Structured Derivations in Education - Experiences from Individual Courses

1. Structured Derivations in Education - Experiences from Individual Courses Tutorial, FM’08 Linda Mannila 26.5.2008

2. Background • Who? • Third year high school students (17-18 year olds) • When? • Fall 2007 • Where? • Pargas svenska gymnasium • Katedralskolan in Turku • What? • Elective course in advanced mathematics: “Logic and number theory”

3. Course contents • Logic • Formalization, connectives • Structured derivations • Truth values • Proofs • Quantifiers • Number theory • Divisibility • Prime numbers • Congruences

4. Research questions • How do students react to structured derivations when introduced in a single course? • First impressions, attitudes • Difficulties learning the approach • Changes during course • What errors do students make when learning structured derivations? • Method related vs. content related • Changes during course

5. Data collection • Diagnostic tests • Pre course • Mid logic part • Exams • Mid term (logic) • Final (number theory) • Questionnaires • Pre, mid and post • New types of assignments • Read and correct • Fill in the blanks • …

6. Done so far • Preliminary analysis of student opinions • mid and post course • Categorized answers to open-ended questions • Benefits and disadvantages of traditional method • Benefits and disadvantages of SD • Spontaneous comments about SD

7. Structured derivations + _

8. Traditional method + _

9. Negative comments • “You don’t always know how to write them.” • “It’s been difficult. Much easier before when you didn’t have to explain what you were doing, with laws etc.” • “When solving relatively simple exercises it feels long-winded.” • “Sometimes difficult to remember writing out all steps, sometimes frustrating.”

10. But… • Of the comments that start out in a negative tone • “It’s time consuming…” • “Solutions become long…” • “Feels unnecessary at times…” • most end up quite positive • “…but you learn more.” • “…but you understand what you’re doing.” • “…but it makes solutions clearer.”

11. Positive comments • “I’ve also noticed that I’ve started writing in a more ordered, clear and structured way in other math courses too.” • “It’s difficult to do a ‘normal’ task with the methods used in this course since you need to explain what you’re doing, and I usually don’t know what I’m doing.” • “I actually liked this course (which is rare when it comes to math), structured derivations made everything much clearer.” • “It becomes clearer and you know what you do. At least for me it’s easier to understand when more details are explained, not just a lot of numbers all over the place.” • “At first I thought writing like that was unnecessary, but now I think it’s a very good way, because now I understand exactly how all exercises are done.” • “It’s also good to once and for all clarify why things are done the way they are.”

12. Conclusions • Students can learn how to write solutions in a structured manner during one single course • Benefits outweigh the drawbacks • But are there any long term benefits if only used in one single course? • In that case, what?