IS MATHEMATICAL BACKGROUND CRUCIAL TO

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1 IS MATHEMATICAL BACKGROUND CRUCIAL TO FRESHMEN ENGINEERING STUDENTS? Edinéia Zarpelon Luis Mauricio Resende Ednei Felix Reis Federal University of Technology Paraná (UTFPR) Brazil Abstract: The objective of this paper is to study the influence of the background in Mathematics on the failure rates in Calculus. Some papers in the literature deal with problems related to teachinglearning in Calculus under different points of view. Among those problems, one that is often cited is the lack of mathematical background presented by freshman-year Engineering students. In order to analyze the impact of this lack of background, a study with 419 first-year Calculus students on two campuses of the Federal University of Technology Paraná (UTFPR) was conducted. The students took a diagnostic test with 15 questions that measured their knowledge on topics that are considered as prerequisite for Calculus. The results of this study showed that there is a moderate correlation and dependency between the background in Mathematics and the final grade in Calculus. Evidently, if the student has a solid mathematical background, there is a higher probability of success in Calculus. However, one can notice that even with lower scores on the diagnostic test, some students managed to obtain passing grades in Calculus. This fact indicates that the background in Mathematics might not be a decisive factor on the performance in Calculus. Keywords: Calculus, mathematical background, failure rates. I. INTRODUCTION The admission at the university marks the beginning of a new cycle in the life of every student. At this time, most of them have to deal with new variables that require fast adaptation and emotional balance: to live in a new place (most times away from family); to develop autonomy to manage their lives; to adapt to different methodologies used by the professors; to organize study hours, are examples of obstacles that the students must face on their academic lives. Moreover, the freshman-year students in Engineering must deal with another variable that may cause periods of intense stress, namely, the heavy load of mathematical contents required to their academic formation. In particular, the freshman-year students must take classes like Calculus, Physics, Analytical Geometry and Linear Algebra, which require mathematical language interpretation and comprehension that are different from those in which they were used to from High School. Therefore, to succeed in those courses constitutes one of the main challenges faced by the students. At the Engineering schools in Brazil, it is not uncommon to find failure rates higher than 50% at those classes. One may add that these statistics are not a particular problem associated to a specific region or institution. Those failure rates are a symptomatic problem faced by several institutions. Moreover those rates generate the retention of students at the first semesters of the course, the evasion from the course, and in worse cases, the evasion from the university altogether. Engineering courses, toghether with Mathematics and Physics courses are among the courses with higher evasion rates. According to [1] this fact occurs mostly because the students cannot follow some of the initial classes of the course, such as Calculus. Consequently, many students fail those classes, leading them to evade the course. Notice that the topics of classes cited above supply the necessary background to develop essential concepts in Engineering. Reference [2] states that Calculus is one of the most relevant classes taught at Engineering schools, since it allows the study and modeling of problems related to several areas in Engineering. However, the author cited above reinforce that once calculus is offered just in the first year, it competes with others problems students have to deal when they are at the university as adaptation, immaturity and some lacks is their formation. Therefore isn t any new the high level of failure or evasion of students from the first year. The understating of the variables that interfere in this context may produce measures that can minimize the failure rates and help the first-year students to overcome some of the difficulties presented to them at the initial period of their academic formation. Therefore, assuming that the lack of mathematical background might be a variable that contributes to higher failure rates in first-year Engineering courses, the objective of this work is to verify whether the lack of mathematical background is decisive to the success of the student in Calculus. To achieve this goal, data from 419 freshmen-year Engineering students from two different campuses of UTFPR were collected and analyzed. Those students were admitted at the university during the second semester of 2013 and the two semesters of The Federal University of Technology Paraná is one of the largest Engineering schools in Brazil, offering 47 Engineering courses in 13 campuses. It is expected that 4,136 new students will be admitted to Engineering programs at the University in 2016.

