Teaching Mathematics: Methodology · Chapter 6

Instructional Materials, TLM, Dale's Cone, Mathematics Laboratory and Club

What to remember

Teaching-learning materials (TLM) make abstract ideas concrete. Mathematics is abstract, so aids matter more here than in most subjects.
Edgar Dale's Cone of Experience runs from concrete direct experience at the base to abstract verbal symbols at the top. It is a guide to how concrete or abstract an experience is, not a rule of percentages.
A mathematics laboratory is a place for learning by doing; a mathematics club is a voluntary pupils' group for enrichment.

Instructional materials and TLM

Instructional materials are all things used by teacher and pupils to make learning clear, interesting and lasting. TLM stands for teaching-learning material. Some are teacher-made, some are pupil-made, and some are bought. Low-cost and no-cost materials (bottle caps, sticks, seeds, cardboard, old calendars, string, paper) are encouraged, and they can be made with local help.

Why they matter: they give concrete experience; they save time in explanation; they keep interest; they help the slow learner; they support retention; they link mathematics to life; and they allow learning through several senses.

Classification

Type	Meaning	Mathematics examples
Audio	Heard	Radio talks, audio recordings, rhymes for tables
Visual	Seen	Charts, models, flash cards, graphs, geoboard, number line
Audio-visual	Seen and heard	Films, television, videos, computer lessons
Projected	Need a projector or screen	Slides, overhead projector, LCD projector
Non-projected	No projection needed	Blackboard, charts, models, objects
Activity/manipulative	Handled by pupils	Abacus, tangrams, algebra tiles, Cuisenaire rods, Dienes blocks

Common mathematics materials

Blackboard and chalk: the commonest aid; used for neat steps, tables and figures.
Charts and posters: tables, squares, cubes, number charts, formulae.
Models: solids (cube, cuboid, cone, cylinder, sphere), clinometer, plane-figure cut-outs.
Number aids: abacus, number line, place-value cards, counters, Napier's bones.
Geometry kits: compass, set squares, protractor, ruler, geoboard, tangram, paper for folding.
Colour strips and fraction kits: strips, circles and bars for fractions.
Cuisenaire rods (Georges Cuisenaire) and Dienes blocks (Zoltan Dienes) for number and place value.
Calculator and computer: drawing graphs, checking results, simulations; free dynamic software like GeoGebra helps with geometry and graphs.
Games and puzzles: magic squares, number cards, dice, snake-and-ladder with sums.
Textbook, workbook and reference books, charts of mathematicians.
Community resources: field visits to a shop, bank or post office, local measures.

Principles for selecting and using a TLM

1. It should serve the lesson objective.
2. It should suit the age and level of the pupils.
3. It should be accurate and clear.
4. It should be simple, durable, safe and low in cost.
5. It should be large enough to be seen by all.
6. It should be shown at the right time, not before or after it is needed.
7. It should invite pupils' activity, not only the teacher's display.
8. The teacher should test it before the class and keep it out of sight until needed.
9. It should be removed or covered after use so that attention does not wander.

Edgar Dale's Cone of Experience

Edgar Dale (American educator, 1946) arranged learning experiences in the form of a cone. The base is the most concrete: learning by direct doing. The tip is the most abstract: words and symbols. The cone is a guide to choosing materials: the more concrete the experience, the easier it is for the learner to understand, especially for young children. It also shows that teachers should move from concrete to abstract.

