Strategies of enhancing students’ mathematics perception and self-efficacy to improve their mathematics achievement

Abstract This study implemented intervention strategies to enhance grade 12 students’ mathematics perception and self-efficacy to improve their mathematics achievement. Conveniently selected 394 students (200 as comparison & 194 as treatment group) were involved. Two FGDs were conducted. Forty items of mathematics achievement test were administered for both groups. Next, interventional training focusing on mathematics perception and self-efficacy enhancement strategy (innovative classroom-based) was given to the treatment groups. Counseling psychologist and mathematics teachers participated in the training that took 12 class-periods within 3 weeks. Math test, perception, and self-efficacy given before were given for the second time for both groups at the end of the training. Quantitative data were double-entered into SPSS version 26, cleaned, and analyzed using descriptive and inferential statistical techniques. Qualitative data were transcribed, coded using Atlas Ti7 software, and analyzed in the form of narration. The finding of this study revealed a statistically significant mean difference in students’ mathematics perception [t (392) = −5.28, p < .001], self-efficacy [t (392) = −3.40, p < .001], and mathematics achievement [t (392) = −6.45, p < .001] between comparison and treatment groups. The effect size result of both of students’ mathematics perception and self-efficacy revealed a small effect (r = .257 & .169), respectively, whereas mathematics achievement revealed a moderate effect (r = .309). Interventional training was given to adjust students’ mathematics perception [t (193) = −3.64, p < .001], and self-efficacy [t (193) = −14.98, p < .001] was effective. There was no statistically significant difference between male and female students in their mathematics achievement post-test result, i.e. [t (192) = .45, p = .66].


Introduction
Teachers could promote students' mathematics perception and their self-efficacy to improve their mathematics achievement.In the process of adjusting students' sense of self-efficacy belief and teaching methodology (Zeldin et al., 2008), teachers could integrate the usual method of instruction with a self-efficacy enhancement strategy (Getachew & Birhane, 2016).The traditional teaching methodologies applied in most mathematics classes such as lectures seem not efficient especially when new concepts and contents need to be delivered.Thus, when students' extent of attention could be improved by means of effective pedagogical methods, it is more likely that these changes could improve students' learning and their mathematics achievement (Getachew & Birhane, 2016).
Self-efficacy beliefs are defined as people's judgments of their capabilities to organize and execute courses of action that require attaining designated types of performances (Bandura, 1992;Pajares, 2002).The social cognitive theory developed by Albert Bandura is the most prominent learning theory, and self-efficacy is an important component of the theory (Sakellariou, 2022;Yesuf et al., 2023).
In this paper, perception refers to the range of students' beliefs, organization, identification, and interpretation of mathematical tasks.Students' perceptions of their mathematics ability vary by gender and seem to influence their mathematics achievement (Gernsbacher, 2015).Jointly, students' self-efficacy and students' perceptions had a positive impact on mathematics achievement.Appiah et al. (2022) recommend in their study that teachers should aid in promoting students' attitudes toward mathematics as well as promoting their self-efficacy in order to accelerate their mathematics performance.
Many students dislike mathematics since they develop negative attitudes and perceive it as a more difficult subject than others (Rodríguez et al., 2020).It is seen as a subject reserved for the selected few outstanding students.It often evokes feelings of stress, anxiety, and fear.Perceptions, beliefs, and efficacy levels about mathematics originate from past experiences, comprising both cognitive and affective dimensions (Mutodi & Ngirande, 2014).From a cognitive point of view, it relates to a person's knowledge, beliefs, and other cognitive representations while from an affective domain, it refers to a person's attitudes, feelings, and emotions about mathematics.This is why both mathematics perception and self-efficacy need to be linked and studied together (Mutodi & Ngirande, 2014).
Bandura developed training materials that can help to enhance students' self-efficacy beliefs.A strong sense of efficacy enhances human accomplishment and personal well-being in many ways.Bandura stated that individuals develop their self-efficacy beliefs by interpreting information from four main sources of influence.People's beliefs in their efficacy are developed by four main sources of influence, including mastery experiences (Özcan & Kültür, 2021), vicarious experiences, social persuasion, and emotional states (Silver et al., 2001).
Bandura's self-efficacy model has been tested and used widely by different researchers (Zeldin et al., 2008).For example, Pangburn (2020) argued that the intervention conducted to enhance Caucasian and African-American students' ages ranged from 16 to 22 years old, with a mean age of 18.29 using Bandura's self-efficacy enhancement strategy model revealing the effectiveness of the model.The study broadly used MyMathLab to examine its effect on students' skills in a collegelevel blended mathematics course: goal setting, time management, help-seeking, mathematics self-efficacy, and self-reflection.Getachew and Birhane (2016) reported the effectiveness of the model in enhancing students' self-efficacy and math achievement.It is considered an innovative classroom-based strategy, because the self-efficacy enhancement strategy was aligned with lecturing, and the intervention was implemented in all math periods with the support of math teachers within the given interventional period.
Research findings reported by different researchers reported different findings by comparing male and female students towards mathematics' self-efficacy, perception, and mathematics achievement (Shone et al., 2023).For example, Michelle (2013) reported that male students achieved higher mathematics self-efficacy than females.However, gender differences were not found to exist regarding mathematics achievement (Michelle, 2013).
The finding of the study by Holenstein et al. (2022) indicated that gender difference was observed in which males preferred mathematics self-efficacy and self-concept than females.However, females were reported to have better grades in mathematics than males.There was no significant difference obtained between male and female students' mathematics self-efficacy and mathematics achievement (Ayotola & Adedeji, 2009).
In order to measure students' mathematics achievement, researchers usually use grades obtained over the academic year, semester grades which were converted to letter grades from semester cumulative average, CGPA, SGPA, cumulative average, semester average, and the result of particular subjects (Holenstein et al., 2022;Ramirez et al., 2018).
Many factors have been studied and identified as affecting students' academic achievement.However, the above-mentioned problems could not be solved once identified and recommended being solved.It needs effort to conduct an intervention so as to realize and practically find possible means (Getachew & Birhane, 2016).Following strong justification of different studies, an intervention was conducted so as to enhance students' mathematics perception and self-efficacy to improve their mathematics achievement.
In particular, two research questions were addressed: (1) What is the effect of intervention on students' mathematics self-efficacy, perception, and mathematics achievement?
(2) What is the extent of the statistically significant mean difference between male and female students in their mathematics achievement after intervention?

