Anxiety to Mathematics and its Relationship with Mathematical Learning

Theoretical Framework

The researchers Ashcraft and Ridley (2005) is one of the most influential authors in this field, who explained how "the American culture abounds with attitudes that promote mathematical anxiety", the most outstanding American researchers on the subject of mathematical anxiety, emerged for the attempt to find solutions to this serious problem at the national and global level. Currently, there is an impulse at the global level to carry out scientific studies in this field of research in response to the growing problems of modern society in the interlearning of Mathematics (Ersozlu & Karakus, 2019).

One of the basic tasks in the inter-learning process of fractions is to compare two of them to decide which has a higher numerical value. Evidence indicates that students with a basic level of knowledge of fractions are predisposed to compare the components of separate fractions (numerator with numerator and denominator with denominator), instead of comparing the numerical values of both fractions; generating that the students could obtain correct answers when comparing 4/9 with 7/9 or 1/3 against 5/7, but they would develop incorrect answers when comparing 2/5 with 2/7 or 4/9 with 2/3, situation that occurs because they do not work with the fraction as a whole but only with its parts (Gómez et al., 2014; Gómez & Dartnell, 2015).

This mathematical notion in fractions is called congruence and the pairs of fractions such as those listed above are known as congruent and incongruent respectively, the relevance of this variable in the interlearning of fractions is supported by studies in adult, adolescent, and child populations, investigations where it is observed that that respondents answer to congruent questions more accurately and quickly than to questions that contain incongruent fractions (Branje et al., 2018). Research indicates that one of the main causes of the variation in the level of learning and academic performance in Mathematics or in other areas of knowledge, are caused by emotional states; affirmation that is supported by the findings that show that most of the learning problems that develop in basic education have an emotional origin and if it is possible to balance the intellectual quotient with the emotional quotient, better results could be achieved in academic performance (Fernández & Sánchez, 2013).

School anxiety is defined as a set of cognitive, psychophysiological and motor responses that an individual emits in situations evaluated as threatening, dangerous or ambiguous, although objectively they are not, the negative consequences in the short and long term when an educator presents anxiety is that it affects school performance, an affective block that could arise due to the lack of an adequate method of inter-learning in Mathematics or the absence of adequate representations to solve problems (Auzmendi, 1992).

That is, at high levels of anxiety, it could possibly lead to making mistakes, since memory processes intervene, making concentration difficult and in general alters the psychological functioning of the student, affecting the performance of any activity that requires attention and concentration (Contreras et al., 2005). Findings guide that mathematics anxiety is a negative affective reaction-action process that a student experiences when performing exercises with numbers, develops mathematical calculation problems, processes that develop sensations of tension that hinder the development in learning arithmetic and hence Mathematics (Eden et al., 2013; Richardson & Suinn, 1972). The students who present anxiety to Mathematics show lower levels of confidence in their mathematical abilities, both constructs are negatively correlated (Ashcraft & Ridley, 2005; Bursal & Paznokas, 2006).

Studies show that there are differences between levels of anxiety according to sex, thus they investigated the presence of anxiety to Mathematics in primary and secondary school students, finding that girls presented higher visible symptoms of nerves, tension and discomfort than boys (Wigfield & Meece, 1988). Other research conducted on high school students reported that women have higher levels of anxiety than men (Eden et al., 2013; Fennema & Sherman, 1976). There is evidence that in teachers, adults and adolescents that anxiety to mathematics affects other cognitive abilities in a negative way, in addition, anxiety to mathematics generated by learning fractions negatively influences reading speed and comprehension, and in some cases it affects in the IC performance of the WAIS test (Hopko et al., 1998; Hopko et al., 2005; Venkatesh & Karimi, 2010).

Mathematics anxiety affects emotions and consequently Mathematics learning (Palacios, 2016) a study carried out by PISA in 2012, indicates that the countries where students had higher levels of anxiety, also their performance academic was less good. The report also indicates that one in three students becomes nervous when they have to solve a mathematical problem and in almost all the participating countries, women presented higher levels of anxiety to mathematics than men (OECD, 2013).

Anxiety to Mathematics is directly related to the perceptions that the students have about their possibilities to solve exercises or mathematical problems and / or when taking an assessment, presenting anxiety reactions that develop low performance, despite the presence of anxiety decreases as feelings of self-efficacy, efficiency and effectiveness improve because it increases personal security, allowing the resolution of evaluations to present better scores improving their performance (Nortes & Nortes, 2014).

