Relationship between prior mathematical knowledge and basic-learning skill

Prior mathematical knowledge in secondary education is thought to be one of the most important premises to obtain good grades in mathematics in the university studies. Many authors have done a lot of research to investigate the association between prior knowledge in mathematics and basic-learning skill in university studies.

Prior knowledge had a positive influence on new learning (Lin & Liou, 2019; Reinholz & Gillingham, 2017). The high preference for mathematics have a much greater influence on the process of acquiring knowledge in mathematics (Lambić & Lipkovski, 2012; Mutodi & Ngirande, 2014). (James, Montelle, & Williams, 2008) found that the secondary school qualifications in mathematics influence their results in the core first-year mathematics, meanwhile (Kizito, Munyakazi, & Basuayi, 2016) found that students' perceptions of their workload emerged as the factor having the greatest impact on student's performance, and (Richardson, Abraham, & Bond, 2012) revealed that the academic self-efficacy, grade goal, and effort regulation generated medium-sized correlations with GPA. The higher levels of mathematical knowledge (Hill, Schilling, & Ball, 2004; Schlomer, 2017) showed a greater focus on student mathematical thinking, meanwhile, limited mathematics thinking impact a smooth transition into mathematics programs (Hourigan & Leavy, 2017). (Flitcroft & Woods, 2018) revealed that teacher behaviors to students' academic performance have both positive and negative impacts, according to their positive or negative attitude, and (King & Cattlin, 2015) found that there are significant negative impacts associated with students’ pass rates with large numbers of students enrolling in degrees. Hence, there is evidence of positive relationship between prior knowledge, mathematics preferences, qualification, workload, higher levels of mathematical knowledge, as well as teacher behaviors and mathematics learning. (Fernández & Figueiras, 2014) found that teachers' knowledge impacts the continuity of mathematics, but (Hilby, Stripling, & Stephens, 2014) revealed that there is no relationship between mathematics ability of teachers. Teachers applying meta-cognitive skills in assisting their learners, and academic service-learning experience influence a shift away toward self-efficacy for teaching by the students (Hollingsworth & Knight-McKenna, 2018; Tachie & Molepo, 2019). (Remillard, 2005) indicated that use of mathematics curriculum materials is an increasingly widespread practice to regulate mathematics teaching, and (Krain, 2016) confirm that case-based materials led students to gain a better understanding of theoretical concepts. (Steele, Johnson, Otten, Herbel-Eisenmann, & Carver, 2015) found that student thinking influence mathematics learning, meanwhile deepened teachers' knowledge influence mathematical knowledge for students (Kaur, 2017), and colleagues’ critical review provided progress of mathematics teachers (Kortjass, 2019). (McLeskey & Waldron, 2004) indicated that knowledge-for-practice, knowledge-in-practice, and knowledge-of practice of teacher learning (Cochran-Smith & Lytle, 1999) influence academic progress of students, and (Baumert et al., 2017) revealed a substantial positive effect of pedagogical content knowledge on students’ learning gains. Thus, it is evidenced that teachers' knowledge, meta-cognitive skills, curriculum materials student thinking, colleagues’ critical review, as well as pedagogical content knowledge impact students’ learning in mathematics. In conclusion, the investigation of the relationship between mathematical knowledge and basic-learning skill, as resulted in previous research, is important. Therefore, based on the above literature review it is hypothesized that:

H # 1: The higher mathematical knowledge scores are associated with higher basic-learning skill scores

