Teaching Mathematics: Methods and Pedagogy

Different Philosophies of Mathematics

A philosophy of mathematics explains the nature, origin and justification of mathematical knowledge — what mathematics is about and how we come to know it. The major philosophies are:

1. Logicism Mathematics is a branch of logic; all mathematical truths can be derived from logical axioms and definitions. Founded by Frege and Russell. Example: The number 2 is defined logically as the class of all pairs — "2 = the class of all two-membered sets." Arithmetic truths like 2 + 2 = 4 follow from logical laws alone.

2. Formalism Mathematics is a game played with symbols according to strict rules; it is about the manipulation of marks on paper, not about meaning. Associated with Hilbert. Example: In algebra, we treat x as a symbol and apply rules (x + x = 2x) without asking what x "really is." Euclidean geometry is treated as a set of axioms and derivation rules.

3. Intuitionism (Constructivism) Mathematics is a construction of the human mind; a mathematical object exists only if it can be constructed step by step. Founded by Brouwer. Example: A proof that "there exists a number with property P" is accepted only when we can actually construct such a number. Intuitionists reject proof by contradiction for existence claims.

4. Platonism (Realism) Mathematical objects exist independently in an abstract, non-physical world; mathematicians discover them rather than invent them. Associated with Plato. Example: The circle is a perfect ideal object; a drawn circle is only an imperfect copy. The truth of the Pythagorean theorem was "already there" before anyone proved it.

5. Empiricism Mathematical knowledge arises from experience and observation of the physical world, not from pure reason alone. Example: Counting, measuring land and trade led to the development of arithmetic and geometry. Children learn number concepts by handling real objects (stones, sticks).

6. Constructivism (in learning) Learners actively build mathematical meaning by connecting new ideas with prior experience, rather than passively receiving it. Based on Piaget and Vygotsky. Example: A pupil discovers that 3 × 4 = 12 by arranging counters in rows and columns, rather than memorising the table.

7. Logico-Empiricism / Fallibilism Mathematics grows by conjecture, proof and refutation; mathematical knowledge is corrigible and not absolutely certain. Associated with Lakatos. Example: The history of the definition of "polyhedron" shows how Euler's formula V − E + F = 2 was revised when counterexamples were found.

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Major Issues of Mathematics Teaching

1. Fear and negative attitude (math anxiety) Many pupils believe mathematics is difficult and only for "clever" students. This fear lowers confidence and achievement. Example: A student who fails once in algebra stops attempting word problems and sits silent in class.

2. Abstract nature of content Mathematical concepts are abstract, while children think concretely at the primary level. Teaching without concrete materials makes learning meaningless. Example: Teaching "fractions" only by writing ½ on the board, without cutting an apple or folding paper.

3. Lack of trained and qualified teachers Many teachers lack subject mastery and modern methods; they rely only on lecture and textbook. Example: A teacher who cannot explain why the sum of angles of a triangle is 180° simply asks pupils to memorise it.

4. Over-dependence on rote learning and examination Teaching aims at marks, not understanding. Pupils memorise formulas and procedures without reasoning. Example: Students can recite (a + b)² but cannot expand (x + 3)².

5. Inadequate teaching materials and aids Rural schools lack charts, models, geometry boxes, ICT tools and even textbooks. Example: Teaching solid geometry with no models of cube, cylinder or cone.

6. Crowded curriculum and insufficient time The syllabus is long and content-heavy, so teachers rush to finish it and cannot give practice or remedial help. Example: A teacher covers three units in a week before the terminal examination.

7. Individual differences among learners Pupils differ in pace, ability and background, but one uniform method is used for all. Example: Slow learners are left behind while fast learners get bored in the same class.

8. Language and terminology problems Difficult mathematical terms and the gap between home language and the medium of instruction create barriers. Example: Words like "denominator," "congruent" and "coefficient" confuse pupils who study in English as a second language.

9. Poor evaluation practices Examinations test memory and computation only, not reasoning, application or problem-solving. Example: A test paper with twenty mechanical sums and no real-life problem.

10. Lack of parental and administrative support Illiterate parents cannot supervise homework, and schools give little time or budget for mathematics activities. Example: No mathematics club, quiz or project work in the school calendar.

Remedial measures: use of concrete-to-abstract approach, activity and play-way methods, locally made teaching aids, in-service teacher training, continuous assessment, remedial classes and a positive, encouraging classroom climate.

Different Learning Theories: Overview and Comparison

1. Behaviourism

Learning is a change in observable behaviour caused by stimulus–response associations. The mind is treated as a "black box".

