The aim of this chapter is to supply a reappraisal of relevant literature on the instruction of Trigonometry and the usage of ICT in the instruction of Mathematics. In peculiar, this chapter addresses the theories behind the usage of ICT, pupils ‘ background cognition, pupils ‘ low accomplishments in the subject and the grounds to utilize ICT as a instruction assistance.
Curriculum Theories in Relation to the Research
Piaget ‘s Stages of Cognitive Development[ 1 ]
Piaget ‘s phase 4 of Cognitive Development is referred to as the Formal Operation phase where the chief focal point is adolescence up till maturity. The pupils at this phase of development should be able to utilize symbols in relation to abstract constructs. They are able to believe about multiple variables in a systematic attack and besides formulate hypotheses.
Piaget ‘s Theory can be adapted into the educational field through the careful and planned version of instructions catered specially to the scholar ‘s developmental degree. The instructions given to the pupils must be consistent throughout the instruction procedure harmonizing to their degree. Piaget besides recommended the usage of concrete “ hands-on ” experience to assist pupils larn such as concrete props or ocular AIDSs.
Discovery Learning non merely allows the pupils to research and experiment on the package but besides promote new apprehensions either on their ain or through the coaction between a little group of pupils of different cognitive degrees. Discovery Learning will supply an chance for the lower ability pupils to progress to a more mature degree of apprehension.
Through exploring and experimenting, the pupils are more likely to retrieve constructs and cognition which they discovered on their ain in contrast to the traditional method where the instructor is the one conveying all the information to the pupils.
Vygotsky ‘s Social Development Theory[ 2 ]
Vygotsky ‘s Social Development Theory is one of the foundations for Constructivism. His theory asserts three major subjects i.e. :
Social interaction plays a cardinal function in the development of knowledge. Vygotsky stated that “ Every map in the kid ‘s cultural development appears twice: first, on the societal degree, and subsequently, on the single degree ; foremost, between people ( interpsychological ) and so inside the kid ( intrapsychological ) . This applies every bit to voluntary attending, to logical memory, and the formation of constructs. All the higher maps originate as existent relationships between persons. ”
More Knowing Other ( MKO )
The MKO refers to anyone who has a better apprehension or a individual who possesses a higher ability degree than the scholar with regard to a peculiar undertaking. The instructor is the usual MKO in a schoolroom context.
Zone of Proximal Development ( ZPD )
The Zone of Proximal Development or ZPD is defined as the distance between a pupil ‘s ability to execute a undertaking under grownup counsel and / or with peer coaction and the pupil ‘s ability to work out the job independently i.e. the scope of accomplishment that can be developed with big counsel or peer coaction exceeds what can be attained entirely.
Vygotsky believed that larning occurred in this zone and that the full development of ZPD depends upon full societal interaction. Harmonizing to Vygotsky, worlds use tools that develop from a civilization, such as address and authorship, to intercede their societal environments. Initially kids develop these tools to function entirely as societal maps, ways to pass on demands. Vygotsky believed that the internalisation of these tools led to higher believing accomplishments.
Vygotsky ‘s Social Development Theory has several cardinal advantages or benefits when applied. Vygotsky ‘s theory helps to advance larning contexts in which pupils play an active function in larning. Therefore by utilizing ICT based activities, the acquisition in the schoolroom is directed towards student-centered acquisition.
The function of the instructor at this point is to join forces with his / her pupils in order to assist ease acquisition in the schoolroom. The instructor will be at that place to help the pupils when necessary. Learning hence becomes a mutual experience for both the pupils every bit good as the instructor.
Dewey ‘s Learning By Making[ 3 ]
Harmonizing to John Dewey, pupils learn by “ directed life, ” with an accent on workshop-type undertakings so that acquisition is combined with concrete activity and practical relevancy. He rejected the pattern of rote acquisition, which was the common manner of direction in his twenty-four hours. Dewey ‘s thoughts lie at the bosom of the constructivist course of study. Students must be engaged in meaningful and relevant activities, which allow them to use the constructs they are endeavouring to larn. Hands-on undertakings are the cardinal to making reliable acquisition experiences.
