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INTRODUCTION Instruction in all content areas must be designed to meet the needs of all students. Teachers must set high, but reasonable expectations that challenge students ability, while matching each students interests, needs and developmental levels (Baroody & Coslick, 1998). Instruction cannot target only the average ability level. Students significantly above and below the average ability level require specialised instruction to meet their needs. When these needs are not met, both gifted and struggling students will lose motivation, and often perform poorly. They need to have the encouragement especially from teachers and often require an individualised pace to complete a variety of mathematical tasks. Many of the characteristics of curricula for the gifted and struggling students can benefit all students learning, but they become especially important with these diverse learners. Remedial programmes are provided to help students who have difficulties in reading, writing and doing arithmetic. These programmes provide alternative activities for students to overcome these basic problems. By implementing remedial activities, it is hoped that students who struggle to cope during normal instruction can achieve a certain acceptable standard of understanding in mathematics. Enrichment programmes, on the other hand, can be considered as a set of extra mathematical tasks provided by teachers who are more complex, yet interesting and challenging. These activities are different from those activities provided during normal instruction. Students involved in enrichment activities are encouraged to apply and expand their knowledge in mathematics acquired during normal instruction.


ACTIVITY 9.1 Imagine a situation in which you have 25 students sitting in front of you. You are to teach the addition of improper fractions. What would be your teaching strategy? Will you use only one method of instruction? If yes, state why. If not, state why. There are students with various achievements, interests and abilities in a classroom. Therefore, as teachers, you need to design different teaching and learning activities to accommodate every one of them. Some examples of such activities are remedial and enrichment activities. Remedial and enrichment activities are important components of the

teaching and learning process in mathematics. We usually provide students with remedial and enrichment activities especially during group work. Groups with gifted students are provided enrichment activities, while groups with students who struggle to cope with the subject matter will be provided remedial activities. Students who are struggling to meet a certain expectation in mathematics require a curriculum to help them understand and be comfortable with mathematical concepts. Many suffer from mathematics anxiety and have a predisposition to dislike mathematics and lack the motivation to change. Teachers need to work on making instruction meaningful and interesting, getting students involved actively, and providing support and encouragement (Baroody & Coslick,1998). This can help teachers to engage students to learn mathematics, motivate them to achieve and establish high expectations for their learning. Once students have overcome their mathematics anxiety and found the concepts meaningful, they often need instruction that builds concepts from the basic. Prior knowledge must be engaged and they typically benefit from concrete representations of the mathematical concepts. The use of mathematical manipulatives allows remedial students to encounter concrete, physical and practical manifestations of the mathematical concepts being taught. Through establishing a very concrete understanding of the concept, they can begin to expand their understanding to more abstract representations and problems. They can also benefit from a more structured curriculum that offers students the support in finding solutions by providing specific methods in doing so. Remedial students often require more repetition, consistent review and a curriculum that spirals back to show the connections between learned concepts. Teaching and learning materials can be made meaningful by carrying out the learning process with concrete representations and repeating the concepts. Remedial students can benefit from a remedial curriculum to reach the expectations that match their potential.

9.1.1 Objectives of Remedial Activities Usually, there will be some students who cannot understand certain mathematical concepts or skills being taught. According to Baroody & Coslick (1998), these remedial students cannot cope with their teachers. In fact, teachers should work on making instruction meaningful and interesting by planning remedial activities in the classroom. The objectives of having remedial activities in the classroom are to:

Help remedial students build mathematical concepts and acquire mathematical skills so that they do not lag behind other students in the class; Guide these students to apply the mathematics skills acquired to solve practical problems in order to build their confidence and interest; Help them correct the mathematical concepts and mathematical skills that they have wrongly learnt; and Guide them so that they will be able to learn mathematics meaningfully and at their own pace.

You should have a better understanding of the concepts and objectives of remedial activities by now. Thus, you should be able to summarise in your own words the importance of remedial activities in your mathematics lesson.

9.1.2 Factors that Affect Learning among Students Students in a classroom usually will differ from each other in terms of their ability, physical build, intelligence, interest and skills. Thus, some of them seem to be very fast in grasping information and knowledge conveyed to them while some others may only be able to understand certain concepts or skills after going through special remedial activities. At the same time, there are also students who cannot learn at all. Generally, students with learning disabilities are those who really need help to make their learning more meaningful and interesting. This can be achieved by ensuring that they are actively involved in classroom activities. In order to design activities for these students, it is necessary to understand and analyse the factors that hinder them from learning. Some of the factors that affect students learning are:


Individual Factors Individual factors are usually related to socio-economic and social background.

Students from poor families may not have a conducive environment to study in. In fact, they may be working to earn extra income for their family which results in their not having time to study at home. Even if they may have time to study, they are just too tired to do so. These students are not motivated at all by their parents. These are some of the situations that may affect students performance.


