#### ATTENDING TO UNCERTAINTY IN THE DESIGN AND IMPLEMENTATION OF DGE TASKS TO ENGAGE MATHEMATICS STUDENTS IN PRODUCTIVE STRUGGLE

## ABSTRACT

Struggling with mathematics is important for several reasons. According to learning theorists and empirical studies, the act of struggling can help students learn mathematics. Also, several important mathematics education documents advocate that students should engage in struggle as part of the problem-solving process. In addition, being able to struggle and persevere through struggle when solving problems is an essential skill for graduates to have as they pursue future employment. However, it may not be clear to mathematics teachers how to engage students in struggle and make and keep that struggle productive. Creating uncertainty for students could possibly be a means for engaging students in productive struggle, because of the close relationship between uncertainty and struggle. This study sought to engage students in productive struggle by engaging them in dynamic geometry tasks that elicit uncertainty. Dynamic geometry environments were chosen for their usefulness in creating and resolving uncertainties due to their exploration and feedback features. Eight mathematical tasks were chosen, modified, or designed in such a way as to elicit uncertainty. Uncertainty was to be created in two ways in the tasks, through competing claims and unknown paths/questionable conclusions. Classroom observations were conducted to observe secondary geometry students working on these tasks using the dynamic geometry environment, GeoGebra. This data was analyzed in such a way as to determine answers to the following questions: (a) Did the task create uncertainty?, (b) Did the uncertainty lead to struggle?, (c) What parts of the struggle were productive and what parts of the struggle were not productive?, and (d) What supported or hindered students’ productive struggle? The tasks did create uncertainty, but this uncertainty did not always lead to students engaging in productive struggle. Even when students were faced with clear contradictions or questionable conclusions, they did not always seek out resolutions for these uncertainties. One possible explanation for why students did not seek out resolutions for these uncertainties is because they did not believe they possessed the mathematical authority to do so.

## TABLE OF CONTENTS

LIST OF FIGURES……………………………………………………………………………………………….. ix

LIST OF TABLES…………………………………………………………………………………………………. xi

ACKNOWLEDGEMENTS…………………………………………………………………………………….. xii

Chapter 1 Rationale ……………………………………………………………………………………………………… 1

Relationship Between Productive Struggle and Problem Solving …………………………… 1

Importance of Productive Struggle ……………………………………………………………………… 3

Learning Theories ………………………………………………………………………………… 3

Policy Documents and NCTM Publications …………………………………………….. 5

Future Employment ………………………………………………………………………………. 7

Difficulty Engaging Students in Productive Struggle ……………………………………………. 8

Suggestions for Engaging Students in Productive Struggle ……………………………………. 11

Technology ……………………………………………………………………………………………………… 12

Technology-Equipped Classrooms …………………………………………………………. 13

Policy Documents and NCTM Publications …………………………………………….. 14

Research Problem …………………………………………………………………………………………….. 15

Chapter 2 Review of the Literature ………………………………………………………………………………… 17

Task Framework ………………………………………………………………………………………………. 18

Designing Tasks: What are the Characteristics of Tasks That Engage Students in

Productive Struggle? ………………………………………………………………………………………… 22

Setting-Up and Implementing Tasks: What are Teaching Strategies That Make and Keep

