Teaching Mathematics: Methodology · Chapter 7

Gifted and Slow Learners; Drill, Assignment, Oral and Written Work

What to remember

Gifted children are those with very high ability (intelligence quotient, IQ, of about 130 and above) and need enrichment, not only more work; slow learners are below average in learning speed and need remedial teaching with patience.
Drill is meaningful, planned practice done AFTER understanding; it fixes skill and speed. It is never blind repetition.
Oral work builds quick thinking and mental calculation; written work gives a permanent record and tests accuracy; assignments extend learning beyond the class.

Individual differences in a mathematics class

No two children learn at the same speed. A class has fast, average and slow learners. A good teacher plans for all three. The mathematics teacher must identify each type early, find the cause of the difficulty and then choose a suitable method.

IQ = (Mental age ÷ Chronological age) × 100. Example: mental age 12 years, chronological age 10 years: IQ = 12/10 × 100 = 120.

The gifted child in mathematics

Characteristics

Learns new ideas very fast and remembers them for a long time.
Asks many "why" and "how" questions; shows curiosity and deep interest.
Likes puzzles, patterns, problems and logical games.
Finishes class work early and then becomes bored or restless.
Has a long attention span on topics of interest and a wide vocabulary.
Can often generalise, find short methods and see relations between ideas.

Identification

Intelligence tests (IQ of about 130 or more).
Achievement tests and past school records.
Teacher's observation and checklists.
Parents' reports and the child's hobbies.
Performance in mathematics talent tests and quizzes.

Provision for gifted children

Enrichment: giving extra, deeper and more challenging material in the same class (puzzles, projects, open-ended problems, mathematics clubs, Olympiads, library work).
Acceleration: allowing the child to move to a higher class or to finish the course faster (double promotion).
Grouping: forming special groups or classes of gifted children for part of the time.
Mentoring: guidance by a teacher or expert; peer teaching, where the gifted child helps others.
Giving responsibility, such as leading a group or running a mathematics laboratory activity.

Enrichment is regarded as the best provision inside a normal classroom, because the child stays with age-mates and develops socially. Acceleration may cause social and emotional problems. Simply giving "more of the same sums" is not enrichment.

The slow learner in mathematics

A slow learner is a child who learns at a speed lower than the class average. The child can learn, but needs more time, more practice and more concrete material. Slow learning is not the same as mental disability. A backward child is one whose achievement is clearly below the level expected for the age or class; backwardness may come from poor environment, illness, absence or poor teaching even when ability is normal. An under-achiever has good ability but performs poorly.

Causes

Low intelligence or slow mental development.
Weak foundation in earlier classes (missing number sense, tables, place value).
Poor health, poor eyesight or hearing, malnutrition.
Fear and anxiety about mathematics, bad earlier experiences.
Irregular attendance, poor home environment, lack of encouragement.
Dull teaching, large classes, too fast a pace.

Characteristics

Needs repeated explanation; forgets quickly.
Poor understanding of abstract ideas; depends on rote memory.
Short attention span, low confidence, avoids answering.
Makes the same errors again and again.

Remedial teaching (steps)

1. Diagnose: use a diagnostic test to find exactly which concept or skill is weak.
2. Plan: prepare simple, graded exercises on that weak point only.
3. Teach: use concrete objects, pictures and activities before symbols; move from concrete to semi-concrete to abstract.
4. Practise: short, frequent practice with immediate feedback.
5. Encourage: praise small success, build confidence, avoid ridicule.
6. Re-test: check progress and repeat the cycle.

Other measures: extra periods, peer tutoring, small groups, activity-based learning, use of the mathematics laboratory, and cooperation with parents. The teacher should never compare the child openly with others.