For Perina (2002) the claims made that the female sex presents higher levels of anxiety to mathematics than men, could be because women are more likely to feel anxiety and not because they are necessarily more anxious, the differences in levels of anxiety to Mathematics by sex were and are investigated trying to cover all educational levels (Ashcraft & Ridley, 2005). It should be considered that there are some findings in male and female secondary school students in which the works report differences in levels of anxiety to Mathematics according to sex, but in these findings, greater anxiety is characterized in men than in women (De la Torre et al., 2009; Eden et al., 2013; Valero, 1999).

For the researcher Maroto (2015) emotions, beliefs and attitudes are transcendental components to: a) Determine what could be the mathematical procedure that future teachers will use, b) Establish how these 3 components influence the Mathematical learning, c) Understand what is the maneuver of the 3 components when it works as a regulatory system, d) Be used as an indicator in the context of inter-learning, e) Influence as inertial forces or resistance to change and f) Function as a vehicle of knowledge. Research carried out confirms the importance of studying how affective control contributes to the interlearning of future teachers (Caballero, 2013; Gómez-Chacón, 2016; Marbán et al., 2016; Nortes & Martínez, 1996; Pérez-Tyteca, 2012; Pérez-Tyteca et al., 2013; Sánchez et al., 2011).

The researchers Palacios et al. (2013) consider that anxiety to mathematics is a feeling of tension and / or fear that leads to presenting behaviors that are not well analyzed and assertive when operating with numbers, evidence that is supported by the study from a sample of 1064 students with a mean age of 13 years and 8 months (53% were male and 47% were female), it was also concluded that the higher the level of anxiety, the lower the mathematical performance of the learners and that the presence of anxiety is one of the indicators for the presence of school rejection.

Another investigation considered evaluating the levels of anxiety depending on the degree to which the situation of the evaluation (written test) is perceived as threatening for the learners, the results oriented that men face the evaluation processes as a challenge, implying that they do not perceive whether or not they are unable to operate mathematically, on the other hand, women face the evaluation situation as threatening, demonstrating behavior influenced by the presence of anxiety (Rosario et al., 2008).

Research indicates that a large number of people have extremely negative attitudes to mathematics, which may possibly lead to severe anxiety (Ashcraft, 2002; Hembree, 1990; Maloney & Beilock, 2012). Another investigation stated that parents who present high levels of anxiety to Mathematics influence that their children can also develop it (Maloney et al., 2015). On the contrary, there are other investigations that show that, if parents constantly support developing high expectations in their children, they would reduce their anxiety levels, increasing their achievement in mathematics (Vukovic et al., 2013).

Findings guide that, if teachers and parents increase the levels of motivation to mathematics of students who present levels of anxiety, supporting them in the analysis processes to decode the keys to solving the math exercises or problems, they could overcome the causes that generated the negative and affective cognitive antecedents of anxiety, managing to increase their performance (Wang et al., 2015). European Economic Organization for Development (OECD), promote first world countries to develop neurocognitive research to benefit the teaching-learning process, supporting research centers such as the Mind, Brain and Education program and the Max Planck Institute, to investigate how traditional teaching methods influence the learning process in the classroom and present educational proposals, based on the findings of Cognitive Neuroscience to improve inter-learning processes (Istance, 2008).

In the learning of Mathematics, one of the most difficult contents to understand is fractions, it is necessary to know that learning rational numbers is the first approach that students have to understand fractions (Gómez et al., 2014), despite since several alternatives are used to teach fractions, there are students who fail to understand their meaning and usefulness, causing students considered averages to not obtain a conceptual understanding when operating them (Fazio & Siegler, 2011); this problem is a challenge to overcome in the basic education curriculum, basing that the learning of fractions is a necessary tool for the development of multiplicative, quantitative thinking of the student (Thompson & Saldanha, 2003).

For this reason, in the field of Mathematics, teachers must have the responsibility of developing cognitive skills in students so that they can compare, value and order fractions, allowing the student to have a clear appropriation of the sense of magnitude and / or exact quantity of a fraction (Capilla, 2016), it is necessary to understand that the interlearning of fractions involves a succession of properties that are different from that of natural numbers and that in order to understand them it is necessary to differentiate that fractions and the numerousness of a rational number manages to be represented through other rational numbers. Understanding this activity would induce students to rethink their conceptual understanding of these numbers (Stelzer et al., 2016).

