TFU and PT Approaches Evaluated Using Kilpatrick's Framework - Essay Sample

Introduction

This study used TFU approach on 50 students and PT approach on some other 50 students. Kilpatrick's Framework explains a lot about the evaluation of training programs and mathematical proficiency. The framework is divided into four strands, namely reaction, learning, behaviour, and results. Reaction means that the lecturer needs to ensure that the training is valuable. He or she tests how he or she contributed to the delivery of knowledge as well as how the students received it. In this strand, the lecturer interacts with the students to try and find out if the learning process as successful and if the students liked the presentation. In this study, the lecturer interacted with the students in the hope of finding out whether the students were engaged in the lesson.

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The second strand is learning, and it is very important in the learning process. This strand focuses on what the students have and what knowledge they have learned. It also tends to find out what the students know and what they are able to do differently. In this study, the lecturer identified the topics that the students would cover. Some of the included finding the slope in a graph as well as how to form a tangent line from the secant line. The lecturer sought to know how much the students knew about the topics before the lesson would begin. This way, it became possible to address various gaps that were identified.

The other strand of the framework is behavior. This strand helps the lecturer to be able to determine how well the students are able to apply their training. This study made use of both the procedural teaching approach and teaching for an understanding approach to determine which between the two is better. To do this, two groups, each consisting of 50 students, were used. This enabled the researcher to realize the behavior of the students based on the level of the style of teaching applied.

The fourth strand of the Kilpatrick's framework is results. This is the level that entails the analysis of the final results of the training or the teaching process. This level tests whether the methods that have been applied differ in any way or whether they were effective or not. In this study, the learners were tested based on the teaching approach applied. They were tested based on their ability to perform different questions. They were also tested on conceptual understanding, procedural fluency, strategic competence, and adaptive reasoning. However, it is essential to mention that these strands are not independent but are intertwined, and it would be difficult to separate one from the others.

Kilpatrick's Competency Framework

From a mathematical perspective, the Kilpatrick's framework is composed of conceptual understanding, procedural fluency, strategic competency and adaptive reasoning. Conceptual understanding refers to he ability of the student to know more than isolated facts and methods. In this strand, the learner understands mathematical ideas and gains the ability to transfer their knowledge into new situations as well as be able to apply it to new contexts. In addition, this strand lies in the ability of the student to be able to construct their own knowledge. In this study, the students under TFU were given questions that required them to develop knowlgde based on what they alrwady knew to be able to answer them. For example, the students were pushed to think on how to find abetter approximation to the velocity from a provided graph. After some moments of calculations and trials, the students were able to apply their knowledge to answer the questions.

The TFU students were asked about wha they noticed about the slopes. They responded, "they are different." The lecturer went ahead and asked how it was possible to find Mr Bolts velocity at A. The students thought for a moment and then responded, "by taking points closer to A." The lecturer agreed with them that getting the points closer to A would defintiley give a better estimation to the slopes at A. Such kind of responses were evidences that the students were able to apply knowledge to answer questions, thanks to TFU approach. Conceptual understanding was observed when the students dragged the secant line nearer until the slope of the secant line becomes same as the slope of the tangent line at a point. This process helped the students to make sense of the infinitely small distance to find the rate of change at a point. The students were provided with the knowledge that limit of slope of secant line approximates to the slope of tangent line when the distance approaches to zero.

Conceptual understanding as also seen from the responses that students gave regarding derivatives. The students were given a task to identify the slope of the tangent line. In this exercise, the conceptual knowledge was able to be identified based on the responses that they gave. When the lecturer asked whether the slope was same throughout the tangent lines, the learners were able to respond in the affirmative. The lectuerer went ahead to ask them about the reason for their saying so. They responded, "because it is a straight line" and "they are same." The lecturer went ahead to ask the slope when the tangent line is horizontal. They responded, "zero." This was evidence that the students had conceptualized the contents that they had been taught.

In the PT group, conceptual understanding was also observed. However, unlike in the TFU group where the students worked alone with the lecturer posing questions and affirming their answrers, in the PT group, the studenets worked together with the lecturer to formulate concepts. The lecturer was forced to do a lot of explanations to ensure that students understood. In some cases, the lecturer would ask questions and respond to them on his own in an effort to ensure that students understood the concept thoroughly. This affected the delivery of content in the PT group. As a result, The conceptual understanding was minimal since the students' were not provided with visual images and technology. The lecturer was demonstrating the concept which limited the conceptual understanding of abstract concepts and skills. The conclusion made from here is that TFU approach is better than PT approach in developing conceptual understanding.

Procedural fluency refers to the ability to have skills in carrying out procedures in an accurate and flexible manner. Procedural fluency fluency is essential in the support of conceptual understanding especially in the analysis of rational numbers. Students need to be efficient as well as acurrate in the performance of basic computations with whole numbers. The students in the TFU group were tested on procedural fluency and they showed competitiveness. The students were able to use the computational skill for evaluating the derivative.

The lecturer wanted to know whether the students could use different derivative rules to solve a problem. In this regard, the lecturer gave the students a question that required them to apply the quotient rule in finding the derivative. At first, the students were not able to get the variable of x in the provided question. This was a show that they had not acquired procedural fluency. However, after a few trials, the lecturer noticed that the students had acquired procedural fluency since they were able to calculate the difference quotient using the labels by interpreting the meaning and linking it to the symbols.

In the PT group, the lecturer was also able to observe procedural fluency among the students. In this group, the lecturer had to work with the students in a bid to establish the different techniques to be followed to find the derivative of functions. Students were able to solve more problems independently which was an indication that they had developed procedural fluency. After solving the problems, the students were able to identify the possible errors and the different techniques for solving the problems accurately and efficiently.

Unlike in the TFU group where procedural fluency was developed by the students working on their own with little assistance from the lecturer, in the PT group, procedural fluency was developed by the lecturer guiding the students step by step on what they were supposed to do. For example, in looking at the functions, the lecturer had to tell the students, "also 6.001 is closer to 6, therefore, limit at x approacheds to 3 is f(x) - 6. This shows that the lecturer had to do more for the students to ensure that they understood.

Strategic competence is the other strand of the Kilpatrick's framework. It is often argued that the relationship between the conceptual knowledge and procedural knowledge indicated the strategic competence. Strategic competence was involved when the students answered the questions while exploring. Students were expected to find the value of x at a particular slope. In the TFU group, the students were able to show their competence by working in groups. By doing this, it was noticed that the students were able to solve problems and verify their answers. For instance, when the students were required to plot a cubic graph fx=2x3-3x2, they were able to do so effectively.

It was also observed that the students in the PT group also exhibited strategic competence. However, unlike in the TFU where the students worked mostly on their wn in exhibiting strategic competence, in the PT group, the students had to work with the lecturer to solve the problems posted. Without the lecturer, perhaps, it would have been difficult for the students in the PT group to exhibit strategic competence.

Based on the adaptive reasoning, it was observed that in the TFU group, the students were able to analyse mathematical conditions required. The students' made logical connections between limit and continuity grasped the concept for, they completed the exercise on how to determine the limit and continuity. Since adaptive reasoning is connected to adaptive reasoning, students were able to work in groups and discussed characteristic features of derivative in order to sketch the first derivative. In the PT group, the students were able to show adaptive reasoning by finding the coordinates of A and B. However, the students faced difficulties in identifying the coordinates of C. Where questions demanded logical thinking, they seemed to experience some problems. This shown that TFU was a better approach than PT in testing adaptive reasoning.

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 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 1...