The Effect of Digital Visualization Tools on Understanding of Non-Linear Functions in Advanced Algebra: A Mixed Methods Study in Georgia Schools, USA

This mixed methods study investigated the impact of digital visualization tools (Desmos and GeoGebra) on high school students’ understanding of non-linear functions in advanced algebra. A sample of 120 students from three high schools participated in a quasi-experimental design with pre- and post-assessments, supplemented by qualitative interviews. Results indicated that students using digital visualization tools demonstrated significantly higher achievement in understanding quadratic (p < 0.001), exponential (p < 0.001), and rational function transformations (p < 0.001) compared to traditional instruction methods. Qualitative findings revealed enhanced conceptual understanding, improved visualization skills, and increased engagement with mathematical concepts. These findings suggest that digital visualization tools are effective pedagogical instruments for teaching non-linear functions in advanced algebra curricula.

Introduction

The integration of technology in mathematics education has transformed traditional pedagogical approaches, particularly in the teaching of advanced algebraic concepts. Non-linear functions, including quadratic, exponential, and rational functions, present significant conceptual challenges for students transitioning from linear to more complex mathematical relationships (Kieran, 2007). Traditional paper-and-pencil methods often fail to provide students with the dynamic visualization necessary to comprehend function transformations and behavioral patterns effectively.

Digital visualization tools such as Desmos and GeoGebra have emerged as powerful educational resources that enable students to manipulate mathematical objects in real-time, observe immediate graphical feedback, and develop deeper conceptual understanding through interactive exploration (Hohenwarter & Lavicza, 2007). These tools address the abstract nature of mathematical concepts by providing concrete visual representations that bridge the gap between symbolic manipulation and conceptual understanding.

Research in mathematics education has consistently emphasized the importance of multiple representations in developing mathematical understanding (Duval, 2006). Digital visualization tools facilitate the seamless transition between algebraic, graphical, and numerical representations, allowing students to observe how changes in function parameters affect graphical behavior instantaneously. This dynamic interaction is particularly crucial for understanding non-linear functions, where parameter changes result in complex transformations that are difficult to visualize through static methods.

The theoretical framework for this study draws from constructivist learning theory, which posits that learners actively construct knowledge through interaction with their environment (Piaget, 1952). Digital visualization tools provide an interactive environment where students can experiment with mathematical concepts, test hypotheses, and construct understanding through guided discovery. Additionally, the study incorporates principles from the Technology Pedagogy Content Knowledge (TPACK) framework, which emphasizes the intersection of technology, pedagogy, and content knowledge in effective teaching practices (Mishra & Koehler, 2006).

Problem Statement

Despite the widespread availability of digital visualization tools in mathematics education, many high school students continue to struggle with understanding non-linear functions in advanced algebra courses. Traditional instructional methods that rely primarily on static representations and algebraic manipulation often fail to develop students’ conceptual understanding of how parameter changes affect the behavior and transformations of quadratic, exponential, and rational functions (Zaslavsky, 1997). This disconnect between procedural knowledge and conceptual understanding has been identified as a persistent challenge in mathematics education, with students frequently demonstrating the ability to perform algebraic operations without truly comprehending the underlying mathematical relationships (Hiebert & Lefevre, 1986).

The problem is particularly acute in the context of non-linear functions, where the abstract nature of concepts such as vertex transformations, exponential growth patterns, and asymptotic behavior requires sophisticated visualization skills that are difficult to develop through traditional paper-and-pencil methods. Research indicates that students who lack adequate visualization skills for non-linear functions experience significant difficulties in advanced mathematics courses and may develop negative attitudes toward mathematical learning (Confrey & Smith, 1995). Furthermore, the increasing integration of technology in educational settings has created a need for empirical evidence regarding the effectiveness of digital visualization tools in addressing these conceptual challenges.

While anecdotal evidence suggests that digital visualization tools such as Desmos and GeoGebra may enhance student understanding of non-linear functions, there is a lack of rigorous empirical research examining their specific impact on student achievement and conceptual understanding in the context of advanced algebra. This gap in the research literature presents a significant problem for educators, administrators, and policymakers who must make informed decisions about technology integration and resource allocation in mathematics education.

Purpose Statement

The purpose of this mixed methods study is to investigate the effect of digital visualization tools (specifically Desmos and GeoGebra) on high school students’ understanding of non-linear functions in advanced algebra courses. This study aims to examine both the quantitative impact on student achievement and the qualitative aspects of student experiences when using these technological tools for learning quadratic, exponential, and rational functions.

