The effect of GeoGebra on STEM students learning trigonometric functions

Abstract This study explores the effectiveness of the GeoGebra on Grade 12 students’ success in making associations between the representations of trigonometric functions and the interpretation of graphs. A technology-oriented classroom is a commanding and accommodating tool in mathematics instruction in understanding mathematical concepts. A non-equivalent control-group pre-test and post-test quasi-experimental design was used. The study sample consisted of seventy-three Grade 12 students enrolled in Science, Technology, Engineering and Mathematics (STEM) from different schools of the Hadiya zone. The research instrument used was a trigonometric representation ability test. The data analysis used was based on the t-test, ANOVA, and the mean of achievement indicators. The results indicated a statistically significant difference between the overall score mean achievements of experimental groups A and B and the control group on making associations between representations of trigonometric functions, and interpretations of the representations of the trigonometric functions, in the indulgence of both experimental groups. As the association between the trigonometric function representations and an interpretation of graphs of the functions are major challenges to many students in the mathematics field, integration of technology into the teaching of the subject simplifies the life of students. We therefore strongly recommend that teachers integrate GeoGebra with the teaching of trigonometric functions and mathematics in general. Hence, we also recommend that teacher training institutions should include courses on how to use dynamics software applications to teach mathematics in teacher-training programmes. This recommendation will also be applied in Wachemo University, Ethiopia, where the research occurred.

, a BTech (Educational Management, 1999) and a BTech in Natural Science (Mathematics & Physics, 2000) also from TUT and a BSc (Hons) (Mathematics Education) from Wits University (2005). His research interests are in Teaching and learning of Mathematical Literacy alongside Mathematics, investigation of learners' misconceptions in understanding mathematics concepts, and the usage of everyday context in the teaching and learning of Mathematics.

PUBLIC INTEREST STATEMENT
In the fourth industrial revolution (4IR), it is expected of individuals to develop a positive attitude toward the learning and teaching methods; and hence they can be accelerated by integrating technology into the education system. The integration of technology into the teaching of the subject simplifies the life of students, and there is a need to train teachers to integrate technology more by using an open assessed GeoGebra into their teaching and learning process. By using the GeoGebra applet in the classroom, teaching STEM students the topic of trigonometric functions, the results indicated a statistically significant difference between the overall mean score achieved by experimental groups A (using teaching with GeoGebra and low teacher scaffolding) and B (using teaching with GeoGebra and high teacher scaffolding), and the control group in associations between trigonometric function representations, and interpretations of the representations of the trigonometric functions, in the indulgence of both experimental groups. The visible difference also occurs in group A and group B students.

