In the inductive method of teaching mathematics, the teacher primarily:

Concept: methods of teaching mathematics are classified by the direction in which the reasoning travels between particular instances and general statements.

Induction is the movement from observed particular cases to a general statement that covers them; deduction is the reverse movement, from an already-accepted general statement down to particular instances. A method takes its name from this direction of travel, not from the material or the activity it happens to use.

Application: in the inductive method the teacher therefore opens with concrete, particular material and withholds the general statement until the learners have produced it. The classroom sequence runs:

1. Present several particular examples of the same kind, worked out fully.

2. Let the learners observe and compare those examples and describe what is common to all of them.

3. Guide them to state that common feature as a general rule or formula, in their own words first.

4. Try the stated rule on a fresh example to confirm it, and only then use it as an established result.

For instance, to reach the identity (a + b)2 = a2 + 2ab + b2, learners first expand (1 + 2)2 = 9 = 1 + 4 + 4, (2 + 3)2 = 25 = 4 + 12 + 9 and (3 + 4)2 = 49 = 9 + 24 + 16 by direct multiplication, notice that every result is the first square, plus twice the product of the two numbers, plus the second square, and only then write the identity in its general form.

Cross-check — contrast the other descriptions by what they say:

• "Begins with abstract theories and applies them to concrete cases" travels from the general to the particular, which is the opposite direction of travel.

• "Focuses on memorizing axioms without practical demonstrations" describes reception of statements on authority; no generalisation is built by the learner at all.

• "Relies solely on standardized tests to validate mathematical concepts" describes an instrument of evaluation, which measures learning after teaching rather than presenting content during it.

Hence the description that fits is the one that leads students from specific examples to general principles and rules.