Mathematical Self-Efficacy: A Pilot Study Exploring Differences

Between Student Groups
Jon Warwick
Faculty of Business, Computing and Information Management
London South Bank University
London SE1 0AA
Email:warwick@lsbu.ac.uk
Abstract
This paper describes the results of a pilot study designed to
investigate differences in mathematical self-efficacy for two
groups of students taking a general mathematics unit as part of
their year 1 computing and IT undergraduate studies. It further
investigates two linear programming models to see whether
mathematical self-efficacy scores can be used to indicate an
appropriate choice of course for certain students on application to
university. The results of the survey give some indication of
differences between the groups and suggest that a larger study may
yield benefits in the selection of students for courses and also the
way mathematical material is taught.
Key words: Linear programming, mathematical efficacy, mathematics teaching.
Introduction
The mathematical ability of students entering UK universities has been a matter of
some concern and debate for a number of years (Henry, 2004) and this concern has
been felt not only in terms of the mathematics required for general university entrance
(usually GCSE mathematics at grade C or better) but also on courses for which
mathematics is a primary requirement (Engineeringtalk, 2001).
At London South Bank University there are, in common with other similar
institutions, concerns also about the variety of qualifications in mathematics which
students exhibit on application (including for example Key Skills in Numeracy and a
range of overseas qualifications) so that trying to match a student’s abilities in
mathematics with university entrance requirements is sometimes difficult. Within the
computing and IT subject domain mathematics plays an important role and all
students taking these subjects at London South Bank University are required to take a
first year unit entitled ‘Concepts of Mathematics and Statistics’ the purpose of which
is to provide students with a basic grounding in key mathematical ideas (number
representation, number bases, basic algebra etc.) as well as provide an introduction to
statistics and probability. The unit is taken by computing and IT students studying for
honours degree (BSc) and also those taking Higher National Diploma (HND) courses.
The entry requirements of these courses are different in terms of the advanced
qualifications required but frequently tutors have commented on how difficult it is to
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decide whether a candidate’s mathematical background (in terms of formal
qualifications) indicates acceptability for the three-year BSc programme, or whether a
longer path to honours degree via the HND would be better for the student, even if the
candidate is technically qualified for either. This is particularly true in the case of
mature applicants who may have been away from education for a while and therefore
not have studied mathematics for a long time or who may be offering knowledge
gained through appropriate work experience in lieu of formal qualifications. Indeed,
there are now courses (Foundation Degrees) which are specifically designed for
students who have work experience but not necessarily many academic qualifications
who wish to ‘formalise’ their knowledge through an appropriate qualification and for
these courses applicants may well not have studied for many years and may have
previously gained few qualifications. The consequences of a student joining an
inappropriate course can be severe (in the event of failure) both personally and
financially so it is important to use all possible means to match the student with the
right level of course. Experience has also shown that mathematics is one area where
many students have severe difficulties in their first year of university study.
This paper describes the results of a pilot study in which a questionnaire was used to
try and elicit differences in computing and IT students’ subjective views of their
previous mathematical experiences (prior to joining the University) to see whether
these differ between HND and BSc students and, if so, how they differ. Further it was
intended to examine whether the data gathered could be used to discriminate between
HND and BSc students. The questionnaire could then be used with course applicants
who have unusual (or few) formal qualifications and could provide information
suggesting which course might be most appropriate.
The Approach Taken
In order to elicit students’ subjective perceptions of their ability to cope with
mathematical study the concept of mathematical self-efficacy was used. Self-efficacy
is a type of personal cognition defined as “people’s judgements of their capabilities to
organise and execute courses of action required to attain designated types of
performance” (Bandura, 1986). This concept has been applied within the field of
educational research to a variety of subject domains (including mathematics) and at a
variety of levels (Phan, 2000, Hall, 2005). An individual’s self-efficacy beliefs are
conjectured to be oriented around four core concepts: ‘performance experiences’,
‘vicarious experiences’, ‘verbal feedback’ and finally ‘physiological and affective
states’. Each of these contributes to the individual’s ability to organise and execute
effective learning and can be tailored to specific subject domains. A little more detail
of these terms is given in Figure 1 below, where the descriptions are taken from Phan
(2000).
Source of Self-Efficacy Description
Performance Experience An indicator of capability based on
performance in past assessments, courses
etc.
Vicarious Experiences A source of evidence based on
competencies and informative
comparison with the attainment of others.
