SUPPLEMENT: Using the course evaluation rubric
Introduction to Mathematical Thinking
Learn how to think the way mathematicians do – a powerful cognitive process developed over thousands of 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.
Number Theory, Real Analysis, Mathematical Logic, Language
In the last lecture, based on what he previous taught, the professor give us the definition of limitation which is the beginning of the Calculus. I wish I taken this course before university.
Great course about mathematical thinking, mathematical logic and proofs. I have a much better grasp on doing rigorous proofs after taking this course. Thanks for putting it together for us!
In Week 2 we continue our discussion of formalized parts of language for use in mathematics. By now you should have familiarized yourself with the basic structure of the course: 1. Watch the first lecture and answer the in-lecture quizzes; tackle each of the problems in the associated Assignment sheet; THEN watch the tutorial video for the Assignment sheet. 2. REPEAT sequence for the second lecture. 3. THEN do the Problem Set, after which you can view the Problem Set tutorial. REMEMBER, the time estimates for watching the video lectures are machine generated, based on the video length. Expect to spend a lot longer going through the lectures sufficiently well to understand the material. The time estimates for completing the weekly Problem Sets (Quiz format) are a bit more reliable, but even they are just a guideline. You may find yourself taking a lot longer.
Lecture 3 - Implication31:34
Lecture 4 - Equivalence24:31
SUPPLEMENT: Using the course evaluation rubric4:41
Dr. Keith Devlin
Co-founder and Executive Director
In this video, I'll say a little about using the course evaluation rubric.
The most common use of an evaluation rubric is to achieve uniformity in grades
when a team of experts is grading a large collection of examinations.
The goal is to assign a reliable grade to each candidate's work.
In particular,
it's assumed that the people doing the evaluation are experienced graders.
In this class, we're using a rubric in a very different way.
Not to help experts produce a reliable grade of a piece of work,
but as a mechanism to help beginners learn about proofs.
If you look at the rubric for the course,
you see that it asks you to consider five different features of proofs.
Logical correctness, clarity,
opening, stating the conclusion and reasons.
I'll come to the sixth category later, by the way.
These are all important features of proofs.
And for pedagogic reasons, are given equal weight.
Here's why.
As I state repeatedly throughout the course,
proofs do much more than simply establish that some statement is true.
Actually, in practice they don't really do that, they simply provide evidence
that convinces a reasonable person with sufficient background knowledge.
But let's leave that issue for now.
If you look at the role played by proofs in mathematics,
you discover that just as important as logical correctness
is the understandability of the proof by an intended reader.
In fact, in practical terms understandability is more important than
logical correctness.
There's probably no modern proof of more than a few pages
that does not initially have a logical flow buried somewhere.
What makes proofs useful is that others can not only detect errors, but
also have enough information to try to correct them.
Proofs are as much about communication as establishing truth.
Because we have to use simple examples to develop proof evaluation skills,
in most of the examples you see, one or
more of the rubric criteria will apply only in a simplistic way if at all.
But in more complicated examples they will all be important.
That's why the five criteria carry equal weight.
When you evaluate arguments, you should view them in terms of
each of the five rubric criteria in turn on an equal footing.
True, experts rarely use rubrics.
They're so familiar with the proof concept that they automatically and
holistically judge in terms of all those perspectives.
For a beginner, the rubric breaks up the task of judging the overall quality of
a proof to judging those five aspects in turn.
Each of those five is considerably simpler to do than the entire task,
but you'll still find it difficult.
After you've judged a proof in terms of the five proof characteristics, the rubric
asks you to bring them all together to produce a single grade for the proof.
This is the number you would enter if your task were to assign a single
numerical grade to someone's proof.
It corresponds to the number an expert working without a rubric would assign.
The rubric is designed to assist you as a beginner to arrive at that final grade.
Incidentally, when you're evaluating the work of another student,
as you would in test flight if you do it.
You do owe it to that student to be as fair and accurate as possible.
But no one expects accuracy from someone who is just beginning to learn about
proofs.
Because everyone's work is evaluated by three fellow students,
all anonymously by the way, with a reported grade computed from those three
grades, plus the individual's own self grade.
It turns out that most of the time, the final grade is pretty good.
Nevertheless, the real purpose of the test flight evaluation process
is that it offers tremendous benefits to the evaluator
in coming to a better appreciation of what makes a good proof.
In simple terms, in this course, the grading is primarily for
the benefit of the grader.