A global approach to improve the mathematical level of engineering students

A GLOBAL APPROACH TO IMPROVE
THE MATHEMATICAL LEVEL
OF ENGINEERING STUDENTS
SUSANA NIETO ISIDRO sni@usal . e s
HIGINIO RAMOS CALLE higra@usal . e s
DEPARTMENT OF APPLIED MATHEMATICS
POLYTECHNIC HIGH SCHOOL OF ZAMORA
UNIVERSITY OF SALAMANCA

Background:
In this work we have analyzed:
The situation of the initial knowledge in mathematics at
the engineering studies, (which is clearly unsatisfactory)
by means of an specifically designed test.
The generation of groups of students with similar
mathematical deficiencies based on the way of entrance
at the University.
The possible solutions to the mathematical problems
found for each group, using different initiatives due the
previous diagnosis.

Background:
Our aim is the development of a global approach to
solve this situation, which includes:
The use of an on-line tool to improve the mathematical
level of some of the students.
The measure of this improvement bbyy mmeeaannss ooff
qualifications and a self-confidence test.
A strategy to prevent the most common mathematical
errors found in written examination, using participative
classroom sessions.

Model
We work taking into account the frame
of the Spanish educational legislation
about who can access at the University:
High School students
•Students from pre-graduate studies
(High School).
•Students from Vocational Training
Modules.
Vocational Training Modules
students

Model
The mathematical background of these
groups is quite different:
•Students from pre-graduate studies
(High School) have a specific and
compulsory mathematical background.
But a group of them (usually the ones
High School students
Maths
exam ?
having the poorest mathematical level)
have avoided to take the final exam
of mathematics in order to maximize
their accessing score.
•Students from Vocational Training
Modules have a bad mathematical
background.
YES NO
VTM
Vocational Training Modules
students

Model
We have analyzed the mathematical level
of the students coming to the Polytechnic
High School of Zamora, which belongs to
the University of Salamanca.
We have six engineering degrees from
different areas: Civil Engineering,
Industrial Mechanics, Computing, Materials,
Technical Architecture and Agriculture.

Model
The first step is to make a diagnosis
of the situation of our students,
as accurate as possible.
D
i
a
g
n
o
s
i
s

Model
This step is especially important because
teachers usually are not conscious of the
real level of the mathematical knowledge
D
i
a
g
of their students.
The official information from the
University does not include data about
the way of access of our students.
n
o
s
i
s

D
i
a
g
Model
To make this diagnosis we have designed
a specific test which includes:
•Some items to measure the knowledge of
n
o
s
i
s
mathematical concepts.
•Some items to measure the knowledge of
mathematical procedures.
•A header with anonymous information
about the way of accessing
at the University.

Model
The diagnosis shows the presence of
three groups of students, which need
to be treated separately, because they
D
i
a
g
YES
have different kinds of problems (and
different kinds of solutions).
Some teachers make initial tests, but the
results are usually treated globally,
as a proof of the “global” knowledge
of that group of students.
n
o
s
i
s
NO
VTM

Model
Nevertheless, the results of the test
are much better understood when
we separate the students among
D
i
a
g
Calculate the
derivative of
GLOBAL RESULTS
Correct
answer
Incorrect
answer
No
answer
these three groups.
f(x)= 1/x 48.2% 23.8% 28.0%
n
o
s
i
s
g(x) =x ex 43.0% 22.8% 34.2%
h(x)=x/(x2+1) 38.9% 25.9% 35.2%

Model
D
i
a
g
Calculate the
derivative of
Correct
answer
Incorrect
answer
No
answer
Calculate the
derivative of
Correct
answer
Incorrect
answer
No
answer
f(x)= 1/x 59.3% 27.1% 13.6%
g(x) =x ex 55.1% 22.0% 22.9%
h(x)=x/(x2+1) 50.0% 26.3% 23.7%
Calculate the
derivative of
Correct
answer
Incorrect
answer
No
answer
GLOBAL RESULTS
YES
f(x)= 1/x 48.2% 23.8% 28.0%
n
o
s
i
s
g(x) =x ex 43.0% 22.8% 34.2%
h(x)=x/(x2+1) 38.9% 25.9% 35.2%
f(x)= 1/x 42,9% 34,3% 22,9%
g(x) =x ex 37,1% 31,4% 34.3%
h(x)=x/(x2+1) 31,4% 34,3% 34,3%
Calculate the
derivative of
Correct
answer
Incorrect
answer
No
answer
f(x)= 1/x 18.9% 5.4% 75.7%
g(x) =x ex 10.8% 18.9% 70.3%
h(x)=x/(x2+1) 8.1% 16.2% 75.7%
NO
VTM

Model
D
i
a
g
YES
The worst results are found for
VocationalTraining Modules
(VTM) group.
So, we focused on this group by
means of a specifically designed
on-line tool.
n
o
s
i
s
Calculate the
derivative of
Correct
answer
Incorrect
answer
No
answer
f(x)= 1/x 18.9% 5.4% 75.7%
g(x) =x ex 10.8% 18.9% 70.3%
h(x)=x/(x2+1) 8.1% 16.2% 75.7%
NO
VTM

Model
D
i
a
g
YES
Calculate the
derivative of
Correct
answer
Incorrect
answer
No
answer
f(x)= 1/x 59.3% 27.1% 13.6%
g(x) =x ex 55.1% 22.0% 22.9%
h(x)=x/(x2+1) 50.0% 26.3% 23.7%
Calculate the
derivative of
Correct
answer
Incorrect
answer
No
answer
n
o
s
i
s
NO
VTM
f(x)= 1/x 42,9% 34,3% 22,9%
g(x) =x ex 37,1% 31,4% 34.3%
h(x)=x/(x2+1) 31,4% 34,3% 34,3%
Nevertheless, the results of the other
groups are not as good as desirable.

