One of the instructional approaches that have the potential to help students develop their problem-solving skills is an open-ended approach (OEA). OEA focuses on finding the correct response rather than offering a single solution to a problem at hand. This study investigated the achievement and problem-solving skills in Mathematics of the Grade 8 students in Binuangan National High School through an open-ended approach (OEA). It sought to: identify the levels of achievement in Mathematics of students who were exposed to OEA and those exposed to non-OEA; determine the problem-solving skills of students who were exposed to OEA and those exposed to non-OEA in terms of: (a) self-confidence about solving problems; (b) putting effort in solving problems; and (c) procedure followed to solve problems; compare the levels of achievement in Mathematics of students who were exposed to OEA and those exposed to non-OEA; and find out the difference on problem-solving skills of students who were exposed to OEA and those exposed to non-OEA. The researchers adopted the quasi-experimental research design in the study. The experimental group and the control group's pretest results showed very poor student achievement; however, after exposure to OEA, the experimental group's posttest results showed great student achievement. Compared to OEA and non-OEA, students exposed to OEA demonstrated greater problem-solving skills. Between the pretest and posttest, OEA considerably raised students' achievement levels. Additional findings showed that students exposed to OEA had significantly better problem-solving skills than students exposed to non-OEA.
Developing students' ability to solve problems in a way they can use in their daily lives is one of the fundamental goals of mathematics education. In other words, students should be able to understand the problem, recognize potential solutions, and demonstrate their answers. Teachers should be able to develop a plan to help students become autonomous learners and problem solvers.
Despite the beautiful vision of mathematics education, students need help with problem-solving. The issues started because more students needed to read clearly and comprehend the problem. According to the researcher's experience, whenever a topic involves solving problems, teachers are supposed to give a hint for the students to construct ideas, as Thinwiangthong, et al. 1 mentioned that reading comprehension skills are a predictor of problem-solving skills. It implies that students' problem-solving skills depend on their reading skills. Low performance of students is observed before the conduct of the study 2.
Despite the diverse school categories, learning through an open-ended method can better develop the students' capacity for mathematical creativity. Additionally, looking at the school category, high school and middle school students who learn using an open-ended method have higher self-esteem than those who understand traditionally 3.
One of the significant problems with high school students' problem-solving abilities is low self-confidence, which negatively impacts their mathematics learning. Mathematics teachers frequently encounter students who are passive in their approach to problem-solving and want to avoid it. The study's findings identified the primary influencing factors: aptitude, drive, tenacity, the feeling of helplessness, and inhibitor 4.
The conceptual, procedural, and problem-solving abilities of students who learn Mathematics using open-ended versus conventional methods differ significantly. An open-ended approach can enhance students' understanding of how to think mathematically to solve problems. The open-ended approach allows students to interact with their friends through discussion, allowing students to learn more about Mathematics 5.
Teaching students to love Mathematics is a significant issue for mathematics teachers. Everyone has the opportunity to succeed and excel in this area, though. By encouraging students to use the accessible internet connections in their homes to learn the subject independently, we can help them improve. They are taking the initiative to conduct research to improve their education 6.
Student mistakes include poor information retention, failure to comprehend how a problem is transformed, and failure to adhere to instructions in reading the text carefully and understanding any weak topics mathematically. The inability of the student to effectively absorb the information, the student's lack of understanding of the so-called problem transformation, the students' incomplete comprehension of the subject matter, the students' weak understanding of the idea of a prerequisite, the students' lack of prior experience solving problems, and the students' carelessness and sloppiness in the construction process are all factors that contribute to errors in the solution of the mathematical system of linear equations involving two variables 7. Rohmah and Sutiarso 7 pointed out that two things prevent students from giving the correct answers, namely, 1) issues with verbal fluency and conceptual understanding and 2) issues with numerical processing ability. The poor-ability students need a plan of action or a method of solving the issue.