Introduction
A technology-oriented classroom is a commanding and accommodating tool (e.g., GeoGebra) in mathematics instruction and understanding mathematical concepts. Science, Technology, Engineering and Mathematics (STEM) Education has become an international issue of consideration over the past decade in which mathematics is an element of the word STEM. In its simplest form, STEM is an acronym for the four independent disciplines of science, technology, engineering, and mathematics, which often involve traditional disciplinary coursework (Fitzallen, 2015). This interest is stimulated by the dynamic global economy and workforce needs that predict a lack of STEM qualified workers and educators worldwide. As the world is moving with unprecedented speed into a high-tech future of the globalized, technology-driven age there is intense concentration on the importance of (STEM) education and how it impacts on a nation's growth and ability to have a skilled and educated manpower to drive their nation forward with economic expansion. Many see the education of current generations in the areas of STEM as the key to shifting the world's population towards having a greater understanding of the importance of achieving economic growth, environmental conservation, and sustainable development, known as the fourth industrial revolution (4IR) in the industry context and fourth education revolution (4ER) in the education context (Schwab, 2016). As Schwab (2016) stresses, the instruction system will be updated with the emergence of the Education 4.0 era, being an adaptation of the development of Industry 4.0. While Artificial Intelligence (AI) is infiltrating all aspects of this planet, education is still overlooked as its secondary subject. Within the education revolution, the teaching and learning of a subject, more specifically mathematics and especially trigonometric functions, is rapidly changing. Astronomers used trigonometry to find the longitude and latitude of stars, as well as the size and distance of the moon and sun (Nejad, 2016). Electricians use trigonometry calculations in their day to day life in bending conduits (Pascual et al., 2020). They also provided feedback that students had no knowledge of trigonometry to utilise with the knowledge of electrical circuits in their study. With the cooperation of trigonometric functions within the STEM program, the needs of astronomers and electricians may be satisfied.
Ethiopia has been a member of SMASE-Africa since 2007, a regional association for the Strengthening of Mathematics and Science Education, where African countries exchange skills, experience and issues in teacher education in mathematics and science (Ministry of Education, Ethiopia (MoE), 2014). Consequently, the Ethiopian government has established the Centre for Strengthening Mathematics and Science Education in Ethiopia (CSMASEE) under the Federal Ministry of Education (MoE), which is responsible for science and mathematics education (Ministry of Education, Ethiopia (MoE), 2014). In contrast, "without doubt, the currently used curriculum in Africa is obsolete and does not capture the changes being ushered in by the Fourth Industrial Revolution" (Fomunyam, 2020, p. 24). It is supposed that STEM approaches that encourage teamwork (vertical and horizontal interaction of cycle model) belong in the modern classroom, e.g., in a well-organized lesson plan and stages of implementation for teaching and learning (see the cycle model in Bedada's thesis [Bedada, 2021]), and are essential for use in the Fourth Industrial Revolution in this century (Fomunyam, 2020). STEM education is the best method/ approach for increasing students' achievement (Karaşah-Çakici et al., 2021). Moreover, the cycle model posits nine stages of implementing the teaching and learning process within the classroom with the help of technology (GeoGebra) in which students are more fruitful in their achievement both in terms of conceptual and procedural knowledge (Bedada, 2021).
Meanwhile, the Ethiopian government's motivation behind this new emphasis is to produce students versed in Science, Technology and Mathematics (STM), with scientific thinking and logical reasoning skills and promote sufficient and quality graduates fit to work in the country's rapidly growing manufacturing economy industry and construction market. To realise these intentions, politicians, education leaders and policymakers at all levels have called for a new emphasis on STM education (STME) in the nation's schools, from the pre-primary level through to higher education. Providing students with sufficient background in literacy, numeracy and science processing skills, backed up with a proper technological knowledge base, is expected to build up a literate and numerate citizenry committed to nation-building and national development. In this respect, there is a strong consensus among key actors that the country needs to have a national strategic policy for STME, which is a means towards achieving the above stated far-reaching goals.
This National Strategic Policy for STME has been developed with the above notion, considering the range of preschool up to higher education. This policy directly links and contributes to the objectives stated in the Growth and Transformation Plan II (GTP-II), the Education Sector Development Programme (ESDP) V and the document Climate Resilience Green Economy (CRGE) and can be further adapted in the form of a strategy and action plan at the national and regional levels.
The Education and Training Policy (ETP) and ESDPs state that education in science and mathematics is a major component determining the prosperity and welfare of the people and the nation.
The Ethiopian government has outlined several step-by-step guidelines and strategies to deal with outstanding issues regarding STME.

Objectives of the policy strategy
The major objectives of the strategic policy are to: • Strengthen the pre-service training, in-service training, and continuous professional development of STM teachers; • Improve the provision of STM resources by identifying and distributing textbooks, laboratory equipment, and other resources to schools; • Provide students with support to improve student achievement through both in-class and supplementary programs in the short-and long term; • Enhance process skills to positively influence the teaching and learning of STM in schools; • Improve the management of STM teaching and learning; and • Build STM as part of the culture.
Institutions of higher education have a significant responsibility in STEM education to provide undergraduate education, run teacher training programs, and present in-service training for teachers. STEM capacities prepare students to be critical thinkers, to communicate and collaborate across actual obstacles, and to solve complex and ever-changing problems in a progressively technological and complex global community. Ethiopian students with these capacities will drive innovations and fuel the global increasingly STEM-based economy. To this end, Sarica (2020) forwarded that the technology-oriented classroom for teaching content (inquiry-based teaching strategy) should be applied to promote teaching at higher levels for strengthening the effect of STEM teaching.