• Classical conditioning (Pavlov): A neutral stimulus paired repeatedly with an unconditioned stimulus comes to elicit the response. Example: A pupil who has often been scolded while reading aloud starts feeling anxious the moment the teacher opens the book.
• Operant conditioning (Skinner): Behaviour is strengthened by reinforcement and weakened by punishment. Example: A child completes homework daily because the teacher gives a star each time; the stars are withdrawn and the habit fades.
• Thorndike's laws: Law of readiness, law of exercise, law of effect. Example: Tables recited daily are remembered better than those recited once.

2. Cognitivism

Learning is the reorganisation of mental structures, not mere habit formation. The learner processes, stores and retrieves information.

• Gestalt theory: The whole is more than the sum of parts; learning occurs by insight. Example: A student suddenly "sees" the pattern in a geometry proof instead of memorising steps.
• Piaget's cognitive development: Learning depends on the stage of mental maturity — assimilation, accommodation and equilibration. Example: A pre-operational child calls every four-legged animal a "dog".
• Bruner's discovery learning: Learners construct knowledge by exploring; spiral curriculum and three modes — enactive, iconic, symbolic. Example: Children learn volume by pouring water into different-shaped vessels.

3. Constructivism

Learners actively build knowledge by linking new experience with prior knowledge; knowledge is constructed, not transmitted.

• Vygotsky's socio-cultural theory: Learning is social; Zone of Proximal Development and scaffolding. Example: A child solves a puzzle with the mother's hints and later solves it alone.
• Ausubel's meaningful learning: New material is anchored to existing cognitive structure; advance organisers help. Example: Teaching "mammals" after pupils already know "animals that feed milk".

4. Humanistic Theory

Learning is self-directed and aims at the growth of the whole person.

• Rogers: Learner-centred education, unconditional positive regard, freedom to learn. Example: A student chooses a project on local festivals and learns history, language and art together.
• Maslow: Needs must be satisfied before higher learning; safety and belonging precede self-actualisation. Example: A hungry, frightened child cannot concentrate on mathematics.

5. Social Learning Theory (Bandura)

Learning occurs by observation and imitation of models, without direct reinforcement.

• Example: Children speak politely because they observe the teacher speaking politely; a pupil imitates an elder brother's study habits.

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How These Theories Differ

Conclusion: No single theory explains all learning. Behaviourism suits habit and skill formation, cognitivism suits problem-solving, constructivism suits conceptual understanding, humanism suits motivation and values, and social learning suits attitude and behaviour formation. A good teacher selects and blends them according to the content, age and needs of the learners.

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Four Characteristics of Piaget's Pre-operational Period (2–7 years)

1. Egocentrism

The child cannot see a situation from another person's point of view and believes everyone thinks and feels as he does.

• Example: A four-year-old hides his face and says, "You can't see me now." In the three-mountain task, he describes the scene as it looks from his own seat, not from the doll's position.

2. Animism

The child attributes life and feelings to non-living objects.

• Example: A child says, "The moon is following me," or scolds the chair that hurt him: "Bad chair, you hit me!"

3. Centration (and lack of conservation)

Attention is fixed on one striking feature of an object and the other features are ignored; the child cannot understand that quantity remains the same when appearance changes.

• Example: Water poured from a wide glass into a tall thin glass is thought to be "more". Two equal balls of clay — one rolled into a long snake — are thought to be unequal in amount.

4. Symbolic (Semiotic) Function and Irreversibility

The child represents objects and events through words, mental images, drawings and pretend play, but cannot mentally reverse a sequence of actions.

• Example: A girl uses a stick as a "horse" and plays "teacher–student". Asked, "If 3 + 4 = 7, then 7 − 4 = ?", she cannot reverse the operation and answer 3.

Other supporting traits: transductive reasoning (from particular to particular), inability to classify by more than one criterion, and rigid, irreversible thought.

Krathwohl (with Bloom and Masia, 1964) classified the affective domain into five levels, arranged from simple to complex. They are:

1. Receiving (Attending)
2. Responding
3. Valuing
4. Organization
5. Characterization by a Value or Value Complex

1. Receiving (Attending) This is the lowest level. The learner becomes aware of, and is willing to attend to, a particular stimulus or phenomenon. Examples: A student listens attentively when the teacher talks about environmental pollution; a child notices the national flag during assembly.

2. Responding The learner actively participates and reacts to the stimulus, showing willingness and satisfaction in responding. Examples: A student voluntarily answers questions in class; participates in a cleanliness campaign; helps a friend in need.

