Teaching Mathematics: Methodology · Chapter 4

Methods of Teaching Mathematics

What to remember

Inductive goes from examples to a rule (particular to general); deductive goes from a rule to cases (general to particular). Analytic goes from the unknown back to the known; synthetic goes from the known forward to the unknown.
Heuristic, laboratory, project and problem-solving methods are activity-based and learner-centred. The teacher guides; the pupil discovers, does and finds.
Polya's four steps for problem solving: understand the problem, devise a plan, carry out the plan, look back.

Choosing a method

A method is the way the teacher organises content and activity to reach an objective. There is no single best method. A teacher picks a method by the objective, the topic, the age of the pupils, the time and the materials. Good teaching often mixes methods, for example the inductive method to find a rule and the deductive method to apply it.

Inductive method

Induction means moving from particular examples to a general rule. The pupil sees many cases, notices a pattern and states the rule herself.

Steps: (1) present several examples, (2) observe and compare, (3) generalise (state the rule), (4) test or verify the rule on new cases.

Example: Find the sum of the first few odd numbers.

1 = 1 = 1²; 1 + 3 = 4 = 2²; 1 + 3 + 5 = 9 = 3²; 1 + 3 + 5 + 7 = 16 = 4².

Generalisation: the sum of the first n odd numbers is n². Verify: 1 + 3 + 5 + 7 + 9 = 25 = 5².

Merits	Limits
Active, meaningful learning	Slow and takes more time
Rule is discovered, not memorised	Conclusions from a few cases may be wrong
Suits primary and middle classes	Not enough for formal proof
Builds reasoning and interest	Needs a skilled teacher

Inductive reasoning is a good way to find a result, but the result still needs proof in higher classes.

Deductive method

Deduction moves from a general rule to particular cases. The rule, formula or definition is given first; then it is applied to examples.

Steps: state the rule, explain it, apply it to problems, practise.

Example: Give the formula (a + b)² = a² + 2ab + b² and then use it for 103² = (100 + 3)² = 10000 + 600 + 9 = 10609.

Merits	Limits
Fast and short	Rote learning and little understanding
Good for practice and revision	Passive pupils
Used in proof	Rule may seem to come from nowhere

Induction and deduction are complementary. Induction discovers; deduction confirms and applies.

Analytic method

Analysis means breaking a problem into parts. We start from the unknown (what is to be proved or found) and move backward, asking "what do we need to know first?" until we reach known facts.

Example: To prove that the diagonals of a rhombus are perpendicular. We need to show the angle at their meeting point is 90°. For that, the two triangles on one side must be congruent. Congruent by SSS, because all sides are equal and the diagonals bisect each other. These facts are known, so the chain is complete.

Analysis is the thinking path; it shows how the solution was found. It follows "unknown to known". It is slower and longer, but it gives reasons for each step.

Synthetic method

Synthesis means putting together. We start from the known (given data and known facts) and move forward step by step to the unknown (the result).

Example: In the rhombus proof above, the synthetic way writes: given a rhombus; its sides are equal; diagonals bisect each other; triangles are congruent by SSS; so the angles are equal and are 90° each.

Synthesis is the presentation path and is shorter and neater. It is how proofs are normally written.

Point	Analytic	Synthetic
Starts from	Unknown (conclusion)	Known (data)
Direction	Backward	Forward
Chain	Unknown to known	Known to unknown
Nature	Shows how a result was found	Shows the result in short form
Time	Longer	Shorter
Best use	Finding a proof or solution	Writing the proof

Analysis and synthesis are two sides of one process: analysis discovers the path, and synthesis writes it forward. Teachers often use analytic first and then present synthetically.

Heuristic method

The word comes from the Greek *heurisko*, "I find". The method was developed by H. E. Armstrong. The teacher puts the pupil in the position of a discoverer. The pupil finds out facts on her own, and the teacher only guides, with questions and hints, never giving the answer directly.

Steps: pose a problem; collect data; examine and experiment; arrive at a result; verify.

Merits: develops curiosity, independence, scientific attitude and retention.

Limits: slow; needs small classes and a skilled teacher; difficult to complete a long syllabus; some topics cannot be discovered.

