The effect of GeoGebra integrated instruction on students’ learning of the quadratic function concept

Problem statement

One of the major challenges facing the Ethiopian education system is the low performance and lack of motivation among students in mathematics.^17–20 Notes that students exhibit low levels of mathematical competency and their interest also shows a declining trend over time. For example, in the university entrance examination, the average scores of students dropped from 41.11 in 2016/17 to 38.55 in 2019/20, highlighting the inadequate mathematics skills of students entering university.

In the study area under discussion, the students’ performance was arguably worse. The Annual Abstract Statistics report by the Silte Zone Education Department (SZED) stated that the percentage of students who scored at least half on the mathematics entrance examination dropped from 18% to 12% in subsequent years. These findings indicate that there is an urgent need to study students’ performance and motivation for mathematics. Consequently, the main areas where students require assistance in understanding are functions and graphs, which are fundamental concepts in advanced mathematics.²¹ Argue that the school, teaching methodology, and the participation of students greatly affect learning outcomes. Thus, we need to investigate better practices that engage students as active participants in the teaching-learning process. In that sense,^17,22 emphasized the design of alternative methods for teaching to improve student performance and motivation in science and mathematics instruction. Instructors should give thought to current student expectations, the country’s development plan, contemporary learning theories, and the role of technology in education.

Ethiopia General Education Curriculum Framework (EGECF) acknowledges that the 21st-century society is characterized by modern advancements in Information and Communication Technology (ICT), digital learning, and any components therewith. Thus, digital literacy shall be part of the general curriculum and applicable at different levels, for which attainment shall be ensured for all learners.²³ As mentioned by,²⁴ since technology is booming with a very short half-life, the traditional instructional approaches are not suitable to attain the 21^st-century educational objectives that demand an active learning process or student-centered learning approach.

In acknowledgment of the benefits of being successful in mathematics, the new EGECF indicated developing a curriculum that emphasizes the significance of mathematics has added value to the teaching of other subjects and hence to the advancement of science and technology. Besides, the curriculum document emphasized the technology-integrated mathematics instruction to smooth the process of teaching-learning for the benefit of learners. Several research studies (for instance,^1,25–28) have suggested levels of technology-integrated instruction either to improve student performance and interest in mathematics or to motivate learners and overcome observed problems.

The driving force for this study is the nationally recognized concerns, i.e. the current practice does not lead to the realization of the country’s development plan and educational goals Therefore, improved performance and motivation require the development of a different approach in science and mathematics instructions.^22,29

There are different mathematical software’s designed to develop student learning attitudes and improving achievement: GeoGebra, Mathematica^®, and MATLAB^® are some of them. These tools are proprietary software developed by the International GeoGebra Institute, Wolfram Research, and MathWorks, respectively.

GeoGebra, given its nature and advancement, can be regarded as the most important mathematical software for it is intended to show two pillars of representation in mathematics; geometric and algebraic, on the same computer screen. Also, it is intentionally designed for educational purposes and enables students to improve their performance and interest in studying the subject.³⁰

According to,²⁴ GeoGebra is not yet widely used to teach mathematics in Ethiopia. What’s more, it’s easy to use; choices of representation that can be made with one’s preferred colors and includes a slider/“Dynamic Movement” types.^28,31 Add that it would be beneficial to find out what impact GeoGebra can have on the acquisition of pedagogical knowledge deemed necessary for effective use of GeoGebra by the learners. Today’s digital technology practice also provides evidence that we should act accordingly.

Thus, this study aims to assess the efficacy that GeoGebra software integrated mathematics teaching has on student achievement and motivation compared to conventional instruction. In doing so, the study tried to answer the following research question: what effect does the use of Geogebra software integrated instruction has on students learning the quadratic function concept in terms of achievement, concept acquisition, and motivation?

Theoretical framework of the study

With the understanding that computers can influence learning, this study is grounded in the Social constructivism theory. It investigates how students utilize virtual manipulatives, specifically the GeoGebra applets, to enhance their learning outcomes and motivation regarding the concept of quadratic functions.

Social Constructivism Approach (SCA): SCA builds on Vygotsky’s Zone of Proximal Development (ZPD) and consists of four levels that explain how students grasp a concept, with the teacher acting as a facilitator. The degree of facilitation varies based on the learner’s current level of understanding the concept under consideration. In this study, the integration of GeoGebra in instruction acts as a scaffolding tool, while peers function as a community that aids in concept formation. Teachers must be aware of their students’ prerequisite knowledge and cognitive levels to provide suitable assistance tailored to their needs at each ZPD level.³² The use of GeoGebra applets was guided by the four-level model proposed by.³³

Level I: At this stage, the teacher is expected to support students as they tackle specific tasks. To prepare them for the next level, the teacher should consistently ask questions to gauge their understanding of fundamental concepts.³² Support can come from the teacher or a more knowledgeable peer.^33–35 This collaborative interaction benefits students of varying abilities. With ongoing practice and support, students can advance to Level II.