Research design
This study used a quasi-experimental method followed by a concurrent triangulation mixed research design.A quasi-experimental design was used because it is not possible to randomly assign students to either the intervention or the comparison group since the students are already assigned and formed into their sections.This design is often used to evaluate the benefits of specific interventions (Harris et al., 2006).

Population and samples
The target populations of this study were grade 12 students.Participants were selected by using a multi-stage sampling method.Accordingly, two schools were selected conveniently.The reason for the convenient selection of the schools was to get schools that have at least four sections from the natural sciences stream so as to use two sections as intervention and two sections as comparison groups from both schools.From the selected two schools, 394 (194 for intervention and 200 for comparison) were selected conveniently.Similarly, four sections from both schools were selected using a convenient sampling technique.At least, two sections are better to be involved for both intervention and comparison in order to minimize the limitations of quasiexperimental practices.In this study, participants involved in intervention and comparison groups were from a similar school.Finally, student participants from the selected sections that sit for the questionnaire and math achievement test were selected by census sampling technique.
Mathematics achievement test that was prepared by subject teachers and approved by the study supervisors was administered for both intervention and comparison groups before the training.The students' mathematics perceptions and self-efficacy questionnaires were filled out by the selected students for intervention.Then, training was given to the intervention groups.The mathematics achievement test and the selected variables were administered for both groups (I & C) for the second time, and the effect was investigated.Finally, two FGDs (one FGD per school) were conducted with purposively selected students.

Study procedures
After getting an ethical letter for the study from the College's ethical review board, the researchers with assistant data collectors met the participants.The intervention was conducted with the support of three professionals.These professionals were one Counseling Psychologist who participated at both interventional schools and two mathematics teachers, one from the first school and one from the second school.The researchers explained the aim of the study and obtained participants' written consent.Participants were briefed about the processes of intervention and told that they could terminate at any time they need.Every effort was made to ensure and maintain the confidentiality of every activity and data they shared for the study.