Laboratory method

Based on learning by doing. Pupils work with objects, models, paper cutting, graphs, and measuring devices in a mathematics laboratory, and find results by experiment and observation. It rests on the idea that the best teacher is experience.

Steps: statement of the problem; planning the experiment; collecting materials; performing the activity; observing and recording; arriving at a conclusion; verification.

Example: verifying that the angle sum of a triangle is 180° by tearing the three corners and fitting them on a straight line.

Merits: concrete, meaningful, interest; suited to young learners and to geometry and mensuration.

Limits: needs equipment, space and time; results by measurement are only approximate and are not proof.

Project method

Associated with William Heard Kilpatrick, a follower of John Dewey. It rests on Dewey's pragmatism and the principle of "learning by living". A project is a wholehearted purposeful activity carried out in a social setting. The pupils themselves choose, plan and complete the project.

Steps: (1) providing a situation, (2) choosing and purposing, (3) planning, (4) executing, (5) evaluating, (6) recording.

Examples in mathematics: preparing a class budget; surveying the heights of pupils and making a chart; designing a school garden with area and perimeter; making a shop and billing.

Merits: real life learning, cooperation, responsibility, correlation with other subjects.

Limits: time-consuming; hard to cover the syllabus in order; it needs materials; only some topics suit it; pupils may be uneven in contribution.

Problem-solving method

Pupils meet a real problem, and the teacher guides them to find a solution through thinking. John Dewey described problem solving in *How We Think*. George Polya, in *How to Solve It*, gave four steps:

1. Understand the problem (what is given, what is asked).
2. Devise a plan (look for a pattern, draw a figure, work backward, try a simpler case, make a table).
3. Carry out the plan (do and check each step).
4. Look back (check the answer, see if there is another way, think about use elsewhere).

Worked example. A rectangle's length is 3 cm more than its breadth, and the perimeter is 26 cm. Find its sides.

Understand: unknowns are length and breadth. Plan: let breadth be b, so length is b + 3 and perimeter is 2(b + (b + 3)). Carry out: 2(2b + 3) = 26, so 2b + 3 = 13, so b = 5, length = 8. Look back: 2 x (5 + 8) = 26, correct.

Merits: builds reasoning and creative thinking; links to life. Limits: needs time, planned problems and teacher skill.

Summary of methods

Method	Key idea	Starts from	Linked name or phrase
Inductive	Examples to rule	Particular	Observe, generalise
Deductive	Rule to examples	General	Apply, practise
Analytic	Break up	Unknown	Backward
Synthetic	Put together	Known	Forward
Heuristic	Discovery	Problem	Armstrong
Laboratory	Doing	Experiment	Learning by doing
Project	Purposeful activity	Situation	Kilpatrick
Problem solving	Four steps	Problem	Polya

Classroom angle

Use induction to teach a new formula, and deduction to apply it.
Use analysis when pupils get stuck on a riders problem ("what do I need first?"), then write the final answer synthetically.
Laboratory work first, project work later in the term, and problem-solving throughout.
Always end an inductive or laboratory result by saying whether it is only a conclusion from cases or a proved result.

Exam traps

Inductive vs deductive: particular to general versus general to particular.
Analytic vs synthetic: analytic is unknown to known; synthetic is known to unknown.
Heuristic vs laboratory: discovery by questioning and guidance versus working with materials.
Project method: Kilpatrick, not Armstrong.
Heuristic method: Armstrong, not Kilpatrick.
Polya's steps: understand, plan, carry out, look back; the order cannot change.
Proof vs verification: a laboratory check by measurement is verification, not proof.
Method vs technique: a method is the whole plan; a technique is one device within it.

One-liners

1. Induction proceeds from particular cases to the general rule.
2. Deduction proceeds from the general rule to particular cases.
3. Analysis starts from the unknown and works back to the known.
4. Synthesis starts from the known and works forward to the unknown.
5. The heuristic method is associated with H. E. Armstrong.
6. "Heuristic" comes from a Greek word meaning "I find".
7. The laboratory method rests on learning by doing.
8. W. H. Kilpatrick developed the project method on John Dewey's ideas.
9. Project steps: situation, choosing, planning, executing, evaluating, recording.
10. Polya wrote *How to Solve It* and gave four steps.
11. The fourth step of Polya is "look back".
12. The sum of the first n odd numbers is n squared.

