## Introduction

This chapter begins by describing how the Teaching for Understanding Approach (TFU) was implemented in the intervention, followed by the data collection, analysis and interpretation methods adopted to answer the research questions.

The objective of the intervention was to determine if the TFU approach is effective for improving the conceptual understanding of the topic of Differential Calculus.

- The research sub-questions guided the qualitative data are as follows:
- How does the relationship between the TFU and PT(Procedural Teaching)approach lead to an improved students' understanding?
- How does the TFU approach help students' to reflect on the topic of Differential Calculus?

Two NC (V) level 4 class of the experimental group with 50 students and control group with 50 students were observed for five days. The TFU approach was then implemented to the experimental group in one campus with the help of technology focussing on the multiple aspects of Differential Calculus whereas the procedural teaching (PT) approach focussed on the conventional method of teaching on another campus.

The usual instruction was replaced by the TFU approach; an approach that is concept-oriented, thereby providing the students with experience on how to calculate the rate of change of variables. Besides, the children were able to solve complex questions thanks to TFU approach.

## Research Methodology

This study used semi-structured interviews to get the right data. Semi structured interviews helps the researcher to ask questions which had not been planned for as new information is bound to present itself during the process of the interview. The researcher sought to determine if the students were able to justify the need for using 3 options to solve and evaluate a problem using the correct formula. Data was collected from 100 students who were divided into two. The first half was the experimental group with 50 students while the second half was the control group and also consisted of 50 students. The researcher found it worthwhile to divide the two groups into two so as to have a balance in the number to ensure that no side is disadvantaged numerically.

To test the level of understanding of the students, the researcher posed different questions which the students were expected to respond to. One of the questions was "Explain how the second derivative is used to predict the nature of stationary points with the help of a diagram?" In this question, the students were able to explain using previous knowledge that at maximum, the first derivative is zero. The TFU students referred to diagrams as a way to answer the questions related to nature of stationary points. One of the students explained that one is supposed to start with the left side of the of the slope and then draw the derivative tangent at the left side. Upon doing this, the result is that at maximum and minimum, then slope is zero. Based on the finding of derivatives, the students reinforced the connection between graphical and symbolic representation in order to mathematise the situation. The students were able enough to handle the reasoning and recognised their ability to identify the concept by conceptual understanding. Using their problem-solving skills, the students were able to solve the required questions effectively.

## Analysis of Qualitative Data

Based on the semi-structured interviews, the power rule and the quotient rule were more prevalent than the product rule. A majority of the students had the ability to apply the derivative rules that other rules. There is a significant number that was confident in applying the power rules. During the interviews, one of the students responded that "I can do it by simplifying and using the power rule."

Another student who applied the power rule successfully stated that "I think I should first remove x from the denominator.'' This was a show that students had the ability to apply the power rule successfully.

The students also proved to be efficient in applying the quotient rule accurately. In using this rule, the learner is supposed to identify the numerator and denominator successfully. The students were able to identify the numerator and denominator correctly. Such responses from the students are a revelation that they have the proficiency in the application of quotient rule.

In addition to the above mentioned rules, the students also proved to be proficient in the product rule. However, only a few of them showed confidence in using the product rule. Drawing on the examples given, a large number of the students faced difficulties when applying the product rule to solve equations. Students made errors in choosing denominator as x instead of 1x to manipulate the derivative rules.

As stated, one of the research questions that the researcher sought to answer was "How does the relationship between the TFU and PT (Procedural Teaching)approach lead to an improved students' understanding?" The researcher identified that a relationship between TFU and PT approach may lead to an improved students' understanding. A student who had pertinent knowledge in one of these approaches is able to apply the other approach more easily than a student who is proficient in only of them. In the procedural teaching approach, only two students were able to use power rule and quotient rule others just applied power rule. Some of these students who were not able to use quotient rule only applied the power rule and stated that they could not find out any other applicable method. One of the students stated that, "I can't see any other method."

## Procedural Teaching and Teaching for Understanding Approach

While both procedural teaching and teaching for understanding approaches are essential in helping the child to learn and understand more, a combination of both leads to a greater understanding. Procedural understanding mostly involves the students hoarding steps and algorithms (Bengson & Moffett, 2011). The students rely on their memories to recall these formula in order to answer questions. In this case, the students rarely make deep connections during instruction.

