MODELLING STUDENTS’ AFFINITY FOR LECTURE ATTENDANCE USING DIFFERENTIAL EQUATIONS: THE CASE OF STUDENTS OF KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY

Research Context: The aim of the study was to investigate students’ affinity for lecture attendance using differential equations and to explore reasons why students absent themselves from lectures. Research Methods: The study employed a quantitative methodology which was adopted in order to allow the researchers to gather more precise and quantifiable information on students’ perception about lecture attendance. Questionnaires were administered continuously to a cross-section of students numbering five hundred to ascertain their affinity for lecture attendance and reasons why they attend or do not attend lectures. Conclusion: A logistic differential equation had been used to model students’ affinity for lecture attendance. The calculated or predicted values of students who regularly attended lectures were obtained by using logistic differential equation. The predicted values were then compared with the observed values and analyzed. From tables 1,2,3 and the graphs in figures 1 and 2 which were obtained by the usage of Matlab, it was seen that the observed values and the predicted values almost converged to the same values, signifying that the model is good for the prediction of students’ preference for lecture attendance. It was interesting to note that the reasons that drive the majority of students to lectures are to find out what they are supposed to learn, not to miss important information and to find out about assessment task. The study also revealed that some students use on-line notes as a substitute for lecture attendance.


INTRODUCTION
A Differential equation in Dolking (2005) is an equation involving an unknown function and its derivatives.
The subject of differential equations is solving problems and making predictions in Dobson (2003, January). In mathematics, the rate at which a quantity changes is the derivative of that quantity.
The tendency towards decreasing lecture attendance by students is a concern for many tertiary institutions. Despite the emphasis on quality and flexibility, and the introduction of new technologies, the lecture remains at the centre of most universities' approaches to teaching and learning in Wadesango (2011) . Poor lecture attendance may have a negative influence on students' performance: as the semester progresses, poor lecture attendees may perform increasingly worse. When students are absent from lectures they miss valuable information resulting from peer-lecturer interaction and the benefits of the specific examples lecturers use to clarify difficult concepts. Absenteeism (Cleary-Hordforth, 2007) on the part of students disturbs the dynamic teaching-learning environment and adversely affects the overall well-being of lectures.
In this problem, our main objective is to develop and analyse a differential equation model based on a given data on students' affinity for lecture attendance and to demonstrate effective way of improving lecture attendance in the case of students of Kwame Nkrumah University of Science and Technology (KNUST).

METHODOLOGY
The study employed a quantitative methodology which was adopted in order to allow the researchers to gather more precise and quantifiable information on students' perception about lecture attendance.
The research design that was used in this study was the descriptive survey method. The descriptive survey is defined as " the method of research that simply looks with intense accuracy at the phenomena of the moment and then describes precisely what the research sees". Descriptive survey is probably the best method which is available to use in collecting data for the purpose of describing a population large enough to observe directly. The main instrument that was used to solicit for information was the questionnaire method. The questionnaire used can be found at the Appendix.
Self-administered questionnaires with open-ended and closed questions were used for data collection.
The data was collected from both undergraduate and post-graduate students in KNUST from all the six (6) colleges during the second semester of the 2011/2012 academic year. In all, five hundred (500) students comprising 218 females and 282 males responded to the questionnaires. The questionnaires were administered continuously until the number was exhausted.
The logistic equation was used to model the data collected and matlab was also used in the data analysis.

LIMITATIONS OF THE STUDY
The main purpose of the study was to determine student affinity for lecture attendance. However, the fact that only those students who were surveyed were present at specific lectures, somewhat challenges the efforts to determine what reasons why students do not usually attend lectures may have provided to substantiate their view of lecture attendance.
Some students were reluctant in providing us with the information needed.