2 II. MATHEMATICAL BACKGROUND: A PREREQUISITE? Some studies were conducted in order to identify the factors that contribute to higher failure rates in Calculus [3, 4, 5, 6] and the most cited cause was the lack of mathematical background presented by the first-year students [7,8, 9, 10, 11, 12, 13, 14] According to [15] not only the students lack the knowledge and the familiarity with the real number system and elementary functions, but also they do not have the desired cognitive structure in order to interpret the mathematical language and understand structural concepts. The author also states that these students reveal difficulties in reflection, research, exploitation and mainly deduction; memorize the technique, but do not incorporate the meaning of concepts, and therefore have great difficulty in adapting to university education system, resulting in large gaps in their performance, especially in performing tasks more open as may be requested by teachers, such as problem solving [15]. Reference [16] believe that the lack of success in Calculus is related to the gap between the topics studied in High School and in the university. The authors explain that until the decade of 1960, there were two options for High School in Brazil, namely, the classic (closely related to the Humaties) and the scientifc, where topics from Calculus such as limits and derivatives were approached. As the years passed, several reformulations happened in High School, driven in part by changes in the admission process of the universities. During these reformulations, several topics in Mathematics and Physics were excluded. However, the curriculum of Engineering courses stayed almost unaltered from a time where there was a well structured continuity between the Scientific High School and the university [16]. In two studies conducted in1999 and 2008, [13] formulated some hypotheses about the students who repeatedly failed in Calculus. One hypothesis was that most of those students presented gaps or misconceptions in their mathematical background, hampering their ability to absorb new conecpts. And what is worse is that according to the author, most students are unaware of those gaps. Reference [17] developed a study with Engineering students enrolled in Calculus. After several surveys were conducted, the authors concluded that: 1) Students consider as good or excellent their performance in mathematics in High School, but their performance in the exam of entrance it was just regular; 2) Students pointed as a factor that contributed the most to their weak performance in Calculus was huge lack in their formation in mathematics; 3) Teachers who participated in this research considered as a main factor to students low performance was a deficient formation in mathematics when their entrance at the university. The authors also observed that the students did not have the required mathematical background to understand the concepts related to Calculus, and this fact led to the evidence that there must be a large gap between the mathematical topics studied in High School and what is required from a Calculus class [17]. A study conducted by [18] also detected deficiencies in the mathematical backgrounds of the students. A diagnostic test applied by the authors showed that 75% of the students obtained grades inferior to 5 out of 10 points. The data collected from the test enabled the implementation of review sessions that occurred independently from the regular lectures. After a certain period of time, the contents of those sessions were incorporated to the main lectures, and the topics were discussed with the aid of handouts developed by the professors of the class. However, there are no published results about the efficiency of those measures. Even though there are several studies related to the lack of mathematical background, it is necessary to measure its impact in the performance in Calculus. It is believed that the lack of background in Mathematics interferes with the performance in Calculus. However, one can infer that those difficulties can be solved with academic commitment, not representing therefore a decisive variable with respect to the performance in Calculus. III. METHODOLOGY In the present study, the assumption to be analyzed is whether the lack of mathematical background is a decisive variable for failing in Calculus. In order to find evidence, which either confirms or refutes this assumption, the UTFPR was chosen to be the place where the study would take place, since the university has one of the largest number of admissions among the universities in Brazil The target audience was a sample of the University's first-year students, constituted by 419 students of two campuses of the institution, from six Engineering courses, all of them enrolled in Calculus. This sample was composed by 269 students from Civil Engineering, Computer Engineering, Electrical Engineering and Mechanical Engineering from campus Pato Branco, where 144 of the were admitted in the first semester of 2014 and 125 were admitted in the second semester of The other 150 students are from Electronic Engineering and Industrial Engineering from campus Ponta Grossa, 27 of them were admitted in the second semester of 2013, 61 in the first semester of 2014, and 62 in the second semester of A diagnostic test was used to gather the data. This test was designed by a Calculus teacher from the University, and it was composed by 15 questions which covered topics that are assumed to be prerequisites of Calculus, namely, operations with rational and real numbers, linear and quadratic equations, simplification of algebric expressions, exponential and logarithmic functions, and trigonometric functions. The students had taken the test at the beginning of their first

3 semester at the University. At the end of the semester, their final grades were analised in order to make a comparison between those grades and the one from the diagnostic test. IV. RESULTS In Table 01 students are separated in groups concerning the number of correct answers obtained by them in the Diagnostic Test (second column), as well it is listed the number of students successful (third column) and failed (fourth column) in Calculus in each group. correct answers, the majority (75%) is successful in the course. Analyzing Fig. 01 it is possible to notice that grades obtained in the Diagnostic Test have a bigger varation than the final grade in Calculus. When comparing these performance it is perceived great differences. For example, in the Diagnostic Test, 75% of the students obtained grades between 0 2,7, whereas in Calculus 75% of the students got from 0 to 6,8. Diagnostic Test Number of answers TABLE I - DATA FROM DIAGNOSTIC TEST Number of Students with correct answers Students successful in Calculus Students failed in Calculus Final Grade in Calculus FIGURE 01 - BOX-PLOT COMPARATIVE BETWEEN PERFORMANCES Frequency of grades obtained in the Diagnostic Tests as well as the Final grades in Calculus are shown in Fig. 02 and Fig Frequency Total Grades FIGURE 2 - HISTOGRAM OF FINAL GRADE IN THE DIAGNOSTIC TEST It is possible to notice that the group of students with ten or more correct answers in the diagnostic test has successful in Calculus. It is clear that this indicate that a solid formation in mathematics contributes to his success in the course. It is also possible to notice that, in the group of students with 3 to 9 correct answers, a major part of them is successful in Calculus. However in the group of students who didn t have any correct answer in the test, 65% of them failed. When the group of students with two correct answers is analyzed, it is possible to notice that 42% of them had success in the course. When it is analyzed students with 3 or more Frequency Grades FIGURE 3 - HISTOGRAM OF FINAL GRADE IN CALCULUS

