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Welcome to Spring Semester 2013
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Introduction to Mathematical Thinking
Keith Devlin
Learn how to think the way mathematicians do - a powerful
cognitive process developed over thousands of years.
Dear FIX,
There's a great discussion about the derivation of the truth table for the conditional in this forum thread.
I think a lot of the difficulty people are having in this course comes from the way schools (and colleges) restrict mathematics to examples that are clean and clear cut, and don't venture into the more messy real world where things are less straightforward. So I suspect many people are trying to interpret much of what we are doing as formal proofs, which is not the case. (One of the course goals is, however, to **develop some of the machinery** to do rigorous proofs. We will be doing some rigorous proofs in Week 5.)
In the title of this course, I use the term "mathematical thinking" rather than "mathematics" to emphasize that this is not about (pure) mathematics. Rather, the course is about developing **a way of thinking** that is heavily influenced by formal mathematics -- a way of thinking that has proved to be very useful.
It fact, it **is** the way (we) mathematicians think when they set about solving a problem -- **even a problem in pure mathematics**. You don't see this kind of thing in their published papers or the textbooks, however, because they clean everything up and make it rigorous.
I think schools do a disservice to students by not lifting the hood of the mathematical automobile and showing how the engine works. That can create the impression that the ability to do mathematics is some rare ability. It's not. There is no big secret. Discounting a small number of individuals who do seem truly gifted, the vast majority of mathematicians are just regular folks who have learned to think a certain way. (I talk about this in my latest MOOCtalk blog post.)
I hope this helps. Now on to Week 4!
-- KD
There's a great discussion about the derivation of the truth table for the conditional in this forum thread.
I think a lot of the difficulty people are having in this course comes from the way schools (and colleges) restrict mathematics to examples that are clean and clear cut, and don't venture into the more messy real world where things are less straightforward. So I suspect many people are trying to interpret much of what we are doing as formal proofs, which is not the case. (One of the course goals is, however, to **develop some of the machinery** to do rigorous proofs. We will be doing some rigorous proofs in Week 5.)
In the title of this course, I use the term "mathematical thinking" rather than "mathematics" to emphasize that this is not about (pure) mathematics. Rather, the course is about developing **a way of thinking** that is heavily influenced by formal mathematics -- a way of thinking that has proved to be very useful.
It fact, it **is** the way (we) mathematicians think when they set about solving a problem -- **even a problem in pure mathematics**. You don't see this kind of thing in their published papers or the textbooks, however, because they clean everything up and make it rigorous.
I think schools do a disservice to students by not lifting the hood of the mathematical automobile and showing how the engine works. That can create the impression that the ability to do mathematics is some rare ability. It's not. There is no big secret. Discounting a small number of individuals who do seem truly gifted, the vast majority of mathematicians are just regular folks who have learned to think a certain way. (I talk about this in my latest MOOCtalk blog post.)
I hope this helps. Now on to Week 4!
-- KD
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