On Quantitative Analysis Model for the Dynamics of the Effect of Parental Background on Pupil/Students Performance in Mathematics

In this paper, using ordinary differential equations, mathematical performance model was formulated to study the dynamics of pupil/students’ performance in mathematics as a function of parental background, incorporating a number of environmental factors. The model considered 8 subgroups, which led to derivation of 8Dimensional dynamic mathematical model for the study of students’ performance in mathematics. Model analysis explored numerical methods and the computational simulations of the model indicated that the proportion of pupil/students’ from parents with probability of transmission of hereditary and acquired intelligence exhibited high performance in the subject. However, under a cozy environmental factors, male pupil/students’ possess more of acquired intelligence in mathematical, whereas, the females exhibited dominance and are sharper via hereditary intelligence. The model therefore, recommended devotion of attention and resources by parents on acquired intelligence of their pupil/students’; as well as both governmental and non-governmental agencies willingness to compliment efforts of parents in the provision of appropriate environment for the enhancement of pupil/students’ performance in mathematics. Furthermore, the optimal control and broader predominant studying parameters for similar model is highly encouraged.


Introduction
Generally, an in-depth view of the subjectmathematics in any dimension cannot be appreciated without a brief mention of modern education. Modern education is known to have been brought to African countries around 1842, notably by the Roman Catholic Missions and Wesleyan Methodist Missionary [1].
In line with several other definitions, education has been defined as an act of training, teaching morally and mentally as well as changing the behavior, feeling, thinking of the learner, see for examples, [2,3,4]. Imbibing the above qualities on any child is a primary function of parents and/or guardians, whose responsibility are to characterize the intelligence of their children while at their formative stage of development (where their brain could take-in easily). This fact was affirmed by [5], who stated that the intelligent of a student can be better predicted from the educational level of his parent. From the above point of view, this paper tends to define education as the training of youngsters between the ages of 6 to 18 years, which spelt the formative stage of any child.
Therefore, mathematics as an offspring of education can be traced as far back as 250 B.C., when the Greek mathematician, Archimedes assumed that the area of a circle was a number; and that the number could be approximated more and more closely by computing the area of regular inscribed polygon of more and more sides [6,7]. Mathematics as the name implies, is the science of magnitude and numbers and all their relations [8].
Furthermore, as a result of the vital role of mathematics in all facets of academic endeavor, emphases by teachers, non-governmental and parents in particularly, have been on pupil/students to study the subject (mathematics) effectively. However, pupil/students intelligence in mathematics among other subjects is known to depend on a number of factors. It's obvious that availability of these factors largely depends on parents, and also, the love for their wards and likeness for the subject -mathematics [5].
The model [11] had studied the psychological imprisonment or intellectual freedom? A longitudinal study of contrasting school mathematics approaches and their impact on adults' lives. The study considered 2 schools in England that taught mathematics very differently, the first author found that a project-based mathematics approach resulted in higher achievement, greater understanding, and more appreciation of mathematics than a traditional approach. This follow-up showed that the young adults who had experienced the 2 mathematics teaching approaches developed profoundly different relationships with mathematics knowledge that contributed toward the shaping of different identities as learners and users of mathematics. This present study motivated by the above assertion is design to investigate using mathematical model, as could be found in [9,10,12], the performance of pupil/students in mathematics as a function of their parental background. Thus, the novelty of this present work lies in the explicit inclusion of the study of the hereditary and acquired intelligence of students' as a function of parental background, which make the recent study outstanding compared with the studies of [1,2,5]. Moreover, the present model is enhanced 8-Dimensional nonlinear differential equations. Furthermore, the classification of the entire study, which clearly defines the performance of the female students', added a peculiar novelty to this present model.
Other notable mathematical models related to the studies of the effect of students performance in mathematics could be found in [13,14,15].
The entire work is subdivided into five sections, with section 1, devoted to the introductory aspect. Material and methods are explore in section 2, which define the model parameters and assumptions of the study with which the equation of the model are derived. Derivation of model equations and transformation of equations is performed in section 3. Computer simulations and analysis of result are conducted in section 4 and finally, conclusion and recommendation of the model are inscribed in section 5.

