Curriculum Principles and Approaches; Qualities of a Mathematics Textbook

What to remember

• Logical order follows the structure of the subject; psychological order follows the learner's mind, interest and readiness. A good mathematics curriculum tries to join both.
• Topical = a topic is taught fully once; concentric = the same topics return with wider scope; spiral = topics return at a higher level and each stage builds on the earlier one (Bruner).
• A textbook is a tool, not the curriculum. A good one is accurate, graded, clear in language, well illustrated and rich in exercises.

Curriculum, syllabus and textbook

A curriculum is the whole plan of learning experiences a school offers: aims, content, methods, activities and evaluation. A syllabus is only the list of topics, with time, for one class or course. A textbook is a book that presents the content of the syllabus in teachable form. The curriculum is the widest of the three.

In India the national framework was NCF 2005; following NEP 2020 it is now the National Curriculum Framework for School Education (NCF-SE), 2023. In Andhra Pradesh the SCERT prepares the state curriculum and textbooks for school mathematics.

Principles of curriculum construction in mathematics

• 1. Child-centred: based on age, interest and ability. Content should be neither too hard nor too easy.
• 2. Utility (life-centred): includes the mathematics needed in daily life, such as ratio, percentage and simple interest.
• 3. Activity principle: the child learns by doing, using objects, measuring and drawing.
• 4. Sequence and gradation: simple to complex, concrete to abstract, known to unknown.
• 5. Integration and correlation: mathematics is linked with other subjects (science, social studies, art) and within itself (arithmetic, algebra, geometry).
• 6. Flexibility: room for local needs and for slow and fast learners.
• 7. Forward-looking: prepares for higher classes and later careers.
• 8. Balance: theory and practice, skill and concept, logical and psychological order.
• 9. Community and cultural link: uses local examples such as markets, farming and measures used locally.
• 10. Minimum essentials: a core that every child should master.

Logical and psychological approaches

Logical approach. The content is arranged as an expert (the mathematician) sees it. The order follows the structure of the subject: definitions first, then theorems, each built on the one before. Example: teaching axioms and postulates of geometry first, then theorems, then riders.

Psychological approach. The content is arranged as the child can best learn it. It follows interest, needs, age, mental level and readiness. Example: introducing geometry through paper folding and a model before formal proofs.

Good practice: begin with a psychological approach in the early classes and move towards the logical approach as pupils grow in maturity.

Topical, concentric and spiral approaches

Topical approach. One topic is taken up and finished completely at one stage, and it is not repeated later. Example: all of "profit and loss" in a single class. Advantage: saves time; no repetition. Weakness: forgetting, because there is no revisiting, and a young child may not grasp the whole topic in one go.

Concentric approach. The same topics are taught again and again in higher classes, each time in a wider circle with more depth. The idea is like concentric circles which grow bigger but have the same centre. Example: the idea of a triangle in primary classes (recognition), then its types and angle sum, and then congruence and similarity. Advantage: revision and gradual growth. Weakness: if the repeat is mere repetition, it becomes boring.

Spiral approach. Proposed by Jerome Bruner. The learner revisits the basic ideas repeatedly, and each time the idea is taught at a higher level of complexity and links to what was learnt earlier. Like a spring rising upward, it never returns to the same point. Bruner stated that any subject can be taught in some honest form to a child at any stage of development. He also described three modes of representing ideas: enactive (by action), iconic (by images) and symbolic (by words and symbols). A spiral curriculum moves through them in that order.

The main difference between concentric and spiral is this: concentric widens the same content, while spiral adds new depth and links each turn with earlier learning. Many school mathematics programmes are close to spiral. Number work, geometry and algebra are met again and again across classes with rising difficulty.

Qualities of a good mathematics textbook

A mathematics textbook is judged under these heads:

The textbook should help the pupil to think, not just to memorise. The teacher should not follow the textbook line by line but should use it with the aids, activities and other resources.

Classroom angle

• Use a psychological start with objects and local examples, then move toward the formal rule.
• Revisit earlier ideas in new settings (spiral): for instance, use a fraction idea again in ratio and then in percentage.
• Plan activities before formal definitions; pupils form the idea and then the symbol.
• Keep the textbook for practice and reference, but build lessons from pupils' experience.
• When selecting or reviewing a textbook, check the list above, one head at a time.