Statement of the problem
The technology-oriented classroom is a powerful and helpful tool in teaching and learning mathematics, in understanding the mathematical concept (Hohenwarter & Jones, 2007). There is a struggle to integrate the technology-oriented classroom into the school's teaching practices, especially during the COVID-19 pandemic. For three consecutive years in my university, Wachemo, Ethiopia, the teaching and learning process for STEM students in the University used traditional methods in traditional classroom-based contexts, but this is non-tenable in the current climate of the 21 st century. In contrast, "a true STEM education should increase students' understanding of how things work and improve their use of technologies" (Bybee, 2010, p. 996). Moreover, there is generally a shift from traditional courses to STEM education, both in Ethiopia's lower and upper secondary schools (Teferra et al., 2018). STEM education should introduce more engineering at the pre-college or university level. However, learning based on the Science, Technology, Engineering and Mathematics (STEM) framework strongly supports the current era of educational development (Öztürk İrtem & Hastürk, 2021). One of the components that support the success of STEM learning is the ability of student's technology-oriented literacy in the field of science.

Objectives of the study
Depending on the background and statement of the study, one of the study's objectives is to investigate the effect of GeoGebra on the STEM students learning trigonometric functions.

Research questions
(1) How do GeoGebra environments help students in understanding trigonometric functions?
(2) To what extent are students' achievements within experimental groups A and B different?
(3) To what extent are students' achievements of trigonometric functions after interaction with GeoGebra different?

Purpose of the study
This study intended to explore Grade 12 students' understanding of trigonometric functions using GeoGebra software. Specifically, the study explored how students used GeoGebra to understand trigonometric functions, construction and theories when exposed to these individually and in groups. The study intended to make recommendations to the university from the study's findings to use for the STEM program for the coming summer holiday 2022/23. Additionally, the study sought to provide recommendations and implications to mathematics education practitioners on using GeoGebra to enhance the understanding of trigonometric functions in line with the objectives of Ethiopia's STEM strategy. The Next Generation Science Standards call for the integration of engineering into mainstream science education.

Definition of STEM
STEM (science, technology, engineering, and mathematics) is an educational methodology that is now accompanied by STEAM, the combination of STEM and arts in terms of students' creativity in and outside the classroom (Aguilera & Ortiz-Revilla, 2021;Bedada, 2021). However, some scholars start with a question to define STEM as "Is STEM a new teaching approach or a philosophy?" It could be both because we still do not have precise methods for practising STEM and clear definitions universally accepted in STEM education (Bybee, 2010). Besides, STEM is not a new concept as the ideas have been promoted since 1990. Bybee (2010) states that having students ask questions like how tools, equipment, or anything we use work, what kind of benefits technology brings into our lives to make them work and motivations that will keep their curiosity alive mean that STEM is being practised properly. Moreover, he also added that engineering is directly involved in making innovations and solving problems. In addition, a scholar (Sarica, 2020) states that STEM is a holistic educational approach that includes science, technology, engineering and mathematics; its profile has been increasing and is very important for countries to compete and stand out in the new world order and economy in 21 st century. In contrast to this scholar, STEM is viewed as an educational approach only in the United States of America (USA), and in many other countries, it is considered to be academic curriculum reform (Lai, 2018).

