Using inquiry-based learning approach in the teaching process of divisibility rules

Abstract

The aim of this study is to reveal the impact of Inquiry-Based Learning (IBL) on students’ ability to come up with the rules of divisibility themselves and on their academic achievement in the 8th grade Mathematics course as well as to obtain students’ views on the learning and teaching process, through the use of a mixed-methods sequential explanatory design. The quantitative part of the study was composed of a one-group pre-test–post-test design, while the qualitative part involved a case study design. The sample group consisted of 20 students studying in the 8th grade of a secondary school in the 2020/21 academic year. At the end of the procedure, interviews were held with four students. The relevant data analysis indicated that the students’ level of readiness regarding the divisibility rules was not at a sufficient level, and that the difference between the pre-test and post-test scores was statistically significant with a high positive effect size on the students’ academic achievement. The results based on student views revealed that the rules forgotten due to memorization turned out to become permanent with the use of the IBL activities for learning the divisibility rules, that the students understood the subject matter better, that they came up with the divisibility rules themselves during the implementation of the activities, and that they enjoyed such a learning process.

Introduction

Memorization is the skill to keep the verbal or written information in mind when provided for an individual in such a way that it can be repeated exactly as it is whenever requested (Sekin, 2008). For that reason, it may be briefly defined as imitating rather than being creative with the information given (Senemoglu, 2013). It is also evident that the knowledge acquired through memorization is not likely to be permanent and it hinders the development of reasoning skills in individuals, resulting in a type of individual who ends up never questioning (Bonyah et al., 2023). Thus, it can be suggested that the main reason why students fail in many exams is such a rote learning system (Acikgul et al., 2015; McGuire, 2007). It is a well-known fact that when students acquire knowledge by inquiring rather than memorizing, they can achieve any meaningful learning permanently (Walker, 2007) and can successfully accomplish the transfer process of the acquired knowledge (Primas, 2011). In this respect, Inquiry-Based Learning (IBL) is an approach that allows students to question in such a way that it plays an active role in the development of their problem-solving and critical thinking skills (Stewart et al., 2013).

Inquiry-based learning

Literature review shows that the IBL is perceived differently among educators and researchers. Some educators such as Apps and Carter (2006), and Gilbert (2009) emphasize that the IBL is based on the philosophy of the constructivist paradigm, whereas Mills and Donnelly (2001) argue that it is a philosophical stance. Moreover, Wilhelm (2007) defines the IBL as a process, while Bell (2010) defines it as a teaching approach. For this reason, the distinction between approach, method and teaching technique is still debated even today, yet used as an approach in this study. The IBL can be defined as a teaching approach that includes many teaching methods in order to ensure permanent learning. Similar to this definition, Kuhn et al. (2000) stated that the purpose of inquiry is to enable children to set goals, realize and comprehend the cause and effect relationship in events, while Jessen (2017) reported that this approach encourages students to actively use available knowledge in the learning process. In this approach where knowledge is actively used, students create new knowledge by making use of previous one they have acquired and are involved in a student-centred learning process based on inquiry (Alberta Education, 1990). In this process where learning is prioritized rather than teaching, inquiry is defined as the development of a research plan, the creation of assumptions by investigating the information in line with the developed plan, the critical experience of the assumptions created using constructivist models, and the willingness of individuals to engage in peer discussions while dealing with problems (Linn et al., 2004). Therefore, the IBL is referred to as the main building block of good learning and can be defined as a learning process in which students continuously ask questions in the process of understanding a concept or a subject matter, conduct research to answer these questions and systematically structure the results of their research (Wood, 2003). It is also considered that the self-confidence levels and achievements of disadvantaged students, in particular, and those with low self-confidence will gradually improve as the number of practices through the use of the IBL in schools increases (Rocard et al., 2007). In a similar sense, Bruder and Prescott (2013) suggested that the use of the IBL approach in mathematics lessons could make the lessons exciting and interesting. Moreover, Barrett et al. (2005) emphasized that the IBL should be interpreted as a comprehensive education strategy. The reason for this could stem from the fact that the IBL approach is based on social constructivism and that knowledge ultimately tends to become permanent.

It is known that in an educational environment in which the IBL approach is engaged, students can simplify complex problems by making systematic observations, can make sense of the validity of the results by structuring hypotheses about the problem situations investigated and can establish relationships and connections in line with the consequences (Primas, 2011). Still, teachers play a critical role in creating such a learning environment (Blanchard et al., 2008). Apart from that, students must also fulfil their duties in this process in order to conduct the IBL process effectively. These tasks are presented as planning, organizing, processing, creating, sharing and evaluating in the IBL process (Alberta Learning, 2004, p. 7). Figure 1 illustrates how the learner should proceed at each stage.

Fig. 1

Student Tasks in the IBL Approach (Alberta Learning, 2004, p. 10)

The inquiry process constitutes one of the fundamental steps of learning, and if the steps presented in Fig. 1 can be fulfilled by students, they are likely to provide important opportunities for students to learn to investigate, to gain scientific thinking skills and to test their ideas both with their peers in the classroom and with their environment (Hofstein & Lunetta, 2004). Considering these positive attitudes towards the IBL approach, its role in mathematics education has been a matter of curiosity and as a result of the examinations, the scope of the studies has been gathered under three subject headings, which can be clarified as follows:

The studies examining the effect of the IBL approach on the competencies of the students have mostly reported that it yields positive results (Davis, 2018; Divrik et al., 2020; Radmehr et al., 2023; Salim & Tiawa, 2015) and the IBL approach, which is used with different teaching methods, is said to exert a significant impact on the students (Inel-Ekici & Ekici, 2022; Kusumawati et al., 2019; Suckoo & Ishizaka, 2022). (2) As a result of the studies in which the IBL approach is used as a tool in the development of the competencies of the teachers, it is often reported that the teachers have weaknesses in the application process of this approach and is, therefore, recommended that they be engaged in activities to improve themselves in this regard (Menezes et al., 2013; Rech et al., 2017; Sahin, 2019). (3) The studies that contribute to the development of the IBL approach by analysing it conceptually draw attention to the difficulties in the implementation processes and areas of the IBL approach (Crabtree, 2004), as well as other issues as regards how to prepare a plan and how to question about the given problem situations (Philippeaux-Pierre, 2009), the uses and benefits of learning theories in the IBL approach (Artigue & Blomhoj, 2013) and the preparation processes of the lesson plans for this approach (Bal Incebacak & Ersoy, 2019). When such classifications are taken into consideration, it can be asserted that the use of the IBL approach in the learning process affects students positively, yet the lack of activities and materials could be the reason why teachers do not master the IBL process (Ko & Mesa, 2014). For the purposes of the present study, original and unique lesson plans and worksheets were prepared for teaching the subject of divisibility rules. The present study is believed to contribute to the literature in the field of mathematics education when assisted by the IBL method. In addition, it has also investigated to what extent the IBL approach is effective on learning the subject of divisibility rules.

