. He concluded, “The big [rules] are that all solutions are odd, like the middle can’t be even to do a cross total.” When thinking about why his rules were true during his think-aloud, James explained his rationale:

I: What are you thinking now? James: I’m thinking about this last part, why they’re true…well I know they have to be true because if you have an even in the middle, like, you add up all the numbers 1-9 and you get 45. But you have

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to count the middle one twice, so if it’s like 4, that gets you 49. But 49 can’t give you a cross total because if you cut it in half you get a 0.5. You need to add an odd number twice so it cuts in half perfectly.

After defending his rules for finding possible and impossible cross totals, in general,

James stated he was finished working on Cross Totals. During his video-reflection interview, James shared his experience regarding these final moments:

I: Can you describe your experiences here as you were trying your own examples and writing your rules? James: I was really excited. It felt good to figure things out. I was definitely on a roll…It just started to happen. Like I was trying and trying and then finally it made sense…I didn’t even think about stopping. Earlier I did, but not here. It made sense now. It would be dumb to stop here because I already did all this work and it was making sense. There’s no reason to stop.

In sum, James’s additional attempt working on Cross Totals demonstrated perseverance in all components of the Additional Attempt Phase of the Three-Phase

Perseverance Framework. Leveraging the conceptualizing scaffold work he had done prior to starting the task, James re-initiated his effort toward both objectives after upon impasse by amending his original plan to now examine the role of the middle number in possible and impossible cross totals. He worked to re-sustain his effort toward both objectives by employing multiple new heuristics, including changing his point of view by first studying the given example, and logically reasoning about number placement and the role of even and odd integers as the middle number. Lastly, he perceived additional

109 mathematical progress as a result of his additional attempt, as evidenced by his explanation of why both of his rules worked.

James’s experience with Triangular Frameworks. Beginning his work with

Triangular Frameworks, James affirmed both components of the Three-Phase

Perseverance Framework Entrance Phase when he said, “I am going to try to find a rule for the possible frameworks and I need to do it for evens and odds…I’ve never done anything like this before. I’m definitely not sure how to do it.” This was evidence that

Triangular Frameworks warranted perseverance for James. Because this task was non- scaffolded and not did explicitly prompt James to initially conceptualize the situation,

James moved directly into the Initial Attempt Phase. It is important to note that James did not decide to record any of his initial ideas about the mathematical situation, as he had done previously with scaffolded tasks.

After several moments of quietly re-reading the task, James began his first attempt at when he shared during his think-aloud, “I guess I’ll just try to see if I can find some other ones that work out.” After noting that the given example showed that “when the longest side is 7, he can make six”, James decided to focus on if the longest side of the triangle was 6 meters long. This is evidence that James is simultaneously attending to both objectives of the task: triangular frameworks with the longest side c as an odd numbered length, and triangular frameworks with the longest side c as an even numbered length. He began to guess and check, the same heuristic first employed during his work with Cross Totals, to help him determine possible frameworks he could make with 6

110 meters as the longest side of the triangle. He initially guessed during his think-aloud that

“he could do a 6-5-5 and a 6-4-4 at first, because 5+5=10 and 4+4=8”, implying that two possible frameworks could have 6 meters as the longest side and 5 or 4 meters as the shorter sides, respectively. He noticed, though, that the given example did not consider isosceles triangle frameworks, and he realized that the parameters of the task specifically stated that 푐 < 푏 < 푎 when he thought-aloud, “Oh, I see, it’s all gotta be bigger.” From a perseverance analysis perspective, James’s mistake of first considering “6-5-5” and “6-

4-4” frameworks was not evidence of him reaching a significant setback because he did not report that he was at an impasse, nor did he report that he did not know how to proceed.

After James resolved the situation, he continued moving forward in his first attempt at solving Triangular Frameworks. Still using his guess and check strategy, James wrote down some potential side lengths for a triangle with a longest side of 6 meters and checked to see if they would work (see Figure 8). He said during this think-aloud, “So he could do a 6-5-4, a 6-5-3, and a 6-5-2. Oh and he could do a 6-4-3, too. And I think that’s it. So he could do four when the longest side is 6.” Coupling his finding that when the longest side was 6 meters, an even number, four frameworks were possible with the given example that when the longest side was 7 meters, an odd number, six frameworks were possible, James concluded, “So it looks like it’s one less [when c is odd] or two less

[when c is even], so the rules might be that.” Before James moved on to start writing his rules explicitly, he looked back at the picture on the task and said, “But…well…let’s

111 see…let me look at something.” Clearly troubled by something, James went on to write

“3” and “2” beneath the “5m-4m” text printed in the given example (see Figure 8). When asked what he was doing or thinking here, James said “I’m just seeing something…yeah so…ok…well then, ok so these are my rules.” He then began writing his rules about the situation, rules that were not mathematically correct, that if c is odd he can make 푐 − 1 frameworks, and if c is even he can make 푐 − 2 frameworks (see Figure 9). Finally, he said “Ok so I’m done” and stopped working on the task.

Figure 8. James’s guessing-and-checking during initial attempt on Triangular Frameworks

Figure 9. James’s rules for Triangular Frameworks

While reviewing the video of the aforementioned moments during his first attempt at solving Triangular Frameworks, James revealed that he was, in fact, at an impasse during the latter stages of his first attempt:

I: What specifically were you doing here? When you wrote this “3” and “2”? [see Figure 8]

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James: I was looking at if 5 was the longest side, just to see what would happen…It was 5-4-3 and 5-4-2, not good, it didn’t work! Like my odd rule. There should have been four of them but there were only two. Like my rule for odds was the number of solutions should be one less, but this one was three less…So I just panicked and pretended like it worked. I: Why did you decide to stop working here? James: Right here I was so lost. I didn’t know what to do. You know how like you go down a path but then at the end of it you find out it’s wrong? Then what? I just didn’t know what to do. I didn’t even know where to start to fix it. And I really wanted to stop. So I was just done.

Admitting that he “was so lost” and “didn’t know what to do” when he realized his rule was incorrect was evidence that James did meet a substantial obstacle while working on

Triangular Frameworks, even if he decided not to reveal this struggle during his think- aloud. Therefore, James had the opportunity to amend his original plan and make an additional attempt to solve the problem, but decided to ultimately “[pretend] like it worked” and stop working instead.

Overall, James’s first attempt at solving Triangular Frameworks exhibited perseverance in all components of the Three-Phase Perseverance Framework’s Initial

Attempt Phase, but not the Additional Attempt Phase. He initiated his effort towards finding rules to generalize the possible triangular frameworks when the longest side is of even or odd length, primarily by making explicit his intentions to search for and study examples that satisfy both conditions. He sustained his effort toward both objectives by studying the given example and guessing and checking for the frameworks possible when the longest side was 6 meters. As a result of this effort, James perceived he was making

113 mathematical progress by better understanding why a framework with a longest side of 7 meters would have more possibilities than one with a longest side of 6 meters. Although he had an opportunity to do so, James did not pursue an additional attempt at solving the problem. Therefore, James’s engagement with Triangular Frameworks did not demonstrate any evidence of perseverance for the components in the Additional Attempt

Phase of the Three-Phase Perseverance Framework.

James’s overall experience. James’s experiences with Cross Totals and

Triangular Frameworks, from my perspective and from his perspective, was illustrative of the ways in which participants persevered while working on scaffolded tasks compared to on non-scaffolded tasks. Similarly to his engagement on Cross Totals, James leveraged his initial conceptualization work while working on Sidewalk Stones and Skeleton

Tower, the two other scaffolded tasks, to help make a quality additional attempt at solving the tasks despite reaching an impasse and affirming frustration. On the other hand, during his work on Table Tiling, the other non-scaffolded task, James did not make an additional attempt at solving. Similar to his work on Triangular Frameworks, James cited overwhelming frustration and feeling disorganized after encountering a perceived impasse during his work on Table Tiling as the primary reason he did not continue to make an additional effort at solving. During his exit interview, James shared his perspective on how initially attending to conceptualizing the mathematical situation had a positive effect on his engagement:

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I: What was your overall experience like, working on these five problems? Anything to share? James: They were all problems I didn’t know how to do. And when that happens, you have to have a plan. Not like a plan of how to solve the problem, but like a plan on what you can try…Like the ones that made me write down what I thought first. That was really good. I used that a lot because sometimes you forget where you’re going in a problem, it’s like chaos – like [on Triangular Frameworks]. But on some other ones, like [on Cross Totals] I got stuck but had a way out of it. It seems easier with that because you don’t have to think of a way out when you’re mad or stuck or something. I learned that about myself I guess. For me, it came down to that panic feeling, like when you made a mistake or something and what you can do next.

James referred to the “chaos” of exploring a mathematical situation without knowing the exact solution pathway. For James, and for almost all participants in this study, responding first to the conceptualization scaffold served as a “life-preserver” of sorts later, when participants were at their most frustrated and most tempted to give up. In these moments, the conceptual thinking recorded after engaging with the scaffold prompt acted as an organized toolbox from which to draw a fresh mathematical idea, or a new connection between ideas, to use to help re-engage with the task upon impasse and to continue to productively struggle to make sense of the mathematical situation. Without doing and/or recording such conceptual thinking on non-scaffolded tasks, participants felt lost and panicked after a discouragement and more often gave up without making an additional attempt at solving.

