**CHARACTERIZING THE NATURE OF STUDENTS’ FEATURE NOTICING ****AND USING WITH RESPECT TO MATHEMATICAL SYMBOLS ACROSS **DIFFERENT LEVELS OF ALGEBRA EXPOSURE

# ABSTRACT

**SULLIVAN, PATRICK. **Characterizing the Nature of Students’ Feature Noticing-andUsing With Respect to Mathematical Symbols Across Different Levels of Algebra Exposure. (Under the direction of Dr. M. Kathleen Heid.)

The purpose of this study is to examine the nature of what students notice about symbols and use as they solve unfamiliar algebra problems based on familiar algebra concepts and involving symbolic inscriptions. The researcher conducted a study of students at three levels of algebra exposure: (a) students enrolled in a high school precalculus course, (b) college students enrolled in a second semester calculus course, and (c) prospective secondary mathematics teachers enrolled in a mathematics teaching methods course and who have completed three semesters of calculus, linear algebra, an introduction to proof and several upper level mathematics courses.

Six students from each level of algebra exposure were asked to reason about a series of novel algebra problems that involved symbolic inscriptions and content typical of a second-year algebra course. Data were analyzed for instances of recognizing, reasoning, and linking. One of the outcomes of the research was the development of a feature noticing-and-using taxonomy. The researcher found that students’ feature noticing-and-using was characterized by three different strategies: manipulative, relational, and linking. Most students reasoned from a manipulative strategy, but it was found that students faced challenges reasoning from each of these strategies. Students at the highest level of algebra exposure were much more likely than the other two levels of algebra exposure to use multiple strategies in their reasoning.

# TABLE OF CONTENTS

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

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

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

Chapter 1 STATEMENT OF THE PROBLEM ………………………………………………… 1

Background …………………………………………………………………………………………….. 1

Importance of Symbol Sense …………………………………………………………………….. 2

Research on Symbol Sense ……………………………………………………………………….. 2

Symbol Sense Frameworks……………………………………………………………………….. 3

Problem Statement …………………………………………………………………………………… 5

Symbol Familiarity ……………………………………………………………………………. 6

Levels of Algebra Exposure ……………………………………………………………….. 7

Purpose Statement …………………………………………………………………………………… 9

Research Questions ………………………………………………………………………………….. 11

Summary ………………………………………………………………………………………………… 12

Chapter 2 FEATURE NOTICING-AND-USING ……………………………………………… 13

Feature Noticing-and-Using Taxonomy ……………………………………………………… 13

Key Terms ……………………………………………………………………………………………… 13

Symbolic Capacity …………………………………………………………………………………… 14

Dimensions of Symbolic Capacity …………………………………………………………….. 15

Feature Noticing-and-Using ……………………………………………………………………… 16

Components of Feature Noticing-and-Using ……………………………………………….. 17

Recognizing ……………………………………………………………………………………… 18

Reasoning ………………………………………………………………………………………… 18

Linking ……………………………………………………………………………………………. 19

Relationship Between Symbolic Capacity and Feature Noticing-and-Using ……. 20 Summary ………………………………………………………………………………………………… 24

Chapter 3 LITERATURE REVIEW …………………………………………………………………. 25

Reasoning about Symbols…………………………………………………………………………. 25

Defining Symbol Sense ……………………………………………………………………………. 26

Characterizing Symbol Sense ……………………………………………………………………. 26

Existing Symbol Sense Frameworks ………………………………………………………….. 27

Noticing …………………………………………………………………………………………………. 30

Expertise ………………………………………………………………………………………………… 35

Meaning of Symbols ………………………………………………………………………………… 36 Recognizing ………………………………………………………………………………………37

Reasoning ………………………………………………………………………………………… 38

Meaning of Equations/Inequalities …………………………………………………………….. 40

Process and Object …………………………………………………………………………………… 41

Meaning of Symbols Linked to Other Representations …………………………………. 45 Summary ………………………………………………………………………………………………… 48