Instrumentation
Questionnaires and FGDs were used to collect data in this study.Standardized questionnaires were adapted for both of the selected variables from recognized sources through the authors' permission.Instruments used for students' perception of their mathematics were adapted from Shin et al. (2021).Students' mathematics self-efficacy and anxiety questionnaire (MSEAQ) was adapted from May (2009).
The students in the intervention and those in the comparison groups sat for both the pre-test and post-test to assess whether or not there were achievement test differences after improvement training on the psychological variables given to the intervention groups.
Data regarding the variables such as students' school, grade level, age, gender, and allimportant variables were collected by using a self-report questionnaire.Students' mathematics achievement test containing 40 objective items was prepared by subject experts.The validity and reliability of the test were checked and approved by purposely selected mathematics subject experts and study supervisors.
The students in the intervention and those in the comparison groups sat for both the pre-test and post-test to assess whether or not there were achievement test differences after improvement training on the psychological variables given to the intervention groups.
In order to get additional information, two FGDs were conducted with purposively selected 14 students sat for intervention.The basic purpose of using FGD was to substantiate, confirm, and corroborate the quantitative data.Two FGDs, one FGD consisting of seven participants, were conducted.Mainly, it was intended to assess students' perception of intervention, the procedures used, challenges, and improvements.

Pilot test
Five purposively selected professionals from diversified disciplines and 10 potential participants were involved in assessing and confirming the validity of the instruments.
Two experts from Measurement and Evaluation, one Language expert, one Counseling Psychologist, and one Mathematician participated in the contextual validation of the instruments.These scholars reviewed and critically conducted item and dimension-level analysis so as to fit the context focusing more on content and construct validity.Likewise, 10 potential participants who were similar to the subjects of the final study participants were used for the validity checks.
In order to check the reliability of the questionnaires, 110 randomly selected grade 12 students were involved.Thus, 110 questionnaires were distributed to two randomly selected sections of the selected school.
Students' mathematics achievement test containing 40 objective items was prepared by subject experts.The validity and reliability of the test were checked and approved by three purposively selected mathematics subject experts and study supervisors.
In order to analyze the pilot test, SPSS version 26 was used.Students' perceptions toward their mathematics revealed Cronbach's alpha result of .83.Since this result (.83) was acceptable as it is, no item was deleted and all adapted items were taken to serve for the final data collection.Students' mathematics self-efficacy revealed .87.Thus, 36 items after the deletion of 3 items from the mathematics self-efficacy scale (due to relatively high Cronbach's alpha if item deleted) reported were used for the final data collection.

Design of the intervention phase
With regard to students' mathematics achievement test, pre-test items prepared were administered both for intervention (I) and control (C) groups similarly.In the meantime, training on the ways to improve students' mathematics self-efficacy and perception was given to intervention groups for four mathematics class periods for both of the schools.The intervention was given for one section for four class periods.Then, eight class periods were used for the application of interventional training.The intervention took 2-months periods.The total time used for the training was 11 hours and 15 minutes.This was based on Munoz (2000) that recommended 8 to 12 days of psychological training to enhance students' academic achievement.
The training was given by one counseling psychologist and two mathematics teachers.The practical application of training was applied by the subject teacher during the normal mathematics content instruction using the activity guide given by the researchers.Training materials were adapted from recognized sources and adjusted to the standardized training manual.Components of self-efficacy training concentrated on the general concept of self-efficacy and Bandura's four main models of internal and external self-efficacy enhancement.These are mastery experiences, vicarious experiences, modeling experiences, and emotional and physical experiences (Bandura et al., 1996).Additionally, four major ways to increase self-efficacy and how to best promote self-efficacy in the classroom were considered.
Students in the experimental groups were taught math subject using innovative classroombased instructional strategies containing self-efficacy enhancement strategy and effective teaching methodology.Students' opportunity to enhance their self-efficacy and the extent of performing math subject could be expected to be applicable to such opportunities for self-efficacy enhancement strategy and effective teaching methodology (Pangburn, 2020).
Additionally, using innovative classroom-based instructional strategies containing self-efficacy intervention management for about 3 to 4 weeks by the classroom teacher could enhance students' math achievement (Getachew & Birhane, 2016).However, students in the control groups were taught with the usual instructional methods.The counseling psychologist and the school director have evaluated the effective implementation of the training using a guide on the checklist.
The outcome of the training was expected to help the student's exploration of the practical know-how of building, maintaining, and improving self-efficacy to aim for better academic achievement and better living.Students' mathematics subject perception training components incorporated behavioral approaches, humanistic and cognitivist approaches, and others contextually.The training components comprise students' self-regulation skills (Pangburn, 2020).These skills comprised time management, motivation, help-seeking, self-efficacy, self-reflection, and strategic planning skills.A confident self-regulated learner knows how to effectively monitor their time management, motivation, help-seeking, self-efficacy, self-reflection, and strategic planning skills (Pangburn, 2020).These skills were chosen because they could be easily delivered, monitored, and implemented in class contexts.These components of self-regulation skills were given directly to the intervention groups of students within the planned time of intervention.
Finally, mathematics self-efficacy, perception, and the same mathematics achievement test given before were administered again for both intervention and comparison groups by modifying some of its features such as order of items, options, etc., at the end of the intervention.

