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Transcript of 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.