Chapter 4 METHOD ……………………………………………………………………………………… 50

Justification for Methodology……………………………………………………………………. 50

Choices of Levels of Algebra Exposure ……………………………………………….. 51

Recruitment ……………………………………………………………………………………… 52

Selection of Students …………………………………………………………………………. 52

Pilot Study ……………………………………………………………………………………………… 53

Choice of Interview Tasks ………………………………………………………………………… 54

Task Novelty …………………………………………………………………………………………… 55

Interview Tasks ………………………………………………………………………………… 56

Feature Noticing-and-Using Potential ………………………………………………….. 57

Expert Panel ……………………………………………………………………………………… 58

Task Decision-Making ………………………………………………………………………. 59

Data Collection ……………………………………………………………………………………….. 61

Human Subjects Compliance …………………………………………………………………….. 62

Data Analysis ………………………………………………………………………………………….. 63

Qualitative Research Standards …………………………………………………………………. 68

Quality and Verification …………………………………………………………………….. 68

Subjectivity Statement ……………………………………………………………………….. 69

Ethical Concerns ……………………………………………………………………………….. 71

Chapter 5 RESULTS……………………………………………………………………………………… 73

Nature of Feature Noticing-and-Using ……………………………………………………….. 75

Claim 1: Different Reasoning Strategies ………………………………………………. 75

Manipulative Strategy …………………………………………………………………. 77

Recognizing Features That Cue Procedures …………………………….. 77

Recognizing Conditions in Which Procedures Can Applied ………. 81

Relational Strategy ……………………………………………………………………… 83

Same Truth Sets …………………………………………………………………… 86

Equivalent Expressions …………………………………………………………. 87

Relationships Between Numbers ……………………………………………. 88

Linking Strategy …………………………………………………………………………. 90

Links from Results of Procedure to Graphical Representation ……. 91

Links from Symbolic Inscription to Graphical Representation …… 93

Links from Graphical Representation to Symbolic Inscription …… 96

Claim 2: Challenges of Manipulation Strategy ……………………………………… 101

Sub-Claim 2a: Lack of Attention to Procedural Conditions …………………….101

Sub-Claim 2b: Results Do Not Meet Expectations ………………………………… 104

Sub-Claim 2c: Lack of Attention to Structural Conditions ……………………… 108

Sub-Claim 2d: Lack of Attention to Mathematical Conditions ………………… 111

Claim 3: Purposeful Movement Between Strategies and Reasoning ………… 121

Attending to Gaps in Reasoning ……………………………………………………. 122

Confirm Reasoning …………………………………………………………………….. 125

Compensating for Errors in Reasoning ………………………………………….. 128

Claim 4: Linking Strategy and Productive Links …………………………………… 131

Instances of Productive Links ………………………………………………………. 132

Lack of Coordination of Meaning …………………………………………………. 136

Source Register to Target Register …………………………………………. 136

Target Register to Source Register …………………………………………. 139

Claim 5: Nature of Meaning and Reasoning from Strategies ………………….. 141

Different Meanings within a Relational Strategy………………………. 142

Different Meanings within a Linking Strategy …………………………. 146

Connected Meaning to Other Links ………………………………………… 151

Claim 6: Forms of Inscriptions and Revealing Features ………………………… 153

Claim 7: Prominent Features are Not Recognized ………………………………… 153

Findings Across Levels of Algebra Exposure ……………………………………………… 156

Claim 1: Limitations in Reasoning of Students with Less Algebra

Exposure ……………………………………………………………………………………. 157

Claim 2: Reasoning of Students with More Exposure ……………………………. 159

Claim 3: Impact of Current Mathematical Experience ……………………………. 162

Claim 4: Purposeful Movement and Students with Highest Exposure ……… 165

Adjustments to Feature Noticing-and-Using Taxonomy ……………………………….. 167

Rationale for Adjustments to Feature noticing-and-using Taxonomy ………. 168

Change in Instance Codes ………………………………………………………………….. 169

Addition of Venn Diagram Structure to Taxonomy ……………………………….. 171

Newt’s Feature Noticing-and-Using Venn Diagram of Task 3 ………………… 173 Summary ………………………………………………………………………………………………… 177

Chapter 6 DISCUSSION ……………………………………………………………………………….. 179

Summary and Discussion of Research Findings ………………………………………….. 179

Nature of Feature Noticing-and-Using ……………………………………………………….. 179

Discussion of Research Question 1 ……………………………………………………… 181

Manipulative Strategy …………………………………………………………………. 181

Relational Strategy ……………………………………………………………………… 185

Linking Strategy …………………………………………………………………………. 189

Feature Noticing-and-Using Across Levels of Algebra Exposure ………………….. 194

Adjustments of Feature Noticing-and-Using Taxonomy ………………………………. 198

Expanding Current Research …………………………………………………………………….. 199 Conclusions……………………………………………………………………………………………..200

