Pariksha SaathiLesson 7 of 8 · Mathematics Pedagogy
Diagnostic and Remedial Teaching in Mathematics: Slow and Gifted Learners
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Why diagnosis comes first
When a child keeps getting one type of sum wrong, teaching the same lesson again, more loudly, rarely helps. Diagnosis means finding exactly what the child cannot do and why, as a doctor finds the cause before prescribing a medicine. Because mathematics is hierarchical, the cause is often an earlier gap: a child failing at subtraction with regrouping may not understand place value, and a child failing at fractions may not understand equal sharing. Diagnosis is the bridge between assessment and teaching.
Tools of diagnosis
The teacher observes children at work, analyses the errors in written work, asks children to explain their steps aloud, talks with them one to one, and uses diagnostic tests. A diagnostic test differs from an achievement test. An achievement test measures how much a child has learnt, gives a score and may rank children; a diagnostic test finds where and why a child is weak. It therefore covers a small unit broken into sub-skills, has several items for each sub-skill arranged from easy to difficult, is usually untimed, and is not used for grading or ranking. For two-digit subtraction, for example, separate items test subtraction without regrouping, with regrouping, with a zero in the minuend and in word problems, so the pattern of wrong items shows the exact step that fails.
The remedial cycle
Remedial teaching is planned instruction to remove the difficulty that diagnosis has found. The cycle runs: identify the children who are struggling, often through formative assessment; diagnose the specific difficulty and its cause; plan the remedial work; teach it; re-assess to check whether the difficulty has gone; and give feedback, diagnosing again if it has not. Good remedial teaching uses a different approach from the one that failed, usually going back to concrete materials and pictures; it works in small steps, starts from what the child can already do, gives immediate feedback and frequent success, and is done individually or in small groups. It must never label children as weak or keep them permanently apart from the class. Section 4 of the RTE Act 2009 likewise gives an out-of-school child admitted to an age-appropriate class the right to special training so that she can catch up.
Causes beyond the concept
Difficulty may also come from language (the child cannot read the word problem), from mathematics anxiety, from long absence, from teaching that gave rules without meaning, or from a specific learning disability. Dyscalculia is a specific learning disability in understanding numbers: despite normal intelligence and teaching, the child may have weak number sense and find it hard to count backwards, remember number facts, read a clock or tell which of two numbers is bigger. The Rights of Persons with Disabilities Act 2016 includes dyscalculia among specific learning disabilities, along with dyslexia, dysgraphia, dyspraxia and developmental aphasia. Such children need multisensory materials, extra time, number lines and other aids, and patience, not punishment.
Peer tutoring and group work
In peer tutoring a child who has understood a concept helps a classmate who has not. It works because a peer explains in familiar language, the learner asks questions freely, and the tutor deepens her own understanding by explaining, which is learning by teaching. It draws on Vygotsky's idea that learning in the zone of proximal development happens with help from a more knowledgeable other, who may well be a classmate. NEP 2020 also suggests peer tutoring as a voluntary and joyful activity under the supervision of trained teachers. Mixed-ability groups, pair work and cooperative games spread the same benefit across the class.
Slow learners
Slow learners can learn the same mathematics but need more time, more concrete experience and more practice. The teacher breaks a concept into small steps, keeps concrete and pictorial supports for longer, links tasks to the child's life, gives shorter assignments with frequent feedback, praises effort and progress, and avoids comparison with faster children. Expectations stay high; only the path and the pace change.
Gifted learners
Mathematically gifted children grasp ideas quickly, spot patterns, prefer reasoning to routine and may get bored by repetitive sums, so giving them more of the same sums is a mistake. Enrichment (horizontal) gives deeper and broader work at the same level: puzzles, open-ended problems, investigations, problems with many solutions, problem posing and projects. Acceleration (vertical) lets them move faster through the curriculum. Both slow and gifted learners are best served in an inclusive classroom through differentiated tasks, in the spirit of the Position Paper's call to engage every student with a sense of success while offering conceptual challenges to the emerging mathematician.
Key facts
- Diagnosis finds what and why; because mathematics is hierarchical, the cause is often an earlier gap, such as place value behind regrouping errors.
- Achievement test: how much is learnt, scored and ranked. Diagnostic test: where and why, small unit, items by sub-skill from easy to hard, usually untimed, not for grading.
- Remedial cycle: identify, diagnose, plan, teach, re-assess, give feedback; diagnose again if the difficulty remains.
- Remedial teaching uses a different approach (often back to concrete materials), small steps, immediate feedback and small groups, and never labels the child.
- RTE Act 2009, Section 4: special training for an out-of-school child admitted to an age-appropriate class.
- Dyscalculia: a specific learning disability with numbers; the RPwD Act 2016 lists it with dyslexia, dysgraphia, dyspraxia and developmental aphasia.
- Peer tutoring: the tutor learns by teaching and the learner gets help in familiar language; based on Vygotsky's ZPD and the more knowledgeable other; suggested in NEP 2020.
- Slow learners: the same goals with more time, smaller steps, longer use of concrete aids, frequent success and no comparison.
- Gifted learners: enrichment (horizontal: puzzles, open-ended problems, projects) or acceleration (vertical: faster pace), not more of the same sums.
Quick revision
Diagnosis: what and why, often an earlier gap; tools: observation, error analysis, interview, diagnostic test (sub-skills, untimed, no ranking) vs achievement test; remedial cycle: identify, diagnose, plan, teach, re-assess; a different method in small steps; RTE Act Section 4 special training; dyscalculia in RPwD Act 2016; peer tutoring (Vygotsky, NEP 2020); slow learners: more time; gifted: enrichment and acceleration.