Mathematics

Focus: Teach the building blocks of mathematics—definitions, operations, and rules.
Example: Learning fractions.
- The student understands what a fraction represents: parts of a whole.
- They memorize the terminology (numerator, denominator) and practice operations like addition, subtraction, and multiplication of fractions.
Outcome: A solid foundation of the "language" of mathematics, enabling them to recognize and describe mathematical elements confidently.

Focus: Develop reasoning skills to explore relationships between mathematical ideas.
Example: Connecting fractions to decimals and percentages.
- The student learns how to convert fractions into decimals and percentages, recognizing that all three are different representations of the same idea.
- They solve problems, such as determining what percentage of a class passed a test, using fractions and conversions.
Outcome: Students see how mathematical concepts interlink, fostering deeper understanding and enabling them to approach problems systematically.

Focus: Teach students to apply their knowledge creatively and articulate their reasoning clearly.
Example: Explaining a real-world problem, such as budgeting.
- The student uses their understanding of fractions, decimals, and percentages to allocate parts of a monthly budget.
- They explain their reasoning: "I allocated 50% of the budget for rent because it’s half of the total income, and I need to keep 10% for savings, which is 0.1 times the total."
Outcome: Students gain the confidence to express their mathematical reasoning effectively, demonstrating mastery and the ability to solve practical problems.

Focus: Teach the building blocks of verbal reasoning—words, relationships, and basic logical structures.
Example: Identifying synonyms, antonyms, and analogies.
- Students learn what verbal reasoning entails: the ability to understand and evaluate relationships between concepts expressed in words.
- They practice identifying relationships in analogies:
  - Example: "Cat is to kitten as dog is to ____ (puppy)."
- They explore basic logical statements: "If all A are B, and C is A, then C is B."
Outcome: Students grasp the "syntax" of verbal reasoning—the rules and structures that govern word relationships and logical connections.

Focus: Develop the ability to reason logically and infer conclusions from premises.
Example: Drawing inferences from a passage.
- Students analyze verbal "proofs," determining whether conclusions follow logically from given premises.
  - Example:
    - Premise: "All mammals are warm-blooded."
    - Premise: "Whales are mammals."
    - Conclusion: "Therefore, whales are warm-blooded."
- They learn to identify faulty reasoning:
  - "If whales are mammals, and dolphins are mammals, then all mammals are whales." (False conclusion based on a misunderstanding of relationships.)
Outcome: Students connect verbal reasoning to the principles of inference, deduction, and pattern recognition, enabling them to analyze arguments critically.

Focus: Teach students to articulate ideas clearly, build convincing arguments, and critique others’ reasoning.
Example: Writing or debating a response to a prompt.
- Students use verbal proofs to support a thesis:
  - Example: Prompt: "Is honesty always the best policy?"
  - Argument: "Honesty fosters trust, which is the foundation of relationships. Without trust, collaboration and progress are hindered. For instance, when scientists share data honestly, discoveries flourish. Therefore, honesty, while not always convenient, is ultimately the best policy."
- They also practice critiquing weak arguments, identifying logical fallacies, and proposing counterexamples.
Outcome: Students master the ability to express their thoughts, defend their ideas, and evaluate others' reasoning, empowering them to thrive in academic, social, and professional contexts.