The levels (from base to top) as usually given:

Level (base to top)	Nature
1. Direct, purposeful experiences	Real doing and handling (measuring a field)
2. Contrived experiences	Models, mock-ups, specimens
3. Dramatised experiences	Role play, plays, simulation
4. Demonstrations	The teacher shows how it is done
5. Field trips (study tours)	Visiting a place and observing
6. Exhibits	Displays, fairs, exhibitions
7. Educational television	Programmes seen on TV
8. Motion pictures	Films
9. Recordings, radio and still pictures	Audio or single pictures
10. Visual symbols	Charts, maps, diagrams, graphs
11. Verbal symbols	Spoken or written words, mathematical symbols

Key ideas:

The cone is not a hierarchy of value. The top is not a bad level; what matters is the right mixture for the lesson.
The cone does not give percentages of how much is remembered. The widely repeated figures (10 percent of what we read, 90 percent of what we do) are not Dale's and have no sound research basis. Do not quote them as facts.
Dale's cone supports the principle "concrete to semi-concrete to abstract". It is related to Bruner's enactive, iconic and symbolic modes.
A good mathematics lesson on area could begin with direct activity (tiling a floor with squares), then a model, then a diagram, and finally the formula (symbol).

Mathematics laboratory

A mathematics laboratory is a room, or a corner, where pupils learn mathematics through activity, using materials and experiment. It is a place for "learning by doing" and for verifying and discovering ideas. A good room has tables for groups, shelves, a display board, storage and a stock of materials.

Aims:

Make abstract ideas concrete.
Give a chance for discovery and verification.
Develop interest, accuracy and manipulative skills.
Encourage cooperative work and thinking.
Suit slow and fast learners with different tasks.

Typical equipment: geometry boxes, geoboards, dotted and graph paper, tangrams, models of solids, charts, abacus, measuring tapes and scales, weighing balance, measuring jars, counters and dice, calculators, computers, magic squares, puzzles, clinometer, pair of scissors, cardboard, paper for folding, mathematical games, and books of recreational mathematics.

Activities (examples):

Verify the angle sum of a triangle by paper tearing.
Verify (a + b)² with a model made of squares and rectangles.
Verify the Pythagoras theorem with cut-out squares.
Find pi by measuring circumference and diameter of circular objects.
Measure heights using a clinometer or a shadow.
Make nets of solids and find faces, edges and vertices.
Collect and graph classroom data.

Procedure for a lab period: the teacher states the task; pupils work singly or in groups with the guide sheet; they observe and record; they form a conclusion; the teacher discusses and corrects it.

Role of the teacher: plans and prepares activities; keeps materials in order; guides with questions, not answers; and records pupils' work and progress. Pupils should keep a laboratory notebook.

Merits: concrete learning; interest and retention; scope for individual pace; practice of observation and recording.

Limits: cost, space, time and trained teachers; measurement is not proof, so formal proofs need separate teaching.

Mathematics club

A mathematics club is a voluntary association of pupils (and teachers) interested in mathematics. It meets after or outside regular class time and works through self-activity, under the guidance of a teacher.

Objectives: to build interest and a positive attitude; to give enrichment beyond the textbook; to develop leadership, cooperation and responsibility; to give scope for talented pupils and also help for the weak; to link mathematics to life and culture.

Organisation

Membership is open and voluntary; classes join as members.
A teacher serves as patron or guide.
Pupils elect office-bearers: president, secretary, treasurer, and members of an executive group.
A fixed schedule of meetings and a simple programme for the year.
Funds come from small member contributions and school grants.
A register records membership, activities and accounts.

Activities: quizzes and contests; puzzles and recreational problems; magic squares; mathematical games; talks on mathematicians such as Aryabhata and Ramanujan; a wall magazine or bulletin board; exhibitions and models; surveys and projects; field visits; making charts and TLMs; celebrating Mathematics Day. In India, 22 December (Ramanujan's birthday) is observed as National Mathematics Day.

Point	Mathematics laboratory	Mathematics club
Main work	Experiments and activities within the course	Enrichment and recreation beyond the course
Place	A room or corner of the school	Meets in a hall or classroom
Membership	All pupils of the class use it	Voluntary
Timing	During regular periods	Outside class time (usually)
Aim	Concrete learning, discovery	Interest, talent, leadership

Classroom angle

Use at least one concrete material in each new concept lesson.
Let pupils help to make low-cost TLM; making is learning.
Plan the use of the laboratory as part of the unit plan.
Use the club to support slow learners (peer help) and gifted learners (challenge problems).
Store, label and check materials so that they are ready for the next class.