Model
D
i
a
g
YES For these students, we have
focused on the most common
mathematical errors found in the
n
o
s
i
s
NO
VTM
written examinations.
We have tried to correct this errors by
means of some classroom sessions
dedicated to fix the attention of the
students on these common mistakes
in order to minimize their presence.

Model
D
i
a
g
YES
A global approach
Classroom
sessions
n
o
s
i
s
NO
VTM
On-line tool

Correction 1: VTM group
This VTM group accessing Engineering studies is
increasing nowadays due to two factors:
The economic crisis, which is leading many workers to
improve their training at the University.
The current Spanish educational lleeggiissllaattiioonn ((22001100))
includes no limitation in the number of VTM students
who can access at the University.
Other characteristic of this group is the big “drop-out”
rate: 25% of them did not perform the final
examination in the academic year 2011-2012.

We have designed an on-line course using
STUDIUM(based on Moodle) which is the official tool of
the “virtual campus” at the University of Salamanca.
It is particularly useful for teachers, since it offers a
wide range of tracking tools, including tools for
generating questionnaires, evaluation tools, etc.
Detailed description in:
Nieto and Ramos (2013) TEEM’13 Proceedings.
Nieto and Ramos (2014) JCIT (in press).

“Drop-out” rates:
Groups 2011-2012 2012-2013
Performing examination 75% 85,7%
Non-performing examination 25% 14,3%
Students passing the exam. 25% 28,6%
Performance at the final examination:
Groups 2011-2012 2012-2013
Averaged qualification 2,6 3,4
Passing the exam 5,5 5,7
Non-passing the exam 1,1 2,4

Test of satisfaction:
Questions Yes No
The course has been useful to understand the
mathematics at the classroom 100% -
The course has given you confidence to perform
78,6% 21,4%
the final examination The course has been useful to improve your
mathematical knowledge 78,6% 21,4%

Correction 1: Conclusions
An on-line course arises as a very useful tool to
give students for Vocational Training Modules the
mathematical contents they need to face their
studies successfully.
The results show an increase of the academic
qualifications and an increase in the percentage of
students performing the final examination.
A satisfaction test filled by the students shows an increase
in their self- confidence in their mathematical knowledge
and capabilities.

Correction 2: Common errors
The second step is to avoid the repetition of the
most common mistakes found in written
examinations.
This kind of ability is especially suitable to be
trained in the classroom (and not bbyy tthhee uussee ooff oonn--
line material) due to the immediate feedback given
by the teacher to the students.
We have analyzed the impact of this initiative over
the performance of the students.

The active-participative classroom session contains
the following schema:
We have compelled the students to complete the solution
of a typical erroneous procedure in an anonymous sheet.
We have mixed the anonymous sshheeeettss aanndd wwee hhaavvee
distributed them again so that each student corrects the
results of one of his/her classmates
The teacher has solved the whole process at the
blackboard paying attention to possible bugs, asking the
students to correct the work of their peers “as if they were
the teacher”.

The impact of this initiative has been measured
comparing the errors committed by the students in
the written examination for two consecutive
academic years: 2012-2013 and 2013-2014.
We have used a classification of errors aanndd ffaaiilluurreess
and have quantified them.
In addition, we have quantified the absence of
answers for the theoretical or practical questions
related with the most “problematic” topic
(diagonalization of a square matrix).

Some results: trained vs. not trained errors
ERRORS/ FAILURES 2012-2013 2013-2014 Relative difference (%)
Matrix rank (not trained) 41,2%. 39,3% Similar (-4,6%)
Zeros of a function (not trained) 19,7%. 20,0% Similar (+1,5%)
Matrix determinant (trained) 34,4% 23,6% -31,4%
Equation systems (trained) 61,9%. 37,0% -40,0%

Some results: questions unanswered
ERRORS/ FAILURES 2012-2013 2013-2014 Relative difference (%)
No answers for
diagonalization questions
44,7% 40% -10,5%
(theoretical)
No answer for the
diagonalization problem
(algoritmic)
23,7% 10% -57,8%

Correction 2: Conclusions
Some active-participative classroom sessions are
an effective tool to improve the mathematical
performance of our students:
The results detected in the written examinations show a
decrease of the main errors or failures which were
specifically trained, while the non-trained errors remain
stable.
The number of questions left unanswered have also
decreased, particularly the ones related with algorithmic
(not theoretical) procedures.

Conclusions:
As mathematics teachers at first university levels,
we must be aware of the shortcomings of our
students and we need to find a way to help them to
improve their mathematical performance.
The accurate diagnosis of the real kknnoowwlleeddggee ooff oouurr
students is a fundamental tool to develop the best
didactic approach for our students.
A global approach witch take into account the
presence of different groups of students can be
seen as a effective strategy to improve students’
learning.

A global approach to improve the mathematical level of engineering students

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