Learning Factors Some learning factors that may contribute to students not performing as expected are

unattractive teaching and learning materials used by the teachers, unsuitable instructional strategies used and sometimes the curriculum itself is not appropriate to the cognitive development of these students. If these learning factors are not addressed accordingly, students will not be interested to learn. They will start skipping lessons and later have a high tendency to become school drop-outs.


IQ Factors IQ factors are closely related to brain activities. There are many instances where

students are slow in giving opinions, slow in reading, forget very quickly what they have learned and sometimes find it difficult to understand what is being taught. These factors might be due to hereditary, abnormal birth, brain damage or even malnutrition.

9.2 IDENTFYING REMEDIAL STUDENTS Providing various activities that suit various categories of students in some ways contribute to the improvement of students performance in mathematics. Thus, before teachers start to plan and implement remedial activities, they must be able to identify the strengths and weaknesses of their students. This could be achieved by evaluating and identifying which students have fully learned and understood what has been taught. The following procedures (not in the order of preference) can help teachers in the process of identifying students who require remedial programmes:


Observation Observation is essentially selective. In other words, teachers must identify the

important criteria that they should be looking for in the observation. Good planning and practice is necessary before they can carry out high quality observation. As teachers, it is helpful to plan what to observe and what to record during the actual observation exercise. For example, if teachers want to observe students doing pair work activities, the following questions might help them to record their findings: Is he/she portraying a dominant character? Is he/she persistent with his/her answer? Does he/she listen to his/her friends views?

Does he/she always accept his/her friends idea?


Questioning The questioning technique will usually complement the observations. Asking

questions is an art. The technique needs to be developed and practised before teachers can master and apply it efficiently and effectively. Questions forwarded or set by teachers are considered efficient if they are clear, concise and with logical expressions. While asking questions, it should be noted that teachers must allow appropriate time for questioning and answering.


Interviewing The interviewing technique is a powerful method to identify students strengths and

weaknesses. It involves a combination of skills used in questioning and observing techniques. Interviews are usually conducted with one student at a time in a quiet place. Teachers can get insights about their respondents thinking through interviews. The significant factors in a successful interview are establishing rapport with the respondent, accepting response without being judgemental and encouraging the respondent to talk and explain his/her views. Before starting to interview the students, teachers need a basic plan of what to investigate, what questions to ask and what form of information to be recorded. Reviewing Samples of Students Work Work samples can include written assignments, homework, projects and other products produced by students. By reviewing the work produced by students, teachers can identify the students mistakes. Teachers may ask them to do corrections and if the mistakes still prevail, then there should be some form of remedial activities planned for the students. In fact, taking a closer look at each students work can reveal their strengths and weaknesses. This will help teachers to monitor the performance of their students.



Tests According to Gagne (1965), test is a systematic way to measure changes in an

individuals behaviour. In a classroom condition, a test is conducted to measure changes in behaviour related to students learning activities. In general, it is a measuring instrument to obtain information about students achievement in the cognitive, psychomotor or affective fields.

Test requires students to show explicitly, in the form of activities such as answering questions orally or in written form or through demonstration, the changes in their behaviour related to the concepts or skills after instruction. Normally, during the testing process, observation and formal tests are usually used as a measuring instrument. Thoughtful, well-constructed and correctly analysed tests can tell a lot about students. It is important to recognise that tests alone will not give a complete assessment of students performance, but they can be considered as an important piece of information to the puzzle. Performance tests used in combination with other forms of evaluations can help teachers identify students who are experiencing difficulties learning mathematics. They can then plan remedial activities for the students identified in acquiring and understanding the mathematical concepts taught.


PLANNING A REMEDIAL ACTIVITY The objectives of remedial instruction can only be achieved if the activities are

planned in a sequential and systematic manner. Good planning of remedial activities should be conducted in a sequence as shown in Figure 9.1.

