Evidence Of Low Achievement On Trigonometry Education Essay

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The chapter will present a literature review of relevant research articles done locally and internationally on the use of ICT in Mathematics education. In particular, this chapter includes information on students' background knowledge, students' low achievements in the topic, and the use of ICT in Mathematics Education including its benefits and drawbacks.

Background Knowledge of Students

Students who wish to pursue Advanced Level Mathematics generally must meet the minimum requirements of obtaining at least a Credit 6 for Mathematics Syllabus 'D' regardless of whether they have done Additional Mathematics or not in their Ordinary Level Cambridge Examinations. In my opinion, such a low 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 every student should have acquired during their Form 5 level would be Trigonometric Rules for a right-angled triangle i.e. SOH CAH TOA. On the other hand, students who have done Additional Mathematics would have a deeper knowledge in Trigonometry compared to those who have not.

Cambridge (2012) requires the students to know the six trigonometric functions of angles of any magnitude namely sine, cosine, tangent, secant, cosecant and cotangent. They should be able to understand the terms "amplitude" and "period", relationships between graphs and sketch graphs of the trigonometric functions. Students who have done Additional Mathematics should be able to use different trigonometric relationships like,andto solve simple trigonometric equations as well as prove trigonometric identities.

Comparison of Trigonometry Syllabus Content between O-Level and A-Level

Similarities between the syllabus content in Trigonometry for both O-Level and A-Level include the sketching of graphs for the sine, cosine and tangent functions, using trigonometric relationships to solve simple trigonometric equations and proving trigonometric identities.

However, students would have to acquire some additional knowledge in the Advanced Level Trigonometry. According to the Advanced Level Mathematics Syllabus from Cambridge (2012), students doing Advanced Level Trigonometry would need to understand the relationship between angles in degrees and radians and be able to sketch graphs in both angular measures. Students would have to use the exact values of the sine, cosine and tangent of,and, and relate these value to other angles, for example. Students should be able to use notations like,andto denote the principal values of the inverse trigonometric relations. Finally, the students should be able to solve trigonometric functions in any specified interval.

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 and Vectors.

Advanced Level Examination Reports by Cambridge (2007 - present) have provided evidence and indications that students have had difficulties in correctly answering questions in Trigonometry. General comments made by the Cambridge examiners about 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]


"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."

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)]


"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]


"… 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."

Based on the extracts taken from the reports provided by Cambridge, it could be seen that majority of students have struggled with Trigonometry as a whole for the past few years. Students were either very weak in the topic or made unnecessary errors in their approach to obtaining the correct solutions.

Students tend to waste their time drawing accurate graphs of trigonometric functions rather than just sketching them which was required in the question. When it comes to proving identities, students have the tendency to skip certain steps of the working.

ICT and Mathematics Education

Ong (2011) also made some recommendations on using Graphing Calculators in the teaching of Trigonometry. One recommendation was that Graphing Calculators can be seen as a catalyst for students' achievements and used as a learning tool with correct and planned implementation.

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.

The researchers found 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 in general, but it depended on a few variables like technologies available at school and home, and dedication from the students during the lesson.

Studies on the Use of ICT in Trigonometry at Advanced Level or Trigonometry as a whole have not been done in Brunei. However, a similar study was conducted by Lim (undated) in Sarawak, Malaysia. The topic of research done by Lim 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 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.

He 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 used in this intervention study. During the study, she interviewed 2 English and 2 Taiwanese teachers and 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.

Benefits and Drawbacks of ICT?

From years' experience of teaching Advanced Level Mathematics, I have to admit that I have been practicing the traditional method of teaching which was "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 ICT 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 be gradually reduced and students are able to work independently once sequenced practice is put into action.

There are reports on the advantages of the use of ICT into teaching. One reports success in using multimedia approach such as graphing software or Power-point presentation to generate the students' conceptual understanding (Ferrer, 2002). Selinger (2004) 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) in her study found that Information Communications Technology (ICT) is commonly used for literacy. She states that teachers should go beyond just using ICT for teaching to ensure that their students have adequate ICT skills and also to cultivate a culture of learning where there is student empowerment. She also stated using ICT to do investigations and explorations allows students' role to change from being 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 give the students a feeling of self-owned knowledge.

