The effects of a multiple solution method in mathematics learning for secondary schools

Solving problems in mathematics does not involve a single method only. Using a multiple solution method can also assist students in seeking a solution. A multiple solution method is a method that can be applied in the process of teaching and learning in which students will learn a variety of methods to be applied in finding solutions for a single question. This paper reports a study conducted to review students’ views of the effects of the use of a multiple solution method in learning mathematics. The study used two approaches: the quantitative approach (Section A, B and C) and qualitative approach (Section D), in a questionnaire for collecting data. 100 students from a secondary school located in Klang, Selangor were selected as respondents in this study. The questionnaire focused on the perceptions of students in using a multiple solution method and consequently the effect towards students’ level of self-regulation. The Likert scale ranging from 1 (strongly disagree) to 4 (strongly agree) was used. All respondents were also instructed to indicate the challenges faced in the use of a multiple solution method in the open-ended questions. The results showed that students have a good perception on the use of a multiple solution method in mathematics learning. A multiple solution method can also affect students’ self-regulation. The findings of the open-ended questions show that which are the challenges faced by students when use multiple solution method are get confused because it has many methods, it is hard because students lack mathematics skill, and also involve time constraints because they need to memorize many methods.


Introduction
Solving mathematical problems varies according to the situation such as in learning mathematics or in a real-life. Students can use a multiple solution method in finding the solution whether it is one or more problems. However, what is a multiple solution method? Generally, the multiple solution method means arriving at a solution by using more than one method. In mathematics learning, a learner needs to present a variety of methods in order to solve a problem. He or she can present the solution using an equation, graph, table, and many more [1]. The multiple solution method actually can improve students' mathematical knowledge. When students try to compare the solution with their friends who come out with different ways to solve the problem, students actually increase their understanding [2] and by IOP Publishing doi: 10.1088/1742-6596/1988/1/012048 2 comparing different solution methods for solving the same problem also will improve mathematics learning [3]. There are many ways to solve the problem but sometimes when students are only on one way solution, students do not have another option to solve the problem. Most of the problems happen when the students present a short solution or only a few steps of the solution, so the selection of student work is generally straightforward [4]. When students are focussed on a single method, students lack flexibility. It is because students cannot present any ideas that are related to find the solution of the problem [5]. The education system is not to be blamed on the one solution method in solving the problem, but if students have the opportunity to solve in many ways, it can help students to be more flexible. How do they become more flexible in finding the solution? If students use multiple solution method, will it give a good effect in their learning process? These issues will affect the learning skills of the students in learning mathematics.

Literature review
The perception of multiple solution is defined as students' feedback about the use of multiple solution method in learning mathematics. Each person has his or her own perception towards multiple solution method. In using the multiple solution in teaching and learning process, it can increase the students' motivation [6]. By applying the multiple solution method, students were motivated to develop more than one solution on the same problem where students are not too focus only in single solution. Besides that, students get motivated in their learning because there are some favourable conditions for students when they use multiple solutions in the classroom [7]. When the teacher has a deep understanding of the contents by topic they are teaching, it can grab students' attention [8]. Learning from multiple solution methods fosters the understanding of solution methods, especially when the learning is supported by self-explanation or by the instructional explanation which compare different solutions [9]. According to Schukajlow and Krug in 2012, not all students are using the multiple solution method in the learning. 58% apply two or more solutions and it showed that if the teacher encourages their students to apply more than one method, as most of the students will do it. Students promote higher self-regulation of their learning when they prompted to develop multiple solution method [10]. There are positive effects of encouraging students to find various methods of solution and competence between their friends during the learning process as well as the students' interest [11].
Students' self-regulation is a contributor of mathematics success learning because it relates to the factor of study environment regulation and seeking of help from friends and teacher [12]. Those students who can regulate their learning are proposed to gain the most out of education because their actions are motivated by learning rather than external rewards [13]. The same goes with other research [14] where the researchers also found that when students work with others, it contribute to students' level of selfregulation. Based on the previous study in 2017, the study stated that there was no difference in students' self-regulatory from the different group where the researcher just pick two group classes with an equal number of students [15]. It can be proved by Leidinger and Perels who found that the achievement from students in grade four increased [16].
The multiple solution method also has challenges in implementation. When students try to apply the multiple solution method in their learning, they can get confused to apply the best method to the question. Students are stuck to apply the multiple solution method because they cannot understand the meaning of some terms of the questions, and it was also difficult to develop the mathematical model into the problem. When students cannot understand what the questions was about, not only multiple solution method but students cannot apply even single method when they get confused [10]. Students lacked mathematics skill such as problem solving, differentiate the question task are the difficulties students faced when learning mathematics. Furthermore, when students are afraid to ask, it can lead them to not attempt answering questions [17]. To add to this view, this study aims to identify students' perception using multiple solution method in mathematics learning. Besides that, it determines the effect of multiple solution method towards self-regulation and the challenges faced by students.