Mathematics and STEM education
Much can be gained in support of the teaching and learning of mathematics through connecting and integrating science, technology, and engineering with mathematics, both in mathematics classes and in STEM activities. STEM has been introduced to secondary and preparatory schools to reinforce the education of Science and Technology in Ethiopia (Teferra et al., 2018). Engineering design, for example, offers an approach that nurtures and supports students' development of their problem-solving abilities, a top priority for mathematics teachers. The design process reinforces and extends how students think about problems and offers tools that can help students creatively expand their thinking about solving problems of all types-the very types of problems and issues that students are likely to encounter in their personal and professional lives. The Ethiopian government has been emphasising mathematics achievement recently. It is noted that students' poor learning achievements such as low competencies in reading and mathematics are challenges that can be nourished by STEM education (Teferra et al., 2018).
However, there is more to mathematics than being part of STEM. The mathematics that students learn in school includes content and thinking that can be used as tools for tackling integrative STEM problems. But it also includes content that might be considered "just math" or might be connected to non-STEM disciplines. Problems involving mathematical models of finance might or might not connect to science (S) or engineering (E) and might or might not involve in-depth uses of technology (T). Likewise, art might be integrated into a mathematics lesson that does not involve either science or engineering. Mathematics goes beyond serving as a tool for science, engineering, and technology to develop content unique to mathematics and apply content in relevant applications outside of STEM fields.
The National Council of Teachers of Mathematics (NCTM) has described appropriate mathematical content and processes for grades K-12 in "Principles and Standards for School Mathematics" (National Council of Teachers of Mathematics (NCTM.), 2000). The standards describe a strong, balanced, comprehensive foundation in mathematical knowledge, thinking, and skills reflected in mathematics standards across all the grades. Essentially every grade gives attention to the kind of mathematical thinking, processes, and practices that student should develop as part of their balanced mathematics experience. Thus, there is strong professional guidance and policy direction for the mathematics that should be taught at each grade level.
Defining inclusive teaching and mentoring practices: GeoGebra for teaching trigonometric functions Teaching mathematics well is an important component of a comprehensive STEM program that can help students develop creativity, reasoning, and problem-solving skills that align with the goals of STEM programs. Trigonometry is one of the subjects in mathematics in which students experience crucial difficulties in learning, and it is also perceived to provide difficulties for teaching as well as learning (Nejad & J. A. Wood, M., B., Turner, E. E., Civil, M. & Eli, 2016). This study determined the efficiency of the computer-assisted instruction method, in which the GeoGebra software is used to teach trigonometric functions and graphs of trigonometric function subtopics.
Trigonometry is integral to mathematics education. Trigonometry plays a crucial role in studying mathematics and its applications. Despite the importance of the subject, students struggle to understand trigonometric constructs such as angle measures. It has also been noted how students generally struggle to understand transformations of functions. Trigonometric functions are the earliest mathematics topics that links algebraic, geometric, and graphical reasoning which can be manipulated by using the dynamic software GeoGebra, that can serve as an important precursor towards the understanding of pre-calculus and calculus as well as college-level courses relating to Newtonian physics, architecture, surveying, and engineering (Weber, 2005).
Trigonometric functions are used frequently in the STEM fields. They are presented in engineering tasks such as digital image processing and finding orthogonal force vectors. Here, the GeoGebra program will be discussed on trigonometric functions under four compiled lessons discussed in the following process sections.

Participants and sample size
The sample of the study of 73 students attending grade twelve and enrolled into the STEM program in their summer holiday in Wachemo University during the 2020/21 academic year was purposively selected. Fifty participants of this sample group were selected into experimental groups A and B, while the other 23 were selected into the control group. All the students were selected from the Hadiya zone by using top students ranked 1 to 3 in their marks/scores in the normal classroom at the school in the 2020/21 academic year(C. Bullock, 2017).

Research Method and Design
The type of research used was quasi-experimental. The research design used a non-equivalent control group design, which consisted of two experimental groups (A and B) and one control group that had cause and effect relationships between the teaching method and students' scores (Bedada, 2021;Mcmillan & Schumacher, 2014). The non-equivalent quasi-experimental design with pre-, intermediate and post-test control groups was employed. Table 1 indicates that the treatment of the three groups was different from each other in such a way that X 1 consists of the use of GeoGebra, student self-scaffolding and cooperative learning and teaching; X 2 consists of the use of GeoGebra, teacher scaffolding and cooperative learning and teaching; and X 3 consists of the use traditional teaching methods. Within the treatment time, I carefully observed the classroom by "seeing as" which mean that seeing as is regarded as one of the tools for expressing relationships between seeing and knowing, with the help of an intermediate test (Kvernbekk, 2000). A classroom observation is a formal or informal observation of teaching taking place in a classroom or other learning environment.

Experimental and control group
This study contained both experimental and control groups and lasted for one month. The study contained two experimental groups [group A and group B]. In group B (27 students), the teaching and learning were posed to the students in a cooperative way with the help of GeoGebra on the topic of trigonometric functions. They cooperated during education and examination time. Group A (23 students) only cooperated using GeoGebra in the learning period. The control group students (about 23 students) were taught only with the traditional teaching methods.