this Struggle Productive? …………………………………………………………………………………… 25

Teaching Principles That Engage Students in Productive Struggle ……………… 27

Valuing Process and Struggle …………………………………………………….. 27

Supporting Students While Keeping Cognitive Demand High ………… 29

Teaching Strategies That Engage Students in Productive Struggle ……………… 33

Relationship Between Productive Struggle and Uncertainty ………………………………….. 33

Three Types of Uncertainty ……………………………………………………………………………….. 35

Competing Claims ………………………………………………………………………………… 35

Unknown Path/Questionable Conclusion …………………………………………………. 38

Non-Readily Verifiable Outcome …………………………………………………………… 39

Summary …………………………………………………………………………………………….. 39

Relationship Between Characteristics of Tasks That Engage Students in Productive

Struggle and Three Types of Uncertainty ……………………………………………………………. 40

High Cognitive Demand ………………………………………………………………………… 40

Novel ………………………………………………………………………………………………….. 41

Solution Is Not Immediately Apparent ……………………………………………………. 41

Worth Solving ……………………………………………………………………………………… 42

Summary …………………………………………………………………………………………….. 42

Dynamic Geometry Environments ……………………………………………………………………… 42

What Is a DGE? ……………………………………………………………………………………. 43

Relationship Between DGEs and Uncertainty ………………………………………….. 45

Creating Uncertainty with DGEs ………………………………………………… 45

Competing Claims …………………………………………………………. 46

Unknown Path/Questionable Conclusion …………………………. 46

Resolving Uncertainty with DGEs ………………………………………………. 46

Competing Claims …………………………………………………………. 46

Unknown Path/Questionable Conclusion …………………………. 47

Non-Readily Verifiable Outcome ……………………………………………….. 47

Summary of DGE Affordances ………………………………………………………………. 48

Vygotsky’s Sociocultural Theory of Cognitive Development ………………………………… 48

Vygotsky’s Sociocultural Theory of Cognitive Development …………………….. 49

Designing Tasks …………………………………………………………………………………… 50

Implementing Tasks ……………………………………………………………………………… 53

Conclusion ………………………………………………………………………………………………………. 55

Chapter 3 Methods ………………………………………………………………………………………………………. 57

Participants ……………………………………………………………………………………………………… 57

Population of Interest ……………………………………………………………………………. 58

Identification of Participants ………………………………………………………. 61

Number of Participants ………………………………………………………………. 62

Benefit of Knowing Participants …………………………………………………………….. 63

Prior Knowledge and Dispositions ………………………………………………. 63

ZPD ………………………………………………………………………………………… 66

Claims about Students and Supporting Evidence ………………………….. 68

GeoGebra ………………………………………………………………………………… 70

Generalizability and Bias …………………………………………………………… 70

Generalizability …………………………………………………………….. 70

Bias ……………………………………………………………………………… 71

Summary …………………………………………………………………………………. 72

Data Collection ………………………………………………………………………………………………… 72

Tasks …………………………………………………………………………………………………… 73

Dividing Land Task …………………………………………………………………… 75

Uncertainty …………………………………………………………………… 77

Approaches to the Task and Struggle ……………………………….. 77

Congruency Paradox Task …………………………………………………………. 79

Uncertainty …………………………………………………………………… 81

Approaches to the Task and Struggle ……………………………….. 81

Sum of Interior and Exterior Angles of a Polygon Task …………………. 82

Uncertainty …………………………………………………………………… 83

Approaches to the Task and Struggle ……………………………….. 84

Lost Square Task ………………………………………………………………………. 85

Uncertainty …………………………………………………………………… 86

Approaches to the Task and Struggle ……………………………….. 87

Pentagon Problem Task ……………………………………………………………… 88

Uncertainty …………………………………………………………………… 89

Approaches to the Task and Struggle ……………………………….. 90

Areas of Triangles Task …………………………………………………………….. 91 Uncertainty …………………………………………………………………… 92 Approaches to the Task and Struggle ……………………………….. 92

