• confidently read and write proofs using the appropriate language, symbols and presentation;
• apply the four fundamental proof techniques to several areas of mathematics;
• create lessons that incorporate appropriate proof techniques in various area of mathematics;
• guide students and ensure that they use appropriate presentation, mathematical language, terminology and symbols when presenting mathematical proofs;
• ensure that students recognise the elegant and rigorous processes involved in establishing and proving mathematical results including the ability to construct proofs of negative statements or counterexamples;
• Guide students to ensure that they are aware of the insolvability of the general quintic equation by radicals showing competence in linking group theory with classical algebraic problems;
• Apply the method of contrapositive reasoning to prove mathematical statements particularly those involving quantified expressions.

• Explain the logic underlying each of the four fundamental proof techniques;
• Define what constitutes a mathematical proof;
• identify, use and interpret logical symbols and quantifiers;
• identify different types of mathematical statements, including theorems, propositions, lemmas, and corollaries;
• identify the difference between mathematical statements, namely: theorems, propositions, lemmas and corollaries;
• Explain the meanings and roles of definitions, axioms/postulates, and conjectures;
• Identify the structure of conditional statements and their logically equivalent contrapositives.

• create and interpret truth tables for simple mathematical statements involving AND, OR, IF…THEN…, and IF AND ONLY IF;
• use logical statements, connectives and quantifiers in constructing arguments and expressing statements;
• write proofs according to standard guidelines in mathematical writing;
• choose the appropriate and most efficient proof technique according to the context;
• apply the four fundamental proof techniques in a variety of mathematical contexts;
• write proofs using the correct approach and presentation for each of the four fundamental techniques;
• construct direct proofs using deductive reasoning to prove statements in various contexts such as algebraic relationships, trigonometric relationships, geometric properties, and number theory;
• apply the method of proof by contradiction to prove statements in various contexts such as number theory and algebraic relationships;
• apply the method of proof by contrapositive to prove statements in various contexts such as number theory and algebraic expressions;
• Use inductive reasoning, the principle of mathematical induction, and apply it to prove statements on number theory, matrices, sequences and series, and inequalities, where the statements involve natural numbers;
• apply appropriate proof techniques to prove some standard theorems and statements such as De Moivre’s Theorem for , Infinity of Prime Numbers, Binomial Theorem for , , sum of the interior angles of a polygon with

This module will be assessed through: Portfolio, Lesson Plan, Presentation.

Core Reading List

1. Study unit notes pack

Supplementary Reading List

1. Schumacher, C. (2004). Chapter Zero: Fundamental Notions of Abstract Mathematics (2nd ed.). Addison-Wesley Publishers.
2. Velleman, D.J. (2019). How to Prove It: A Structured Approach (3rd ed.). Cambridge University Press.

Papers:

• Arzarello, F., & Soldano, C. (2019). Approaching proof in the classroom through the logic of inquiry. In G. Kaiser, & N. Presmeg (Eds.), Compendium for Early Career Researchers in Mathematics Education, ICME-13 Monographs (pp. 221-243). Springer Open. https://doi.org/10.1007/978-3-030-15636-7_10
• Bertran-San Millán, J. (2021). Frege, Peano and the Interplay between Logic and Mathematics. Philosophia Scientiæ, 25(1), 15-34. https://doi.org/10.4000/philosophiascientiae.2831
• Stylianides, A. J. (2019). Secondary students’ proof constructions in mathematics: The role of written versus oral mode of argument representation. Review of Education, 7(1), 156-182. https://doi.org/10.1002/rev3.3157