Relationship between comprehensive learning approach and basic-learning skill

The comprehensive learning approach is assumed to be one of the important variables that impact basic-learning skill in mathematics at the university. A lot of research is carried out to investigate the association between comprehensive learning approach in mathematics and basic-learning skill at university. (Karagiannopoulou & Christodoulides, 2005) found that the learning is a stronger predictor of academic achievement, and (Jenkins, 2017) indicated that problem-based learning impacts positively the students’ proficient level in mathematics. (Wyse & Soneral, 2018) revealed that the cognitive complexity is a contributor of the learning process, and (Gilstrap, 2019) reports a strong correlation between the standards for learners and the approaches to learning. (Crawford & Wang, 2015) revealed an increasingly significant performance by prior academic performance, and (Johnson, Johnson, & Johnson, 2017) found that comprehensive learning impact student achievements. (Yildirim, 2017) pointed out that flipped classroom increases the learning, and (Rimbey, 2013) revealed significant differences in classroom practice as a result of the treatment using the learning mathematics for teaching. Thus, it is evidenced that problem-based learning, cognitive complexity, standards for learners, prior academic performance, comprehensive learning, flipped classroom correlate positively with basic-learning skill. (Tsouccas & Meletiou-Mavrotheris, 2019) indicated a positive impact of utilizing mobile apps as an instructional tool, and (Holmes & Hwang, 2016) showed that the students benefited greatly from the problem- based learning in mathematics. (White & Nitkin, 2014) demonstrate the positive impact of the program on students learning, meanwhile (Alkhateeb, 2003) found that surface learning and deep learning approach accounting for 34.6% of variance to learning mathematics. (Barakat, 2005) showed that differences in the teaching methods yielded no statistically significant differences in skills, but (Schaub & Baker, 1991) revealed that instructional methods yield higher mean classroom knowledge. (Kogan & Laursen, 2014) pointed out that the impact of inquiry-based learning on students' grades is sizable and persistent. Hence, there is evidence that mobile apps, the program, deep learning approach, instructional methods, and inquiry-based learning influence basic-learning skill. In conclusion, the investigation of the relationship between a comprehensive learning approach and basic-learning skill, as resulted by previous research, indicates the scientific and practical importance. Therefore, based on the above research it is hypothesized that:

H # 2: The higher comprehensive learning approach scores are associated with higher basic-learning skill scores

Relationship between problem-based teaching and basic-learning skill

Problem-based teaching used by math teachers is supposed to be one of the most important variables that impact basic-learning skill in mathematics. A lot of research has been done to investigate the association between problem-based teaching in mathematics and basic-learning skill at university.

Collaborative problem-solving performance is positively related to performance in the core PISA subjects (science, reading and mathematics), but the relationship is weaker than that observed among those other domains (OECD, 2019). The mathematics teaching predicted achievement, quality of the learning, and learning experience (Giles, Byrd, & Bendolph, 2016; Huntley, 2013; Karagiannopoulou et al., 2005; Serra, Bikfalvi, Masó, Carrasco, & Garcia, 2017). From the other point of view, (Toetenel & Rienties, 2016) found no positive correlation between learning design and student outcomes. Virtual manipulatives displayed considerable growth in students’ learning, and better performance in mathematics (Demir, 2018; Lee & Chen, 2014); but (Wilder & Berry, 2016) indicated that the traditional approach to instruction was equally effective in improving student mathematics knowledge. Students who received e- problem-based learning instruction (Baele, 2017) self-reported significantly greater engagement than those who received traditional instruction, and face-to-face courses (Derr, 2017). Therefore, as above work evidenced, the collaborative problem-solving performance, mathematics teaching, virtual manipulatives, e- problem-based learning instruction correlate positively with basic-learning skill.

(Maciejewski, 2016) pointed out that how a student conceives the nature of a subject affects their learning outcome, and (Kelly, 2002) found that an equitable classroom climate in mathematics influence students' formation with knowledge. Teachers’ mathematical knowledge, a computerized instructional management system, and simulated teaching environment explained significant gains in math achievement (Hill, Blazar, & Lynch, 2015; Hill et al., 2004; Ma et al., 2016; Ysseldyke et al., 2003). (Kearns & Fuchs, 2018) found that cognitively focused instruction accelerated low-achieving students' academic progress, but (Chung, 2004) revealed that students from both constructivist and traditionalist approach improved their skills. (Burns, 2005) indicated that solving mathematics problems often requires searching for new approaches, and there is a significant improvement in mathematics’ achievement influenced by the learning style approach (Al-Balhan & Soliman, 2019; Burns, 2005). Hence, it is evidenced the positive association between conceives the nature of a subject, teachers’ mathematical knowledge, computerized instructional management system, simulated teaching environment, cognitively focused instruction, and basic-learning skill. In conclusion, the investigation of the relationship between problem-based teaching and basic-learning skill, as resulted in previous research, is important. Therefore, based on the previous research it is hypothesized that:

H # 3: The higher problem-based teaching scores are associated with higher basic-learning skill scores

Relationship between assessment impact and basic-learning skill

Knowledge assessment of students is one of the important variables that are related to basic-learning skill in mathematics. A lot of research is carried out to investigate the association between assessment impact in mathematics and basic-learning skill in the university studies.

Assessment methods and matriculation examination score were the predictors of students’ progress (Karagiannopoulou et al., 2005; Kizito et al., 2016). (Duzhin & Gustafsson, 2018) found that online homework with immediate feedback was found to be even more effective than clickers, as well as (Abushammala, 2019) indicates that the use of multiple in-class activities and digital technology are important to enhance students' performance. (Lees & Anderson, 2015) pointed out that formative assessment may improve student performance, and (Brosvic, Dihoff, Epstein, & Cook, 2006) revealed that feedback facilitates the acquisition of mathematics knowledge. Mathematics teachers' judgments and information literacy assessment were highly predictive of students' performance (Chen, 2006; Pinto & Fernández-Pascual, 2017). Thus, assessment methods, matriculation examination score, online homework with immediate feedback, multiple in-class activities, digital technology, formative assessment, mathematics teachers' judgments, as well as information literacy assessment are associated with basic-learning skill.

Assessment support the advancement of teaching and learning (Cao, Jung, & Lee, 2005; Pinto et al., 2017). (Prevost, Vergara, Urban-Lurain, & Campa, 2018) found the relationship between the assessment and the teaching and learning, and (Pomplun & Omar, 2000) found that invariance of a mathematics assessment was supported for factor loadings and intercepts. Assessing student learning support academic progress (Poskitt, 2014), although, students from different countries show the different academic ability of mathematics achievement (Randel, Stevenson, & Witruk, 2000). Thus, as above work evidenced, assessment support, assessment, as well as invariance of a mathematics assessment influence basic-learning skill. In conclusion, the investigation of the relationship between assessment impact and basic-learning skill, as resulted by literature review, indicates the research and practical importance. Therefore, it is hypothesized that:

H # 4: The higher assessment impact scores are associated with higher basic-learning skill scores.

Relationship between p rior knowledge, comprehensive learning approach , problem-based teaching, assessment impact , and student's basic-learning skill in mathematics

Many researchers have done a lot of work to investigate the impact of prior knowledge, comprehensive learning approach, problem-based teaching, and assessment impact on basic-learning skill in mathematics. Approaches to instruction and curriculum design, formative assessment data and feedback to students (Gersten et al., 2009), as well as prior knowledge (Fries, DeCaro, & Ramirez, 2019) have their impact on students’ learning. But, from the other point of view (Gruendler, 2018) found that none of the previous grades had a significant impact on the students' actual earned grade. A positive effect of pedagogical knowledge on students’ learning gains was mediated by the provision of cognitive activation and individual learning support (Baumert et al., 2017), as well as by teachers’ mathematical knowledge (Hill et al., 2015). (Gearing & Hart, 2019) revealed that problem-based lessons increased the students’s ability, and (Hakyolu & Ogan-Bekiroglu, 2016) found a positive relationship between knowledge of students and arguments during a scientific argumentation. On-going professional development for teachers are positively associated with student achievement (Fox, 2014; Visser, Juan, & Hannan, 2019), but (Benken, Ramirez, Li, & Wetendorf, 2015) reported that merely the number of years of mathematics in high school does not necessarily imply that students are prepared for the level of rigor expected in postsecondary institutions. Hence, it is evidenced that approaches to instruction and curriculum design, formative assessment data, feedback to students, prior knowledge, pedagogical knowledge, individual learning support, teachers’ mathematical knowledge, problem-based lessons, on-going professional development for teachers impact student's basic-learning skill in mathematics. (Garet et al., 2016) revealed that math content knowledge and instructional practice were generally not correlated with student math achievement, but (Lampinen & McClelland, 2018) pointed out that pedagogical materials affect learning. Project-based learning instruction, as well as diagnostic assessment information, influenced student achievement in mathematics (Han, Capraro, & Capraro, 2015; Linsell et al., 2012). (Nguyen, 2016) found that students with math preference in high school, as well as teaching approach influence the academic outcome (Nguyen, 2016; Santagata & Yeh, 2014), and teaching techniques influence students’ achievement (Kilion, 2016; Rimbey, 2013). Thus, there is evidence that pedagogical materials, project-based learning instruction, diagnostic assessment information, students with math preference in high school, teaching techniques influence student's basic-learning skill in mathematics. In conclusion, the investigation of the relationship between prior mathematics knowledge, comprehensive learning approach, problem-based teaching used by the teacher, assessment impact, and knowledge conceived in mathematics, as confirmed by previous research, is important. Therefore, based on the above previous research it is hypothesized that:

H # 5: The higher prior knowledge, comprehensive learning approach, problem-based teaching, and higher assessment impact scores predict higher basic-learning skill scores

Prior knowledge

Prior knowledge’ frequencies indicates that most of the respondents (46.7%) of the experimental groups and 40.5% of the control groups passed in the previous academic year; 33.3% of the experimental groups and 39.7% of the control groups achieved good results; meanwhile 20% of the experimental groups and 19.8% of the control groups achieved high results. Central tendency values for experimental groups (M= 2.733, SD = .774), as well as for control groups (M= 4.00, SD = .897), indicate the same tendency for values as measured by frequencies. Hence, there are small differences of prior knowledge (pass: 6.2%; good: -6.4%; high: 0.2%) between the experimental and control groups of students. Therefore, there are small differences of prior knowledge between the experimental and control groups of students.

Prior knowledge vs current knowledge (pre-post-test)

Comparing prior knowledge from previous academic year with current knowledge of experimental group has resulted that: (1) 5.93 % more students fail; (2) 2.96% more students pass, (3) 4.44% less students achieved good; (4) 4.44% less students achieved outstanding results in mathematics. Comparing prior knowledge from previous academic year with current knowledge of control group scores has shown that: (1) 9.09 % more students fail; (2) 6.61% more students pass, (3) 8.27% less students achieved good; (4) 7.43% less students achieved outstanding results. Hence, there is a difference between experimental and control group of students, especially in fail and outstanding levels achieved in mathematics.

Comprehensive learning approach

Comprehensive learning approach’ frequencies indicates that most of the respondents (53.4%) of experimental groups and 47.1% of control groups include full information from class reading material, practical, etc. during learning; 13.3% of the experimental groups and 26.4% of the control groups include a little; meanwhile 33.3% of the experimental groups and 26.4% of the control groups are undecided. Central tendency values for experimental groups (M= 3.46, SD = .808), as well as for control groups (M= 2.79, SD = .751) indicate the same tendency for values as measured by frequencies. Therefore, there are small differences of comprehensive learning approach (fully include: 6.3%; a little include: -13.1%; undecided: -6.9%) between the experimental and control groups of students.

Problem-based teaching

Problem-based teaching’ frequencies indicates that most of the respondents (66.7%) of the experimental groups and 19.8% of control groups have acquired fully support by lecturers in every lesson during teaching; 6.7% of the experimental groups, and 60.4% of the control groups a little support; meanwhile 26.7% of the experimental groups and 19.8% of the control groups are undecided. Central tendency values for experimental groups (M= 3.86, SD = .887), as well as for control groups (M= 3.27, SD = .930), indicate the same tendency for values as measured by frequencies. Hence, there are substantial differences of problem-based teaching (fully support: 46.9%; a little support: -53.7%; undecided: 6.9%) between the students to whom problem-based teaching is used compared to the students to whom the traditional approach is used.

