Beliefs of Secondary students’ of Coelemu, Chile and its rural areas about mathematics and problem solving

The aim of this study was to analyze the beliefs of secondary students from Domingo Ortiz de Rozas High School of Coelemu, Chile and its rural areas regarding the subject of mathematics and problem solving due to the teaching and learning process. The research was developed under a quantitative approach of quasi-experimental descriptive design survey type (n=156). As a basis was used the closed survey of Barrantes (2008), adapted based on the feedback provided by expert judgment. The results show statistically significant differences related to origin and academic performance in their beliefs about mathematics and problem solving when considering the usefulness of the contents in the learning process and how this subject allows the development of skills. According to gender, it was observed differences in their beliefs about mathematics, however related to problem solving, no statistically significant differences were found. It was concluded that the learning experiences about mathematics and problem solving from rural sector students are related to the development of adverse attitudes towards the subject, since they do not find use and purpose of its applications in their everyday life.


INtrODUctION
In 2015 in Chile, the new curricular bases were implemented for seventh grade. Which seek students to develop the skills of: arguing and communicating, modeling, representing and problem solving. To achieve these proposed objectives, it is fundamental the learning activities designed by the teacher, since they will allow students understand the importance of mathematics in life.
According to Thompson (1984), during the mathematic teaching process, it is possible to identify three types of teachers: a) one that focuses on routine procedures, b) those that give relevance to the learning of concepts and logic of mathematical processes and c) the one that cares about addressing problem solving. According to Sepúlveda, Oyarzún, Díaz and Opazo (2017) these differences can help or hinder the learning process.
This new curriculum is a challenge for teachers when designing activities since according to Aravena (2001), the teaching of this subject in Chile, focuses on repetition of algorithms. Consequently, Polya (1945) in Echeñique (2006) notes that the learning activities during this process, are fundamental, as these can awake interest in mathematics.
Therefore, Callejo and Vila (2003) considered important to study the development of beliefs in the area of mathematics, allowing to understand what students understand by learning and applying mathematics.
Based on these arguments it is important to analyze, what are the beliefs that second year, high school students have on the subject of mathematics and problem solving. Question that will provide us with information about students' learning experiences, usefulness and relevance that they give to their own learning process.

thEOrEtIcAl rEfErENcE beliefs in the subject of mathematics and problem solving
The Ministry of Education of Chile (MINEDUC) implemented in 2015 the new curricular bases for seventh grade which included new learning objectives. For teachers, this new proposal is a challenge, since they must formalize new and complex learning objectives and work together with students' beliefs regarding this subject. Aravena and Caamaño (2007), suggest that the development of these beliefs is due to the teaching model, which is generally of a traditionalist nature, phenomenon that according to these authors occurs more frequently in municipal establishments, where low learning results are accompanied by a low willingness to work in this subject. In this same direction Barrantes (2008) conducts a study to determine the beliefs that Costa Rican middle school students possess.
To compile the information Barrantes (2008) applied a questionnaire to eighth and tenth year high school Costa Rican students and their respective teachers of mathematics.
Finally, Barrantes (2008) concludes that, according to the educational practices of the mathematics teacher, students develop beliefs, regarding what it means to know mathematics, what mathematical problem is and what its characteristics are.
Regarding with the above, Lampert (1992) cited by Schoenfeld (1992), states that these constructions on the meaning of mathematics are consolidated during the teaching and learning process, in which learning mathematics only means following the rules that the teacher indicates for solving an exercise or simply memorize knowledge.
According to Callejo and Vila (2003), the beliefs about the subject of mathematics that predominates in students is that of a rigid, boring and mechanical science, since they do not find meaning in its learning. Therefore, Pérez and Beltrán (2011) consider relevant to study the beliefs that students have about mathematics, because allow us to understand the performance and effort that a student will make during the teaching and learning process.
That is why the research of Barrantes (2008), together with the new curricular bases proposed by the MINEDUC (2015), which seek that students understand the usefulness of mathematics in life, it is essential to analyze the beliefs that second year of high school students have at the Domingo Ortiz de Rozas High School in Coelemu Region of Ñuble and understand how during the teaching and learning process received, they have whether or not applied their knowledge in urban and rural contexts.

MEthODOlOGy rESErch APrOAch AND DESIGN
The research is assigned to a quantitative approach, which according to Monje (2011) seeks to quantify and objectify the results from a sample. The type of study corresponds to a descriptive quasi-experimental design of the survey type, for Hernández, Fernández, and Batista (2010), this type of study seeks to measure or collect information.