Divisibility rules

In the base-ten number system, each digit has a value independent of its position as well as a value dependent on its position (Chan et al., 2014). If each digit takes a value based on its position within the number, it represents the place value; if it takes a value independent of its position, it represents the number value (Vareles & Becker, 1997). For example, in the number 456, the digit 5 is in the tens place. Thus, it holds the value 50 (5 × 10) based on its position, while independently of its position in the number, it holds the value 5 (5 × 1). The first value is equal to the place value of the number 5; the second value is equal to the number value of the number 5. Therefore, understanding the value of each digit in a multi-digit number is essential both for solving mathematical problems and mastering arithmetic (Chan et al., 2014; Wearne & Hiebert, 1994). The concept of decomposition refers to analysing each digit based on its position within the number (Zazkis & Campbell, 1996). For example, the place values of the digits in the number 456 are calculated as 400 (4 × 100) for the 4, 50 (5 × 10) for the 5 and 6 (6 × 1) for the 6. The combination of these values, written as 400 + 50 + 6, represents the decomposed form of the number 456. Consequently, a thorough understanding of the concepts of number value, place value and decomposition can aid in dispelling misconceptions regarding divisibility rules (Posamentier, 2003; Potgieter & Blignaut, 2018).

Divisibility rules are crucial for comprehending the multiplicative structure of natural numbers and their interconnections (Zazkis & Campbell, 1996). When the elementary mathematics curriculum of the Ministry of National Education (MoNE) (2018a) is examined, it can be seen that the subject of divisibility rules is to be taught for the first time at the elementary level in Grade 6 under the sub-unit of Multipliers and Multiples contained in the unit of Numbers and Operations. As a result of examining the ways of instructing the rules of divisibility on the basis of intended learning outcomes (ILOs), no other outcome has been identified except for the following: “M.6.1.2.2 Students will be able to explain and use the rules of divisibility by 2, 3, 4, 5, 6, 9 and 10 without remainders”, and two conditions for the identified ILO are stated: “(a) It is to be taken into account that the rule of divisibility by 6 without a remainder can be developed by using the rule of divisibility by 2 and 3 without a remainder. (b) Letter expressions are not used for the rules of divisibility”. When the secondary mathematics curriculum of the MoNE (2018b) is examined, it can be understood that the subject of divisibility rules is aimed to be taught for the first time at the high school level in Grade 9 under the sub-unit of the Equations and Inequalities contained in the unit of Numbers and Algebra. The examination of the subject matter of the rules of divisibility on the basis of the ILOs shows that there is no other ILO involved at the high school level except for the following: “9.3.2.1. Students will be able to solve problems related to the rules of divisibility in whole numbers”, and the requirement for this outcome is as follows: “The divisibility rules of numbers such as 2, 3, 4, 5, 8, 9, 10, and 11, and 6, 12, 15 obtained from these numbers are to be taken into account” A comparison of the elementary mathematics curriculum and that of the secondary on the basis of the subject of divisibility shows that the divisibility rules for the numbers 2, 3, 4, 5, 9 and 10 are common at both grade levels. The difference is that the high school curriculum includes the divisibility rules of the numbers 8 and 11 and the divisibility rules of the numbers in the form of the product of different primes other than the number 6. In addition, while there is a condition that letter expressions are not included at the secondary school level, it is not required at the high school level.

When the 6th Grade mathematics textbook by the MoNE (2020a) is examined, it can be seen that the rules of divisibility are presented through making generalizations from specific examples to all numbers. For example, it has been verified that the numbers whose last digits are 0, 2, 4, 6, 8 can be divided by 2 without remainder using the given number 38, that the numbers whose sum of digits is 3 or its multiple can be divided by 3 without remainder using the number 48, and that the numbers that are both divisible by 2 and 3 without remainder can be divided by 6 without remainder using the number 96. The validity of each rule is shown with the specific examples used. Moreover, the 9th grade mathematics textbook of the MoNE (2020b) includes the information that the decomposition can be performed by using letter expressions for teaching the divisibility rules of the numbers 3, 4, 8 and 11. Since the rules devised from the analysis of three (ABC) and four (ABCD) digit numbers are generalizable to all situations, any rule is supported with the necessary examples once it is devised. In addition, the validity of the divisibility rules for the numbers 2, 5, 9 and 10 are presented with examples. In addition, the divisibility rules of the relatively prime numbers are supported with examples by adhering to the following statement: “A natural number that is divisible by each of its prime factors is also exactly divisible by the product of these numbers”. However, when both the elementary mathematics curriculum and the 6th Grade mathematics textbook are examined, it is seen that the traditional teaching method is used for teaching the divisibility rules and students try to learn through memorizing the given examples.

As can be found in the literature on divisibility rules, students’ memorization often seems to result in weaknesses in the conceptual knowledge to explain the reason for the given situation, as well as seeing and explaining relationships (Togrul, 2014), leading them to believe that the letters used in algebra also have a place value (Akkaya & Durmus, 2006). Research has also shown that there could be a number of misconceptions in the process of division (Roche & Clarke, 2013; West, 2014), that the students are poor in grouping numbers with similar rules and writing such rules (Aytas & Ugurel, 2016; Potgieter & Blignaut, 2018), and that they over-generalize and apply the divisibility rules incorrectly (Zazkis & Campbell, 1996). The relevant literature review on the subject of divisibility rules has also indicated that most studies have been focussed on the sub-learning domain of multipliers and multiples (Lee et al., 2012; Togrul, 2014), and others have been conducted to present some divisibility rules only (Ibrahimpašić et al., 2011; Peretti, 2015; Young-Loveridge & Mills, 2012; Zazkis, 1999). Posamentier (2003) emphasizes that these rules should not be presented as something that needs to be memorized and that students should follow a process that allows them to make sense of the basis of the rules. However, it appears that students are not given enough opportunities to develop their own strategies to come up with the applicable rules (West, 2014).