Methods of data analysis
Quantitative data were double-entered into SPSS version 26 and were cleaned.Descriptive and inferential statistical techniques were used to analyze the quantitative data.Frequencies and percentages were used for the proportion of responses on students' mathematics perception, selfefficacy, and achievement.Throughout the study, .05alpha level and 2-tailed were used.Mediation analysis assumptions were checked, and all assumptions of parametric statistics were confirmed.
Independent samples T-test was used to examine the difference between comparison and treatment groups of students.Similarly, the T-test was used to examine gender differences in mathematics perception, self-efficacy, and mathematics achievement before and after the intervention.Eta-square was used to examine the difference in effect size.Dependent samples T-test was used to investigate the impact of the intervention (pre-posttest) on students' mathematics perception, self-efficacy, and achievement independently before and after training for treatment groups.Cohen's d formula was used to investigate the effect size difference of the treatment group of students' pre-post mathematics perception, self-efficacy, and achievement after the intervention.Additionally, the qualitative data obtained through FGD were transcribed and coded using Atlas Ti7 software.Next, data were analyzed in the form of narration in line with the objectives of the study.

Impact of intervention on students' mathematics perception, self-efficacy, and their mathematics achievement before and after intervention
The comparison groups of students' perception and self-efficacy towards mathematics revealed (M = 2.845 & SD = .538)and that of the treatment groups revealed (M = 2.776 & SD = .569),respectively.The comparison groups of students' mathematics self-efficacy before intervention were found (M = 3.206 & SD = .712),and the treatment groups were found (M = 3.199 & SD = .682),respectively.It could be inferred from this result that there was a slight mean difference between the two groups.
The two groups of comparison and treatment revealed no statistically significant mean difference in their mathematics perception before the intervention, [t (392) = 1.23, p > .22].Similarly, there was no statistically significant mean difference in their mathematics self-efficacy before the intervention, [t (392) =.10, p > .92].This result shows that both groups of students' perception and self-efficacy towards mathematics before intervention were almost similar and it was possible to conduct an intervention to see the effect of interventional training (see Table 1).
The two groups of students categorized as comparison and treatment revealed statistically significant mean differences in their perception of mathematics after the intervention [t (392) = −5.28,p < .001].Similarly, there was a statistically significant mean difference in their mathematics self-efficacy after the intervention [t (392) = −3.40,p < .001].This result refers as both groups of students' mathematics perception and self-efficacy were different after intervention.There was an effect of interventional training in which the treatment groups of students' mathematics perception and self-efficacy were enhanced than the comparison groups (see Table 2).
Even though our t-statistic is statistically significant, this does not mean the effect is important in practical terms.To discover whether the effect is substantive, it is recommended to use effect sizes.
The value of the effect size difference of comparison and treatment group of students' mathematics perception was revealed as r = .257.The benchmark for effect sizes this represents was small to medium.Therefore, as well as being statistically significant, this effect is relatively medium and so represents a substantive finding.Moreover, this result revealed that the effect of intervention brought a moderate effect on the treatment groups in enhancing their perception of mathematics compared to the comparison groups.
Likewise, the value of the effect size difference of comparison and treatment groups of students' mathematics self-efficacy was revealed (r = .169).The benchmark for effect sizes this represents was small.Therefore, as well as being statistically significant, this effect is small, but it can represent a substantive finding.This result revealed that though the effect of intervention seems small, it brought an effect on the treatment group in improving students' self-efficacy towards mathematics than in the comparison groups.
A qualitative study was conducted to substantiate, confirm, and corroborate the quantitative data.Focus group discussions were conducted for this purpose.The first FGD question was intended to assess their perception towards intervention.The results of the two FGDs were summarized and discussed as follows.
The student respondents reported that they had a serious problem with fear of mathematics subject before intervention.Due to this fear of mathematics, they reported that they hate their mathematics teacher.However, after the intervention was conducted, students reported that the trainers' training delivery techniques were very attractive.Discussants responded that the actual practical application of the training result was more than what they had expected to get improvements from the training.Student respondents have seen improvements in their perception of their mathematics teachers' teaching methods, subject matter knowledge, and self-efficacy towards mathematics subject and achievement.
Moreover, the treatment groups of respondents showed a progressive change in their mathematics achievement test due to the intervention.The comparison and treatment groups of students' differences in their perception and self-efficacy towards mathematics after the intervention were assessed qualitatively, and the result of qualitative data was almost similar to the quantitative result.
The above table also depicted the most important result that revealed as the t-value is a negative number for both variables that reveals the first condition (pre-test result) had a smaller mean than the second (post-test result).This posttest result refers to the effectiveness of the training that enhanced students' perception towards their mathematics teachers' teaching methodology and their mathematics self-efficacy and anxiety.Therefore, it is possible to conclude that the interventional training given to the treatment groups of students to adjust students' perception towards their mathematics was effective [t (193) = −3.64,p < .001].Similarly, the interventional training given to the treatment groups of students to improve their mathematics selfefficacy was effective [t (193) = −14.98,p < .001].
Similarly, the result of the pre-posttest intervention effect size of the students' mathematics self-efficacy and anxiety revealed a difference of 1.076 that showed a very large effect.This result infers that the intervention conducted to minimize the extent of students' mathematics anxiety and boost their self-efficacy shows improvement (see Table 3).