SAMPlE StUDy AND SElEctION crItErIA
To select the study sample, the fundamental criterion was that the teaching and learning process would have to be under the guidelines of the new curricular bases for mathematics. In this area, the second year, high school students corresponding to 2018 meet this requirement, since they have worked for four years on this new work plan.
The sample (N) consisted of 156 students from the second year from Liceo Domingo Ortiz de Rozas, located in Coelemu, Ñuble Region. Their age range from14 (14.10%), 15 (69.87%) to 16 years (16.03%). This high school receives teenagers from the city of Coelemu and its rural areas, which are characterized by agricultural, winegrowing and forestry activities.
The characteristics seen in the sample are that 109 students belong to the town of Coelemu (69.87%) and 47 to the rural areas (30.13%). According to gender, 76 students are male (48.72%) and 80 female students (51.28%). Finally, in relation to academic performance in the subject of mathematics, the averages fluctuate between grades 2.1 to 7.0, the average of the sample 5.4

INfOrMAtION cOllEctION INStrUMENt
The instrument in the first instance collects demographic information: class, age, gender, origin and academic performance in the subject of mathematics.
The second part, allows to obtain information referring to the beliefs dimension in mathematics, which is composed of three sub dimensions and the beliefs dimension about problem solving, which is composed of four sub dimensions. Mathematics in the class Relation of the contents presented in classes with the context of the student.

Beliefs on problem solving
What is a mathematical problem? Concept that predominates around what is a mathematical problem.
What characteristics do mathematical problems possess?
Characteristics that mathematical problems have.

Usefulness of a mathematical problem
Usefulness of mathematical problems.
Mathematical problems in the teaching of mathematics Understand the assessment and meaning given to mathematical problems (2008) to Costa Rican students.

INStrUMENt VAlIDAtION
The closed belief survey of Barrantes (2008) was used as the basis for the construction of the instrument and to gather the information.
To validate the new questionnaire, it was submitted using the proposal by Lawshe (1975) modified by Tristan (2008), which states that if an instrument is validated by 14 experts, it must obtain an CVI (Content Validity Index) of 0.51. This new instrument, obtained a CVI of 0.67.
To determine the reliability, it was analyzed by means of Cronbach's alpha, obtaining a coefficient of 0.89. Which according to the proposal by Hernández, Fernández, and Batista (2010) reflects a high degree of reliability in their results.

cOllEctION tEchNIQUES AND INfOrMAtION ANAlySIS
Considering ethical aspects of research, the consent of the parents was requested to apply the questionnaire.
Subsequently the instrument was applied in the math class schedule, it was indicated that the answers were anonymous and for research purposes only.
Students were informed that they had 60 minutes to answer the questionnaire and circle their degree of agreement on each of the statements presented on a Likert scale, which contained values from 5 (Strongly agree) to 1 (Strongly disagree).
To analyze the answers, the information was entered in the SPPS 23.0 statistical software. Seven tables were constructed according to the sub-dimensions of the study.
In them were calculated descriptive statistics of central tendency (mean), dispersion (standard deviation), calculation of frequencies and percentages, in order to determine the beliefs that predominated in the students.
Finally, and with the objective of establishing statistically significant differences, between gender, origin and academic performance, a t-test was used for independent samples, with an α = 0.05

rESUltS
The results are presented in tables with the sub-dimensions corresponding to the beliefs about mathematics and problem solving. Subsequently, a comparison analysis of means according to gender, origin, academic performance and results by dimension is performed.

SUb-DIMENSIONS ON bElIEfS IN thE SUbJEct Of MAthEMAtIcS AND PrOblEM SOlVING
The mean values and the standard deviation of the reagents corresponding to the beliefs in mathematics are presented in Tables 2, 3 and 4 and the beliefs about problem solving in Tables 5, 6, 7 and 8 for the total sample (N = 156).

Source: Prepared by the authors based on the obtained results. What is to know mathematics?
In the sub-dimension, what is to know mathematics? The belief of higher value was that mathematics allow to solve any problem related to the subject of study (M = 4.13; SD = 0.914), where 40.4% say they strongly agree. In the case of beliefs that indicate that knowing mathematics corresponds to knowing the fundamental theorems (M = 3.56; SD = 0.991), deciding the importance of a mathematical concept (M = 3.68; SD = 0.990) and applying creative processes (M = 3.73; SD = 1.012), it is observed in each of the three statements that 38.5% of the students express their agreement. With a slightly lower percentage of 35.9% and 36.5%, the students agreed to determine that knowing mathematics is knowing procedures by heart (M = 3.88; SD = 1.018) and solving problems related to the topic of study without difficulty (M = 3.52; SD = 1.032). Finally, in the statement referring to knowing too many definitions (M = 3.38; SD = 1.018), a positively low belief is observed, since 42.3% express neither agree nor Disagree. In the sub-dimension purpose of teaching and learning of mathematics in secondary education, there is a high appreciation, in the beliefs that states that the purpose of this subject is to provide knowledge (M = 4.14; SD = 0.933) and enhance skills (M = 4.09; SD = 0.904) to face everyday problems.
In the statements referring to the development of skills to determine the solution to contextualized problems (M = 4.01; SD = 0.984) or routine ((M = 3.87; SD = 0.930), in both statements 37.8%, of the students agree.