Other principles linked to the mathematics curriculum

• Concrete to abstract: begin with objects and pictures; end with symbols and rules.
• Known to unknown and simple to complex: each new idea rests on one the pupil already holds.
• Part to whole and whole to part: some topics (fractions) begin with parts; others (a figure) are first seen whole and then analysed.
• Pre-requisites: a topic is placed only after the ideas needed for it. Division of fractions, for instance, needs multiplication of fractions and the idea of a reciprocal.
• Learner diversity: the curriculum should serve slow learners with support and gifted learners with challenge problems.
• Continuous review: the curriculum is reviewed and revised from time to time, in the light of feedback from teachers, pupils and research.

Reading a textbook chapter critically

A teacher who reviews a chapter asks: Does it begin with an activity or a familiar situation? Is each term defined before it is used? Are the solved examples varied and complete? Do the exercises move from easy to hard? Is there a place for pupils to think and discuss? Are the figures neat and labelled? Are there answers or hints? Is the language right for the class? Such a check list turns the list of qualities into a working tool.

Exam traps

• Curriculum vs syllabus: the curriculum is all planned experiences; the syllabus is the topic list.
• Concentric vs spiral: circles of the same content versus an upward spring with new depth.
• Topical vs concentric: topical has no return; concentric has a return.
• Logical vs psychological: subject structure versus child's readiness.
• Bruner's modes: enactive (action), iconic (image), symbolic (words); do not swap the order.
• Textbook vs curriculum: a book is only one tool of the curriculum.
• Principle of utility vs principle of activity: life use versus learning by doing.
• Correlation vs integration: correlation links subjects; integration merges them into a unified whole.

One-liners

• 1. The curriculum is wider than the syllabus.
• 2. The logical approach follows the structure of the subject.
• 3. The psychological approach follows the learner's interest and readiness.
• 4. In the topical approach, a topic is not repeated after it is taught.
• 5. The concentric approach repeats topics at increasing breadth.
• 6. The spiral approach is linked to Jerome Bruner.
• 7. Bruner's three modes are enactive, iconic and symbolic.
• 8. Early classes lean toward the psychological order; later classes toward the logical.
• 9. A graded exercise moves from easy to hard problems.
• 10. Answers to exercises and solved examples are marks of a good mathematics textbook.
• 11. Language in a textbook must suit the class level.
• 12. The SCERT prepares the school curriculum framework and textbooks in Andhra Pradesh.

Practice questions

1. A curriculum differs from a syllabus because the curriculum

   1. Is only a list of topics
   2. Is a single textbook
   3. Is the question paper
   4. Includes aims, content, methods, activities and evaluation
   Answer

   D. Includes aims, content, methods, activities and evaluation

   The syllabus is a topic list; the curriculum is the whole plan.

2. Arranging content as the subject expert sees it, from definitions to theorems, is the

   1. Incidental approach
   2. Psychological approach
   3. Logical approach
   4. Spiral approach
   Answer

   C. Logical approach

   Logical order follows the structure of the subject.

3. Introducing geometry through paper folding before proofs is an example of

   1. Topical order
   2. Logical order
   3. Psychological order
   4. Axiomatic order
   Answer

   C. Psychological order

   Concrete experience first suits the learner's mind.

4. In the topical approach, a topic is

   1. Repeated each year in wider form
   2. Taught fully at one stage and not repeated
   3. Taught in a rising spiral
   4. Taught only through projects
   Answer

   B. Taught fully at one stage and not repeated

   Topical means complete treatment at one stage.

5. The concentric approach can be pictured as

   1. Circles of growing size about one centre
   2. A pyramid of proofs
   3. A rising spring
   4. A straight line of topics
   Answer

   A. Circles of growing size about one centre

   Same centre, wider circles.

6. The spiral curriculum is associated with

   1. Edgar Dale
   2. Kilpatrick
   3. Herbart
   4. Jerome Bruner
   Answer

   D. Jerome Bruner

   Bruner proposed the spiral curriculum.

7. Bruner's three modes of representation, in order, are

   1. Enactive, symbolic, iconic
   2. Iconic, enactive, symbolic
   3. Enactive, iconic, symbolic
   4. Symbolic, iconic, enactive
   Answer

   C. Enactive, iconic, symbolic

   Action first, then image, then symbol.

8. Which approach adds new depth each time and links to earlier learning?

   1. Topical
   2. Concentric
   3. Logical
   4. Spiral
   Answer

   D. Spiral

   Spiral builds on the previous turn at a higher level.

