Chapter 1 — Mathematical reasoning
Lesson map
Lesson 01 published. Map, example, EX001.
flowchart TD
N0["Learning outcomes"]
N1["1.1 A course built around schools"]
N2["1.2 Rules as logic"]
N3["1.3 Hidden assumptions"]
N4["1.4 Base rates"]
N5["Synthesis"]
N0 --> N1 --> N2 --> N3 --> N4 --> N5
N5 --> EX["Worked example"]
EX --> Q["Exercise EX001"]
Logical drawing
flowchart LR
N0["Learning outcomes"]
N1["1.1 A course built around schools"]
N2["1.2 Rules as logic"]
N3["1.3 Hidden assumptions"]
N4["1.4 Base rates"]
N5["Synthesis"]
N0 --> N1 --> N2 --> N3 --> N4 --> N5 --> Q["EX001"]
[Learning outcomes]
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v
[1.1 A course built around schools]
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v
[1.2 Rules as logic]
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v
[1.3 Hidden assumptions]
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v
[1.4 Base rates]
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v
[Synthesis]
|
v
[Worked example] --> [Exercise EX001]
Learning outcomes
You will (1) use logic to state privacy rules precisely; (2) spot hidden assumptions in arguments about data; (3) reason about proportions and base rates.
1.1 A course built around schools
Elena Rossi begins a four-year bachelor's course at the fictional Fairhaven College, tutored by Dr. Kwame Asante. Her placement partner is the fictional Brindle Valley Schools Trust, which runs twelve primary schools with about 6,000 pupils. Schools hold names, addresses, health notes, special educational needs records and photographs of children. Dr. Asante explains that privacy and data protection will be Elena's main subject and that every foundation course will be read through the trust's work, starting with mathematical reasoning. Her first task is counting pupils per school.
1.2 Rules as logic
Privacy rules are often written loosely. The trust's policy says staff may share pupil data with apps if it is necessary and parents agree. Elena rewrites the rule as a logical statement: sharing is permitted only if necessity holds and a lawful basis exists. She notices that "parents agree" is not the only possible basis; a school may rely on a public task for teaching. Writing the rule precisely exposes the confusion. She learns that logic is a tool for making policies testable before they are applied to real children.
1.3 Hidden assumptions
Elena examines an argument from a teacher: the reading app is safe because it is popular with schools. The hidden assumption is that popularity implies careful data handling. She lists other hidden assumptions she meets: that a vendor's privacy page reflects its practice, that anonymised data cannot identify children, and that parents read consent forms. Each assumption can be checked. Dr. Asante asks the class to underline assumptions in every argument they read, a habit that will help them evaluate vendors and policies throughout the course.
1.4 Base rates
Elena studies base rates. If one in a hundred apps mishandles data and a vetting check catches nine in ten bad apps but wrongly flags five in a hundred good ones, most flagged apps are actually fine. She works through the arithmetic: of 1,000 apps, ten are bad and nine are flagged; of 990 good apps, about fifty are flagged. So only about one flagged app in six is bad. The lesson is that checks must be designed with base rates in mind, or staff will lose trust in them.
Worked example
Elena rewrites a sharing rule as logic, exposes hidden assumptions about popular apps and shows that a vetting check flags mostly good apps when bad ones are rare.
Synthetic case. Fairhaven, Brindle Valley and all figures are synthetic.
Errors and limits
Base-rate figures here are illustrative; real rates must be estimated.
Exercises
- EX001 — Necessity and basis. How does Elena rewrite the sharing rule? Rubric: 3 = sharing permitted only if necessary and a lawful basis exists;; 2 = mentions necessity;; 1 = keeps original;; 0 = none.
- EX002 — Popular app assumption. What hidden assumption lies in "the app is popular, so it is safe"? Rubric: 3 = popularity implies careful data handling;; 2 = names an assumption;; 1 = vague;; 0 = none.
- EX003 — Flagged apps. Roughly what share of flagged apps is actually bad in Elena's example? Rubric: 3 = about one in six;; 2 = a minority;; 1 = most;; 0 = none.
Synthesis
Chapter 1 sharpens reasoning. Chapter 2 looks at computing systems.