Curriculum Theories In Relation To The Research Education Essay

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Introduction

The objective of this chapter is to provide a review of relevant literature on the teaching of Trigonometry and the use of ICT in the teaching of Mathematics. In particular, this chapter addresses the theories behind the use of ICT, students' background knowledge, students' low achievements in the topic and the reasons to use ICT as a teaching aid.

Curriculum Theories in Relation to the Research

Piaget's Stages of Cognitive Development [1] 

Piaget's stage 4 of Cognitive Development is referred to as the Formal Operation stage where the main focus is adolescence up till adulthood. The students at this stage of development should be able to use symbols in relation to abstract concepts. They are able to think about multiple variables in a systematic approach and also formulate hypotheses.

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Piaget's Theory can be adapted into the educational field through the careful and planned adaptation of instructions catered specially to the learner's developmental level. The instructions given to the students must be consistent throughout the teaching process according to their level. Piaget also recommended the use of concrete "hands-on" experience to help students learn such as concrete props or visual aids.

Discovery Learning not only allows the students to explore and experiment on the software but also encourage new understandings either on their own or through the collaboration between a small group of students of different cognitive levels. Discovery Learning will provide an opportunity for the lower ability students to advance to a more mature level of understanding.

Through exploring and experimenting, the students are more likely to remember concepts and knowledge which they discovered on their own in contrast to the traditional method where the teacher is the one transmitting all the information to the students.

Vygotsky's Social Development Theory [2] 

Vygotsky's Social Development Theory is one of the foundations for Constructivism. His theory asserts three major themes i.e.:

Social Interaction

Social interaction plays a fundamental role in the development of cognition. Vygotsky stated that " Every function in the child's cultural development appears twice: first, on the social level, and later, on the individual level; first, between people (interpsychological) and then inside the child (intrapsychological). This applies equally to voluntary attention, to logical memory, and the formation of concepts. All the higher functions originate as actual relationships between individuals."

More Knowledgeable Other ( MKO )

The MKO refers to anyone who has a better understanding or a person who possesses a higher ability level than the learner with respect to a particular task. The teacher is the usual MKO in a classroom context.

Zone of Proximal Development ( ZPD )

The Zone of Proximal Development or ZPD is defined as the distance between a student's ability to perform a task under adult guidance and / or with peer collaboration and the student's ability to solve the problem independently i.e. the range of skill that can be developed with adult guidance or peer collaboration exceeds what can be attained alone.

Vygotsky believed that learning occurred in this zone and that the full development of ZPD depends upon full social interaction. According to Vygotsky, humans use tools that develop from a culture, such as speech and writing, to mediate their social environments. Initially children develop these tools to serve solely as social functions, ways to communicate needs. Vygotsky believed that the internalization of these tools led to higher thinking skills.

Vygotsky's Social Development Theory has several key advantages or benefits when applied. Vygotsky's theory helps to promote learning contexts in which students play an active role in learning. Therefore by using ICT based activities, the learning in the classroom is directed towards student-centered learning.

The role of the teacher at this point is to collaborate with his / her students in order to help facilitate learning in the classroom. The teacher will be there to assist the students when necessary. Learning therefore becomes a reciprocal experience for both the students as well as the teacher.

Dewey's Learning By Doing [3] 

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According to John Dewey, students learn by "directed living," with an emphasis on workshop-type projects so that learning is combined with concrete activity and practical relevance. He rejected the practice of rote learning, which was the common mode of instruction in his day. Dewey's ideas lie at the heart of the constructivist curriculum. Students must be engaged in meaningful and relevant activities, which allow them to apply the concepts they are endeavoring to learn. Hands-on projects are the key to creating authentic learning experiences.

Kolb's Experiential Learning Theory [4] 

David A. Kolb's Experiential Learning Theory is build upon the earlier works of John Dewey and Kurt Levin, he believed that " Learning is the process whereby knowledge is created through the transformation of experience ". His theory sets out four-stage learning styles which are based on a four-stage learning cycle. Kolb's theory offer a great deal of benefit for researchers because it offers both a way to understand individual people's different learning styles and also an explanation of a cycle of experiential learning that apply to everyone.

The four-stage learning cycle and learning styles can be summarized into a two-by-two matrix as shown below.

Doing

( Active Experimentation)

Watching

( Reflective Observation)

Feeling

( Concrete Experience )

Accommodating

Diverging

Thinking

( Abstract Conceptualization )

Converging

Assimilating

The four-type definition of learning styles can be briefly explained as follows.

Diverging ( Feeling and Watching )

This means that students who diverge learn better when allowed observing and collecting a wide range of information from different perspectives. These students prefer to watch rather than do. Such students prefer to work in groups.

Assimilating ( Thinking and Watching )

The assimilators learn better when presented with sound logical theories to consider. The students who fall under this learning style require good clear explanation rather than practical opportunity.

Converging ( Thinking and Doing )

They who are able to converge learn better when provided with practical applications of concepts and theories. The students prefer technical tasks which allow them to use their learning to find solutions.