Implications for Teaching …………………………………………………………………… 200

Manipulative Strategy …………………………………………………………………. 200

Relational Strategy ……………………………………………………………………… 202

Linking Strategy …………………………………………………………………………. 203

Suggestions for Future Research …………………………………………………………. 204

Limitations of the Study …………………………………………………………………………… 205

Summary ………………………………………………………………………………………………… 206

References …………………………………………………………………………………………………….. 208

Appendix A—High School Recruitment Script …………………………………………………. 217

Appendix B—College Recruitment Script ………………………………………………………… 219

Appendix C—Letter to Parents ………………………………………………………………………… 221

Appendix D—Informed Consent (Under 18) …………………………………………………….. 223

Appendix E—Informed Consent (Over 18) ……………………………………………………….. 227

Appendix F—Panel Task Evaluation………………………………………………………………… 231

Appendix G—Interview Schedule ……………………………………………………………………. 237

Appendix H—Task Summaries ……………………………………………………………………….. 242

Appendix I—Student Task Completion…………………………………………………………….. 306

Appendix J—Analysis of Newt’s Reasoning …………………………………………………….. 307

Appendix K—Task 3 Narrative ……………………………………………………………………….. 338

**Chapter 1 STATEMENT OF THE PROBLEM**

Many of the challenges of learning algebra are related to the symbols that give meaning to the subject. The activity of algebra involves generating, transforming, and utilizing strings of algebraic symbols, known commonly as symbolic representations. Being competent in algebra requires one to interact with symbolic representations in different ways. Not only must one be able to see symbolic representations as structured strings of symbols, objects in their own right, but they may also be seen as descriptors connecting some reality or situation familiar to the individual (Pimm, 1995). The complexity of these interpretations of symbolic representations is of great difficulty for those learning algebra (Rubenstein & Thompson, 2001). Choosing the appropriate interpretations or meanings of a symbolic representation that assist the student in solving a particular problem only adds to this complexity. This ability to discern different meanings and make interpretations of symbolic representations that are helpful in solving problems has been described in the research literature as *symbol sense*.

# Background

What follows is an argument for the importance of a study that examined the nature of students’ reasoning about symbols. This argument also involves making a case for the need for a taxonomy to classify the nature of this reasoning.

# Importance of Symbol Sense

Symbol sense is considered by some the heart of algebraic competency (Arcavi, 1994). Symbol sense involves the core aspects of algebra, symbolizing generalizations and syntactically guided reasoning about generalizations expressed in conventional symbol systems (Kaput, Blanton, & Moreno, 2008). Picciotto and Wah (1993) suggested that symbol sense is the true prerequisite for further work in mathematics and science and should be the primary purpose of algebra.

# Research on Symbol Sense

Over the past two decades headway has been made in better understanding some of the challenges students have in understanding the meaning of symbolic representations (MacGregor & Stacey, 1997). While much of the research focus has been on early- or beginning-algebra students’ efforts to use symbolic representations to express relationships (Carraher & Schliemann, 2007; Kieran, 2007; Lee, 1996) there is a growing body of research focused on articulating and describing symbol sense.

Several have described symbol sense in general terms (Arcavi, 1994; Arzarello & Robutti, 2010; Keller, 1993; Kinzel, 2001; Zorn, 2002), whereas others have proposed a set of characteristics describing specific elements of symbol sense (Arcavi, 1994; Fey, 1990). Fey (1990), from a multiple representation perspective, characterized symbol sense as the ability to (a) scan an algebraic expression to make rough estimates of the patterns that emerge in numeric or graphic representations, (b) make informed comparisons of magnitudes of functions, (c) scan a table of function values or a graph to interpret verbally stated condition to identify the likely form of an algebraic rule that expresses the appropriate pattern, (d) inspect algebraic operations and predict the form of the result, or (e) determine which of several equivalent forms might be most appropriate.

