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 2014.
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.
978-1-4799-8706-1/15/$31.00 ©2015 IEEE
20-24 September 2015, Florence, Italy
Proceedings of 2015 International Conference on Interactive Collaborative Learning (ICL)
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 2014. 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 2014.
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
978-1-4799-8706-1/15/$31.00 ©2015 IEEE
20-24 September 2015, Florence, Italy
Proceedings of 2015 International Conference on Interactive Collaborative Learning (ICL)
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
TABLE I - DATA FROM DIAGNOSTIC TEST
Students successful
in Calculus
Students
failed in
Calculus
15
-
-
-
14
-
-
-
13
1
1
-
12
4
4
-
11
2
2
-
10
1
1
-
9
9
8
1
8
15
13
2
7
11
9
2
6
26
17
9
5
21
15
6
4
18
12
6
3
44
33
11
2
64
27
37
1
94
34
60
0
109
28
81
Total
419
204
215
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
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. 03.
Grades
FIGURE 2 - HISTOGRAM OF FINAL GRADE IN THE DIAGNOSTIC
TEST
Frequency
Number of
Students with
correct
answers
Frequency
Number of
answers
Grades
FIGURE 3 - HISTOGRAM OF FINAL GRADE IN CALCULUS
978-1-4799-8706-1/15/$31.00 ©2015 IEEE
20-24 September 2015, Florence, Italy
Proceedings of 2015 International Conference on Interactive Collaborative Learning (ICL)
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].
So, initially Spearman’s coefficient of correlation
( rs ) was used to analyze relation between two variables
t=
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 0.5185, 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 ( H1 ) the followed statement
“There is a significant correlation between the variables”.
1− rs2
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 in Diagnostic test and final grade in Calculus).
Values from rs range from -1 to 1, and the closer the
rs n − 2
Final
Grade ≥ 6
Diagnostic test
17
Calculus
204
Final
Grade < 6
402
215
It was tested the following hypothesis:
H o : grades are independent as
H1 : 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.
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
0,09586. 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 rs under the null hypothesis, by the
variable
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.
978-1-4799-8706-1/15/$31.00 ©2015 IEEE
20-24 September 2015, Florence, Italy
Proceedings of 2015 International Conference on Interactive Collaborative Learning (ICL)
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
[6]
REFERENCES
[17]
[1]
[2]
[3]
[4]
[5]
SIMMAS, A. As graduações campeãs de desistência. URL:
http://www.gazetadopovo.com.br/educacao/vida-nauniversidade/ufpr/as-graduacoes-campeas-de-desistencia26khijqty1gurtas1veawhyz2 [27 April 2015]
GOMES, E. Ensino e Aprendizagem do Cálculo na Engenharia: um
mapeamento
das
publicações
nos
COBENGEs.
URL:
http://matematica.ulbra.br/ocs/index.php/ebrapem2012/xviebrapem/pape
r/viewFile/533/365 [12 April 2015]
BARBOSA, G. O. Raciocínio lógico formal e aprendizagem em Cálculo
Diferencial e Integral: o caso da Universidade Federal do Ceará. M.Sc.
Thesis. Universidade Federal do Ceará: Brazil.1994.
BARBOSA, M. A. O insucesso no ensino e aprendizagem na disciplina
de Cálculo Diferencial e Integral. M.Sc. Thesis. Pontifícia Universidade
Católica do Paraná: Brazil. 2004.
MURTA, J. L. B.; MÁXIMO, G. C. Cálculo Diferencial e Integral nos
cursos de Engenharia da UFOP: estratégias e desafios no ensino
aprendizagem, in Proceeding of XXXII COBENGE - Congresso
Brasileiro de Ensino de Engenharia, Brasília: Brazil. 2004.
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[18]
[19]
[20]
[21]
[22]
[23]
GARZELLA, F. A. C. A disciplina de Cálculo I: a análise das relações
entre as práticas pedagógicas do professor e seus impactos nos alunos.
Ph.D. Thesis. Universidade Estadual de Campinas: Brazil. 2013.
SOARES DE MELLO, J. C. C. B; SOARES DE MELLO, M. H. C.;
FERNANDES, A. J. S. Mudanças no ensino de Cálculo I: Histórico e
Perspectivas, in Proceeding of XXIX COBENGE - Congresso Brasileiro
de Ensino de Engenharia, 2001, Porto Alegre: Brazil.