Research highlights that a large number of students at all educational levels experience difficulty learning rational numbers and within these fractions (Vamvakossi & Vosniadou, 2010). The confusion between the concept of integer and rational number originates when talking about magnitudes, to avoid this difficulty in learning fractions, the student must understand that the integer has its own value, that is, five is worth five (5) and it is not difficult to say five (5) is greater than four (4) and less than six (6), on the other hand in a rational number when saying 7/3 complicates recognizing if it is greater than 9/5 or if is less than 6/2, this bias develops because students often confuse concepts and procedures for operating whole numbers with those used for fractions (Ni & Zhou, 2005).

Currently, Cognitive Neuroscience provides a sustainable basis for understanding how the brain works in teaching-learning processes, for which some sciences dedicated to education incorporated these findings into curriculum design, didactics, and evaluation of learning (Puebla & Talma, 2011). Research shows that most of the mistakes when learning fractions do not arise from the incompatibility between the concept of fraction and the cognitive architecture of the student, but is due to current methods of inter-learning because they lack efficiency and because they miss the neurocognitive and perceptive skills of the students (Lewis et al., 2015).

In Ecuador, the evaluations carried out by INEVAL from 2013 to 2018 to students in the 4th, 7th, 10th, 11th years of basic education and the third year of high school, in public educational establishments, fiscal (Catholic educational establishments with the financial support of the state) and private, it was determined that achievement levels should be divided into 4 categories (Insufficient = 400 to 699; Elementary = 700 to 799; Satisfactory = 800 to 949 and Excellent = 950 to 1000).

Although it is true that there is vast information on the relationship between anxiety to Mathematics and Mathematical learning, it is still an unresolved field in this line of research, therefore this research contributes from the point of view that was evaluated a Mathematics content that was the knowledge of fractions through a standardized test (Ministry of Education) and a computerized questionnaire for comparing fractions, for which two different evaluation systems were applied (one written and the other computerized), correlating them with the Mathematics anxiety questionnaire.

In table 1 it can be seen that in 5 years of evaluation the results of the performance of the students in the area of Mathematics are worrisome since when performing the arithmetic mean of the first two categories (Insufficient and elementary) they exceed 70% (Ineval, 2016; Ineval, 2017a; Ineval, 2017b; Ineval, 2018; Ineval, 2019; Ministerio de Educación del Ecuador, 2014). According to the results obtained in the official measurements, some elementary assumptions of inter-learning that guide the process of curricular design and strategies in the teaching of Mathematics should be reviewed and within this, that of the fractions since this theme is which reports the lowest level of performance, which is used by the teacher in Ecuador; considering that the learning of fractions is a basic tool for the development of multiplicative, quantitative and mathematical thinking of the student (Lamón, 2007; Thompson et al., 1992). For this reason, it is necessary for teachers to develop cognitive skills in the students so that they can compare, value and order fractions, since these skills allow the student a clear appropriation of the sense of magnitude and / or exact quantity that a fraction represents, knowledge that develops an understanding of the correct meaning that mathematics assigns to fraction (Capilla, 2016).

Homework

Anxiety to Mathematics MASC

The MASC test developed by Chiu & Henry (1990) stated that the students had to answer the questionnaire in writing, which had 22 questions, evaluating each according to a 4-point scale in terms of how much anxiety they feel: 4 represents very very nervous, 3 represents very nervous, 2 represents a little nervous, and 1 represents not nervous at all; this test is applicable from the fourth grade of primary (Fifth year of basic education in Ecuador), the students had a time of 30 minutes to answer it, after the delivery they were coded to save the identity of the evaluated student, Cronbach's Alpha was calculated to determine its reliability, which was 0.93.

Proof of knowledge of fractions

The students were evaluated by applying a test containing 10 questions which had 71 sub-items; for the preparation of the test, the researchers collected the questions from the teacher's book delivered by the Ecuadorian Ministry of Education, which at the end of each unit has evaluation questions, the test was carried out in writing, for which the students had 80 minutes to solve it, before starting the evaluation it was explained to them that it should be done individually and that if they had any doubts, they should raise their hand and an evaluator would come to solve the concern, after delivery they were coded to save the identity of the evaluated student.