Source: Prepared by the authors based on the obtained results. Mathematics in the class
In the sub-dimension mathematics in the class, a smaller percentage (10.2%), considers that mathematics has nothing to do with reality, while over 50% of students (63.4%) agree with this statement (M = 3.79; SD = 1.064).

Source: Prepared by the authors based on the obtained results. What is a mathematical problem?
According to the sub-dimension, what is a mathematical problem? There is a high acceptance regarding that a mathematical problem can solve a real situation (M = 4.03; SD = 1.031), that encourages the development of new skills (M = 4.15; SD = 0.958), that motivates the learning of new procedures (M = 3.99; SD = 0.984) It can also be observed positive perceptions in beliefs that express that a mathematical problem motivates learning new concepts (M = 3.84; SD = 1.038) and that a mathematics exercise allows the application of a formula (M = 3.87; SD = 1.014).
A lower percentage of 32.1% and 37.2%, agree with that a mathematical problem is a possibility to discuss (M = 3.60; SD = 1.082) and to discover new knowledge (M = 3.77; SD = 1,002) In the sub-dimension characteristics of the mathematical problems, it is observed that 62.2% and 59.6% of the students claim that a mathematical problem has only one correct answer (M = 3.76; SD = 1.287), which the teacher should always know (M = 3.66; SD = 1.346).

Source: Prepared by the authors based on the obtained results. Usefulness of a mathematical problem
In the sub-dimension usefulness of a mathematical problem, a positively low value is found (M = 3.25; SD = 1.173), considering a problem as a technique to motivate a class, however in beliefs about the usefulness of a problem to teach mathematics (M = 4.33; SD = 0.886), to develop skills (M = 4.18; SD = 0.823) and to learn mathematics (M = 4.39; SD = 0.847), were positively valued.
Finally, 39.1% and 34% express to agree with the usefulness of a mathematical problem allows theory to be put into practice (M = 3.91; SD = 0.890) and to make new discoveries (M = 3.77; SD = 0.983). In the sub-dimension mathematical problems in mathematics teaching, it is observed that the reagents to consolidate mathematical knowledge (M = 4.19; SD = 0.820), encourage logical thinking (M = 4.00; DT = 0.909), develop skills to apply concepts (M = 3.97; SD = 0.894) and manage to execute the theorems presented in classes (M = 3.90; SD = 0.844), obtained a high rating.
For the statements that express that mathematical problems allow to develop critical reasoning (M = 3.70; SD = 0.933), creative thinking (M = 3.70; SD = 1,138) and prepare people for life (M = 3.87; SD = 1.003) were positively valued.

StAtIStIcAlly SIGNIfIcANt DIffErENcES IN thE bElIEfS Of MAthEMAtIcS AccOrDING tO GENDEr.
From the information collected, a t-student test was applied for equality of means in the dimensions: gender, origin and academic performance with the purpose of determining statistically significant differences, which are presented in the following tables. Table 9 shows statistically significant differences with respect to their beliefs in mathematics, since female students make a positive assessment of the reagents referred to the purpose of teaching and learning mathematics allows to develop skills (M = 4.26; SD = 0.759) and enhance skills to face real-life problems (M = 4.38; SD = 0.786).
On the other hand, male students value these reagents positively, but with a lower score (M = 3.75; SD = 1.111) and (M = 3.79; SD = 0.928).

Group statistics T-test for equality of means
To develop mathematical skills to creatively face the solution of contextualized problems.