Procedure
A four-week course that contained twelve main GeoGebra activities and many other practices about the stated achievements has been planned according to the Ethiopian Grade 11 mathematics curriculum. After that, the activities were constructed with GeoGebra for the experimental groups (Bedada, 2020). The GeoGebra prepared activities aimed to make the subject more dynamic, concrete, measurable and visual. GeoGebra software was introduced to students in the introductory class in the first hour of the course. In all the other sessions, the GeoGebra prepared activities were shared with the students with visual and dynamic features. Furthermore, examples and drawings on the textbooks were constructed with the GeoGebra during the sessions. The teaching of trigonometric functions and graphs of trigonometric functions subtopics from eleventh grade takes a total of 16 hours with three different learning outcomes. These include the definition of trigonometric functions, inverse trigonometric functions and amplitude, phase shift and period of a trigonometric function and related problems.

Week1: Lesson one (Definition of trigonometric function)
Trigonometric functions is a branch of mathematics that studies the relationships between the angle and sides of the triangles. During this week, during the first session of the lesson, the researcher (the teacher) introduced the GeoGebra mathematical software to the students according to the steps of the cycle model (Bedada, 2021). So, after identifying the environment or laboratory classroom of the cycle model (first step) have been one at the beginning, the whiteboard, projector, and laptops were used to introduce trigonometry concepts and representations through graphs. The teacher then introduced GeoGebra orally and through a PowerPoint presentation with its component. This was then followed by the students working on the computers both individually and in pairs. The applet was used to introduce trigonometric functions. In this lesson, the teacher introduced six trigonometric functions with the help of GeoGebra mathematical software by visualization on one GeoGebra window, as shown in Figure 1.
From Figure 1, the students can simply identify the domain and range of each trigonometric function.
In the experimental groups students were arranged in the classroom in groups of three to four to keep the university COVID-19 protocol. Each student in the groups was ordered to sketch the graph of each trigonometric function by using GeoGebra by being in front of students in the classroom, and their results were shown on the board with the help of a PowerPoint representation. The teacher used the scaffolding stages of the cycle model while teaching, and in other cases group students can scaffold with each other, known as self-scaffolding.

Week2: Lesson two (Inverse trigonometric function)
A trigonometric function is a periodic function (the range value is not limited for its domain value), leading to the function not having an inverse. So, the inverse trigonometric function is obtained by restricting the domain of each trigonometric function. The restriction of the domain is mandatory as the trigonometric function is not a one-to-one function to pass the horizontal line test criteria. The definition behind the inverse function is: • If the point a; b ð Þ is an element of the function f , then the point b; a ð Þ is the element of the inverse function f À 1 ; • The domain of f ¼ the range of f À 1 , and vice versa.
Students can also visualize and identify the domain and the range of the six inverse trigonometric functions by using GeoGebra. From Figure 2, one can see both the types of the six inverse trigonometric functions from an algebraic view and their graphs from a graphics view with their congruency colours. For instance,sin À 1 x ð Þ was represented with a blue colour, and its graph was also sketched with a blue colour. As the software is dynamic, we can change the colour of the functions as we want. The domain and range of each inverse trigonometric function can be visualized and identified by students within the classroom. For instance, the domain of tan À 1 x ð Þ is all the real numbers and the range of tan À 1 x ð Þ is À π 2 ; π 2 À � . From Figure 2 above, the horizontal broken line stands for the open interval that means π 2 is not a range of tan À 1 x ð Þ; as tan π 2 À � is not defined at all. Also, while sketching the graphs of inverse trigonometric functions, we need to convert the x-axis to the radian measures. To do this, we put our cursor on the y-axis and right-click on it and go to the graphics properties and select the y-axis and click on distance and finally select radians πorπ=2.