Figure 9.1: Steps in planning remedial activity

9.3.1 Screening and Diagnostic Tests Planning and administering screening test can be considered to be the first step in planning remedial activity. The ultimate role of this test is to identify students who must undergo remedial activity for a particular skill and task identified. Teachers can administer the screening test casually by implementing any one of the techniques discussed in Section 9.2 above (such as observation, questioning, interviewing or by looking at students daily exercises). Once teachers have gathered students who are struggling in learning mathematics, the next task is to identify the specific nature of their difficulties. The diagnosis of the nature of problems faced by students can be done using one or more of the procedures mentioned earlier or by using a specific diagnostic test. Some of the important guidelines for diagnosis in mathematics are as follows: Teachers have to make sure that students noticeable lack of specific mathematical understanding or skills is in fact a true lack of such understanding or skills. Making wrong diagnosis of students abilities would result in high consequences. Teachers must always keep in mind that every one student progresses through several stages of development before reaching an adult conceptual level. This shows that the difficulties faced by students could be caused by the misconceptions gathered through early education. Teachers must not deny the fact that as teachers they might unintentionally have contributed to the complications experienced by students. They must sometimes reflect on their mode of instruction and see if it could be rectified. Teachers must also pay attention to the emotional effects of students in the diagnosis. Their feelings must not be affected and their expectations must be addressed. Teachers must always be patient and sometimes flexible in gathering relevant information to form an accurate and precise picture of students performance. Only then they can plan a suitable remedial activity for the identified difficulty. Teachers must make sure that students are receptive to their method of diagnosis. In other words, teachers must maintain a climate of acceptance while conducting the diagnosis. Teachers must be careful in distinguishing between errors that occur randomly and those that occur more systematically among students. If the errors made during diagnostic test are systematic and not random, they can identify the patterns in the

errors made which will connect to some underlying skills that have not been mastered or some conceptual understandings that have not been properly absorbed.

The following is some of the important guidelines to follow while constructing a diagnostic test: Identify a particular task or skill that students are very weak in (for example, subtraction involving three digits). Identify the sub-tasks or sub-skills involved and arrange them in a hierarchy of increasing complexity (from simple to more complex sub-task). Construct four (4) items for each sub-tasks or sub-skills that has been identified. Distribute the above mentioned items randomly in a test. Do not impose a time limit. Actually, a diagnostic test is to determine students strengths and weaknesses in a particular skill of mathematics and not how fast they could complete the task. A student who answered incorrectly for more than one of the four tested items for a particular sub-task or sub-skill may need to go for remedial activity on that sub-task or sub-skill concerned. Therefore, it is important for teachers to carefully analyse students responses for each test.

9.3.2 Planning Remedial Activity Based on the analysis made during the diagnosis test, teachers can plan a remedial activity to help students overcome the difficulties that they face. In order to implement an effective remedial activity, teachers must follow a few basic principles. These basic principles can help teachers to plan mathematical remedial teaching: The remedial instruction should only focus on weak areas identified. Steps in the presentation should be developed from concrete to abstract, from simple to complex, according to the capability and experience of the students. Teachers should use various teaching aids to help students understand the basic concept that should be mastered. Teachers should use symbols and mathematical terms that should be related to students experience. This will make it easier for students to identify their errors. Teachers should give oral and written exercises after every lesson. They should construct questions that are from simple to more complex questions. At the early

stage, teachers can allow students to use teaching aids to solve problems, but towards the end, students should be able to solve the problems without the teaching aids. Teachers should give an evaluation activity after every written activity in the class. They should plan and prepare written mathematical problem. This will determine whether the teaching objective is achieved.

Table 9.1: An example of a remedial lesson plan Step Set Induction Remedial Activity Flash cards are used to get students to give spontaneous responses.

Lesson Development (Step I)

1. Match sticks are tied in bundles of ten each. 2. The following question is asked: If 10 sticks are removed from 20 sticks, how many sticks are left? 3. Students are guided into solving the problem by showing the process in a Mahjong paper prepared by the teacher. 20 (llllllllll) (llllllllll) - 10 (llllllllll) = _____ The activity is repeated with three more examples.

(Step II)

Students are guided to answer similar questions without using bundles of match sticks.


1. Students are given worksheets with 5 similar examples. 2. Students solve the questions without any guidance from the teacher.

9.3.3 Evaluating Achievement This stage is important for teachers to determine whether students have acquired the skills after going through the remedial activity. Teachers can evaluate students performance based on the answers given in their exercises after the remedial activity.

9.3.4 Making Decision After the evaluation activity, teachers can make a reliable decision. If students achievement is satisfactory, the students can be given the chance to learn mathematics with other students who have acquired that particular skill. If the achievement is not satisfactory, then teachers can ask a special teacher trainer for remedial teaching to help them to teach the particular students in a special remedial class.


CONCEPT AND OBJECTIVES OF ENRICHMENT PROGRAM Gifted students have unique needs that need to be addressed in a mathematics

curriculum. They can be bored doing easy activities repeatedly. Instead, these students will benefit from a curriculum of enrichment. They often approach mathematics with inquisitiveness. They will challenge the purpose of material or the rationale of procedures. Just like remedial activity, enrichment should also provide opportunities for students to explore materials in meaningful ways. Meaning can be achieved by applying concepts to authentic contexts. Gifted students should be encouraged to solve problems in creative manners. Instruction for enrichment should involve projects that require application and extension of the content (Baroody & Coslick, 1998). Instruction should challenge students to apply learning in innovative and purposeful ways and ask them to be reflective upon their choices. Fostering gifted students inquisitiveness by asking them to reflect upon, apply, and extend content in meaningful ways allows them to reach high expectations while gaining mathematical power. Differentiating instruction in an integrated classroom can be a very difficult task. One way of providing the specialised instruction needed by these groups of students involves the use of mathematics learning centres. These centres will allow students to work with specialised materials at an individualised pace. The mathematics learning centre will provide students a structure for their own learning that does not require teacher supervision. The centre also provides facilities needed for any student who either wishes to extend their learning or gain further practice regardless they are gifted or remedial students.


