Question 1: Mathematics in Society (Q1.1–Q1.5)
What to do:
Define each term in your own words, then attach a short classroom or real-life
example.
Keep explanations in full sentences (the brief asks for paragraph format).
For history items, name the civilisation → name the contribution → the real-world
problem it addressed.
Scaffolds
1.1 (three views):
o Instrumentalist/toolbox: math as a set of tools (algorithms, formulas) to
solve problems. Example: Using long division to share 84 cupcakes
among 6 groups.
o Platonist: math objects exist independently; we “discover” them. Example:
π was “there” before we measured circles.
o System view: math as a coherent, connected human-made system of
definitions and theorems. Example: Fractions, decimals, and percentages
are interlinked representations of rational numbers.
1.2 (Babylonian, Egyptian, African):
o Babylonian: base-60 place-value system → solved time/astronomy/land
measurement issues.
o Egyptian: unit-fraction methods and geometry → solved
surveying/floodplain area after Nile floods.
o African: Lebombo/Ishango bones; fractal design in architecture and arts →
supported counting, calendars, spatial design in settlements.
1.3 (Math as cultural human activity, Grade 4):
o Hook with local patterns (weaving, beadwork, rhythms).
, o One short historical story (e.g., how calendars tracked seasons for
farming).
o Mini activity: learners find patterns in bead strings and describe the rule.
1.4 (Worked example you can learn from): Babylonian 2,30,30 (base-60) 2 ×
602 + 30 × 60 + 30 = 2 × 3600 + 1800 + 30 = 9030 . (Use this method for any
sexagesimal number.)
1.5 (Lebombo bone – 29 notches): Discuss how tally marks show early
counting, patterning (e.g., lunar cycle ~29.5 days), and the shift from concrete
marks to abstract number.
Question 2: Teaching & Learning Mathematics (Q2.1–Q2.5)
What to do:
Always contrast terms (define → compare → classroom implication).
Give one concrete classroom move per proficiency/component.
Scaffolds
2.1 Procedural vs conceptual:
o Procedural: knowing rules/steps (e.g., the standard algorithm).
o Conceptual: knowing why they work and how ideas connect.
o Which is more desirable? Aim for both, with emphasis on conceptual to
sustain transfer and error checking.
o Short anecdote from your learning/teaching.
2.2 Five components of proficiency (Kilpatrick et al.):
o Conceptual understanding, procedural fluency, strategic
competence, adaptive reasoning, productive disposition.
o For each, add a concrete routine (e.g., number talks for conceptual;
worked-example→completion for procedural; open problems for strategic;
What to do:
Define each term in your own words, then attach a short classroom or real-life
example.
Keep explanations in full sentences (the brief asks for paragraph format).
For history items, name the civilisation → name the contribution → the real-world
problem it addressed.
Scaffolds
1.1 (three views):
o Instrumentalist/toolbox: math as a set of tools (algorithms, formulas) to
solve problems. Example: Using long division to share 84 cupcakes
among 6 groups.
o Platonist: math objects exist independently; we “discover” them. Example:
π was “there” before we measured circles.
o System view: math as a coherent, connected human-made system of
definitions and theorems. Example: Fractions, decimals, and percentages
are interlinked representations of rational numbers.
1.2 (Babylonian, Egyptian, African):
o Babylonian: base-60 place-value system → solved time/astronomy/land
measurement issues.
o Egyptian: unit-fraction methods and geometry → solved
surveying/floodplain area after Nile floods.
o African: Lebombo/Ishango bones; fractal design in architecture and arts →
supported counting, calendars, spatial design in settlements.
1.3 (Math as cultural human activity, Grade 4):
o Hook with local patterns (weaving, beadwork, rhythms).
, o One short historical story (e.g., how calendars tracked seasons for
farming).
o Mini activity: learners find patterns in bead strings and describe the rule.
1.4 (Worked example you can learn from): Babylonian 2,30,30 (base-60) 2 ×
602 + 30 × 60 + 30 = 2 × 3600 + 1800 + 30 = 9030 . (Use this method for any
sexagesimal number.)
1.5 (Lebombo bone – 29 notches): Discuss how tally marks show early
counting, patterning (e.g., lunar cycle ~29.5 days), and the shift from concrete
marks to abstract number.
Question 2: Teaching & Learning Mathematics (Q2.1–Q2.5)
What to do:
Always contrast terms (define → compare → classroom implication).
Give one concrete classroom move per proficiency/component.
Scaffolds
2.1 Procedural vs conceptual:
o Procedural: knowing rules/steps (e.g., the standard algorithm).
o Conceptual: knowing why they work and how ideas connect.
o Which is more desirable? Aim for both, with emphasis on conceptual to
sustain transfer and error checking.
o Short anecdote from your learning/teaching.
2.2 Five components of proficiency (Kilpatrick et al.):
o Conceptual understanding, procedural fluency, strategic
competence, adaptive reasoning, productive disposition.
o For each, add a concrete routine (e.g., number talks for conceptual;
worked-example→completion for procedural; open problems for strategic;
