Mathematical modelling of the grade point average (GPA) system of mathematics students of Federal University, Lafia

Introduction

Mathematical Model according to Abam, Oladejo and Atanyi is an expression and a description of a real life system with the use of mathematical symbols or concepts and languages. It is the representation of abstract things with the use of symbols in real life situations. Grade Point Average (GPA) is a computational technique for public and private universities that helps students, staff and parent to have an idea about the performance of a student per semester and at the end of every session. This is also the ratio between the total grade points (TGP) of a student in a semester with the total credits registered (TCR). That is, the sum of the product of credit units with the grade points of the student in each course registered divided by the total credits registered. The GPA's role and effect in the life of a student is vital throughout the students' stay in the school as well as after graduation because it is the proponent factor that determines what class of degree the student will graduate with and also if the student could gain admission or be admitted into a higher degree with the use of the GPA. In the light of the above, haven realized the usefulness and importance of the GPA to a student, parent, school and the society generally, it is expedient to assess and formulate a Mathematical Model for the Grade Point Average (GPA) System of students of Federal University of Lafia (FU Lafia) taking into consideration, the students of the Department of Mathematics of FU Lafia.

Model formulation

The model seeks to understand the GPA system of grading and the variables or parameter that influences high and low GPA of students using a system of differential equations. Since the course credit system is employed, six compartments involving cumulative credit units at each level from 100 to 400 level, first spill-over (FSO) and second spill-over (SSO) is considered. We defined 𝐶 𝑖 (𝑡) as each level, 𝑖 = 1, 2, ⋯ , 6 as the cumulative total credit registered.

A student will normally graduate at 400 level (𝑖 = 4) if he or she was able to pass all prescribed course registered for within the first four years. For all students admitted for a 4-year course, a minimum of 120 credit units must be earned before graduation, hence 𝐶 4,5,6 (𝑡) ≥ 120. Where a student was unable to graduate in the 4th year, he will spill over to the next year, refer to as first spill over (FSO) year and subsequently if he fails to graduates moves to the second or last spill over year. According to the NUC regulations, a four year course programme will have an additional two year grace period. Hence it is expected that students admitted for a 4-year programme, must graduate within the 4th and the 6th year. This allowance will take care of certain factors beyond the students' control, like sickness, accidents etc. and factors within his control like failure in previous year and carelessness. The student progress to the next level at the rate 𝛽 𝑖 , where 𝛽 𝑖 is the current cumulative grade point average (CGPA) of the second semester of level 𝑖. A student probates if his CGPA is less than one. It is assume that a student can only probate at 100 level, 200 level and 300 level at a rate 𝜂 𝑖 * , where, 𝑖 = 1, 2, 3. The total number of credit units a student fails (also refers to carryovers) in the previous year (first and second semester examination inclusive) is denoted by 𝜂 𝑖 , where 𝑖 = 1, 2, ⋯ , 5. We assume that for student on probation, 𝜂 𝑖 = 𝜂 𝑖 * . The probability of a student graduating is denoted by 𝛼 𝑖 , where 𝑖 = 4, 5, 6. A student withdraws if he could not pass all the prescribed courses at the end of the second spill over year and was unable to earned a minimum of 120 credit units. All parameters used in the formulation of the model are assumed to be non-negative. Given the above description, the relationship between students' academic progress and GPA is governed by the following system of ODEs. The following diagram describes the dynamics of student academic progress and will be used in the formulation of the model equation

In the table below, variables and parameters used in the model are defined.

Mathematical analysis

We discuss the existence and uniqueness of the student progress free equilibrium (SPFE) of the model and their stability analysis in this section. The SPFE of the model is obtained by setting the left hand side of equation ( 1) -( 6) to zero and solving the resulting system of equations simultaneously. The system of equation ( 1) -( 6) has a unique student progress free equilibrium (𝑆𝑃𝐹𝐸 𝐸 0 ) = (𝐶 1 , 𝐶 2 , 𝐶 3 , 𝐶 4 , 𝐶 5 , 𝐶 6 ) = (0, 0, 0, 0, 0, 0)

Lemma 1

Let 𝑅 be a commutative subring of 𝐹 𝑛 𝑛 , where 𝐹 is a field (or a commutative ring), and

) where A, B, C, D are 𝑛 𝑥 𝑛 block matrices over 𝐹, then 𝑑𝑒𝑡 𝐹 𝑀 = 𝑑𝑒𝑡 𝐹 (AD -BC), whenever at least one of the blocks A, B, C, D is an 𝑛 𝑥 𝑛 zero matrix.