Material and Methods
In this section, we define the statement of the problem and formulation of the model equation, guided by a number of assumptions.

Statement of the Problem
In proposing the study for the dynamics of pupil/students' performance in mathematics as a function of parental background, the model explores experimental population consisting of 8 subgroups from which the following model parameters (functions) deduce and define as: 3. Population consist of only pupil/students at their formative stage (i.e. 1 ( ) 18 n t ≤ ≤ ,years); 4. They exist parents with acquired educational background in mathematics; 5. They exist parents with hereditary educational background in mathematics; 6. Age-structure is ignored; and 7. Emigration and immigration do not occur in the population. From the system parameters and enlisted limitations, we construct our model schematic flow-chart as seen in Figure 1.

Model Formulation
From Figure 1, and the model parameters as well as the limitations (assumptions) of the model, we drive the model equations as follows: 1 .

Equation Transformation and Model Derivation
In order to reduce the number of equations for explicit analysis of the mathematical implication of both students hereditary and acquired intelligence of mathematics, it becomes obvious to transform our initial models equations (2.1)-(2.8) into proportions i.e. ; ; ; Therefore, using the above proportions, the general rate of both hereditary and acquired intelligence for both male and female pupil/students can be defined as follows: (2.14) Figure 1. Schematic flow-chart for the effect of parental background in pupil/students' performance in mathematics Thus, further application of above derived proportions and substitution of equations (2.9)-(2.14) into equations (2.1)-(2.8), leads to the derivation of desired equations of the system as seen below: ) .
Thus, equations (2.15)-(2.22) define the equations of the model. We next illustrate the derived model.

Numerical Simulations and Discussion
We denote this section to the numerical illustrations and analysis of our derived model equations, followed by the analysis and discussion of the resulting outcome of our investigation.

Numerical Illustrations and Analysis
In simulating the derived model, a number of illustrative examples are established. The simulations explore the classical in-built Runge-Kutter of order of precision 4 in a Mathcad environment, using a set of deduced established numerical data with varying source [1,2,5] and set of model variables as seen in Table 1 & Table 2, below: with initial values of all characteristics remaining constant as in Table 2 below:  The numerical experiments are meant to study the dynamics of the effect of parental background on the performance of pupil/students' in mathematics particularly at their formative stage. Furthermore, it is worth to note that in simulating our model for the investigation of students intellectual performance in mathematics, we took into account some parameters, which were considered as predominantly the decisive functions of the parental background. These includes: conducive learning environment, e ; availability of textbooks, µ ; qualify teachers, q ; students/teachers relationship, r ; teachers welfare, w t .
Thus, from a systematic simulation of variants 1-3, of Table 1 with the model variables as in Table 2 above, the following results were established: Figure 2   year and through the duration of experimental investigation. Finally, for some zero predominant parameters, we conduct as in Figure 2(h) above, the female performance for their level of intelligence of acquired knowledge in mathematics. Result shows a near zero knowledge of mathematics particularly at the formative stage. Gradual extinction of mathematical knowledge is seen after 6 years and through the experimental period. The implication here is that female acquired intelligence in mathematics exists spatially at the formative stage and ceases to exist after the 10 th year.
Next, while we allow other parameter values of variant 1, we intensify our experiment with the activation of some  Table 2, we investigate from Figures 3(a-h) below, the varying intellectual behavior of pupil/students in mathematics. Figure 3(a) below represents the simulation of male parents' background in mathematics taking into account the conditions given above. Result indicates initial sharp appreciation of parental knowledge of mathematics to a value of 2.11% and then diminishes gradually through the early 6 years. Parents' knowledge of mathematics gained momentum after 6 years through 12 years exhibiting high tendency of mathematical intelligence at 18 -21 years. Loss of concentration is seen at the 23 rd year, which then spurs up with a peak value 5.51% at the 27 th year before exhibiting further undulating knowledge of the subject.
Observing similar conditions of Figure 3(a), we simulate from Figure 3 Figure 3(c) below, with similar environmental studying factors as in Figure 3(b) below, we conducted an investigation for only hereditary intellectual behavior of male pupil/students in mathematics. Result vilifies slight knowledge of mathematics at formative stage from 1-7 years and then experience deteriorating knowledge of mathematics within the age range of 8-21 years. Rejuvenation of mathematical intelligence is observe again after 21 years with a peak value of 461.89% at 27 th year and remain stationary thereafter. Subjection of Figure 3(d) to similar structural learning factors as in Figure 3(c) below, shows that for a male population of 0.15 m u = , the acquired intelligence of this class of pupil/students in mathematics is low at the formative stage through the 11 th year. Gradual spanning of mathematical intelligence is seen from the 12 th year with mix oscillated cluster intelligence of peak value For a population of 0.2 m y = , we study as in Figure 4(b) below, the intelligence performance of male pupil/students with both hereditary and acquired knowledge in mathematics. Here, result shows that mathematical intelligence is quite low and tends to remain static along the formative stage through 12 years and then deteriorates through 21 years. Knowledge of mathematics resurfaces from the 22 nd year and tremendously appreciated with apex value of On the other hand, we observe the behavioral changes of the female population in mathematics, given similar conditions as their male counterparts. From Figure 4(e) below, we simulate as in variant 3 of Table 1, the intellectual background of female parents in mathematics. Result show that for a population proportion of 0.1 f x = , the general mathematical intelligence is low at their initial stage through 12 years of study. This is followed by slight deterioration of mathematical intelligence after 12 years through 18 years. Female parental knowledge of mathematics is visible after 18 years through 30 years with inclinative value of 4 2.82 10 % × . Maintaining similar predominant studying environment for the rest of the varying intelligence quota for the female population, we investigate as in Figure 4(f) below, the hereditary and acquired intelligence of female students' performance in mathematics taken a population sample of 0.1 y = . Result shows stationary growth in knowledge for mathematics through 13 years before set of deterioration down to as low as 6 4.59 10 − × seen from 14-27 years. Knowledge of mathematics is seen to appreciate thereafter to an apex value 6 6.18 10 × from 27 through 30 years.
The hereditary intelligence of female population in mathematics is further investigated as presented by

