Abstract Nctm

8/6/2019 Abstract Nctm

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Abstract

The study sought to investigate high school mathematics teachers' conceptual knowledge

regarding topics on calculus. Forty high school mathematics teachers in the directorate of

education in Riyadh, Saudi Arabia were involved in the study. The instrument used was

chosen questions from "Assessing conceptual understanding in the calculus sequence." The

test consisted of 14 open-ended items. The findings revealed that mathematics teachers' level

of conceptual knowledge is low- to-average. The teachers displayed that they were unable to

use simple facts and relations regarding concepts of calculus when solving conceptual tasks

as presented in new context. The mathematics teachers tended to often see different concepts

of calculus as separate ones, and were not able to often link between these concepts to reach

logical conclusions.

Introduction

Over the past two decades, the question of teachers' knowledge of a subject matter

they teach has become a focus of interest to policy makers and educators (Heather, Brain, and

Deborah, 2005; Shulman, 1986). Teachers need to acknowledge and thoroughly understand

the mathematical concepts that they teach (Zakaia, Zaini, 2009). Studies have shown that

most teachers do not possess a good understanding of content in the subjects that they teach

(Frykholm, 2000; Ibrahim, 2003; Wilson et al. 2001). The teachers must deliberately

encourage their students to solve problems in different ways in order to develop connected

mathematical knowledge, and allow students to present their own multiple solutions to a task

even if this type of activity was not planned. Teachers who understand different

representations of mathematical concepts are able to use these representations to deepen

students' understanding of these concepts (AlSalouli, 2005; Leikin, Leava-Waynberg, 2007).

Moreover, deep knowledge of the content would help teachers to search for non-traditional

solutions to the tasks they bring to their students, and solve these tasks in several ways in


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which a lot of creativity and innovation (Leung & Park, 2002). On the other hand, Silver et

al. (2005) argue that the knowledge of teachers might limit use of multiple solutions in the

classroom. Furthermore, teachers who have little knowledge of mathematics content usually

provide incomplete or distorted concepts, and focus on the procedures more focused on

deepening the understanding of mathematical concepts (Leung & Park, 2002).

According to Faulkenbray (2003), conceptual knowledge refers to the knowledge that

is rich in relationships and refers to the underlying structure of mathematics and connecting

between ideas that explain and give meaning to mathematical procedures. In this regard refers

Kifoat (Kifowit, 2004) that knowledge is the conceptual highlighting by the ability of learners

to access the circulars through a variety of special situations, and apply mathematical ideas in

new situations, and the link between old ideas and new ideas, and the ability to solve math

problems more than way (algebraically, numerically, visually ,....).

Theoretical and literature

In the view of just (Toh, 2009) that the difficulties faced by teachers in the concepts of

calculus (Calculus) may be formed to have are students, and in this sense, the study of the

current difficulties faced by students in the concepts of calculus may be a good introduction

to the study of the difficulties faced by teachers in this concepts. According to Godson and

Nahmora (Judson & Nishimori, 2005) that the lack of clarity of the concept of function

(Function) for many of the students may cause misunderstanding of the matter in resolving

the issues related to applications of calculus. Indicates Iuskn (Uskin, 2003) that students can

improve their achievement in calculus if they were given the concepts of Palmtbainat and

totals (Summation) and other algebraic concepts at an early stage of their studies. Indicates

Akkok and Hailt (Akkoc & Huillet, 2005) that mathematics teachers have misconceptions


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about the concept of the end, largely due to the presence of a clear gap between the definition

of the concept of the end and is held by teachers about the concept. This is confirmed Hillat

(Huillet, 2005), who believes that mathematics teachers have many difficulties related to the

concepts of end and contact functions.

And confirms Morris (Morris, 1999) that secondary teachers are usually focused on the

actions during their teaching to the issue of calculus, and therefore not surprising that ignores

students the conceptual differentiation and focus on the actions and accounts, and finish their

studies and they do not have very little understanding of the conceptual to this thread . In this

context, considers just (Toh, 2007) that the concepts of Baltfadil such as the concept of the

derivative (Derivative) are the concepts are very important even for people to non-specialists

in mathematics, and suggests that mathematical knowledge important for students of non-

athletes (Non-Mathematics) is the knowledge of the conceptual and not procedural

knowledge.