Level II: Here, students engage in specific tasks with less dependence on their teacher³⁵ and draw on their prior knowledge to complete tasks independently.³⁶ However, some tasks may still pose challenges for students to finish on their own³⁷ as students’ autonomy is not yet fully developed.³⁸ For instance, in the implementation of GeoGebra, at this level, they actively engage with GeoGebra applets and regularly request assistance from either the teacher or more capable colleagues for help.

At Level III, external support becomes less beneficial; students’ performance becomes internalized, developed, and automated.³³ This stage marks the transition from the ZPD to a more advanced level of competence for a specific task.³²

At Level IV, students begin to reverse their execution steps, leading to revisions of their work. They apply insights gained from past experiences organized through the ZPD. Lasting learning involves cycling through the same controlled ZPD stages, moving from assistance to self-sufficiency repeatedly to enhance untapped abilities.³³

Overview of the GeoGebra software

GeoGebra, developed by Markus Hohenwarter,^30,39 is an open-source dynamic mathematics program that seamlessly integrates algebra, statistics, calculus, and geometry into a single, user-friendly interface. The name GeoGebra reflects how its interface displays each mathematical object: once in the graphic (geometry) window and once in the algebra window. This software enables students to visualize mathematical processes and provides intuitive guidance. With its various tools, students can explore a wider range of function types by linking symbolic and visual representations.⁴⁰ The immediate changes that occur when an object is modified in one of these windows enhance the learner’s ability to recognize key cognitive relationships. While participants in this study utilize this software for various applications, they specifically engage with lessons created using it.

Figure 1 displays a screenshot of the GeoGebra window, illustrating the connection between geometry and algebra. As shown in the figure, multiple functions can be easily demonstrated on the same screen. The software includes an algebra window, a menu bar, a navigation bar, an input field, a geometry window or working area, and a toolbar. Although its primary focus is on teaching geometry, it also offers excellent features for teaching algebra, particularly in the areas of functions and graphs.

Figure 1. Screenshot from a GeoGebra Window.

Functions can be defined algebraically and then modified dynamically. For instance, when an equation is entered, the corresponding graph is generated immediately in the geometry area, while the algebraic representation appears in the algebra window at the same time. This tool is user-friendly, reduces the effort needed to manage processes, and offers options for representation with preferred colors and types of slider.³¹ These features help minimize the abstraction needed to grasp concepts, which is a significant challenge for students learning about quadratic functions.¹²

The impact of GeoGebra on students’ achievement and motivation has been examined in various contexts. Some notable benefits include graphic displays that clarify abstract concepts and boost motivation to learn, instant feedback that can validate or challenge a student’s actions, and the enhancement of both domain-specific knowledge and digital skills.^41–43 These studies demonstrate that instruction integrated with GeoGebra leads to improved learning outcomes for students.

The intervention

For the intervention, the control group received instruction based on traditional methods, while the experimental group was taught using GeoGebra software to explore concepts across three themes. After a two-day introduction to using GeoGebra (which corresponds to level I of the theoretical framework), instruction continued with the help of an overhead projector. The second researcher delivered ten different lesson episodes to both groups. Each episode of the experimental group was designed to address the three concepts as identified in Table 1. Figure 2 illustrates one such episode.

Figure 2. An example of the teaching episode.

In the computer lab, twenty-one computers were set up for the experimental group in pairs to encourage collaboration. The instruction included explanations, examples, demonstration activities, two summary worksheets, and presentation tasks, all carefully designed to meet the various levels of the theoretical framework. The researcher took on a facilitating role, correcting and checking the tasks saved in folders named after each student on their desktop. In contrast, the control group completed the same tasks but did not use the GeoGebra software.

Through the manipulation of the GeoGebra applets, the participants got a chance to see the effect of the variation of one variable on the other variable. Besides, the learners were free to learn according to their learning preference as suggested in the selected theoretical frameworks, and left free to communicate with their colleagues through their ways of thinking and language.

Theoretical framework integration

GeoGebra was employed as a scaffolding tool. The teacher (in this case, the second author) used it to help students letting them move through the four stages of ZPD: at Level I, the teacher showed students how to use GeoGebra and walked them through basic exercises to make sure they got the main ideas and their implementation. At Level II, students started to work with quadratic functions on their own using GeoGebra but asked for help when they needed it. At Level III, students were proficient (and those who have a gap consult their peers or the teacher) in the concepts, seeing and manipulating functions using GeoGebra independently of the teacher. Finally, at Level IV, students strengthened what they learned through practice and used it to tackle harder problems from a worksheet and the textbook.