Kolb ‘s Experiential Learning Theory[ 4 ]
David A. Kolb ‘s Experiential Learning Theory is build upon the earlier plants of John Dewey and Kurt Levin, he believed that “ Learning is the procedure whereby cognition is created through the transmutation of experience “ . His theory sets out four-stage acquisition manners which are based on a four-stage acquisition rhythm. Kolb ‘s theory offer a great trade of benefit for research workers because it offers both a manner to understand single people ‘s different acquisition manners and besides an account of a rhythm of experiential acquisition that apply to everyone.
The four-stage acquisition rhythm and acquisition manners can be summarized into a two-by-two matrix as shown below.
( Active Experiment )
( Brooding Observation )
( Concrete Experience )
( Abstract Conceptualization )
The four-type definition of larning manners can be briefly explained as follows.
Diverging ( Feeling and Watching )
This means that pupils who diverge learn better when allowed observing and roll uping a broad scope of information from different positions. These pupils prefer to watch instead than make. Such pupils prefer to work in groups.
Assimilating ( Thinking and Watching )
The learners learn better when presented with sound logical theories to see. The pupils who fall under this learning manner necessitate good clear account instead than practical chance.
Converging ( Thinking and Doing )
They who are able to meet larn better when provided with practical applications of constructs and theories. The pupils prefer proficient undertakings which allow them to utilize their acquisition to happen solutions.
Accommodating ( Feeling and Doing )
Obligers are those that learn better when provided with “ hand-on ” experiences. This learning manner is utile in functions that require actions and enterprises. Students under this class of larning manner prefer to work in squads to finish undertakings.
Background Knowledge of Students
Students who wish to prosecute Advanced Level Mathematics by and large must run into the demands of obtaining at least a Credit 6 for Mathematicss Syllabus ‘D ‘ regardless if they have done Additional Mathematics or non in their Ordinary Level Cambridge Examinations. In my sentiment, the minimal demand is fundamentally giving the pupils an feeling that Advanced Level Mathematics is easy, which is non the instance at all.
The basic Trigonometry cognition that a pupil should hold acquired during their Form 5 degree, is fundamentally the Trigonometric Rules for a right-angled trigon i.e. SOH CAH TOA.
Evidence of Low Achievement on Trigonometry
Having taught Advanced Level Mathematics since 2004 has allowed me to detect and observe pupils ‘ larning troubles in the Advanced Subsidiary Level ( AS Level ) subjects like Functions, Trigonometry, Series Expansion, Vectors etc. I have described the jobs or troubles that pupils have encountered in the acquisition of Trigonometry in Chapter 1.
5Advanced Level Examination Reports by Cambridge ( 2007 – nowadays ) have provided sufficient grounds and indicants that pupils have had troubles in right replying inquiries under Trigonometry. General remarks made by the Cambridge testers for Trigonometry chiefly focused on the pupils ‘ ability to work out trigonometric equations, graph sketching and turn outing trigonometric individualities.
With respects to work outing trigonometric equations, I have selected a inquiry that bulk of the pupils have troubles in, for case:
Solve the equationfor.
[ November 2009, Paper 11, Q1 ]
Although the bulk of campaigners right equatedwith, many weaker campaigners automatically replacedbyor by. A surprising figure addedtorather than subtract, and a ample minority of all campaigners merely considered angles in the first quarter-circle.
With respects to chart sketching, bulk of pupils were unable to right chart the given equations or reply the sub-questions such as:
Sketch, on the same diagram, the graphs ofandfor. [ November 2011, Paper 12, Q5 ( I ) ]
Despite many curves being excessively consecutive orbeing parabolic, most campaigners obtained some Markss for this portion and most obtained the right end-points for both curves. Many campaigners unnecessarily drew accurate graphs and wasted a considerable sum of clip.
With respects to turn outing trigonometric individualities, I have selected a inquiry which the bulk of pupils have troubles in have been chosen.
Prove the individuality. [ June 2009, Paper 1, Q1 ]
aˆ¦ but many others in which campaigners were unable to obtain a right common denominator or to obtain a right numerator for the two fractions on the left-hand side of the identityaˆ¦ It should be pointed out to campaigners that it is non sufficient to expressaˆ¦without the intermediate line of working.