Verbal Persuasion This refers to verbal feedback from
teachers or adults.
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Physiological and Affective States Judgements of capability, strength and
vulnerability to dysfunction.
Figure 1: Self-Efficacy Sources
An individual’s judgements regarding self-efficacy can be elicited by questionnaire
(Betz, 1983) and in this case a sample of 48 first year computing and IT students (28
BSc and 20 HND) taking the first year mathematics unit were selected for the pilot
shortly after the start of the new academic year. Each student was given a
questionnaire consisting of 16 statements relating to mathematical self-efficacy and
asked to indicate the extend to which they agreed with the statement on a 7-point
Likert scale ranging from 1 (not true) through to 7 (very true). There were four
questions relating to each of the four sources of self-efficacy and examples of these
are shown in Figure 2.
Source of Self-Efficacy Example Statement
Performance Experience “I am not good in mathematics”
“I am generally please with my
mathematics results”
Vicarious Experiences “I don’t have anyone to help me with
mathematics”
“I have a close friend who is good in
mathematics”
Verbal Persuasion “I like to get verbal feedback from my
teacher”
“When my teacher praises me I want to
do well in mathematics”
Physiological and Affective States “Mathematics is interesting”
“I am always worried about
mathematics”
Figure 2: Sample Statements from the Pilot Questionnaire
For each respondent, the four scores for each self-efficacy source were added to give
a total score, and the total scores averaged over each of the four sources for the two
courses. The four pairs of averages were then compared using a small sample t-test to
see which of the sources of self-efficacy significantly differentiate between the two
groups. The four sources were then to be used to investigate the possibility of
establishing a mathematical function, using linear programming, that could suggest
which course would be most appropriate based on a student’s score on the
mathematical self-efficacy questionnaire.
Partitioning Data Using Linear Programming
In many other studies involving self-efficacy, researchers have used complex
statistical methods such as linear discriminant analysis to try and partition data over
several variables. There are difficulties with these procedures though since results are
often difficult to interpret and the statistical assumptions implicit within the
techniques are not necessarily valid for small samples of data. Other non-statistical
approaches have been used quite successfully and one classical example is that of
Linear Programming (LP). LP has been one of the most versatile operational research
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techniques with applications covering a range of topics including resource allocation,
product mix, transportation and location problems (Pidd, 1996). A well known
application within the field of medicine was that of breast cancer diagnosis
(Mangasarian, 1995) in which an LP was used to partition a set of data relating to
breast tumours into benign and malignant sets based on the measured characteristics
of individual cancer cells. The approach taken was to find a hyperplane that best
partitioned the two sets of data by minimising the average distance of misclassified
points from the hyperplane. Other approaches using ellipsoidal separation have also
been used (Konno, 2002).
More formally, let us suppose that A and B represent our two groups to be classified
and that ai (i = 1, …, m) and bj (j = 1, …, n) are the vectors of data (in our case this
would be the self-efficacy data for each student) to be used for the classification
process. Using the notation of Konno (2002) the method is to find a vector (c, c0)
which will partition the points exactly so that cT
ai > c0 for all members of set A and
c
T
bj < c0 for all members of set B and (c, c0) defines the hyperplane that separates the
sets. Now, of course, it is unlikely that complete separation will be possible so we
introduce additional variables yi and zj (for sets A and B respectively) which measure
the distance of misclassified points from the hyperplane. The objective function is to
minimise the weighted average of these distances. The LP formulation is, after
normalisation:
Minimise: (1-λ)
m
1 ∑=
m
i
y
1
i + λ
n
1 ∑=
n
j
z
1
j
Subject to:
cT
ai + yi ≥ c0 + 1 i = 1, …, m
cT
bj – zj ≤ c0 – 1 j = 1, …, n
yi ≥ 0 i = 1, …, m
zj ≥ 0 j = 1, …, n
In the above λ is just a weight used to express the relative importance of the two sets
for classification purposes. This classical formulation has been very successful in
partitioning the large data sets within the context its original medical application and
it was felt to be an approach that could well be useful in partitioning the rather smaller
data sets within this pilot study and any subsequent main study. However, given the
smaller data sets generated here, a similar but alternative formulation was developed
in which the objective was to find the hyperplane that correctly classifies the
maximum number of data points and this entailed the formulation of a 0-1 integer
linear programme (ILP). The ILP formulation is given below:
Maximise: ∑=
m
i
i v
1
+ ∑=
n
j
wj
1
Subject to:
cT
ai – c0 ≥ i v – i ( )M i = 1, …, m 1− v
cT
bj – c0 ≤ -wj + (1-wj )M j = 1, …, n
i v , wj are 0-1 variables for all i and j.