Specifically, this research seeks to determine whether students who receive instruction incorporating digital visualization tools demonstrate significantly greater improvement in their understanding of non-linear function concepts, transformations, and behavioral characteristics compared to students receiving traditional instruction methods. The study will measure student achievement through comprehensive assessments of conceptual understanding and procedural knowledge, while also exploring student perceptions, attitudes, and experiences through qualitative data collection methods.

The intended outcomes of this research include: (a) providing empirical evidence regarding the effectiveness of digital visualization tools for teaching non-linear functions, (b) identifying specific aspects of student understanding that are most significantly impacted by technology integration, (c) exploring student perspectives on the use of digital tools in mathematics learning, and (d) informing educational practice regarding optimal implementation strategies for technology-enhanced mathematics instruction.

This study addresses a critical need in mathematics education by providing rigorous empirical evidence to guide decision-making regarding technology integration in advanced algebra curricula. The findings will contribute to the broader understanding of how digital visualization tools can be effectively utilized to enhance student learning outcomes and address persistent challenges in mathematics education.

Literature Review

Digital Visualization Tools in Mathematics Education

The effectiveness of digital visualization tools in mathematics education has been extensively documented in recent literature. Zenginet al. (2012) conducted a comprehensive study on the impact of GeoGebra on students’ understanding of function concepts, finding significant improvements in conceptual understanding and problem-solving abilities. Similarly, Roschelleet al. (2010) demonstrated that technology-enhanced mathematics instruction resulted in substantial learning gains compared to traditional methods.

Research specifically focusing on function understanding has shown that digital tools facilitate the development of function sense, defined as the ability to think flexibly about functions and their representations (Carlsonet al., 2002). Students using digital visualization tools demonstrate improved ability to connect algebraic expressions with their graphical representations, a crucial skill for understanding non-linear functions.

Non-Linear Functions in Advanced Algebra

Non-linear functions present unique pedagogical challenges due to their complex behavioral patterns and transformation properties. Quadratic functions, characterized by their parabolic shape and vertex form transformations, require students to understand how parameter changes affect graph positioning and orientation (Zaslavsky, 1997). Exponential functions introduce concepts of growth and decay that are often counterintuitive to students accustomed to linear relationships (Confrey & Smith, 1995). Rational functions, with their asymptotic behavior and discontinuities, represent the most complex category of non-linear functions typically encountered in advanced algebra.

Traditional instruction methods often emphasize algebraic manipulation at the expense of conceptual understanding, leading to procedural knowledge without deep comprehension of function behavior (Hiebert & Lefevre, 1986). Digital visualization tools address this limitation by providing immediate visual feedback that connects algebraic operations with graphical consequences.

Technology Integration and Student Achievement

Meta-analyses of technology integration in mathematics education have consistently shown positive effects on student achievement. Cheung and Slavin (2013) analyzed 74 studies and found that technology integration resulted in moderate to large effect sizes for mathematics achievement. Specifically, studies focusing on visualization tools showed the strongest positive effects, particularly for topics involving graphical representations.

The effectiveness of technology integration depends on several factors, including teacher preparation, implementation quality, and alignment with curriculum objectives (Ertmer & Ottenbreit-Leftwich, 2010). Research indicates that successful technology integration requires careful planning, appropriate teacher training, and ongoing support to maximize educational benefits.

Research Questions

This study addresses the following research questions:

1. Primary Research Question: To what extent do digital visualization tools (Desmos and GeoGebra) improve high school students’ understanding of non-linear functions (quadratic, exponential, and rational) compared to traditional instruction methods?

2. Secondary Research Questions:

∘ How do digital visualization tools affect students’ ability to identify and describe function transformations?

∘ What are students’ perceptions of using digital visualization tools for learning non-linear functions?

∘ How do achievement gains vary across different types of non-linear functions?

Research Methodology

Research Design

This study employed a mixed methods approach combining quantitative and qualitative data collection and analysis. The quantitative component utilized a quasi-experimental design with pre- and post-assessments to measure changes in student understanding. The qualitative component included semi-structured interviews and classroom observations to explore student experiences and perceptions.

Participants

The study sample consisted of 120 students enrolled in advanced algebra courses at three suburban high schools in the Midwest United States. Participants were in grades 9–11 (ages 14–17) and represented diverse socioeconomic and ethnic backgrounds. Schools were selected through convenience sampling based on administrative approval and teacher willingness to participate.

Experimental Group (n = 60): Students receiving instruction using digital visualization tools (Desmos and GeoGebra) integrated into regular curriculum activities.

Control Group (n = 60): Students receiving traditional instruction methods including textbook exercises, paper-and-pencil graphing, and teacher-led demonstrations.

