Archive for July, 2008
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!
Taught: A Pascal’s Triangle wall-display
“What are those numbers on the wall, sir?”
“An interesting question. What do you think they are?”
This isn’t a single taught lesson, but something which impacted many of my lessons for at least a week. It’s the story of one of the first pieces of decoration I put up in my new classroom. A wall-display… to be precise, this one:
(as with most images on this site, click for full size)
Background
Pascal’s Triangle has an enormous number of properties, and can be the starting point for a large number of investigations. I think it should be mandatory for all maths classrooms to feature it in a prominent place!
The Story of the Wall-Display
The above image isn’t how the display started, though. When I first put the (individually cut out and laminated) numbers on my wall, there were no questions, no header, and several rows missing. All the questions which subsequently appeared were ones which had been asked by students, mainly during the ‘dead time’ at the start of lessons, while waiting for a critical mass to arrive.
Before the questions arrived on the board, I gave the numbers their heading, but only after every group I teach had had the chance to see the numbers ‘naked’. These groups vary both in age and ability, but the initial questions asked by the students were all very much the same (and notice that I say ‘asked by the students’ rather than ‘asked of the students’).
Most of the sets figured out the generating relationship very quickly, but the colours took a little extra time, particularly as the focus of the lessons being taught that week was never on the display itself. Several of the students I discussed it with quickly saw that the white numbers were those divisible by three, but were unsure what the other colours meant. For several, it was their first exposure to the ideas which will lead into modular arithmetic, which is not actually in any UK GCSE or A-level syllabus.
Although it took a significant amount of time to cut out and laminate each number individually, I feel it was worth it for the increase in flexibility you get: you can easily remove several numbers, or a row, and ask students to put them back; you can remove a diagonal column, and lay it out somewhere else for inspection, so that students can see (for example) the trianglular numbers without getting distracted by the rest of the triangle; you can slowly build up the triangle over several days, so that the display isn’t seen as something static and fixed from the very start.
The display remained on my wall for the rest of the year for several reasons — one being that the numbers are actually useful in several different contexts. In statistics, for example, they give the number of ways of choosing k elements from an n-element set, and in pure mathematics they form the coefficients needed in the Binomial theorem. It also looks pretty!
The Quest for Ownership
By the time this display arrived, I had had the following poster on my walls for several weeks:
As you can see, it contains essentially the same information as my wall display, although in a slightly less colourful format. It actually goes much further, revealing a large number of patterns in the Triangle which my students never guessed at. Despite this, not a single student demonstrated any interest in Pascal’s Triangle until my display appeared… and I certainly didn’t get the impression that anyone in any of my sets had read anything on this poster.
To me, this emphasises the importance of getting students engaged in the material by feeling that they are making a contribution to something dynamic and developing. The static, over detailed poster with all the information already there ellicited no interest, while the simple, changing, colourful display (which didn’t intially have a title imposted on it) was very successful at engaging a wide variety of pupils.
Future Development
Next year, I would love to use this wall display as the basis of a mathematical investigation with one of my groups. It might lead to combinatorics, or to fractals (via the Serpinski gasket), or to the Stirling numbers, or to elementary number theory… I need to spend some time over the summer thinking about how best to structure the activity.
I also feel there is a need to put this in some historical context, both European (who was Pascal, and what else did he do?) and international: Indian commentaries on poems, Chinese diagrams with rod numerals, etc. This historical context is something that I will be spending much more time on with all the areas I teach next year (expect a post on this in the future!).
Sources
Here are the files I used to create the wall display:
![[SVG]](http://joningram.org/blog/wp-content/uploads/2008/07/pascals-triangle-questions.svg){kind=link}
Both of these were originally created using the excellent free vector drawing program Inkscape, which I plan to write about in the future. Inkscape’s native format is .SVG, and I have uploaded in this format for anyone wishing to edit the files.
Through the magic of embedded media, you can also browse them:
Tags: classroom, Pascal's Triangle, reflection, Resources, wall-display
Learnt: Make your room your own.
(Update: This post is featured in the 180th Carnival of Education. Check it out!)
Apologies for those of you who wish to read this post, but it has been removed.
Tags: displays, Lessons Learnt, preview, The Classroom