THE LOGISTIC MODEL OF GROWTH
Exponential growth may be a good assumption provided it is not constrained either by space or by the availability of food or some other resource. But of course all populations will be so constrained once they reach some size. A more 'realistic' model is given by introducing some limiting effects as the stock grows. One of the simplest ways of doing this is through the logistic model which is given by The equation (1) is called the logistic equation. Its solution is known as the logistic function and its graph is known as the logistic curve. The model of population growth embodied in the logistic equation is called the logistic model. Here r is the 'instantaneous' growth rate when stock, P, is very low, and K is the 'carrying capacity', the maximum stocks the environment is capable of supporting. This is a differential equation which can be solved explicitly.
The equation can be split up to give dP rp rP P dt K   The first part on the right hand side, , rP is the same as the exponential function. The second part gives the reduction in growth rate as stock rises. When P = K, stock is at the carrying capacity, and, as the equation shows, the growth rate is zero.
In ecological terms, the equation can be considered as consisting of a 'density independent' rate r and a 'density dependent' rate -.

rp K
One way to think of this is in terms of a rate at which individuals grow when there is no competition and plentiful resources, being reduced by a rate reflecting competition or some other negative interaction as population increases towards the carrying capacity.
The usage of nonlinear differential equations is employed. Let   P t be the size of a population at time, t and 0 k  is a constant [2]. Then the specific rate is defined by dP dt P It is assumed that the rate at which a population grows or declines is dependent only on the current number. It would not be dependent on any time-dependent system such as situations bordering on seasons. This can then be stated as The equation (3) is called the density-dependent hypothesis in Zill (2001). With some assumptions equation (3) becomes (1). It is assumed that   f P is linear and so   The solution of (4) is bounded as . t  

Separation of variables is used to solve (4). Thus
  dP dt P r bP   Use partial fractions on the LHS to get: It was assumed that the rate at which the affinity for lecture attendance was developed was proportional to the number of students. All the five hundred (500) students were observed through administering questionnaires to individual students within the space of fifteen days (See Table 1).

Theorem
If f is a continuous real -valued function whose domain   D f is an interval, then Using the convergence method it shows as t approaches infinity the number of students converges to 500 and this depicts all being equal lecture can be stabilized.
The graphs in Figures 1 and 2  The logistic equation has proved to be a very useful model in assessing students affinity for lecture attendance in that should when they are made to know the essence of lectures they will make good use of it. Lecture here could involve video lectures.

CONCLUSION
A logistic differential equation had been used to model students' affinity for lecture attendance in Kwame Nkrumah University of Science and Technology, Kumasi. The calculated or predicted values of students who regularly attended lectures were obtained by using logistic differential equation. The predicted values were then compared with the observed values and analyzed. From tables 1,2, 3 and the graphs in figure 1 and 2 which were obtained by the usage of matlab, it was seen that the observed values and the predicted values almost converged to the same values, signifying that the model is good for the prediction of students' preference for lecture attendance. The study was also to gain insight into students' reasons to attend or not to attend lectures in tertiary education environment in KNUST. Students were provided with a list of reasons for attending lectures and stated whether they applied to them or not. The result to this question indicates that, the reason that drive the majority of students to lectures are to find out what they are supposed to learn, not to miss important information and to find out about assessment tasks. However, trying to learn by oneself and expecting to be at lectures seem to be rarely encountered reasons. On the reasons why students do not attend lectures, majority of the respondents cited too long lecture periods, lectures integrated with tutorials, uninteresting courses and unexciting lecturing style usually make lectures boring.
Out of the 500 students who were interviewed in this study, 495 students were found to be students who attend lectures regularly and this shows affinity students of KNUST have for lecture attendance in spite of many internal and external factors that can have negative influence on lecture attendance The KNUST students' preference for lecture attendance buttresses the notion that the lecture remains at the centre of most universities' approaches to teaching and learning despite the emphasis on quality and flexibility, and the introduction of new technologies. To this end, it is therefore necessary to encourage lecture attendance in universities and other tertiary institutions.
In spite of many internal and external factors that have negative influence on lecture attendance, our study found out that, most of the students were profoundly appreciative of the relentless efforts lecturers and other stakeholders of tertiary education put in to improve lecture attendance so as to encourage students to attend lectures regularly.

RECOMMENDATIONS
The following recommendations were made:  Students should be encouraged not to consider on-line lecture notes as a substitutes for lectures but use them as complements for lectures.
 Lectures and tutorials should be offered separately so as to avoid long lecture periods.