4 From the 419 students analyzed, only 17 (4%) obtained a final grade superior to 6,0, but 204 of them (49%) had success. To explain this difference of performance, it can be considered that students, even with a weak background in mathematics, can solve their difficulties along the course, and be successful in it. In analyzing Fig. 2 and 3, it is possible to deduce that data don t have a normal distribution. This affirmative can be verified by a test of normality of Ryan-Joiner. So, to data with a non normal distribution, it is necessary to use nonparametric test in order to analyze or evaluate some hypothesis to the population, from the sample s data. It is relevant to observe that these tests neither require that the variable in question is a non numerical variable nor assumptions about the distribution of this variable [19]. r s So, initially Spearman s coefficient of correlation ( ) was used to analyze relation between two variables (final grade in Diagnostic test and final grade in Calculus). Values from r s range from -1 to 1, and the closer the coefficient is from -1 or 1 higher is the intensity of correlation. The closer is the coefficient to zero, the lower is the intensity of correlation, indicating a non-correlation between variables. In this work, the value found to Spearman s coefficient was , what indicates a moderate correlation, once the value is between 0.30 and 0.60 [20]. After calculated the Spearman s coefficient, it is possible do determine if correlation between variables is significant [21], taking as null hypothesis ( H o ) the statement there is no correlation between the variables and as the alternative hypothesis ( H 1 ) the followed statement There is a significant correlation between the variables. If sample size is bigger than 30, critical value can be found by the equation ±z n 1 where z correspond to the level of significance. When applied this test to a significant level of 0.05, it was observed that critical value is ± 0, This value indicate a significant correlation between grades obtained by the students in the Diagnostic test and the final grade in Calculus. There is another way to verify the significance of a value obtained to r s under the null hypothesis, by the variable t = r s n 2 1 r s 2 where distribution is approximately equal to distribution t, with n-2 degrees of freedom [22]. Using data from this work to test the null hypothesis (there is no correlation between the variables), it is found t =12,38. To the level of significance of 0.05, where the critical value is 1.96, we have 12,38>1,96, what indicates that the null hypothesis can be rejected. Thus, it is concluded that there is a significant association between performance in the Diagnostic test and Calculus. In order to verify any dependence between variables, Kendall test was performed, using Table II as values of contingence. TABLE II - Frequency of students performance at the Diagnostic test and final grade in calculus Final Grade 6 Final Grade < 6 Diagnostic test Calculus It was tested the following hypothesis: H o : grades are independent as H 1 : grades are dependent This test showed that there are evidences to reject H o, i.e., there are indicatives that there is dependence between the variables. V. FINAL CONSIDERATIONS The high rate of failure in Mathematics and Physics classes in engineering courses is one of the steepest obstacles faced by several universities. Consequently, to obtain passing grades in classes like Calculus, Physics, Analytical Geometry and Linear Algebra is one of the biggest challenges presented to the freshman-year students at UTFPR. Therefore, if one understands the variables that affect the failure rates, it is possible to adopt measures aiming to minimize this problem. One of the most cited variable related to higher failure rates in Calculus is the lack of mathematical background. The present study was built aiming to either verify or refute the statement that the lack of mathematical background is a decisive variable in the context of failure rates in Calculus.

5 Based on comparisons made between the performance at a diagnostic test and the final grades in Calculus, it was possible to verify that there is a significant correlation between the two variables. Moreover, the tests showed a certain dependency between those variables. Therefore there is evidence that the students with a more solid mathematical background tend to have a better performance in Calculus. However, even the students with gaps in their background overcame their initial difficulties and obtained a passing grade in Calculus. Students with serious background problems represent the majority of the students who fail in Calculus. If we eliminated the students whose score was inferior or equal to two points from the sample, the failure rate would decrease from 51% to 24%. Therefore, it seems that a basic background in Mathematics would be sufficient to obtain a passing grade in Calculus. However, it is important to mention that the correlation coefficient, independently from its value, does not give information about the cause-and-effect relationship or a more complex structure of factors [23]. In other words, one can speculate that the lack of mathematical background, the methodologies adopted by the professor, the structure of the class, the lack of academic commitment are factors that may cause the higher failure rates in Calculus, but this speculation cannot be answered only using the observed correlation. Therefore, it is necessary other studies in order to consider whether the relationship between those variables is caused by a third variable, or maybe by a combination of several other factors, which indicates the necessity of new studies in the area REFERENCES [1] SIMMAS, A. As graduações campeãs de desistência. URL: universidade/ufpr/as-graduacoes-campeas-de-desistencia- 26khijqty1gurtas1veawhyz2 [27 April 2015] [2] GOMES, E. 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