Pythagorean Theorem Task ………………………………………………………… 92

Uncertainty …………………………………………………………………… 93

Approaches to the Task and Struggle ……………………………….. 94

Circle Theorems Task ……………………………………………………………….. 94

Uncertainty …………………………………………………………………… 97

Approaches to the Task and Struggle ……………………………….. 97

Common Characteristics of the Tasks ………………………………………….. 97

Curriculum Calendar …………………………………………………………………. 101

Number of Classroom Observations ……………………………………………………….. 103

GeoGebra ……………………………………………………………………………………………. 103

Logistics ……………………………………………………………………………………………… 104

Teaching Strategies That Engage Students in Productive Struggle ……………… 104

Valuing Process and Struggle …………………………………………………….. 105

Supporting Students While Keeping Cognitive Demand High ………… 107

Artifacts ………………………………………………………………………………………………. 109

Data Analysis ………………………………………………………………………………………………….. 110

Creating Uncertainty …………………………………………………………………………….. 111

Engaging in Productive Struggle …………………………………………………………….. 111

Conclusion ………………………………………………………………………………………………………. 113

Chapter 4 Results …………………………………………………………………………………………………………. 114

Competing Claims Tasks …………………………………………………………………………………… 115

Creating Uncertainty …………………………………………………………………………….. 115

Engaging in Productive Struggle …………………………………………………………….. 127

Task Design ……………………………………………………………………………… 128

Task Implementation …………………………………………………………………. 131

Teacher Actions ……………………………………………………………. 131

Group Work …………………………………………………………………. 146

GeoGebra …………………………………………………………………….. 153

Content Knowledge ……………………………………………………….. 157

Cognitive Demand ………………………………………………………… 160

Conclusion …………………………………………………………………………………………… 162

Unknown Path/Questionable Conclusion Tasks …………………………………………………… 163

Creating Uncertainty …………………………………………………………………………….. 163

Engaging in Productive Struggle …………………………………………………………….. 173

Task Design ……………………………………………………………………………… 173

Task Implementation …………………………………………………………………. 177

Teacher Actions ……………………………………………………………. 178

Group Work …………………………………………………………………. 187

GeoGebra and Cognitive Demand …………………………………… 194

Content Knowledge and Reasoning …………………………………. 198

Conclusion …………………………………………………………………………………………… 201

Chapter 5 Discussion and Implications …………………………………………………………………………… 203

Creating Uncertainty ………………………………………………………………………………………… 203 Engaging in Productive Struggle ………………………………………………………………………… 204 Uncertainty and Productive Struggle Disconnect …………………………………………………. 206

Examples of the Disconnect …………………………………………………………………… 207

Mathematical Authority as an Explanation for the Disconnect …………………… 208

Alternative Explanations ……………………………………………………………………….. 212

Content Knowledge …………………………………………………………………… 212

Beliefs About Mathematics ………………………………………………………… 215

Implications …………………………………………………………………………………………………….. 216

Limiting the Authority of External Entities ……………………………………………… 216

Building Students’ Authority …………………………………………………………………. 222

Classroom Norms ……………………………………………………………………… 222

Group Work ……………………………………………………………………………… 226

Tasks as “Problems” ………………………………………………………………….. 229

GeoGebra as a Tool …………………………………………………………………… 231

Conclusion ………………………………………………………………………………………………………. 233

References ………………………………………………………………………………………………………………….. 236

Appendix Recruitment Letter ………………………………………………………………………………………… 248

**Chapter 1 **

### Rationale

As a secondary mathematics teacher, I all too often hear comments from my students such as “I don’t get this,” “This is too hard,” “I’ll just look in the back of the book,” “I give up,” or “I’ll ask someone for the answer later.” Students do not like to struggle with mathematics; many give up at the first sign of difficulty rather than persevering. I am interested in how I can get my students to engage in struggle and persevere through that struggle, or in other words, “struggle productively”. My interest in getting students to engage in struggle stems from the importance of struggle in learning mathematics.

Many learning theorists state that struggle is necessary for learning (Dewey, 1910; Festinger, 1957; Hatano, 1998; Piaget, 1960). This idea is backed by empirical studies that have found that students learn more when they struggle (Kapur, 2014; Silver & Stein, 1996), yet there is evidence showing it may not be clear to teachers how to engage students in productive struggle (Warshauer, 2015a). This topic is important to me because, as a secondary mathematics teacher, I want to know how I can engage my students in productive struggle and help other teachers engage their students in productive struggle.

I start by defining productive struggle, and then follow with a theoretical, empirical, and practical argument for the importance of productive struggle. Next, evidence from empirical studies is given to show that teachers may not know how or simply do not want to engage students in struggle. The use of technology, specifically dynamic geometry environments (DGEs), is discussed as a possible means to engage students in struggle and make that struggle productive. Last, this chapter closes with the research question and purpose of the present study.

### Relationship Between Productive Struggle and Problem Solving

Struggling with mathematics involves delving deeply into mathematical ideas and trying to figure out things that are not immediately apparent (Hiebert & Grouws, 2007). When trying to engage students in struggle, the process of solving problems is emphasized rather than focusing on coming up with the correct answer as quickly as possible (NCTM, 2014).

Productive struggle involves being able to work through impasses that arise when working on problems. By problems, I mean genuine problems that students do not immediately know how to solve and that have a high level of cognitive demand. Problems that have a high level of cognitive demand require thinking processes such as conjecturing, justifying, interpreting, and attending to concepts and understanding. These types of problems are in contrast to routine exercises that students immediately know how to solve. In order to struggle productively, students cannot give up when they encounter difficulties. Instead, they need to figure out a way to move forward with the problem and persevere without becoming overly frustrated or giving up. One does not need to reach a correct answer or reach any answer for struggle to be productive. If one is remaining engaged with the intent of solving the problem, then the struggle can be productive. In this study, student struggle will be categorized as productive if students are actively engaged with relevant mathematics at a high level of cognitive demand with the intent of resolving uncertainty while keeping a positive disposition.