As it is widely recognized that students’ prediction skills improve in learning processes in which the IBL is used (Primas, 2011), this study employed the approach so that the students will be able to learn the basics and nature of the subject of divisibility rules by avoiding memorization, develop their metacognitive knowledge and skills and consciously manage their own learning processes.

For this reason, it is suggested that the approach used in the study will produce better outcomes than those obtained when students memorize a set of given rules. Considering the importance and difficulty of teaching the divisibility rules, it is believed that the present study is important in that it will significantly contribute to the literature.

Aim of the study

As mentioned above, this study aimed to reveal the impact of Inquiry-Based Learning on students’ ability to come up with the divisibility rules by themselves and on their academic achievement in the 8th grade mathematics course as well as to reveal their views on the learning and teaching process. Based on the general research question of this study, which reads, “What is the impact of Inquiry-Based Instruction on students’ ability to come up with the divisibility rules by themselves and on their academic achievement in the 8th grade mathematics course, and what are their views on the learning and teaching process?”, the relevant sub-questions to be analysed can be listed as follows:

• 1. What are the students’ levels of readiness regarding the divisibility rules?

• 2. Is there a statistically significant difference between the pre-test and post-test mean scores of the students in the “Achievement Test in Learning the Divisibility Rules in the Secondary School 8th Grade Mathematics Course”?

• 3. What are the students’ levels of achievement as a result of using the worksheets for coming up with the Divisibility Rules through Inquiry-Based Learning?

• 4. What are the students’ views on the mathematics course regarding the teaching of divisibility rules) and the learning and teaching process?

Method

This section provides information about the model and design of the present study, the sample group, data collection tools used in the evaluation of the students in the sample group, and how the data were analysed, in addition to the development processes of the tools.

Research model and design

This study employed a sequential exploratory design—one of the mixed research method designs-, which consists of two stages: quantitative and qualitative process. The second stage is formed by following the results of the quantitative data collected in the first stage. The results of the qualitative data obtained in the second stage help to explain the results of the quantitative data in the first stage (Creswell & Plano Clark, 2017). The aim of this design is to explain the relationships covered by quantitative data in a qualitative phase. For that reason, in the first stage, the researchers prepared a process that enabled the collection and analysis of quantitative data. In the quantitative part, the one-group pre-test and post-test design was utilized as one of the experimental designs; the students were administered the “Achievement Test in Learning the Divisibility Rules” (ATDR) as a pre-test and post-test.

The reason for choosing this research design is that—due to the COVID-19 pandemic—the students voluntarily came to the school where the study was carried out and they did not want to be included in the study due to health measures. In order to overcome the weaknesses of the design, another process was planned as a second phase, which guided the collection and analysis of qualitative data through quantitative findings. The qualitative part consisted of student opinions, by employing a case study design. A case study is used to answer the questions of ‘how’ and ‘why’ on current situations (Yin, 2009).

Sample group

With the outbreak of COVID-19 (SARS-Cov-2) in Turkey on 11 March 2020, education at all levels was suspended on 16 March 2020. In this process, education processes started to be carried out remotely with the 'EBA/Education Information Network' system in order not to interrupt education. As the effects of the pandemic decreased in Turkey, coeducation (face-to-face and distance) was gradually introduced in grades 2, 3, 4, 8 and 12 (MoNE, 2020c). Therefore, a number of reasons were taken into consideration in determining the study sample: (i) due to the COVID-19 pandemic conditions, only the 8th grade students in secondary schools receive face-to-face education, (ii) the introduction of the subject of divisibility rules in the relevant Elementary Mathematics Curricula, (iii) the relationship between the examples given in the relevant mathematics textbooks on the divisibility rules and the grade level and (iv) the further instruction of the subject matter of “decomposition”, which plays an active role in proof-finding for the divisibility rule of any number, at the last grade level of secondary school. It was, therefore, deemed appropriate that the students in the study sample would be at the 8th Grade level. For this purpose, interviews were held with the secondary school administrators and class teachers in Ortaca district of Muğla, Türkiye. The school administrator was interviewed at the school of a mathematics teacher who had agreed to participate in the study. In line with the interviews, the school administrators stated that “only those who volunteered came to the school due to health precautions, but their access to additional reinforcement processes such as tutoring and training courses was limited and that there would be two active 8th Grade classes (28 and 14 students, each respectively) during the implementation process”.

It was decided that the study sample would be the class with 28 students as the participating mathematics teacher mentioned being “the classroom teacher for the one with 28 students for four years, which could help obtain more information about the achievement of the students”. Then, the students in the selected classroom were informed about the content, duration and implementation of the study and were distributed the Participant Consent Forms. In addition, due to the COVID-19 pandemic, Parent Approval Forms were sent to the parents with the students. Given the feedback received, it was thought that students should not lag behind educationally during the COVID-19 pandemic, and the study was, therefore, conducted in a secondary school with a middle socioeconomic level in Ortaca district of Muğla, Türkiye, in the 2020/21 academic year. The sample group consisted of a total of 20 students: 8 girls and 12 boys, all studying in the 8th grade. All of the students were included in the quantitative part of the study, but the interviews in the qualitative part were conducted with a total of 4 students (A–B–C–D): 2 girls (A–D) and 2 boys (B–C), with the completion of the worksheets prepared in line with the IBL process. Maximum diversity sampling was used as one of the purposive sampling methods in order to select 4 people who participated in the interview from among 20 people. In this context, each individual in the sample of 20 students was first subjected to ranking in terms of achievement (A–B–C–D, with A being the most successful-…-D the least successful), through which the quantitative data were collected. As a result of the letter order, the groups were divided into five students, one of whom was selected from each group. The rationale behind this choice was to understand the effects of the topic in terms of students at different achievement levels.