9. The chief difference between concentric and spiral approaches is that the spiral

   1. Builds new complexity on earlier learning
   2. Never revisits a topic
   3. Covers only one stage
   4. Avoids concrete examples
   Answer

   A. Builds new complexity on earlier learning

   A spiral rises; concentric mainly widens.

10. A major weakness of the topical approach is

    1. Heavy cost
    2. Forgetting due to no revisiting
    3. Too much repetition
    4. No logical order
    Answer

    B. Forgetting due to no revisiting

    With no return, retention may fall.

11. A major risk of the concentric approach is

    1. No revision at all
    2. No sequence
    3. No link with life
    4. Boredom if the repeat is mere repetition
    Answer

    D. Boredom if the repeat is mere repetition

    Poorly planned repetition becomes dull.

12. Teaching 'triangle' as shapes in primary classes, then angle sum, then congruence and similarity, follows mainly the

    1. Laboratory method
    2. Project method
    3. Concentric or spiral approach
    4. Topical approach
    Answer

    C. Concentric or spiral approach

    The idea returns in wider and deeper form.

13. Which principle of curriculum construction asks that learning be by doing?

    1. Activity principle
    2. Forward-looking principle
    3. Flexibility principle
    4. Utility principle
    Answer

    A. Activity principle

    Learning by doing is the activity principle.

14. Including ratio, percentage and simple interest because daily life needs them follows the

    1. Principle of integration
    2. Principle of utility
    3. Principle of gradation
    4. Principle of balance
    Answer

    B. Principle of utility

    Life use is the utility principle.

15. Moving from concrete to abstract and easy to hard is the principle of

    1. Flexibility
    2. Integration
    3. Sequence and gradation
    4. Community link
    Answer

    C. Sequence and gradation

    Gradation gives a simple-to-complex order.

16. Linking mathematics with science and art follows the principle of

    1. Rote learning
    2. Isolation
    3. Specialisation
    4. Correlation
    Answer

    D. Correlation

    Correlation links a subject with other subjects.

17. Keeping room for slow and fast learners and local needs is the principle of

    1. Flexibility
    2. Sequence
    3. Utility
    4. Forward-looking
    Answer

    A. Flexibility

    Flexible curriculum adapts to the learner and place.

18. Which is a quality of a good mathematics textbook?

    1. Heavy use of difficult words
    2. Exercises only at the end of the book
    3. Graded exercises with solved examples
    4. Examples taken only from foreign contexts
    Answer

    C. Graded exercises with solved examples

    Graded exercises and solved examples are marks of quality.

19. Which language feature is best for a Class 6 mathematics textbook?

    1. Long, complex sentences
    2. Simple short sentences and exact terms
    3. Old words rarely used
    4. Poetic phrases
    Answer

    B. Simple short sentences and exact terms

    Language must suit the class level.

20. Which belongs to the mechanical features of a textbook?

    1. Accuracy of facts
    2. Clear print and strong binding
    3. Graded exercises
    4. Order of units
    Answer

    B. Clear print and strong binding

    Print, paper and binding are mechanical features.

21. A textbook is best described as

    1. The whole curriculum
    2. One tool of the curriculum
    3. The syllabus
    4. An evaluation tool only
    Answer

    B. One tool of the curriculum

    The textbook serves the curriculum but is not the curriculum.

22. Which is the best use of a textbook by a teacher?

    1. Reading it aloud every period
    2. Skipping all exercises
    3. As one resource with activities and aids
    4. Using only the answers
    Answer

    C. As one resource with activities and aids

    A teacher should not tie lessons to the book line by line.

23. Study the statements. 1. The logical approach follows the learner's interest. 2. The psychological approach follows the learner's readiness. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    Statement 1 describes the psychological order, so only statement 2 is correct.

24. Study the statements. 1. In the topical approach, a topic is not revisited. 2. In the concentric approach, a topic is revisited at wider level. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both statements are standard.

25. Study the statements. 1. A curriculum is wider than a syllabus. 2. A syllabus is a list of topics with time. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both statements are standard.

26. Study the statements. 1. The spiral approach was proposed by Bruner. 2. The spiral approach never revisits any topic. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    A spiral revisits ideas repeatedly, so statement 2 is false.

27. Study the statements. 1. Early classes lean toward the psychological order. 2. Later classes can move toward the logical order. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both statements reflect good practice.

28. Study the statements. 1. A good textbook has graded exercises. 2. A good textbook should be free of gender and caste bias. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are qualities of a good textbook.