Assessment impact

Assessment impact’ frequencies indicates that most of the respondents (73.3%) of experimental groups and 66.9% of control groups have fully learned from mistakes and study better next time if they don’t do well in an exam; 6.7% of the experimental groups and 6.6% of the control groups have a little learning; meanwhile 20.0% of the experimental groups and 26.4% of the control groups are undecided. Central tendency values for experimental groups (M= 2.73, SD = .774), as well as for control groups (M= 3.40, SD = 1.311), indicate the same tendency for values as measured by frequencies. Therefore, there are small differences of assessment impact (fully learned: 6.4%; a little learned: 0.1%; undecided: -6.4%) between the experimental and control groups of students.

Basic-learning skill

Basic-learning skill’ frequencies indicates that most of the respondents (53.4%) of experimental groups and 53.7% of control groups are fully able to learn the basic concepts taught in the math course; 6.7% of the experimental groups and 6.6% of the control groups are a little able; meanwhile 26.7% of the experimental groups and 26.4% of the control groups are undecided. Central tendency values for experimental groups (M= 4.00, SD = .897), as well as for control groups (M= 3.80, SD = .832), indicate the same tendency for values as measured by frequencies. Hence, there are small differences of basic-learning skill (fully able: -0.3%; a little able: 0.1%; undecided: 0.3%) between the students to whom problem-based teaching is used compared to the students to whom the traditional approach is used.

Test of hypothesis

Table 2 and Table 3 shows Pearson correlations outputs of the relationships between variables

H # 1: As shown in Table 2 and Table 3, there is a negligible correlation between prior knowledge and basic-learning skill variables, r² = -.008, n = 135, p > .005 for experimental groups, as well as for control groups, r² = .003, n = 121, p > .005. Furthermore, there is not a statistically significant relationship between prior knowledge and basic-learning skill by students, thus a random chance could explain the result.

H # 2: As shown in Table 2 and Table 3, there is a significant positive correlation between comprehensive learning approach and basic-learning skill variables, r² = .417, n = 135, p < .005 for experimental groups as well as for control groups: r² = .443, n = 121, p < .005. Hence, high scores of comprehensive learning approach are associated with high scores of knowledge conceived.

H # 3: As shown in Table 2 and Table 3 , there is a significant positive correlation between problem-based teaching and basic-learning skill variables, r² = .319, n = 135, p < .005 for experimental groups, meanwhile there is a low negative correlation between these variables for control groups: r² = -.048, n = 121, p < .005. Therefore, high scores of problem-based learning teaching’ strategies are associated with high scores of knowledge conceived. Meanwhile, high scores of traditional learning problem-based teaching are associated with low scores of knowledge conceived.

H # 4: As shown inTable 2 and Table 3, there is almost a negligible correlation between assessment impact and basic-learning skill variables, r²= .029, n = 135, p < .005 for experimental groups as well as for control groups: r² = .106, n = 121, p < .005. Hence, it is expected that the scores of assessment impact are not likely to associate with the scores of basic-learning skill.

H # 5: As shown in Table 4, the total variance of basic-learning skill levels explained by prior knowledge, problem-based teaching, comprehensive learning approach and assessment impact (the model) is 50.8%, F (4, .933), p < .005 for experimental groups, and 54.4%, F (4, .799), p < .005 for control groups. The model reaches statistical significance (Sig. = .000; this means p <.0005). The F value, that is the ratio of the mean regression sum of squares- an estimate of population variance that accounts for the degrees of freedom indicates that null hypothesis is false (regression coefficients are different from zero).

For experimental groups, beta value for prior knowledge is-.154, for comprehensive learning approach, is .658, for problem-based teaching is .217, and for assessment impact is -.315. The largest beta standardized coefficient is .658, which is for comprehensive learning approach.

For control groups, beta value for prior knowledge is.115, for comprehensive learning approach is .629, for the traditional approach is -.320, for assessment impact, is .194. The largest beta standardized coefficient is .629, which is for comprehensive learning approach.