CURY, H. N. Análise de erros em Cálculo Diferencial e Integral:
resultados de investigação em cursos de Engenharia in Proceeding of
XXXI COBENGE - Congresso Brasileiro de Ensino de Engenharia,
2003, Rio de Janeiro: Brazil.
PASSOS, F. G. dos; DUARTE, F. R.; LEITE, A. A. M.; PEREIRA, T.
N.; LEITE, T. N.; DONZELI, V. P. Análise dos índices de reprovações
nas disciplinas Cálculo I e Geometria Analítica nos cursos de
Engenharia da UNIVASF, in Proceeding of XXXV COBENGE Congresso Brasileiro de Ensino de Engenharia, Curitiba: Brazil. 2007.
CAVASOTTO, M. Dificuldades na aprendizagem de cálculo: o que os
erros podem informar. M.Sc. Thesis. Pontifícia Universidade Católica
do Rio Grande do Sul: Brazil. 2010.
CAVASOTTO, M; VIALI, L. Dificuldades na aprendizagem de cálculo:
o que os erros podem informar. Boletim GEPEM, nº 59, p.15-33,
jul./dez. 2011.
MENESTRINA, T. C.; MORAES, A. F. Alternativas para uma
aprendizagem Significativa em Engenharia: Curso de Matemática
Básica. Revista Brasileira de Ensino de Engenharia, v.30, n.1, pp.52-60,
2011.
KESSLER, M. C.; DE PAULA, C. G; LEMOS, R. S. M. PROMA: em
busca de respostas para as repetências sucessivas em Cálculo
Diferencial, in Proceeding of XXXIX COBENGE - Congresso
Brasileiro de Ensino de Engenharia, Blumenau: Brazil. 2011.
REHFELDT, M. J. H.; NICOLINI, C. A. H.; QUARTIERI, M. T.;
GIONGO, I. M. “Investigando os conhecimentos prévios dos alunos de
Cálculo do Centro Universitário Univates”. Revista de Ensino de
Engenharia, v.31, n.1, p.24-30, 2012.
LACAZ, T. M.; CARVALHO, M. T. L. Implicações das dificuldades
dos alunos na aprendizagem da disciplina de Cálculo Diferencial e
Integral I da FEG/UNESP para as práticas pedagógicas, in Proceeding of
XXXV COBENGE - Congresso Brasileiro de Ensino de Engenharia,
Curitiba: Brazil. 2007.
SANTAROSA, M. C. P.; MOREIRA, M. A. “O Cálculo nas aulas de
Física da UFRGS: um estudo exploratório”. Investigações em Ensino de
Ciências, v. 16(2), p.317-351, 2011.
SANTOS, R. M.; NETO, H. B. (2005) Avaliação do desempenho no
processo de ensino-aprendizagem de Cálculo Diferencial e Integral I (o
caso da UFC). URL: http://www.multimeios.ufc.br/arquivos/pc/artigos
[02 May 2015].
TEIXEIRA, K. C. B.; PEREIRA, A. C. C. Os conhecimentos prévios de
Matemática trazidos pelos alunos ingressantes nos cursos de
Engenharias da UNIFOR: atual cenário, in Proceeding XL COBENGE Congresso Brasileiro de Ensino de Engenharia, Belém: Brazil. 2012.
VIEIRA, S. Bioestatística: tópicos avançados. 2nd. ed. Rio de Janeiro:
Elsevier, 2004.
CALLEGARI-JACQUES, S. M. Bioestatística: princípios e aplicações.
Porto Alegre: Artmed, 2003. 255p.
LARSON, R.. FARBER, B. Estatística Aplicada. 4th. ed. São Paulo:
Pearson Prentice Hall, 2010.
HOFFMANN, R. Estatística para Economistas. 4th ed. São Paulo:
Cengage Learning, 2011.
WITTE, R. S.; WITTE, J. S. Estatística. 7th ed. Rio de Janeiro: LTC,
2005.
978-1-4799-8706-1/15/$31.00 ©2015 IEEE
20-24 September 2015, Florence, Italy
Proceedings of 2015 International Conference on Interactive Collaborative Learning (ICL)