Mathematics: Kilpatrick's Framework for Learning - Essay Sample

Introduction

Kilpatrick's competency framework has been applied various disciplines but it is in mathematics where it seems to have a huge influence. From a mathematical perspective, the Kilpatrick's framework is composed of conceptual understanding, procedural fluency, strategic competency and adaptive reasoning. Conceptual understanding refers to the ability of the student to know more than isolated facts and methods (Taber & Bricheno, 2009). 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.

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In addition, this strand lies in the ability of the student to be able to construct their own knowledge. Additionally, 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 study, the students under TFU were given questions that required them to develop knowledge based on what they already knew to be able to answer them. For example, the students were pushed to think on how to find a better 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 what 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 definitely 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. 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. 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. It is seen through the ways that a student is able to connect discrete bits of information to answer questions. As students are taught, they are introduced to different concepts which they will need to join together to be able to answer related questions. This means that every bit that the student is taught on is very important. The students were given a task to identify the slope of the tangent line. Conceptual understanding 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. 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 lecturer 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 TFU group, a file of angry birds was also used to advance conceptual understanding. The file of angry birds was used to five a dynamic representation on how the tangent line if formed from the secant line. In this exercise, the students were expected to visually interpret the changing of the slope of the secant line as the points came closer, an exercise that they were able to perform perfectly. The ability of the students to do this was a proof of conceptual understanding. Besides, conceptual understanding in the TFU group 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. To show the conceptual connection of the way in which the average rate is linked to the instantaneous rate of change at point x by bringing the points closer which is same as the slope of the tangent line, the GEOGEBRA file came in handy.

In the conceptual understanding, the lecturer asked the students questions that they responded to. The lecturer drew a graph and asked the students whether the point he touched was a straight line and the students responded that it was not a straight line but a parabola. Additionally, the lecturer chose another point which was h distance away from a point, since we need two points to calculate the slope of a secant line. He then asked the question, "Can you calculate the slope of secant line using the labels given? "The students were able to calculate the slope on their own. This was an indication of conceptual understanding.

In the TFU group, the use of conceptual activities was necessary in the contexualisation of the learning process. The students were able to recognise that the limit of average rate of change at smaller and smaller intervals and were able to conclude derivative as the small-scale behaviour at a point. Conceptual understanding was also evident when the students worked as groups to identify various derivative graphs. The lecturer allowed the students to type their own function in the input and learn how to draw a derivative graph by using the slope of the original graph. After drawing the graph, the students in the TFU group were able to type trigonometric and polynomial functions to observe the shape of the derivative graph related to the original graph. The lecturer held a conversation with the students that was meant to test the level of understanding of the students. The lecturer took the students through the process of tracing a point in the original graph using the trace tool.

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. The use of visual images and technology is both helpful in making it easier for the lecturer to teach and also enhancing better understanding of the concepts.

Procedural fluency refers to the ability to have skills in carrying out procedures in an accurate and flexible manner. Procedural fluency is essential in the support of conceptual understanding especially in the analysis of rational numbers. Students need to be efficient as well as accurate 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 approached 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. Strategic competence 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.

The TFU students were also encouraged to work in groups so that they could benefit from each other. The rationale behind this is that when students work in groups, they are able to help each other understand certain concepts as well as remember those that they had already learnt and be able to apply them. When the students did the assigned questions, the compared their answers to their peers and this improved their ability to solve various problems assigned to them. Group work was found to be a good way of enhancing strategic competence since the students are able to clarify issues among themselves even without involving the lecturer. Still in those groups, the students were able to explain their procedure as well as verify their answer. In the PT group, the strategic competence was exhibited when the students were able to formulate the idea that at maximum and minimum, the derivative is zero.

As an additional means of testing their strategic competence, the students were tested on the application of differentiation. The students revised as they had been advised to do and tried formulate the procedures that they were to use in the sketching of the graph of cubic function. As stated earlier, strategic competence refers to the ability of the student...