3. Valuing The learner attaches worth or importance to a particular object, behaviour or idea, and shows commitment to it. Examples: A student values honesty and refuses to cheat in an examination; shows respect for elders and national heroes; takes pride in Nepali culture.

4. Organization The learner organizes different values into a consistent system, resolves conflicts between them, and gives priority to some values over others. Examples: A student balances study time and social service; gives more importance to honesty than to personal benefit; develops a personal code of conduct.

5. Characterization by a Value or Value Complex This is the highest level. Values are internalized so deeply that they become part of the learner's personality and consistently guide behaviour. Examples: A person who is always truthful and disciplined in all situations; a teacher who is devoted to duty and social service throughout life; a citizen who consistently works for national welfare.

Note: The levels proceed from simple attention → active participation → commitment → systematization → total internalization, i.e., from the lowest to the highest level of affective learning.

Principles and Standards Advocated by NCTM

The National Council of Teachers of Mathematics (NCTM), USA, put forward a framework known as Principles and Standards for School Mathematics (2000). It consists of six principles and five content standards along with five process standards.

A. Six Principles

1. Equity Principle — All students, regardless of background, must have access to a high-quality mathematics education with adequate support and high expectations.

2. Curriculum Principle — The mathematics curriculum should be coherent, focused on important mathematics, and well articulated across the grades.

3. Teaching Principle — Effective mathematics teaching requires understanding what students know, what they need to learn, and then challenging and supporting them to learn it well.

4. Learning Principle — Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.

5. Assessment Principle — Assessment should support the learning of important mathematics and furnish useful information to both teachers and students.

6. Technology Principle — Technology is essential in teaching and learning mathematics; it influences the mathematics taught and enhances students' learning.

B. Five Content Standards

1. Number and Operations — Understanding numbers, ways of representing them, relationships among numbers, and number systems.
2. Algebra — Understanding patterns, relations and functions; representing and analysing mathematical situations.
3. Geometry — Analysing characteristics and properties of two- and three-dimensional shapes and developing spatial reasoning.
4. Measurement — Understanding measurable attributes of objects and the units, systems and processes of measurement.
5. Data Analysis and Probability — Collecting, organising and analysing data, and understanding basic concepts of probability.

C. Five Process Standards

1. Problem Solving — Building new mathematical knowledge through solving problems arising in mathematics and other contexts.
2. Reasoning and Proof — Recognising reasoning and proof as fundamental aspects of mathematics.
3. Communication — Organising and consolidating mathematical thinking through communication.
4. Connections — Recognising and using connections among mathematical ideas and to contexts outside mathematics.
5. Representation — Creating and using representations to organise, record and communicate mathematical ideas.

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Process of Curriculum Development in Nepal

Curriculum development in Nepal is carried out mainly by the Curriculum Development Centre (CDC), Sanothimi, under the Ministry of Education, Science and Technology. The process generally follows these steps:

1. Need Identification and Pre-planning

Problems in the existing curriculum, national needs, social demands, global trends and feedback from stakeholders are studied. A decision is taken to revise or develop a new curriculum.

2. Formation of Committees

Subject committees, subject experts, teachers, subject specialists and representatives of concerned agencies are appointed. A Curriculum Development Committee and subject sub-committees are formed.

3. Determination of Objectives and Scope

National goals of education are translated into level-wise and subject-wise objectives. The scope, sequence and weightage of contents are decided.

4. Drafting the Curriculum

Subject committees prepare the draft curriculum including objectives, content, teaching methods, learning activities, instructional materials and evaluation scheme.

5. Try-out / Pilot Testing

The draft is tried out in a limited number of representative schools. Feedback is collected from teachers, students and experts.

6. Revision and Improvement

On the basis of try-out results and expert suggestions, the draft is revised and improved.

7. Approval

The revised curriculum is approved by the Curriculum Development Council and the Ministry of Education.

8. Implementation

The approved curriculum is implemented throughout the country. Teacher training, textbooks, teacher guides and other materials are prepared and distributed.

9. Monitoring and Evaluation

The implemented curriculum is monitored continuously. Feedback from the field is collected and used for further revision. Thus curriculum development is a continuous and cyclical process.

Conclusion: In Nepal, curriculum development moves through need identification, committee formation, drafting, try-out, revision, approval, implementation and evaluation — making it a systematic, participatory and ongoing process.