Proof

As a first case suppose that 𝐵 = 𝐶 = 0, be the 𝑛 𝑥 𝑛 zero matrix, so that 𝑀 = ( A 0 0 D ) be a block matrix. It is a well-known fact that

Equation ( 7) is the result of the Lemma above in the special case when 𝐵 = 𝐶 = 0.

To prove (7) we use the Laplace extension of 𝑑𝑒𝑡 𝐹 M by the first 𝑛 rows, which give the result immediately. A more elementary proof is describe thus: Generalized to the case where A is 𝑟 𝑥 𝑟, but D is till 𝑛 𝑥 𝑛. The result is now obvious if 𝑟 = 1, by expanding by the first row. Using induction on 𝑟 and expanding by the first row to perform the inductive steps gives the same result as (7) if we know only that B = 0. We obtain

Taking the transpose or by repeating the proof using columns instead of rows, we also obtain the result when C = 0, namely

To proof the multiplicative property of equation ( 7) that is

We make the following assumptions a) Adding a multiple of one row (respectively, column) to another row (respectively, column) of a matrix does not alter the determinant. b) Multiplying a matrix on the left (respectively, right) by a unitriangular matrix corresponds to performing a number of such operations on the rows (respectively, columns), does not alter the determinant. (A unitriangular matrix is a triangular matrix with all diagonal entries equal to 1). c) We shall assume that 𝑑𝑒𝑡 𝐹 I 𝑛 = 1, where I 𝑛 is the 𝑛 × 𝑛 identity matrix.

), since the first three matrices on the left of (11) are unitriangular. From ( 8) and ( 9), it follows from this that

Hence the second matrix on the left is unitriangular, so taking determinants and using (8) and the first part of (12) we have

Since 𝑑𝑒𝑡 𝐹 I 𝑛 = 1, the multiplicative law (18) for determinants in 𝐹 𝑛 𝑛 , follows.

Lemma 2

The eigenvalues 𝜆 𝑖 of the

If 𝜆 1 and 𝜆 2 are eigenvalues of A (not necessarily distinct), then we have that

Comparing ( 13) and ( 14), we have the following identity

Since the characteristic polynomial is quadratic the eigenvalues is given by

Let ∆ = 𝑇𝑟𝑎𝑐𝑒 (𝐴) 2 -4𝐷𝑒𝑡 (𝐴) be the discriminant, then the nature of the root is determined by the sign of the discriminant, which invariably satisfy the result of the lemma. Therefore 𝜆 1 𝜆 2 = Det 𝐴 > 0 and 𝜆 1 + 𝜆 2 = Trace 𝐴 < 0 for real 𝜆 1 𝜆 2 and 𝜆 1 𝜆 2 = Det 𝐴 < 0 and 𝜆 1 + 𝜆 2 = Trace 𝐴 > 0, for imaginary 𝜆 𝑖 . Hence the proof.

Stability analysis of student progress free equilibrium (SPFE) state

We now proceed to show the stability of the equilibrium states. To study the behaviour of the system of ODEs ( 1) -( 6) around the student progress free equilibrium states, we applied the linearized stability principles.

At the student progress free equilibrium state (𝑆𝑃𝐹𝐸 𝐸 0 ), we evaluate the partial derivative of the system ( 16) -(21) to get the Jacobian matrix below:

Using the result in Lemma 1 and 2, we partition the matrix (22) as follows:

Where A, B, C, and D are block matrices defined as follows

Theorem 1

The student progress free equilibrium (SPFE) of equation ( 1)-( 6) is stable if 𝛽 𝑖 > 0, 𝛼 𝑖 > 0 and 𝜂 𝑖 ≥ 0.