Supporting Points of View: Tom and Marcia. James’s perseverance with scaffolded tasks and his point of view on the significance of the scaffolds was not unique.

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Many participants leveraged their initial conceptualization work in Cross Totals,

Sidewalk Stones, and Skeleton Tower, which were the scaffolded problems, to help stay productively engaged after reaching an impasse. At similar moments of frustration during work on Table Tiling and Triangular Frameworks, which were the non-scaffolded tasks, the same participants did not stay productively engaged. Participants Tom and Marcia were indicative of this general finding in their work across all tasks, and shared their overall points of view on the role of the scaffold in their respective exit interviews. For instance, Tom explained:

When you work real hard and then you haven’t gotten anywhere, all you think about is stopping. And you have to get yourself up again, keep going and not stop. You can just tell I did better with those ones with the brainstorming. Like I still got stuff wrong with those, but I didn’t think about stopping as much. The ones where I didn’t do that…I couldn’t stop thinking about stopping.

Also reflecting on her overall experiences during the exit interview, Marcia shared, “I was actually impressed that I used those little notes for myself! They help you sometimes, you know? They give you something to do when you have nothing. I should start doing that all the time!”

Both Tom and Marcia echoed James’s perspective on the role of the scaffolds, specifically noting how revisiting their initial conceptualization of the mathematical situation helped them continue to make progress when they were stuck, or “when you have nothing.” Tom also introduced the idea of momentum during moments when perseverance is required. For him, and many other participants, the feeling of wanting to

116 stop working in these uncertain situations was always prevalent. Yet, with the option of revisiting scaffold work during moments when frustration is at a maximum, it was easier for participants to quell the urge to quit. Instead, they more often chose to explore a new mathematical idea to continue their productive struggle.

Result 2: Perseverance Improved in Quality over Time

Participants’ perseverance on the three scaffolded tasks improved in quality over time as evidenced by increasing mean Three-Phase Perseverance Framework scores (see

Figure 10). These three tasks were administered in a random order for each participant, so

Cross Totals, Sidewalk Stones, and Skeleton Tower could have been encountered either first, second, third, or fourth overall for any participant (recall, Triangular Frameworks was always encountered fifth). Comparing perseverance on the first two scaffolded tasks encountered, all participants’ (10 out of 10) total perseverance score on their second scaffolded task was greater than or equal to their total perseverance score on their first scaffolded task. All but one participants’ (9 out of 10) total perseverance score on their third scaffolded task was greater than or equal to their total perseverance score on their second scaffolded task. Similarly, every participants’ (10 out of 10) total perseverance score on their third scaffolded task was greater than or equal to their total perseverance score on their first scaffolded task.

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Figure 10. Mean Three-Phase Perseverance scores over time

A simple linear regression was calculated to predict participants’ perseverance scores on their second scaffolded task based on their first scaffolded task (see Table 11).

A significant regression equation was found (F(1, 8) = 58.593, p < .001), with an R2 of

.880, indicating participants’ perseverance with their first scaffolded task explained 88% of the variance in their perseverance scores on their second scaffolded task. Also, participants’ perseverance on their first scaffolded task was a significant predictor of their perseverance on their second scaffolded task, with a one-point increase on their first scaffolded task predicting a .727-point increase on their second scaffolded task. A multiple linear regression was calculated to predict participants’ perseverance scores on their third scaffolded task based on their first and second scaffolded tasks (see Table 11).

A significant regression equation was found (F(2, 7) = 10.741, p = .007), with an

118 adjusted R2 of .684, indicating that a participants’ perseverance on their first and second scaffolded tasks, together, conservatively explained 68.4% of the variance in their perseverance scores on their third scaffolded task. Apart, participants’ perseverance on their first or second scaffolded task were not significant predictors of their perseverance on their third scaffolded task.

Table 11

Summary of Regression Analyses for Participants’ Three-Phase Perseverance Framework Scores

R2 Adjusted B SE(B) Sig. (p) Comparison R2 1st → 2nd .880 .865 .727 .095 < .001 Scaffolded .007 Tasks 1st/2nd → 3rd .754 .684 .385/.234 .348/.449 (.305/.618) Non- Scaffolded 1st → 2nd .553 .497 .680 .216 .014 Tasks

Participants’ perseverance on the two non-scaffolded tasks also improved in quality over time, in general, demonstrated by increasing mean Three-Phase Perseverance

Framework scores (see Figure 10). However, as explained in the methods, these two tasks were not encountered randomly. The first encountered non-scaffolded task was always

Table Tiling, and the second encountered non-scaffolded task was always Triangular

Frameworks. Table Tiling was administered randomly to participants, encountered first, second, third, or fourth overall. Triangular Frameworks, a post hoc task addition to increase the pool of non-scaffolded tasks, was always encountered fifth. Comparing

119 perseverance on the non-scaffolded tasks, most participants’ (8 out of 10) total perseverance score on their second non-scaffolded was greater than or equal to their total perseverance score on their first non-scaffolded task, but less of an improvement compared to perseverance over time with scaffolded tasks.

A simple linear regression was also calculated to predict participants’ perseverance scores on their second non-scaffolded task based on their first non- scaffolded task (see Table 11). A significant regression equation was found (F(1, 8) =

9.879, p = .014), with an R2 of .553, indicating participants’ perseverance with their first non-scaffolded task explained 55.3% of the variance in their perseverance scores on their second non-scaffolded task. Also, participants’ perseverance on their first non-scaffolded task was a significant predictor of their perseverance on their second non-scaffolded task, with a one-point increase on their first non-scaffolded task predicting a .680-point increase on their second non-scaffolded task.

In addition to the regression outcomes, the descriptive statistics of perseverance over time on scaffolded tasks and non-scaffolded tasks (as illustrated in Figure 10) suggests participants may have been improving in their perseverance more greatly when working on scaffolded tasks compared to working on non-scaffolded tasks. On average, the slope of participants’ perseverance scores on scaffolded tasks was 1.2 points per task.

This was three times greater than the slope of participants’ perseverance scores on non- scaffolded tasks, which was 0.4 points per task.

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During their think-aloud, video-reflection, and exit interviews, participants stated they noticed improvements, over time, in their engagement with the challenging mathematical tasks. Most participants (8 out of 10) mentioned they thought they were getting better, somehow, as they had more practice with these types of problems. Several participants (6 out of 10) mentioned their improved work on tasks specifically prompting them to conceptualize the situation prior to starting, i.e., the scaffolded tasks. Arguably the most noticeable improvement, from the participants’ point of view, was affective in nature. Many participants (7 out of 10) explained in their exit interview that they felt like they were getting better at handling the stress of the situation as they reached impasses within their work on mathematical tasks they did not know how to solve. Some participants (4 out of 10) reported cognitive gains, believing the way they were thinking about the mathematics was changing for the better, over time, and that their problem- solving skills were improving.

Next I will share representative participant perspectives describing how perseverance was improving over time. I will first unpack Sandra’s experiences during her work across the scaffolded tasks and detail specific areas of her perseverance improvement. For Sandra, as well as most other participants, this improvement included generally increasing Three-Phase Perseverance Framework scores and participant perceptions of cognitive and affective progressions over time. Then, I will share the overall exit interview perspectives of Laurie and Dennis, two participants who echoed several of the themes of perseverance improvement apparent in the sample.

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Sandra’s Perspective: “Getting Better at Sticking with It.” Unpacking

Sandra’s experiences across her work on the three scaffolded tasks represents how perseverance improved over time, especially by supporting participants in making a more quality additional attempt at solving within their work on scaffolded tasks. For Sandra’s work on scaffolded tasks, she encountered Cross Totals first (first overall), then Sidewalk

Stones (second overall), and then Skeleton Tower (fourth overall). Sandra earned a

Three-Phase Perseverance Framework score of 6 for her perseverance on Cross Totals, a score of 9 on Sidewalk Stones, and a score of 12 on Skeleton Tower (see Table 12).

Sandra, like many other participants, made no additional attempt at solving while working on her first task, but progressively improved her perseverance after a setback in the Additional Attempt Phase as she had more experiences with tasks necessitating productive struggle. When reflecting on her work across these scaffolded tasks, Sandra revealed that she was thinking about mathematical generalization in a different, better way the more she practiced with these problems. She also shared regulatory gains, such as improvements in stress management over time. For comparison, Sandra earned Three-

Phase Perseverance Framework scores of 6 for her work on both non-scaffolded tasks:

Table Tiling (third overall) and Triangular Frameworks (fifth overall). In addition,

Sandra did not mention any perceived improvements as a result of her work with the two non-scaffolded tasks.

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Table 12

Sandra’s Three-Phase Perseverance Framework scores for Cross Totals, Sidewalk Stones, and Skeleton Tower

Cross Totals (S, Sidewalk Stones Skeleton Tower INITIAL ATTEMPT PHASE 1st) (S, 2nd) (S, 4th) Evidence of Perseverance Obj. 1 Obj. 2 Obj. 1 Obj. 2 Obj. 1 Obj. 2 Initial Engaged with the Effort task       Used problem- Sustained solving heuristics to Effort       explore the task Outcome Made mathematical of Effort progress       ADDITIONAL ATTEMPT Cross Totals (S, Sidewalk Stones Skeleton Tower PHASE 1st) (S, 2nd) (S, 4th) Evidence of Perseverance Obj. 1 Obj. 2 Obj. 1 Obj. 2 Obj. 1 Obj. 2 Initial Engaged with the Effort task       Used problem- Sustained solving heuristics to Effort       explore the task Outcome Made mathematical of Effort progress       Total Perseverance Score 6 9 12 Notes: S = Scaffolded Task; Obj. = Objective; 1st = First task overall; 2nd = Second task overall; 4th = Fourth task overall

Sandra’s experience with Cross Totals. After affirming clarity around the

objectives of the task, yet uncertainty about how to achieve such objectives, Sandra

began her work on Cross Totals by responding to the scaffold prompt (see Figure 11).