Arcavi (1994) characterized symbol sense in ways that were helpful in thinking about aspects of this study. He stated symbol sense includes an (a) understanding of how and when symbols can be and should be used in order to display relationships, (b) ability to abandon symbols in favor of other approaches in order to make progress in solving a problem, (c) ability to manipulate and to “read” symbolic expression as complementary aspects of solving algebraic problems, (d) awareness that one can engineer symbolic relationships that express the verbal or graphical information needed to make progress in solving a problem, or (e) ability to select a possible symbolic representation of a problem.

Other researchers have focused on students’ understanding of and difficulty with particular mathematical entities and ideas that are connected to aspects of students’ symbol sense abilities. The current study examined students’ reasoning about different elements of students’ symbol sense in the context of problems that involved symbolic inscriptions. Elements of symbol sense discussed in this study include structure (Hoch & Dreyfus, 2004; Menghini, 1994; Pomerantsev & Korosteleva, 2003; Vaiyavutjamai,

Ellerton, & Clements, 2005), linking representations (Knuth, 2000; Pierce, 2001; Sfard &

Linchevski, 1994) , equivalence (Knuth, Stephens, McNeil, & Alibali, 2006; Linchevski & Herscovics, 1996) and meanings of letters (e.g., unknowns, variables, and parameters)

(Bloedy-Vinner, 1994; Furinghetti & Paola, 1994).

# Symbol Sense Frameworks

These studies shed light on the aspects of symbol sense that are challenging for students as they reason about symbols. These studies also suggest that there are many complexities as students reason about symbols. Krutetskii (1976) argued that any genuinely scientific approach to the study of a complex phenomenon requires an analysis of its structure and an isolation of its components. Understanding the nature of students’ symbol sense, reasoning about symbols, is complex, and there is a need for organizational structures such as frameworks and taxonomies to examine the nature of students’ reasoning about symbols and understand what this reasoning entails.

In more recent studies (Kenney, 2008; Pierce & Stacey, 2001), using elements of symbol sense as identified by Arcavi (1994), researchers have created frameworks that have been used to analyze the structure and isolate the components of students’ symbol sense capabilities. Pierce and Stacey (Pierce, 2001; Pierce & Stacey, 2001) created a framework, Algebraic Insight, that characterizes elements of symbol sense in a computer algebra system (CAS) environment. Algebraic Insight, as described by Pierce and Stacey (2001), is the algebraic knowledge necessary for correctly entering expressions in a CAS, efficiently scanning for possible errors, and interpreting the output as conventional mathematics. The limitation of Pierce and Stacey’s algebraic insight framework as a framework for describing students’ reasoning about symbols is that it is designed to apply only to elements of symbol sense for which the CAS is helpful, the stage of solving a formulated problem. It does not provide a means to describe the activity in other stages of problem solving, such as formulating the problem and interpreting the solution.

Kenney (2008), in her dissertation study, expanded the work of Pierce and Stacey by incorporating their framework into a framework for identifying students’ uses and understandings of symbolic structures in other stages of problem solving. While

Kenney’s framework does seem to provide an organization framework to study symbol sense, it has a few limitations. First, her framework does not address the back-and-forth movement between representations that seems to be characteristic of students’ reasoning about symbols. Her framework categorizes reasoning from symbolic representations to graphical representations in broad terms, linking symbolic and graphical representations, without consideration for the direction of the link and the importance of this directionality. Second, findings from her study suggests that while her framework was helpful in categorizing aspects of symbol sense, it did not provide a lens to examine some of the challenges in students’ reasoning about symbols that she saw in her data. Particular areas of difficulties that she mentioned include linking different representations, reasoning about symbol meaning in the context of the problem, and understanding the objects represented by the symbols (p. 302). The current body of research seems to need a tool to classify the nature of students’ reasoning about symbols with an eye toward the difficulties that Kenney. Developing a tool that addresses this issue seems to be an important contribution to the body of research in the area of symbol sense.

# Problem Statement

The problem(s) addressed in this study related to the nature of students’ reasoning about symbolic representations in the context of solving problems presented using symbolic inscriptions. A *symbolic inscription* is defined as a symbol string. This study will account for several important aspects related to symbol sense that have not been addressed in prior studies. In particular, it will (a) account for the application of symbol sense to unfamiliar algebra problems based on familiar algebra concepts and involving symbolic inscriptions, (b) examine students’ feature noticing-and-using about symbols across different levels of algebra exposure, and (c) describe the nature of students’ feature noticing-and-using about symbols in a range of problem settings.

