Problem solving and critical thinking: the child as investigator

Give a group of eight-year-olds 12 laddoos to share among 4 friends and they will solve it long before they know the word "division". Some deal them out one at a time, some make piles of three, some draw pictures. Problem solving is not a skill school has to install. It is something children already do, and school either builds on it or buries it under drill.

The CTET syllabus names this directly under Learning and Pedagogy: "child as a problem solver and a 'scientific investigator'". Questions test the steps of problem solving, the levels of Bloom's taxonomy, the difference between convergent and divergent thinking, and, most often, which teacher action encourages children to think rather than recall.

What counts as a problem

A problem is a situation where you want to reach a goal and the way to get there is not obvious. "7 × 8 = ?" is not a problem for a child who knows the table; it is recall. "Find three different ways to make ₹50 using ₹10 and ₹5 coins" is a problem, because the child has to work out a method.

Critical thinking is the habit of asking for reasons and evidence before accepting a claim: How do we know? What else could explain this? Is this fair? It is closely linked to problem solving, because good solutions come from testing ideas rather than accepting the first one.

Steps in problem solving

Two classic models appear in teacher-education books.

Dewey's steps of reflective thinking

John Dewey, in How We Think (1910), described reflective thinking as a sequence:

1. A felt difficulty: the learner notices a problem.
2. Locating and defining the problem.
3. Suggesting possible solutions (hypotheses).
4. Reasoning out what each suggestion implies.
5. Testing the chosen solution by observation or experiment, and accepting or rejecting it.

Pólya's four steps

The mathematician George Pólya set out four steps in How to Solve It (1945), still used in maths teaching:

1. Understand the problem.
2. Devise a plan.
3. Carry out the plan.
4. Look back: check the answer and ask whether the method works elsewhere.

The fourth step is the one children skip and the one teachers most need to teach. A child who gets "a bus carries 40 people; how many buses for 130 people?" as 3.25 buses has not looked back at whether the answer makes sense.

Ways of solving

Convergent and divergent thinking

J. P. Guilford distinguished two kinds of thinking.

Both matter. CTET questions usually ask you to identify an open-ended, divergent question as the one that "promotes creativity" or "encourages thinking". See gifted and creative children for more on creativity.

Bloom's taxonomy

Benjamin Bloom and colleagues published a classification of educational objectives in the cognitive domain in 1956. Lorin Anderson and David Krathwohl revised it in 2001, turning the levels into verbs and placing "create" at the top.

Rows 5 and 6 are where the two lists differ. The revision renamed Synthesis as Create and moved it to the top, above Evaluate. The top three levels (analyse, evaluate, create) are called higher-order thinking skills.

What blocks problem solving

• Mental set: sticking to a method that worked before even when a simpler one exists. Luchins's water-jar experiments showed people repeating a long formula after a short one became possible.
• Functional fixedness: seeing an object only in its usual use. Duncker's candle problem is the classic example: people fail to see a box of tacks as a shelf.
• Fear of being wrong, which stops children from trying ideas aloud.
• Rote learning, which gives children answers but not methods.
• Too little time: a teacher who answers their own question after three seconds teaches children to wait.

How a teacher builds problem solvers

• Ask open-ended questions: "Why do you think…?", "What would happen if…?", "Is there another way?"
• Give wait time after a question before taking answers.
• Ask children to predict before an experiment, and to explain afterwards why the result matched or did not.
• Accept multiple methods and ask children to compare them.
• Use real problems from the child's life: sharing, shopping, planning a class trip.
• Ask for evidence: "How do you know?"
• Let children pose problems as well as solve them.

Worked classroom scenarios

Scenario 1

After an experiment on evaporation, a teacher asks, "Would the water dry faster on a sunny day or a cloudy day? Why?" This question mainly develops:

(a) the ability to apply and reason from what was observed
(b) rote memory
(c) handwriting
(d) obedience

Best answer: (a). The child has to transfer what was seen to a new situation and give a reason.

Scenario 2

A child solves a word problem and gets 3.25 buses. The teacher's best response is to:

(a) mark it wrong
(b) give the correct answer, 4
(c) ask the child whether 3.25 buses makes sense and what should happen to the extra people
(d) give the child ten more division sums

Best answer: (c). This is Pólya's "look back" step. The child already has the arithmetic; what is missing is checking the answer against the situation.

Scenario 3

Which question best promotes divergent thinking?

(a) What is 15 × 4?
(b) Who wrote the national anthem?
(c) In how many different ways can you arrange 12 chairs in equal rows?
(d) What is the boiling point of water at sea level?

Best answer: (c). It has several correct answers (1 × 12, 2 × 6, 3 × 4, and the reverses), so children must generate and check possibilities.

Scenario 4

A child keeps using counting on fingers for 9 + 8 even after learning the "make 10" strategy (9 + 1 + 7 = 17). This persistence with an old method is an example of:

(a) insight (b) divergent thinking (c) functional fixedness (d) mental set

Best answer: (d). The child sticks to a method that worked before.

Practice MCQs

Q1. In the revised Bloom's taxonomy, the highest level is:

(a) create (b) analyse (c) evaluate (d) apply

Q2. "Look back" is a step in problem solving described by:

(a) Pavlov (b) Erikson (c) Skinner (d) Pólya

Q3. Divergent thinking is characterised by:

(a) many possible answers (b) one correct answer (c) memorising (d) copying

Q4. Which teacher behaviour best supports problem solving?

(a) giving the method and then practice sums
(b) answering doubts only at the end of the term
(c) asking children to predict, test and explain
(d) insisting on one method for every child

Q5. Failing to see that a coin could be used as a screwdriver is an example of:

(a) mental set (b) functional fixedness (c) insight (d) transfer

Q6. The first step in Dewey's reflective thinking is:

(a) testing a hypothesis (b) collecting data (c) drawing a conclusion (d) a felt difficulty

Q7. "Explain why we have seasons" belongs to which level of the revised Bloom's taxonomy?

(a) remember (b) understand (c) create (d) evaluate

Answers

1. (a).
2. (d).
3. (a). Guilford's distinction.
4. (c).
5. (b). Seeing an object only in its usual function.
6. (d).
7. (b). The child explains an idea in their own words.

What to do next

• Write Pólya's four steps and Dewey's five from memory.
• Rewrite one closed textbook question as an open-ended one.
• Put the six revised Bloom levels in order, with one verb each.
• Read constructivism and then emotions and learning, because fear is the biggest block to a child trying out ideas.