But a way of knowing what?
This is a set of resources that are predominantly linked to the IB Diploma Theory of Knowledge course, and about Mathematics
There is a presentation with some links to it at the end. The other resources can be used before or after doing the presentation or at any time really. They are just interesting and in some cases fun!
Any discussion of Maths should begin with an episode of Numberwang. Feel free to watch several more episodes to get you thinking. I recommend starting each lesson with one. Work up to The History of Numberwang with its special appearance by Bertrand Russell and of course The Numberwang code.
Mitchell and Webb at their crazy best.
These two could be homework
The More or Less podcast on the art of counting
This BBC series is a wonderful look at current statistics and their fallibility, too frequent inaccuracy and ability to mislead. This short clip interviews the writer of a book on the art of counting, indicating that what you count and how you count it matters more than the numbers you end up with. Any statistic requires you to ask who is doing the counting, what are they counting and why.
If you only watch the first minute and a half of this wonderful BBC documentary on Andrew Wiles, you will see the passion of discovery and never again believe people who say that Maths is unemotional. I still well up in tears before the opening credits! And for good measure, you even see the great John Conway if you go on a bit as well!
The invention of complex numbers
An excellent video on the history of the search for the solution to the cubic and Bombelli’s decision. The people who made this are called Veritasium. They also have an excellent one on Godel’s incompleteness theorem
The next two are activities I suggest you do in class before or after the introductory session
This investigates, in a very simple way, what we mean by the things we use in mathematics. Are they actually real?
This is a great activity which I recommend for everyone. Can you count? What is counting? Why Mathematical truth has absolutely nothing to do with evidence. Possibly the only are of knowledge not requiring evidence.
… It’s the way that you prove it
This may be the most important link in the list. We are often faced with assertions that the character of Maths is that it is irrefutably correct. As the play shows in trivial examples, that is not the case. It is an evolving growing changing area of knowledge. I think this play also shows it is more of an activity than a body of knowledge. The misconception often crops up when non-mathematicians ask if it is invented or discovered.
This is the full play, in both acts. I freely credit Imre Lakatos with the inspiration, but this version is designed for High School students. It should be fun, understandable and thought-provoking. I always used to read it through in class with the students taking the main roles. I have also staged it in various schools with the Maths teachers taking the roles of students. Fun!
The next few are referred to in the session on Maths as a way of knowing. If you have inquisitive students who want to know what a 2 ½ sided shape is or why there are more irrationals than rationals, then these are relatively simple activities and explanations.
This shows you that the Pentagram is actually a 2 ½ sided shape and invites you to explore other shapes with sides that are rational rather than whole numbers
This is an exercise to prove that the Golden Ratio is not a number in Pythagorean language, or is an irrational number in modern language. There is a case to be made that this was the great leap into formal abstraction, and indeed abstract thought itself. In that sense, I would be prepared to argue that it was the dawn of civilisation.
This is a very short intuitive proof that there are infinitely many more irrational numbers than there are rational numbers. If you look at the set of all numbers there are almost no whole numbers and fractions
This is a video I made during lockdown about another case of mathematicians finding the current numbers that we have are insufficient and so creating new ones. In this case the development of Transfinite numbers by Georg Cantor. It was highly controversial at the time and he was called a madman and worse. But it is now accepted as one of the most beautiful areas of mathematics. It is introduced through the story of The Infinite Hotel.
Those last few pieces are all on my website www.mathsthoughts.com along with lots more different and interesting mathematical ideas.
This is the clip from the film.
Related to this, and clearly the inspiration for the film is the case of the Twins described by Oliver Sachs in his book: “The Man Who Mistook His Wife For A Hat”.
This is the chapter
This is the link to the Prezi file for Windows and for Mac.
The part where I do the Maths using other programmes is on a video here.
The presentation for the Maths and Ethics part is here.