# Symbol Familiarity

An assumption made in this research study is that feature noticing-and-using about symbols develops over time. It may take a significant amount of time for students to become comfortable enough with algebraic forms/notations to extract meaningful information from them (Arcavi, 1994). Likewise, Gray and Tall (1994) argue that it takes time working with new content for students to step back and reason about the symbolic representation in a conceptual manner. In the two prior studies (Kenney, 2008; Pierce & Stacey, 2001) that attempted to characterize symbol sense using a framework, the study participants were interviewed on content related to the class they were taking, calculus and precalculus, respectively.

Kenney’s study involved students enrolled in a college precalculus class who were asked in an interview setting to reason about problems that involved precalculus content. This may have been too early to examine symbol sense capacities because the students may have been unfamiliar with what the symbolic representation revealed. For example, Kenney asked interviewees to reason about the following task, solve for *x: *

. Results from her study suggest students had difficulty attending to the

denominator of the rational expression to solve the problem (p. 71). One possible hypothesis regarding their difficulty is that solving rational equations was recently taught to students so it may have been too early for students to step back and engage the symbolic representation in a more conceptual manner, which would suggest a higher degree of symbol sense.

The current research will be able to contribute to the field by examining students’ feature noticing-and-using, an aspect of symbol sense, on unfamiliar algebra problems involving symbolic inscriptions in which the algebra content is familiar to them. Students participating in this study will be given tasks involving content typical of a second-year high school algebra course, a level of content with which all students in the study are expected to be familiar, but the tasks themselves are designed to challenge students to reason beyond simply performing the steps to a procedure.

# Levels of Algebra Exposure

Another issue to consider is the study population. There is evidence within the literature to suggest that students across a range of levels of algebra exposure face challenges when reasoning about symbols. Specific levels of algebra exposure described within the literature include advanced high school (Hoch & Dreyfus, 2004), undergraduate mathematics students (Crowley, 2000; Kenney, 2008; Pomerantsev &

Korosteleva, 2003), and prospective mathematics teachers (Pomerantsev & Korosteleva,

2003; Vaiyavutjamai et al., 2005). It seems important to examine the nature of students’ feature noticing-and-using from symbols across levels of algebra exposure in order to better understand the nature of and challenges in feature noticing-and-using both within a level of algebra exposure and across levels of algebra exposure.

Different levels of algebra exposure were chosen because one would expect that students’ exposure to algebra would have some effect on their feature noticing-and-using. Precalculus students were chosen as the base level because it would be expected that these students would be comfortable reasoning about problems involving content typical of a second-year algebra course. Of all three levels of exposure their experience with second-year algebra course content would have been most recent. Calculus II students were chosen because of their assumed increased exposure working with algebraic symbols. One could argue that the Calculus II content involves more extensive symbol manipulating and graphical analysis (Stewart, 2008), and requires applications of symbolic manipulations in applied settings. The last level of exposure, prospective secondary mathematics teachers, were chosen not only because of their different exposure, but because they were prospective secondary mathematics teachers. These experiences include participating in at least 3 semesters of calculus, a linear algebra course, a proof course, two mathematics teaching methods courses, and possibly practice in teaching secondary mathematics classes. Each of these courses requires students to make mathematical arguments involving algebraic symbols.

Each level of algebra exposure represents an increased level of algebra exposure as well as a higher degree of mathematical importance with respect to career goals. The highest degree of importance lies with prospective secondary mathematics teachers. They have chosen a career that involves mathematics and are being trained to help others learn mathematics. It could be argued that this level of student represents those with a perceived high level of expertise who have at the very least chosen a career path that involves mathematics on a regular basis. In other words, mathematics is integral to their career choice. Prospective secondary mathematics teachers also, as part of their mathematics exposure, participated in methods courses in which there was presumed explicit attention given to multiple representations (including symbolic representations).

At the next level of importance, Calculus II students, there is a reasonable expectation that they have chosen a field of study that requires at least two semesters of calculus. It cannot be discerned that, unlike prospective mathematics teachers, their chosen career path will involve mathematics. The precalculus level represents the level where the importance of mathematics to the student could be the least. Many precalculus students take additional courses in mathematics while many others do not. Since the purpose of this study was to examine the nature of students’ feature noticing-and-using, the goal was to examine the reasoning of students at levels that represent a range of exposures to algebraic symbols.

