DEGREE LEVEL COURSE Mathematical Thinking

1. A degree-level course in mathematical thinking is a university-level course that teaches students how to think like mathematicians. This includes developing the ability to:

   

   
   

   • Identify and solve problems using mathematical methods
   • Represent problems in mathematical terms
   • Use logical reasoning to prove theorems and solve problems
   • Communicate mathematical ideas effectively

   Degree-level courses in mathematical thinking are typically offered in mathematics departments, but they may also be offered in other departments, such as computer science, engineering, and physics.

   Some examples of degree-level courses in mathematical thinking include:

   • Introduction to Mathematical Thinking
   • Discrete Mathematics
   • Mathematical Logic
   • Proof Techniques
   • Real Analysis
   • Complex Analysis
   • Number Theory
   • Topology
   • Abstract Algebra

   Proofs and problem solving are two essential skills in mathematics. Proofs are used to establish the truth of mathematical statements, while problem solving is used to find solutions to mathematical problems. Proofs A proof is a logical argument that establishes the truth of a mathematical statement. It is a sequence of statements, each of which is supported by one or more previous statements. The first statements in the sequence are typically axioms, which are statements that are assumed to be true. The remaining statements in the sequence are deduced from the previous statements using the rules of logic. There are many different types of proofs, but they all share the common goal of establishing the truth of a mathematical statement. Some of the most common types of proofs include: Direct proofs: These proofs start with the given hypotheses and use the rules of logic to deduce the conclusion. Indirect proofs: These proofs start with the negation of the conclusion and use the rules of logic to deduce a contradiction. Proofs by contradiction: These proofs are essentially indirect proofs, but they start with the assumption that the statement is false and use the rules of logic to deduce a contradiction. Proofs by induction: These proofs are used to prove statements that are true for all positive integers. They start by proving the statement for the base case (usually n = 1) and then prove that if the statement is true for n = k, then it is also true for n = k + 1. Problem solving Problem solving is the process of finding solutions to mathematical problems. It is a complex skill that involves a variety of different strategies and techniques. Some of the most common problem-solving strategies include: Guess and check: This strategy involves making a guess at a solution, testing the guess, and then revising the guess as needed. Working backwards: This strategy involves starting with the desired outcome and working backwards to find a solution. Breaking the problem down into smaller parts: This strategy involves breaking a complex problem down into smaller, more manageable problems. Looking for patterns: This strategy involves looking for patterns in the problem that can be used to find a solution. Using analogies: This strategy involves using analogies to other problems that you have solved to find a solution to the current problem. Ideas for proofs and problem solving Here are some ideas for proofs and problem solving in mathematics: Proofs: When trying to prove a theorem, it is often helpful to start by drawing a diagram or creating a table. This can help you to visualize the problem and to identify any patterns. Try to break the theorem into smaller, more manageable pieces. This can make it easier to come up with a proof. If you are stuck, try to come up with a counterexample to the theorem. If you can find a counterexample, then the theorem is false and you do not need to prove it. Problem solving: When trying to solve a problem, it is often helpful to start by reading the problem carefully and identifying the key information. Try to come up with a plan for solving the problem. This may involve breaking the problem down into smaller parts, identifying patterns, or using analogies. If you are stuck, try to come up with a simpler version of the problem. Once you have solved the simpler problem, you may be able to apply the same strategy to solve the original problem. Do not be afraid to try different strategies and techniques. There is no one right way to solve a problem. The best way to learn how to prove theorems and solve problems is to practice. Try to find problems that are challenging but that you can still solve with some effort. As you solve more problems, you will develop your problem-solving skills and learn new strategies and techniques. ourses typically cover a wide range of mathematical topics, but they all emphasize the development of mathematical thinking skills.

   Degree-level courses in mathematical thinking are beneficial for students who are interested in pursuing a career in mathematics, computer science, engineering, or other fields that require strong mathematical skills. They are also beneficial for students who are simply interested in learning how to think more logically and analytically.

   WEEK 1	Numbers, Sets and Functions
   WEEK 2	Language and Proofs
   WEEK 3	Induction, Bijections and Cardinality.
   WEEK 4	Combinatorial Reasoning.
   WEEK 5	Divisibility, Modular Arithmetic.
   WEEK 6	The Rational Numbers.
   WEEK 7	Two Principles of Counting, Recurrence Relations. Prescribed Books The following are the suggested books for the course: “Mathematical Thinking: Problem-solving and Proofs”, John D’Angelo and Douglas West, Pearson, 2000 (2nd edition) “Mathematical Proofs: A Transition to Advanced Mathematics”, G. Chartrand, A. D. Polimeni, P. Zhang, Pearson, 2012