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Strengths of Formative Feedbacks with Line by Line Hints & Repeated Parallel Question on Promoting Students’ Mathematics Achievement

Francis Ian F. Cabang*, Dennis B. Roble

University of Science and Technology of Southern Philippines, Lapasan Highway, Cagayan de Oro City, Philippines

Abstract: Feedback is an indispensable component of formative assessment for it helps students cope their learning problems. However, students are rarely engaged in the feedback process. This experimental study explored the different feedback methods to promote students’ mathematics achievement. This study utilized the traditional, line by line hints and repeated parallel question as a form of written feedback. The participants were the 13-14 years old grade 8 students of Alae National High School during the school year 2021-2022. It employed a Quasi-experimental Pretest-Post Test Control Group design. A 48-item validated teacher-made achievement test in Algebra was the main instrument used in this study. Descriptive statistics such as mean and standard deviation along with one-way analysis of covariance (ANCOVA) were used to analyze the effect of different feedback methods on students’ mathematics achievement. Bonferroni pairwise posteriori test was used to determine which method made a significant difference in the students’ mathematics achievement. Results of the analysis revealed that students’ level of mathematics achievement was in the approaching proficiency level when exposed to formative feedback with line-by-line hints as well as those with repeated parallel question as compared to the control group who used the traditional feedback. Furthermore, the formative feedback with repeated parallel question was effective in improving students’ achievement as it provides higher degree of assistance and error-correction mechanism. Hence, the researchers recommend that mathematics teachers across all levels may employ this method to foster better understanding and promote students’ active engagement in the feedback process. Also, in order to have generalizability of the results, a comprehensive study might be done to address the limitation of this method towards improving students’ mathematical flexibility especially in solving real-life mathematical problems.

Keywords: formative feedback, repeated parallel question, line by line hints, written feedback, mathematics achievement

Received January 03, 2022; Revised February 05, 2022; Accepted February 11, 2022

Citation: Francis Ian F. Cabang, Dennis B. Roble, Strengths of Formative Feedbacks with Line by Line Hints & Repeated Parallel Question on Promoting Students’ Mathematics Achievement. American Journal of Educational Research, 10 (2). 73-80. January 2022. doi:10.12691/education-10-2-1

1. Introduction

Feedback is an indispensable component of assessment for learning [1]. The information within the feedback allows the teachers to identify students’ misunderstandings and help them correct their errors. Furthermore, feedback coupled with corrective activities challenges the students to cope their learning problems [2]. Unfortunately, students received minimal quality of descriptive comments from teachers. In fact, teachers spend less time in giving feedback with corrective activities to students [3]. Specifically, studies in the context of Filipino students reveal that low performance in mathematics of the students can be traced on teachers’ conventional and mechanistic methodologies [4]. For many years, this low performance is evident in local, national and international assessments such as the National Achievement Test (NAT), Program for International Student Assessment (PISA) and Trends in International Mathematics and Science Study (TIMSS). This pose a great challenge to mathematics teachers to explore the various methods of giving feedback to help students become independent learners. Hence, choosing an innovative feedback method to foster learning must be given importance.

Due to COVID-19 pandemic, public secondary schools in the Philippines are shifting to modular instruction. In this modality, teachers’ comments on students’ outputs are valuable since there are rarely experts at home who can help them improve their mistakes. But, it has been observed that teachers’ written feedback is one-sided mechanistic process of which learners are simply informed of the correct answers and are not challenged to improve their mistakes. This is alarming since students are more likely to progress without understanding. This contradicts to the contemporary view of which effective feedback should promote self-regulated learning and actively involve both the teacher and the learner in the feedback process [5]. The limitations of the existing method can be addressed by the formative feedback with repeated parallel question. This new method stems from the notion that the amount of transfer of learning depends on the identical elements present in both learning situations. In this study, analyzing a worked-out solution of the missed item and providing a parallel question actively engage the learners in the feedback process.

Furthermore, researchers have argued that corrective feedback is one of the strategies to engage the learners in the formative assessment [6]. Repeated parallel question as a form of corrective feedback stems from the concept of worked examples. Studies revealed that presenting first worked problems to novice learners improves students learning [7,8]. Also, students with low prior knowledge of the correct solution strategies benefited from studying worked problems [9]. Specifically, analyzing a worked – out solution after solving a mathematical task can foster better learning [10]. However, previous studies have not utilized this method as a corrective feedback. Being aware of this gap, this present study seeks to determine the strengths of different written feedback methods in mathematics. Specifically, it aims to determine the influence of formative feedback with repeated parallel question on students’ achievement in mathematics.