According to Rahayuningsih et al. 8, "highly creative" students could use cognitive flexibility and fluency to solve open-ended tasks. Students who could think creatively and flexibly when handling open-ended problems were the other kind of students. According to the findings, the open-ended problem-solving exam helped measure students' mathematical creativity in terms of outcomes and processes. It was able to identify the two types of students' mathematical creativity, very creative and creative, in terms of results, and it was able to define the distinctions between the two types' cognitive profiles in terms of processes.
A substantial correlation exists between the open-ended and direct learning techniques to students' learning outcomes, and significant influences include thinking styles and mathematics learning results for students. The results show that students' learning outcomes in mathematics who possess divergent thought processes do better than learners' results with convergent thought habits 9.
The Open Ended Teaching and Learning Approach is suggested for secondary school Mathematics teachers to increase students' willingness to learn mathematics and improve their academic achievement 10.
The 1989 Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) expressed a distinct understanding of the central role of problem-solving in Mathematics 11. The 2000 Principles and Standards for School Mathematics also reaffirm the importance of problem-solving at all mathematical levels. Being a skilled problem solver has several benefits in real-world situations at work 12. Students should learn problem-solving to develop ways of thinking, habits of persistence and curiosity, and confidence in novel situations. These skills will be helpful to them inside and outside the mathematics classroom.
To develop students' problem-solving skills, NCTM 13 suggested an open-ended approach as a teaching strategy in which students are given open-ended problems to solve before being given multiple opportunities to do so, giving them practice in learning something new.
A lot of investigations have been conducted to determine factors associated with students’ achievement such as teachers’ skills and competencies 14, 15, 16, 17, teachers’ awareness, perceptions, and challenges 18, 19, 20, 21, 22, contemporary pedagogies 23, 24, 25, 26, 27, 28 and others 29, 30, 31, 32, 33, 34, 35. Meanwhile, only a few studies showed a significant increase in students' achievement through the integration of an open-ended approach 36, 37, 38.
Group-to-group strategy and an open-ended approach assist in improving knowledge, reasoning, and mathematical problem-solving skills 39. The way class time is used is another issue 40. In a classroom context, using open-ended activities that involve practical work and group projects takes up a lot of time, not to mention the requirement for a discussion before implementing those activities. Here in the Philippines, a mathematics class typically lasts three to five hours weekly. With less time for the students, this amount is now sufficient solely for the teacher to fully and thoroughly explain any mathematics idea. This only applies to one particular class period. Consider how much more there would be for a grade level competency for an entire academic year. The teacher's ability to provide an imprecise closure during a regular mathematics lesson can increase with the amount of instructional time. Segumpan and Tan 40 also explained the situation of mathematics education in the Philippines. The creation and familiarization of open-ended activities, including practical work in every classroom lesson, is something that a Filipino mathematics teacher must always keep in mind. They continued by saying that to optimize each lesson's advantages, Filipino teachers must get ready to organize and manage cooperative learning groups. Sadly, in most cases, the teacher was still primarily responsible for explaining concepts and posing questions to the class. Because they cannot maintain the conversation and make it more valuable, the teacher must introduce and conclude the subject. Exposure, practice and consolidation, and conversation are the most often used methods in math instruction.
The study assessed the students' achievement and problem-solving skills in mathematics through an open-ended approach at Binuangan National High School for SY 2021-2022. The study utilized a quasi-experimental research design. This type of design used two intact classes: the experimental group (exposed to an open-ended approach) and the control group (exposed to a non-open-ended approach). The two heterogeneous classes were subjected to the same pretest and posttest to determine the significant difference in the achievement and inventory questionnaire to determine the problem-solving skills.
