Archive for the ‘Books’ tag
Visualisation: More rotating squares
While we’re rotating polygons, here’s another nice visualisation, this time of a linkage of squares that make up a larger square, and rotate around without self-intersecting as you move the slider.
(Source: rotating_square_linkage.ggb)
Just as with the previous visualisation, you can see this just as a source of pretty pictures, like
or you can start asking yourself questions. For example:
- What other ways can we find to connect the small squares together so that they will expand and collapse back into a square without any of the small squares intersecting? How many of these have rotational or reflectional symmetry?
- How can we describe the movement (the locus) and the amount of rotation of the different squares?
- Can we do something similar with other polygons?
- More generally, can we do something like this to move smoothly between different shapes? And what does ’something like this’ mean?
For the second question, we can use the ‘trace’ feature of Geogebra (and all other dynamic geometry programs) to follow the path of particular points, which might give us some idea about how to derive the equations of the points.
On the last point, there are a large category of dissections which involve taking an object, slicing it into a finite number of shapes, and reassembling them into a different shape. There are several in Amusements in Mathematics, a puzzle book by Henry Dudeney from 1917 which I had a part in digitizing for Project Gutenberg. See this section, for example, for a great introduction to dissection puzzles.
Moving further afield, we can do something very similar to this example to demonstrate Pythagoras’ Theorem. That visualisation may appear later this week :).
Summer Reading: Part 1 — Books for me
Like many other people, I find the summer a great time for kicking back and catching up on some reading. My wife is off on a quick break at the moment, and in preparation bought seven trashy crime fiction novels. My choice of summer reading is a little more eclectic, and would probably have been impossible if Amazon didn’t exist.
I’ve broken down the material I plan to read over the summer into several different categories. In this first part, I’ll include the books which I want to read. Another large category will be books that I ought to / need to read, which will include all the textbooks for the modules I will be teaching that term. I’m not thinking about that too much at the moment…
Anyhow, this selection of reading matter is a combination of new books that I haven’t even opened yet, older books which I want to reread, and one relic which I think will give me an insight into the ‘new math’ of the 1960s. There’s no crime fiction :).
(click for bigger)
From left to right, we have…
1. The Calculus Gallery; Masterpieces from Newton to Lebesgue by William Dunham.
This book arrived yesterday, and I’ve already read 3/4 of it… expect a detailed review of it (and all the other books on this photo) when I’m done! It’s a whirlwind tour through some classics of calculus and real analysis, done in chronological order, focusing on the mathematicians and the key questions that motivated them.
2. The Colossal Book of Mathematics, by Martin Gardner. I’ve read loads of bits and pieces of Gardner over the years, and felt that this summer would be a good opportunity to properly engage with what is, according to the blurb, ‘Gardner’s 50 best articles for Scientific American’. I’m hoping to find quite a lot of interesting material to use in lessons, or maths-based activity sessions.
3. Some Lessons in Mathematics; A Handbook on the Teaching of ‘Modern’ Mathematics, by Members of the Association of Teachers of Mathematics. This one is out of left-field! It’s not an Amazon purchase, but something I picked up for free at this year’s joint ATM/MA Easter conference. It’s a collection of lesson ideas and resources for the teaching of ‘new maths’, published in 1962.
4. Trigonometric Delights, by Eli Maor. Several books by this author have appeared on my ‘recommendations’, and I chose this one as a sampler for several reasons: first, I’m looking for ways to enliven my teaching of trigonometry, and this sounds like it has a lot of interesting content; second, it approaches the subject from a historical perspective (I’m guessing the Rhind papyrus appears prominently, given the cover!), something I am quite receptive to at the moment; third, it has a lot of positive reviews on Amazon. Let’s see!
5. Quirkology; The Curious Science of Everyday Lives, by Richard Wiseman. I’m a sucker for ‘Freakonomics’-style popular-science books which look at the intersection of mathematics with economics, or psychology, etc. This one sounds like it is full of interesting psychological studies, some of which I may be able to replicate with my students next year.
6. Euler; The Master of Us All, by William Dunham (the author of book (1)). I’ve always liked the image of Euler, both as a man and as a mathematician, and my connection to Euler will get closer next year, as our head of department is naming all our classrooms after famous mathematicians, and mine will be called Euler! Given this, I thought it would be a really good idea to find out more about the mathematics which Euler actually did, perhaps with the aim of making some of it accessible to my students.
7. The Psychology of Learning Maths, by Richard R. Skemp. This is a classic in the field of mathematical education, and a book which I bought (but didn’t properly read) back when I took a course called ‘The Development of Mathematical Concepts’ during my undergraduate days back in the last century. It’s something I’ve been meaning to reread for ages, and this summer is as good a time as any.
8. Where Mathematics Comes From; How the Embodied Mind Brings Mathematics into Being, by George Lakoff and Rafael Nunez. This was a bit of an impulse buy, as I was intrigued by the writeup given to the book in several recent blog posts (see for example this one at Wild About Math). In an initial skim-through, it looks like it has the potential to be an enormously interesting book, even getting close to Godel, Escher Bach territory… or it could just decend into linguistic nonsense. I look forward to finding out!
Actually, looking at my Amazon page, I have two more books ordered which are due to drop through my letter-box soon:
9. Writing to Learn Mathematics; Strategies that Work, by John Countryman. The synopsis says: “This text demonstrates that the use of journals, learning logs, letters, autobiographies, investigations, and formal papers can improve the reasoning abilities of maths students“. I was led to it by several blog posts about ‘learning logs’, and other experiments in getting students to write about mathematics. It’s American, but I’m hoping the content will be transferrable over to a UK context.
10. Mathematical Footprints: Discovering Mathematical Impressions All Around Us, by Theoni Pappas. I’ve bought several books by Pappas in the last year, including ‘Joy of Mathematics‘ and ‘The Adventures of Penrose, the Mathematical Cat‘, and think they are an excellent entry point into some quite advanced maths for students of a wide age range. Footprints intrigued me as I want to bring more context, background, and history into my teaching. I look forward to reading it.
That should get me through the next couple of weeks!