2. Literature Review

Previous studies that support the current study are reviewed as regards to feedback, line by line hints, repeated parallel question and achievement of students.

Feedback represents the information communicated to the learner that is intended to modify the learner’s thinking or behavior for the purpose of improving learning [11]. Specifically, feedback confirms correct responses, tells the student how well the content is being understood, and allows the learner to correct their errors [12]. This correction function is probably the most important aspect of feedback. Specifically, it guides the learners to improve and reconceptualize their work. As a result, it closes the learning gap between the actual and desired performance.

The implications of feedback in instructional design was examined in the study of Cohen [13]. Results revealed the information within the feedback may address the accuracy of a response to a problem or task and may additionally touch on particular errors and misconceptions. The main aim of formative feedback is to increase students’ knowledge, skills, and understanding in some content area or general skill such as solving a specific mathematical task. However, Agricola et al. [14] suggested that students often misunderstand teachers’ written feedback. This is worrisome, since written feedback is the main form of feedback in education. The researchers of this study proposed interventions in organizing feedback conversations through feedback request forms and verbal feedback to prevent misunderstanding of written feedback. In this study a 2 × 2 factorial experiment (N = 128) was conducted to examine the effects of a feedback request form (with vs. without) and feedback mode (written vs. verbal feedback). Results showed that verbal feedback had a signiﬁcantly higher impact on students’ feedback perception than written feed- back; it did not improve students’ self-efficacy, or motivation. Feedback request forms did not improve students’ perceptions, self-efficacy, or motivation.

Over the years, studies have focused on integrating feedback through hints with various learning platforms and modality. A comprehensive study of Corbalan et al. [15] investigated the three forms of feedback in solving mathematical problems. In this study, feedback was given on the final solution step, on all the solution steps at once, and on all the solution steps successively. In particular, the researchers determined whether these methods had an effect on students’ views towards mathematics, learning, and motivation. It was found out that on all solutions steps was perceived more helpful than feedback on the final solution step. Furthermore, effective learning and higher motivation was achieved when students used the on all problem-solving steps type of feedback.

A survey conducted by Mulliner and Tucker [16] compared the perceptions of students and academics with regards to feedback practice. This study was conducted at the School of the Built Environment, Liverpool John Moores University. This study focused on perceptions towards academics and students’ preferences for different types of feedback, timeliness of feedback, students’ engagement and interest in feedback, quality feedback and satisfaction. This study suggested that feedback should be timely, constructive, encouraging, provide detailed direction or hints for future improvement, be linked to a marking scheme and be specific about the failings of the work. However, the study revealed that students with low prior knowledge have problems in engaging with the feedback.

The effects of feedback types and opportunities to change answers on the learners’ achievement and their ability to solve physics problems was explored by Wancham and Tangdhanakanond [17]. This study compared three methods of feedback, namely static feedback with hints, reducing feedback with hints, and knowledge of response, and two types of opportunities to change answers, namely (a) no answer changing allowed and (b) answer changing allowed. Each participant was randomly assigned to take one of the six experimental conditions. It was found that participants provided with static feedback with hints and those provided with reducing feedback with hints had significantly higher achievement and ability to solve physics problems than those provided with knowledge of response feedback. Moreover, participants who were allowed to change the answers had significantly higher achievement and ability to solve physics problems than those who were not allowed to change the answers. But, there was no interaction between feedback types and opportunities to change answers on achievement and ability to solve physics problems.

Furthermore, researchers have argued that corrective feedback is one of the strategies to engage the learners in the formative assessment. The experiment conducted by Fyfe et al. [9] examined the effects of 1 form of guidance, feedback, during exploratory mathematics problem solving for children with varying levels of prior domain knowledge. In 2 experiments, 2nd- and 3rd-grade children solved 12 novel mathematical equivalence problems and then received brief conceptual instruction. After solving each problem, they received (a) no feedback, (b) outcome feedback, or (c) strategy feedback. In both experiments, prior knowledge moderated the impact of feedback on children's learning. The study revealed that children with little prior knowledge of correct solution strategies benefited from feedback during exploration, but children with some prior knowledge of a correct solution strategy benefited more from exploring without feedback.

