The nature of mathematics and logical thinking for CTET

Every CTET maths section opens its pedagogy half with questions such as "Mathematics is a subject that primarily deals with…" or "According to NCF 2005, the main goal of mathematics education is…". They look like definitions to memorise. They are really asking something more practical: if mathematics is this kind of subject, how should it be taught, and what should a teacher value in a child's work?

The syllabus heading for both papers is "Nature of Mathematics/Logical thinking; understanding children's thinking and reasoning patterns and strategies of making meaning and learning". This guide takes it in that order: what mathematics is, how it reasons, what NCF 2005 says it is for, how children actually think, and what all that means in a classroom.

What kind of subject is mathematics?

Inductive and deductive reasoning

Mathematics as a finished body of knowledge is deductive: results are proved from definitions and axioms. But mathematics is often discovered inductively, by noticing examples. School maths needs both.

Induction alone can mislead. The rule "n² + n + 41 is always prime" works for n = 0, 1, 2 and far beyond, but at n = 40 it gives 1,681 = 41 × 41. A single counter-example disproves a general claim. At primary level, "All primes are odd" is disproved by 2, and "Multiplying always makes a number bigger" by ½ × 8 = 4.

Mathematisation and the NCF 2005 position paper

NCF 2005 was supported by position papers from national focus groups, including one on the teaching of mathematics, published by NCERT. Its central statement: the main goal of mathematics education in schools is the mathematisation of the child's thinking. Mathematisation means the ability to think and reason mathematically: to see the maths in a situation, handle abstraction, pursue assumptions to their logical conclusion and solve problems.

The position paper distinguishes two aims:

• Narrow aim: useful capabilities, especially numeracy: numbers, operations, measurement, decimals and percentages.
• Higher aim: developing the child's inner resources to think and reason mathematically.

Its vision for school mathematics, in paraphrase, is that children:

• learn to enjoy mathematics rather than fear it;
• learn important mathematics, which is more than formulas and mechanical procedures;
• see mathematics as something to talk about, communicate, discuss and work on together;
• pose and solve meaningful problems;
• use abstractions to see relationships and structure, and to argue whether statements are true or false;
• understand the basic structure of mathematics;

and that teachers expect to engage every child in the class.

It also names the main problems: many children fear mathematics and feel they fail at it; the curriculum disappoints a talented minority and the non-participating majority at the same time; crude assessment makes maths look like mechanical computation; and teachers lack preparation and support. It stresses mathematical processes such as problem solving, use of heuristics, estimation and approximation, visualisation, representation, reasoning and proof, making connections and mathematical communication.

For how this sits in the curriculum as a whole, see maths in the curriculum.

How children think with numbers

Children do not wait to be taught one method. Given room, they invent strategies, and these show how they make meaning:

A move from counting all to counting on is real progress in thinking, even though both give 8. This is the constructivist view: children build knowledge from experience and interaction, so the teacher's job is to understand their reasoning, not to overwrite it. Piaget's concrete operational stage explains why objects matter in the primary years; Jerome Bruner's enactive, iconic and symbolic modes underlie the "concrete, pictorial, abstract" sequence; and Vygotsky's zone of proximal development explains the value of scaffolding and peer discussion. See Piaget and constructivism.

Pólya's four steps of problem solving

George Pólya set out four steps in How to Solve It (1945):

1. Understand the problem. What is known, what is asked, what are the conditions?
2. Devise a plan. Draw a picture, make a table, look for a pattern, work backwards, try a simpler case, guess and check.
3. Carry out the plan, checking each step.
4. Look back. Check the answer against the problem, and ask whether there is another method or a wider use.

Worked example. A farm has hens and cows: 20 heads and 56 legs. How many cows?

1. Understand: each animal has one head; a hen has 2 legs and a cow 4.
2. Plan: suppose all 20 are hens, then adjust.
3. Carry out: 20 hens would have 40 legs. There are 16 extra legs, and each cow adds 2 extra, so there are 16 ÷ 2 = 8 cows and 12 hens.
4. Look back: 8 × 4 + 12 × 2 = 32 + 24 = 56 legs, and 8 + 12 = 20 heads.

For more on problem solving across subjects, see problem solving and critical thinking.

What this means in the classroom

• Move from concrete to pictorial to abstract, and let children move back when they need to.
• Value reasoning and explanation as well as correct answers.
• Accept multiple methods and ask children to compare them.
• Use errors as information about thinking, not only as things to mark wrong.
• Give problems from children's lives so that mathematisation has something to work on.
• Encourage estimation and conjecture: "What do you think the answer will be? Is that always true?"

Scenario questions with reasoning

1. According to the NCF 2005 position paper on mathematics, the main goal of mathematics education is: (a) the mathematisation of the child's thinking (b) developing computational speed (c) preparing for competitive exams (d) memorising formulas
2. Class 4 children notice that 3 + 5, 7 + 9 and 11 + 13 are all even, and say "two odd numbers always add to an even number". This is: (a) deductive reasoning (b) analogy (c) rote learning (d) inductive reasoning
3. A child adds 9 + 6 by thinking "9 + 1 = 10, and 5 more is 15". The teacher should: (a) insist on the textbook method (b) recognise a valid "making ten" strategy and ask the child to share it (c) mark it wrong because the steps are not written (d) give more drill
4. Which statement best reflects the nature of mathematics? (a) Maths is mainly about quick calculation (b) Maths has only one correct method for each problem (c) Maths involves abstraction, logic and finding patterns (d) Maths can only be learnt by memorising
5. A student solves a word problem correctly but does not check whether the answer makes sense. Which of Pólya's steps is missing? (a) Understanding the problem (b) Devising a plan (c) Carrying out the plan (d) Looking back
6. A teacher says "Is this always true? Can you find a case where it fails?" This mainly encourages: (a) memorisation (b) conjecture and reasoning about generalisations (c) speed (d) neat work
7. Which is a "higher aim" of teaching mathematics in the NCF 2005 position paper? (a) Knowing multiplication tables (b) Calculating percentages (c) Developing the child's resources to think and reason mathematically (d) Measuring length

Answers and reasoning

1. (a). The position paper's central statement.
2. (d). A general rule drawn from examples is induction. A proof using 2m + 1 and 2n + 1 would be deductive.
3. (b). Children's own strategies show understanding; sharing them builds a classroom where methods are compared.
4. (c). The other options describe misconceptions about the subject.
5. (d). Looking back includes checking the answer against the situation.
6. (b). Asking for counter-examples builds mathematical reasoning.
7. (c). Tables and percentages belong to the narrow aim of useful capabilities.

What to do next

• Write the narrow and higher aims, and four points of the position paper's vision, in your own words.
• Make a two-column note on inductive and deductive reasoning with one example each.
• Solve two word problems by writing out all four of Pólya's steps.
• Continue with maths in the curriculum and evaluation in mathematics.