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Second, a survey questionnaire to measure the problem-solving skills of the participant adapted from the study of Devrim Erdem-Keklik (2013) with Cronbach alpha reliability coefficients for subscale scores ranged between 0.74 and 0.82. This problem-solving skills questionnaire by Uysal (2007) consists of twenty-eight (28) items. The researchers utilized the five-point Likert scale to analyze students' problem-solving skills in Mathematics. Interpretation of data followed the scale below:
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The study used descriptive statistics such as the means, percentages, and standard deviation to analyze the data obtained from the questionnaire results. Analysis of Covariance (ANCOVA) was used to compare the effects of the open-ended approach on students' mathematics achievement levels and t-test for independent mean to find out any significant difference in problem-solving skills of students exposed to open-ended approach and those exposed to non-open-ended approach.
This section presents the results and interpretation of the data of this study. The presentation followed the study's objectives in order. Simple data analysis was provided using tables and other figures.
3.1. Students’ Achievement in MathematicsTable 1 presents the pretest mean scores of students exposed to the open-ended approach and those exposed to the non-open-ended approach. It shows that the pretest mean percentage scores of students exposed to open-ended and non-open-ended approaches are 24.85% and 25.92%, respectively, indicating "Did Not Meet Expectations." The students exposed to the open-ended approach have a lower mean percentage score than those exposed to the non-open-ended approach. This result shows that the pretest scores of students exposed to open-ended and non-open-ended approaches failed to meet the standards set by the Department of Education. They need to have a deeper understanding of the lesson on the topics before receiving formal instructions through an open-ended approach.
The result of the study shows that both groups had a very low level of achievement. It supports the study of Saligumba and Tan 2 when they found out that the students' performance in Mathematics in terms of pretest before exposure to the Gradual Release of Responsibility Instructional Model was very low. It also backs up other research findings, which demonstrated that students' levels of mathematics performance before exposure to any interventions is at the beginning level, suggesting a deficient performance 30, 33, 35.
Table 2 shows students' posttest results when exposed to open-ended and those exposed to non-open-ended approaches. They have a mean percentage score of 80.23%, indicating "very satisfactory," and 63.63% indicating "fairly satisfactory”, respectively.
Interestingly, there was an increase in students' mean percentage scores (MPS) for both groups. For students exposed to an open-ended approach, there were 9 (34.62%) who were "outstanding," 11 (42.31%) reached "very satisfactory," 1 (3.85%) came "satisfactory," 2 (7.69%) students got "fairly satisfactory” and there were 3 (11.54%) students who did not reach the 75% which means “did not meet expectations." The posttest overall mean percentage score (MPS) is 80.23%, indicating "very satisfactory." This finding implies increased students' academic achievement after exposure to the strategy used.
On the other hand, there was also an increase in the mean percentage score of students exposed to the non-open-ended approach. There was 1 (3.70%) student who achieved "outstanding," 4 (14.81%) students were "Very Satisfactory," 8 (29.63%) students were "satisfactory," 4 (14.81%) students got "fairly satisfactory," and 10 (37.04%) did not reach 75% which means "did not meet expectations." The posttest overall mean percentage score (MPS) is 63.63%, indicating "fairly satisfactory."
The students have gained scores on their posttest, but there is high academic achievement among students exposed to an open-ended approach. These results imply that students exposed to the open-ended approach have higher scores after the intervention than those exposed to the non-open-ended approach. The result indicates that only a few students in the non-open-ended based approach could get better scores.
The high mean scores of the students exposed to the open-ended approach also indicate that the students understand the concepts deeply compared to students under the non-open-ended approach. More so, the posttest scores also imply that the academic gains of students exposed to the open-ended approach are very satisfactory. This group of students meets the standard set by the Department of Education based on DepEd Order No. 8, s. 2015 is the Policy Guidelines on Classroom Assessment on the K to 12 Basic Education Curriculum. On the other hand, the mean score of students in the non-open-ended approach has increased, and they are reasonably satisfactory. But the group needed to meet the standard set by the department.
The result embraces the study of Widiartana 37, which showed the same effect on the performance in the pretest. However, upon exposure to an open-ended approach, the students' mathematical reasoning level in the experimental group was better than in the control group. Moreover, the results parallel the study of Irawan and Surya 36 that students' achievement in the experimental group increased compared to the control group after the open-ended approach.