She recorded ideas about her initial conceptualization about the mathematical situation,

including that she “probably only need[s to use] addition”, that there is probably “a

123 pattern about how to put the numbers in, like an even distribution thing”, and that

“put[ting] numbers into pairs [that sum to] 10” might be relevant.

Figure 11. Sandra’s initial conceptualization for Cross Totals

Next, Sandra started her first attempt at solving Cross Totals. In her think-aloud, she shared how she could initiate her effort by reasoning about the placement of greater and lesser integers within the lines of the cross to help determine possible cross totals.

Using logical reasoning as her primary heuristic, Sandra sustained her effort by exploring with this idea and said, “the high and low numbers should be distributed evenly because if one side was all the low numbers and the other side was all the high numbers then obviously they wouldn’t add up equally.” Sandra then began reasoning about the parameters of the different sides within the cross, determining that “one side of the cross can’t be less than 15…and the opposite for 35.” She then began to construct her own example of a possible cross total using 5 in the middle and incorporating her initial ideas of “pairs [that sum to] 10” as a method of “even distribution”. During her think-aloud she said, “if you put numbers in pairs of 10 and then added them to get the cross totals so they would be equal. And you could think of these as impossible rules too, if it’s not possible then it’s impossible.” At this point it became apparent that Sandra was thinking about both task objectives, rules for possible and impossible cross totals, during her first

124 attempt. Soon thereafter, Sandra constructed an example of a possible cross total of 25

(see Figure 12).

Figure 12. Sandra’s own example during initial attempt on Cross Totals

However, after finding an example of a possible cross total, Sandra affirmed she reached an impasse. During her think-aloud she said, “I know this one works, but it’s just one example. These things aren’t for all cross totals. It wants me to get it in general. I don’t know.” When reflecting on this moment during her video-reflection interview,

Sandra shared:

I mean, I felt good at first that I had some ideas down, and that they helped me figure things out. Like, stuff was working and made sense for the example I had. I was better understanding how the crosses worked or didn’t work. But in this moment I was just overwhelmed I guess. I felt any other rules, like general rules, would be a lot more complicated to get and I kind of just ran out of ideas at this point.

Sandra’s comments illustrated that she did perceive she was making mathematical progress toward better understanding possible and impossible cross totals, but was overwhelmed by the idea of generalizing those mathematical ideas. At this point during her think-aloud, Sandra decided to write down some rules that applied to her constructed example (see Figure 13), but then said, “I know these rules won’t help me find all

125 possible and impossible cross totals though, but I guess I’m done.” Sandra decided not to continue her effort and make an additional attempt at solving.

Figure 13. Sandra’s rules for Cross Totals

Overall, with Cross Totals, Sandra demonstrated evidence of perseverance in all components of the Three-Phase Perseverance Framework’s Initial Attempt Phase. During her first attempt, she initiated her effort toward both objectives by stating her plan to reason about the magnitude of integers, and she sustained her effort toward both objectives by exploring the logical ramifications of distributing different integers within the cross. As evident by her rules, Sandra perceived she was making mathematical progress in better understanding the situation by which cross totals may be possible or impossible. Although she recognized she had more work to do to better generalize the situation, Sandra decided not to make an additional attempt at exploring the problem.

Next, I will review Sandra’s experience with her second scaffolded task (second task overall), Sidewalk Stones, and explain the ways in which her perseverance with Sidewalk

Stones showed improvement compared to her perseverance with Cross Totals.

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Sandra’s experience with Sidewalk Stones. Sandra began her experience with

Sidewalk Stones by insisting she understood the task objectives, yet she was “not sure how to even begin.” She then moved on to her scaffold work, writing down her ideas about the mathematical relationships at play and how to explore them (see Figure 14).

Sandra believed she could make progress on both objectives of Sidewalk Stones, generalizing rules for the number of gray and white stones needed to build Pattern #n, by making “a chart listing the number of grays and whites for different patterns…and one for the total number of blocks”, as well as search for patterns amongst the data and “write an equation from that.” Additionally, Sandra revealed during her think-aloud that she was planning ahead when she said, “And like if I was still stuck, or couldn’t find a rule in general, I could make up examples and check them.”

Figure 14. Sandra’s initial conceptualization for Sidewalk Stones

Sandra then moved into her Initial Attempt Phase and started engaging with the problem. She stated in her think-aloud that she would initiate her effort by making “that table about gray and white and the total number [of stones].” Moving forward with the problem-solving heuristic of making a table (see Figure 15), Sandra sustained her effort by studying data and searching for patterns amongst the different types of stones.

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Eventually she noticed there was “a similar ratio between gray and white…oh and gray always has one more that the white.” By searching for patterns between the gray and white stones within the sidewalk patterns, it was clear that Sandra was attending to both objectives of the task – generalizing rules for both types of stones.

Figure 15. Sandra’s table of values during initial attempt on Sidewalk Stones

At this point during her think-aloud, Sandra believed she was making mathematical progress toward one objective of Sidewalk Stones because she was

“starting to see how things are working, with the gray blocks at least. With the white ones

I don’t really know what’s going on.” Despite her perceived progress with the patterns of gray stones, Sandra began to get frustrated about the objectives of generalization. After studying her table for several moments, she said, “But like how do I know about Pattern

#n? This general stuff…this is hard.” Sandra had reached a perceived impasse as a result of her first attempt at solving Sidewalk Stones. During her video-reflection interview, she clarified her point of view at this moment of frustration:

I: What was happening in this moment? Sandra: Up to here I was feeling good because I found two things that explained the pattern, and I was starting to see the gray patterns, but I still didn’t have a clue about Pattern #n. I: So what were you thinking about doing?

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Sandra: Honestly, I was thinking about quitting. Like, I got farther than I thought, but I didn’t think I was gonna get it, you know? The general stuff again, it’s hard, like finding #n. I didn’t know what to do. I was stumped.

Sandra’s first attempt at solving Sidewalk Stones demonstrated perseverance in 5 out of 6 Initial Attempt Phase components of the Three-Phase Perseverance Framework.

She initiated and sustained her effort toward both objectives by making a table of values and searching for patterns within the data. Further, by her own admission, Sandra perceived she was making mathematical progress understanding how the gray stones were changing from Pattern to Pattern, but she did not think she better understood how the white stones were changing as a result of her first effort.

After Sandra shared that she “was stumped” during her think-aloud and revealed in her video-reflection interview that she was “thinking about quitting” at this moment, she decided to keep working by amending her current plan and changing strategies. She said during her think-aloud, “Well, I guess I can make another example like I said I would.” Sandra was referring to her scaffold work, in which she stated that if she “was still stuck” with the objective of generalization, she could “make up examples and check them.” When reflecting on this decision to continue with Sidewalk Stones during her video-reflection interview, Sandra explained:

I still wanted to quit, but like I already had that idea down about what to do in this situation. Like if I was stuck. So since I already thought about it ahead of time, it made sense to do it. I’m kinda surprised I kept going here, because I was annoyed about it. Like I didn’t know what I was doing

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really. But I decided to try that idea I had anyway. Maybe it would help me with the general part.

Thus, drawing from her earlier conceptualization work, prompted by the scaffold, Sandra decided to make an additional attempt at solving Sidewalk Stones.

Sandra re-initiated her effort by amending her first problem-solving plan to now focus on drawing a diagram of Pattern #4, still considering both objectives of the task.

During her think-aloud she said, “So Pattern #4 would have extra blocks here and here

(drawing on Pattern #3).” Sandra was drawing extra stones of both types along the edges of Pattern #3 to help her visualize the construction of Pattern #4 (see Figure 16). With this new Pattern in mind, Sandra continued, “So that would make 145 and 146, 289 total

[stones]…I’ll put that in the table…one more [stone] again, that’s what I thought.”

Sandra amended her original table to include the stones of Pattern #4 (see Figure 17), and studied it to search for patterns that may help her generalize the situation. After a few moments of study, Sandra said during her think-aloud, “I still don’t really know how to get a #n rule…yeah I think this will be it. I am going to write up what I found as my rules.” Sandra moved on to writing down her observations as a result of her efforts, stating that “there will always be 1 more gray block than white block” and that if one knew the total number of stones in a Pattern, they could use this fact to determine the number of gray and white stones needed (see Figure 18). After writing these rules, Sandra stopped working on Sidewalk Stones.