Instrumentation

Non-Linear Functions Assessment (NLFA)

A researcher-developed instrument consisting of 30 items assessing understanding of quadratic, exponential, and rational functions. Items were designed to measure both procedural knowledge and conceptual understanding, including function identification, transformation recognition, and behavioral analysis. The instrument demonstrated acceptable reliability (Cronbach’s α = 0.85) through pilot testing.

Technology Perception Survey (TPS)

A 20-item Likert-scale instrument measuring student attitudes toward technology use in mathematics learning. Items addressed perceived usefulness, ease of use, and impact on learning outcomes.

Semi-structured Interview Protocol

A standardized protocol exploring student experiences with digital visualization tools, including perceived benefits, challenges, and impact on understanding.

Procedures

The study was conducted over an eight-week period during the spring semester. Both groups received equivalent instructional time (40 hours) covering quadratic, exponential, and rational functions. The experimental group used Desmos and GeoGebra for exploration activities, homework assignments, and assessment tasks, while the control group used traditional methods.

Pre-Assessment

Both groups completed the NLFA and TPS during the first week of instruction to establish baseline measures.

Intervention Implementation

The experimental group participated in structured activities using digital visualization tools, including parameter manipulation, transformation exploration, and real-world application modeling. The control group engaged in equivalent activities using traditional methods.

Post-Assessment

Both groups completed the NLFA and TPS during the final week of instruction. Additionally, 20 students from the experimental group participated in individual interviews.

Data Analysis

Quantitative Analysis

Descriptive statistics, independent samples t-tests, and ANCOVA were used to analyze pre- and post-assessment data. Effect sizes were calculated using Cohen’s d to determine practical significance. Statistical significance was set at α = 0.05.

Qualitative Analysis

Interview transcripts were analyzed using thematic analysis following Braun and Clarke’s (2006) six-phase approach. Data were coded inductively to identify patterns and themes related to student experiences with digital visualization tools.

Results

Quantitative Results

Pre-Assessment Comparison

Independent samples t-tests revealed no significant differences between experimental and control groups on pre-assessment measures, confirming group equivalence at baseline (see Table I).

Table I. Pre-Assessment Comparison Between Experimental and Control Groups

Measure	Experimental group	Control group	t-value	p-value
Total NLFA score	M = 42.3, SD = 8.7	M = 41.8, SD = 9.2	0.28	0.78
Quadratic functions	M = 14.2, SD = 3.1	M = 13.9, SD = 3.4	0.45	0.65
Exponential functions	M = 13.8, SD = 3.2	M = 13.7, SD = 3.0	0.16	0.87
Rational functions	M = 14.3, SD = 3.8	M = 14.2, SD = 3.6	0.14	0.89
Technology perception	M = 3.2, SD = 0.8	M = 3.1, SD = 0.9	0.61	0.54

Post-Assessment Results

Significant differences emerged between groups on post-assessment measures, with the experimental group demonstrating superior performance across all categories (see Table II).

Table II. Post-Assessment Comparison Between Experimental and Control Groups

Measure	Experimental group	Control group	t-value	p-value	Cohen’s d
Total NLFA score	M = 67.4, SD = 7.3	M = 52.1, SD = 8.9	9.87	<0.001	1.89
Quadratic functions	M = 22.8, SD = 2.4	M = 17.3, SD = 3.1	10.45	<0.001	2.01
Exponential functions	M = 22.1, SD = 2.7	M = 17.6, SD = 3.2	7.89	<0.001	1.51
Rational functions	M = 22.5, SD = 3.0	M = 17.2, SD = 3.6	8.32	<0.001	1.60
Technology perception	M = 4.3, SD = 0.6	M = 3.2, SD = 0.8	8.14	<0.001	1.56

Learning Gains Analysis

Analysis of learning gains (post-assessment minus pre-assessment scores) revealed substantial improvements for the experimental group across all function types (see Table III).

Table III. Learning Gains by Function Type and Group

Learning gains comparison
	Experimental group	Control group
Quadratic functions	+8.6	+3.4
Exponential functions	+8.3	+3.9
Rational functions	+8.2	+3.0

ANCOVA results, using pre-assessment scores as covariates, confirmed significant treatment effects for all function types: quadratic functions, F(1, 117) = 89.2, p < 0.001, η² = 0.43; exponential functions, F(1, 117) = 67.4, p < 0.001, η² = 0.37; rational functions, F(1, 117) = 74.6, p < 0.001, η² = 0.39.