Why Use ICT?
From old ages ‘ experience of learning Advanced Level Mathematics, I have to acknowledge that I have been practising the traditional method of learning i.e. Chalk and Talk method. Throughout the old ages, consequences obtained by my pupils have been above norm but non up to my outlooks. Students have frequently confided in me that lessons taught were reasonably easy to understand but they are merely unable to set it to paper.
With the readily available educational package every bit good as the easy usage of Internet to seek for educational resources, many pedagogues have created and produced activities that can assist heighten the pupils ‘ apprehension of Mathematical constructs. Using such package will let the pupils to detect the sketching of graphs and understand constructs with the visual image obtained from charting package.
The usage of a engineering resource could supply the instructors with extra support in the signifier of sequencing and scaffolding ( John and Catherine, 2009 ) . John and Catherine explained scaffolding as giving the pupils impermanent support by supplying larning stuffs which enable them to accomplish a coveted public presentation. Scaffolding will bit by bit cut down and pupils are able to work independently one time sequenced pattern is put into action. In add-on to the above, ICT will increase the instructors ‘ willingness to supply redress once pupils could independently finish the assigned undertakings.
There are studies on the advantages of the usage of ICT into learning. One reported the success in utilizing multimedia attack such as charting package or Power-point presentation to bring forth the pupils ‘ conceptual apprehension ( Ferrer, 2002 )[ 6 ]. Selinger ( 2004 )[ 7 ]claimed the quality of instruction with the usage of ICT can be improved as multimedia helps in exemplifying and explicating hard constructs which could non be done utilizing the traditional instruction attack.
Arellano ( 2002 )[ 8 ]in her survey found that Information Communications Technology ( ICT ) is normally used for literacy. Teachers should travel beyond merely utilizing ICT for learning to guarantee that their pupils have equal ICT accomplishments but besides to cultivate a civilization in larning where there is student authorization. She besides stated utilizing ICT to make probes and geographic expeditions allow pupils ‘ function to alter from simply inactive receiving systems of cognition to active processors of cognition.
Making so will besides assist equip pupils with the capableness of brooding thought including the ability to believe critically and creatively as they make sense of their lessons. By making so will merely so give the pupils a feeling of self-owned cognition.
However the sum of advantages of utilizing ICT into the instruction, Leong ( 2008 )[ 9 ]warned the danger of instructors holding to exchange from the traditional “ chalk-and-talk ” method with “ drill-and-practice ” to “ show-and-tell ” with “ synergistic drill-and-practice ” which could go “ drill-and-kill ” . This statement can besides be supported by researches being done on the Bruneian instructors ‘ perceptual experience of learning their lessons utilizing ICT.
Assurance in utilizing ICT to learn seemed to be a major issue as to why there is a deficiency of execution of ICT into schoolroom instructions ( Chong & A ; Wong, 2004 )[ 10 ]. Sallimah ( 2005 )[ 11 ]found that 27 % of discrepancy in purpose to implement ICT into schoolroom instructions was affected chiefly by the instructor ‘s attitudes such as liking, basking etc.
She besides stated in her research that the ICT execution was straight influenced by sensed behavioural control such as being able to utilize ICT, success and effectiveness etc.
Susilawati and Kyeleve ( 2005 )[ 12 ]reported through her interview with Mathematics instructors in Brunei, although most schools have two or three ICT research labs with LCD units and synergistic whiteboards, most instructors said that there was a deficiency of specific package for learning Mathematicss, logistical jobs in utilizing the ICT research labs, deficiency in readying clip and cognition.
ICT and Mathematics Education
Information Communications Technology ( ICT ) tools will besides supply larning chances for the pupils to develop the conceptual foundation required to better understand Trigonometry.
A research survey on “ Effectiveness of Information and Communication Technology ( ICT ) in Teaching Mathematicss at Secondary Degree ” was carried out by research workers in Pakistan ( Safdar, Yousuf, Parveen and Behlol, 2011 ) .