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Figure 16. Sandra’s new diagram during additional attempt on Sidewalk Stones

Figure 17. Sandra’s amended table of values during additional attempt on Sidewalk Stones

Figure 18. Sandra’s rules for Sidewalk Stones

During her video-reflection interview, Sandra shared her perspective on these final moments of her additional attempt at solving Sidewalk Stones:

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Sandra: I thought that drawing a new Pattern would help me with the general rule, but it didn’t. It was just the same as the rest of [the Patterns]. Like last time [on Cross Totals] I got really stuck with the general part, so this time I thought I was ready, but yeah it didn’t help much. I: What do you mean you thought you were ready? Sandra: Like I thought I had planned to do better this time. Like planning for what to do when I was stuck. I thought that would help me, but it didn’t really. Like I didn’t figure out anything new from drawing the new pattern and adding to the table. The stuff I had figured out the first time didn’t change after doing this new stuff. And the rules I figured out by this point, they were true, but they really weren’t answering the question.

Sandra explained that although she better prepared during her scaffold work for encountering a possible obstacle while problem-solving, her amended plan ultimately did not help her progress toward finding rules that explained how to build Pattern #n. Instead, her rules explained how to find the number of gray and white stones if the total number of stones were known. In Sandra’s opinion, she did not make any new mathematical progress as a result of her additional effort on Sidewalk Stones.

In total, Sandra’s additional attempt working on Sidewalk Stones demonstrated perseverance in 4 out of 6 Additional Attempt Phase components of the Three-Phase

Perseverance Framework: the initial and sustained effort components toward both objectives, but not in the mathematical progress component. She re-initiated her effort by choosing a different problem-solving heuristic, drawing a diagram, and re-sustained her effort by amending her table of values and searching for a pattern amongst all available data. By her own admission, however, Sandra believed she did not better understand the mathematical situation as a result of her additional efforts. Compared to her engagement

132 with Cross Totals, however, Sandra’s perseverance with Sidewalk Stones was much improved primarily because she included in her initial conceptualization work a plan for if she got “stuck.” Sandra’s backup plan helped her make an additional attempt at solving the problem upon a perceived impasse, an additional attempt she did not make one week earlier with Cross Totals. In the next section, I will review Sandra’s experience with her third scaffolded task (fourth task overall), Skeleton Tower, and detail the ways in which her perseverance with Skeleton Tower showed improvement compared to her perseverance with Sidewalk Stones.

Sandra’s experience with Skeleton Tower. With Skeleton Tower, the final of

Sandra’s scaffolded tasks, she began her think-aloud by affirming her understanding of both objectives and admitting “I have never seen this [problem before], I don’t know what I’ll do with it.” During her work with the scaffold prompt, Sandra recorded her conceptualization of the mathematical situation (see Figure 19). She thought that “adding up all the squares” and “knowing the area of a square, s2” would help her make progress with the generalization objectives. Further, she planned to “use the diagram and add [the squares] up in a table and maybe get an equation or something.” During her video- reflection interview, Sandra explained how she was preparing for generalization objectives when responding to the conceptualization prompt:

Well here I was just doing the stuff for [the scaffold prompt]. And I thought that maybe the s2 would help me get an equation…like a general answer with an equation. Something I could just plug height into. The general stuff has been really hard in all these problems so I thought the variables might help. I wasn’t sure, though.

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Figure 19. Sandra’s initial conceptualization for Skeleton Tower

Next, Sandra began her first attempt at solving Skeleton Tower. During her think- aloud, she shared that she planned to initiate her first effort by “look[ing] for a pattern with all the cubes, and with the side cubes.” Sandra sustained her first effort by counting the total number of cubes in the tower, “66”, as well as the number of cubes in each “leg” of the tower, “15” (see Figure 20).

Figure 20. Sandra’s counting of cubes during initial attempt toward first objective on

Skeleton Tower

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Paying specific attention to the generalization objectives, Sandra then began reasoning about the quantities in the tower to discern algebraic expressions, namely that

“the length of the leg on the ground is 5 and the height is 6, so it’s like n-1 and n.” Sandra began visualizing piecing two legs of the tower together, stating in her think-aloud, “it’s like two squares put together if you flip [two of four the legs].” Thinking about how to algebraically represent that action, Sandra concluded, “I think (n-1)2 x 2 could be my rule for the total number of blocks because that’s like two squares made up of the legs.”

Shortly thereafter, though, Sandra mentally tested her equation for a skeleton tower of height n=6, and concluded “That would be 50, that doesn’t work because it should be

66…Hmm, I guess I’m stuck.” When reflecting on these moments during her video- reflection interview, Sandra shared her perception of mathematical progress, as well as her disappointment in her error:

Sandra: Well here I ended up thinking the equation (n-1)2 x 2 would work, but it wasn’t working. I was thinking if I were to flip that leg and make it into a square that it would be like the leg squared. It was annoying here. I was annoyed that it wasn’t working. I: Why? Sandra: Because I really thought I was doing it right. Like I had variables and an equation and I was putting pieces [of the tower] together. It was making sense. But it wasn’t exactly right. It was annoying. I didn’t know what to do after that because I had already tried my ideas and I thought I was doing it right.

Thus, Sandra had reached an impasse as a result of her first effort and had an opportunity to continue her perseverance by amending her plan and making an additional attempt at solving.

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At this point, Sandra had been working toward one objective of Skeleton Tower, to figure out a rule that describes the number of cubes needed to build a tower n cubes high, and had demonstrated perseverance in 3 out of 3 Initial Attempt Phase components of the Three-Phase Perseverance Framework for Objective 1. She still had the opportunity to work later toward the second objective of the task, to rewrite her rule in a different way. She initiated and sustained her effort by looking for patterns within the cubes of the tower and modeling the construction of the tower with algebraic expressions.

Sandra noticed her own mathematical progress, specifically that algebraically representing the different pieces of the tower was crucial for her to generalize the situation, even if she had made a mistake in doing so.

After sharing during her think-aloud that she was “stuck”, Sandra said, “I’m not really sure how to keep going here.” She spent the next several moments staring at her paper and then ultimately pointed at one of the legs of the Skeleton Tower and said to herself, “How could this work…like how could this squared work?” During her video- reflection interview, Sandra clarified what she was thinking in those moments:

Sandra: I wanted to know how the legs could piece together and how I could use a formula to show it. I was annoyed that it wasn’t working, but I thought there might be another way to think about it if I just tried to. I: Why? Sandra: Well I had that plan to use the area of a square (points to scaffold work), and I saw it here and really thought it was a good idea, so I decided to try it again. To see if I could think of another way to use it.

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Sandra’s earlier conceptualization work of Skeleton Tower helped her rethink about deconstructing the tower into squares and algebraically modeling the situation by considering the expression s2. During this time when she was most “annoyed”, it was the revisiting of her scaffold work that encouraged her to make an additional attempt at solving the problem.

As she continued to work, Sandra began to better understand how to deconstruct the tower into rectangles, not squares. During her think-aloud she said, “Oh! If you piece

[the legs] together it’s not a square! If I draw it, it takes less squares to fill it up.” Next,

Sandra began drawing her own example about how to use blocks from the tower to construct a rectangle (see Figure 21). As she drew, she explained:

See if you were to take a leg and then you would only need, you would need fewer squares [than another leg] to finish it off. That piece to finish the square is smaller than the other legs. So really what’s happening is the length of this leg on the ground is n-1, and if you put another leg into it, it will give you the height of the tower. Like, it’s a rectangle, not a square. It’s six by five.

Sandra then realized that she could build two of these 6 x 5 rectangles from the four legs of the tower, and still have the center column of blocks remaining. She summarized her rule during her think-aloud when she said, “so n-1 times n times 2 gives you the

[rectangles] for all the legs, and then add the n for the middle column.” As she spoke during her think-aloud, she wrote down her rule (see Figure 22) and confirmed the output of 66 blocks when n = 6. When reflecting on this moment during her video-reflection interview, Sandra shared, “This was big for me. I was so upset that I didn’t have it earlier,

137 but then it finally worked and made sense…having an equation in mind from the start was a big help.”

Figure 21. Sandra’s diagram during additional attempt toward first objective on Skeleton Tower

Figure 22. Sandra’s first rule for Skeleton Tower

Regarding her additional attempt toward the first objective of Skeleton Tower, to figure out a rule that describes the number of cubes needed to build a tower n cubes high,

Sandra demonstrated perseverance in 3 out of 3 Additional Attempt Phase components of the Three-Phase Perseverance Framework for Objective 1. Revisiting her earlier conceptualization recorded during her scaffold work, Sandra re-initiated her effort toward generalizing the situation by trying to change her original perspective on deconstructing the tower to form squares. She worked to re-sustain her effort by drawing a diagram of a rectangle built from two legs of the tower, and ultimately used an equation to model the scenario. Since she affirmed her rule “made sense” and because she explained the ways in which her model represented the construction of a skeleton tower, it was evident that

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Sandra perceived she was making mathematical progress. Because she had only worked toward the first objective of the task thus far, Sandra still had the opportunity to make a first and, if necessary, an additional effort toward the second objective of Skeleton

Tower, to rewrite the rule in a different way, and demonstrate perseverance in the remaining components of the Three-Phase Perseverance Framework.

During her think-aloud, Sandra began working toward the second objective of

Skeleton Tower when she planned to initiate her effort by making a table of values. She said, “Now I need another rule for the same thing…I could try the table idea I wrote back here (points to scaffold work), I haven’t done that yet.” Next, Sandra sustained her effort by making a table for how many cubes were required to construct a tower of various heights. To help visualize towers of different heights, Sandra used her finger to cover up the bottom layer(s) of the given example. After she had filled in her table for three different tower heights (see Figure 23), she began looking for patterns amongst the data.