Research methodology
This study involved both approaches which are the qualitative approach and quantitative approach. This study used a descriptive design for the quantitative approach. The quantitative approach subscribes to a specific empirical approach to knowledge, trusting that by measuring precisely enough we can make claims about the objective of study [18]. The quantitative approach was applied to identify students' perception using the multiple solution methods in mathematics subject and to determine the effect of multiple solution method towards self-regulation. As for the qualitative approach, data were collected by using open-ended questions in order to examine the challenges of using the multiple solution method. The target population of this study were 100 respondents from a secondary school, located in Klang, Selangor.
This study analyses using a non-probability sampling which is convenience sampling to select the respondents. Before questionnaires had been distributed to respondents, the pilot study was tested with the 10 respondents. SPSS was used to check the reliability of the questionnaire. Table 1 shows the result of pilot study where the value of Cronbach's Alpha is greater than 0.7. For Section A, it only focus on the demographic part consist of 2 items which are race and class of respondents. After questionnaire was distributed to respondents, they need to tick the box that are relevant to them. The aim for section B is to identify students' perceptions using the multiple solution method in mathematics subject. In this part, it consists of nine related items. Next, Section C is to determine the effect of multiple solution methods toward self-regulation. It consist of 5 items. For Section B and Section C, the respondents need to rate honestly for each item in scale from 1 to 5 where 1= strongly disagree, 2= disagree, 3= neutral, 4= agree and 5= strongly agree. Lastly, Section D consists of the open-ended questions which was the respondents need to answer the question 'What are the challenges on using the multiple solution methods? and it will classified the respondents answer into several categories to find the differences and similarities of patterns to answer the questions where the challenges faced by students when using the multiple solution method.

Section A: the demographic part
The finding for section A, the demographic part consists of race and class as shown in table 2 and table  3.  Table 3. The distribution of the respondents by class.
It can be seen in table 3, out of 100 respondents, the highest respondents came from 4 Nifty which is 36 (36%) respondents, then followed by 4 Optimistic which is 20 (20%) respondents. Next, 15 (15%) respondents are from 4 Tolerance and 13 (13%) respondents are from 4 Charismatic. Furthermore, 9 (9%) respondents are from 4 Noble and the lowest numbers of respondents are from 4 Efficient which is 7 (7%) respondents.

Section B: students' perception using multiple solution method
The finding for Section B is shown in table 4. The overall results for students' perception using the multiple solution methods in mathematics subject (Mean=4.0078, SD= 0.64179). Most of the respondents think that using multiple solution method can increase their knowledge since it gives the highest good perception of using the multiple solution method in mathematics learning (Mean= 4.24, SD= 0.842). Other than that, they think it is good to use the multiple solution method than the single method because they can understand better about the solution and relate it with the problem as well as perform better when using the multiple solution method. In addition, they also think it is easy to find the answer using the multiple solution method. However, they have a moderate perception towards the presentation of the multiple solution method foster learning outcome compared to the presentation of single solution method and it give the lowest mean of respondents' perception (Mean= 3.94, SD= 1.013). Next, to determine the significant difference between classes on perceptions of using multiple solution methods in mathematics subject. The hypothesis to be tested is: H0: There is no significant different between classes on perceptions of using the multiple solution methods in mathematics subject. H1: There is significant different between classes on perceptions of using the multiple solution methods in mathematics subject.  The F test as shown in table 5 showed the significant difference in the perceptions of using the multiple solution methods in mathematics subject by class of respondents (F(5, 94)=1.484, p-value=0.202 > 0.05). Thus, this simply indicates that we accept H0, there is no significant difference between classes on perceptions of using the multiple solution methods in mathematics subject.

Section C: the effect of multiple solution methods toward self-regulation
The finding for Section C is shown in table 6. The overall results for the effects of the multiple solution methods (Mean=3.7260, SD=0.60879) in table 6 indicate a moderate mean towards self-regulation. Most of the respondents give a moderate scale of the effect of the multiple solution method towards selfregulation. The respondents accept that having a flexibility in solving the problems in mathematics are the most effect of multiple solution method since it gives the highest mean towards self-regulation (Mean=3.92), followed by reflection on various solution method help them be more flexible and effective. Other than that, the respondents also have a moderate mean on constructing the multiple solution method can improve self-regulation. Last but not least, not all of the respondents compare their work with friends, and it give the lowest mean of effect of this method towards self-regulation. Next, to determine the significant difference between Classes on the effects of the multiple solution methods towards self-regulation in mathematics subject. The hypothesis to be tested is: H0: There is no significant difference between Classes on the effects of the multiple solution method towards self-regulation in mathematics subject. H1: There is significant difference between Classes on the effects of the multiple solution method towards self-regulation in mathematics subject.  Figure 2. The descriptive statistics on the effects of multiple solution method towards self-regulation.
As shown in figure 2, the highest mean of respondents on the effects of the multiple solution method towards self-regulation by class was 4 Noble (Mean= 4.067) where it shows a good mean from 4 Noble students. Then, follow by 4 Efficient (Mean = 4.029) also gives a good response on the effects of multiple solution method towards self-regulation. Then, other four classes give a moderate perception of using multiple solution methods which are 4 Optimistic (Mean=3.88), 4 Charismatic (Mean= 3.785), 4 Tolerance (Mean = 3.64), and the mean for 4 Nifty (Mean= 3.511) was the lowest mean. Therefore, ANOVA test was conducted to determine the significant difference. The F test as shown in table 7 showed significant difference effects of the multiple solution method towards self-regulation by classes of respondents [ F (5, 94) = 2.286, p-value = 0.052 > 0.05). This simply indicates that we accept H0 which are there is no significant difference between Classes on the effects of the multiple solution methods towards self-regulation in mathematics subject. Although it is done in previous study on the effect of the multiple solution method towards self-regulation, but comparison on the effect between classes is still in its infancy.

Section D: open-ended questions 'what are the challenges on using the multiple solution methods?'
In order to answer the research question in Section D, the challenges on using the multiple solution method, open-ended questions are conducted. 100 students are selected as respondents. The respondents