The problem of the study:

From the previous view is clear the importance of studying the extent to which teachers of

mathematical knowledge on the content they teach because of its clear impact on their

practices and direct teaching, and assessment methods they use, and in many cases to collect

their students. The new challenges facing the teacher He had a deep knowledge of sports

content over the proceedings, to dive into the concepts. And in coordinating our efforts to

deepen teachers' knowledge content sports, it is important to define clearly the point at which

stopped at teachers in their understanding of this content, and provide them with

opportunities to move forward in their understanding. Since many of the studies are signs that

teachers lack understanding of the conceptual (Conceptual Understanding) for many of the

topics in mathematics, and show less interest in the development of the knowledge concept to


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their students, and they spend the most time in the teaching of skills, algorithms, and

procedures (Porter, 1989; Ball, 1990; Coony, 1994; Attorps, 2003), it raises an important

question about the extent to which teachers of the mathematical knowledge of the

mathematical content they teach.

From this point of this study is trying to explore the conceptual knowledge (Conceptual

Knowledge) on the subject of the calculus mathematics teachers in secondary school.

Sample:

The sample consisted of 40 high school mathematics teachers; all of whom were male

respondents. Twenty five of the teachers had master's degrees and fifteen had bachelor's

degrees. The teachers had mathematics backgrounds and they are teaching mathematics at

high schools. They teach general mathematics to tenth and eleventh and as well as calculus to

twelfth grade. This group of teachers was selected because they were having a training

session in the directorate of education for male in Riyadh, Saudi Arabia, and the researchers

took this opportunity to give them this test.

Instrument:

The conceptual knowledge test by Kifowit (2004) was used to assess conceptual

understanding in the calculus. This test had 38 questions. The researchers chose only 14

questions to be given to the teachers, believing these questions were almost included in the

textbooks they teach. These items had been translated by the researchers. The test consisted

of conceptual questions regarding calculus such as " If lim ( ) 50x

f x

and ( )f x is positive for

allx , what is lim ( )x

f x

? ( Assume this limit exist). Explain your answer with a picture."


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The test went through several piloting stages where a number of mathematics

professors and three expert teachers who have more than 10 year's experience in teaching,

commented on the clarity of the contents of the test, linguistic confusion, as well as the test

format. A few changes were made based upon the comments and suggestions in order to suit

the textbook used for teaching mathematics in Saudi Arabia. Each question was given a score

of 0 to 4 according to a rubrics modified from Faulkenberry (2003). The teachers scored in

the range of 0 to 56. The reliability coefficient was found to be 0.84. Thus, the test was found

to be reliable.

Findings of the study

In general, 3 (7%) teachers achieved a score of 45 to 56 and were categorized as having high

levels of conceptual knowledge. Meanwhile, 22 (55%) were considered average, with score

ranging from 37 to 44, and 22 (27.5%) scored 23 to 36 and were considered low achievers,

while 4 (10%) were considered very low, with scores from 19 to 22 out of all score of 56.

Table (1) teachers' level of conceptual understanding

Level of conceptual knowledge Number in sample

High ( 45-56)

Average (37-44)

Low (23-36)

Very low (19-22)

3 (7%)

22 (55%)

11 (27.5)

4 (10%)

The teachers achieved an overall high-low average, with score 36.67 out of 56. It was found

that the highest mean scores, 3.7 of a possible 4, which requires the teachers to find limit of a

function at a point through drawing the function curve. The second highest mean score was

3.63, which requires the teachers to find the points on the function curve when the tangent is

horizontal. Meanwhile, the lower item had a mean score of 1.18, which requires the teachers

when giving an amount of things to determine which of them represent a specific amount.

Discussion:


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The response given by the teachers in the conceptual knowledge test produced a low to

average performance. The mean score was 36 out of a total of 56 points. The teachers could

not be able to use the simple facts and relations when solving conceptual tasks as they

presented in a different context. The following item showed clearly the teachers' performance

on this test. For instance, "The following figure shows the graph of a function and its

derivative. Which is which? Give at least two reasons to support your conclusion." This item

requires the use of the relation between the signal of first derivative of a function and the

interval of increasing and decreasing. The majority (62%) of the teachers could not use the

simple relations to determine which of the curves represent the function and which represents

the derivative. These findings are similar to that of Toh (2009) that show the teachers require

deep understanding and build their conceptual knowledge in order to have impact on the

students.

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