Data collection tools

The available quantitative data were collected with the data collection tool called “Achievement Test in Learning the Divisibility Rules (ATDR)”, and qualitative data were collected with a “Semi-Structured Interview Form” and “Worksheets for Inquiry-Based Learning”.

Achievement test in learning the divisibility rules (ATDR)

The ATDR developed by Teke (2021) was used to evaluate the effectiveness of the instruction and to reveal the achievements of the students. In the test, there are 28 items in total for five intended learning outcomes related to the subject of divisibility rules. A pilot study involving 378 8th grade secondary school students was conducted to assess the validity and applicability of the ATDR. Based on the students’ answers, the item discrimination and item difficulty indices of the questions in the test were examined in such a way as to determine the questions to be included in the final test. In conclusion, five items with an item discrimination index lower than 0.39, as well as three items classified as 'difficult' due to the inclusion of letter expressions in the divisibility rules of the numbers 8 and 12, were removed from the test. At the end of the process, the ATDR consisted of 20 items, with item discrimination indices (rj) ranging from 0.41 to 0.91 and item difficulty indices (pj) ranging from 0.41 to 0.85. The average discrimination value of the test was 0.67, the average difficulty value was 0.62 and the KR-20 value calculated for the reliability of the test was 0.84. In the scoring of the ATDR, incorrect responses were scored as “0”, while correct ones as “1”. The analysis results revealed that the internal consistency of the test was good, the effectiveness of the teaching method was easy and its discriminatory power was high. Additionally, as a result of the pilot study, students completed the test in an average of 30 min. Therefore, a 30-min duration was deemed appropriate for the final version of the ATDR. Two sample items in the ATDR are given below (Fig. 2).

Fig. 2

Sample Items in the ATDR

Interview form for evaluation of inquiry-based learning process (IFEIBL)

A total of 10 questions were prepared for the draft version of the IFEIBL prepared by the researcher. The draft form was then presented to a team of six experts consisting of one professor, two associate professors, two Mathematics teachers and one Turkish teacher, three of whose expertise is in mathematics education, two in elementary mathematics teaching and one in Turkish teaching. The experts were asked to provide feedback on the comprehensibility of the texts in the IFEIBL, the compatibility of the questions in line with the purpose of the study, and whether or not the number of question items was adequate. As a result of the feedback, the number of questions increased to 13 after corrections were made, due to which the interview form was again submitted to expert opinion in line with the same criteria and finalized in accordance with the feedback. After the IBL activities were completed, the interviews were conducted one-on-one with the students in the teachers’ room of the relevant school. The interviews, which lasted approximately 20–25 min, were recorded using a voice recorder with the consent of the participants; the recordings were then transcribed and analysed. Four sample questions in the IFEIBL are given below (Fig. 3).

Fig. 3

Sample questions in the IFEIBL

Worksheets and lesson plans for inquiry-based learning

A total of three worksheets were prepared by the researchers in order to measure the inquiry skills of the students on the subject of divisibility rules (a sample worksheet is given in Annex 1). In order to test the validity and applicability of the worksheets prepared in the draft form, expert opinions were obtained from an associate professor who is an expert in the field of mathematics education, two elementary mathematics teachers and a Turkish teacher. The experts were asked to provide feedback on the comprehensibility of the worksheets, the compatibility of the questions prepared in line with the purpose of the study and how the processes designed in each worksheet could be improved. By taking the feedback into consideration, we decided that the question stem of some questions needed to be changed, some linguistic corrections needed to be made and the fictionalized processes needed to be narrated. As a result of the expert opinions, the worksheets were finalized before the activities by making the necessary corrections in line with the linguistic, fictional and study purposes; lesson plans were prepared by the researchers for the worksheets. In this process, the same experts who tested the validity and applicability of the worksheets were asked to give feedback on the linguistic adequacy and compatibility of these plans with the purpose of the study, and on as to how the six stages presented by Alberta Learning (2004, p. 10) should be expressed, including how students should follow these stages. As a result of the feedback, the resources were organized, the process phase was explained in a clear way, and the questions were specified to inform students about which ones to share during the sharing phase.

Procedure

In the preparation phase of the ATDR, the ILO that writes “M.6.1.2.2: Students will be able to explain and use the divisibility rule for 2, 3, 4, 5, 6, 9, and 10 without a remainder”, was analysed, considering the subject of Divisibility Rules in the Mathematics Curriculum (MoNE, 2018a). Based on this, the following ILOs were prepared:

• Students will be able to come up with the divisibility rules for 2 and 5 and solve related problems.

• Students will be able to come up with the divisibility rule for 4 and solve the related problems.

• Students will be able to come up with the divisibility rules for 8 and 11 and solve the related problems.

• Students will be able to come up with the divisibility rules for the numbers that can be written as the product of two different prime numbers and solve the related problems.

Before proceeding to the preparation stage of the Worksheets for Inquiry-Based Learning, the relevant literature was reviewed as well as books and online web-sites. At the end of the study, an idea emerged about how the Worksheets for Inquiry-Based Learning should be prepared. In this context, three worksheets were prepared by the researchers and made ready for expert opinion. The drawbacks in the worksheets submitted to the expert opinion were corrected after the feedback and given its final form before the implementation. In addition, lesson plans were prepared for the three worksheets. The plans were compared with the sample lesson plans in the related literature (Bal Incebacak & Ersoy, 2019) and given their final form.

Before proceeding to the preparation stage of the Semi-Structured Interview Form, research similar to the preliminary research conducted for the preparation of the worksheets was conducted. After the quantitative data were collected, a Semi-Structured Interview Form was prepared by the researchers in order to complete the quantitative data in accordance with the sequential explanatory design.

Prior to the procedure, the necessary permissions were obtained and the school where the procedure would be carried out was selected. After determining the school, the relevant school administration, the mathematics teacher and the students were given the necessary information about the process. The dates/days, course hours and activities deemed appropriate by the school administration are presented below in Table 1.

Table 1 Timeline of the study

Related curricula, textbooks and daily plans of the classroom mathematics teacher were examined in view of the divisibility rules. It was seen that four lesson hours at the 6th grade level and five lesson hours at the 9th grade level were assigned for teaching the rules of divisibility. Therefore, a total of 12 lesson hours were deemed sufficient for the purpose of this study.

