Maths teaching aids and manipulatives for CTET

A child can recite "one ten equals ten ones" and still write 34 + 28 = 512. The words are there; the idea is not. A teaching aid closes that gap by letting the child hold the idea in their hands: ten loose cubes really do line up to make one rod of ten. That is why CTET keeps asking which aid suits which concept, and why the "correct" option is almost never "the teacher demonstrates while children watch".

Questions usually give a concept (place value, area, fractions, shapes) and ask for the best aid, or describe a teacher using an aid and ask what it develops. This guide covers the aids you must know and how to use them well.

Why concrete materials matter

Primary children are, in Piaget's terms, mostly in the concrete operational stage. They reason well about things they can see and handle, and struggle with ideas presented only as symbols. Jerome Bruner described three modes of representation: enactive (doing with objects), iconic (pictures and images) and symbolic (numbers and signs). Good maths teaching moves through them in order, which is where the familiar sequence comes from:

Concrete → pictorial → abstract (symbolic)

The aid is not the goal. It is a bridge to the symbol, and the child should eventually cross it and leave it behind. For the wider picture of how children think in maths, see the nature of mathematics.

The aids you must know

Dienes (base-ten) blocks

Named after Zoltan Dienes, the Hungarian-born mathematician who developed structured blocks for teaching number bases. The base-ten set has four pieces:

They teach place value, regrouping (carrying and borrowing), and the four operations with whole numbers. Their strength is that they are proportional: a long is physically ten times the size of a unit, so the child sees why ten ones make a ten.

Worked example: 34 + 28 with blocks.

1. Show 34 as 3 longs and 4 units; show 28 as 2 longs and 8 units.
2. Put them together: 5 longs and 12 units.
3. Twelve units is more than nine, so exchange ten units for one long.
4. Now there are 6 longs and 2 units: 62.

The "carried 1" on paper is exactly that exchange. A child who has done it with blocks knows what the little 1 means.

The abacus

A spike abacus has vertical rods for ones, tens, hundreds and so on, with beads placed on them. To show 243, put 3 beads on the ones rod, 4 on the tens rod and 2 on the hundreds rod. When a rod reaches ten beads, the child removes them and puts one bead on the next rod to the left.

The abacus teaches place value and the positional nature of our number system, including zero as a placeholder: 305 has an empty tens rod. Unlike Dienes blocks, it is non-proportional. Every bead looks the same; its value depends only on its position. That makes it a more abstract model, so it works best after proportional materials such as bundles of sticks or base-ten blocks.

The geoboard

A square board with pegs or nails in a grid, on which children stretch rubber bands to make shapes. It is widely associated with the educator Caleb Gattegno. It teaches:

• Shapes and their properties: make a triangle with one right angle, a square, a shape with exactly one line of symmetry.
• Area by counting unit squares, and perimeter by counting unit lengths along the edges.
• The difference between area and perimeter.

Worked example. Using unit squares, a child makes rectangles with area 12. A 4 × 3 rectangle has perimeter 2(4 + 3) = 14; a 6 × 2 has perimeter 2(6 + 2) = 16; a 12 × 1 has perimeter 2(12 + 1) = 26. Same area, three different perimeters. This one activity breaks the common belief that shapes with equal area must have equal perimeter.

The tangram

A square cut into seven pieces: two large triangles, one medium triangle, two small triangles, one square and one parallelogram. Children rearrange all seven to make figures, or fit them back into the square. It develops spatial reasoning, shape recognition, and the idea that area is conserved when pieces are rearranged.

It also teaches fractions of a whole. If a small triangle has area 1, then the square, the parallelogram and the medium triangle each have area 2, and each large triangle has area 4. The total is 1 + 1 + 2 + 2 + 2 + 4 + 4 = 16, so a large triangle is 4/16 = 1/4 of the whole square, and a small triangle is 1/16. Two small triangles can be arranged to cover the square, the medium triangle or the parallelogram exactly.

Other aids in brief

Low-cost and no-cost materials

CTET favours material from the child's surroundings: bottle caps, seeds, pebbles, sticks, bangles, matchboxes, old calendars and cardboard. It is cheap, familiar, and children can take it home. Fraction strips cut from old notebook covers are as valid as a boxed kit.

Using aids well, and how they go wrong

Good use:

• Every child handles the material, individually or in small groups. Watching a teacher demonstrate is not the same experience.
• Talk while doing. Ask "What happened when you had twelve units?" so the child puts the action into words.
• Record alongside. Write the symbols next to the action: 5 longs 12 units, then 6 longs 2 units, then 62.
• Fade the aid. Move to drawings of blocks, then to symbols alone, as understanding grows.
• Match the aid to the idea. A tangram cannot teach regrouping; Dienes blocks do little for symmetry.

Common problems:

• The aid becomes a new rote procedure: the child moves beads mechanically without linking them to the number.
• The aid is used only for display on the teacher's table.
• The aid is withdrawn too early, before the child can explain without it, or kept too long, so the child cannot work without it.
• A model is used that is too abstract for the stage, such as the abacus before any grouping experience.

For using the same materials in remedial teaching, see children's errors and remedial teaching.

Scenario questions with reasoning

1. Which aid best shows a child why ten ones make one ten? (a) abacus (b) clock model (c) tangram (d) Dienes blocks
2. A tangram has: (a) 5 pieces (b) 6 pieces (c) 7 pieces (d) 8 pieces
3. A teacher asks children to make different rectangles of area 16 square units on a geoboard and record their perimeters. The activity mainly helps children see that: (a) shapes with the same area can have different perimeters (b) area and perimeter always increase together (c) perimeter is found by counting squares (d) squares have the largest area
4. The abacus is called a non-proportional model because: (a) it has no beads for zero (b) it can show only two-digit numbers (c) the value of a bead depends on its position, not its size (d) it is used only in Paper II
5. A teacher shows fraction strips on the blackboard while children copy the drawings. The main weakness is that: (a) fractions should not be taught with strips (b) drawings are better than strips (c) strips are too costly (d) children are not handling the material themselves
6. A child can add with Dienes blocks but struggles with the same sums on paper. The teacher should: (a) record each block action in symbols alongside, then move through drawings (b) remove the blocks at once (c) give more written drill (d) switch to a calculator
7. The best reason to use bottle caps and sticks as counters is that they are: (a) required by NCF 2005 (b) familiar, low-cost and easy for every child to handle (c) more accurate than blocks (d) easy to keep in the cupboard
8. In the tangram, if the whole square has an area of 16 units, one large triangle has an area of: (a) 2 (b) 4 (c) 6 (d) 8

Answers and reasoning

1. (d). Dienes blocks are proportional: a long is visibly ten units.
2. (c). Two large, one medium and two small triangles, a square and a parallelogram.
3. (a). For area 16: 4 × 4 gives perimeter 16, 8 × 2 gives 20, 16 × 1 gives 34.
4. (c). Position, not size, carries the value.
5. (d). The power of a manipulative lies in the child's own handling.
6. (a). This is the concrete-to-pictorial-to-symbolic bridge; the aid is faded, not snatched away.
7. (b). Familiar, cheap material that every child can use.
8. (b). A large triangle is one quarter of the tangram square.

What to do next

• From memory, write the four Dienes pieces with their values and draw 34 + 28 as block exchanges.
• List the seven tangram pieces and their areas taking a small triangle as 1.
• For each of place value, fractions, area, time and symmetry, name one aid and one activity.
• Move on to problems of teaching maths and the geometry guide.