Todays topics Teaching Discrete Mathematics Active Learning in - PDF document

Today’s topics • Teaching Discrete Mathematics • Active Learning in Discrete Mathematics Discrete Mathematics • Educational Technology Research at UW • Big Ideas: Complexity Theory Richard Anderson University of Washington 7/1/2008 IUCEE: Discrete Mathematics 1 7/1/2008 IUCEE: Discrete Mathematics 2 Highlights from Day 1 Website • http://cs.washington.edu/homes/anderson – Home page • http://cs.washington.edu/homes/anderson/iucee – Workshop websites – Updates might be slow (through July 20) • Google groups – IUCEE Workshop on Teaching Algorithms 7/1/2008 IUCEE: Discrete Mathematics 3 7/1/2008 IUCEE: Discrete Mathematics 4 Re-revised Workshop Schedule Wednesday • Monday, June 30, Active learning and • Wednesday, July 2, Data Structures • Each group: instructional goals – Morning – Morning • Data Structures Teaching (2hrs) • Welcome and Overview (1 hr) • Data Structure Activities (1 hr) – Design two classroom activities for your classes. • Introductory Activity (1 hr). Determine – Afternoon background of participants • Group work (1.5 hrs) • Active learning and instructional goals (1hr) in • Content Lecture: (1.5 hr) Average Case Identify the pedagogical goals of the activity. Discrete Math, Data Structures, Algorithms. Analysis – Afternoon • Thursday, July 3, Algorithms • Group Work (1.5 hrs). Development of – Morning activities/goals from participant's classes. • Five of the groups will give progress report to • Algorithms Teaching (2 hrs) • Content lectures (Great Ideas in Computing): (1.5 hr) Problem mapping • Algorithms Activities (1 hr) – Afternoon • Tuesday, July 1, Discrete Mathematics the class • Activity Critique (.5 hr) – Morning • Research discussion (1 hr) • Discrete Mathematics Teaching (2 hrs) • Theory discussion (optional) • Activities in Discrete Mathematics (1 hr) • Overnight each group should prepare ppt • Friday, July 4, Topics – Afternoon – Morning • Educational Technology Lecture (1.5 hrs) • Content Lecture (1.5 hrs) Algorithm • Content Lecture: (1.5 hrs) Complexity Theory slides implementation • Lecture (1.5 hrs) Socially relevant computing – Afternoon • Follow up and faculty presentations (1.5 hrs) • Thursday there will be a feedback/critique • Research Discussion (1.5 hrs) session June 30, 2008 IUCEE: Welcome and Overview 5 June 30, 2008 IUCEE: Welcome and Overview 6

University of Washington Thursday and Friday Course • Each group will develop a presentation on CSE 321 Discrete Structures (4) Fundamentals of set theory, graph theory, enumeration, and algebraic how they are going to apply ideas from structures, with applications in computing. Prerequisite: CSE 143; either MATH 126, MATH 129, or MATH 136. this workshop. • Thursday • Discrete Mathematics and Its Applications, – Two hours work time Rosen, 6-th Edition • Ten week term • Friday – 3 lectures per week (50 minutes) – Three hours presentation time – 1 quiz section • 15 minutes per group with PPT slides – Midterm, Final June 30, 2008 IUCEE: Welcome and Overview 7 7/1/2008 IUCEE: Discrete Mathematics 8 Analyzing the course and Course overview content • Logic (4) • What is the purpose of each unit? • Reasoning (2) – Long term impact on students • Set Theory (1) • What are the learning goals of each unit? • Number Theory (4) – How are they evaluated • Counting (3) • What strategies can be used to make • Probability (3) material relevant and interesting? • Relations (3) • How does the context impact the content • Graph Theory (2) 7/1/2008 IUCEE: Discrete Mathematics 9 7/1/2008 IUCEE: Discrete Mathematics 10 Broader goals Overall course context • First course in CSE Major • Analysis of course content – Students will have taken CS1, CS2 – How does this apply to the courses that you – Various mathematics and physics classes teach? • Broad range of mathematical background of • Reflect on challenges of your courses entering students • Goals of the course – Formalism for later study – Learn how to do a mathematical argument • Many students are not interested in this course 7/1/2008 IUCEE: Discrete Mathematics 11 7/1/2008 IUCEE: Discrete Mathematics 12

