Sabados Literarios
Introduction to Mathematical Thinking
Keith Devlin
Learn how to think the way mathematicians do - a powerful cognitive process developed over thousands of years.
Course goals and syllabus
The goal of the course is to help you develop a valuable mental ability – a powerful way of thinking that our ancestors have developed over three thousand years.
Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today’s world. This course helps to develop that crucial way of thinking.
The primary audience is first-year students at college or university who are thinking of majoring in mathematics or a mathematically-dependent subject, or high school seniors who have such a college career in mind. If that is you, you will need mathematical thinking to succeed in your major. Because mathematical thinking is a valuable life skill, however, anyone over the age of 17 could benefit from taking the course.
The course starts on Monday September 17 and lasts for seven weeks, five weeks of lectures (two a week) followed by two weeks of monitored discussion and group work, including an open book final exam to be completed in week 6 and graded by a calibrated peer review system in week 7.
Lecture videos are released at 10:00AM PDT on Mondays and Fridays. (Week 1 is slightly different: The Monday video release (listed as Lecture 0) is a short welcome and course description, the Wednesday and Friday releases are of lectures 1 and 2.)
The lecture topics are (in addition to the initial Instructor’s welcome on the first day of class):- Introductory material
- Analysis of language – the logical combinators
- Analysis of language – implication
- Analysis of language – equivalence
- Analysis of language – quantifiers
- Working with quantifiers
- Proofs
- Proofs involving quantifiers
- Elements of number theory
- Beginning real analysis
Course structure
Basic elements of the course
- Lectures – videos presented by the instructor.
- In-lecture quizzes – simple multiple-choice questions that stop the lecture, designed to assist you in pacing and monitoring your progress.
- Assignment sheets – downloadable PDF files to work through in your own time at your own pace, ideally in collaboration with other students. Not graded.
- Problem sets – in-depth problems like those on the assignment, but with a deadline for submitting your answers (in a multiple choice format). Graded.
- Tutorial sessions – the instructor distributes (PDF) or explains (video) answers to some of the previous week’s assignment problems.
- Reading assignments – downloadable PDFs files providing important background information.
- Final exam – a downloadable PDF file that you will have one week to complete before participating in a peer review process. Required to be eligible for a grade of completion with distinction.
Lectures
Lecture videos are released at 10:00AM PDT on Mondays and Fridays. (Week 1 is slightly different: The Monday video release (listed as Lecture 0) is a short welcome and course description, the Wednesday and Friday releases are of lectures 1 and 2.) The lecture videos vary in length from about 25 to 50 minutes if played straight through, but completing the embedded progress quizzes will extend the total duration by a few minutes, and you will likely want to stop the playback several times for reflection, and sometimes you will want to repeat a section, perhaps more than once. So you can expect to spend between one hour and ninety minutes going through each lecture, occasionally perhaps more.
In-lecture quizzes
Each lecture is broken up by short multiple-choice “progress quizzes”. These in-lecture quizzes are essentially punctuation, providing a means for you to check that you are sufficiently engaged with the material.
Slightly modified versions of the quizzes will also be released as standalones at the same time as the lecture goes live, so if you do not have a good broadband connection and have to download the lecture videos to watch offline, you can still take the quizzes. In which case, you should do so as close in time to viewing the lecture as possible, to ensure gaining maximim benefit from the quizzes in monitoring your progress. The standalone quizzes are grouped according to lecture.
Completion of all the quizzes is a requirement (along with watching all the lectures) for official completion of the course, but we do not record your quiz scores, so quiz performance does not directly affect your final grade. If you complete the quizzes while watching the lecture (the strongly preferred method, as it helps you monitor your progress in mastering the material), you do not need to complete the standalone versions.
Course assignments
An assignment will be released at the end of each lecture, as a downloadable PDF file. The assignment is intended to guide understanding of what has been learned. Worked solutions to problems from the two weekly assignments will be demonstrated (video) or distributed (PDFs) in a tutorial session released the following Wednesday (so in weeks 2 through 6). The tutorial sessions will be released at 10:00AM PDT.
Working on these assignment problems forms the heart of the learning process in this course. You are expected to form or join a study group, discuss the assignment problems with others in the group, and share your work with them. You are also strongly urged to assess one another’s answers. A structured form of peer review will be used for the final exam, when you will be graded by, and grade the work of, other students, randomly (and blindly) assigned, so it will help to familiarize yourself beforehand with the process of examining the work of others and providing (constructive) feedback.
Problem sets
At the end of each week, following the Friday lecture, a for-credit Problem Set will be posted, with submission due the following Tuesday. The scores on these problem sets will count toward the course grade. Though the Problem Set has a multiple-choice quiz format, these questions are not the kind you can answer on the spot (unlike most of the in-lecture quizzes). You will need to spend some time working on them before entering your answers.
Though you are strongly encouraged to work with others on understanding the lecture material and attempting the regular assignments, the intention is that you work alone on the Problem Sets, which are designed to give you and us feedback on how you are progressing.
Tutorial sessions
The tutorial sessions are more than mere presentations of solutions to the previous week's assignments and problem session. They are really lectures based on problems that the student has already attempted. You can expect to expand your knowledge of the course material beyond the Monday and Friday lectures. Not all questions on the assignments sheets and problem set will be considered in the tutorial session.
Final exam
Though the lectures end after week 5 (apart from a tutorial on the final assignment), the final two weeks are intended to be highly active ones for any students seeking a grade of distinction, with considerable activity online in the various forums and discussion groups. This is when you are supposed to help one another make sense of everything.
At the start of week 6, an open-book exam will be released, to be completed by the end of the week. Completed exams will have to be uploaded as either images (or scanned PDFs) though students sufficiently familiar with TeX have an option of keyboard entry on the site. The exam will be graded during week 7 by a calibrated peer review system. The exam will be based on material covered in the entire course.
As with the weekly Problem Sets, the intention is that you work alone in completing the final exam.
NOTE: The process of peer reviewing the work of others (throughout the course, not just in the final exam) is intended to be a significant part of the learning experience and participating in the formal peer review procedure for the final exam is a requirement for getting a grade of distinction. In principle, it is during week 7 that stronger students will make cognitive breakthroughs. (Many of today’s professors really started to understand mathematics when, as graduate student TAs, they first helped others learn it!)
Course completion and final grade
There are two final course grades: “completion” and “completion with distinction”. Completion requires viewing all the lectures and completing all the (in-lecture) quizzes and the weekly problem sets. Distinction depends on the scores in the problem sets and the result of the final exam.
Pacing
The pacing of the lecture releases is designed to help you maintain a steady pace. At high school, you probably learned that success in mathematics comes from working quickly (and alone) and getting to the right answer as efficiently as possible. This course is about learning to think a certain way – the focus is on the process not the product. You will need time to understand and assimilate new ideas. Particularly if you were a whiz at high-school math, you will need to slow down, and to learn to think and reflect (and ideally discuss with others) before jumping in and doing. A steady pace involving some period of time each day is far better than an all-nighter just before a Problem Set is due.
No comments:
Post a Comment