Difference between treatment and comparison groups of students' mathematics test result before and after intervention
Two groups of students assigned as comparison and treatment sat for mathematics pre-test before intervention.Students' mathematics pre-test revealed almost similar results (M = 26.89& 27.75) for comparison and intervention, respectively.
The two groups of students' mathematics pre-test out of 40 items revealed no statistically significant mean difference [t (392) = −1.28,p > .20].This result refers that both groups of students' mathematics test results were almost similar and possible to conduct an intervention so as to see the effect of interventional training (see Table 4).
Two groups of students assigned as comparison and treatment sat for mathematics post-test after the intervention.Accordingly, their mathematics posttest revealed a fairly large difference result (M = 27.095& 31.087) for comparison and intervention, respectively.Two points could be observed from these pre-test and post-test results for both groups.The first was that there was a very slight mean difference for the comparison groups from pre-test to post-test results (M = 26.890& 27.095), respectively.However, there was a large students' mathematics mean difference from the treatment group from pre-test to post-test results (M = 27.757& 31.087),respectively (see Table 5).Independent samples T-test was computed to see if there was a statistically significant mean difference among comparison and treatment groups of students' mathematics post-test results or after the intervention.Accordingly, the two groups of students mathematics posttest out of 40 items revealed tha there was a statistically significant mean difference, i.e. the significance level of the T-test was [t (392)= −6.45, p < .001].This result infers that there was a fair effect of interventional training in which the treatment groups of students achieved relatively better results than the comparison groups.
In addition to the t-statistic significant difference, it needs to assess the extent of their difference by using effect size calculation.Accordingly, the value of the effect size difference of comparison and treatment groups of students' mathematics achievement test revealed (r = .309).The benchmark for effect sizes this represents was medium.Therefore, as well as being statistically significant, this effect is moderate and it can represent a substantive finding.This result infers that the effect of intervention moderately increased the treatment groups of students' mathematics test achievement (see Table 5).

Impact of intervention on students' mathematics achievement after training for treatment groups
Students' mathematics pre-test and post-test results out of 40% were M = 27.757 and 31.087,respectively.This mean result infers that students' mathematics test result was improved after the interventional training.
The value of Pearson's r data of the experimental conditions of students' mathematics achievement revealed a relatively moderate correlation coefficient (r = .303)and significantly correlated, p < .05.Moreover, the t-value is a negative number that reveals that the first condition (pre-test result) had a smaller mean than the second (posttest result) and so the posttest result refers to the effectiveness of the training.Therefore, it could be concluded that the interventional training given to the treatment groups of students was effective at improving students' mathematics test [t (193) = −6.70,p < .001](see Table 6).

001
.481 The mean difference is significant at p < .05 The effect size of students' mathematics test achievement was estimated using the raw score mean difference and the standardized mean difference.The raw score mean difference between the preposttest interventions of students resulted in a difference of .481.This effect size could be judged as relatively moderate.In this study, therefore, the interventional training conducted to enhance the students' perception of their mathematics teachers' teaching methodology and mathematics anxiety and self-efficacy produced a positive attitude and revealed a better result.This positive index and moderate effect size show that the students who sat for intervention as treatment groups from pre-to post-test showed progress in their perception, self-efficacy, and mathematics achievement.