Data analysis

Traditional descriptive statistical (frequency count, percentage, mean, and standard deviation) as well as inferential statistical techniques (independent sample test selected based on the assumptions the data satisfied) were used. The analysis was focused on checking whether there is a significant difference or not in students’ achievement and motivation to learn between the groups. The hypotheses corresponding to each research question were tested at a 95% confidence interval using a two tailed test.

Ethical issues

In the study, the researchers endeavored to satisfy all admissible requirements for safeguarding and guaranteeing the rights of all those involved. The research obtained permission from Wachemo University Department of Mathematics DGC (Department Graduate Committee) on 27 March 2013 (Date 18/7/2015E.C, Ref. No WCU/CNCS/Math/0145/2015), and the researchers accepted the approval letter on April 2023 from the department. The university letter was submitted to SZED to obtain letters to the sample schools. The letters were submitted to the relevant school principals. The students were also informed in full about their involvement in the study and of their participant rights. Besides, the students signed a written consent form prepared in English and translated to the local language (Amharic) to express their agreement. The consent form was part of the proposal approved by the DGC. The study was conducted from May 8 to May 26, 2023. The participants’ responses were coded and their identities were not revealed in the study report.

Pre-test result

The pre-test results indicated that the average (M) and standard deviation (SD) of the control group students’ achievement (M = 10.36; SD = 2.693) were nearly identical to those of the experimental group students (M = 10.36; SD = 3.399). Additionally, the T-test results confirmed that both groups had similar levels of prerequisite knowledge before the treatment (p-value 0.998 ≥ 0.05). Therefore, any differences observed in the understanding of quadratic functions after the intervention can be attributed to the treatment itself.

Post-test result

Effect of GeoGebra-integrated Instruction on Students’ Achievement

While Table 3 provides a summary of the statistics for both groups, Table 4 presents the results of the independent sample t-test comparing the two groups. The findings in Table 3 indicate that the control group students had a lower achievement level (M = 17.60; SD = 4.614; SE = 0.688) compared to the experimental group students, who had a higher average achievement (M = 27.90; SD = 2.844; SE = 0.688).

According to the data in Table 4, Levene’s test shows a significant result (p = .00, which is less than 0.05). This indicates that there is a significant difference in variance.⁴⁷ In both scenarios, the outcome remains consistent, demonstrating a significant difference in the performance of the two groups based on their mean values (at an alpha level of 0.05, two-tailed probability), with the experimental group performing better. Therefore, we can conclude that the GeoGebra integrated instruction had a positive impact on students’ understanding of quadratic function concepts.

To strengthen the conclusion (to see magnitude of the effect), we have also calculated the effect size (r) of the result. Accordingly, we used the formula $r=\sqrt{\frac{t^2}{t^2+\mathit{df}}}$ ⁴⁷ where $t$ and $\mathit{df}$ as in Table 3. Thus, $r=\sqrt{\frac{{(-12.63)}^2}{{(-12.63)}^2+73.959}}$ = 0.83 which implies the effect is large (Cohen, 1992 as in⁴⁷).

Effect of GeoGebra integrated instruction on students’ conceptual understanding

For items related to outcome 1, the overall percentage of correct responses for the control group was 59.45%, while the experimental group achieved 83.93%. This suggests that the intervention helped the experimental group better understand the IUVFQF concept. Additionally, the control group had a mean percentage of 69.23%, compared to 89.16% for the experimental group in answering items related to outcome 2. This further indicates that the experimental group outperformed the control group in grasping the IUSFQF concept. Figure 3 displays scanned images of students’ test scripts, with the left side representing the control group and the right side showing the experimental group, for items across all three outcomes.

Figure 3. A scanned image of the works of students.

The students’ test results show that the control group achieved a mean percentage of 58.15% correct responses for items in outcome 3, while the experimental group had a significantly higher rate of 91.07%. This indicates that the experimental group outperformed the control group. Figure 4 displays scanned images of the students’ test scripts, with the left side representing the control group and the right side representing the experimental group for items in outcome 3.

Figure 4. A scanned image of the works of students for items in outcome 3.

The introduction of the applet significantly enhanced students’ understanding and eased the challenges associated with learning quadratic functions in grade 9. This improvement can be attributed to several factors, including a focus on grasping the conceptual aspects, a reduction in the effort needed for routine calculations, the ability to view multiple representations on a single screen, and the immediate feedback provided to verify the accuracy of their work through GeoGebra’s integrated instruction. Furthermore, the concepts and techniques learned by using applets like GeoGebra are lasting and can be seamlessly incorporated into students’ existing cognitive frameworks, making them more manageable.⁴⁸ Figure 5 illustrates the frequency distribution of the post-test results for the two groups.