Comparison table

Point	Gifted learner	Slow learner
Learning speed	Very fast	Slow
Approximate IQ	About 130 and above	Below average (roughly 70 to 90)
Main need	Enrichment, challenge	Remedial help, patience
Common problem	Boredom, restlessness	Lack of confidence, repeated errors
Teaching aid	Puzzles, projects, open problems	Concrete material, charts, models
Tool to identify	IQ and achievement tests	Diagnostic test

Drill work

Meaning: Drill is organised, repeated practice of a skill or fact (tables, formulae, number facts, procedures) so that it becomes automatic. It develops speed, accuracy and retention.

Principles

Give drill only after the child has understood the idea (understanding first, then practice).
Keep it short, regular and varied; avoid boredom.
Make it purposeful and interesting through games, quizzes and competitions.
Give immediate feedback and correct mistakes at once.
Match difficulty to the child's level; spread practice over time (distributed practice) rather than cramming.

Forms of drill: oral drill, written drill, mental drills, number games, flash cards, tables chanting, mathematical quizzes, computer or app-based practice.

Merits: fixes facts and skills; increases speed and accuracy; builds confidence. Demerits: may become mechanical and dull; if used before understanding it leads to rote learning; does not by itself develop thinking.

Assignment work

An assignment is a task given to a student to be completed on their own, in or out of class. It should be clear, related to the lesson, of suitable difficulty and not too long.

Type	Meaning
Individual assignment	Each student works alone (home work, exercise)
Group assignment	A group works together on one task (project, survey)
Short-term	To be finished in a day or a few days
Long-term	Projects or data collection over weeks
Practice assignment	To fix skills already taught
Enrichment assignment	Extra, deeper tasks for fast learners
Remedial assignment	Easy tasks to repair weak points

Good assignment practice: explain the task clearly, give it at the right time, check it regularly, give feedback, and differentiate it for gifted and slow learners. Assignments should reduce, not increase, stress, and should not be used as punishment.

Oral work

Oral work means questions and answers spoken in class. In mathematics it includes mental arithmetic, oral tables, quick questions, explaining a method aloud and discussion.

Advantages: quick; keeps the whole class alert; builds speed and mental ability; tests understanding of the thinking process; gives instant feedback; helps shy and weak students if questions are graded. It improves mathematical language and communication.

Limitations: no permanent record; shy children may not take part; hard to check each child; needs careful planning.

Written work

Written work includes class exercises, notebooks, tests and homework. It gives a permanent record, trains neat and logical presentation of steps, and lets the teacher analyse errors.

Good written work practice: show every step; write neatly in order; use proper units and symbols; check answers. The teacher should check promptly, mark errors, discuss common mistakes and ask students to correct them (correction work). Marking should be positive and give helpful comments.

Oral versus written work

Point	Oral work	Written work
Speed	Fast	Slower
Record	None	Permanent
Main skill	Mental ability, expression	Accuracy, presentation
Feedback	Immediate	Delayed until checked
Best use	Warm-up, revision, mental maths	Practice, tests, assessment

Both are needed. A balanced lesson uses oral work first to build understanding and written work later to fix it.

Exam traps

Enrichment is not acceleration: enrichment widens and deepens the work in the same class; acceleration moves the child ahead.
Slow learner is not the same as a mentally disabled or a backward child; backwardness can arise from environment even when ability is normal.
Under-achiever means good ability, poor performance; slow learner means low learning speed.
Drill comes after understanding, not before.
Diagnosis comes before remedial teaching.
Oral work gives no permanent record; written work does.
More sums of the same kind is not enrichment for a gifted child.
Gifted children need guidance too; they cannot be left to learn alone.

One-liners

1. Gifted children usually have IQ of about 130 or more.
2. IQ = (Mental age ÷ Chronological age) × 100.
3. Enrichment is considered the best provision for gifted children in an ordinary class.
4. Acceleration means double promotion or faster completion of the course.
5. Slow learners learn at a speed lower than the class average.
6. Remedial teaching begins with a diagnostic test.
7. Concrete to semi-concrete to abstract is the right order for slow learners.
8. Drill develops speed, accuracy and retention.
9. Drill should follow understanding.
10. A long-term assignment such as a project may run for weeks.
11. Oral work includes mental arithmetic and quick questions.
12. Written work gives a permanent record of the child's steps and errors.