On the first day of the procedure, the ATDR was administered as a pre-test to determine the readiness levels of the students regarding the rules of divisibility. After the procedure, students were instructed about the concepts of “decomposition, number value, and place value”, which are the basis of divisibility rules. Afterwards, the activities prepared for those concepts were carried out with the participation of all students who were also given homework at the end of the lesson. The first day was then completed.

After the completion of the first day, the responses in the ATDR were analysed. The students were then ranked according to their ATDR scores, report card grades and teacher opinions (A–B–C–D, A being the most successful-…-D the least successful). Instead of students’ names, a code was assigned to each student. Table 2 below presents the contents of the groups.

Table 2 Group content created as to the IBL application

On the second day of the procedure, the assignments were checked first. Then, the classroom was organized so that the worksheet activity for the IBL, which included the divisibility rules for the numbers 2 and 5, could be conducted. At the beginning of the lesson, the students were handed out the worksheets so that they could fill them in under the guidance of the researcher. On the third and fourth days of the procedure, just like the second day, the IBL-related worksheets with the divisibility rules for the numbers 3 and 9 and the one with the divisibility rule for the number 4 were distributed to the groups so that they could fill them in again under the guidance of the researcher. The researcher acted as a guide in this process and also collected research data. A sample picture of the procedure of the IBL-related worksheets is given below (Fig. 4).

Fig. 4

Administering the worksheets for inquiry-based learning

At the end of the procedure, the ATDR was administered to the students as a post-test in order to determine the impact of the IBL on the ability of students to come up with the divisibility rules by themselves and to specify the level of students’ achieving the intended learning outcomes. After the activity, four students were selected for semi-structured interviews, and the interviews were recorded with the consent of the participants.

Data analysis

Providing an explanation of the triangulation strategy employed in the study prior to analysing both quantitative and qualitative data strengthens confidence in the findings (Merriam, 2009). A triangulation strategy refers to the integration of different data sources, theoretical perspectives, researchers, or methodologies to address a research question. Specifically, it involves (1) collecting data from different times, places or people (data triangulation), (2) combining various theoretical perspectives (theory triangulation), (3) involving multiple researchers in data collection or analysis (investigator triangulation) and (4) applying different methodologies to the same topic (methodological triangulation) (Denzin, 1978). In this study, investigator triangulation was used. The aim here is to verify the data by cross-checking and to eliminate the possibility of bias in the results (Thurmond, 2001).

Analysis of quantitative data

The analysis of the quantitative data was conducted on the pre-test and post-test scores collected through the ATDR. Statistical operations performed during the final version of the ATDR used in the present study were carried out with the TestAn programme developed by Aydin and Ergun (2015) after obtaining the necessary permissions. For using this valid and free-of-charge programme, a special login code was first created for the users. Descriptive analyses of the question items in the test were made through the programme, and item difficulty, item discrimination and reliability calculations were made with option analyses for each item separately.

During the application process, the pre-test and post-test scores of the ATDR were entered into the IBM SPSS 23.0 package programme so as to test whether or not they showed a normal distribution. Since the number of people in the study sample was less than 50 (20), whether or not the results showed a normal distribution was determined by examining the Shapiro-Wilks test (Buyukozturk et al., 2016) and the value obtained by dividing the kurtosis-skewness coefficients by the standard error. In general, the fact that the value obtained by dividing the kurtosis and skewness values by the standard error is between ± 1.96 indicates that there is a normal distribution (Kim, 2013). The Shapiro–Wilk test results and descriptive statistics of dividing the kurtosis and skewness values by the standard error values are provided below (Table 3).

Table 3 ATDR normality analysis results

As seen in Table 4, the value obtained by dividing the kurtosis and skewness values of the ATDR pre-test-post-test score distributions by the standard error is ± 1.96. For this reason, the pre-test and post-test data of the ATDR were found to have a normal distribution as a result of considering the Shapiro–Wilk test results and z-score values. Thus, the dependent samples t test was used to identify whether there was a significant difference between the pre-test and post-test mean scores in the ATDR. If there was a significant difference between the mean scores of the pre-test and post-test, the effect size of this difference on student achievement was calculated using Cohen's d formula.

Table 4 ATDR descriptive statistics values

Analysis of qualitative data

Content analysis technique (Cohen et al., 2007) was used in the analysis of student views on the IBL process. For such a purpose, reliability is normally ensured by including direct quotations while explaining the interview data (Yildirim & Simsek, 2013). The interviews were carried out one-on-one in the teachers’ room of the relevant school; they lasted approximately 20–25 min and were recorded using a voice recorder with the consent of the participants. The recordings were then listened to and transcribed by the researchers. Data with similar features in the analysis of the interviews were combined under the themes and categories developed by the researcher and given codes accordingly. In order to ensure reliability in the coding process, cross-coding was conducted along with a professor who is an expert in the field of mathematics education, after which the negotiation approach was adopted. At this stage, the “Inter-Rater Reliability Formula” developed by Miles and Huberman (1994) was used to calculate the reliability, which can be presented as follows:

$${\text{Reliability}} = \frac{{{\text{Agreement}}}}{{{\text{Agreement}} + {\text{Disagreement}}}}$$

The consistency between the coders using the formula was examined and the reliability was found to be 92%. According to Miles and Huberman (1994), a reliability value of 90% or more is sufficient. In this sense, the qualitative analysis reliability score of 92% indicates that the study is reliable.

Validity and reliability

The concept of validity has been approached and debated in distinct ways within both quantitative and qualitative research methodologies. For example, Pallant (2011) characterizes validity in quantitative research as the alignment of the research process with the specified aims, whereas Creswell (2007) referred to validity in qualitative research as accuracy or credibility. Within this framework, validity pertains to the alignment of the research with its stated objectives, its trustworthiness and the veracity of its findings (Zohrabi, 2013). For this reason, researchers have analysed validity as internal validity and external validity. Internal validity is the degree to which research findings accurately reflect the true state of affairs (Heiman, 2001), whereas external validity concerns the generalizability of findings to other populations, contexts or constructs (Gay et al., 2009).