Exam traps

Dale's cone: base concrete, tip abstract; not "percent remembered".
Direct purposeful experience vs contrived experience: real doing versus models and mock-ups.
Visual symbols vs verbal symbols: charts and diagrams versus words and mathematical symbols.
Audio vs visual vs audio-visual: heard, seen, both.
Projected vs non-projected aids: with projector versus without.
Mathematics lab vs mathematics club: a lab is a learning place within the course; a club is voluntary enrichment.
Verification vs proof: a laboratory result by measurement is only verification.
TLM vs textbook: TLM is the wider set of aids; a textbook is one printed aid.

One-liners

1. TLM stands for teaching-learning material.
2. Edgar Dale proposed the Cone of Experience in 1946.
3. The base of the cone is direct, purposeful experience.
4. The tip of the cone is verbal symbols.
5. Dale's cone moves from concrete to abstract.
6. Dale did not give percentage figures of retention.
7. An abacus, tangram and geoboard are manipulative materials.
8. Cuisenaire rods are linked to Georges Cuisenaire.
9. Dienes blocks are linked to Zoltan Dienes.
10. The laboratory approach rests on learning by doing.
11. A mathematics club is voluntary and managed with teacher guidance.
12. National Mathematics Day is on 22 December.

Practice questions

TLM stands for
1. Time and learning module
2. Teacher-led method
3. Test and learning measure
4. Teaching-learning material
Answer
D. Teaching-learning material
TLM means teaching-learning material.
The Cone of Experience was proposed by
1. Edgar Dale
2. W. H. Kilpatrick
3. Benjamin Bloom
4. Jerome Bruner
Answer
A. Edgar Dale
Dale arranged experiences in a cone.
The base of Dale's cone is
1. Visual symbols
2. Motion pictures
3. Direct purposeful experiences
4. Verbal symbols
Answer
C. Direct purposeful experiences
The base is the most concrete level.
The top of Dale's cone is
1. Exhibits
2. Verbal symbols
3. Demonstrations
4. Direct experiences
Answer
B. Verbal symbols
Words and symbols are the most abstract.
Dale's cone is arranged from
1. Abstract to concrete
2. Teacher to pupil
3. Easy to hard tests
4. Concrete to abstract
Answer
D. Concrete to abstract
The base is concrete and the tip abstract.
Which level lies immediately above 'direct purposeful experiences' in the cone?
1. Demonstrations
2. Dramatised experiences
3. Field trips
4. Contrived experiences
Answer
D. Contrived experiences
The second level is contrived experiences.
Charts, maps and graphs belong to which level of the cone?
1. Visual symbols
2. Verbal symbols
3. Contrived experiences
4. Demonstrations
Answer
A. Visual symbols
Charts and diagrams are visual symbols.
Role play and simulation fit which level?
1. Exhibits
2. Dramatised experiences
3. Verbal symbols
4. Field trips
Answer
B. Dramatised experiences
Acting out is dramatised experience.
Which statement about Dale's cone is correct?
1. It says abstract experience is always bad
2. It states exact percentages of what is remembered
3. It does not give reliable retention percentages
4. It was proposed by Piaget
Answer
C. It does not give reliable retention percentages
The percentages often quoted are not Dale's.
A radio talk on the history of numbers is a
1. Manipulative aid
2. Audio aid
3. Visual aid
4. Projected aid
Answer
B. Audio aid
It is heard only.
A chart of squares is a
1. Audio-visual aid
2. Projected aid
3. Audio aid
4. Visual aid
Answer
D. Visual aid
A chart is seen only.
A film on the life of Ramanujan is an
1. Audio-visual aid
2. Non-projected visual aid
3. Audio aid
4. Manipulative aid
Answer
A. Audio-visual aid
A film is seen and heard.
The abacus, tangram and geoboard are examples of
1. Verbal symbols
2. Projected aids
3. Manipulative materials
4. Audio aids