This survey was carried out by the research workers to find the effectivity of Information and Communication Technology ( ICT ) as compared to the traditional method of learning on the pupils ‘ academic accomplishments in Mathematicss at secondary degree in Pakistan. The research workers besides aimed to analyze the effectivity of ICT in contrast to traditional method on academic accomplishments of pupils in Mathematicss in both public and private sectors at secondary degree.
A sample consisted of one hundred and 20 pupils analyzing mathematics in category IX at two selected schools each from the populace and private sectors. The selected schools have the same Mathematics course of study content at secondary degree and had computing machine lab installations.
The research workers besides concluded that pupils from the private sectors showed better public presentation than pupils from the public sectors after ICT was used as a instruction scheme because of:
The technological installations were readily available at school / place.
More dedication during the lesson.
The research workers have decidedly found the replies to their research inquiries sing how effectual was ICT on the pupils ‘ academic accomplishments in contrast to the traditional method of learning. Consequences obtained showed the effectivity of ICT in the instruction of Mathematics as a general context but it all depended on a few variables like engineerings available at school and place, dedication from the pupils during the lesson etc.[ 13 ]
Surveies on the Use of ICT in learning Sine and Cosine Graphs in the Advanced Level or Trigonometry as a whole have non been done in Brunei. However, a similar survey was conducted by Y. T. Leong in Sarawak, Malaysia. The subject of research done by Leong was entitled “ Effective and Affectional Teaching of Trigonometric Graphs Using ICT and Acting ” .
The chief aims of the research were to measure the assorted effectual possible solutions in get the better ofing the jobs through the usage of moving and different ICT plans and besides to analyze its effectivity in bettering the competence and involvement of the pupils in the sketching of Trigonometric Graphs in Additional Mathematics.
The sample chosen was Form 5 Additional Mathematics pupils and the instruments used by the research worker to roll up the informations were pre-test, post-test and questionnaire which was given to the pupils.
A few of the package used by the research worker for his survey were Microsoft Excel, Microsoft Power-Point, Geometer Sketchpad and Hot-potatoes.
The research worker stated the benefits of implementing ICT into the instruction by analysing the questionnaire given to the pupils which included instruction and larning go more effectual and student-centered, utilizing ICT makes larning fun etc.
Finally, the research worker concluded that such an execution will do the bringing procedure in the classroom/computer room more effectual and affectional because of its benefits to both the pupils and the instructor.
Lu ( 2008 ) did a survey on “ Associating Geometry and Algebra: A Multiple-case Study of Upper Secondary Mathematics Teachers ‘ Concepts and Practices of Geogebra in England and Taiwan ” . Geogebra is a dynamic geometry package. During the survey, she interviewed 2 English and 2 Chinese instructors where she concluded that instructors viewed Geogebra as an educational tool for assorted intents. These intents included readyings of learning stuffs and appraisals, presentations, geographic expeditions, probes and many more.
The instructors from Taiwan idea of Geogebra as an amusement tool used chiefly for presentation intents and acted as an amusement tool for pupils.
Datas obtained by Lu categorized the instructors into three different types:
These instructors are non exposed to utilizing engineering and would merely utilize Geogebra to fix learning lessons and presentations.
These instructors are adept in utilizing engineering and willing to spread out their cognition on the Geogebra package.
Geogebra Advanced-skilled Teachers
Classified as the experts who prepared synergistic lessons that encouraged pupils to look into and research with the activities given. Teachers involved in the survey besides commented that Geogebra is non applicable to be implemented in all subjects such as subjects which require 3D visual image. Other preferable pick of package included Archimedes 3D, Sketchup or Autograph.[ 14 ]
I have decided to reexamine on these three literatures / diaries because my research exercising is similar in footings of aims which will be farther discussed in Chapter 3.
Majority of pupils have troubles in Trigonometry in the Advanced Level. Students are required to understand the constructs and use them into assorted facets such as graph sketching, work outing trigonometric and turn outing individualities. The usage of mathematical package and on-line resources that are readily available will assist heighten the quality of instruction and acquisition in the schoolrooms and eventually better the pupils ‘ accomplishments.