The concept of reliability, like that of validity, has been approached and debated in distinct ways within both quantitative and qualitative research methodologies. For instance, Pallant (2011) defines reliability in quantitative studies as the consistency of results, whereas Ihantola and Kihn (2011) describe reliability in qualitative studies as the consistency between what is intended to be measured and what is actually measured. Reliability in this sense means that research findings are consistent, can be replicated and are trustworthy (Zohrabi, 2013). For this reason, researchers have analysed reliability as internal reliability and external reliability. Internal reliability refers to the consistency within the research process, from data collection to interpretation, while external reliability emphasizes the reproducibility of the research findings (Nunan, 1992). Table 5 presents the studies on validity and reliability conducted during the data analysis phase of this research.

Table 5 Validity and reliability in data analysis

An application was made to the Ethics Committee on 18.08.2020 for research compliance, which was then accepted with the decision number 2020/530 on 26.08.2020.

Results

Results related to the first sub-questions of the study

In order to determine the readiness level of students regarding the divisibility rules, relevant questions were specified in the ATDR related to the following ILO as stated by the Ministry of National Education (2018a), which reads “M.6.1.2.2: Students will be able to explain and use the divisibility rules for 2, 3, 4, 5, 6, 9, and 10 without remainder”. Afterwards, the correct answer rates [item difficulty indices (pj)] for the questions among the pre-test answers collected from the ATDR were calculated. The relevant literature has shown that the values of pj estimated between 0.70 and 1.00 indicate that the related question is likely to be answered correctly (Atılgan et al., 2016). For this reason, the pj mean value was determined as 0.70 in order to consider the readiness level of the students as high on the subject matter of divisibility rules.

As shown in Table 6, the readiness levels of the students in Item 1, Item 7, Item 10 and Item 17 (simple questions requiring knowledge of divisibility rules) in the ATDR were high, but such levels in Item 4, Item 5, Item 11 and Item 15 (questions requiring inquiry skills) remained below the accepted limit. The pj values of the eight identified questions were calculated and the mean pj value was found to be 0.66. Since this value was below the accepted limit, it was determined that the students’ level of readiness for the ILO, which reads “M.6.1.2.2: Students will be able to explain and use the divisibility rules for 2, 3, 4, 5, 6, 9, and 10 without remainder”, was not at a sufficient level.

Table 6 Readiness levels of the students in the sample group

Results regarding the second sub-questions of the study

Dependent samples t test was used to determine whether or not there was a statistical significant difference between the pre-test and post-test mean scores on the ATDR.

Table 7 shows that the difference between the pre-test and post-test scores of the students was statistically significant (p < 0.05), and that the mean post-test scores (78.00) was considerably higher than that of the pre-test scores (52.75). The administration of the IBL activities to students with low level of readiness for the subject of divisibility rules affected their achievement in the divisibility rules. When the mean scores are examined, this difference seems to be in favour of positive post-test scores. Thus, it can be suggested that the use of IBL activities for learning the divisibility rules has a positive effect on students’ achievement. In order to determine the extent of this impact, the effect size was calculated with Cohen’s d formula (d = 1.16 > 0.80). In the interpretation of the calculated effect size value, the effect size can be defined as weak if d ≤ 0.2, medium if 0.2 < d < 0.8 and high if d ≥ 0.8 (Cohen, 1988). As a result, it can be asserted that the application of the IBL activities for teaching the rules of divisibility has a high effect size on students’ achievement.

Table 7 Dependent samples t test results on ATDR

Results regarding the third sub-questions of the study

In order to determine the level of students’ achievement in relation to five ILOs prepared for teaching the divisibility rules, the item difficulty indices of 20 questions were found from the pre-test and post-test data collected from the ATDR. The calculated values were determined as the level of attainment of the relevant ILO by the students in the sample group. Research has shown that the level of attainment for any relevant ILO should be at least 0.75 (Baykul, 2000). The level of attainment of the students’ achieving an ILO was also determined as 0.75 in the present study.

As shown in Table 8, the pj pre-test and pj post-test scores of the students for the ILO, which reads “Students will be able to come up with the divisibility rules for 2 and 5 themselves and solve the related problems” were 0.40 and 0.90, respectively; those for the ILO, which reads “Students will be able to come up with the divisibility rule for the number 4 by themselves and solve the related problems” were 0.35 and 0.75, respectively; those for the ILO, which reads “Students will be able to come up with the divisibility rules for 3 and 9 themselves and solve the related problems” were 0.40 and 0.95, respectively; for the ILO, which reads “Students will be able to find out the divisibility rules for 8 and 11 themselves and solve the related problems” were 0.35 and 0.75, respectively, and for the ILO, which reads “Students will be able to express the divisibility rules of the number being in the form of a multiplication of two different primes, and solve related problems” were 0.25 and 0.90, respectively. As a result, the students ended up achieving all the ILOs at a level of 0.75 and above.

Table 8 Students’ attainment levels in the given ILOs

Results regarding the fourth sub-questions of the study

Table 9 presents the results of the analysis of the students’ views reflecting their attitude towards the mathematics lesson.

Table 9 The analysis of the students’ views reflecting their attitude towards the mathematics lesson

As seen in Table 9, in the teaching of mathematics lessons, the teachers first explained the subject and then, a number of questions were answered by putting the students in the centre through reinforcements such as homework and tests in an effort to reinforce the subject. From the student feedback, it can be considered that the lesson process in the classroom where the study sample took part was carried out in accordance with the traditional teaching method. When the students were asked how they learned mathematics best, S3 and S4 indicated that they learned better through inquiry in the mathematics lesson and that the reason for choosing the inquiry method was that the information could easily be forgotten when memorized.

Table 10 provides the results of the analysis of the students’ views on the application of the inquiry-based learning approach.

Table 10 The analysis of students’ views on the application of inquiry-based learning approach

As Table 10 shows, when the IBL was put forward, the students appeared to believe that they had tried memorization before but did not use inquiry at all and said that they were able to learn unknown things by inquiring, and thus they were happy by making new explorations. From this feedback, it is thought that students associated the approach with expressions they had not learned before. When the students were asked what they thought about the use of the IBL in mathematics lessons, some students said that it should be used, but S2 said that it should be used at a moderate level. S3 and S4, on the other hand, stated that they liked the approach very much since it ensured permanent learning. Furthermore, when the students were asked whether they wanted the IBL approach to be engaged in all courses, some students said that they wanted the approach to be implemented as they could find out where a result came from thanks to questioning, and that if memorized, information is immediately forgotten but can be easily remembered thanks to the IBL. However, S2 did not want the approach to be employed in all courses. Although students appeared to believe that the IBL had such positive effects as enabling the permanence of learning, it is to be noted that some students may be negatively affected by the process as a result of long-term use of the approach.