Gender difference regarding students' mathematics achievement after intervention
The final objective of this study was to scrutinize whether or not there exist gender differences regarding students' mathematics test after the intervention.Males achieved 31.242 and females scored 30.912 out of 40 items.This result refers to the presence of some differences in which males scored slight test differences than females.
Furthermore, the result of this test revealed that there was no statistically significant mean difference between males and females in their mathematics post-test result [t (192) = .44,p = .66].This result infers a similarity of the effectiveness of intervention among male and female students (see Table 7).
Following the quantitative result, qualitative part, the selected student participants for FGD were asked to express their feelings regarding the rate of the extent of gender difference towards students' mathematics achievement after the intervention.The general summary of male and female students who participated in the intervention study reported that it seems relatively males are better than female students in terms of their mathematics perception, self-efficacy, and achievement test.However, the difference might not be considered as the very determinant.
The majority of students who sat for FGD concluded that they were very happy with the intervention, enhanced their perception towards mathematics subject, and learned many skills by which they could adjust and flexibly follow their mathematics teachers' teaching methodology.The majority of the respondents replied that they have developed their mathematics self-efficacy and independence to learn and achieve better results in mathematics.Moreover, they reported and assured that their mathematics achievement test improved after they got interventional training.
Generally, the result of both quantitative and qualitative intervention study revealed that there was slight students' mathematics perception, self-efficacy, and achievement difference among gender.However, the difference might not be considered a big difference.

Discussion
The aim of this study was to conduct an intervention so as to improve students' mathematics perception and self-efficacy to improve their mathematics achievement in the selected secondary schools.
The finding of this interventional study revealed that the two groups of students categorized as comparison and treatment groups revealed no statistically significant mean difference towards their mathematics perception, self-efficacy, and mathematics achievement before the intervention, [t (392) = 1.23, p > .22],[t (392) = .10,p > .92],and [t (392) = −1.28,p > .20],respectively.However, the two groups of students categorized as comparison and treatment groups revealed statistically significant mean differences in their mathematics perception, self-efficacy, and mathematics achievement after the intervention, i.e. [t (392) = −5.28,p < .001],[t (392) = −3.40,p < .001],and [t (392) = −6.45,p < .001],respectively.Effect size results of comparison and treatment groups of students' mathematics perception revealed (r = .257),mathematics self-efficacy (r = .169),and mathematics achievement revealed (r = .309).Therefore, interventional training was statistically significant, and brought a relatively small effect that could bring moderate improvement in students' mathematics achievement..737 The mean difference is significant at p < .05 Supporting the finding of this study, Quaye and Pomeroy (2022) reported in their study that the students' mathematics achievement can be improved when their perception and self-efficacy towards mathematics subject could be enhanced.The self-confidence that students can get after improving their self-efficacy and perception has been identified by many researchers as a good determinant of success in mathematics.Students having higher mathematics selfefficacy also had higher mathematics achievement (Michelle, 2013;Simamora et al., 2018).The more students positively view and perceive learning mathematics, the better they get confidence and the better they achieve.Students are encouraged to learn mathematics at a higher level of motivation and achieve better results.Similarly, when students' beliefs in their self-efficacy and perception are positive and enhanced by teachers, their attitude and interest in learning increase, and this in turn improves their mathematics achievement (Rodríguez et al., 2020).
Moreover, students need assistance in developing preferred mathematics learning strategies for better understanding (Shone et al., 2023).Mathematics learning strategies are specific techniques used to promote and enhance mathematics learning.Mathematics teachers are required to teach using and applying mathematics in practical tasks and in real-life problems so as to help students contextualize mathematical calculations (Ramirez et al., 2018).This is because mathematics teachers' teaching effectiveness is a core issue to help students achieve better marks though there is little consensus about the most important teaching method.
The effectiveness of teaching has remained a "black box" to adapt students by attending their classes and performing to their level best (Hill & Chin, 2018).When there is positive teacherstudent interaction, students' engagement in math learning increases and this could improve secondary school students' math achievement (Ramirez et al., 2018;Yang et al., 2021).This is because teachers play a relevant role in guiding, encouraging, motivating, and emotions generated in the students, including issues that determine decision-making in the students' learning and mathematics performance (Trigueros et al., 2020).
Similarly, Getachew and Birhane (2016) reported the existence of a statistically significant difference between the experimental and control groups of students in their mathematics achievement.However, their study revealed that there was no statistically significant mean difference between the experimental and the control groups on the mean score of self-efficacy belief in mathematics.
Academic self-efficacy mediated the relationship between teachers' emotional support and the mathematical performance of Chinese boys and girls.Also, mathematics behavioral engagement mediated the relationship between teachers' emotional support and the mathematics performance of Chinese students (Yang et al., 2021).Similar to arguments in previous studies selfefficacy contributes to academic achievement and is more than a simple reflection of prior performance (Holenstein et al., 2022;Zakariya, 2021).
Finally, the outcome of the T-test revealed that there was no statistically significant mean difference between male and female students towards their mathematics post-test result, i.e. [t (192) =.45, p = .66].This result infers a similarity of the effectiveness of intervention among male and female students.Similar to this study, an attempt to improve students' mathematics achievement through enhancing students' mathematics achievement was successful (Schöber et al., 2018).Michelle (2013) reported that although male students' reported higher mathematics self-efficacy than females, gender differences were not found to exist in their mathematics achievement.Contrary to the above findings, the result of the study conducted by Holenstein et al. (2022) indicated that gender differences were observed in which males preferred mathematics selfefficacy and self-concept than females.However, female students reported having better grades in mathematics than males.