Week3: Lesson three (amplitude, phase shift and period of trigonometric function)
Several technology applications offer an opportunity for students to discover the nature of trigonometric graphs and understand and visualize why they are periodic (Kissane & Kemp, 2009). One of the technologies that can do this is GeoGebra mathematical software. In employing this software, the students differentiated the graphs of the trigonometric functions of the form f For the function of the form f x ð Þ ¼ asin kx þ p ð Þ, we can sketch the graph by simply shifting the function f x ð Þ ¼ asin kx ð Þ À p k units in the x-direction and c units in the y-direction. For instance, the graph of the function f x ð Þ ¼ 3 sin 2x À π 3 À � À 2 created by using the GeoGebra applet is depicted in Figure 3.
From Figure 3, we saw that if the value of c was negative, the graph of the function was shifted up c units in the y-direction and if the value of p k <0, then the graph shifted to the negative x-axis and otherwise to the positive x-axis. Here students can visualize and understand the definition by using the GeoGebra applet.

Week4: Lesson four (evaluation of Lessons 1, 2 and 3)
In this week, the researcher (teacher) identified the activities of week one, two and three. Within this week, the intermediate test with class observation and post-test was delivered to the three groups.

Achievement test
Examining the target behaviours determined by the Ethiopian Ministry of Education (MoE, Ethiopia) for the unit trigonometric functions and graphs of trigonometric function subtopics and the pretest, intermediate test, and post-test led to eight trigonometric functions test items that were developed by the researchers. The items' reliability was .654, which is acceptable if one item Q1 was deleted as it correlates negatively with the total scale and probably should be reversed (Revelle, 2021). The Rasch model software verified the misfit of Question 1 by Table 2. The researcher reversed this problematic item with arrangements to prepare for the main study.
In Table 2, Infit means inlier-sensitive or information-weighted fit. This is more sensitive to the pattern of responses to items targeted on the person and vice-versa. These patterns can be hard to diagnose and remedy, whereas Outfit means outlier-sensitive fit. This is more sensitive to responses to items with difficulty far from a person, and vice-versa. For example, Outfit reports overfit for imputed responses, underfit for lucky guesses and careless mistakes. These are usually easy to diagnose and remedy. The table shows that the mean square value of infit and outfit are 1.7 and 1.8, respectively, which is found between 1.5 and 2, indicating that item Q1 is unproductive for constructing a measurement but not degrading (Linacre, 2002). In line with the targets of the given unit, an achievement test consisting of eight multiple-choice questions with a graphic exploration of the given question with justification prepared by using student books was designed. The achievement test was designed to measure the objectives that the students in three groups were expected to achieve during the study. The test was prepared by the researchers and checked by two mathematics teachers; all were mathematics educators. The test was piloted with Grade 12 students. The main purpose of the pilot was to determine students' difficulties in understanding the tasks used in the test and prepare open-ended explorations in the main study.

Data analysis
Quantitative data analyses were done using independent-samples t-tests and a one-way ANOVA test. These tests were used because the data were normally distributed. The observation part was analysed through thematic analysis (Saldaña, 2013). Moreover, the t-test is the most frequently used test in mathematics education that involves two groups of small sample sizes (De Winter, 2013); in addition, the t-test is more favourable for a two-grouped sample. Computationally, the statistical software package, Statistical Package for the Social Sciences (SPSS) (version 27), as well as Ministep and Jamovi software for the Rasch model were used.

Findings
The results of this study are presented according to the research questions of evaluating whether any significant difference existed between the pre-test, intermediate test, post-tests of both the control and treatment groups A and B.