In procedural teaching, the teacher implores on the students to get a better glimpse of the formula used to solve various questions. Once the learner has memorized the formula, it becomes very easy for them to apply them when solving questions. Procedural teaching involves giving the learners the steps that they are supposed to follow when doing mathematical questions. The students have to follow these steps failure to which they will not get the right answer.

For example, from the graph below, the learners were expected to calculate the velocity.

The students' worked on the activity to calculate the average rate of change of AB, AC, AD and AE using the information provided in the graph. The students' used their prior knowledge to calculate the slope of the secant line and linked it to the average rate of change to understand the concept instantaneous rate of change. Based on the calculations obtained, the average rate change of velocity from A to B was 0.6, A to C, 0.55, A to D, 0.45, A to E 0.4.

Upon using the formula they had learnt previously and calculating, the students compared their results and found out that they had got similar answers. The fact that they had obtained similar results was a proof that they had applied similar formula to work out the questions. Since the figures were provided for in the graph, the role of the students was to apply them to the formula that they had been taught.

The researcher developed graphs which were supposed to measure different aspects. One of these graphs was supposed to measure Mr. Bolt's speed. A graph of distance against times was plotted. The distance was expressed in kilometres while time was expressed in minutes.

The students' worked on the activity to calculate the average rate of change of AB, AC, AD and AE using the information provided in the graph. The students' used their prior knowledge to calculate the slope of the secant line and linked it to the average rate of change to understand the concept instantaneous rate of change. Based on the calculations obtained, the average rate change of velocity from A to B was 0.6, A to C, 0.55, A to D, 0.45, A to E 0.4.

Velocity versus time graphs are also essential for testing the knowledge of learners. In a velocity versus time graph, the slope of the line is equal to the acceleration of the object. For example, given that the object moves with an acceleration of 4m/s/s, it means that its velocity changes by 4 m/s per second. Thus, the slope of the line be equals to 4m/s/s. In the same way, if an object is moving at -3m/s/s, the slope of that line will be -3m/s/s. Additionally, if the object has a velocity of 9m/s/s, then the slope will be 9m/s/s. A student of Differential Calculus must have this knowledge to make him or her have the ability to calculate velocity and slope in any given question.

In the above graph, it is observed that the line is sloping upwards to the right. However, in a mathematical sense, one can be asked to determine by how much the line slopes upwards for every 1 second along the time or horizontal axis.

Based on the above understanding, the slope can be calculated using the formula;

Slope = DyDx= y2-y1x2-x1 = riserunTo ensure that such content is understood by the students, it is essential to use both TFU and PT teaching.

The research was also to find out that Teaching for Understanding Approach helps in the understanding of learners. In this method, learners are trained to understand a certain topic or idea progressively and be able to apply that knowledge elsewhere. In mathematics, a learner may be shown how to use a certain formula to do certain questions but still be able to apply the same knowledge in solving other questions. Thus, in this approach, the biggest emphasis is on how to understand the working of the process rather than cramming the formula.

Understanding is a process that takes time and the teacher need to recognize this and spend as much time as possible in ensuring that the content he or she teaches has been understood. The best way for testing whether the children have understood is by assessing them by giving them tests or questions to solve. The questions should demand the students to use some of the knowledge learnt to answer them. The student needs to create connection between the content learnt and other things that he or she experiences in the world (Kishan, 2007).

In this study, the students were required to derive the definition of the derivative. Since this was a complex process, the students were expected to apply the previous knowledge that they had learnt to solve the question. The students reflected on the procedure for finding the definition of derivative by using the slope and then the limit tool. To bring their knowledge together, the learners first worked in groups before working in the procedure. The purpose of working in groups was essential for the students to bring the knowledge that they had together.

The success of this was seen when the students worked successfully in the reconstruction of the definition of the derivative given in different labels to exhibit procedural fluency. This activity invited the students to think deeply about how concepts are linked and interrelated as opposed to memorising. In the PT group the procedural fluency was observed when the lecturer worked together with the students to establish the different techniques to be followed to find the derivative of functions. The procedure is followed based on the concepts formulated by the students through conceptual understanding.

Using the knowledge learnt, the students were able to solve the problems and also be able to identify possible errors as well as the different techniques for solving t...

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