Problem solving involves many components, such as possessing proper prior knowledge and utilizing appropriate problem-solving techniques. I would argue that productive struggle is also an essential part of problem solving and it is unlikely that one will truly engage in problem solving without encountering uncertainty and struggle. If one is solving a new and difficult problem, then one will most likely face obstacles and struggle. It is important for the learner to be able to persevere and struggle by way of using and evaluating different strategies and then having a willingness to abandon certain strategies and try others in order to move forward and be productive (Pasquale, 2015). If a student cannot struggle productively, it is unlikely that they will be a successful problem solver.

### Importance of Productive Struggle

Struggling with mathematics is important for three reasons. First, the act of struggling can help students learn important mathematics. Second, struggle, as part of problem solving, is advocated in mathematics education policy documents and publications from the National Council of Teachers of Mathematics (NCTM). Third, being able to struggle and persevere through struggle when solving problems is an important skill for graduates to have as they pursue future employment. These three factors are elaborated, in turn, in the discussion below.

#### Learning Theories

Several learning theorists support the claim that struggling is necessary for learning. For example, Dewey (1910) believed that deep understanding begins with perplexity, confusion, and doubt. Festinger (1957) proposed that if there are inconsistencies in one’s cognition, this cognitive state is called cognitive dissonance. The individual then seeks ways to resolve these inconsistencies, which can result in cognitive development. Hatano (1988) referred to cognitive incongruity in his theory of learning. Hatano wrote that not having an adequate understanding, or cognitive incongruity, would lead individuals to seek out knowledge, which, subsequently, would lead to learning. Piaget (1960) claimed that some instances cannot be accounted for by one’s current cognitive structure, which causes a perturbation and results in a cognitive state of disequilibrium. Learners seek to resolve their state of disequilibrium by adapting their current cognitive structure, and this is when learning has occurred.

All of these authors include an aspect of learning consistent with the notion of struggle. More specifically, all of the aforementioned theorists state that the process of learning starts with some sort of cognitive inconsistency that one must struggle through. The development of knowledge occurs through the struggle, as these inconsistencies are resolved, and the result is new cognitive structures.

Hiebert and Grouws (2007) cite many of the above learning theories when discussing the importance of struggle in learning. The authors relate struggle to the learning of a certain type of knowledge, namely, conceptual knowledge. They state that having students struggle with mathematics is an important factor of teaching that promotes conceptual understanding. The authors define conceptual understanding as “mental connections among mathematical facts, procedures, and ideas” and this understanding grows “as mental connections become richer and more widespread” (p. 380). Hiebert and Grouws seem to equate conceptual understanding to a certain depth of knowledge. I agree with Star (2005) that conceptual knowledge does not refer to a *quality* of knowledge, but instead refers to a *type* of knowledge. Two types of knowledge are procedural and conceptual, and one can have either a superficial or deep quality of either type of knowledge. I suggest, in accordance with the authors mentioned above, that engaging students in struggle could promote both deep conceptual understanding and deep procedural knowledge.

While it is argued that struggle is necessary for learning both conceptual and procedural knowledge, students should not struggle endlessly resulting in needless frustration. Problems that students are given should be challenging, but within reach. Vygotsky (1978) used the term “zone of proximal development”, or ZPD, to describe the difference between what an individual is capable of doing independently and what he or she is capable of doing with the assistance of a more knowledgeable other. The ZPD is the difference between an individual’s actual development and his or her potential development. Students should be given tasks that are within their ZPD so that they have the opportunity to struggle with a challenge, but yet, the struggle is not despairing.

Empirical evidence supports the theoretical notion that struggle is necessary for learning. Many studies have shown that when students struggle, they experience deeper learning. Silver and Stein (1996) discovered that when middle school students engaged in tasks with higher cognitive demand, it led to greater conceptual development and the more challenging the problem, the more conceptual understanding. This study illustrates that when students were faced with great difficulty, and one could assume also engaged in struggle, then deeper learning occurred.