Comparison of the three groups' achievement in the pre-test, intermediate test, and post-test
(1) Table 3 shows the comparison of the groups' (the mean scores of experimental groups A and B and the control group) achievements in the pre-test, post-test, and intermediate test.
In all question items, students in experimental groups A and B scored better in the post-test in associations between the representations of trigonometric functions in terms of algebraic views and an interpretation of graphs (graphic views) of the function than the control group, except in question Q1. Table 3 shows that the overall mean scores of experimental group B are greater than both the control group and experimental group at the pre-test, intermediate test, and post-test achievement of trigonometric functions. Table 3 further shows that the mean score of the control group is initially greater than the experimental group A, but still, there is no visible significant difference (see , Table 4). After the intervention has been employed with the help of the GeoGebra mathematical software, the mean scores of Experimental Group A are greater than the control group. This result shows that the intervention with the GeoGebra applet influences students' achievements in trigonometry. Table 4 below show that there was a statistically significant difference in students' achievement in trigonometric functions at the pre-test, the intervention (F (2,70) = 22.5, p = .00 < 0.05) with effect size Eta-squared (η 2 ¼ :391),post-test, the intervention (F (2,70) = 23.059, p = .00 < 0.05) with effect size Eta-squared (η 2 ¼ :397), and the intermediate test, the intervention (F (2,70) = 43.616, p = .00 < 0.05) with effect size Eta-squared (η 2 ¼ :555). The effect size etasquared is characterised as small, medium or large effects if it possesses the values .01, .06 and .14, respectively (Stevens, 2009). In the three tests, the variables indicate high effect-size that show there are high statistically significant differences between the students' trigonometric achievement at pre-, intermediate and post-test. However, there is no evidence about which pair of groups differ statistically on these study variables. Hence, the post hoc Tukey LSD test was used for the trigonometry achievement test (intermediate test), and trigonometric achievement posttest as the Levene test was not significant, and so the assumption of homogeneity of variance was not violated.
The post hoc Tukey LSD test indicated that in the trigonometric intermediate and post-test, the three groups differed significantly in their achievement (p < .05, d = 2.060) with a high effect size.
At the identification stage of student ability of the cycle model, the students' ability does not show a visible significant difference (Cohen et al., 2018). Thus, the difference comes with the help of employing the GeoGebra applet by bearing in mind some prior students' ability on the trigonometric function from their elementary schools. Table 5 below shows the differences in students' achievement on the three tests for the three groups.
Tables 4 and 5 inferred that the difference came about because of the interventions. In other words, the students in the experimental groups A and B have a better understanding of the associations between the representations of trigonometric functions and an interpretation of graphs of the function than the control group.
The means scores of students' achievement can be visualized as depicted in Figure 4.   Figure 4 shows that experimental group A increased in the mean score from 42.4% to 73.4%, the control group decreased from 52.13% to 50.5%, and experimental group B increased from 74.13% to 85.61%. There are visible percentage differences in the three groups in which experimental group B, which utilised cooperative and scaffolding teaching methods, performed better than the other two groups. Figure 4 indicates that experimental group A's students scored low mark on intermediate test and students in experimental group B scored high marks compared to experimental group A; this may be because of the presence of scaffolding in the case of experimental group B.
A paired-samples t-test from Table 6 Cohen et al., 2018;Stevens, 2009). Students' trigonometric achievement after the intermediate test had been conducted and observation of the class by the teacher had been employed increased dramatically in the post-test. This result has been achieved because of the scaffolding and identification of student ability at the first stage of the cycle model, and the result suggests that GeoGebra does affect students' ability to make associations between representations of trigonometric functions. Specifically, the result suggests that when students are taught trigonometry using GeoGebra, their ability to make associations between representations of trigonometric functions will likely be better than when they are taught using the traditional chalk and talk method. The study reveals that the students taught with the help of teacher scaffolding through GeoGebra and cooperative learning are preferable to teaching only the topic with assistance on the GeoGebra (self, mentor scaffolding) without giving sufficient help for the students and cooperative learning, leaving the students in this space to initiate and/or engage in self-directed learning (Self-directed learning is based on the concept that individuals can take control of their learning by deciding what, when, where, and how to learn) around teaching and learning development simply by watching others (Mueller & Schroeder, 2018).

