Problems of teaching mathematics and problem-solving pedagogy for CTET

A child who helps at a family vegetable stall can total a customer's bill in their head, give correct change, and round prices sensibly. The same child may freeze when the teacher writes 45 + 38 on the board. The mathematics is the same. What changed is the setting: fear of being wrong in public, a procedure learnt without meaning, and a problem that has nothing to do with the child's life. That, in one scene, is the "problems of teaching" topic in CTET.

The maths pedagogy syllabus for both papers lists "problems of teaching", "community mathematics" and "language of mathematics". Questions usually describe a classroom difficulty and ask for its cause or the best response. This guide covers the main problems, the causes behind them, and problem solving as a way of teaching, which is the remedy CTET most often points to.

The main problems and what lies behind them

The NCF 2005 position paper on the teaching of mathematics names several of these directly: many children fear maths and feel they fail at it, the curriculum disappoints both a talented minority and the non-participating majority, and crude assessment makes mathematics look like mechanical computation. See the nature of mathematics for the paper's aims.

Maths anxiety

Anxiety is not a lack of ability. It is a feeling that uses up the working memory a child needs for thinking, so an anxious child performs below what they know.

Causes a teacher can control:

• Punishing or mocking wrong answers.
• Timed drills with public results, such as making the slowest child stand.
• Labelling children as "weak" or "good at maths".
• Only one accepted method, so a different correct approach is marked wrong.

What helps:

• Treat errors as information about thinking, not as failure. See children's errors and remedial teaching.
• Praise strategy and effort, not speed.
• Let children work in pairs or small groups before answering alone.
• Start from success: tasks the child can do, then small steps forward.
• Use games, puzzles and concrete material.

The language of mathematics

Maths has its own vocabulary, and much of it clashes with everyday words. "Table" means a piece of furniture at home, "difference" means a quarrel, "odd" means strange, "product" is something sold in a shop and "volume" is how loud the radio is. A child reads a word problem through everyday meanings unless the teacher builds the maths meanings through talk.

Children also learn to spot "key words": "more" means add, "left" means subtract. This shortcut fails often.

Worked example. "A child has 12 marbles. That is 5 more than a friend has. How many marbles does the friend have?"

A key-word reader sees "more" and adds: 12 + 5 = 17. But the friend has fewer. The child has 12, which is 5 more than the friend, so the friend has 12 − 5 = 7. Check: 7 + 5 = 12.

The remedy is not a longer list of key words. It is making sense of the situation: act it out, draw it, retell it in the child's own words, and let children write their own word problems for a given sum. Children who come to school speaking a different home language benefit from discussing the problem in that language first.

Community mathematics

Community maths means using the mathematics already present in children's homes and neighbourhoods: shopping and change, measuring cloth or grain, reading bus timetables, patterns in rangoli and weaving, calendars, estimating quantities in cooking, and the counting and measuring done by farmers, masons and tailors.

It serves three purposes: it makes maths meaningful, it values the knowledge children bring from home, and it gives the teacher real contexts for problems. A class survey of how families measure things (in handspans, kilograms, litres, bighas) is both community maths and data handling.

Teaching through problem solving

A problem is a task where the method is not immediately known. An exercise repeats a method just taught. "Add 234 and 158" is an exercise; "Find three consecutive numbers that add up to 48" is a problem for a Class 4 child. (Answer: 15, 16 and 17, because the middle number is 48 ÷ 3 = 16.)

Teaching through problem solving means the problem comes first and children build methods while solving it, instead of learning the method and then applying it. George Pólya's four steps give the structure: understand the problem, devise a plan, carry out the plan, look back. The "look back" step, checking and asking whether the method would work for other numbers, is the one classrooms most often skip.

Strategies children can learn

Check the examples: ₹20 can be paid as 10 + 10, 10 + 5 + 5 or 5 + 5 + 5 + 5, so three ways. The two numbers are 12 and 8. Working backwards, 17 − 3 = 14 and 14 ÷ 2 = 7. For handshakes among 5 people, the first shakes 4 hands, the next 3 new ones, then 2, then 1: 4 + 3 + 2 + 1 = 10.

Open-ended problems

A closed question has one answer: "What is 6 × 4?" An open-ended question has several: "Twenty-four chairs are to be arranged in equal rows. Find all the ways." A child can find 1, 2, 3, 4, 6, 8, 12 or 24 rows, which is 8 arrangements. Every child can start, strong children can go further, and the teacher sees their thinking. That is why open-ended tasks are the usual CTET answer for mixed-ability classes and for assessment of understanding.

Problem posing, where children make up their own problems, is a closely related move and one of the best checks of understanding. For the general psychology of problem solving, see problem solving in CDP.

Scenario questions with reasoning

1. A child freezes during oral tables practice and later says, "I am bad at maths." The most likely cause is: (a) fear created by public, timed drill (b) low intelligence (c) lack of homework (d) poor handwriting
2. "A child has 12 marbles, 5 more than a friend." Most children answer 17 for the friend's marbles. This shows: (a) weak addition facts (b) careless copying (c) reliance on key words without understanding the situation (d) dyscalculia
3. Which is an open-ended question? (a) What is 25 + 17? (b) Write 36 in words (c) What is half of 40? (d) Name two numbers whose sum is 42
4. Community mathematics mainly means: (a) maths taught by community members only (b) group tests (c) using the mathematics of children's everyday life and surroundings (d) maths for adult literacy
5. In Pólya's model, checking whether the answer is sensible and whether the method would work elsewhere belongs to: (a) understanding the problem (b) devising a plan (c) carrying out the plan (d) looking back
6. A teacher marks a correct answer wrong because the child did not use the method taught in class. This practice: (a) discourages children's own strategies and reasoning (b) builds discipline (c) is required for board exams (d) improves speed
7. The best way to help children who struggle with word problems because of language is to: (a) give a list of key words (b) let them act out, draw and retell the problem, and pose their own (c) avoid word problems (d) give only numerical sums
8. Problem solving as a pedagogy means: (a) giving problems only after teaching the method (b) using problems as the starting point from which children build methods (c) solving textbook exercises fast (d) the teacher solving problems on the board

Answers and reasoning

1. (a). The fear comes from the setting; ability is not the issue.
2. (c). "More" triggered addition, but the friend has fewer: 12 − 5 = 7.
3. (d). Many pairs work, such as 20 and 22 or 40 and 2.
4. (c). It connects school maths to maths already in children's lives.
5. (d). Looking back includes checking and generalising.
6. (a). It tells children that only one route counts, which feeds rote learning and anxiety.
7. (b). Understanding the situation beats key-word rules.
8. (b). The problem comes first and the method grows from it.

What to do next

• Write the problems table from memory with one cause and one remedy for each.
• List five maths words with a different everyday meaning.
• Solve each strategy example above without looking, then write one new open-ended question.
• Revise maths teaching aids and the maths revision sheet.