Conclusions
The result of the intervention study revealed no statistically significant mean difference among the comparison and treatment groups of students during the pre-test study regarding their mathematics perception, self-efficacy, and their mathematics achievement test.However, the intervention study result revealed that the treatment group of respondents' score of their perception towards mathematics was increased from the pre-test to post-test study.There was a statistically significant effect of the intervention and a moderate effect that enhanced the treatment groups of participants' perception towards their mathematics subject more than the comparison groups.Similarly, the intervention study revealed that there was an increment score of the treatment group of students towards their mathematics self-efficacy from the pre-test to posttest study.There was a relatively small effect of the intervention.However, statistically significant mean differences existed among the treatment and comparison groups of students regarding their mathematics self-efficacy.
Following the treatment groups of participants' improvement of their perception and selfefficacy towards mathematics, they were given 40-item mathematics achievement tests.The result of the intervention study revealed that there was an improvement of the treatment groups of students towards their mathematics achievement test from pre-test to post-test.Similarly, there was a moderate effect size and statistically significant difference between the treatment and comparison groups after the intervention.Similar to the quantitative result, the qualitative result reported enhancement of treatment groups of students' mathematics perception and self-efficacy due to the interventional training.Following the enhancement of students' mathematics perception and self-efficacy, their mathematics achievement test was also improved.
Finally, the intervention study revealed that though male students' mean result was relatively more than female students, this difference was not statistically significant among genders in their mathematics posttest results.This result infers that both males and females attended the intervention relatively in a similar way and similarly benefitted from the interventional training.

Implications of the study
The general conclusion of this intervention study revealed that effective interventional training can enhance students' mathematics perception and self-efficacy.This was because teachers are responsible in playing a great role in producing students with better mathematical perception and self-efficacy by applying innovative classroom-based teaching strategies.This implies the need for effective interventional training for the teachers that equip them to enhance students' mathematics perception and self-efficacy so as to improve their mathematics performance.Moreover, this intervention study suggested future intervention studies that attempt to apply advanced mechanisms that could enhance students' mathematics perception and self-efficacy so that their achievement could be improved.Lastly, the way intervention could be applied to enhance students' perception and self-efficacy to improve their other subjects than mathematics like natural sciences, languages, and social sciences have been forwarded.

Limitations
There are several limitations of this study that should be considered alongside the findings.
There could be limitations with regard to interventional training materials used in this interventional study.Information dissemination could have been made due to the presence of the two groups of intervention (treatment and control) from one school which could be the barrier not to giving a relatively strong recommendation.Concerning the procedure, the possible basic limitation is that the mathematics teachers' training was rather short.It would be interesting to investigate whether the intervention would have a greater impact on students' learning and achieving more.This could be successful if mathematics teachers had more extensive training in internalizing and advancing their teaching in plenty of psychological and pedagogical methodologies.

Table 3 .
Paired samples Correlations and test statistics of impact of interventional training on treatment groups of participants (size estimate by Cohen's d, The mean difference is significant at p < .05