29. Study the statements. 1. Bruner's iconic mode uses images and diagrams. 2. Bruner's enactive mode uses symbols and words. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Enactive is through action; symbolic is through words and symbols, so statement 2 is false.

30. Study the statements. 1. Integration means merging subjects into a unified whole. 2. Correlation means linking subjects while they remain separate. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both describe the usual distinction.

31. Match: (a) Topical (b) Concentric (c) Spiral (d) Logical with (p) Circles about one centre (q) Topic taught once only (r) Structure of the subject (s) Rising spring with deeper links.

    1. a-q, b-p, c-r, d-s
    2. a-s, b-p, c-q, d-r
    3. a-p, b-q, c-s, d-r
    4. a-q, b-p, c-s, d-r
    Answer

    D. a-q, b-p, c-s, d-r

    Topical is once only, concentric is circles, spiral is a rising spring, logical follows subject structure.

32. Match: (a) Enactive (b) Iconic (c) Symbolic with (p) Words and symbols (q) Action with objects (r) Images and diagrams.

    1. a-q, b-p, c-r
    2. a-r, b-q, c-p
    3. a-p, b-r, c-q
    4. a-q, b-r, c-p
    Answer

    D. a-q, b-r, c-p

    Enactive is action, iconic is image, symbolic is words and symbols.

33. Match principle with example: (a) Utility (b) Activity (c) Correlation with (p) Measuring a classroom with a tape (q) Percentage for shop discounts (r) Using geometry in art.

    1. a-q, b-p, c-r
    2. a-p, b-q, c-r
    3. a-q, b-r, c-p
    4. a-r, b-p, c-q
    Answer

    A. a-q, b-p, c-r

    Shop discounts show utility; measuring is activity; art links show correlation.

34. A teacher teaches all of 'mensuration' in Class 7 and never returns to it in later classes. This is the

    1. Inductive approach
    2. Spiral approach
    3. Concentric approach
    4. Topical approach
    Answer

    D. Topical approach

    A single full treatment without return is topical.

35. Class 6 teaches fraction as parts of a whole; Class 7 adds operations; Class 8 uses fractions in rational numbers. This is best seen as

    1. Drill
    2. Spiral approach
    3. Heuristic method
    4. Topical approach
    Answer

    B. Spiral approach

    Each return builds on earlier learning at a higher level.

36. A textbook explains each new term in simple words before using it. This is a quality of

    1. Mechanical features
    2. Cover design
    3. Price
    4. Language and presentation
    Answer

    D. Language and presentation

    Careful definition of terms belongs to language quality.

37. A book that gives only theory and no solved examples is weak mainly in

    1. Examples and exercises
    2. Cover
    3. Binding
    4. Price
    Answer

    A. Examples and exercises

    A good mathematics book supports concepts with examples.

38. A teacher begins fractions by folding paper strips, then draws pictures, then writes 1/2 and 1/4. This follows Bruner's order of

    1. Enactive, iconic, symbolic
    2. Iconic, symbolic, enactive
    3. Symbolic, iconic, enactive
    4. Symbolic, enactive, iconic
    Answer

    A. Enactive, iconic, symbolic

    Folding is action, pictures are iconic, and written fractions are symbolic.

39. A topic is taught in Class 6 and then taught again in wider form in Classes 7 and 8. How many times is it revisited after first teaching?

    1. 2
    2. 3
    3. 1
    4. 0
    Answer

    A. 2

    It returns in Class 7 and again in Class 8, so twice.

40. Which choice shows a weakness of a purely logical order for Class 5 children?

    1. It is too practical
    2. It uses too many activities
    3. It has too many local examples
    4. It may seem dull and abstract
    Answer

    D. It may seem dull and abstract

    A subject-structured order may not fit young learners.

41. Balanced curriculum design means giving proper place to

    1. Only theory
    2. Only skill drills
    3. Theory and practice, skill and concept
    4. Only activities
    Answer

    C. Theory and practice, skill and concept

    Balance combines both sides.

42. Which feature of a textbook best helps revision?

    1. Thick paper
    2. Summary and review exercises at chapter end
    3. A long preface
    4. Large cover picture
    Answer

    B. Summary and review exercises at chapter end

    Summaries and review exercises help revision.

43. Using local examples such as markets and farming in word problems follows the principle of

    1. Logical order
    2. Community and cultural link
    3. Isolation
    4. Forward-looking
    Answer

    B. Community and cultural link

    Local examples link mathematics to the community.