Practice questions

Gifted children are usually identified by an IQ of about
1. 70 to 90
2. 130 and above
3. 110 to 120
4. 90 to 110
Answer
B. 130 and above
A common standard is an IQ of about 130 or more.
The formula for intelligence quotient is
1. (Mental age ÷ Chronological age) × 100
2. (Chronological age ÷ Mental age) × 100
3. Mental age × Chronological age ÷ 100
4. (Mental age + Chronological age) ÷ 100
Answer
A. (Mental age ÷ Chronological age) × 100
IQ = (MA ÷ CA) × 100.
A child has mental age 12 years and chronological age 10 years. The IQ is
1. 100
2. 130
3. 120
4. 83
Answer
C. 120
12/10 × 100 = 120.
Giving gifted children deeper and more challenging work in the same class is called
1. remediation
2. detention
3. acceleration
4. enrichment
Answer
D. enrichment
Enrichment widens and deepens the work without moving the child to another class.
Double promotion of a gifted child is an example of
1. remedial teaching
2. acceleration
3. enrichment
4. drill
Answer
B. acceleration
Acceleration moves the child faster through the course or to a higher class.
Which provision is generally considered best for gifted children inside a normal classroom?
1. Giving more sums of the same type
2. Keeping them idle after work
3. Punishing early finishers
4. Enrichment programmes
Answer
D. Enrichment programmes
Enrichment keeps the child with age-mates and gives challenge.
Which of the following is NOT a characteristic of a gifted child?
1. Needs repeated explanation of simple ideas
2. Shows curiosity and long attention on interesting problems
3. Learns new ideas quickly
4. Asks many deep questions
Answer
A. Needs repeated explanation of simple ideas
Needing repeated explanation is typical of slow learners.
The first step in remedial teaching of a slow learner is
1. giving a final examination
2. increasing homework
3. diagnosing the exact difficulty
4. grouping with gifted children
Answer
C. diagnosing the exact difficulty
Diagnosis of the weak concept comes before remedial work.
The correct order of teaching a slow learner is
1. abstract, semi-concrete, concrete
2. semi-concrete, abstract, concrete
3. abstract, concrete, semi-concrete
4. concrete, semi-concrete, abstract
Answer
D. concrete, semi-concrete, abstract
Objects first, then pictures, then symbols.
A child with good ability who performs poorly in class is called
1. an under-achiever
2. a gifted child
3. a prodigy
4. an over-achiever
Answer
A. an under-achiever
Under-achievement means performance below ability.
Drill work in mathematics should be given
1. before the concept is explained
2. only as punishment
3. only at the end of the year
4. after the concept is understood
Answer
D. after the concept is understood
Practice should follow understanding or it turns into rote learning.
Which is a demerit of drill work?
1. It improves speed
2. It increases accuracy
3. It may become mechanical and dull
4. It fixes number facts
Answer
C. It may become mechanical and dull
Too much repetition can be boring and mechanical.
Chanting multiplication tables in class is an example of
1. a diagnostic test
2. oral drill
3. a long-term project
4. an anecdotal record
Answer
B. oral drill
Repeated spoken practice of facts is oral drill.
A project carried out over several weeks is a
1. placement test
2. oral drill
3. long-term assignment
4. short-term assignment
Answer
C. long-term assignment
Projects and surveys need time and are long-term assignments.
Which statement about assignments is best?
1. They should be clear, related to the lesson and checked with feedback
2. They should be used as punishment
3. They should be very long to keep children busy
4. They need not be checked
Answer
A. They should be clear, related to the lesson and checked with feedback
Good assignments are purposeful, suitable and followed by feedback.
Assignments in which a group of students works together on one task are called
1. individual assignments
2. group assignments
3. oral assignments
4. remedial tests
Answer
B. group assignments
Group assignments share one task among members.
A main advantage of oral work in mathematics is
1. neat presentation of work
2. a permanent record of steps
3. less need for teacher planning
4. immediate feedback and development of mental ability
Answer
D. immediate feedback and development of mental ability