While working, she said, “Ok so six is 66, five is 45, and four is 28…it might have a linear relationship, like as it goes down by one maybe it goes down by the same cubes.”

Sandra tested her hypothesis and found that “from six to five it loses 11, but from five to four it loses 17…no that doesn’t work. Hmm, I’m not sure what to do here.”

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Figure 23. Sandra’s table during additional attempt toward second objective on Skeleton Tower

When watching these moments during her video-reflection interview, Sandra explained her frustration:

This part was really hard because I had made it so far on a really hard problem and it was like I was back at the beginning. I was stuck here. I had figured out it wasn’t linear, but I didn’t know what else to do. I really wanted to stop at this point.

Therefore, Sandra had reached a perceived impasse during her first attempt toward the second objective of Skeleton Tower.

As Sandra worked to try to satisfy the remaining task objective and write her generalization rule in a different way, she demonstrated perseverance in 3 out of 3 Initial

Attempt Phase components of the Three-Phase Perseverance Framework for Objective 2.

She initiated and sustained her effort by making a table of values for towers of different heights and looking for patterns to describe the relationship between tower height and the total number of cubes. Sandra also had demonstrated awareness of mathematical progress when she determined the relationship was not linear.

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Although she had reached an obstacle as a result of her first effort and “wanted to stop at this point”, Sandra decided to make an additional attempt at solving the problem by reconsidering the role of area. She said during her think-aloud, “Well maybe it could be another area formula.” During her video-reflection interview, Sandra reflected on this moment when she decided to amend her original plan to continue to persevere and explained, “I was just about to give up, but then I thought about my s2 idea again. This idea here (points to scaffold work). I thought maybe I could piece the tower together in a different way.” Much like she did while working toward Objective 1, Sandra leveraged her earlier conceptualization work to continue to productively struggle with Skeleton

Tower.

Next, Sandra began thinking about different ways she could model the construction of the tower. She thought-aloud, “Maybe I could take the middle piece as well as one of the legs to make a bigger square and then a smaller square…like n2 + (n-

1)2…wait no, they wouldn’t both be squares!” Sandra immediately realized her mistake, a similar mistake to the mistake she made earlier regarding visualizing pieces of the tower as squares, and continued with her plan to deconstruct the tower into more manageable shapes. Then, she progressed to generalize the situation in a different way. She said during her think-aloud:

It can be the middle piece attached to a leg and another leg, and then also two legs together as a rectangle. That would be a 6 by 6 and a 5 by 6 for this one…so n2 + n(n-1)…yeah that will work I think. You can see how you could put the tower together that way.

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Sandra wrote down her new rule and checked it with the values in her table (see Figure

24).

Figure 24. Sandra’s second rule for Skeleton Tower

During her video-reflection interview, Sandra clarified her satisfaction with her effort on

Skeleton Tower, and the prominent role of recording her initial conceptualization of the mathematical situation to help her:

Sandra: I was really happy to figure this out. I just kept trying things I thought might help and I got it eventually. I just kept going and I got it. I: So what were you thinking or feeling about your rule? Sandra: I mean I can see how I could put the tower together this way, you know? And my equation matches that, and the numbers worked out, too…and this one feels good because it was like from start to finish, I had to use all sorts of beginning ideas to get it. Like all my beginning plans really helped me. Those were good. I’m happy about that.

Sandra’s additional attempt toward the second objective of Skeleton Tower, to write her earlier rule in a different way, demonstrated perseverance in 3 out of 3

Additional Attempt Phase components of the Three-Phase Perseverance Framework for

Objective 2. She re-initiated her effort by changing her point of view to visualize the construction of the tower and re-sustained her effort by exploring the ways in which the tower could be broken down into more manageable shapes and modeled using

142 expressions of area. Further, Sandra recognized her own mathematical progress because she was able to visualize, describe, symbolically model, and numerically test her second rule for Skeleton Tower. Compared to her work with Sidewalk Stones, her previous encounter with a scaffolded task, Sandra’s perseverance with Skeleton Tower was much improved mainly because she planned to work with variables from the outset, during her scaffold work, to generalize the situation. Sandra’s more-refined initial planning helped her make an additional attempt when she was at her most frustrated and go on to make progress and solve the problem.

Sandra’s overall experience. Sandra’s experiences with all tasks, but especially with her scaffolded tasks, helped illuminate the ways in which participants were improving in their perseverance. Such improvements were apparent by Three-Phase

Perseverance Framework scores increasing over time, as well as by participants noticing themselves improving in particular ways. Sandra’s perseverance gains were most noticeable in the improved quality of her additional attempt at solving a task after reaching an impasse. Sandra’s scaffold work – recording her conceptualization of the mathematical situation at hand – seemed to play a role in why her perseverance changed because the types of ideas she recorded changed over time as well.

In the beginning, on Cross Totals, Sandra did not explicitly attend to or record specific strategies to support generalizing a mathematical situation. But as she engaged with more tasks, she started specifically planning in her scaffold work for how she could make better progress on generalizing the mathematics. On Sidewalk Stones, Sandra

143 included in her initial conceptualization a way to explore the situation more generally if she “got stuck”, specifically by drawing a new Pattern of stones and searching for relationships across all the Patterns. On Skeleton Tower, Sandra responded to the conceptualization prompt even more specifically by incorporating variable expressions into her problem-solving plan. Sandra had seemingly learned from her prior struggles and was thinking about how modeling with variables from the outset could help her better generalize the situation.

In addition to my own analyses, Sandra explained during her exit interview how she thought she was improving in the way she approached these scaffolded tasks:

I: What was it like working on these five problems? Anything to share? Sandra: Well I think I got better at writing out my ideas and steps I could take. Well if you look at most of them I did. Like the ones that asked for it, you can see there’s just so much more writing in the beginning. I: Getting better how? Sandra: When you keep doing that you get better at the brainstorming or planning stuff, at least for me. Like all the problems ask for a general rule, and I started to learn about how to do that, even though all the problems were different. You just get better at it I guess.

Sandra also mentioned how she was better handling the stress of the situations:

Sandra: I was getting better at sticking with it, too. At like figuring it out, or trying to, when you don’t know what to do. Like in the beginning it was overwhelming, but by the end it wasn’t as bad to stick with it, you know? Like making a plan and sticking with it, coming back to it, stuff like that. That got easier. Like the problems weren’t easy, but I could bounce back easier. I: What do you mean bounce back? Sandra: Like after getting stuck or making a mistake. You know, we don’t do this stuff. We’d have a formula already. So like you just kind of

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have to get into the problem and start to make some sense of it. And maybe even make some mistakes to figure it out. And that got easier for me I guess. Like not getting too annoyed or frustrated after mistakes.

Sandra’s overall perspective is indicative of how both cognitive and affective changes impacted participants’ perseverance improving over time for scaffolded tasks.

Sandra was “getting better at sticking with it” because she better prepared strategies for generalization objectives and better regulated her frustration at her most stressful moments during her problem-solving. While working on her non-scaffolded tasks, Sandra earned Three-Phase Perseverance Framework scores of 6 for both Table Tiling (third task overall) and Triangular Frameworks (fifth task overall). While working on these two tasks, Sandra did not demonstrate any evidence of specifically preparing for mathematical generalization, nor did she report in any of her interviews specific ways she was changing how she prepared. Also, Sandra did not mention if she felt better about handling the stress during work on Table Tiling and Triangular Frameworks, even though they were the third and fifth task, respectively, with which she worked. This suggests that exposure to such tasks warranting struggle did not alone improve Sandra’s perseverance in specific ways, but responding to the initial scaffold prompt, in which she attended explicitly to conceptualizing the situation, played an influential role in her perseverance improvement.

Supporting Points of View: Dennis and Laurie. The ways in which Sandra’s perseverance improved over time on Cross Totals, Sidewalk Stones, and Skeleton Tower

145 was representative of all participants’ engagement with scaffolded tasks in any order. In general, participants’ perseverance improved over time on scaffolded tasks, most notably in the ways in which they prepared to generalize the situation of each mathematical task, as well as how they managed their frustration when they reached an impasse. Participants

Dennis and Laurie demonstrated these findings in their experiences with the scaffolded tasks.

For example, Dennis’ perseverance on Skeleton Tower (second task overall),

Cross Totals (third task overall), and Sidewalk Stones (fourth task overall) improved over time, as evidenced by his Three-Phase Perseverance Framework scores of 8, 11, and 12, respectively. Like Sandra, Dennis learned from his early struggles with his first scaffolded task and began planning for ways to explore how to generalize a mathematical situation during his responses to the conceptualization prompt for his remaining scaffolded tasks. Additionally, Dennis affirmed how this type of planning made it easier for him to continue to persevere when he encountered impasses. Dennis explained his perspective during his exit interview:

Dennis: I feel like I did better over time, with these three [scaffolded tasks] anyway. I feel like I gradually got better and better and doing these different problems. I: Why? Dennis: I don’t know, just doing something with problems like this. Doing it over and over with different tasks, you find different ways to solve. Like these problems are all kinda similar to me because they all involve like visualizing and patterns and rules and stuff. And I think I’m learning more that can help me. Like I got stuck in all of these, but I think I got better at dealing with it and trying things to help figure it out.

