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

Background Who? When? Where? What? 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”

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

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

Data collection Diagnostic tests Exams Questionnaires 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 …

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

Structured derivations + _

Traditional method + _

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.”

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.”

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.”

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?