Whakapapa & Mathematical Thinking
Genealogy, Networks & Exponential Growth Through Cultural Connection
Exploring how whakapapa (genealogy) reveals fundamental mathematical concepts including exponential growth, network theory, and the mathematics of human connection.
Mathematics Lives in Our Connections
Whakapapa is far more than a list of names - it's a sophisticated mathematical network that reveals exponential growth, probability theory, and the stunning complexity of human relationships. Every person's whakapapa contains the same mathematical patterns that govern population dynamics, genetics, and network theory.
"Ko wai au? He uri nō wai au?"
Who am I? Whose descendant am I? - Questions that lead to exponential mathematics
The Mathematics of Ancestry - Exponential Growth
The Exponential Pattern
Each person has exactly 2 biological parents, 4 grandparents, 8 great-grandparents... This creates a perfect exponential sequence: 2¹, 2², 2³, 2⁴, 2⁵...
Whakapapa Tree - Mathematical Structure
2⁰ = 1 person
2¹ = 2 people
2² = 4 people
2³ = 8 people
Whakapapa Exponential Formula
Ancestors = 2ⁿ (where n = generations back)
Example: 10 generations back = 2¹⁰ = 1,024 ancestors!
Whakatōpū - Mathematical Investigations
Investigation 1: Your Personal Exponential Growth
Calculate Your Ancestry
Using the formula 2ⁿ, calculate how many ancestors you have at each generation:
| Generations Back (n) | Power | Calculation | Number of Ancestors |
|---|---|---|---|
| 3 (Great-grandparents) | 2³ | 2 × 2 × 2 | 8 |
| 5 | 2⁵ | ____ | ____ |
| 8 | 2⁸ | ____ | ____ |
| 10 | 2¹⁰ | Use calculator! | ____ |
| 15 | 2¹⁵ | Use calculator! | ____ |
Critical Thinking Questions
- At what generation does the number of ancestors exceed the current population of New Zealand (5.1 million)?
- If we go back 1000 years (approximately 40 generations), how many ancestors should you theoretically have?
- Why is this number impossible? What does this tell us about shared ancestry?
Investigation 2: Whānau Network Mathematics
Scenario: A whānau reunion brings together descendants of a couple who lived 200 years ago.
Family Growth Pattern
Assumptions for modeling:
- Original couple: 2 people
- Each generation has an average of 3 children per couple
- 25 years between generations
- 200 years = 8 generations
Mathematical Challenges
- Generation 1: 2 people have 3 children = __ people in Gen 2
- If pattern continues, how many people in Generation 4?
- What formula could model this growth?
- How many people might attend the reunion?
Advanced Challenge: Network Connections
If 50 people attend the reunion, and everyone greets everyone else once, how many handshakes occur? (Hint: This is a combinations problem: C(50,2) = 50!/(2!(50-2)!))
Investigation 3: Whakatōhea & Mathematical Decision Making
Cultural Context: In traditional Māori society, important decisions were made collectively by whānau, hapū, and iwi. Understanding the mathematics of representation helps ensure fair participation.
Hui Representation Problem
A hui needs to make a decision that affects three hapū:
- Hapū A: 150 members
- Hapū B: 300 members
- Hapū C: 450 members
Challenge: If the hui has 30 representative positions, how should they be distributed fairly?
Mathematical Solutions:
- Equal representation: Each hapū gets 10 representatives. Is this fair? Why/why not?
- Proportional representation: Calculate what proportion each hapū should receive based on population.
- Formula: (Hapū population ÷ Total population) × 30 positions = Representatives for that hapū
| Hapū | Population | Calculation | Representatives |
|---|---|---|---|
| Hapū A | 150 | (150 ÷ 900) × 30 = | ____ |
| Hapū B | 300 | ____ | ____ |
| Hapū C | 450 | ____ | ____ |
Whakangā Ake - Advanced Mathematical Extensions
Logarithmic Relationships
If you have 1,024 ancestors 10 generations back, how many generations would give you approximately 1 million ancestors? Use: n = log₂(ancestors)
Probability in Genealogy
What's the probability that two random New Zealanders share a common ancestor within 20 generations? Research the mathematical concept of "degrees of separation."
Graph Theory
Family trees are mathematical graphs. Research how graph theory mathematics is used in genetics, social networks, and DNA analysis.
Population Genetics
Investigate the Hardy-Weinberg principle: how do gene frequencies change over generations? What mathematical models predict genetic diversity?
Ā Muri, Ā Mua - Mathematical Applications Today
Where Whakapapa Mathematics is Used
DNA Analysis & Genealogy
- 23andMe and AncestryDNA use exponential mathematics to calculate relationship probabilities
- Genetic algorithms use the same mathematical principles as family trees
- Population genetics models predict disease inheritance patterns
Social Networks & Technology
- Facebook's "People You May Know" uses network mathematics
- LinkedIn connection recommendations use graph theory
- Contact tracing for disease outbreaks uses the same mathematical models
Māori Cultural Applications
- Iwi Registration: Mathematical verification of whakapapa connections for iwi membership
- Settlement Distribution: Using population mathematics to determine fair distribution of treaty settlements
- Cultural Preservation: Network analysis to understand how cultural knowledge spreads through whānau networks
- Language Revitalization: Mathematical modeling to predict optimal strategies for Te Reo Māori transmission
Whakaaro - Mathematical Thinking Reflection
Essential Understanding Check
- Explain exponential growth: Why does the number of ancestors grow so rapidly as you go back in time?
- Mathematical impossibility: Why can't you actually have 2²⁰ ancestors 20 generations back?
- Cultural connections: How does understanding the mathematics of whakapapa change your perspective on human connection?
- Real-world applications: Give two examples of how exponential growth mathematics appears in current technology or science.
Extended Challenge
Research the mathematical concept of "most recent common ancestor" (MRCA). Using mathematical models, approximately when did all humans share a common ancestor? How does this mathematical finding connect to the concept of whakatōhea (collective responsibility)?