Discussion
In this paper, we had considered and implemented a mathematical model that accounted for 8-Dimensional mathematical differential equations for male and female performance in mathematics as a function of their parental background in the subject. The model explored quantitative ordinary differential equations (ODEs) and the analysis conducted using numerical computation. The model clinically analyses the varying intelligence levels of parental and students' performance in mathematics.
For simplicity, evaluation of the overall performance is considered in four major categories: parental knowledge of mathematics under varying studying environmental factors; males and females mathematical intelligence under zero availability of predominant parameters (i.e. , , , ,  To an apex value of

Conclusion
In this paper, ordinary differential equations were used for the formulation of an 8-Dimensional mathematical dynamic model for the study of pupil/students' behavioral intelligence in mathematics as a function of parental background. The method of analysis explored numerical quantitative approach, which classified resulting outcome into four categories: parental knowledge of the subject at varying accessibility of learning environment; pupil/students mathematical intelligence under zero availability of studying predominant parameters; pupil/students behavioral intelligence in mathematics under intermediary (semi-) availability of predominant parameters and pupil/students' performance in mathematics given enhanced predominant parameters.
From our computational analysis, its viewed that male parents with combined hereditary and acquired intelligence in mathematics were sharper and brighter compared with the female counterparts under enhanced accessibility of studying predominant parameters and was far worse when predominant parameters was completely zero. The result depicted the intelligence of pupil/students' performance in mathematics as the population with combined hereditary and acquired intelligence exhibited brilliant knowledge of the subject after 18 years with the females showing more better understanding as time dependent. Furthermore, result showed general low appreciation of the subject at formative stage by both male and female pupil/students and as well, revealed that under intermediary accessibility of predominant parameters, females with only hereditary intelligence were sharper from 18 years compared to their male counterparts whereas, the males with acquired intelligence were brighter than their female counterparts. The model therefore, suggest the inevitable need for more collaboration of parents, government and governmental agencies to foster more endurable and sustainable studying environment for the upgrade of pupil/students' intelligence in mathematics. Furthermore, a broader model with more improved predominant studying parameters is thus encouraged.