Accommodating ( Feeling and Doing )

Accommodators are those that learn better when provided with "hand-on" experiences. This learning style is useful in roles that require actions and initiatives. Students under this category of learning style prefer to work in teams to complete tasks.

Background Knowledge of Students

Students who wish to pursue Advanced Level Mathematics generally must meet the requirements of obtaining at least a Credit 6 for Mathematics Syllabus 'D' regardless if they have done Additional Mathematics or not in their Ordinary Level Cambridge Examinations. In my opinion, the minimum requirement is basically giving the students an impression that Advanced Level Mathematics is easy, which is not the case at all.

The basic Trigonometry knowledge that a student should have acquired during their Form 5 level, is basically the Trigonometric Rules for a right-angled triangle i.e. SOH CAH TOA.

Evidence of Low Achievement on Trigonometry

Having taught Advanced Level Mathematics since 2004 has allowed me to observe and note students' learning difficulties in the Advanced Subsidiary Level (AS Level) topics like Functions, Trigonometry, Series Expansion, Vectors etc. I have described the problems or difficulties that students have encountered in the learning of Trigonometry in Chapter 1.

5Advanced Level Examination Reports by Cambridge (2007 - present) have provided sufficient evidence and indications that students have had difficulties in correctly answering questions under Trigonometry. General comments made by the Cambridge examiners for Trigonometry mainly focused on the students' ability to solve trigonometric equations, graph sketching and proving trigonometric identities.

With regards to solving trigonometric equations, I have selected a question that majority of the students have difficulties in, for instance:

Solve the equationfor.

[November 2009, Paper 11, Q1]

Report:

Although the majority of candidates correctly equatedwith, many weaker candidates automatically replacedbyor by. A surprising number addedtorather than subtract, and a sizeable minority of all candidates only considered angles in the first quadrant.

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With regards to graph sketching, majority of students were unable to correctly graph the given equations or answer the sub-questions such as:

Sketch, on the same diagram, the graphs ofandfor. [November 2011, Paper 12, Q5(i)]

Report:

Despite many curves being too straight orbeing parabolic, most candidates obtained some marks for this part and most obtained the correct end-points for both curves. Many candidates unnecessarily drew accurate graphs and wasted a considerable amount of time.

With regards to proving trigonometric identities, I have selected a question which the majority of students have difficulties in have been chosen.

Prove the identity. [June 2009, Paper 1, Q1]

Report:

… but many others in which candidates were unable to obtain a correct common denominator or to obtain a correct numerator for the two fractions on the left-hand side of the identity… It should be pointed out to candidates that it is not sufficient to express…without the intermediate line of working.

Why Use ICT?

From years' experience of teaching Advanced Level Mathematics, I have to admit that I have been practicing the traditional method of teaching i.e. Chalk and Talk method. Throughout the years, results obtained by my students have been above average but not up to my expectations. Students have often confided in me that lessons taught were fairly easy to understand but they are just unable to put it to paper.

With the readily available educational software as well as the easy use of Internet to search for educational resources, many educators have created and produced activities that can help enhance the students' understanding of Mathematical concepts. Using such software will allow the students to observe the sketching of graphs and understand concepts with the visualization obtained from graphing software.

The use of a technology resource could provide the teachers with additional support in the form of sequencing and scaffolding (John and Catherine, 2009). John and Catherine explained scaffolding as giving the students temporary support by providing learning materials which enable them to achieve a desired performance. Scaffolding will gradually reduce and students are able to work independently once sequenced practice is put into action. In addition to the above, ICT will increase the teachers' willingness to provide remediation once students could independently complete the assigned tasks.

There are reports on the advantages of the use of ICT into teaching. One reported the success in using multimedia approach such as graphing software or Power-point presentation to generate the students' conceptual understanding (Ferrer, 2002) [6] . Selinger (2004) [7] claimed the quality of education with the use of ICT can be improved as multimedia helps in illustrating and explaining difficult concepts which could not be done using the traditional teaching approach.

Arellano (2002) [8] in her study found that Information Communications Technology (ICT) is commonly used for literacy. Teachers should go beyond just using ICT for teaching to ensure that their students have adequate ICT skills but also to cultivate a culture in learning where there is student empowerment. She also stated using ICT to do investigations and explorations allow students' role to change from merely passive receivers of knowledge to active processors of knowledge.

Doing so will also help equip students with the capability of reflective thinking including the ability to think critically and creatively as they make sense of their lessons. By doing so will only then give the students a feeling of self-owned knowledge.

However the amount of advantages of using ICT into the teaching, Leong (2008) [9] warned the danger of teachers having to switch from the traditional "chalk-and-talk" method with "drill-and-practice" to "show-and-tell" with "interactive drill-and-practice" which could become "drill-and-kill". This statement can also be supported by researches being done on the Bruneian teachers' perception of teaching their lessons using ICT.