A quantitative study was conducted by Van Gog and Kester [10] exploring the effects of providing worked examples on students’ problem-solving skills. Participants of the study were the forty (40) Dutch university students and were randomly assigned to one of the conditions: (a) study only: Worked examples study followed by an immediate (5 min) and delayed (1 week) retention test or (b) study–test: Example–problem pairs followed by an immediate (5 min) and delayed (1 week) retention test. Analysis of the findings have shown equal performance in both conditions on an immediate retention test after 5 min, but the SSSS condition outperformed the STST condition on a delayed retention test after 1 week. Results showed that studying a worked – out solution after solving a mathematical task can foster better learning. However, the testing effect did not develop problem-solving skills from studying worked examples.

Meanwhile, Mclaren and Isotoni [18] have compared the effects alternating worked examples / tutored problems, all tutored problems, and all worked examples. In particular, the study was conducted with 145 high school students in the domain of chemistry. It was hypothesized that the alternating condition would lead to better results (i.e., better learning and/or learning efficiency) than either all examples or all tutored problems. However, the hypothesis was not confirmed. While all three conditions learned roughly the same amount, the all worked examples condition took significantly less time and was a more efficient learning treatment than either alternating examples/tutored problems or all tutored problems. The researchers suggested that providing worked-examples enabled the students to concentrate more on building cognitive schemas as it prepared them to solve similar tasks. In the present study, worked example coupled with a parallel question will be given as a form of feedback in an attempt to improve students’ solution to a task.

However, some researchers have different findings on corrective feedback with task repetition. Noureen [19] explored the effect of descriptive feedback and corrective feedback on academic achievement of VII grade in mathematics. Only post-test control group design was used. There were three groups. Group 1 received descriptive feedback, group 2 received corrective feedback and group 3 served as a control group. All groups contained 53 students each. The results showed that descriptive feedback has more positive effect on students’ achievements as compared to corrective feedback. Students taught through corrective feedback performed better than control group. This study also suggest that feedback strategies can be used to enhance mathematical understanding and improves learning.

Finally, feedback practices were explored by of Stovner and Klette [8]. The study deals with sixteen lessons taught by five teachers and were purposefully sampled from a larger video study (172 lessons) as lessons with high-quality feedback according to a standardized observation instrument. This study has shown that feedback provided by mathematics teachers usually addresses procedural skills and, to a much lesser extent, other competencies such as conceptual understanding or engagement in mathematical practices. Also, study revealed that feedback on conceptual understanding and mathematical practices was provided in situations when students were especially challenged, and entailed a series of complex decisions, thereby placing demands on the teachers to manage both the students’ understanding and behavior. Likewise, the present study focuses on actively engaging the learners in the feedback cycle using the repeated parallel question.

In summary, providing feedback that is specific and clear for conceptual and procedural learning tasks is a reasonable general guideline. This practice may depend on variables such as methods of feedback and achievement.

3. Methods

This study employed a quasi-experimental pretest-posttest control group design. Mean and standard deviation was used to examine student’s pretest and posttest scores while one-way ANCOVA was used to analyze the effect of different feedback methods on students’ achievement. Bonferroni pairwise was used as post-hoc procedure to determine which method made a significant difference in the students’ achievement.

The participants of the study were the one hundred eleven (111) grade 8 students of Alae National High School during the first quarter of the school year 2021 - 2022. The participants’ age ranges between 13 to 14 years old and were randomly selected from the three intact classes. One intact class was employed with feedback using line by line hints and another intact class was given with the feedback with repeated parallel question. The remaining intact class was the control group which was treated using the traditional method. These sections were handled by the same teacher to ensure proper execution of the different feedback methods. Thus, the confounding variables that may contaminate the results were prevented.

The main instrument of this study was a 48 items teacher – made test. The test was prepared with a table of specification covering the topics from Factoring to Linear Equations in two variables. The competencies and objectives of the table of specification were based on the K-12 Most Essential Learning Competency for the first quarter of mathematics 8. In addition, the test was highly reliable with a reliability coefficient of 0.9399. Five levels of mastery were used to describe students’ achievement.

At the onset of the experiment, all groups were given the pretest and learning contract. The learning contract emphasized the duties and responsibilities of the teacher, students and parents during the modular instruction.