Practice questions

Moving from particular examples to a general rule is the
1. Deductive method
2. Analytic method
3. Synthetic method
4. Inductive method
Answer
D. Inductive method
Induction goes from particular to general.
Moving from a general rule to particular cases is the
1. Project method
2. Inductive method
3. Heuristic method
4. Deductive method
Answer
D. Deductive method
Deduction goes from general to particular.
The analytic method proceeds from
1. Unknown to known
2. Known to unknown
3. Simple to complex only
4. Concrete to symbolic only
Answer
A. Unknown to known
Analysis works backward from what is to be found.
The synthetic method proceeds from
1. Result to data
2. Known to unknown
3. Rule to exception
4. Unknown to known
Answer
B. Known to unknown
Synthesis builds forward from the given data.
The heuristic method of teaching is associated with
1. Edgar Dale
2. W. H. Kilpatrick
3. George Polya
4. H. E. Armstrong
Answer
D. H. E. Armstrong
Armstrong developed the heuristic method.
The project method was developed by
1. Herbart
2. W. H. Kilpatrick
3. H. E. Armstrong
4. George Polya
Answer
B. W. H. Kilpatrick
Kilpatrick built it on Dewey's ideas.
The four steps of problem solving were given by
1. George Polya
2. Jerome Bruner
3. Bloom
4. Mager
Answer
A. George Polya
Polya's How to Solve It gives four steps.
The correct order of Polya's steps is
1. Plan, understand, look back, carry out
2. Look back, plan, understand, carry out
3. Understand, plan, carry out, look back
4. Understand, carry out, plan, look back
Answer
C. Understand, plan, carry out, look back
The fixed order is understand, devise a plan, carry out, look back.
The word 'heuristic' comes from a Greek word meaning
1. I find
2. I teach
3. I measure
4. I prove
Answer
A. I find
Greek heurisko means 'I find'.
The laboratory method rests on the idea of
1. Learning by listening
2. Learning by doing
3. Learning by copying
4. Learning by memory
Answer
B. Learning by doing
The lab method is activity-based.
Verifying that the three angles of a triangle sum to 180 degrees by tearing corners is an example of the
1. Deductive method
2. Synthetic method
3. Analytic method
4. Laboratory method
Answer
D. Laboratory method
It is an experiment with materials.
Which is a weakness of the inductive method?
1. Conclusions from a few cases may be wrong
2. It is too fast
3. It is limited to proofs
4. It makes pupils passive
Answer
A. Conclusions from a few cases may be wrong
General rules drawn from few cases can fail.
Which is a merit of the deductive method?
1. It needs no rules
2. It always builds deep understanding
3. It is short and fast for practice
4. It is purely discovery based
Answer
C. It is short and fast for practice
Deduction saves time and suits practice.
Writing a proof from the given data forward to the conclusion in short form is the
1. Project method
2. Synthetic method
3. Analytic method
4. Heuristic method
Answer
B. Synthetic method
Proofs are normally written synthetically.
Finding how a proof was discovered, by asking 'what do I need first?', uses the
1. Synthetic method
2. Deductive drill
3. Laboratory method
4. Analytic method
Answer
D. Analytic method
Analysis supplies the path from the conclusion.
Which is a step of the project method?
1. Cramming
2. Providing a situation
3. Reciting
4. Dictation
Answer
B. Providing a situation
The first step is to provide a situation.
Preparing a class budget by pupils with planning and recording is an example of the
1. Project method
2. Deductive method
3. Drill
4. Lecture method
Answer
A. Project method
It is purposeful social activity planned by pupils.
Which method best suits discovering that the sum of the first n odd numbers is n squared?
1. Synthetic method
2. Lecture method
3. Inductive method
4. Deductive method
Answer
C. Inductive method
Pupils observe cases and generalise.
Using the formula (a + b)² = a² + 2ab + b² to compute 103 squared by writing 103 as 100 + 3 is the
1. Inductive method
2. Heuristic method
3. Project method
4. Deductive method
Answer
D. Deductive method
A given rule is applied to a case.
1 + 3 + 5 + 7 + 9 = 25. In the inductive pattern, 25 equals
1. 4²
2. 5²
3. 5 x 6
4. 6²
Answer
B. 5²
The sum of the first 5 odd numbers is 5 squared = 25.
The sum of the first 12 odd numbers is
1. 144
2. 132
3. 156
4. 121
Answer
A. 144
12 squared = 144.
Using (100 + 3)², the value of 103² is
1. 10306
2. 10900
3. 10609
4. 10069
Answer
C. 10609
10000 + 2 x 100 x 3 + 9 = 10609.
A rectangle has breadth 5 cm and length 3 cm more than breadth. Its perimeter is
1. 16 cm
2. 40 cm
3. 21 cm
4. 26 cm
Answer
D. 26 cm
Length = 8 cm, perimeter = 2 x (5 + 8) = 26 cm.
The length of a rectangle is 3 cm more than its breadth and the perimeter is 26 cm. The breadth is
1. 6 cm
2. 8 cm
3. 5 cm
4. 4 cm
Answer
C. 5 cm
2(2b + 3) = 26 gives b = 5.
In Polya's fourth step on the rectangle problem above, a pupil checks 2 x (5 + 8). The value checked is
1. 16
2. 13
3. 40
4. 26