Proof

Consider the Jacobian matrix (22), the required criteria for stable equilibrium (that is, of having the trace of the Jacobian matrix negative and the determinant positive) give the conditions for stability. From the Jacobian matrix (22), we have that Det (𝐽 𝑆𝑃𝐹𝐸 ) = (𝜂 1 + 𝛽 1 )(𝜂 2 + 𝛽 2 )(𝜂 3 + 𝛽 3 )(𝜂 4 + 𝛽 4 + 𝛼 4 )(𝜂 5 + 𝛽 5 + 𝛼 5 )(𝛽 6 + 𝛼 6 ) > 0 and Trace (𝐽 𝑆𝑃𝐹𝐸 ) = -(𝛽 1 + 𝛽 2 + 𝛽 3 + 𝛽 4 + 𝛽 5 + 𝛽 6 + 𝜂 1 + 𝜂 2 + 𝜂 3 + 𝜂 4 + 𝜂 5 + 𝛼 4 + 𝛼 5 + 𝛼 6 ) < 0 provided that 𝛽 𝑖 > 0, 𝜂 𝑖 > 0 and 𝛼 𝑖 > 0

Numerical experiments

The model equations ( 1) -( 6) were all solved numerically using Maple 15 (Maplesoft, Waterloo Maple 2012). The parameter chosen were in consonance with the threshold values obtained in the stability analysis of the student progress free equilibrium states of the models. The data use were from four students from the Nasarawa State University, Keffi, Nigeria, whose result from year one to the year of their graduation were gotten from the examination officer of the Mathematical Sciences Department of the University.

Results

We present the following results from the numerical experiments.

Discussion of results

Table 2 presents a summary of four students' computed result used for the numerical experiments, while Table 3 shows the parameter values used. These values were found to meet the criteria for stability of the student progress free equilibrium (SPFE) states of the model. The grade point average (GPA) system allows courses registered to be weighted by a combination of grade (A, B, C, D, E, F) and grade points (5, 4. 3, 2, 1, 0) to the number of credits given for a particular course. The Cumulative Grade Point Average (CGPA) is a number that represent the average of a student's grade during his/her time in the university. From figure 2 above, we observed that the students has low grade point resulting from low grade score like (D = 2, E = 1, F = 0). Failing a course implies that the grade point is zero and when this course is registered in the next year, it increases the cumulative credit registered C (t) as compare with students in the same level who passed the course with high grade in the previous year. Since the GPA system account for every semester in a consistent manner through cumulative calculation, it then has a way of increasing a student stay in the university. In Figure 2, the student graduate after six years due to probation (GPA < 1.00) in year two and many carry overs. From the result in figure, we conclude that having repeatedly failing one or more courses in the semester system has a way of increasing one stay in the university.

From figure 3, we observed that the total cumulative units registered increases for student with consistent carryovers (in this case the total credit for all repeat courses was 31). This is very high. When a carryover course is a pre-requisite course to one or more courses, the student will not be able to register such courses until he has re-written and passed the carryover courses in the current year. This student also was not able to graduate at the 400 level official year of graduation; rather he graduated at the 6th year. The student result progress shown in Figure 4 indicates that the student had carryover once at 200 level (a total of 9 credit units), but had a recurrent mid-grade (like C = 3, D = 2) on a consistent basis. The student graduated at the fourth year though the probability of not graduating could still be high.

From figure 5, we observed that the student had high grade point in a consistent manner leading to a consistent high cumulative grade point average (CGPA). The student had no carryover and as such graduated at 400 level which is the official graduating year. The level where a student had carryover, no matter the total number of credits involves (small or high), the probability of not graduating at the official year could still be high.

The GPA system and the course credit system must be clearly understood by undergraduate students, so that they can set realistic goals and developed good study habits. Student must understand the negative effect of failing any course irrespective of the level.

The GPA system gives the students the great opportunity to decide which class of degree he/she will come out with even at the first examination in the university, so students must take the 100 level very serious as they would do when in their final year.

Conclusion

We proposed a mathematical model that captures students' academic progress via the GPA evaluation system. Our purpose is to understand the effect of failing a course (carryovers), low grade point and probation on the overall academic progress (CGPA) of students. From the result of this work, there is a direct relationship between high GPA and high grade point, and low GPA and low grade point. Students with high CGPA on a consistent manner tend to graduate in the fourth year while students with low grade and failing courses graduate in the fifth or sixth year. The GPA system goes hand in hand with the semester and course credit system, hence the need for students to understand the functionalities of the impact of the credit system on his overall academic progress. The GPA is of great importance to the student present and future aspiration, hence the need to find suitable and specific strategies to help improve once current GPA and subsequent CGPA. The probability of graduation at the 4th, 5th or 6th year is directly related to the student academic performance, mostly in having good grade on a continuous manner and not failing any course. Student must developed good study pattern, control environmental factors within his learning situation and external factors from family or society, if he is to have high CGPA and graduate at the official graduation year.

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