In the distribution of modules, six (6) modules were utilized. The content and coverage of these modules were based on the K-12 Most Essential learning competencies. The researcher facilitated the distribution of modules for the three (3) groups to ensure the proper implementation of the experiment. In particular, each module was given on a weekly basis to avoid contamination of the answers of the tasks from the three groups. All participants were given only one (1) week to accomplish the tasks. The duration of 1 week was based on the guidelines of the Department of Education in the conduct of modular instruction.

In answering the modules, each participant was given an answer sheet that was purposely designed for this study. Students were asked to perform first the tasks without any form of feedback. All the answers and revisions of the participants as well as the feedback of the teacher were written on this answer sheet.

The outputs of the experimental groups were checked and provided with feedback immediately. Afterwards, the researcher returned it for revisions. Both experimental groups were given only 1 week to improve their mistakes. However, the outputs of the control group were not checked immediately for it represents the traditional method of giving feedback during the modular instruction.

In giving of feedback, the experimental group 1 received the line by line hints. The hints were provided in each line of the solution where the error occurred. The participants used the hints in revising their solutions. Meanwhile, the experimental group 2 was given the formative feedback with repeated parallel question. Here, participants received a worked-out solution of the previous task for them to analyze. Afterwards, a parallel question was given to re-assess their skill. The control group was given the traditional method where the teacher simply put a check mark and correct solutions were given on the part where the error occurred.

This modality was used throughout the 9-weeks duration of the conduct of the study. A posttest was administered at the end of the experimental period.

The performance of the students in the pretest and the posttest were evaluated and described in terms of mean and standard deviation. The variation, as well as the significant effect of different feedback methods, were measured and analyzed by one-way analysis of covariance (ANCOVA). In testing the hypotheses, alpha is set at a 0.05 level of significance.

4. Results and Findings

The results of this study were presented in the following tables:

Table 1. Mean and Standard Deviation of Students’ Critical Thinking Skill in Mathematics

Table 1 shows the mean and standard deviation in the pretest and posttest scores for both the control and experimental groups. The pretest means scores for the three groups were as follows 13.35, 13.47 and 13.42 respectively. However, data revealed that all groups were still at the developing level of proficiency. This further suggests that the levels of student’s abilities in mathematics from both groups were relatively close and therefore comparable prior to the conduct of this study. Moreover, the result shows a wider dispersion in the scores of the students in the experimental group 2 while the scores of the other groups were closer to the mean. In particular, the standard deviation of the experimental group 2 indicates that some students got high and low scores in the pretest while the remaining groups had almost similar scores.

On the posttest, it can be observed that the control group who received the conventional method of feedback still remained at the developing level of proficiency. This indicates that the increase on the achievement scores of the students in the control group did not improve their level of proficiency. This further implies that they were starting to cope up and develop knowledge but the fundamental skills were not strengthened and fully acquired to aid their understanding of the concepts. Conversely, both experimental groups who received corrective feedback improved and obtained an approaching level of proficiency.

Figure 1 displays the solution of the student who received hints as a form of feedback. The misconceptions encountered by the student were addressed using the line by line hints. Specifically, the student was assisted with chunks of information to complete the task. In fact, the student responded positively to the feedback as he weakened his misconceptions and presented an organized solution. This study has shown that the information within the line by line hints addressed the accuracy of a response to a task and can improve particular errors. Furthermore, the present study suggest that this method is an active two-way process. First, the teacher provided the hints in logical manner which challenges the learner to reflect and analyze it. Second, the learner responded to the suggestions and used the hints in revising his solutions. This claim was supported by the study of Wang [11] suggesting that the step by step hint is a reflection-in-action process of which new knowledge was gained through self-reflection. This study has shown that hints challenge the students to interact with the feedback by reflecting on their understanding of the concept and ability to solve.

However, there were instances wherein students were not able to revise correctly their solutions despite the presence of hints. Another example from the experimental group 1 was shown below.

Clearly, the figure above showed that the student was not able to utilize the hints effectively. It can be inferred that 1) The student ignored the feedback 2) The student did not comprehend the hints and 3) The foundational concepts of arithmetic and algebra were not acquired. Although, it was a rare event that occurred during the experiment. Nevertheless, these findings were relevant. This study has shown that students with low prior knowledge have problems in engaging with this type of feedback. Mulliner and Tucker [16] further argued that these problems arose due to students’ level of satisfaction and interest to respond towards the feedback. Most importantly, this method shows only single edit at a time, and lacks context and details needed to interpret the suggestion. This study has shown that students with low prior knowledge are more likely to be confuse with the hints. In particular, Aleven [20] further suggests that some students often misused hints allowing them to progress without understanding. Evidently, this study showed that students’ prior knowledge and learning experience must be considered for it affects student’s ability to analyze and interpret the hints.