Answer
D. 26
2 x 13 = 26 matches the given perimeter.
Using (100 - 1)² = 10000 - 200 + 1, the value of 99² is
1. 9999
2. 9801
3. 9601
4. 9901
Answer
B. 9801
10000 - 200 + 1 = 9801.
A teacher shows squares of 11, 101 and 1001 as 121, 10201 and 1002001, and asks for the pattern. This is the
1. Inductive method
2. Analytic method
3. Lecture method
4. Deductive method
Answer
A. Inductive method
Pupils generalise from particular cases.
A problem has 4 steps in Polya's model. If 6 minutes are given to each step equally in a 24-minute activity, the time per step is
1. 8 minutes
2. 4 minutes
3. 12 minutes
4. 6 minutes
Answer
D. 6 minutes
24 divided by 4 = 6.
Study the statements. 1. Induction goes from general to particular. 2. Deduction goes from general to particular. 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 is false; induction goes from particular to general.
Study the statements. 1. Analysis starts from the unknown. 2. Synthesis starts from the known. 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.
Study the statements. 1. The heuristic method was developed by Kilpatrick. 2. The project method was developed by Armstrong. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
D. Neither 1 nor 2
The names are reversed: Armstrong for heuristic, Kilpatrick for project.
Study the statements. 1. The laboratory method uses objects and experiments. 2. A measurement in the laboratory is the same as a formal proof. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Measurement gives approximate verification, not formal proof.
Study the statements. 1. Induction can find a rule but does not prove it. 2. Deduction is used in proof. 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 correct.
Study the statements. 1. Analytic method is generally shorter than synthetic. 2. Synthetic method gives the neat final presentation. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Analytic is longer, so statement 1 is false.
Study the statements. 1. In the project method pupils choose and plan the work. 2. A project is a purposeful activity in a social setting. 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 defining features.
Study the statements. 1. The heuristic method is quick and suits covering a long syllabus. 2. The heuristic method develops curiosity. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
The heuristic method is slow, so statement 1 is false.
Match: (a) Inductive (b) Deductive (c) Analytic (d) Synthetic with (p) General to particular (q) Particular to general (r) Unknown to known (s) Known to unknown.
1. a-q, b-p, c-r, d-s
2. a-p, b-q, c-r, d-s
3. a-r, b-s, c-q, d-p
4. a-q, b-p, c-s, d-r
Answer
A. a-q, b-p, c-r, d-s
Inductive is particular to general; deductive is the reverse; analytic is unknown to known; synthetic is known to unknown.
Match: (a) Heuristic (b) Project (c) Problem solving with (p) Kilpatrick (q) Armstrong (r) Polya.
1. a-q, b-p, c-r
2. a-q, b-r, c-p
3. a-p, b-q, c-r
4. a-r, b-p, c-q
Answer
A. a-q, b-p, c-r
Armstrong, Kilpatrick and Polya respectively.
Match: (a) Understand (b) Devise a plan (c) Carry out (d) Look back with (p) Check the answer (q) Work backward or draw a figure (r) Find what is given and asked (s) Do each step and check it.
1. a-p, b-q, c-s, d-r
2. a-r, b-s, c-q, d-p
3. a-q, b-r, c-s, d-p
4. a-r, b-q, c-s, d-p
Answer
D. a-r, b-q, c-s, d-p
Polya's steps in order.
A teacher gives the formula for the area of a circle first and then asks pupils to solve five problems. She is mainly using the
1. Heuristic method
2. Deductive method
3. Project method
4. Inductive method
Answer
B. Deductive method
A rule is stated first and then applied.
A teacher lets pupils cut circles, divide into sectors, rearrange and find that the area is about pi times r squared. She is mainly using the
1. Dictation method
2. Pure lecture
3. Rote drill
4. Laboratory (and inductive) approach
Answer
D. Laboratory (and inductive) approach
Pupils experiment with materials and generalise.
Pupils stuck on a riders problem are asked 'What must I know to show these sides are equal?' This reflects the
1. Analytic method
2. Project method
3. Synthetic method
4. Drill method
Answer
A. Analytic method
Working backward from the conclusion is analysis.
In an inductive lesson, after pupils state the rule, the teacher should next
1. Erase the board
2. Start a new topic
3. Test it on new cases
4. Give the proof only
Answer
C. Test it on new cases
Verification is the last inductive step.
Which method is most time consuming and hard to use for a long syllabus?
1. Lecture method
2. Drill method
3. Project method
4. Deductive method
Answer
C. Project method
Projects need long and flexible time.
A project in mathematics on measuring the plots of a school garden for area and perimeter mainly links to
1. Correlation with real life
2. Rote memorising only
3. Dictation
4. Isolation of topics
Answer
A. Correlation with real life
Projects bring real situations into mathematics.