Discussion
The GeoGebra mathematical software can be used as a facilitation tool in the teaching and learning of mathematics both in groups or individuals by scaffolding, and more specifically of trigonometric functions, as there was a significant difference in the mean achievement of experimental groups A and B students on trigonometric functions as compared to the control group. In Table 2, Infit means inlier-sensitive or information-weighted. This approach proved to be more beneficial to the mean achievement than the cases of both experimental group A, in which only cooperative teaching and learning was employed, and control group, where the traditional teaching methods were employed. The use of the GeoGebra software not only increased student scores but it was also observed that the software enabled the realisation of an animated classroom where cooperative and collaborative activities of learning were evident in both experimental groups A and B. This finding is supported by various other researchers (Bakar et al., 2010;Doğan & Içel, 2011;Zengin et al., 2012). This improvement can be attributed to the experimental group's social constructivist learning environment, which stimulated students to interact, make conjectures, and eventually construct knowledge (Vygotsky & Cole, 1978). This becomes visible when an intermediate test is incorporated with the help of evaluative observations (Mueller & Schroeder, 2018). On associations of trigonometric functions, both experimental groups were found to have scored higher than the control group. Here we can deduce that the use of GeoGebra effectively improves the students' ability to make associations between different representations and contexts of trigonometry functions.
Most of the students in experimental groups in this study managed to plot the graphs by pen on the examination paper. However, in experimental group B this part occurs more. This result indicates that cooperative learning and teacher scaffolding together do more for creative learning and proves the proverb "together we can". This part was also investigated during the intermediate test in the classroom by an observation checklist. Concerning the interpretation and analysis of trigonometric functions, both experimental groups A and B was found to have performed significantly better than the control group with visible effect size d = 2.060. Here the use of GeoGebra was effective in improving the students' ability to interpret and analyse trigonometric functions. During the lesson in the experimental groups, the students only needed to type in equations that produced different trigonometric graphs using one GeoGebra window by hiding and unhiding objects in the algebraic view. This approach gave them time to explore, investigate and interpret the properties of the different graphs. This was unlike situations where students would have to draw graphs manually from point to point and then analyse them with pen and paper.

Conclusion
The results of this study showed that GeoGebra assisted instruction in the teaching of trigonometric functions and had significant effects on students' achievement. Specifically, the GeoGebra assisted instruction was more effective than the traditional chalk and talk method of teaching in improving students' achievement in making connections between representations of trigonometric functions and their interpretation of trigonometric function graphs. GeoGebra helped the students in the experimental group to better understand representations of trigonometric functions and graphs. The GeoGebra assisted instruction made the students gain more knowledge through exploration and active participation than the traditional teaching method. The findings of this study suggest several implications for teaching and learning mathematics in general and trigonometric functions. The current study is the first intervention study at Wachemo University. The study aimed to investigate the effect of GeoGebra in the teaching and learning of trigonometry with a focus on associations between representations and interpretations of trigonometric functions. Finally, the findings strengthened the principles and Standards for School Mathematics which states that effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well (Wilson et al., 2005).
To sum up, due to the cooperative learning and iterative nature of the intervention (experimental group A) with low teacher scaffolding, the learning of trigonometry known as migrant learning (Kong, 2012) and the cooperative and iterative nature of the intervention (experimental group B) with high teacher scaffolding. Also, due to the power of the GeoGebra software and its ease of use, all STEM students in the experimental groups have improved their trigonometric function achievement more than the control group and made more use of GeoGebra in their classes, shifting from teacher-centred to student-centred use of GeoGebra.

Recommendations
Based on the findings of this study and other studies, the author recommends the following: • First, we recommend that more studies be conducted on the effect of GeoGebra assisted instruction on students' achievement with a larger sample of students, at different grade levels and on different topics in mathematics. The findings from such large sample studies may be used to corroborate the findings of this study.
• Second, we recommend that teachers integrate GeoGebra with the teaching of trigonometric functions and mathematics in general. For teachers to integrate GeoGebra into mathematics teaching, they may need to be trained on general technology-oriented classroom skills and how to use GeoGebra in teaching (Bedada, 2020).
• Third, many teachers do not use a technology-oriented classroom because they do not know how to incorporate a technology-oriented classroom into teaching (Bedada & Machaba, 2022).
• Fourth, as the findings of the study were fruitful in improving the achievement of students in the STEM program, we recommend that for the coming summer holiday of 2022/23, all selected subjects in the STEM program may use the GeoGebra mathematical software to train students at Wachemo University, Ethiopia.

Availability of data and materials
The data and materials used in the current study are available from the corresponding author upon request.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Authors' contributions
Tola Bekene Bedada: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data and wrote the paper. Prof. France Machaba: Conceived and designed the experiments and more deeply coaches/supervise about the study and wrote the paper.