The study agrees with Jehadus et al. 39 that the open-ended approach enhanced students' mathematical communication skills, supplementary to increased problem-solving abilities.
A Filipino mathematics teacher must constantly consider the construction and familiarization with open-ended activities involving practical work in every classroom lesson. However, Grouws et al. (2010), cited by Segumpan and Tan 40, said that using open-ended activities in a classroom setting that demand practical work and group projects takes up much time, not to mention the need for a discussion before those activities are implemented. They explained that Filipino teachers must prepare to set up and oversee cooperative learning groups to maximize each class's benefits.
1.1. Students’ Problem-solving SkillsThe students' problem-solving skills on self-confidence in solving problems is shown in Table 3. The result of the group exposed to the open-ended approach has a mean range of 3.11 which indicates "sometimes," and the group exposed to the non-open-ended approach has a mean range of 3.06 which means "sometimes."
The group exposed to the open-ended approach has the highest mean range of 3.85, indicating "often" in the statement "I believe that if I make enough efforts, I will solve any problems" and the lowest mean range of 2.65, indicating "sometimes" in the statement "I have no confidence in resolving the problem I face." On the other hand, the group exposed to the non-open-ended approach has the highest mean range of 3.59, indicating "often" in the statement "I have trouble solving problems" and lowest mean range of 2.37, indicating "rarely" in the statement "I can't produce a lot of solutions for a problem."
The mean range of the students exposed to the open-ended approach is higher than those exposed to the non-open-ended approach because they were given real-life activities that helped them think logically and deeply understand the concepts.
These findings confirmed the results of the study of Fatah et al. 3 on the open-ended approach: an effort to cultivate students' mathematical creative thinking ability and self-esteem in Mathematics.
The study of Behzadi et al. 4 pointed out that the leading concern in mathematics education among high school students is low self-confidence in problem-solving, which managed to reduce the level of learning mathematics.
The result of the students' problem-solving skills on putting effort in solving problems is shown in Table 4. The group exposed to an open-ended approach has a mean range of 3.99, indicating "often," and the non-open-ended approach has a mean range of 3.23, meaning "sometimes". The group exposed to the open-ended approach has the highest mean range of 4.50, indicating "often" in the statement "I plan what I'm going to do before I start solving the problem." Lowest mean range of 3.23, indicating "sometimes" in the statement "When I can't solve the problem, I don't need help with others." While the group exposed to non-open-ended approach has a highest mean range of 4.41 indicating "often" in the statement "I use existing strategies to solve the problem" and lowest mean range of 2.41 indicating "rarely" in the statement "After I've solved the problem, I'll review what I've done for the solution, and I check the transactions before making a definitive judgment". The mean range of the students exposed to the open-ended approach is greater than those exposed to the non-open-ended approach.
These results reinforced the study of Azis 5 that using open-ended questions helps improve students' understanding of how to utilize mathematics to think through situations to solve problems. Coronel and Tan 6 opined that teachers ought to encourage students to use the accessible internet connections in their homes to learn the subject through their effort so that they can advance. To improve their learning, they take the initiative to do research.
Table 5 shows the result of the students' problem-solving skills on the procedure followed in solving problems. The group exposed to an open-ended approach obtained a mean range of 3.61, indicating "often," and the non-open-ended approach had a mean range of 3.30, meaning "sometimes."
The students exposed to an open-ended approach had a higher mean range result. The group exposed to the open-ended approach obtained the highest mean range of 4.23, which indicates "often" in the statement "I use my own strategies to solve the problem," and the lowest mean range of 2.92, which shows "sometimes" in the statement "I can adapt similar situations from everyday life to the problem to facilitate the solution" while the group exposed to non-open-ended approach obtained a highest mean range of 4.48 which indicates "often" in the statement "I try to guess how it will work before I implement the solution" and a lowest mean range of 2.67 which indicates "sometimes" in the statement "I can easily test the solutions and think about the accuracy."