Oral work is quick and gives instant feedback.
A limitation of oral work is that it
1. cannot be used for mental maths
2. leaves no permanent record
3. takes too long to check
4. gives only delayed feedback
Answer
B. leaves no permanent record
Spoken answers are not recorded.
Which type of work gives a permanent record of a student's steps and errors?
1. Written work
2. Oral work
3. Mental arithmetic
4. Class discussion
Answer
A. Written work
Written work can be reviewed later for error analysis.
Mental arithmetic mainly develops
1. map reading
2. neat handwriting
3. speed and mental ability
4. graph skills
Answer
C. speed and mental ability
Quick calculation without writing builds mental ability.
Correction work after marking a notebook is meant to
1. reduce the teacher's work
2. punish careless students
3. avoid discussion
4. help students learn from their errors
Answer
D. help students learn from their errors
Corrections turn mistakes into learning.
Which tool is most suitable to locate the exact weak concept of a student?
1. Diagnostic test
2. Annual examination
3. Placement test
4. Cumulative record
Answer
A. Diagnostic test
A diagnostic test identifies the specific difficulty.
Peer tutoring, where a gifted child helps a slow learner, is useful because
1. it removes the need for the teacher
2. it is a form of acceleration
3. both children gain in understanding and confidence
4. it prevents any diagnosis
Answer
C. both children gain in understanding and confidence
Explaining to others strengthens the helper and supports the weaker child.
Which of the following is a possible cause of slow learning in mathematics?
1. Very high IQ
2. Weak foundation in earlier classes
3. Over-enrichment
4. Early completion of the syllabus
Answer
B. Weak foundation in earlier classes
Missing basics, poor health and anxiety are common causes.
The best attitude of a teacher towards a slow learner is
1. scolding for mistakes
2. patience, encouragement and extra practice
3. frequent comparison with others
4. ignoring the child
Answer
B. patience, encouragement and extra practice
Praise and patience build confidence.
Distributed (spaced) practice in drill means
1. one long cramming session
2. practice spread over short periods across days
3. practice only before examinations
4. practice without feedback
Answer
B. practice spread over short periods across days
Short, regular practice gives better retention.
A backward child is best described as one whose
1. attendance is always perfect
2. speed of work is always the fastest in class
3. achievement is clearly below the level expected for age or class
4. IQ is above 130
Answer
C. achievement is clearly below the level expected for age or class
Backwardness refers to low achievement, which may have environmental causes.
A mathematics teacher finds a bright child finishing every task early and becoming restless. The most suitable step is to
1. give open-ended problems and puzzles for enrichment
2. give the same sums again
3. reduce the difficulty of tasks
4. ask the child to sit quietly
Answer
A. give open-ended problems and puzzles for enrichment
Challenging, open tasks keep the gifted child engaged.
A student repeatedly makes mistakes while regrouping in subtraction. The teacher should first
1. assign a project
2. lower the child's marks
3. skip the topic
4. analyse the errors with a diagnostic exercise
Answer
D. analyse the errors with a diagnostic exercise
Error analysis reveals the exact misconception.
Which statement correctly differentiates enrichment from acceleration?
1. Enrichment is for slow learners; acceleration for gifted
2. Enrichment deepens learning in the same class; acceleration moves the child ahead
3. Both mean repeating the same class
4. Enrichment is promotion; acceleration is detention
Answer
B. Enrichment deepens learning in the same class; acceleration moves the child ahead
Enrichment widens work, acceleration speeds progress.
Consider the statements on drill work. 1. Drill should come after understanding of the concept. 2. Drill should be long and without variety to ensure learning.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Statement 1 is a principle of drill; drill should be short, varied and interesting.