Answer
C. Manipulative materials
Pupils handle them to learn.
Cuisenaire rods are used mainly for
1. Teaching trigonometry only
2. Number relations and early arithmetic
3. Map reading
4. Land survey
Answer
B. Number relations and early arithmetic
Coloured rods show lengths and number relations.
Dienes blocks are especially used to teach
1. Place value
2. Probability
3. Trigonometry
4. Pythagoras theorem
Answer
A. Place value
They show units, tens, hundreds and thousands.
Georges Cuisenaire is linked with
1. Dienes blocks
2. The geoboard
3. Cuisenaire rods
4. The abacus
Answer
C. Cuisenaire rods
The rods carry his name.
Which is a low-cost TLM for teaching counting?
1. A satellite channel
2. A film projector
3. A computer lab
4. Bottle caps and sticks
Answer
D. Bottle caps and sticks
Local, cheap objects work well.
Which is a principle for selecting a TLM?
1. It should be shown a week before
2. It should serve the lesson objective
3. It should be too small for all to see
4. It should be as costly as possible
Answer
B. It should serve the lesson objective
A TLM must match the objective.
Free dynamic software such as GeoGebra mainly helps to
1. Draw and explore geometric figures and graphs
2. Print textbooks
3. Write attendance
4. Replace all teachers
Answer
A. Draw and explore geometric figures and graphs
It supports geometry and graphs.
A mathematics laboratory is based on
1. Learning by punishment
2. Learning by dictation
3. Learning by rote
4. Learning by doing
Answer
D. Learning by doing
The laboratory is activity-based.
Verifying (a + b)² with a model of squares and rectangles is best done in the
1. Mathematics laboratory
2. Library only
3. Playground only
4. Staff meeting
Answer
A. Mathematics laboratory
A model-based verification suits the lab.
Verification of a result by measurement in the laboratory is
1. Unfit for children
2. An exact formal proof
3. Approximate and not a formal proof
4. Always wrong
Answer
C. Approximate and not a formal proof
Measurement has errors; proof needs reasoning.
Which is an aim of a mathematics laboratory?
1. To replace the textbook
2. To avoid all theory
3. To make abstract ideas concrete
4. To reduce pupil activity
Answer
C. To make abstract ideas concrete
Concrete learning is the central aim.
A mathematics club is
1. A compulsory class
2. A staff council
3. A school board
4. A voluntary pupils' group guided by a teacher
Answer
D. A voluntary pupils' group guided by a teacher
Membership is voluntary.
Which is a normal activity of a mathematics club?
1. Quiz and puzzles
2. Only rote drills
3. Collecting fees
4. Marking the register
Answer
A. Quiz and puzzles
Quizzes and puzzles give enrichment.
Which are the usual office-bearers of a mathematics club?
1. Collector, judge and clerk
2. President, secretary and treasurer
3. Inspector and examiner
4. Warden and cook
Answer
B. President, secretary and treasurer
A simple executive is chosen by members.
National Mathematics Day in India is observed on
1. 14 November
2. 5 September
3. 22 December
4. 2 October
Answer
C. 22 December
It marks Ramanujan's birthday.
A mathematics club differs from the laboratory because the club is
1. Compulsory and for tests
2. Only for teachers
3. Only for the weak
4. Voluntary and for enrichment
Answer
D. Voluntary and for enrichment
A lab is part of regular learning; a club is voluntary.
Study the statements. 1. The base of Dale's cone is the most concrete. 2. The tip of Dale's cone is the most abstract. 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 true.
Study the statements. 1. Dale gave exact percentages of what pupils remember. 2. Dale's cone moves from concrete to abstract. 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; only statement 2 is correct.