StAtIStIcAlly SIGNIfIcANt DIffErENcES IN thE bElIEfS AbOUt MAthEMAtIcS AccOrDING tO thE OrIGIN.
In Table 10 it is possible to determine statistically significant differences, since the students that come from the town of Coelemu highly value the reagents referred to know mathematics is to know procedures by heart (M = 4.01; SD = 0.938) and solve any problem related to the subject of study (M = 4.24; SD = 0.870).
On the contrary, the students from the rural sector for the previous reagents make a lower assessment (M = 3.57; SD = 1.137) and (M = 3.87; SD = 0.969).
The reagent that expresses that to understand mathematics, corresponds to know the fundamental theorems, although both groups obtain low scores, a better evaluation is observed by the students from the Town of Coelemu (M = 3.67; SD= 0.953), than those of the rural places (M = 3.32; SD = 1.045).

StAtIStIcAlly SIGNIfIcANt DIffErENcES IN thE bElIEfS ON PrOblEM SOlVING AccOrDING tO thE PrOcEDUrE.
In Table 11, are observed statistically significant differences, since the students from the Town of Coelemu positively value the reagent referring to the fact that the usefulness of a mathematical problem is only a mean to apply theory to practice (M = 4.01; SD = 0.897), on the other hand, students in the rural area show a more downward valuation compared to this statement (M = 3.68; SD = 0.837).

StAtIStIcAlly SIGNIfIcANt DIffErENcES IN bElIEfS AbOUt
MAthEMAtIcS AccOrDING tO AcADEMIc PErfOrMANcE.
In Table 12, it is possible to determine statistically significant differences, since students who have an academic performance above the average of the sample make a positively high assessment in the reagents referred to that knowing mathematics is knowing by heart many procedures that serve to solve exercises (M = 4.14; SD = 0.850).
On the contrary, those students who are below the average of the sample have a lower assessment with respect to the same statement (M = 3.66; SD = 1.097).
For the sub-dimension purpose of the teaching and learning of mathematics, statistically significant differences are also observed, since students with an academic performance above the average of the sample value the statements that express the purpose of the subject positively, the statement that the purpose of the subject is to develop skills (M = 4.24; SD = 0.765) and enhance skills to face real-life problems (M = 4.38; SD = 0.724), however, despite statistically significant differences, students with academic performance below the average of the sample positively value the previous reagents (M = 3.82; SD = 1.104) and (M = 3.85; SD = 0.970).

StAtIStIcAlly SIGNIfIcANt DIffErENcES IN thE bElIEfS AbOUt PrOblEM SOlVING AccOrDING tO AcADEMIc
PErfOrMANcE. Table 13 shows statistically significant differences, since students with an academic performance above the average of the sample value highly reagents referred to that the usefulness of a mathematical problem is a tool to teach (M = 4.49; SD = 0.715) and a method to learn mathematics (M = 4.61; SD = 0.597).
However, in spite of the existence of statistically significant differences, students with an academic performance below the average of the sample, also value highly the previous statements (M = 4.19; SD = 0.957) and (M = 4,21; DT = 0.977).
In the case of the sub-dimension mathematical problems in the teaching of mathematics, statistically significant differences are also observed, because students whose academic performance is above the average of the sample, make a positively high assessment considering that these activities allow the development of logical thinking (M = 4.18; SD = 0.762) and prepare people for life (M = 4.06; SD = 0.876). However, despite observing statistically significant differences, students with an academic performance below the average of the sample, also make a positive assessment of these statements (M = 3.85; SD = 0.994) and (M = 3.72; SD = 1,130), but with a lower score.

INtErPrEtAtION Of rESUltS frOM bElIEfS IN MAthEMAtIcS
From the data obtained, it is possible to determine that students conceive this subject as a useful tool to solve everyday problems, an objective that has been exposed by the Ministry of Education MINEDUC (2015) in the new curricular bases for secondary education, this due to students point out that mathematics is related to everyday topics, that is, students can identify relationships, properties, data and unknowns that are linked to their life (Salinas and Sgreccia, 2016).
The results in the t-tests indicate statistically significant differences, regarding the gender of the students, about their beliefs in mathematics, since female students express that this subject allows to develop skills and enhance skills to face problems. Therefore, it is possible to complement the conclusions of Radovic (2018) who, in his research carried out in Santiago-Chile, establishes that women with a low socioeconomic level, such as the sample of this research, exhibit unfavorable attitudes compared to men in the subject of mathematics, which is a relevant antecedent to determine how the context influences the development of beliefs.
In respect of the origin, it is possible to establish that the students from the town of Coelemu, express beliefs in the subject of mathematics associated with the ones indicated by Erazo and Aldana (2015, p.163), that is to say "perceive mathematics as a subject useful given its importance, but mechanical, which is learned through repetition".
In the group of rural origin, it is possible to verify what was pointed out by Pérez and Beltrán (2011, p.75), who establish that "mathematical beliefs allow us to understand the performance and effort that an individual will make", which is evidenced in the low scores and assessments provided by students in the rural area, together with the low academic performance presented by students in this group, which as a whole is below the sample average ((x) ̅ = 5.4).