Consider the statements on slow learners. 1. Slow learning always means mental disability. 2. They can learn if given more time, practice and concrete material.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Slow learners can learn; slow learning is not necessarily disability.
Consider the statements on oral and written work. 1. Written work leaves no permanent record. 2. Oral work gives immediate feedback.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Written work does give a permanent record; statement 1 is wrong.
Consider the statements on gifted children. 1. Simply giving more sums of the same type is enrichment. 2. Enrichment gives deeper and wider experiences.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Enrichment is quality, not just quantity.
Consider the statements on assignments. 1. Assignments may be individual or group. 2. Assignments should differ in difficulty for gifted and slow learners.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are correct good practice.
Consider the statements on acceleration. 1. It always involves giving more homework at the same level. 2. It may be done by double promotion.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Acceleration is about faster progress; statement 1 is wrong.
Consider the statements on remedial teaching. 1. Remedial teaching should begin with a final summative test. 2. Diagnosis should come before remedial teaching.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Diagnosis first; statement 1 is wrong.
Consider the statements on written work. 1. Marking should be only negative so that students are afraid. 2. It helps in analysing the child's errors.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Marking should be positive and helpful; statement 1 is wrong.
Consider the statements on under-achievers. 1. They have ability above their performance. 2. They are the same as children with low IQ.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Under-achievers perform below their ability; low IQ is a different case.
Match the term with its meaning. P. Enrichment Q. Acceleration R. Remedial teaching S. Drill 1. Planned repair of weak points 2. Faster promotion 3. Organised repeated practice 4. Deeper extra material in the same class
1. P-4, Q-1, R-2, S-3
2. P-2, Q-4, R-3, S-1
3. P-1, Q-2, R-4, S-3
4. P-4, Q-2, R-1, S-3
Answer
D. P-4, Q-2, R-1, S-3
Enrichment: deeper material; acceleration: faster promotion; remedial: repairing weak points; drill: repeated practice.
Match the learner with the need. P. Gifted Q. Slow learner R. Under-achiever
1. P-challenge, Q-motivation, R-remedial help
2. P-motivation, Q-challenge, R-remedial help
3. P-challenge, Q-remedial help, R-motivation and guidance
4. P-remedial help, Q-challenge, R-motivation
Answer
C. P-challenge, Q-remedial help, R-motivation and guidance
Gifted need challenge, slow learners need remedial help, under-achievers need motivation and guidance.
Match the work with its feature. P. Oral work Q. Written work R. Long-term assignment
1. P-permanent record, Q-quick feedback, R-project
2. P-project, Q-quick feedback, R-permanent record
3. P-quick feedback, Q-project, R-permanent record
4. P-quick feedback, Q-permanent record, R-project
Answer
D. P-quick feedback, Q-permanent record, R-project
Oral gives instant feedback, written leaves a record, long-term tasks include projects.
A teacher gives a child a task to finish alone at home and bring next day. This is a
1. long-term group assignment
2. oral drill
3. short-term individual assignment
4. diagnostic test
Answer
C. short-term individual assignment
Single child, one day: short-term individual.
A teacher uses flash cards for number facts and then checks speed. This mainly helps the child to
1. fix number facts through drill
2. produce a project
3. avoid understanding the concepts
4. become a gifted learner
Answer
A. fix number facts through drill
Flash cards are a drill device that builds speed and accuracy.
Which situation is a clear case of poor drill practice?
1. Short quiz after a concept lesson
2. Game-based tables practice
3. Immediate correction of errors
4. Making children copy the same sum fifty times before the idea is explained
Answer
D. Making children copy the same sum fifty times before the idea is explained
Mechanical repetition before understanding is poor drill.

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