Study the statements. 1. A TLM must suit the age of the pupils. 2. A TLM should be shown only when needed. 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 principles of using aids.
Study the statements. 1. A mathematics club is compulsory for all pupils. 2. A mathematics laboratory is based on learning by doing. 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 club is voluntary, so statement 1 is false.
Study the statements. 1. Low-cost materials can serve as good TLM. 2. Expensive electronic aids are always necessary. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Simple local materials are often enough.
Study the statements. 1. In Dale's cone 'visual symbols' are placed below 'verbal symbols'. 2. 'Contrived experiences' are placed above 'verbal symbols'. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Verbal symbols are at the top, so visual symbols lie below them (statement 1 true); contrived experiences lie near the base, below them, so statement 2 is false.
Study the statements. 1. A laboratory result by measurement is only a verification. 2. A mathematics lab needs materials, space and trained teachers. 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 correct.
Study the statements. 1. Projected aids need a projector or screen. 2. A blackboard is a projected aid. 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 blackboard is non-projected.
Match: (a) Audio (b) Visual (c) Audio-visual (d) Manipulative with (p) Video (q) Radio (r) Chart (s) Tangram.
1. a-r, b-q, c-p, d-s
2. a-q, b-r, c-s, d-p
3. a-p, b-q, c-r, d-s
4. a-q, b-r, c-p, d-s
Answer
D. a-q, b-r, c-p, d-s
Radio is audio, chart is visual, video is audio-visual, tangram is manipulative.
Match: (a) Direct purposeful (b) Contrived (c) Demonstration (d) Verbal symbols with (p) Teacher shows how (q) Measuring a field (r) A model (s) Written formula.
1. a-r, b-q, c-p, d-s
2. a-q, b-r, c-p, d-s
3. a-q, b-p, c-r, d-s
4. a-s, b-r, c-p, d-q
Answer
B. a-q, b-r, c-p, d-s
Measuring a field is direct; a model is contrived; showing is demonstration; formula is symbolic.
Match: (a) Dale (b) Cuisenaire (c) Dienes (d) Bruner with (p) Cone of Experience (q) Rods (r) Blocks (s) Spiral curriculum.
1. a-p, b-r, c-q, d-s
2. a-q, b-p, c-r, d-s
3. a-s, b-q, c-r, d-p
4. a-p, b-q, c-r, d-s
Answer
D. a-p, b-q, c-r, d-s
These are the standard associations.
A teacher teaches area of a rectangle by tiling the floor with squares, then a figure, then the formula. This follows
1. Rote method
2. Dictation
3. Abstract to concrete
4. Concrete to abstract
Answer
D. Concrete to abstract
Direct activity, then picture, then symbol.
A teacher takes pupils to a market to find prices and make a bill. In Dale's cone this is closest to
1. Verbal symbols
2. Field trip or direct experience
3. Visual symbols
4. Recordings
Answer
B. Field trip or direct experience
Real contact with the situation is concrete.
A teacher has 24 cardboard squares and 6 pupils working in equal groups. Each group gets
1. 3 squares
2. 4 squares
3. 8 squares
4. 6 squares
Answer
B. 4 squares
24 divided by 6 = 4.
A club has 40 members and fees of Rs 5 per member for the year. Total fees collected are
1. Rs 200
2. Rs 100
3. Rs 45
4. Rs 400
Answer
A. Rs 200
40 x 5 = 200.
A club meets for 2 hours every week. The total hours it meets in 20 weeks are
1. 10
2. 22
3. 40
4. 60
Answer
C. 40
2 x 20 = 40.
A lab has 36 pupils working in groups of 4. The number of groups is
1. 8
2. 12
3. 10
4. 9
Answer
D. 9
36 divided by 4 = 9.

Page 1 of 1

Chapter quiz · 10 questions

0/10 answered

On Mathematics, from the live question bank. 🔒 Answers and solutions are shown to registered students (free).
अध्याय पढ़ने के बाद अभ्यास करें।

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?

10 left. You can check any time.