Computerized fraction comparison questionnaire

The computerized fraction comparison questionnaire was applied, which consisted of 48 exercises, classified with common components (Congruent and Incongruent) and without common components (Congruent, Incongruent and Neutral), the program used to evaluate was Python and PyGame. Each student was asked to verify that their computer was working properly and the students were explained what the evaluation consists of, after the general indications they were asked to write the data requested by the computer program, indicating them to read carefully what came out on the screen.

It was highlighted that to choose the answer that they considered correct, they should press the Q or P key to select the largest fraction, if they considered that the fraction on the left was greater, press the Q key and if they considered that the largest fraction was that of the Right press the P key, they were also told that they had seconds to give an answer and that after the answer was issued, they could not rectify, the data was stored in a pen drive, and then they coded it with the aim of saving the identity of the evaluated student.

Descriptive Analysis

The descriptive data of the sample in relation to the year of basic education and the sex of the participants is presented below.

In the table 2 , it was observed that, of the 330 participants, 178 belonged to the sixth and 152 to the seventh year; 172 were male, of which 93 belonged to the sixth year and 79 to the seventh year and the remaining 158 were female, classified in 85 belonging to sixth and 73 of seventh.

Fraction Knowledge Test

Next, the descriptive analysis of the Mathematics anxiety questionnaire and the Knowledge test of fractions were carried out and the level of hits that they had in each of them related to the evaluative scale of the learning results used in Ecuador

Table 3 shows that of the 330 students, 207 presented anxiety to mathematics and 123 did not, of the 207 students who presented anxiety, 121 belonged to the sixth and 86 to the seventh year, and of the 123 students who did not present anxiety, 57 belonged to sixth and 66 to seventh year of basic education, in relation to performance and anxiety there were 106 of the 207 students who were on the scales (PAAR and AAR), as for the 123 students who did not present anxiety to Mathematics, 62 were on the scales (PAAR and AAR). In addition, it was evident that 205 students did not achieve the required learning (NAAR and PAAR) and only 2 achieved the required learning.

When calculating the value of chi2 we had the value of p = .036 but two cells had a value less than 5, so we proceeded to calculate the value of the contingency coefficient that had a value of p = .982, that was, the value of the correlation is 98.2%.

Computerized fraction comparison questionnaire

The following table performs the descriptive analysis of the computerized fraction comparison questionnaire, which characterizes the percentage of the hit rate according to the type of fraction (Congruent with common components and Incongruent with common components) and the level of hits that they had in each of them related to the evaluative scale of the learning results that was used in Ecuador.

Table 4 shows that of the 330 students who compared fractions with congruent common components, 247 were on the scales (NAAR and PAAR) and the remaining 83 were on the scales (AAR and DAR) and that when they compared fractions with incongruent common components 284 students were on the scales (NAAR and PAAR) and the remaining 46 were on the scales (AAR and DAR).

Below, the descriptive analysis of the computerized questionnaire for comparing fractions in reference to fractions with no common components (Congruent, Incongruent and Neutral) can be found and the level of hits they had in each of them related to the evaluative scale of the results of learning that was used in Ecuador.

Table 5 shows that of 330 students who compared fractions, 247 who compared Congruent Common Non-Components fractions were found on the scales (PAAR and NAAR) and 83 were on the scales (DAR and AAR) but when the same number of students investigated compared Incongruent Non- Common Components fractions, 251 were on the scales (PAAR and NAAR) and 79 on the scales (DAR and AAR); It was also evident that when those 330 students compared Neutral Non-Common Components fractions, 258 were found on the scales (PAAR and NAAR) and 72 learners on the scales (DAR and AAR).

The following table presents the descriptive analysis of the computerized questionnaire for comparing fractions, the level of success they had when comparing fractions with common components (Congruent and Incongruent) related to the evaluative scale of learning results used in Ecuador and according to sex and the presence or not of anxiety to Mathematics.

Table 6 shows that the male sex showed 106 students who presented anxiety to Mathematics, of which 29 students were on the learning scales (AAR and DAR) and 77 were on the scales (NAAR and PAAR), on the other hand, there were 66 students who did not present anxiety, of which 20 were on the scales (AAR and DAR) and 46 were on the scales (NAAR and PAAR); Instead, when they compared fractions with Incongruent Common Components, 13 children presented anxiety to Mathematics and were found on the scales of (AAR and DAR) and 93 were found on the scales (NAAR and PAAR), in addition it was observed that 15 children who did not present anxiety were found on the scales (AAR and DAR) and 51 were found on the scales (NAAR and PAAR).