Fatah et al. 3 concluded that even at the low school level, an open-ended approach helps develop pupils' capacity for mathematical creativity. However, it's crucial to evaluate how teachers assist their students in solving difficulties using proper procedure techniques while considering their learning styles.
As mentioned by Rohmah and Sutiarso 7, two factors make the students unable to produce correct answers, namely: problems in the fluency of languages and understanding concepts and problems with process skill of Mathematics (understanding, transformation errors, process skill, and writing answers)". The low-ability students need a strategy or procedure to solve the problem.
Table 6 shows the result of the overall summary for problem-solving skills. The group exposed to the open-ended approach obtained a mean range of 3.58, indicating "often," and the non-open-ended approach had a mean range of 3.27, meaning "sometimes."
On the three subscales, the group exposed to the open-ended approach and non-open-ended approach has a high mean range on putting effort in solving problems at 3.99, indicating "often" and procedure followed to solve problems at 3.30 showing "sometimes," respectively, followed by the procedure to solve problems of 3.61 showing "often" and putting effort in solving problems of 3.23 indicating "sometimes." Moreover, both groups have low mean range on self-confidence about solving problems of 3.11 and 3.06, indicating "sometimes" in favor of students exposed to the open-ended approach.
Those open-ended problem-solving assessments efficiently measured students' mathematical creativity regarding outputs (products) and processes. This finding implies that the group exposed to the open-ended approach has higher problem-solving skills than those exposed to the non-open-ended approach. It is aligned with the study of Rahayuningsih et al. 8.
3.3. Analysis of Covariance on Students’ AchievementTable 7 shows the analysis of covariance (ANCOVA) on students' achievement. The pretest was used as a covariate to compare unrelated predictive variables which may affect the analysis statistically. The table shows a mean score of 40.12 for students exposed to an open-ended approach and 31.82 for students in a non-open-ended approach. The mean scores appeared so close to each other that both groups had almost the same achievement after the treatment.
The computed F-value of the covariate (pretest) is 64.84 with a p-value of 0.000, indicating significant implying that there was a statistically significant difference between the groups when controlling for the covariate, which means that the two groups were not comparable before the conduct of the study. Conversely, the computed F-value between groups is 51.14 with a p-value of 0.000, indicating a highly significant difference. It is observed that the control group has a higher percentage mean score as compared to the experimental group.
As a result, the null hypothesis that there is no significant difference in the mathematics achievement levels of students exposed to the open-ended approach and those exposed to the non-open-ended approach is rejected. This finding is attributed to the students' exposure to the open-ended approach used as a pedagogical approach. Students had shown a deeper understanding of the concepts presented with the instructional strategy employed by the teacher. Hence, the open-ended approach enhances academic performance and promotes conceptual understanding of the lessons discussed in class.
The recent results showed that an open-ended approach is more effective in enhancing students' achievement in mathematics. These results conform to the findings of several studies 9, 36, 37, 38 that the open-ended learning approach significantly influences students' learning outcomes.
As suggested in the study of Ulep (2006) as cited 39, in mathematics education in the Philippines, a Filipino mathematics teacher must constantly keep in mind the development and hands-on experience with open-ended activities involving practical work in every classroom lesson.
It is also recommended that mathematics teachers in secondary schools use open-ended teaching and learning approaches to enhance students' motivation in learning mathematics concepts, which will improve their academic performance 10.
Moreover, consistent with the present study sharing mathematical concepts to establish a common understanding and move toward a common objective, students in mathematics classes who use open-ended approach study mathematics more meaningfully on their own 1.