Confidence in using ICT to teach seemed to be a major issue as to why there is a lack of implementation of ICT into classroom teachings (Chong & Wong, 2004) [10] . Sallimah (2005) [11] found that 27% of variance in intention to implement ICT into classroom teachings was affected mainly by the teacher's attitudes such as liking, enjoying etc.

She also stated in her research that the ICT implementation was directly influenced by perceived behavioral control such as being able to use ICT, success and effectiveness etc.

Susilawati and Kyeleve (2005) [12] reported through her interview with Mathematics teachers in Brunei, although most schools have two or three ICT laboratories with LCD units and interactive whiteboards, most teachers said that there was a lack of specific software for teaching Mathematics , logistical problems in using the ICT laboratories, lack in preparation time and knowledge.

ICT and Mathematics Education

Information Communications Technology (ICT) tools will also provide learning opportunities for the students to develop the conceptual foundation required to better understand Trigonometry.

A research study on "Effectiveness of Information and Communication Technology (ICT) in Teaching Mathematics at Secondary Level" was carried out by researchers in Pakistan (Safdar, Yousuf, Parveen and Behlol, 2011).

This study was carried out by the researchers to determine the effectiveness of Information and Communication Technology (ICT) as compared to the traditional method of teaching on the students' academic achievements in Mathematics at secondary level in Pakistan. The researchers also aimed to examine the effectiveness of ICT in contrast to traditional method on academic achievements of students in Mathematics in both public and private sectors at secondary level.

A sample consisted of one hundred and twenty students studying mathematics in class IX at two selected schools each from the public and private sectors. The selected schools have the same Mathematics syllabus content at secondary level and had computer lab facilities.

The researchers also concluded that students from the private sectors showed better performance than students from the public sectors after ICT was used as a teaching strategy because of:

The technological facilities were readily available at school / home.

More dedication during the lesson.

The researchers have definitely found the answers to their research questions regarding how effective was ICT on the students' academic achievements in contrast to the traditional method of teaching. Results obtained showed the effectiveness of ICT in the teaching of Mathematics as a general context but it all depended on a few variables like technologies available at school and home, dedication from the students during the lesson etc. [13] 

Studies on the Use of ICT in teaching Sine and Cosine Graphs in the Advanced Level or Trigonometry as a whole have not been done in Brunei. However, a similar study was conducted by Y. T. Leong in Sarawak, Malaysia. The topic of research done by Leong was entitled "Effective and Affective Teaching of Trigonometric Graphs Using ICT and Acting".

The main objectives of the research were to evaluate the various effective possible solutions in overcoming the problems through the use of acting and different ICT programs and also to examine its effectiveness in improving the competency and interest of the students in the sketching of Trigonometric Graphs in Additional Mathematics.

The sample chosen was Form 5 Additional Mathematics students and the instruments used by the researcher to collect the data were pre-test, post-test and questionnaire which was given to the students.

A few of the software used by the researcher for his study were Microsoft Excel, Microsoft Power-Point, Geometer Sketchpad and Hot-potatoes.

The researcher stated the benefits of implementing ICT into the teaching by analyzing the questionnaire given to the students which included teaching and learning become more effective and student-centered, using ICT makes learning fun etc.

Finally, the researcher concluded that such an implementation will make the delivery process in the classroom/computer room more effective and affective because of its benefits to both the students and the teacher.

Lu (2008) did a study on "Linking Geometry and Algebra: A Multiple-case Study of Upper Secondary Mathematics Teachers' Conceptions and Practices of Geogebra in England and Taiwan". Geogebra is a dynamic geometry software. During the study, she interviewed 2 English and 2 Taiwanese teachers where she concluded that teachers viewed Geogebra as an educational tool for various purposes. These purposes included preparations of teaching materials and assessments, presentations, explorations, investigations and many more.

The teachers from Taiwan thought of Geogebra as an entertainment tool used mainly for demonstration purposes and acted as an entertainment tool for students.

Data obtained by Lu categorized the teachers into three different types:

Unskilled Teachers

These teachers are not exposed to using technology and would only use Geogebra to prepare teaching lessons and demonstrations.

Technology-skilled Teachers

These teachers are skillful in using technology and willing to expand their knowledge on the Geogebra software.

Geogebra Advanced-skilled Teachers

Classified as the experts who prepared interactive lessons that encouraged students to investigate and explore with the activities given. Teachers involved in the study also commented that Geogebra is not applicable to be implemented in all topics such as topics which require 3D visualization. Other preferred choice of software included Archimedes 3D, Sketchup or Autograph. [14] 

I have decided to review on these three literatures / journals because my research exercise is similar in terms of objectives which will be further discussed in Chapter 3.

Summary

Majority of students have difficulties in Trigonometry in the Advanced Level. Students are required to understand the concepts and apply them into various aspects such as graph sketching, solving trigonometric and proving identities. The use of mathematical software and online resources that are readily available will help enhance the quality of teaching and learning in the classrooms and finally improve the students' achievements.