INtErPrEtAtION Of rESUltS frOM thE bElIEfS ASSOcIAtED WIth SOlVING MAthEMAtIcAl PrOblEMS
Regarding the beliefs that second-year students have about solving mathematical problems, it is possible to determine that a belief predominates, in which this type of activity is pointed out as a method to learn mathematics that allows new skills to be developed.
The previous results express the purpose and usefulness that students give to solve mathematical problems during their learning, therefore, it is possible to verify that beliefs about mathematics affect the way students understand problem solving. Barrantes (2008).
The information analyzed in the t tests indicates that there are no statistically significant differences in beliefs about the resolution of mathematical problems according to gender, however, in the origin and academic performance of the sample, statistically significant differences are observed, regarding their beliefs in problem solving.
Based on the answers provided, it is observed that the students of the town of Coelemu and those who find themselves with an academic performance above the average of the sample, express beliefs associated with the usefulness and application of the resolution of mathematical problems to solve everyday situations, which, according to Barrantes (2006, p.4) as well as Sepulveda, Medina and Jauregui (2009), these learning experiences "affect the way a student behaves when facing a mathematical problem", that is, this group of students presents a better willingness to work on problem solving.
In the same way it is observed that the students of the rural area and that are below the average of the sample obtained a lower score in the beliefs about problem solving, which for Martinez (2013) is due to the unfavorable attitudes developed towards the learning of this subject, since they do not see the usefulness of applying the contents that are taught, which in the course of time according to Hidalgo, Maroto and Palacios (2004) generates a "downward trend in the liking and interest in mathematics as they advance in educational levels." cONclUSIONS From the objectives proposed at the beginning of this investigation, it is possible to conclude that: 1. With respect to the objective that determined the beliefs about mathematics and problem solving that secondary students of the Domingo Ortiz de Rozas High School had, it is possible to determine that there are beliefs in accordance with those proposed by the Ministry of Education (MINEDUC) in the new curricular bases, which are developed due to the experiences that foster the usefulness of mathematics to acquire new knowledge and thus favor problem solving.
2. Regarding the objective that categorized the beliefs in mathematics and problem solving in the teaching and learning process, it is possible to recognize that the second year of secondary students in the sub-dimension referred to What is knowing math? They establish that such belief corresponds to solving any problem related to the topic being studied. With respect to the sub-dimension associated with the purpose of the teaching and learning of mathematics, the belief that mathematics provides the knowledge to face intelligently practical real-life problems predominates. In the case of the sub-dimensions on problem solving, specifically in the one referred to What is a mathematical problem? here the belief that allows to develop new skills and a method to learn mathematics predominates. For the characteristics of a mathematical problem there is a belief, in which this type of activity has only one correct answer, which the teacher must always know. Finally, in the last sub-dimension it is possible to determine that students point out that the role of a mathematical problem in the teaching process is to develop the theorems presented in class.
3. In relation to the objective that corresponds to establish statistically significant differences, according to the gender of the students, it is possible to determine that there are statistically significant differences, regarding their beliefs in the subject of mathematics, when considering its purpose in the process of teaching and learning in secondary education and in the case of beliefs about problem solving, no statistically significant differences were observed. 4. Regarding the objective referred to establish statistically significant differences, according to the origin of the second year students, it is possible to determine that there are statistically significant differences, with respect to their beliefs in the subject of mathematics and problem solving , since, according to the information collected in this research, students belonging to the town of Coelemu manage to see the usefulness of the contents taught in classes, a situation that does not occur with students in the rural area, which explains the lack of disposition of this group to work in activities of this subject, as well as the low learning results evidenced by the internal evaluations applied by the establishment to these students. 5. In relation to the objective associated with establishing statistically significant differences according to the academic performance of the sample, it is possible to determine statistically significant differences, regarding their beliefs in the subject of mathematics and problem solving, since those with an academic performance above the average of the sample, see the utility of the subject of mathematics and resolution of problem to develop skills necessary to face problematic situations, however those that are below the average of the sample do not show greater interest in this type of affirmations , such results are the reflection of the experiences of each of the individuals that conform the analyzed groups, since they do not make sense of the contents addressed in classes, which explains the learning experiences they have.

cONtrIbUtIONS Of thE AUthOrS
Each of the authors contributed to the tabulation, analysis, interpretation and conclusion of the data analyzed in each of the tables presented.