Figure 5. Frequency distribution of post-test results of the two groups.

In general, Figure 5 illustrates that the overall performance of students in the post-test differed between the two groups. Specifically, a greater number of students in the experimental group achieved higher scores, while more students in the control group scored at lower levels compared to those who performed at higher levels.

Motivational effect of GeoGebra-integrated instruction

To explore the motivational impact of the GeoGebra integrated instruction, a questionnaire was developed that included six motivational attribute items and eight perception attribute items. Figure 6 illustrates the percentage distribution of these motivation attributes among the respondents based on the items in the motivation questionnaire.

Figure 6. Percentage distribution of motivation attributes of respondents.

As shown in Figure 6, the results indicated that out of the 42 students in the experimental group, 40 (95.24%) responded ‘Yes’ to the question: Are you motivated to learn about quadratic functions and their graphs? The experimental group demonstrated a strong motivation to learn these concepts. All students reported that the GeoGebra-integrated instruction encouraged their participation in the learning process, and 40 (95.24%) confirmed that it helped them maintain focus throughout the lesson.

Regarding the enjoyment of lessons, 41 (97.62%) of the students stated that learning about quadratic functions and their graphs using GeoGebra was very enjoyable. Additionally, 38 (90.48%) of the students who were asked if the instruction enabled them to answer every test item responded “yes,” indicating that the method effectively helped them grasp the concepts in the material. Furthermore, all respondents recommended the use of the GeoGebra integrated instructional approach for teaching concepts related to quadratic functions. Lastly, participants noted that their background in computer skills influenced their level of benefit, motivation, and satisfaction with the instruction.

Concerning the eight Likert scale perception questionnaire, the result is summarized in Table 2. As indicated in Table 2, all students have shown positive perception towards the GeoGebra-integrated instruction consistently through the eight items. Thus, despite the slight difference in ratings, the aggregated mean (M = 3.78, SD = .752) indicates that the respondents positively perceived the new experience. This is promising, if not absolutely a permit, to get the attention of students on hard concepts in particular and hard sciences in general.

Overall, the findings of this study suggest that using GeoGebra-integrated instruction is beneficial for teaching mathematical concepts at the secondary school level. The applet adds value by enhancing student achievement, facilitating knowledge acquisition, and increasing motivation to learn through features such as alternative representations (geometric, algebraic, various color, font, line, and letter style options), immediate feedback, and a dynamic platform, which together reduce the time and attention required for routine procedures. In line with this, the findings of⁴⁹ indicated that the traditional paper-and-pencil teaching approach becomes more effective when used alongside technology-assisted instruction in the same classroom.

The post-test results indicate a notable difference in graphing quadratic functions, with the experimental groups performing better than the control groups. The applet enables students to visualize the effects of altering parameters, link concepts with visual aids, and tackle mathematical problems relevant to the secondary school curriculum.⁵⁰ Noted that low-income countries should focus on curriculum and instruction that incorporate technology. Additionally, their findings revealed that the use of technology significantly enhances students’ motivation and problem-solving skills. Conversely,²⁶ found no relationship between the time spent using technology and the procedural knowledge gained in mathematics learning.

Increased participation and motivation for learning were noted in classrooms that integrated GeoGebra. Supporting this, a study aimed to investigate students’ motivation through the use of technology for teaching transformations, as highlighted by,⁵¹ which reached a similar conclusion. Their findings confirm that utilizing GeoGebra significantly enhances students’ learning motivation. Other research, such as,^52,53 also indicated that technology-integrated instruction boosts students’ commitment to learning.

The results of this study indicate that, although GeoGebra-integrated instruction generally has a positive impact on students’ performance, understanding of concepts, and motivation to learn, it is important to note that less impact was seen for certain specific concepts, such as writing the vertex form of a quadratic function, identifying the parameter that shifts the graph of a quadratic function vertically, and defining the axis of symmetry of a parabola.

Limitations of the study

This research provided valuable insights into teaching and understanding mathematical concepts. However, it has some limitations that could influence the results. Firstly, there are several complicating factors to consider, such as the students’ intelligence levels, gender, and the specific nature of the concepts being taught. Secondly, both groups were instructed by the same teachers (the second author), which may have impacted the results. Additionally, the first author consistently guided the teacher on how to implement each activity according to the design. Consequently, these factors could affect the study’s conclusions in various ways. Therefore, further research that incorporates additional confounding variables, such as gender, students’ intelligence levels, and the nature of the concepts, as suggested by,⁵⁶ is strongly recommended. It is also advisable to conduct a follow-up longitudinal study to assess the durability of the performance and how the motivational effects evolve.