Logic Goals • Begin by motivating the entire course • Understanding boolean algebra – “Why this stuff is important” • Connection with language • Formal systems used throughout computing – Represent statements with logic • Propositional logic and predicate calculus • Predicates • Boolean logic covered multiple time in – Meaning of quantifiers curriculum – Nested quantification • Relationship between logic and English is hard for the students – implication and quantification 7/1/2008 IUCEE: Discrete Mathematics 13 7/1/2008 IUCEE: Discrete Mathematics 14 Why this material is important Propositions • A statement that has a truth value • Language and formalism for expressing • Which of the following are propositions? ideas in computing – The Washington State flag is red – It snowed in Whistler, BC on January 4, 2008. – Hillary Clinton won the democratic caucus in Iowa • Fundamental tasks in computing – Space aliens landed in Roswell, New Mexico – Translating imprecise specification into a – Ron Paul would be a great president – Turn your homework in on Wednesday working system – Why are we taking this class? – If n is an integer greater than two, then the equation a n + b n = c n has no – Getting the details right solutions in non-zero integers a, b, and c. – Every even integer greater than two can be written as the sum of two primes – This statement is false – Propositional variables: p, q, r, s, . . . – Truth values: T for true, F for false p → q p q p → q Compound Propositions ¬ p • Negation (not) • Implication p ∧ q – p implies q • Conjunction (and) p ∨ q – whenever p is true q must be true • Disjunction (or) – if p then q p ⊕ q • Exclusive or – q if p p → q • Implication – p is sufficient for q p ↔ q • Biconditional – p only if q

English and Logic Logical equivalence • Terminology: A compound proposition is a • You cannot ride the roller coaster if you – Tautology if it is always true are under 4 feet tall unless you are older – Contradiction if it is always false than 16 years old – Contingency if it can be either true or false – q : you can ride the roller coaster p ∨ ¬ p – r : you are under 4 feet tall ( p ⊕ p ) ∨ p – s : you are older than 16 p ⊕ ¬ p ⊕ q ⊕ ¬ q ( r ∧ ¬ s ) → ¬ q ( p → q ) ∧ p ¬ s → ( r → ¬ q ) ( p ∧ q ) ∨ ( p ∧ ¬ q ) ∨ ( ¬ p ∧ q ) ∨ ( ¬ p ∧ ¬ q ) Logical Proofs Statements with quantifiers Domain: ∃ x Even( x ) Positive Integers • • To show P is equivalent to Q Even( x ) – Apply a series of logical equivalences to ∀ x Odd( x ) Odd( x ) • Prime( x ) subexpressions to convert P to Q Greater( x , y ) ∀ x (Even( x ) ∨ Odd( x )) • Equal( x , y ) • To show P is a tautology – Apply a series of logical equivalences to ∃ x (Even( x ) ∧ Odd( x )) • subexpressions to convert P to T ∀ x Greater( x+ 1, x ) • ∃ x (Even( x ) ∧ Prime( x )) • Statements with quantifiers Prolog • ∀ x ∃ y Greater ( y , x ) • Logic programming language For every number there is some number that is greater than it • Facts and Rules ∃ y ∀ x Greater ( y, x ) • RunsOS(SlipperPC, Windows) Later(x, y) :- RunsOS(SlipperTablet, Windows) Later(x, z), Later(z, y) • ∀ x ∃ y (Greater( y , x ) ∧ Prime( y )) RunsOS(CarmelLaptop, Linux) NotLater(x, y) :- Later(y, x) OSVersion(SlipperPC, SP2) NotLater(x, y) :- OSVersion(SlipperTablet, SP1) SameVersion(x, y) • ∀ x (Prime( x ) → (Equal( x , 2) ∨ Odd( x )) OSVersion(CarmelLaptop, Ver3) MachineVulnerable(m) :- LaterVersion(SP2, SP1) OSVersion(m, v), LaterVersion(Ver3, Ver2) • ∃ x ∃ y (Equal( x, y + 2) ∧ Prime( x ) ∧ Prime( y )) VersionVulnerable(v) LaterVersion(Ver2, Ver1) VersionVulnerable(v) :- CriticalVulnerability(x), Version(x, n), NotLater(v, n) Greater(a, b) ≡ “a > b” Domain: Positive Integers

Nested Quantifiers Quantification with two variables Expression When true When false • Iteration over multiple variables • Nested loops ∀ x ∀ y P(x,y) • Details – Use distinct variables ∃ x ∃ y P(x,y) • ∀ x( ∃ y(P(x,y) → ∀ x Q(y, x))) – Variable name doesn’t matter • ∀ x ∃ y P(x, y) ≡ ∀ a ∃ b P(a, b) ∀ x ∃ y P(x, y) – Positions of quantifiers can change (but order is important) • ∀ x (Q(x) ∧ ∃ y P(x, y)) ≡ ∀ x ∃ y (Q(x) ∧ P(x, y)) ∃ y ∀ x P(x, y) Reasoning Goals • Students have difficulty with mathematical • Understand the basic notion of a proof in a proofs formal system • Attempt made to introduce proofs • Derive and recognize mathematically valid proofs • Describe proofs by technique • Understand basic proof techniques • Some students have difficulty appreciating a direct proof • Proof by contradiction leads to confusion 7/1/2008 IUCEE: Discrete Mathematics 27 7/1/2008 IUCEE: Discrete Mathematics 28 Reasoning Proofs • “If Seattle won last Saturday they would be • Start with hypotheses and facts in the playoffs” • Use rules of inference to extend set of • “Seattle is not in the playoffs” facts • Therefore . . . • Result is proved when it is included in the set