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Laurie’s experience with her scaffolded tasks was indicative of these general improvement trends, as well. Her perseverance on Skeleton Tower (first task overall),

Sidewalk Stones (second task overall), and Cross Totals (fourth task overall) improved over time, demonstrated by her Three-Phase Perseverance Framework scores of 3, 5, and

7, respectively. Laurie specifically mentioned the role of planning while responding to the scaffold prompts as the primary reason for her improvement. She explained her point of view during her exit interview:

Laurie: Of the planning ones, the first one seemed the hardest and then it seemed to get easier each time. I mean looking at them now all these problems seem hard, like all similar kinda. But it definitely seemed to get easier. Or I got better. I: What do you mean? Laurie: Well I really like to plan. Like whenever I write an essay I always plan out what I’m going to do before I do it. But I don’t usually do that in math, like we just get an equation and use it for stuff. But for these problems, the planning stuff helps me…it definitely seemed to get easier, or I got better at handling the stress when I didn’t know what to do.

Dennis and Laurie reiterate Sandra’s point of view on how engaging with the scaffolded tasks helped improve their perseverance in specific ways, over time. All participants reached an impasse when working on these tasks, yet, as they had more experience in situations requiring perseverance and requiring planning, they persevered more and noticed cognitive and affective improvements. Participants did not report noticing such improvements as a result of working on non-scaffolded tasks. Through this practice with scaffolded tasks, participants were repeatedly noticing past shortcomings in

147 their initial problem-solving plans to generalize a mathematical situation. As they encountered new scaffolded tasks with new opportunities to record their initial conceptualizations, participants reflected on their past struggles and learned from their mistakes, often planning more refined strategies to explore the mathematical patterns embedded in each problem. The ways in which they were thinking about generalizing the mathematics was changing, and by recording these changes when responding to the scaffolding prompt, participants were better able to manage the stress of the situation when they reached an obstacle and persevere to make additional mathematical progress on the task at hand.

Summary of Results

In the context of 10 ninth-grade students working on a series of challenging mathematical tasks, the results show that participants persevered more when working on scaffolded tasks than on non-scaffolded tasks. Arguably the most notable difference between perseverance on scaffolded and non-scaffolded tasks occurred when participants encountered an impasse. Engaging with the scaffold prompts encouraged participants to re-initiate and re-sustain their productive struggle when at their most frustrated by revisiting their initial conceptualization of the situation. This resulted in participants making a productive attempt at solving a task after impasse more than twice as often on scaffolded tasks than on non-scaffolded tasks. In addition, participants reported in-the- moment cognitive and emotional support by way of attending to their mathematical conceptualization as prompted by the scaffolds. Related, participants reported feeling

148 disorganized and overwhelmed without being prompted to engage in any conceptualization work in the non-scaffolded tasks.

Additionally, the results show the quality of participants’ perseverance improved over time, more so when working on scaffolded tasks than on non-scaffolded tasks.

Specifically, participants demonstrated more-refined planning during their initial conceptualization work over the course of working on their three scaffolded tasks.

Participants often leveraged that planning to make higher quality additional attempts at solving upon impasse, compared to their previous work on past problems a week(s) before. Also, participants reported affectual improvements as a result of their engagement with the scaffolded tasks, citing better stress management at moments of impasse due to their improved conceptualization work. No such improvements were reported as a result of working on non-scaffolded tasks.

Discussion

This study was designed to investigate ways of supporting students’ perseverance through scaffolds embedded in challenging mathematics tasks. More specifically, I aimed to address less-explored areas in the literature by explicitly examining ways to support student perseverance after moments of perceived impasse and by attending to how student perseverance can improve over time. Findings from the study suggest that prompting students to record their initial conceptualization of a task’s mathematical situation, prior to beginning work on the task, can support students’ perseverance by engaging them in a cyclical process of conceptual thinking while problem-solving. Also,

149 findings suggest that students showed signs of perseverance improvement, especially with scaffolded tasks, because they were engaging in cycles of deliberate practice with appropriate support.

Conceptual thinking scaffolds, like being prompted to record one’s initial conceptualization of a situation, are designed to help students engage with a challenging task by clarifying possible connections to prior knowledge and specific solution strategies

(Anghileri, 2006). In the context of problem-solving, engaging with such scaffolds can help students better understand the problem and devise a plan, which, in turn, can support the perseverant actions of initiating and sustaining productive struggle as students carry out and reflect on the effectiveness of their plan (Anghileri, 2006; Pólya, 1971) (see again

Figure 3 for this theoretical framework for scaffolding perseverance in problem-solving).

However, the results of this study show how a student can revisit his or her initial conceptualization of a mathematical situation to support continued perseverance upon a perceived impasse, after he or she had productively struggled with his or her first plan of attack.

As illustrated by representative participants James (on Cross Totals) and Sandra

(on Sidewalk Stones and Skeleton Tower), revisiting one’s earlier recorded ideas for what mathematics might be relevant or what strategies might be useful for solving the problem can help students navigate moments when they are at their most frustrated and most tempted to give up – when they realized their first attempt at solving an unfamiliar problem was not successful. In these moments, students’ original recorded

150 conceptualization, now an artifact of their conceptual thinking at the time, can act as an organized toolbox from which to draw an unexplored mathematical idea, or a new connection between ideas, to use to help re-engage with the task upon impasse and to continue to productively struggle to make sense of the mathematical situation. Using

Pólya’s (1971) terms, students revisiting an artifact of their original ideas while reflecting on the effectiveness of their initial attempt can help them understand the problem better by recognizing the exact shortcomings of their first plan and devise a new plan to address those shortcomings. Better understanding the problem and devising a more focused plan acts as a way of re-conceptualizing the situation after a failed attempt at solving and encourages re-initiating and re-sustaining productive struggle via an additional attempt.

Thus, as a result of initially engaging with a conceptualization scaffold, students can persevere in their problem-solving in a cyclical manner, with each additional attempt after impasse representing a new opportunity to productively struggle supported by their own conceptual ideas (see Figure 25). Without explicitly attending to one’s initial conceptual thinking, as suggested by findings from participants working on non- scaffolded tasks, students may find it more difficult to revisit how they first conceptualized the problem upon impasse, and thus more often give up without making an additional attempt at solving.

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Figure 25. Scaffolding perseverance in problem-solving: A cyclical representation

Developing students’ perseverance for solving challenging mathematics tasks may be possible through deliberate practice with appropriate support, as suggested by greater improvements across student experiences with scaffolded tasks compared to non- scaffolded tasks in this study. The process of deliberate practice, first posited by Ericsson,

Krampe, and Tesch-Romer (1993), describes concerted, systematic efforts to improve performance in a specific domain. The major tenets of deliberate practice, for both the environment and the person, include: (a) well-defined and specific goals toward which effort is directed, (b) setting goals beyond a person’s immediate and current abilities, (c) appropriate fundamental knowledge and previously acquired skills, (d) full conscious attention and self-control, (e) opportunities for feedback and modifications of efforts, and

(f) repetition under similar conditions (Ericsson, 2016). In mathematics education, deliberate practice has been studied primarily in the context of helping students learn

152 specific skills and developing competencies in particular mathematical domains, like numeracy (Fuchs et al., 2010) or geometry (Pachman, Sweller, & Kalyuga., 2013, 2014).

Yet, findings in this study suggest mathematical practices like perseverance, in addition to domain-competencies, are malleable and able to improve through processes of deliberate practice, especially with support systems in place that encourage initial conceptual thinking.

When working on scaffolded tasks, the data in this study show that the students were engaging in processes akin to those of deliberate practice – practicing to improve their perseverance performance over time. Although the non-scaffolded tasks still warranted perseverance, it was apparent that students in this study were not engaging in all tenets of deliberate practice, as defined by Ericsson (2016), when they worked on non- scaffolded tasks. Therefore, the embedded scaffold prompting students to record their conceptualization of the mathematical situation seemed to encourage engagement in deliberate practice.

Students were deliberately practicing to improve their perseverance in the following ways. First, they worked on challenging mathematical tasks with well-defined objectives. By way of the Entrance Phase of the Three-Phase Perseverance Framework, the goals of tasks were clear to the student, yet they did not know how to immediately achieve them. Also, students had demonstrated in their pre-test the pre-requisite knowledge necessary to initially engage with these tasks. Then, largely specific to the scaffolded tasks, students invested their full effort and attention to make progress on

153 these tasks, relying heavily on self-control to not give up at moments of impasse and continue to persevere. Engaging with the embedded conceptualization prompts were particularly supportive of such effort and self-control. Responding to the scaffolds helped students initiate and sustain a productive first attempt at solving in an uncertain situation, be a self-source of feedback when recognizing a mistake or setback, and modify their effort at moments of impasse to make a second attempt through revisiting their initial conceptual thinking. Students generally did not engage in these aspects of deliberate practice while working on the non-scaffolded tasks. Lastly, this study’s design gave students an opportunity to repeat and practice working through these processes every week as they worked on a problem. With the scaffolded tasks, this repeated opportunity ultimately helped students refine their strategies over that time to learn to persevere in more effective ways. Although students also had a chance to repeat the processes of deliberate practice with non-scaffolded tasks, the data showed that they did not refine their strategies in the same high-quality ways as they did with scaffolded tasks.