# Purpose Statement

It is the intent of this study to examine the nature of students’ reasoning about symbols, an aspect of symbol sense, as they solve problems involving symbolic inscriptions. Recall that a *symbolic inscription* is defined as a symbol string. The nature of students’ reasoning was examined in fine-grained detail by identifying specific features of symbolic inscriptions noticed by students and how those features were used using a taxonomy designed by the researcher called *feature noticing-and-using *(this will be discussed in Chapter 2)*. Feature noticing-and-using*, a term defined by the researcher, is the action triggered by noticing a feature of a symbolic inscription and using the feature to reason about a problem involving symbolic inscriptions. A *feature *is a form, characteristic, or structure of a symbolic inscription or an object or relationship represented by the symbolic inscription. The feature is said to be “noticed” when it is the focus of the student’s attention, as inferred through verbal statements and/or written work, when he or she attends to the symbolic inscription during the solving process. In summary, this study not only examined the features noticed by students as they solve problems that involve symbolic inscriptions, but it also examined how they use these features in their reasoning and the meaning they attached to these features as they used them to solve problems.

This examination of feature noticing-and-using was conducted with students across different levels of algebra exposure. Specifically, the purpose of this dissertation was to answer questions—via analysis of semi-structured task-based interviews—about the nature of students’ feature noticing-and using as students solve problems involving symbolic inscriptions. This analysis of students’ feature noticing-and-using occurred across three levels of algebra exposure with the goal of determining the nature of similarities and differences in feature noticing-and-using across these levels of exposure.

*Level of algebra exposure* is defined as the nature of students’ mathematical experiences beyond a second- year course in algebra. Different levels of algebra exposure were attained through involvement of students from the following three groups: (a) students enrolled in a high school precalculus course, (b) college students enrolled in a secondsemester calculus course, and (c) prospective secondary mathematics teachers who had taken several mathematics courses beyond three semesters of calculus and were enrolled in a secondary mathematics teaching methods course.

It was informative to the field of mathematics education to examine feature noticing-and-using across levels of algebra exposure because it was expected that this would look different as students gain more exposure. In addition to the fact that these three specific levels are described in the research as levels at which symbol sense difficulties have been identified, there are two other reasons it seemed important to examine the nature of feature noticing-and-using across levels of algebra exposure. First, to consider whether additional exposure to reasoning about symbolic inscriptions in the context of a calculus courses would impact the nature of students’ feature noticing-andusing as compared to those students with less exposure. Second, to consider whether students’ feature noticing-and-using at a particular level of exposure in which the career choice involves mathematics and the exposure to symbolic inscriptions is more extensive is different than those who have less exposure and who have not made the choice to teach secondary mathematics.

The current research contributes to the field by examining, in detail, the nature of students’ feature noticing-and-using across different levels of algebra exposure. The questions addressed in this study are:

- What is the nature of students’ feature noticing-and-using as they solve unfamiliar algebra problems based on familiar algebra concepts and involving symbolic inscriptions?
- Across levels of algebra exposure what is the nature of the similarities and differences in students’ feature noticing-and-using as they solve unfamiliar algebra problems based on familiar algebra concepts involving symbolic entities?
- What is a taxonomy that describes the nature of feature noticing-and-using as evidenced in students’ reasoning about symbolically presented unfamiliar algebra

problems that are based on familiar algebra concepts?

The analysis of students’ feature noticing-and-using focused on what features of the symbolic inscription students notice as they solve problems, how they used those features, and the meaning they attach to the features.

# Summary

It has been posited that students’ feature noticing-and-using with respect to symbolic inscriptions, an aspect of symbol sense as claimed by the researcher, is an important issue and that there are many challenges students face in reasoning about features of symbolic inscriptions. Although the research community has identified specific features in which difficulties may lie, there have been few studies that have examined the nature of students’ feature noticing-and-using and the meaning they attach to these features of symbolic inscriptions. Part of this difficulty has been due to the lack of an organizational structure, or taxonomy, that captures the different meanings students associate with features of symbolic inscriptions. This study addressed these issues by providing a taxonomy that captured the nature of students’ feature noticing-and-using with respect to symbolic inscriptions as well as how this may look similar or different across levels of algebraic exposure.