• We should be able to defend our ideas and expect that others will do the same.
• The opponent challenges (argument) and the proponent defends (counter-argument).
• Avoid being aggressive, argumentative, or emotional.
• Separate the argument from the person making it.
• We easily miss our own mistakes.
• The goal is improvement and better results.

• The art of analyzing and evaluating thinking, with an aim of improving it.
• “Thinking about thinking”.
• Why?
• Empowers to understand and critique information.
• Provides basis for justification.
• Allows us to develop our thinking.
• Encourages responsibility.
• Encourages consistency.
• Helps with optimal decisions.

• Extraordinary claims require extraordinary evidence.
• Falsifiability.
• Occam's razor (parsimony).
• Recognizing fallacies and avoiding them.
• Balancing induction and deduction.

• Usual path in teaching maths:
• Exposition → Examples → Exercises → Applications.
• Real life problems:
• A quantitative question with no mathematical context, no data, and no direct means of answering.

You have a barn 20 m by 25 m. Estimate how many cows can sleep there.

a) 5 b) 50 c) 500 d) 5000

A. Real situation — pick relevant info.

B. Real model — mathematize the scenario.

C. Math model — do the calculations.

D. Math results — revise or return to real situation to choose a reasonable answer.

Real situation: barn with cows.

Real model: dimensions, purpose, possible numbers.

Math model: compute barn area, estimate space per cow.

Math results: area per cow and number of cows per m².

Return to real situation to choose a reasonable answer.

Useful skills:

Maths content.

Heuristics / strategies / rules of thumb.

Meta-cognition:

What are you doing?

Why are you doing it?

How will it help?

Beliefs and attitudes.

Step 1: Understand the problem.

Step 2: Devise a plan.

Step 3: Carry out the plan.

Step 4: Reflect on what you’ve achieved.

How to Solve It — George Polya (1945).

Step 1: Understand the problem.

Do you understand all words?

Can you restate it?

What are you asked to find?

Would a diagram help?

Is the given information enough?

Step 2: Devise a plan (often hardest).

Relate to similar problems.

Use examples.

Simplify or modify.

Break into smaller parts.

Work backwards.

Use known strategies.

Step 3: Carry out the plan.

Be patient.

Pay attention to details.

Don’t give up too quickly.

Revise if the plan fails.

Step 4: Reflect.

Can you check the answer?

Could you solve it differently?

Does the result tell you something interesting?

What other problems could be solved with this method?

How many 1L bottles can I drink per day?

(Recommendation: <400 mg caffeine per day.)

Party with 10 people.

Each eats 2/3 of a pizza.

Deal: 2 pizzas for $12.

How much will I spend?

Should I drive to Uni or take the tram?

Refund: 10c each.

Recycling point is 85 km away.

Car burns 6 L / 100 km.

Fuel cost: $2 per liter.

How many cans to break even?

What’s their volume?

How much can I earn?

You want healthier lunches without meal prep.

Comparing microwave meals:

Healthy brand vs supermarket brand.

Which option is better?

(Links removed; text only.)

Protein: 0.8 g/kg body mass.

Active people: 1.2–2 g/kg.

Avoid saturated fat and simple sugar.

Fibre: 20–40 g/day.

Calories: ~2000 kcal (varies).

Sodium: <2300 mg.

Supermarket meal: $6.80.

Healthy meal: $9.99.