Table 11 below illustrates the results of the analysis of the students’ views on the learning process of divisibility rules.

Table 11 The analysis of students’ views on the learning process of divisibility rules

As can be seen in Table 11, all students stated that they had previously studied the divisibility rules by memorization. In particular, S3 emphasized having difficulty in such a memorization stage, and S4 stated that memorizing the rules was wrong. When asked about which concept the divisibility rules might be based on, all students indicated that the concept could be decomposition. In addition, S4 appeared to like the maths operations performed with the concept of decomposition. When the students were asked to explain the divisibility rule of any number by giving an example by considering the activities in the study process, S1 preferred to explain the divisibility rule by 2. In the explanation part, S1 started the process by solving the three-digit ABC number and then analysed whether or not each digit of the number was divisible by 2. Further, S2 explained the divisibility rule of the number 4. In fact, this student analysed the 3-digit ABC number and separated the number 10B as 8B + 2B, adding that 100A and 8B are a multiple of 4, so if 2B + C is a multiple of 4, then the number ABC can be divided by 4. Thus, it seems clear that S2 came up with a rule by herself, although such a rule had never been mentioned in textbooks before, even for the purpose of memorization. When the students were asked to group the divisibility rules by 2, 3, 4, 5 and 9, all students said that the divisibility rule by 2 and 5 was similar when the last digit of any number was looked at, that the divisibility rule by 3 and 9 is similar based on the sum of their values, and that the divisibility rule by 4 can be considered separately.

The students stated that the memorization method had previously been used to teach them the divisibility rules and that they had difficulties in that process. However, it was observed that they came up with the rules of divisibility and established relationships between the rules when instructed with the IBL approach. Therefore, it can be considered that the approach played an important role in developing students’ inquiry skills.

Table 12 below provides the results of the analysis of the students’ views on the use of IBL activities for teaching the divisibility rules.

Table 12 The Analysis of Students’ Views on Utilizing Inquiry-Based Learning Activities for Teaching the Divisibility Rules

As can be seen in Table 12, all students stated that using the IBL activities for the purpose of learning divisibility rules were beneficial. In their answers, some students stated that they kept forgetting the divisibility rules when they memorized them, yet that thanks to the IBL method, the divisibility rules became memorable and permanent. As a result of the analysis of the students’ views on the method choices for the purpose of teaching the subject of divisibility rules, all students ended up having chosen the IBL approach. Based on the relevant student views, S1 in particular favoured the IBL approach more, just like S2, who also liked the IBL approach more than the memorization technique. When asked about their feelings while proving where the divisibility rules came from, S1, for example, said that she would not forget the divisibility rule for any number once she found it; S2 said that when she learned to carry out inquiry, she also learned how numbers were formed, S3 asserted that he learned new formulas thanks to the inquiry method, and that they would become memorable because he found out the formulas himself. The students stated that the approach used was more effective than memorization when learning the divisibility rules. The reason for such statements may be that teachers often choose the memorization method to teach students the related subject, and therefore, students experience difficulties in the process and end up forgetting the rules over time.

Discussion

As a result of the analysis of the first sub-problem of the study, it was concluded that the students’ readiness level for the ILO for Grade 6 was not at a sufficient level as their readiness level turned out to be below the acceptable limit. The reason for this result can be attributed to the fact that memorization method is often chosen to teach the divisibility rules, as stated by the students during the interviews. The relevant literature review points to the presence of a number of studies showing that memorization in teaching divisibility rules has negative consequences. In this connection, Zazkis and Campbell (1996) reported that students over-generalized and misapplied the rules as a result of memorization; Togrul (2014) found that students lacked conceptual knowledge about explaining the reason for the given situation and seeing relationships; West (2014) found that students included some misconceptions in the division process, and Aytas and Ugurel (2016) concluded that students made mistakes in grouping the numbers with similar divisibility rules. Such findings support the results of the current study.

In this study, the difference between the students’ ATDR pre-test and post-test scores was statistically significant to the extent that was in favour of the post-test scores. It was observed that the students in the study sample did not have adequate readiness levels in the questions related to the 6th Grade ILO (Question 4, Question 5, Question 11 and Question 15 in the ATDR), which they were assumed to have previously learned at the beginning of the process, but as a result of the implementation of the IBL activities, they seemed to have achieved all the learning outcomes and even succeeded in the 9th grade learning outcomes that they were not assumed to have previously learned. It can thus be suggested that the use of IBL activities for teaching the subject of divisibility rules has a significant impact on increasing students’ achievement. Given the students’ opinions regarding this conclusion, it is clear that the students did not make inquiries before the implementation of the IBL activities, but they did so during the implementation of the activities, a situation which they appeared to enjoy. Most students stated that this approach should be used in mathematics and other courses, emphasizing the necessity of adopting the IBL approach for teaching the divisibility rules rather than memorization, and stating that they were excited and happy while proving the divisibility rules and that they would not forget the process as the approach helped permanent learning. The literature review related to these results also shows that the IBL approach exerts a positive impact on students’ ability to participate in research and group discussions (Laudano et al., 2019; Radmehr et al., 2023), on their creative thinking (Sreejun & Chatwattana, 2023) and problem-solving and problem-posing skills (Divrik et al., 2020), that it increases their achievement levels more than those of some other groups with whom they were compared (Li et al., 2009; Salim & Tiawa, 2015; Sen et al., 2021), that it positively affects their academic achievement in different courses, their attitudes towards these courses, and the permanence of learning (McCarthy, 2005; Park, 2015; Rubio & Conesa, 2022; Sonsun et al., 2023; Suits, 2004), and that it develops positive attitudes towards the applied approach (Karademir & Akman, 2021; Laudano et al., 2019; Lenz, 2015; Li et al., 2009; Sen et al., 2021).