Figure 1. Student Exp1M5 Solution and Feedback to the Mathematical Task (Task: What is the graph of y=2x+2y?)

Figure 2. Student Exp1F16 Solution and Feedback to the Mathematical Task (Task: What is the slope of a line passing through the points (0,5) and (3,10)?)

On the other hand, students who were given the repeated parallel question as a method feedback were able to improve their solutions. A sample of student’s solution on the mathematical task was shown below.

Figure 3. Student Exp2F12 Solution and Feedback to the Mathematical Task (Task: What is the sum when is added to ?)

The figure above displays the solution of Student Exp2F12 provided with repeated parallel question as a form of feedback. It is worth to note that the student responded positively to the feedback as she was able to solve the parallel question. Also, it can be gleaned from her solution that the student followed the feedback or solution presented to her. This study has shown that analyzing a worked – out solution coupled with reassessment through a parallel question can foster better learning. Most importantly, the transfer of learning was possible as the student applied the same content and method in solving the parallel task.

Unfortunately, it was observed that the student purely imitated the concept of the feedback (worked-out solution) in solving the similar task. Although, the cognitive schema of the solution procedure for the parallel question was actively constructed by the student. Nevertheless, it was constructed based on example study rather than generation of an alternative solution. It was quite alarming for it hinders student’s mathematical flexibility. Specifically, this feedback method may lead to a misconception to the students that the solution presented to them was the only way to present and solve the task where in fact there were many ways of doing it. The present study has shown that feedback with repeated parallel question potentially undermines student’s mathematical flexibility. However, a comprehensive study might be conducted in the future to support this claim. This study further suggests that students must be encouraged to present an alternative solution whenever they were provided with this method of feedback.

Table 2. One-Way ANCOVA Summary of Students’ Achievement

Table 2 shows that there was a significant difference in the posttest scores when employed with different methods of feedback as indicated by the F-ratio of 13.998 and p-value of 0.000 which led to the rejection of the null hypothesis. Based on the result, students treated with different methods of feedback had a better performance than those students who were exposed to the traditional method. Particularly, students’ active engagement in the feedback process improve their performance. Moreover, the significant difference between groups were further analyzed.

Table 3. Comparison of Students’ Achievement Between Groups

The post hoc analysis above revealed that a significant difference on student’s achievement score was observed between the control and experimental group 2 with a mean difference of ±10.042 and p-value of 0.000. Also, data showed a significant difference between the achievement scores of experimental group 1 and experimental group 2 with a mean difference of ±5.789 and p-value of 0.008.

The aforementioned results have shown the strengths of formative feedback with repeated parallel question in improving students’ mathematics achievement. Particularly, this method provided an interactive and friendly environment as manifested in student’s active participation in the feedback process. Furthermore, this method provided students the opportunity to refine understanding, visualize and adopt the feedback in order to solve a similar task. The researchers believed that when this method will be coupled with effective pedagogy and learning modality, then this might have a different effect on the learning gains of the students.

5. Conclusions

Based on the findings of the study, the researchers conclude that the students’ level of achievement was approaching proficiency and needs improvement despite the integration of the three (3) methods of feedback. Specifically, the limitations of each method must be given importance for it affects students’ engagement with the feedback process and future performance. Nevertheless, the formative feedback with repeated parallel question was effective in improving students’ achievement as it provides higher degree of assistance and error-correction mechanism. The researchers then recommend that mathematics teachers across all levels may utilize this method to promote students’ active engagement in the feedback process. Moreover, daily time-pressured quiz with formative feedback can be used by mathematics teachers to prepare students from high stake examinations [21]. Finally, in order to have generalizability of the results, a comprehensive study might be done to address the limitation of this method towards students’ mathematical flexibility especially in solving real-life mathematical problems.

Acknowledgements

The researchers would like to express their heartfelt gratitude to Alae National High School (ANHS), led by Dr. Teresa Marie B. Getueza, for enabling them to conduct the study. The researchers are also grateful for the College of Science and Technology Education (CSTE) headed by Dean Dr. Laila S. Lomibao, the chair of the Department of Mathematics Education headed by Dr. Rosie G. Tan of the University of Science and Technology of Southern Philippines for their approval and support of the conduct of this study.

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