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In this formulation, the 0-1 variables take the value 1 if the data point is correctly
classified and 0 otherwise. The constant, M, is just a number selected to be large
enough to ensure feasibility of the ILP for data points incorrectly classified.
We illustrate the two approaches to data classification by considering a simple 2-
dimensional example. Figure 3 below shows two data sets containing 7 points for set
A (represented by black squares) and 7 points for set B (represented by black
diamonds) which cannot be uniquely partitioned by a straight line. In both this
example and the application to real data described later a value of 0.5 was used for λ
since there was no reason to treat either group as more highly weighted for
classification purposes.
0
1
2
3
4
5
6
7
01234
Variable x
Variable y
Figure 3: Sample Data Set with Partitioning Lines
The results of running both the LP and ILP formulations are shown, with the solid line
representing the partitioning line given by the LP and the dashed line that given by the
ILP. The LP misclassified 2 items from set A and 1 from set B whilst the ILP only
misclassified 1 item from each set.
It was decided to use both the LP and the ILP formulations on the data set to see
whether there was any indication that one would give an improved performance over
the other. It must be remembered that with data sets larger those being used here an
ILP can be difficult to solve (one 0-1 variable would be required for each of the data
points, i.e. there would be i + j of them in total) since a search process such as the
branch and bound method would need to be employed rather than the more efficient
simplex method that would be used for a standard LP.
The student sample was randomly divided into two sets so that the first set of
responses (18 BSc students and 10 HND students) could be used for investigation and
for ‘training’ the classification hyperplane, and the second set (10 BSc students and
10 HND students) used for testing the classification hyperplane.
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Results of the Pilot Study
Table 1 below shows the average score achieved by students within each group for
each of the four mathematical self-efficacy sources together with the result of the two
sample t-test for the difference of means.
Source Group Mean Standard
Deviation
Significance
Level
Combined
Group
Mean
Performance BSc Group 16.06 5.56
Experience HND Group 11.6 5.1
p = 0.044 14.46
Vicarious BSc Group 20.06 3.7
Experiences HND Group 14.8 5.45
p = 0.017 18.18
Verbal Persuasion BSc Group 15.33 4.52
HND Group 17.50 4.79
p = 0.257 16.12
Physiological and BSc Group 19.44 4.75
Affective States HND Group 18.90 7.35
p = 0.836 19.25
Table 1: Questionnaire Results – Descriptive Statistics by Group
We can see that there are only two of the sources that produce significantly different
results between BSc and HND students: ‘previous accomplishment’ and ‘vicarious
experience’. The result for ‘previous accomplishment’ would seem to indicate that
the BSc students have had significantly better experiences in terms of their previous
success in mathematical studies when compared to the HND students – a not
unsurprising result given that BSc students will generally have a larger number of
advanced qualifications which might indicate a more successful record of academic
achievement. Interestingly, the mean for the combined group was lowest for this
source possibly indicating that, generally, the students did not feel themselves to be
good in mathematics as judged by their previous accomplishments in mathematics.
The other significant difference between the groups was in terms of ‘vicarious
experience’ and this leads us to the conclusion that the HND students have not had the
same mathematical study support network (in terms of friends or relatives) which the
BSc students have had or that working within any network that they did have was not
a positive experience for the student. This would have implications for the type of
support that needs to be offered to HND students within the university domain –
support that might have been non-existent before and which might be a source of
positive experiences for the students.
It is interesting that the two remaining factors showed no significant difference
between the two groups. For ‘verbal persuasion’, the mean for the HND group was
higher than that for the BSc group indicating that the HND group felt a stronger
interaction with feedback from tutors (both in positive and negative terms) which,
again may have implications for the way mathematics is taught and supported on the
personal level with HND students. The final source, ‘physiological and affective
states’, again showed no significant difference of means between the two groups but
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this source scored highest in terms of combined average. This suggests that the
students were by no means indifferent towards mathematics i.e. that there was a
motivation and incentive in many students to seek ways of improving their
mathematical abilities.