Preparation of a geoboard of $10\times 10$ size

A geoboard is a square board with nails fixed in a regular grid, used to teach geometry with rubber bands. To prepare a $10\times 10$ geoboard:

1. Materials required: a square plywood board (about 25 cm $\times$ 25 cm), thin smooth nails (about 100–121), a ruler, a pencil, a hammer, sandpaper, and coloured rubber bands.
2. Smooth the board: rub the plywood with sandpaper so that the surface is smooth and rubber bands do not tear.
3. Draw the grid: using the ruler and pencil, mark a square grid on the board. For a $10\times 10$ geoboard, mark 10 equal intervals along each side (interval about 2 cm), giving $11\times 11$ intersecting points.
4. Mark the nail points: at every intersection of the horizontal and vertical lines, mark a dot with the pencil. These dots are the positions of the nails.
5. Fix the nails: hammer one thin nail at each marked dot so that the nails stand upright and only about 1 cm of each nail remains above the board. Keep all nails of equal height.
6. Check the spacing: measure the distance between neighbouring nails; it must be equal in both directions. Label the rows and columns (1 to 10) if desired.
7. Finish: clean the board and keep coloured rubber bands ready for use. The geoboard is now ready.

Demonstration of $h^2=p^2+b^2$ (Pythagoras theorem) with a rubber band

1. On the geoboard, stretch a rubber band to form a right-angled triangle whose perpendicular sides lie along the grid lines. For example, take the base $b=4$ units and the perpendicular $p=3$ units, so the vertices are at $(0,0)$, $(4,0)$ and $(0,3)$.
2. The third side (hypotenuse) is stretched by the rubber band from $(4,0)$ to $(0,3)$. Count the grid squares it crosses: it passes through exactly $5$ units, so $h=5$.
3. Verify by counting unit squares:
   • Square on $p$: $3\times 3=9$ small squares.
   • Square on $b$: $4\times 4=16$ small squares.
   • Square on $h$: $5\times 5=25$ small squares.
4. Now $p^2+b^2=9+16=25=h^2$. Hence $h^2=p^2+b^2$ is verified.
5. Repeat with other triples such as $(6,8,10)$ or $(5,12,13)$ to confirm the result holds for every right-angled triangle formed on the geoboard.

Thus the geoboard gives a concrete, activity-based proof of the Pythagoras theorem.

Take one situation from the Grade X geometry (circle and mensuration) unit and build one subjective question for each of the four higher levels of Bloom's taxonomy: Application, Analysis, Evaluation and Creation.

Situation: A square garden $ABCD$ of side $14$ m has a circular pond built inside it so that the pond touches all four sides.

(a) Application (2 marks)

Question: Using $\pi = \frac{22}{7}$, find the area of the pond and the area of the garden left around it.

Model answer: The pond is inscribed in the square, so its diameter equals the side, $2r = 14$ m and $r = 7$ m.

$\text{Area of pond} = \pi r^2 = \frac{22}{7}\times 7^2 = 154 \text{ m}^2$

$\text{Area of garden left} = 14^2 - 154 = 196 - 154 = 42 \text{ m}^2$

(b) Analysis (2 marks)

Question: Find the ratio of the pond area to the garden area. Does the ratio change if the side of the square changes? Explain.

Model answer: For side $s$, $r = \frac{s}{2}$, so

$\frac{\text{pond}}{\text{garden}} = \frac{\pi (s/2)^2}{s^2} = \frac{\pi}{4}$

For $s = 14$ this is $\frac{154}{196} = \frac{11}{14}$. The ratio does not change, because $s^2$ cancels. Every inscribed circle covers the same fraction, $\frac{\pi}{4}\approx 78.5\%$, of its square, whatever the size.

(c) Evaluation (1 mark)

Question: A student claims that a rectangular garden $14$ m $\times$ $10$ m can also have a pond touching all four sides. Judge the claim with reasons.

Model answer: The claim is wrong. The centre of a circle touching two opposite sides lies midway between them, and the radius is half their distance apart. Here that gives $r = 7$ m for the longer pair and $r = 5$ m for the shorter pair, which cannot both hold. The largest circle that fits has $r = 5$ m and touches only the two longer sides. A circle can touch all four sides only when the figure is a square.

(d) Creation (2 marks)

Question: Design a square garden with an inscribed circular pond of area $616$ m$^2$. State the side of the garden and the area left around the pond.

Model answer:

$\pi r^2 = 616 \Rightarrow r^2 = 616\times\frac{7}{22} = 196 \Rightarrow r = 14 \text{ m}$

Side of garden $= 2r = 28$ m, so the garden area is $28^2 = 784$ m$^2$ and the area left is $784 - 616 = 168$ m$^2$. Check: the distance from the centre to each side is $14$ m, which equals $r$, so the pond touches all four sides.