Qualitative Results

Thematic analysis of interview data revealed four primary themes describing student experiences with digital visualization tools:

Theme 1: Enhanced Conceptual Understanding

Students reported that digital visualization tools helped them develop deeper understanding of function concepts. Representative quotes include:

“When I could move the sliders and see how the graph changed immediately, I finally understood what the ‘a’ in the equation actually did to the parabola” (Student 7).

“With GeoGebra, I could see how the exponential function grew differently than I imagined. It made the concept real” (Student 14).

Theme 2: Improved Visualization Skills

Participants described enhanced ability to visualize function behavior and transformations:

“I used to have trouble picturing what a rational function looked like, but now I can see the asymptotes and how the function behaves near them” (Student 3).

“The tools helped me connect the equation to the graph in a way I never could before” (Student 11).

Theme 3: Increased Engagement and Motivation

Students expressed greater interest and engagement when using digital tools:

“Math class became more interesting when we could explore and discover things ourselves instead of just watching the teacher” (Student 8).

“I looked forward to using Desmos because it felt like playing with math instead of just doing problems” (Student 16).

Theme 4: Collaborative Learning Enhancement

Digital tools facilitated peer collaboration and discussion:

“We could share our graphs and explain to each other what we discovered. It made us talk more about math” (Student 5).

“Working together on the computer helped us learn from each other’s mistakes and discoveries” (Student 12).

Discussion

Interpretation of Findings

The results of this study provide compelling evidence for the effectiveness of digital visualization tools in enhancing student understanding of non-linear functions. The large effect sizes observed across all function types (d = 1.51–2.01) indicate not only statistical significance but also practical importance for mathematics education.

The superior performance of the experimental group can be attributed to several factors inherent in digital visualization tools. First, the dynamic nature of these tools allows students to manipulate function parameters and observe immediate graphical feedback, facilitating the development of intuitive understanding of function behavior. This aligns with constructivist learning theory, which emphasizes the importance of active exploration in knowledge construction.

Second, digital tools provide multiple linked representations that support conceptual understanding. Students can simultaneously observe algebraic expressions, graphical representations, and numerical tables, enabling them to make connections between different mathematical representations. This multi-representational approach is particularly beneficial for understanding complex non-linear functions.

The qualitative findings complement the quantitative results by providing insight into student experiences and perceptions. The themes identified through thematic analysis suggest that digital visualization tools address several key challenges in mathematics education, including engagement, conceptual understanding, and collaborative learning.

Implications for Practice

The findings have several important implications for mathematics educators and curriculum designers:

1. Technology Integration: Schools should prioritize the integration of digital visualization tools into advanced algebra curricula, with particular emphasis on non-linear function instruction.

2. Teacher Professional Development: Educators need comprehensive training in the effective use of digital visualization tools to maximize their pedagogical potential.

3. Curriculum Design: Curricula should be redesigned to incorporate exploration-based activities that leverage the dynamic capabilities of digital tools.

4. Assessment Practices: Assessment methods should reflect the enhanced conceptual understanding facilitated by digital tools, moving beyond procedural skills to include conceptual reasoning.

Limitations

Several limitations should be considered when interpreting these findings:

1. Sample Limitations: The study was conducted in three suburban schools, limiting generalizability to urban or rural contexts.

2. Duration: The eight-week intervention period may not capture long-term retention effects.

3. Teacher Variables: Differences in teacher enthusiasm or technological proficiency may have influenced results.

4. Technology Access: The study assumed consistent access to digital tools, which may not reflect all educational contexts.

Future Research Directions

Future research should address the following areas:

1. Longitudinal Studies: Investigate long-term retention and transfer effects of digital visualization tool use.

2. Diverse Populations: Examine effectiveness across different socioeconomic, cultural, and geographic contexts.

3. Specific Tool Comparison: Compare the effectiveness of different digital visualization tools for specific mathematical concepts.

4. Teacher Preparation: Investigate optimal professional development models for technology integration.

Conclusion

This mixed methods study demonstrates that digital visualization tools significantly enhance high school students’ understanding of non-linear functions in advanced algebra. The substantial learning gains observed across quadratic, exponential, and rational functions, combined with positive student perceptions and experiences, provide strong evidence for the educational value of these technologies.

The integration of digital visualization tools addresses fundamental challenges in mathematics education by providing dynamic, interactive environments that support conceptual understanding, engage student interest, and facilitate collaborative learning. As mathematics education continues to evolve in the digital age, these tools represent essential resources for helping students develop deep, meaningful understanding of complex mathematical concepts.

The findings suggest that successful implementation requires careful attention to teacher preparation, curriculum design, and ongoing support for technology integration. With appropriate implementation, digital visualization tools have the potential to transform mathematics education and improve student outcomes in advanced algebra and beyond.