In order to see whether the mathematical self-efficacy data could be used to indicate
an appropriate course of study for a student the data was used as training data to
determine a hyperplane capable of partitioning the two groups of students. The
original LP formulation by Mangasarian was applied using data for all four sources of
mathematical self-efficacy and the LP model solved using the Solver facility in
Microsoft Excel. It was able to correctly partition 21 out of the 28 cases (75%). Table
2 below shows the training data results by course with the percentage of correctly and
incorrectly classified cases indicated in brackets.
Predicted Course
HND BSc
Actual Course HND 5 (50.0%) 5 (50.0%)
BSc 2 (11.1%) 16 (88.9%)
Table 2: Training Data Results for Mangasarian LP Model
From Table 2 we can see that the LP is able to correctly partition 16 of 18 BSc
students but is much less successful with HND students. When we apply the fitted
hyperplane to the testing data then we get the results shown in Table 3:
Predicted Course
HND BSc
Actual Course HND 8 (80.0%) 2 (20.0%)
BSc 4 (40.0%) 6 (60.0%)
Table 3: Test data Results for Mangasarian LP Model
This gives an overall success rate of 14 out of 20 (70%) correctly classified which is
slightly less than with the training data but the number of cases used is small in this
pilot study.
Turning to the ILP formulation we get training data results as shown in Table 4
below:
Predicted Course
HND BSc
Actual Course HND 6 (60.0%) 4 (40.0%)
BSc 0 (0.0%) 18 (100.0%)
Table 4: Training Data Results for the ILP Model
This model gives a slightly higher level of correct classifications (85.7%) and the
same hyperplane applied to the testing data gives the results as shown below:
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Predicted Course
HND BSc
Actual Course HND 8 (80.0%) 2 (20.0%)
BSc 2 (20.0%) 8 (80.0%)
Table 5: Test Data Results for the ILP Model
This shows that in the test data 16 out of 20 (80%) of cases were correctly classified.
Conclusions
This paper has presented some results from a pilot study which sought to investigate
differences between HND and BSc students in terms of their mathematical selfefficacy
and whether mathematical self-efficacy scores can assist tutors in deciding
whether new students would be best suited to HND or BSc level study. Naturally,
this is not the only determinant of academic course selection, but is an important
contributing factor.
We have found that there are significant differences between HND and BSc students
in terms of two sources that contribute to mathematical self-efficacy and that these
lead to some preliminary conclusions.
Firstly, students who are admitted to the BSc course have, on average, better previous
experiences of mathematics in terms of their achievements in mathematics than HND
students. Note that this does not just imply recorded marks or qualifications gained,
but is a personal view of their performance and progress. This would support the
author’s personal observations that HND students often lack confidence and also
some basic skills that BSc students seem to have.
Secondly, HND students seem to have had significantly worse vicarious experiences
which relate to the degree to which they have (or have not had) a network of friends
who they can work with, and how they view their mathematical abilities in relation to
their friends and classmates. One could argue that this is important in that if such
students are mixed with BSc students for purposes of mathematics teaching (if such a
module is common to both groups) then this could further weaken self-efficacy in
HND students as they establish their mathematical abilities in relation to other (BSc)
students in the class. Furthermore, in teaching HND groups, it would seem to suggest
that these support groups need to be encouraged among similarly skilled students
through group work and practical projects so that vicarious experience can be seen as
a positive one by students.
Thirdly, neither group of students seem to ‘not care’ about mathematics. Some
students may love the subject and others be worried about it but students do have
motivation to improve their mathematics (if only to relieve the stress and worry that
some students have about learning mathematics) and tutors need to get to know
students to see which type of support and encouragement would be best for individual
students.
Turning to the use of LP and ILP methods, we have demonstrated the mathematical
self-efficacy scores can be used to separate the two groups of students with a
reasonably high degree of success. This would imply that for cases of, say, mature
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students with few formal qualifications, collection of this type of data at interview
might well provide useful further information on which admissions tutors may base
course recommendations. Of the two methods investigated, the ILP gave a higher
rates of correct classification but there are two caveats to be applied. One is that these
pilot data sets are small and the methods would need to be tried on larger groups and,
secondly, the Mangasarian LP method has been further developed by re-applying the
method to cases where the classification is in doubt so as to further refine the process.
This was not possible in our case as the data sets were too small.
The author has learned much from this pilot study and will now be proceeding to a
much larger study to be undertaken with the next intake of new students to these
courses.
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