In contrast, in the female sex, 21 girls presented anxiety to Mathematics and were on the learning scales (AAR and DAR) and 13 who did not present anxiety were on the same scale when comparing fractions with Congruent Common Components; in contrast, the same sex when comparing fractions with incongruent Common Components, 14 girls presented anxiety and were on the scales of (AAR and DAR) and there were 4 girls who did not present anxiety and were on the same learning scales.

The following table presents the descriptive analysis of the computerized questionnaire for comparing fractions, the level of hits they had when comparing non-common components fractions (Congruent, Incongruent and Neutral), related to the assessment scale of the learning results used in the Ecuador and according to sex and the presence or not of anxiety to Mathematics.

Table 7 shows that of the 330 students who compared Non- Common Component fractions (Congruent, Incongruent and Neutral), 172 belonged to the male sex and 158 to the female; when children compared Congruent Non-Common Components fractions 30 were found on the scales (AAR and DAR) and 76 on the scales (PAAR and NAAR) instead of the 66 children who did not present anxiety 18 were found on the scales (AAR and DAR ) and the remaining 48 were on the scales (PAAR and NAAR); on the other hand, when the same group of children (172) compared Incongruent Non-Common Components fractions, 23 presented anxiety to Mathematics and were found on the scales (AAR and DAR) and 83 were on the scales (PAAR and NAAR), instead the 66 who did not present anxiety 13 were on the scales (AAR and DAR) and the remaining 53 were on the scales (PAAR and NAAR) and when this group of children compared Neutral Non- Common Components fractions, 28 children who presented anxiety were on the scales ( AAR and DAR) and the 78 on the scales (PAAR and NAAR) instead of the same group that did not present anxiety to Mathematics 17 were on the scales (AAR and DAR) and the remaining 49 were on the scales (PAAR and NAAR ). In relation to the same group, when they compared Neutral Non-Common Components fractions, it was found that 28 children were on the AAR and DAR scales and 78 were located on the scales (PAAR and NAAR), instead of the 66 children who did not present anxiety to Mathematics, 17 were located on the scales (AAR and DAR) and 49 on the scales (PAAR and NAAR).

Instead of the 158 girls, 101 presented anxiety to Mathematics and 57 did not, and when they compared Congruent Non- Common Components fractions, 25 were found on the scales (AAR and DAR) and 76 on the scales (PAAR and NAAR) instead of the 57 girls who did not present anxiety 10 were placed on the scales (AAR and DAR) and the remaining 47 were on the scales (PAAR and NAAR); on the other hand, when the same group of girls (158) compared Incongruent Non- Common Components fractions, 29 presented anxiety to Mathematics and were located on the scales (AAR and DAR) and 72 were on the scales (PAAR and NAAR), whereas the 57 who did not present anxiety 14 corresponded to the scales (AAR and DAR) and the remaining 43 were on the scales (PAAR and NAAR) and when this group of girls compared Neutral Non- Common Components fractions, 15 girls who presented anxiety were found on the scales (AAR and DAR) and the 86 on the scales (PAAR and NAAR,) instead the same group that does not present anxiety to Mathematics 12 were on the scale (AAR) and the remaining 45 were located on the scales (PAAR and NAAR).

Limitations and advantages of the study

In the present investigation, it was not considered to control other variables that could possibly influence the results, such as academic performance in other subjects, reports of the presence of specific learning disorders that the results obtained respond to a single educational institution.

Research contribution

The contribution of this research is the application of two different types of evaluations on the content of fractions, the first evaluation shows the level of general knowledge of fractions that a student has, in which the student can perform calculations in a paper and repeat mechanical and memory processes to give an answer to a question posed, compared to the second evaluation that consisted of comparing the value of a fraction in relation to another fraction in a computerized program, an activity that makes it possible to use basic intuitive processes of the student to indicate the answer that he considers correct. These two forms of evaluation could guide teachers to understand which are the biggest problems that students present when comparing fractions.

Declaration of conflict of interest

The researchers who carried out this study declare that they have no conflict of interest or with public and / or private institutions, as well as with natural and / or legal persons.