MTE1501
Assignment 4
Due 31 August 2025
,Mathematics 1 for Teachers
Question 1: Mathematics in Society (20)
1.1 Three perspectives on mathematics (6 marks)
Instrumentalist / “toolbox” view: Mathematics is treated as a set of techniques
for solving problems. Example: Using long division to calculate how many groups
of 12 can be formed from 1 236 learners.
Platonist perspective: Mathematics is seen as the uncovering of absolute truths
that exist independently of humans. Example: Demonstrating that the interior
angles of any Euclidean triangle always add up to 180°, regardless of context.
Mathematics as a human activity (system view): Mathematics is regarded as
a cultural, evolving system created by people to interpret and organise the world.
Example: Learners compare whether a linear or exponential model better
explains cellphone data usage, then refine the choice based on observed results.
1.2 Contributions of ancient civilizations (6 marks)
Babylonians: Developed a base-60 positional system, which enabled detailed
astronomical records and time measurement. Application: Dividing hours into
minutes and seconds, and recording celestial cycles.
Egyptians: Used unit fractions and applied geometry in land surveying after
annual Nile floods. Application: Re-establishing property boundaries and
calculating grain distribution.
African societies: Evidence from counting tools like the Lebombo and Ishango
bones, as well as advanced systems such as the Yoruba base-20 numeration.
Application: Managing trade, calendars, rituals, and tracking lunar cycles.
1.3 Teaching “mathematics as a cultural human activity” in Grade 4 (4 marks)
Begin with real artefacts (images of tally sticks or rope-stretchers) and ask
learners: “Why do you think people invented these?”
, Rotate through learning stations comparing Zulu/Yoruba numerals, Roman
numerals, and modern base-10 writing of numbers 1–30.
Connect history to practice: explain how Egyptians used a rope with 3-4-5 knots
to create right angles. Learners then use the same method outdoors to mark a
soccer penalty box.
1.4 Converting the Babylonian numeral 2,30,30 (2 marks)
Positional calculation in base-60:
2×602+30×60+30=2×3600+1800+30=90302 \times 60^2 + 30 \times 60 + 30 = 2 \times
3600 + 1800 + 30 = 90302×602+30×60+30=2×3600+1800+30=9030
So the value in base-10 is 9 030.
1.5 Significance of the Lebombo bone (2 marks)
The 29 notches suggest it tracked lunar months (≈29.5 days). This indicates that early
mathematics arose from observing patterns and keeping tallies related to natural cycles,
long before symbolic notation was formalised.
Question 2: Teaching and Learning Mathematics (28)
2.1 Procedural vs conceptual knowledge (5 marks)
Procedural knowledge: Knowing the steps of a process (e.g., applying the long
division algorithm).
Conceptual knowledge: Understanding the reasoning behind processes,
recognising relationships (e.g., viewing division as sharing, as the inverse of
multiplication, or as a rate).
Balance in teaching: Both are important, but conceptual understanding should
lead, as it helps students transfer skills and reduces mechanical errors.
Classroom illustration: Before introducing the formal written method for 1 236 ÷
12, learners first use base-10 blocks or grouping strategies to build meaning,
then transition to the algorithm.
Assignment 4
Due 31 August 2025
,Mathematics 1 for Teachers
Question 1: Mathematics in Society (20)
1.1 Three perspectives on mathematics (6 marks)
Instrumentalist / “toolbox” view: Mathematics is treated as a set of techniques
for solving problems. Example: Using long division to calculate how many groups
of 12 can be formed from 1 236 learners.
Platonist perspective: Mathematics is seen as the uncovering of absolute truths
that exist independently of humans. Example: Demonstrating that the interior
angles of any Euclidean triangle always add up to 180°, regardless of context.
Mathematics as a human activity (system view): Mathematics is regarded as
a cultural, evolving system created by people to interpret and organise the world.
Example: Learners compare whether a linear or exponential model better
explains cellphone data usage, then refine the choice based on observed results.
1.2 Contributions of ancient civilizations (6 marks)
Babylonians: Developed a base-60 positional system, which enabled detailed
astronomical records and time measurement. Application: Dividing hours into
minutes and seconds, and recording celestial cycles.
Egyptians: Used unit fractions and applied geometry in land surveying after
annual Nile floods. Application: Re-establishing property boundaries and
calculating grain distribution.
African societies: Evidence from counting tools like the Lebombo and Ishango
bones, as well as advanced systems such as the Yoruba base-20 numeration.
Application: Managing trade, calendars, rituals, and tracking lunar cycles.
1.3 Teaching “mathematics as a cultural human activity” in Grade 4 (4 marks)
Begin with real artefacts (images of tally sticks or rope-stretchers) and ask
learners: “Why do you think people invented these?”
, Rotate through learning stations comparing Zulu/Yoruba numerals, Roman
numerals, and modern base-10 writing of numbers 1–30.
Connect history to practice: explain how Egyptians used a rope with 3-4-5 knots
to create right angles. Learners then use the same method outdoors to mark a
soccer penalty box.
1.4 Converting the Babylonian numeral 2,30,30 (2 marks)
Positional calculation in base-60:
2×602+30×60+30=2×3600+1800+30=90302 \times 60^2 + 30 \times 60 + 30 = 2 \times
3600 + 1800 + 30 = 90302×602+30×60+30=2×3600+1800+30=9030
So the value in base-10 is 9 030.
1.5 Significance of the Lebombo bone (2 marks)
The 29 notches suggest it tracked lunar months (≈29.5 days). This indicates that early
mathematics arose from observing patterns and keeping tallies related to natural cycles,
long before symbolic notation was formalised.
Question 2: Teaching and Learning Mathematics (28)
2.1 Procedural vs conceptual knowledge (5 marks)
Procedural knowledge: Knowing the steps of a process (e.g., applying the long
division algorithm).
Conceptual knowledge: Understanding the reasoning behind processes,
recognising relationships (e.g., viewing division as sharing, as the inverse of
multiplication, or as a rate).
Balance in teaching: Both are important, but conceptual understanding should
lead, as it helps students transfer skills and reduces mechanical errors.
Classroom illustration: Before introducing the formal written method for 1 236 ÷
12, learners first use base-10 blocks or grouping strategies to build meaning,
then transition to the algorithm.