Kapur (2014) echoed the idea that struggling leads to greater understanding. He found that learners who engaged in problem solving and struggled with problems before being taught concepts demonstrated significantly greater conceptual understanding and the ability to transfer their understanding to novel problems compared to those who were taught the concept first and then attempted to solve problems. Kapur calls this productive failure, because the initial problem solving might result in errors, but in the end, it is more productive. I would argue that productive failure is an essential part of productive struggle. Part of struggling productively involves being able to make use of various strategies and abandon those strategies when they result in failure.

Learning and progress is made possible through these failures.

As discussed above, many learning theorists would argue that engaging in struggle is important and necessary to develop understanding of mathematics. These theories are supported by empirical studies. Not only can engaging in struggle help one learn, but K – 12 policymakers and employers would agree that problem solving and the ability to overcome struggle are important skills for students of all education levels to possess.

#### Policy Documents and NCTM Publications

** **NCTM’s (2014) statement that “Solving problems is not only a goal of learning mathematics, but also a major means of doing so,” makes it clear that problem solving is a skill that students should possess and that engaging in the process of problem solving is also beneficial. Several other policy documents and NCTM publications not only advocate for problem solving, but either specifically include the importance of struggle or imply it. For example, NCTM’s *Principles to Actions* specifically mentions that teachers should engage students in productive struggle (NCTM, 2014). The Common Core State Standards for Mathematics (CCSSM) implicitly includes the idea of struggle by stating that students should persevere through problems (NGA, 2010).

Productive struggle and problem solving are not equivalent constructs. They are related. Problem solving often involves encountering obstacles and in order to successfully problem solve, one must overcome these obstacles. Struggle occurs during the problem-solving process when difficulties arise. Those who are able to deal with struggle and find ways to progress past the struggle (i.e., struggle productively) will have more success with problem solving. If a publication advocates for problem solving, one might assume that it is implicitly advocating for productive struggle as well.

The CCSSM includes Standards for Mathematical Practice that span across grade levels and also specific grade level content standards. The Standards for Mathematical Practice are described as “varieties of expertise that mathematics educators at all levels should seek to develop in their students” (NGA, 2010, p. 6). The first mathematical practice is “Make sense of problems and persevere in solving them.” Students who engage in this practice are able to find an entry point to a problem. As they are working on a problem, they continually monitor and evaluate their progress. Central to persevering in problem solving is the ability to overcome struggles (i.e., struggle productively) when they occur. Part of this mathematical practice is that if necessary, students need to be able to change their course and consider an alternate approach (NGA, 2010).

Problem solving is also one of NCTM’s five process standards in its Principles and Standards document (NCTM, 2000). When describing this process, NCTM uses words and phrases such as “grapple with”, “effort”, “reflect”, and “persistence”, which are all important actions when encountering struggle. They add that while problem solving is important for those in mathematics, it is also a useful skill outside of mathematics. This is another indication of the conviction that students should have the opportunity to struggle with mathematics when problem solving.

Included in NCTM’s *Principles to Actions* document is a list of eight Mathematics Teaching Practices that promote deep learning of mathematics (NCTM, 2014). Two of these practices for effective teaching and learning are “Implement Tasks that Promote Reasoning and Problem Solving” and “Support Productive Struggle in Learning Mathematics”. The first implies and the second directly states when students are working on these problem-solving tasks, they should have the opportunity to struggle with mathematics (NCTM, 2014).

#### Future Employment

** **It is not surprising that the K – 12 documents in the previous section include problem solving and perseverance as important skills for students. These documents are created with college and career readiness in mind. Polls and other studies show that employers believe that problem solving is an important skill for college graduates to possess and part of problem solving involves being able to persevere when solving problems and facing obstacles.

In recent years, problem solving has continually been at the top of the list as far as attributes that employers look for. It was tied for third in 2016, second on the list in 2017, and tied for first in 2018. The National Association of Colleges and Employers (NACE) conducts a yearly job outlook survey to determine what employers look for in future employees. In 2017, 77% of employers responded on the survey that problem solving is an attribute that they would like to see on a future hire’s resume (NACE, 2016). This is up from 70% in 2016 (NACE, n.d.). And in 2018, problem solving tied for first as one of the most desired attributes, with 82.9% of respondents claiming that problem solving is a desired attribute on a candidate’s resume (NACE, 2017). One can assume that employers do not only want employees that engage in problem solving, but that they want employees to do so in a way that is successful or productive. In order to engage in successful problem solving, one needs to be able to preserve through struggles and obstacles that one encounters.