3.4. Independent t-test on Students’ Problem-solving SkillsTable 8 shows the independent t-test on students' problem-solving skills. The mean range of the problem-solving abilities was 3.58 for students exposed to the open-ended approach and 3.27 for those exposed to the non-open-ended approach. The computed t-value between groups is 5.41 with a p-value of 0.000, indicating a highly significant difference. All these results are due to no pretest and posttest. Students exposed to the open-ended approach have higher problem-solving skills than those exposed to the non-open-ended approach.
Thus, the null hypothesis that there is no significant difference in problem-solving skills of students exposed to open-ended and non-open-ended approaches is rejected.
The result of the study is aligned with the 2000 Principles and Standards for School Mathematics as cited 11 that reiterate the fundamental place of problem-solving at all levels of Mathematics as follows: by learning problem-solving in Mathematics, students should acquire the ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that will serve them well outside the mathematics classroom 12.
Irawan and Surya 36 urged educational institutions to implement this open-ended learning approach as a stand-alone learning strategy that may be applied to mathematics learning activities since it can provide students the freedom to think critically and creatively while solving a problem.
Based on the findings of the study, the following conclusions are drawn:
The achievement of students exposed to the open-ended approach demonstrated a better increase from "did not meet expectation" to "very satisfactory" than those exposed to the non-open-ended approach, which is from "did not meet expectations" to "fairly satisfactory”.
The students' problem-solving skills under the open-ended approach indicated "Often" results than those of the non-open-ended approach, which indicated "Sometimes" results. Of the three perceived problem-solving skills, putting effort into solving problems obtained a high mean range showing "Often," followed by the procedure followed to solve problems which also obtained a range indicating "Often," while self-confidence about solving problems obtained a range indicating "Sometimes."
A highly significant difference in achievement occurred when students' performed better in open-ended than non-open-ended approaches.
The students under the open-ended approach exhibited better problem-solving skills than those in the non-open-ended approach.
Based on the summary of findings and conclusion of the study, the following recommendations are put forward:
School administrators and teachers are encouraged to perform diagnostic tests at the beginning of every topic to identify what will be emphasized in the learning delivery. Parents should monitor their students' performance in school to help them understand and learn the concepts.
School administrators and teachers may use an open-ended approach to develop students' problem-solving skills. Teachers determine the level of problem-solving skills of the students to design a more effective teaching strategy. Also, parents may know their students' problem-solving skills for them to assist in developing their strengths and weaknesses in solving.
Teachers may try an open-ended approach to increase students' achievement in Mathematics, as evident in the study. Teachers can develop learning tasks through open-ended problems for the students to learn the concepts more meaningfully since they are the ones who develop the answer.
Teachers are advised to let students solve problems through an open-ended approach. Parents must help their students develop their reading and comprehension skills to analyze and solve the presented concept.
Finally, future researchers may find other teaching approaches to improve students' achievement and problem-solving skills. The application of this study is highly significant, and the use of this approach is highly recommended.
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| In article | View Article | ||
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| In article | |||
Published with license by Science and Education Publishing, Copyright © 2023 Gerald C. Bayarcal and Denis A. Tan
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| [36] | Irawan, A. & Surya, E. (2017). Application of the open-ended approach to mathematics learning in the sub-subject of rectangular. International Journal of Sciences: Basic and Applied Research. 33(3), 270-279. | ||
| In article | |||
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| In article | |||
| [38] | Rahayu, R., & Ulya, H. (2021). Mathematical disposition of students in open-ended learning based on ethnomathematics. Journal of Education Technology. 5(3), 339-345. | ||
| In article | View Article | ||
| [39] | Jehadus, E., Murni, V., Ndiung, S., Nendi, F., & Tamur, M. (2021). The influence of open-ended approach with group-to-group strategy on the improvement of mathematic communication skills for high school students. European Alliance for Innovation. 482-488. | ||
| In article | View Article | ||
| [40] | Segumpan, L. L. B., & Tan, D. A. (2018). Mathematics performance and anxiety of junior high school students in a flipped classroom. European Journal of Education Studies. 4(12). | ||
| In article | |||