Therefore, deliberately practicing with challenging mathematics tasks that warranted perseverance and supported initial conceptual thinking seemed to help students improve the ways in which they persevered, over time, in this study.

Limitations and Future Research

Although I took several precautions to help ensure each task used in this study was of similar difficulty (e.g., vetted by MARS (2015), considered experts’ opinions, considered participants’ opinions), I still cannot be completely sure that Result 1 (Higher

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Quality Perseverance on Scaffolded Tasks than on Non-Scaffolded Tasks) was a function of participants engaging with the scaffolds and not the tasks themselves. It is possible the tasks that were randomly chosen to be scaffolded were less difficult than the non- scaffolded tasks, thus partially explaining why students found it easier to persevere on the scaffolded tasks. To help eliminate the likelihood of this alternate explanation, future research should randomize the assignment of scaffolds to tasks for each participant to better control for this potential bias, and seek to replicate the results found here.

Regarding Result 2 (Perseverance Improved in Quality over Time), it would be inappropriate to generalize the effect of deliberate practice for improving students’ perseverance for solving challenging mathematics tasks. My study of 10 ninth-grade students’ experiences across 50 problem-solving sessions speaks only to this particular context for this relatively short amount of time. For true gains in perseverance, a study of deliberate practice should span years (Ericsson et al., 1993), carefully tracking the ways in which one’s perseverance improves and stabilizes in appropriately challenging problem-solving contexts. However, my findings should motivate future research to investigate longitudinally how concerted, systematic efforts within a supportive deliberate practice cycle can improve students’ perseverance with challenging mathematics.

Next, of practical concern is the clinical nature of this study. Participants were likely willing to work harder and tolerate uncertainty for longer in the one-on-one interview setting compared to a normative classroom setting. Future research should

155 examine challenges of how to implement scaffolds to support student perseverance at the whole class level.

Lastly, an important area of future research is an investigation into the productivity of the perseverance described here, namely conceptual learning. In this study, perseverance itself was the prime focal-point, and no claims are being made about participants’ conceptual learning as a result of it. Although I considered participants’ perceived mathematical progress as a component of their productive struggle, future work should look to make more explicit exactly how participants’ conceptual knowledge is being developed through their perseverance.

Conclusions and Implications

Perseverance is fundamentally important for learning mathematics, but difficult and uncomfortable for students to initiate and sustain in the moment. Thus, encouraging and supporting students’ perseverance with challenging mathematics tasks is of utmost importance in mathematics education. This study found that perseverance could be supported via scaffolds that prompted students to record their initial conceptualization of a mathematical situation prior to working on a task. Most notably, students often revisited their initial conceptualizations at moments when they most wanted to give up and were able to amend their original ideas and continue to persevere. This study also found that consistent engagement with tasks warranting perseverance, especially scaffolded tasks, helped students improve the methods by which they persevered over time. This included

156 student improvements in planning problem-solving strategies and managing stress and frustration in the moment.

Although there is more work to be done on supporting perseverance with challenging mathematics, this study offers some immediate implications for teachers seeking to help nurture student perseverance in their classroom. First, teachers can provide consistent opportunities for students to attend explicitly to their initial conceptualization of a task. Encouraging students at the outset to write in their own words the mathematical ideas that seem important or the exploratory strategies that may be advantageous can pay large dividends later when students encounter a substantial obstacle during problem-solving. These opportunities could be embedded in the task itself via scaffolds, as they were in my study, for a relatively unobtrusive intervention that affects all students in the classroom.

Second, teachers can strive to normalize student engagement with tasks that warrant perseverance. In my study, as voiced by representative participants Sandra and

Laurie in their exit interviews, the tasks with which they engaged during their think-aloud interviews were novel; that is, they were rarely put in a position during their regular math classes to work on a task for which they did not already know a solution pathway. Yet, many participants mentioned how valuable these novel experiences were, and how they could feel themselves improving in particular ways that describe what it means to persevere, even after just five weeks. As such, teachers providing regular opportunities for students to practice with tasks for which solutions are not immediately apparent can

157 pay big dividends in developing student perseverance over time. Of course, these tasks need to be appropriate for perseverance, i.e., have a perceived familiarity yet complex structure with opportunities for support to encourage engagement but maintain cognitive demand.

Last, the above two implications for teachers are only useful if the classroom culture is one that values perseverance. Valuing perseverance means to prize the doubt, to celebrate and recognize how struggle and discomfort are imperative to the process by which mathematics is learned for understanding. It is undoubtedly a complex process, but teachers can work to establish this culture by normalizing several classroom practices.

For instance, consistently asking students to articulate their thinking while wrestling with challenging ideas can help move away from a culture of answers and toward one of process (Kress, 2017). Attentively preserving the cognitive demand of tasks and activities, through careful scaffolding and hesitation to relieve student discomfort, can illustrate a pedagogy focused on appreciating productive struggle (Stein, Grover, &

Henningsen, 1996). Rewarding students for their efforts that lead to mistakes can help exemplify how setbacks, discouragements, and even perceived impasses are part of the learning process and should be celebrated if arising from one’s full effort (Warshauer,

2014). Taken together and amongst others, learning environments like these can help students start to see the usefulness of perseverance and help them tolerate the accompanying discomfort, and also can help teachers better learn how to develop and offer support for their students’ perseverance with challenging mathematics.

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Appendix A

PRETEST ITEMS

1. Mrs. Lucas’ class has a 2-hour science lab. She gives each student a dish with one cell in it. She tells the class that in 20 minutes the cell will divide into two cells, and each 20 minutes after that each cell in the dish will divide into two cells.

(a) Complete the second row in the table below to show how the number of cells increases during the lab.

Time 0 20 40 60 80 100 120 (minutes) Number of cells Number of cells as a power of 2

(b) Olan says that the number of cells can be written in the form 2푛. Complete the third row in the table above to show how the number of cells can be written in this form.

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2. Below are four graphs, four equations, four tables, and four rules. Your task is to match each graph with an equation, a table, and a rule (Graph A doesn’t necessarily go with Equation A, for example). Write your answers in the provided table. Do not use a graphing calculator.

Graph Equation Table Rule

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Appendix B

PARTICIPANTS’ TIME SPENT ON EACH TASK

Triangular Cross Sidewalk Skeleton Table Participant Frameworks Totals (S) Stones (S) Tower (S) Tiling (NS) (NS) Marcia 23 12 11 17 9 Sandra 15 17 21 12 17 James 45 25 23 24 11 Tom 18 24 24 20 13 Laurie 19 19 16 8 8 Dennis 21 15 12 10 7 Darya 25 10 9 6 5 Elizabeth 25 13 13 9 10 Kristen 30 16 22 14 27 Nick 29 14 16 37 21 Notes: Times are rounded to the nearest minute; S = Scaffolded Task, NS = Non- Scaffolded Task

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Appendix C

THINK-ALOUD INTERVIEW PROTOCOL

Interview Questions before Participant Started Working on Task:

1. Now that you’ve read the problem, what will you be trying to accomplish today, in

your own words?

2. Do you know exactly how to accomplish that?

Interview Questions during Participant’s Initial Effort on Task:

1. What are you thinking?

2. Can you describe what you’re doing?

Interview Questions during Participant’s Additional Effort on Task:

1. What are you thinking?

2. Can you describe what you’re doing?

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Appendix D

MATHEMATICAL TASKS

Task: Cross Totals

Crosses follow two rules:

 They must contain all the numbers from 1 to 9

 The horizontal and vertical totals (sums) must be equal (we call this the Cross

Total)

In the cross shown above, the Cross Total is 27 because:

 Adding horizontally: 2 + 1 + 9 + 8 + 7 = 27

 Adding vertically: 6 + 4 + 9 + 3 + 5 = 27

Task Goal: Your goal while working on this task is to figure out any rules that describe which cross totals are possible, and which cross totals are impossible, in general.

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1. Before you start, what mathematical ideas do you think might be important for solving this problem? Write down all of your ideas in detail.

2. Write down any rules you found that describe which cross totals are possible, and which cross totals are impossible, in general.

3. Explain how you know your rules are correct.

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Task: Sidewalk Stones

In Rome, some sidewalks are made of small square blocks of stone. The blocks are in different shades (gray blocks and white blocks) to make patterns that are in various sizes.

The diagram above shows three possible patterns using gray blocks and white blocks.

Task Goal: Your goal while working on this task is figure out some rules that describe how many blocks of each shade Pattern #푛 will need, in general.

1. Before you start, what steps do you think you might take to solve this problem? Write down all of your ideas in detail.

2. Write down any rules you found that describe how many blocks of each shade Pattern

#푛 will need, in general.

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3. Explain how you know your rules are correct.

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Task: Skeleton Tower

The skeleton tower shown above is built with 3-D cubes.

Task Goal: Your goal while working on this task is to figure out a rule that describes how many cubes are needed to build a tower that is 푛 cubes high.

1. Before you start, do you know what you are trying to accomplish while working on this task? Circle yes or no.

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2. Write down a rule you found that describes how many cubes are needed to build a tower that is 푛 cubes high.

3. Explain how you know your rules are correct.

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Task: Table Tiling

Maria makes square tables and then sticks tiles to the top.

She uses three types of tiles:

Half tiles Quarter tiles Whole tiles

The sizes of the square tabletops are all multiples of 10 cm. Maria only uses quarter tiles in the corners and half tiles along the edges of the table. Here is one tabletop:

This square table uses 5 whole tiles, 4 half tiles, and 4 quarter tiles.