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Chapter quiz · 10 questions

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1.Vinod, Kajal and Arpit invest ₹1,040, ₹1,390 and ₹1,890, respectively, to start a business. If the profit at the end of the year is ₹1,440, then what is the share of Arpit in the profit? 6077
2.In a quadrilateral WXYZ, ∠ W = 74° and ∠ X = 58°. The bisectors of ∠ Y and ∠ Z meet at O. What is the measure of ∠ ZOY?
3.In a quadrilateral MNST, ∠ M = 63° and ∠ N = 99°. The bisectors of ∠ S and ∠ T meet at O. What is the measure of ∠ TOS?
4.In △ ABC, BD ⟂ AC at D and ∠ DBC = 80°. E is a point on BC such that ∠ CAE = 37°. What is the measure of ∠ AEB?
5.In △ ABC, BD ⟂ AC at D and ∠ DBC = 72°. E is a point on BC such that ∠ CAE = 71°. What is the measure of ∠ AEB?
6.In △ ABC, BD ⟂ AC at D and ∠ DBC = 21°. E is a point on BC such that ∠ CAE = 27°. What is the measure of ∠ AEB?
7.In a quadrilateral IJKL, ∠ I = 17° and ∠ J = 51°. The bisectors of ∠ K and ∠ L meet at Z. What is the measure of ∠ LZK?
8.In a quadrilateral ABCD, ∠ A = 32° and ∠ B = 58°. The bisectors of ∠ C and ∠ D meet at O. What is the measure of ∠ DOC?
9.In quadrilateral PQRS, the bisectors of ∠ R and ∠ S meet at the point T (inside the quadrilateral) and ∠ STR = 33°. If the ratio of ∠ P to ∠ Q is 3: 8, then what is the difference between the measures of ∠ Q and ∠ P?
10.In quadrilateral PQRS, the bisectors of ∠ R and ∠ S meet at the point T (inside the quadrilateral) and ∠ STR = 18°. If the ratio of ∠ P to ∠ Q is 2: 7, then what is the difference between the measures of ∠ Q and ∠ P?

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