Considering the analysis of the student views, the students appeared to be of the opinion that the divisibility rules, which they had forgotten due to memorization, became permanent, thanks to the IBL activities. In fact, while explaining the divisibility rule for the number 4, one student (T2) created a rule different from the rules expressed both in the relevant textbooks and by the teachers, a situation similar to what was reported in Chakraborty’s (2007) study. Grundmeier et al. (2022) emphasize that the IBL plays an active role in the formation of the awareness of proof in students and that it can enable students to enjoy mathematics owing to being able to establish the relationship of prediction-hypothesis-synthesis, which is the basis of mathematics (McLoughlin, 2008a, 2008b). It is mentioned in the literature that there is a positive correlation between proof and mathematics, which has a positive contribution to learning mathematics (Conner & Krejci, 2022; Kogce, 2013). In this respect, it can be suggested that students’ proof-making skills for the related subject may have improved, a result which will make a positive contribution to mathematics teaching in line with the data obtained from the interviews.

Limitations and future research

The present study has some limitations. The first limitation is the effects and consequences of the COVID-19 pandemic, which coincided with the period of the study. The study was, therefore, designed according to a one-group pre-test–post-test design and conducted with a study sample consisting of a total of 20 students. In order to overcome this limitation, it is thought that conducting a full experimental study in the future by increasing the number of the study sample and including a control group that learns the divisibility content in traditional ways will highlight the pre- and post-test results.

The second limitation is that the way of selecting the participants in the quantitative part was not the same as that of the qualitative part. In order to overcome this limitation, in the future, the number of students to participate in the quantitative part can be increased and the selection criteria be expanded. The third limitation is the short duration of the total lesson hours devoted to teaching the divisibility rules. In order to overcome this limitation, similar studies can be conducted in the future in the sub-learning area of Multiples and Multipliers, which covers the subject of divisibility rules.

Conclusion and implications

This study drew upon data collected from 20 secondary students in order to analyse the students’ inquiry-based learning outcomes. The effect of the IBL approach on the students’ process of learning the divisibility rules was investigated and it was aimed to make sense of the process through interviews. The results revealed that the students preferred the approach rather than memorization for learning the rules of divisibility. The reason for such a preference could be that they believed, by way of questioning, which they also expressed during the interviews, permanent learning could be ensured for the rules of divisibility.

Even though there are many studies in the literature showing that the IBL increases academic achievement in other fields, the number of studies using the IBL approach in mathematics teaching is quite low. For this reason, it can be suggested that the effect of this approach be examined at different grade levels and in different subjects in the field of mathematics in particular. In addition, since there are not many studies in the literature on how to prepare the IBL activities in the field of mathematics, it is thought that the study will contribute to the literature.

Annex 1: A sample worksheet

One day, Pelin the Adventurer loses her way while wandering in the forest. As she tries to find her way out, an old statue under a large fig tree catches her attention.

When she comes near the statue, she sees that a piece is missing and starts looking for it. As the hours pass, she loses hope of finding the missing piece, gets tired and sits in the shade of a tree. As she slowly closes her eyes, a giant creature suddenly appears before her. Then the following conversation takes place between the adventurer and the creature.

• Creature: “Hello, Miss. My name is Pan. I am a ranger in charge of the forest. I know where everything is in the forest.”

• Adventurer: “Hello, my name is Pelin. I got lost while wandering in the forest and suddenly I came across a fig tree the size of which I had never seen before.”

• Pan: “Yes, Miss, I have been watching you, and I know how much you want to find the missing piece. If you pass my test by answering the questions I ask, I promise to lead you to the missing piece.”

Pelin agrees and Pan starts asking her some questions.

Let’s imagine that you are all the Adventurer in the story. How would you answer Pan’s questions?

Question 1: Is the number 18 exactly divisible by 2?

Answer 1:

Question 2: Why//Why not? Can you explain?

Answer 2:

Question 3: If the number was 19, would it still be exactly divisible by 2? Please explain your answer.

Answer 3:

Question 4: Which digit of the numbers did you look at to decide whether or not the two numbers are exactly divisible by 2?

Answer 4:

Question 5: Would we also know that a number is exactly divisible by 2 by looking at the tens digit?

Answer 5:

Question 6: Can you explain your answer to me?

Answer 6:

Question 7: So, is the number 256 exactly divisible by 2?

Answer 7:

Question 8: Why/Why not? Can you explain?

Answer 8:

Question 9: Then, which digit of the given numbers do we look at in the rule of exact divisibility by 2?

Answer 9:

Question 10: How can you explain whether 300 + 50 + 7 is exactly divisible by 2?

Answer 10:

Question 11: If this number were 3.100 + 5.10 + 7, how would you decide whether it is exactly divisible by 2? Why?

Answer 11:

Question 12: OK, adventurers! Consider the number 256. Does the concept of place value matter for this number to be exactly divisible by 2?

Answer 12:

Question 13: Why/Why not? Can you explain?

Answer 13:

Question 14: Very good… So, if our number was 4.1000 + 7.100 + 3.10 + 8, how would you decide whether it is exactly divisible by 2? Why?

Answer 14:

Question 15: Is there a relationship between the concept of place value and the concept of decomposition?

Answer 15:

Question 16: Then which concept do we use to come up with the divisibility rule for the number 2?

Answer 16:

Question 17: Using the concept of ‘decomposition’, how can you tell us the rule of exact divisibility by 2 for the three-digit number ABC?

Answer 17:

Question 18: Using the concept of ‘decomposition’, how can you generalize the rule of exact divisibility by 2 to numbers with 4 or more digits?

Answer 18:

Question 19: Is the number 45 exactly divisible by 5? Why/Why not?

Answer 19:

Question 20: Is the number 140 exactly divisible by 5? Why/Why not?

Answer 20:

Question 21: Is there a similarity between the divisibility rule for the number 2 and the one for 5? Can you explain?

Answer 21:

Question 22: So let’s see, using the concept of decomposition, how can you explain to me the rule of exact divisibility by 5 over the three-digit number ABC?

Answer 22:

Pan gives the missing piece to our adventurer who has answered all the questions correctly, as promised. The adventurer who gets the missing piece immediately completes the statue.