Task Goal: Your goal while working on this task is to figure out any rules that describe how many tiles of each type she needs for larger, square tabletops, in general.

1. Before you start, what mathematical ideas do you think might be important for solving this problem? Write down all of your ideas in detail.

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2. Write down any rules you found that describe how many tiles of each type Maria needs for larger, square tabletops, in general.

3. Explain how you know your rules are correct.

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Task: Triangular Frameworks

Tom uses metal rod to make triangular frameworks in which each side has a different length. He buys metal rods which have lengths of 1 meter, 2 meters, 3 meters, and so on.

He always keeps one rod of each length in stock.

The diagram above shows one of Tom’s triangular frameworks.

Given information:

 푎, 푏, 푐 are all integers (the side lengths are always whole numbers)

 푐 > 푏 > 푎 (푐 is the longest side and 푎 is the shortest side)

 푐 < 푎 + 푏 (the sum of the short sides is always longer than the longest side)

Tom figures out that he can make six different triangular frameworks when the longest side is 7 meters (m). The six frameworks he can make have side lengths of: 7m-6m-5m,

7m-6m-4m,

7m-6m-3m, 7m-6m-2m, 7m-5m-4m, and 7m-5m-3m.

Task Goal: Your goal while working on this task is to consider other values for 푐, and to figure out some rules that describe how you can find the number of possible different

191 frameworks Tom can make, in general.

1. Before you start, what steps do you think you might take to solve this problem? Write down all of your ideas in detail.

2. Write down any rules you found that describe how many different frameworks that exist for a value of 푐.

3. Explain how you know your rules are correct.

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Appendix E

VIDEO-REFLECTION INTERVIEW PROTOCOL

Interview Questions prior to Watching Video:

1. From start to finish, how would you describe how you worked on this problem?

2. Can you explain what you wrote on your paper?

Interview Statement to Transition to Watching Video:

1. Now we are going to watch some video clips of you solving today’s task. As you

watch these clips, I am going to ask you some questions. Try your best to answer

these questions by reconstructing what you were thinking and how you were feeling

in-the-moment while working on this problem. It’s also ok to answer my questions in

a reflective sense, as you are looking back on the work you just did.

Interview Questions while Watching Video:

1. Can you describe your experience here?

2. What was happening in this moment?

3. Why were you doing this?

4. What were you thinking here? Why?

5. What emotions were you feeling here? Why?

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Appendix F

EXIT INTERVIEW PROTOCOL

1. Is there anything you’d like to share about your overall experience working with

these five problems?

2. What was it like working on these problems?

3. How difficult did you think these problems were? Were any easier or harder than the

others?

4. [Share my observations of participant’s engagement with each task] Was your

experience different in any way from my observations? How was it different?

5. Is there anything else you’d like to add to help me better understand your experience

working on these five problems over the last month or two?

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Appendix G

PROBLEM-SOLVING HEURISTICS

 Change your point of view – Approaching a problem from another angle when a

previous approach is not effective.

 Draw a diagram – Doing a sketch based on the available information to visually

represent the problem.

 Guess and check – Making a reasonable guess of the answer and then checking the

result to see if it works. If necessary, repeating the procedure to find the answer, or at

least a close approximation.

 Logical reasoning – Demonstrating that if a statement is accepted as true, then other

statements could be shown as true based on it.

 Look for a pattern – Identifying patterns in the givens based on careful observation of

common characteristics, variations, or differences about numbers, shapes, etc. in then

problem.

 Make a systematic list – Constructing an organized list containing all the possibilities

for a givens situation and finally to find the answer.

 Make a table – Organizing data into a table and then using the entries in the table to

solve the problem.

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 Make suppositions – Making a hypothesis, and then based on the givens and

hypothesis, finding out the relationship between the known and unknown, and finally

solving the problem.

 Restate the problem – Rephrasing the original problem so that the statement of the

problem becomes familiar and hence more accessible.

 Simplify the problem – Changing the complex numbers or situations in the problem

into simpler ones without altering the problem mathematically.

 Solve part of the problem – Dividing a problem into several sub-problems, then

solving them one by one, and finally solving the original problem.

 Think of a related problem – Using methods/results of a related problem, or recalling

a related problem, or considering a similar problem solved before in order to solve the

problem.

 Use a model – Creating visual representations (e.g., using points, lines, or other easy-

to-understand symbols) to model the information on quantities or relationships or

changes that are involved in the problem.

 Use an equation – Using letters as variables to represent unknown quantities in a

problem, and establishing and solving equations or inequalities to get the answer.

 Use before-after concept – Listing information given before and after action, and

observing the change between the two situations (from before to after) to find the

solution.

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 Work backwards – Approaching a problem from its outcomes or solutions backwards

to find what conditions they eventually need to meet.

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Appendix H

SOLUTIONS TO MATHEMATICAL TASKS

Cross Totals:

 Possible cross totals have an odd number in middle, impossible cross totals have even

in middle. This is true because 1+2+…+8+9=45, so cross totals will sum to 45 + the

middle number. 45 is odd, odd + odd is even, so the middle number must be odd.

Sidewalk Stones:

 Pattern n needs 8푛2 + 4푛 white blocks and 8푛2 + 4푛 + 1 gray blocks. Accept

algebraically equivalent rules.

Skeleton Tower:

 A tower n cubes high needs 2푛2 − 푛 cubes. A second rule is accepted if they describe

how their rule works in words, or if they come up with another algebraic expression

and explain why it’s equivalent, or if they write a recursive rule somehow.

Table Tiling:

푛  For n x n table tops (n is a multiple of 10cm), you need 4 ( − 1) half tiles, and you 10

푛 2 푛 2 need ( ) + ( − 1) whole tiles. Accept algebraically equivalent rules. 10 10

Triangular Frameworks:

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(푐−3)(푐−1)  If c is even, then there are different frameworks. If c is odd, then there are 4

(푐−2)(푐−2) different frameworks. Accept algebraically equivalent rules. 4

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Appendix I

INSTITUTIONAL REVIEW BOARD APPROVAL LETTERS

DATE: March 21, 2017

TO: Joseph DiNapoli FROM: University of Delaware IRB

STUDY TITLE: [1039271-1] Investigating the Nature of Perseverance

SUBMISSION TYPE: New Project

ACTION: APPROVED APPROVAL DATE: March 21, 2017 EXPIRATION DATE: March 20, 2018 REVIEW TYPE: Expedited Review

REVIEW CATEGORY: Expedited review categories 6 and 7

Thank you for your submission of New Project materials for this research study. The University of Delaware IRB (HUMANS) has APPROVED your submission. This approval is based on an appropriate risk/benefit ratio and a study design wherein the risks have been minimized. All research must be conducted in accordance with this approved submission.

This submission has received Expedited Review based on the applicable federal regulation.

Please remember that informed consent is a process beginning with a description of the study and insurance of participant understanding followed by a signed consent form. Informed consent must continue throughout the study via a dialogue between the researcher and research participant. Federal regulations require each participant receive a copy of the signed consent document.

Please note that any revision to previously approved materials must be approved by this office prior to initiation. Please use the appropriate revision forms for this procedure. All SERIOUS and UNEXPECTED adverse events must be reported to this office. Please use the appropriate adverse event forms for this procedure. All sponsor reporting requirements should also be followed.

200

Please report all NON-COMPLIANCE issues or COMPLAINTS regarding this study to this office.

Please note that all research records must be retained for a minimum of three years.

Based on the risks, this project requires Continuing Review by this office on an annual basis. Please use the appropriate renewal forms for this procedure.

If you have any questions, please contact Nicole Farnese-McFarlane at 302-831-1119 or [email protected]. Please include your study title and reference number in all correspondence with this office.

201

DATE: February 20, 2018

TO: Joseph DiNapoli FROM: University of Delaware IRB

STUDY TITLE: [1039271-2] Investigating the Nature of Perseverance

SUBMISSION TYPE: Continuing Review/Progress Report

ACTION: Approved for Data Analysis Only APPROVAL DATE: February 20, 2018 EXPIRATION DATE: March 19, 2019 REVIEW TYPE: Expedited Review

REVIEW CATEGORY: Expedited review category # (6,7)

Thank you for your submission of Continuing Review/Progress Report materials for this research study. The University of Delaware IRB has APPROVED your submission. This approval is based on an appropriate risk/benefit ratio and a study design wherein the risks have been minimized. All research must be conducted in accordance with this approved submission.

This submission has received Expedited Review based on the applicable federal regulation.

Please note that any revision to previously approved materials must be approved by this office prior to initiation. Please use the appropriate revision forms for this procedure.

All SERIOUS and UNEXPECTED adverse events must be reported to this office. Please use the appropriate adverse event forms for this procedure. All sponsor reporting requirements should also be followed.

Please report all NON-COMPLIANCE issues or COMPLAINTS regarding this study to this office.

Please note that all research records must be retained for a minimum of three years.

202

Based on the risks, this project requires Continuing Review by this office on an annual basis. Please use the appropriate renewal forms for this procedure.

If you have any questions, please contact Nicole Farnese-McFarlane at (302) 831-1119 or [email protected]. Please include your study title and reference number in all correspondence with this office.

203