Mathematics Content and Pedagogy
Mathematics items divide between content (arithmetic, algebra, geometry, mensuration, data handling) and pedagogy (how concepts are sequenced, represented and diagnosed when a child goes wrong).
Content anchors
- Euclid's division lemma a = bq + r; HCF by prime factorisation; for two numbers LCM × HCF = product of the numbers.
- Percentage, profit and loss, discount; SI = PRT/100; CI = P(1 + R/100)^T − P.
- Identities: (a + b)², a² − b², (a + b)³; factorisation by splitting the middle term.
- Geometry: angle sum of a triangle 180°, congruence by SSS, SAS, ASA, RHS; circle terms (chord, secant, tangent, arc).
- Mensuration: cylinder CSA = 2πrh, TSA = 2πr(r + h); distinguish open and closed containers.
Pedagogy anchors
- Van Hiele levels: visualisation → analysis → informal deduction → formal deduction → rigour. At visualisation a child judges shape by appearance, so a rotated square stops being a square.
- Polya's four steps: understand the problem, devise a plan, carry it out, look back.
- Concrete → pictorial → abstract sequencing; Dienes base-ten blocks, geoboard, fraction kit, tangram, abacus.
- Error analysis beats re-teaching: a wrong answer reveals the rule the child is actually using.
Worked example
A trader marks goods 40% above cost and allows 25% discount. Profit? Let CP = ₹100, so MP = ₹140. Discount always applies to MP: SP = 140 × 0.75 = ₹105. Profit = 105 − 100 = ₹5, i.e. 5% on cost. The answer is not 40 − 25 = 15%, because the two percentages sit on different bases - that base error is exactly what the item is testing.
Traps that cost marks
- Subtracting discount per cent from markup per cent.
- Using LCM when the question says "greatest" and HCF when it says "least common multiple".
- Taking π = 22/7 and reporting an exact decimal answer.
- Using TSA when a pipe or open tank needs CSA.