Pariksha SaathiLesson 5 of 8 · Mathematics Pedagogy
Teaching Approaches and Materials: CPA, Play-Way, Manipulatives and Problem Solving
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Concrete-pictorial-abstract
The CPA sequence rests on Bruner's three modes of representation, enactive, iconic and symbolic. At the concrete stage children handle objects: they make 3 groups of 4 bottle caps. At the pictorial stage they draw or see the same thing: 3 rows of 4 dots, a bar model or equal jumps on a number line. At the abstract stage they write 3 × 4 = 12. The stages are linked, not just passed through: the teacher keeps asking children to connect the symbol with the picture and the object ('Show me where the 3 is in your drawing'), and a child who is stuck at the symbolic level is taken back to pictures or objects, not given more symbols.
Play-way and activity-based learning
Play is a young child's natural way of learning, and the Position Paper says all learning at the pre-primary stage occurs through play. Froebel's kindergarten used gifts (balls, blocks and shapes), Montessori designed self-correcting materials such as number rods and the spindle box, and Henry Caldwell Cook's book The Play Way (1917) gave the method its name. In mathematics, play means games with a purpose: snakes and ladders for counting on, dice and card games for number facts, a classroom shop for money, hopscotch for number order, and puzzles that need reasoning. Activity-based learning, following John Dewey's idea of learning by doing, has children measure the classroom, sort leaves, survey their friends or fold paper to find fractions, and then talk about what they found.
Manipulatives and what each teaches
Dienes blocks (multibase blocks, developed by Zoltan Dienes) model place value and regrouping; Dienes also designed attribute blocks for classification and logic. The abacus shows place value with beads on rods. Cuisenaire rods, 10 coloured rods of lengths 1 to 10 cm devised by the Belgian teacher Georges Cuisenaire and spread by Caleb Gattegno, show number relations such as 3 + 7 = 10, and also fractions. The geoboard, a board with a grid of pegs on which rubber bands make shapes, was also popularised by Gattegno and serves for shapes, area and perimeter. The tangram is a Chinese puzzle of 7 pieces (5 triangles, 1 square and 1 parallelogram) for spatial sense and composing shapes. Other useful materials are the number line, fraction strips, place-value cards, a pan balance, a clock, a measuring tape, mirrors for symmetry and no-cost items such as seeds, matchsticks, bangles and bottle caps. Materials do not teach by themselves: understanding comes from acting on them and reflecting, and the teacher must deliberately link each action to the written symbol.
Problem solving with Polya
George Polya's How to Solve It (1945) describes four steps. Understand the problem: what is known, what is asked, can you say it in your own words? Devise a plan: choose a strategy (heuristic) such as drawing a picture, making a table, looking for a pattern, guess and check, working backwards or solving a simpler problem. Carry out the plan, checking each step. Look back: check the answer against the question, try another method and ask whether the method works for other problems. A real problem is one where the way to the answer is not known in advance; an exercise only practises a known procedure. Children need both, and problem posing, making up their own problems, goes a step further.
Projects and other methods
The project method of William Heard Kilpatrick (1918), based on Dewey's ideas, is a purposeful activity carried out in a social setting. Its steps are usually given as providing a situation, choosing (purposing), planning, executing, evaluating (judging) and recording. Mathematics projects include surveying the class's favourite fruits and drawing a bar graph, planning a class picnic within a budget, measuring the school ground, or keeping a family expenditure record for a month. The laboratory method uses hands-on investigations in a mathematics lab or corner; the heuristic or discovery method (Armstrong; Bruner's discovery learning) makes the child a finder; and cooperative group work lets children compare strategies. Lecture and demonstration have a place, but should not be the main method.
Choosing a method
The best method depends on the concept and the children. Concrete materials and play suit young children and new concepts; pictorial models bridge to symbols; problem solving and projects develop the higher aims; and practice is still needed, but after understanding and in meaningful forms such as games. In CTET, an option that starts with objects or a real situation and moves to symbols, or that lets children discover a rule, is usually preferred over one that begins by stating the rule or formula.
Key facts
- CPA: concrete (3 groups of 4 caps), pictorial (3 rows of 4 dots or a bar model), abstract (3 × 4 = 12); based on Bruner's enactive, iconic and symbolic modes.
- Play-way: Froebel's gifts, Montessori's number rods and spindle box, Caldwell Cook's The Play Way (1917); snakes and ladders teaches counting on.
- Dienes blocks (unit, long, flat, cube) model place value and regrouping; Zoltan Dienes also made attribute blocks for classification.
- Cuisenaire rods: 10 coloured rods of 1 to 10 cm by Georges Cuisenaire, spread by Caleb Gattegno; used for number relations and fractions.
- Geoboard: pegs and rubber bands for shapes, area and perimeter. Tangram: 7 pieces (5 triangles, 1 square, 1 parallelogram) for spatial sense.
- Polya, How to Solve It (1945): understand, devise a plan, carry out the plan, look back; strategies include drawing, a table, a pattern, guess and check, working backwards and a simpler problem.
- A problem has no known path to the answer; an exercise practises a known procedure. Problem posing is making up one's own problems.
- Kilpatrick's project method (1918): situation, purposing, planning, executing, evaluating, recording; for example a class fruit survey shown as a bar graph.
- Materials do not teach by themselves: the teacher links each action on the material to the written symbol.
Quick revision
CPA from Bruner: objects, pictures, symbols, linked both ways; play-way: Froebel gifts, Montessori materials, Caldwell Cook 1917; activity-based learning by doing (Dewey); Dienes blocks for place value, abacus, Cuisenaire rods (Gattegno), geoboard, tangram 7 pieces; Polya 1945: understand, plan, carry out, look back; problem vs exercise; Kilpatrick project method 1918; discover the rule before stating it.