Beaton (2017) listed agility as one of the four skills top employers report that millennials need. Beaton defined agility as “the ability to overcome”. Possessing the skill of agility means having grit and being able to overcome setbacks and persevere. Similarly, in a 2013 TED Talk, Duckworth defined “grit” as “passion and perseverance for very long-term goals” and claimed it to be a significant predictor of success. Agility and grit are important traits for millennials to possess, yet Beaton states that these are things they struggle with. If something does not work out right away for millennials, they want to move on rather than search for alternate pathways to a solution. Subsequently, teachers should help students learn to be agile, or in other words, teach them how to engage in struggle and make that struggle productive. However, Duckworth pointed out that while we need to teach our students how to be “gritty” it is not yet clear how to do this. This study examines teaching strategies for how to make and keep struggle productive, which may provide some insight as to how we can teach our students how to be “gritty”.

### Difficulty Engaging Students in Productive Struggle

The call to engage students in struggle by learning theorists, policymakers, and future employers is clear and loud. However, that does not mean that it will be immediately apparent to teachers how to do so, which is made evident by many empirical studies.

Warshauer’s (2015a) study was focused specifically on productive struggle in the classroom. She explored the types of struggles that students engage in, and teacher responses to those struggles. In only 39 class sessions, there were 186 instances of student struggle while working on mathematical tasks. It is clear that struggle occurs in mathematics classrooms. And, this was only the number of times that the struggle was visible to the teacher or other students. The types of struggle ranged from not knowing how to start a problem, not knowing how to carry out a process, uncertainty in explaining and sense-making, or expressing misconceptions or errors. There were a variety of teacher responses to student struggles, which influenced whether or not these struggles were productive.

Warshauer (2015a) identified the struggle as productive if the interaction with the teacher “(1) maintained the intended goals and cognitive demand of the task; (2) supported students’ thinking by acknowledging effort and mathematical understanding and (3) enabled students to move forward in the task execution through student actions” (p. 390). Only 42% of the student struggles were resolved productively after interacting with the teacher. It is important to keep in mind that additional instances of struggle may have occurred yet were not apparent to the researcher. Similarly, it is possible that additional student struggles were resolved, but done so in such a way that was not apparent to the researcher. Warshauer’s (2015a) study illustrates that it might not be clear to the teacher how to make struggle productive.

Part of engaging students in productive struggle involves engaging students with challenging tasks that have a high level of cognitive demand. Tasks that have a high level of cognitive demand should be challenging for students. When working on challenging tasks that have a high level of cognitive demand, students will most likely encounter struggle. Henningsen and Stein (1997) sought to determine what factors caused the cognitive demand of tasks to decline during task implementation. Although their study was not directly about productive struggle, one can assume that if the cognitive demand of the task was no longer at a high level, then there was no longer an opportunity for students to struggle. At times, the cognitive demand of the task was reduced to purposely avoid student struggle.

Stein, Grover, and Henningsen (1996) determined that the most common factor responsible for the decline of cognitive demand was that the challenge of the problem was reduced, which resulted in less effort and thinking required by students. This challenge could be reduced by students pressuring the teacher to provide answers or the teacher willingly jumping in to complete parts of the problems for the student and consequently, rescuing the student from struggle.

Stein, Grover, and Henningsen’s (1996) study illustrates that teachers often like to rescue students from struggle. Both Stein, Grover, and Henningsen (1996) and Henningsen and Stein (1997) cite Doyle (1988) when they give a possible reason that teachers do this. The reason is that tasks with a high level of cognitive demand can be more ambiguous or risky for students and cause anxiety, so either students pressure teachers to provide assistance or teachers willingly step in as a way of alleviating some of that anxiety.

Reinhart (2000), a middle school mathematics teacher, echoed these same sentiments when reflecting on his early teaching and his experiences transitioning from a direct-instruction model to a more student-centered approach. Reinhart reported that he thought it was his job to explain things as well as possible for students to understand, but later realized that he needed to make students more responsible for their own learning. In order to do so, he did less for them, allowing them to struggle with the mathematics. Similarly, Schoenfeld (1988) describes a classroom from a yearlong study that appeared to have everything right. In some ways, the class was successful and in other ways, a failure. The classroom was well run, in terms of managing behavior and following the curriculum, and the students did well on standardized tests, yet students did not have a full understanding of the subject matter and developed harmful beliefs about mathematics. For example, some students developed the belief that mathematics problems should be solved quickly, and if the problems were unable to be solved quickly, then the problems were deemed impossible by those students. This type of belief impedes students from persevering and engaging in productive struggle. When discussing how teachers can support productive struggle, the authors of *Principles to Actions* (NCTM, 2014) add that when students struggle, sometimes teachers take this as an indication that they have failed their students and as a result, jump in to save them.