Because students can work on projects in the centres without prompting, teachers remain free to interact with individual students to assess, clarify, and personalise instruction. The centres should be well-equipped with materials, have instructions, examples, and materials for specific projects relating to the content of instruction, and should provide space for tasks to be completed at the centre. The centers should be used to supplement and complement other forms of instruction. Activities can be adapted to incorporate cooperative learning or can be centred on individual activities. Students should have the opportunity to engage in multimodal learning in the centres and activities in the centres should incorporate authentic uses of mathematics.

9.4.1 Objectives of Enrichment Program Through enrichment activities, students will have the opportunities to take part in value-added activities so as to widen their learning experience, to enhance their interest, creativity, and also to inculcate self-discipline, and leadership qualities. Students performance in the classroom is determined by their individual differences. Students who have achieved a certain level of mastery should be given an opportunity to try on the enrichment activities prepared by teachers. Enrichment activities, therefore, do not only provide students with additional, more sophisticated and thought provoking work, but still keep them with their peers in the classroom.

9.4.2 Principles in Conducting Enrichment Activities Enrichment activities prepared by teachers should have the following principles: In the form of self-learning activities and have clearly defined instructions so that students will do the activities by themselves, whenever they feel that they are ready for it; Interesting, challenging and diversified according to students ability, so that there is a kind of motivation for students to try it by themselves; In the form of daily project which last for one (1) to two (2) weeks so that it does not make students think that it is a dragging project or it is too easy to complete; and Activities which are related to students learning skills so that they can see the application of the skills in their daily life.



TYPES OF ENRICHMENT ACTIVITIES Enrichment activities can be clustered into two (2) different categories: A. Horizontal Enrichment Activities All groups will be doing the same type of activities but each groups activity will

differ according to their cognitive level. For example, if the enrichment activity given to students in class is tangrams, each group with different level of achievement will be given different sets of tangrams which varies according to the level of difficulties.

B. Vertical Enrichment Activities Enrichment activity is provided for students who have achieved the required level of understanding of a topic that has been discussed in class. The activity planned and conducted here would be of high cognitive level and very challenging. This activity which caters for high achievers will help them to widen their horizon in the topic being taught. While these high achievers are engaged in such activities, we could plan, implement and monitor relevant activities for the rest of the students in the class.

Some of the common types of enrichment activities that can be given by teachers in the classroom are: Games : Snake and ladder, domino, selling and buying activity. Puzzles : Solving puzzles of numbers, time and basic operations. Project : Drawing of patterns using geometrical shapes, constructing threedimensional shapes, drawing picture graphs. Competition : Mathematical quiz, magic map. Experiment : Finding formulas or equations when given a few specific examples. Worksheets : Solving interesting and yet challenging problems.


CONDUCTING ENRICHMENT ACTIVITIES Enrichment activities can be conducted in classrooms through three (3) stages, which

depend on the instructional level and also the students performance. The three (3) stages are as follows: The first stage is after students have mastered one basic skill;


The second stage is after students have mastered all the basic skills for one mathematical topic; and The third stage is after students have mastered all the basic skills for a few related mathematical topics. As a whole, enrichment activities are best provided to all students in the classrooms,

either individually or in groups. Since there are bound to be differences among students, it is good to provide varied enrichment activities which would suit different individuals or groups. Generally, students of higher ability should be given more complex and challenging tasks, while those who are struggling should be given less challenging, but interesting tasks. It should be emphasised that enrichment activities for different groups should be based on the same mathematical concept or skills, but at different difficulty level. Students selected for enrichment activities are usually those who have mastered the required basic skills. Teachers can identify such students through informal tests, such as observations, oral questioning or by checking their daily work. They can even be identified through screening tests in which teachers can segregate those who have mastered the skills from those who have not. Teachers can provide the gifted students with enrichment activities, while the low ability students with remedial activities. When teachers plan the enrichment activities they have to make sure that students have mastered the skills taught, at least at a satisfactorily level. The content of the activities should be suitable to the students capabilities. Make sure that they are capable of carrying out the activity on their own with minimum guidance from their teachers.


In order to carry out remedial activity, teachers should know how to identify students who need remedial activities. This can be done by carrying out screening and diagnostic tests.

Teachers should then plan a remedial programme, which is followed by implementing remedial teaching.

As for gifted students and those who are high achievers, teachers should plan and implement enrichment activities.