Lim (undated) stated the benefits of implementing ICT into the teaching by analyzing a questionnaire given to the students which implied that teaching and learning becomes more effective and student-centered and using ICT makes learning fun.

However, whilst acknowledging the advantages of using ICT in the teaching, Leong (2008) warned of the dangers 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".

Ong (2011) did research on "The Effectiveness of Using Graphing Calculators in the Teaching and Learning of Trigonometry with Year 10 Students in Brunei Darussalam" and concluded that the use of graphing calculators did not improve students' performance. The findings did not support the use of Graphing Calculators in the enhancement of Trigonometry at Year 10 Level. She also stated that the reason behind the ineffectiveness may be due to the short period of implementation and students not having enough practice in using the Graphing Calculators.

Lu (undated) reported that although students' learning could be sped up through the use of ICT, teachers have to be aware that alternatively, misconceptions may be created along the process. She suggested that teachers have to make sure that the students perform each small step carefully, which included shaping and reshaping their mathematical thoughts, and that the dynamic graphical representation was to be used as cognitive prop.

She reported that despite the benefits of Autograph software used in her study, students did not fully understand the graphs due to misunderstandings that had occurred. She recommended that teachers inform their students about the possibilities of technical problems in ICT, and they should be able to think critically whether a graph is correct or otherwise rather than fully trusting the software.

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). Sallimah (2005) found that 27% of variance in the intentions to implement ICT into classroom teachings was affected mainly by the teacher's attitudes such as liking and enjoying.

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

Bingimlas (2009) states that the teachers' lack of confidence and competence and lack of access to teaching resources are some barriers to the successful integration of ICT in teaching and learning environments. Hew and Brush (2007) have analyzed experimental studies ranging from 1995 to 2006 and reported that barriers include knowledge, skills, institution, attitudes, beliefs, assessment and culture.

Susilawati and Kyeleve (2005) through their interviews with Mathematics teachers in Brunei, reported that 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 and lack of preparation time and knowledge.

Kam (2006) stated from his classroom observations and interviews that there was great resistance in the use of ICT by teachers. He questioned this resistance and concluded that the answer lied in the hands of policy makers and school administrators. He also concluded that the success of implementing ICT into teaching should be through a collaborative effort that involves all parties i.e. teachers, principals, policy makers and relevant government agencies.

Dynamic Geometry Software - Autograph

Autograph software is a powerful and interactive software package that operates in three different modes: 1-Dimension (1D), 2-Dimension (2D) and also 3-Dimension (3D). This software package promotes the teaching of different Mathematical aspects such as Statistics, Calculus and many more. It is also the perfect tool that allows teachers to introduce various concepts like graph sketching, transformations, vectors and statistical data representations. The program uses colour and animations that promotes a fun and engaging learning environment for students. (See illustration: Figure 1, overleaf)

Figure 1. Sample of the Autograph Application


This chapter has reviewed the literature and drawn the following conclusions. Comparing the syllabus content for Trigonometry in both the O-Levels and A-Levels has shown a few similarities in terms of graph sketching and solving trigonometric equations. Students doing Advanced Level Mathematics would need to know a more in-depth understanding in the topic in terms of relating the exact values of trigonometric functions for angles like,and to other angles likeand. In addition to that, students were required to be able to solve trigonometric equations in any specified intervals.

Based on the examination reports from Cambridge, the majority of students have difficulties in Trigonometry at Advanced Level. Students need to understand the concepts and apply them into various aspects such as graph sketching, solving trigonometric equations and proving identities.

Researches have provided evidence that 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. However the amount of benefits that could be gained from such implementation, there were evidence that warned teachers on the use of ICT-based activities or software applications as misconceptions may arise during the lessons.

The reviews also provided evidence that teachers avoided using ICT in their lessons due to various reasons which included lack of confidence, time constraints and lack of technical support.