Stein, Grover, and Henningsen’s (1996), Reinhart’s (2000), and Schoenfeld’s (1988) studies all illustrate the idea that some teachers believe it is their job to make things easy for students and struggle should be avoided. This directly contrasts what is suggested by learning theorists. In addition, Warshauer’s (2015a) study showed that even when struggle is present, teachers might not know how to capitalize on that struggle and make it productive for students. The present study seeks to rectify this problem by examining what can be done by a teacher to engage students in struggle and keep that struggle productive for students.

### Suggestions for Engaging Students in Productive Struggle

When Hiebert and Grouws (2007) define struggling they include trying to figure out things that are not immediately apparent (Hiebert & Grouws, 2007). In other words, there is a certain amount of uncertainty involved when struggling. Uncertainty occurs when students are unsure about an answer to a problem or how to proceed with a problem.

Zaslavsky (2005) gives three types of uncertainty associated with mathematical tasks. The first is competing claims. This occurs when a student is confronted with a misconception or two conflicting claims during a task, which leads to uncertainty. The second type of uncertainty is unknown path or questionable conclusion, which will be written as unknown path/questionable conclusion for the sake of clarity in writing. This type of uncertainty involves open-ended problems where a solution is not immediately clear to the learner. The third type of uncertainty is non-readily verifiable outcome. This type of uncertainty involves solutions for which the learner does not currently have a way of checking a solution.

For the first two types of uncertainty, it would be useful for the learner to be able to explore the problem in order to resolve uncertainties. If one is trying to resolve uncertainty where competing claims is involved, then one could explore both of the claims to determine if one of the claims is valid or if one is invalid. For example, one could generate examples that would support or refute the claims. Zaslavsky (2005) defines unknown path/questionable conclusion as the kind of uncertainty associated with exploration since the learner must explore or search for findings all the while unsure of what those findings will be. For the third type of uncertainty, a learner does not have a way of checking a solution and exploring the problem further would not change that.

I propose using a dynamic geometry environment (DGE) as a tool for exploration. A DGE is computer software that allows one to construct geometric objects. These constructions mimic Euclidean constructions done with a straight edge and a compass. A defining feature of DGEs is the dynamic nature of the environment. Users have the ability to interact with and manipulate the geometry objects that they construct. They do so using a set of provided tools. One can drag elements of the geometric object and the object will update in real-time. When doing so, the underlying geometric properties of the object remain unchanged. Examples of

DGEs are Cabri, Geometer’s Sketchpad, and GeoGebra.

A DGE is an ideal tool for exploration because it allows students to make conjectures; do some checking; and then reject, modify, or confirm their conjecture. The exploration features, along with the instant feedback provided by the software, should be very useful when faced with uncertainty in a mathematical task. For example, students can use the DGE to generate examples and obtain feedback about those examples, such as measurements, in order to refute or confirm competing claims. They can also use a DGE to explore tasks with unknown paths. For example, in an existence task, students must determine if a certain case of something exists. This requires exploration and it is natural to use a DGE to do so, because students can check numerous cases in a short amount of time and be sure of the precision of their work.

In summary, one way to engage students in productive struggle would be to engage them in mathematical tasks that create uncertainty. In addition, a DGE would be a useful tool for navigating this uncertainty in order to progress through the task and struggle productively.

### Technology

Using technology, specifically DGEs, to engage students in productive struggle is an appealing solution because of the exploration and feedback features of a DGE. It is an appealing solution for two additional reasons. The first reason being that classrooms are becoming more and more equipped with various technology and yet, teachers are not entirely sure how best to use these tools to engage students in meaningful mathematics. The second reason is that policy documents and NCTM publications have called for technology to play an integral role in mathematics education. The next sections elaborate on the availability of technology and the inclusion of technology in policy documents and NCTM publications. I propose that teachers use this technology as a tool to engage students in productive struggle. One purpose of this study is to determine how teachers might do so.

#### Technology-Equipped Classrooms

** **I am currently teaching in a one-to-one computing environment. Each of my students is equipped with a laptop provided by the school. Since the technology is readily available, I have questioned if I can somehow find out ways to use these computers to help students engage in struggle and also to help them persevere through struggle. I believe one answer could be DGEs.

I believe figuring out ways to engage students in struggle using computers, specifically DGEs, would be of interest to other teachers as well, since many teachers also find themselves in one-to-one computing environments. Front Row Education surveyed over 2,500 teachers as part of their 2017 Technology in the Classroom Survey and over 50% of those teachers reported that they are in a one-to-one computing environment (Sharp, 2016).

Despite many teachers having these devices available, the presence of computers in the classroom is not enough to improve student learning. What matters is how teachers engage students in mathematics using this technology. When the Education Week Research Center analyzed the National Assessment of Educational Progress survey data, they discovered that despite the abundance of computers in the classroom, students are using these computers for rote practice as opposed to using these computers to engage in meaningful mathematics (Education Week, 2017). According to the National Assessment of Educational Progress survey data, in 2015, 49% of 4^{th} graders were using computers for practice or drill compared to only 15% using computers for making charts or graphs, which might require more critical thinking. Wachira and Kenngwe (2011) found that teachers reported a lack of pedagogical knowledge in using technology appropriately, and that this lack of knowledge is a barrier to integrating technology in their classrooms. Consequently, despite having the technology readily available, teachers may not be using the technology as often as they should because they do not know how to use it in such a way that will engage students in meaningful mathematics. This study would be of use to teachers, because it would add to the literature ways in which computers can be used to improve student learning by engaging students in struggle.

#### Policy Documents and NCTM Publications

** **Equipping schools with technology is likely a response to the emphasis of technology in many K–12 policy documents. For example, the CCSSM’s mathematical practice “Use appropriate tools strategically” involves students’ ability “to use technological tools to explore and deepen their understanding of concepts” (NGA, 2010, p. 7). In addition, NCTM has included Tools and Technology as one of its six Guiding Principles for School Mathematics in its *Principles to Actions* document. The principle states that, “An excellent mathematics program integrates the use of mathematical tools and technology as essential resources to help students learn and make sense of mathematical ideas, reason mathematically, and communicate their mathematical thinking” (NCTM, 2014, p. 78).

The documents and standards that advocate for the inclusion of technology in the classroom are the same documents and standards that advocated for problem solving and struggle. These documents recommend that both problem solving and technology be present in today’s classrooms. I propose that the two should supplement one another and that technology, specifically DGEs, could be used to support students’ engagement in problem solving and productive struggle.

### Research Problem

This chapter has defined productive struggle, argued for the importance of productive struggle, asserted that despite its importance, it may not be easy to engage students in productive struggle, and lastly, suggests using DGEs to engage students in productive struggle.

Despite learning theorists and policymakers telling us that students should learn by struggling through problems, I have found that my students do not like to do so. Simply put, students do not like struggle. They want to give up at the first sign of trouble. I am interested in how I can get my students to engage in struggle and persevere through that struggle, all the while keeping the struggle productive. Secondarily, since I am in a one-to-one computing environment, I am curious how I can use computers to engage students in struggle and make that struggle productive.

Existing literature provides insights and tentative answers to these questions. Mathematical tasks can be used to create uncertainty, which could engage students in struggle and DGEs can be a useful tool for navigating uncertainty, which can help students make their struggle productive. In Jones’s (2002) review of the DGE literature, a key message was that despite all of the affordances that a DGE can offer, the software alone will not lead to student gains. It has the potential to lead to student gains, but there are many factors that influence whether or not students will learn geometry using a DGE. One of these factors is the type of tasks given to students by teachers and also how teachers implement these tasks. Healy and Hoyles (2001), among others, stress that a DGE alone is not enough and that tasks and the teacher are important.

The question remains, how does one design DGE tasks that create uncertainty and support productive struggle? And what should a teacher do during implementation of these tasks to ensure that struggle is productive for students and it remains productive for the duration of the task? This brings us to the research question for this study:

Do dynamic geometry environment (DGE) tasks designed to create uncertainty using competing claims and unknown paths/questionable conclusions lead to uncertainty for secondary mathematics students and what is it about the design and implementation of these types of tasks that supports or hinders students’ engagement in productive struggle?