Lessons taught; Lessons learnt

A confession — I knew very little about the theory of splines before writing this week’s introduction to Bezier curves and splines. I’ve been trying to do some catch-up reading in the last few days, and have found quite a few interesting links I thought I should share:

• The Algorithmist has a very interesting series of articles on the history, theory, and calculation of splines. They are well explained and easy to read.
• Tim Lambert has an interesting page giving interactive examples of many different splines: the natural cubic spline, Bezier splines, B-splines and Catmull-Rom splines. Very interestingly for me, he gives the source code for the interactive Java applets — something for me to read through over the weekend!
• Grady Wright also has an interesting interactive applet demonstrating cubic splines.
• It shouldn’t come as a surprise that Mathematica has excellent integrated support for splines — see the “Splines” entry at Mathworld for links to the Mathematica routines. If only I had a spare grand for a copy of it!

If you know any other good resources on the theory or calculation of splines, then please leave a comment. Any recommendations for books on this area would also be appreciated — I need something to keep me occupied this summer…

It’s close to the end of term — why not use part of a lesson to take your students for a tour through a document that introduced one of the most important symbols in mathematics? The document is Robert Recorde’s ‘Whetstone of Witte’, first published in 1557, and the symbol is the equals sign (=).

The passage where equals is introduced is quite well known. See for example this interesting article on 360, which puts the Whetstone into some historical context.

Transcribed:

Howbeit, for easie alteratiô of equations. I will propounde a fewe exâples, bicause the extraction of their rootes, maie the more aptly bee wroughte. And to auoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke vse, a paire of paralleles, or Gemowe lines of one lengthe, thus: ====, bicause noe .2. thynges, can be moare equalle. And now marke these nombers.

Transcription note: the first word of this is often transcribed ‘nowbeit’, but the very next page of this has a very clear example of a capital ‘N’ and a capital ‘H’ — it’s definitely an ‘H’!

There is much more to the book than this paragraph — for a start, it was the first book in English to use the + and – signs. Even just looking at the page where this paragraph appears is a very useful thing to do. Here is a scan of the double page spread where the equals sign is first used:

(Source: whetstone-of-witte-equals-sign.pdf )

It’s a very useful exercise, particularly if you’ve never tried to read a book published in the 16th century before, to spend five minutes or so trying to decipher this text.

On the bottom of the left-hand page we have six examples of things that look a lot like equations — but what are the odd looking symbols?

With a bit of squinting, you might be able to convince yourself that these are symbols that evolved into modern-day ‘x’, ‘y’ and ‘z’, so equation 5 for example would say ‘18x+24y=8z+2x’. That’s a good guess, but it’s not quite right. These were in fact symbols for powers of ‘the unknown’, including a symbol for ‘no unknown involved’ (in other words, a constant term). Earlier in the book Recorde gives (in quite painful detail) tables of symbols to use for various powers of the unknown, using quite a nifty powers-of-primes-style notation system. More on that another day!

Translated into modern notation, the six equations above are:

If you are giving this to a class, then you might find it useful, after making them stare at the original for a while, to give them the following transcription:

(Source: robert-recorde-equals-sign.doc)

This preserves all the idiosyncratic spelling and typography of the original. If you’re feeling particularly nice to your classes, you can always give them a version with modernised spelling (and also some slight grammatical editing).

You can then discuss how Recorde explains what to do with the first equation. In modernised form:

In the first there appears 2 numbers, that is equal to one number, which is . But if you mark them well, you may see one denomination on both sides of the equation, which never ought to stand. Wherefore abating the lesser, that is out of both the numbers, there will remain . That is, by reduction, .

In the rest of this chapter, Recorde goes into great detail about how to rearrange these equations into the correct form (for him, the highest power of the unknown should be on its own on the left, with everything else on the right), and then gives quite a few examples of word problems, with worked solutions. These are very interesting in their own right — look out for some 16th century word problems in the next week or so!

Looking into the future, it is now only 48 years until the 500th anniversary of the equals sign. If any of us are still around in 2057 (fingers crossed: I’ll be in my late 70s), I hope to see the contribution of Robert Recorde properly recognised — I think a week or so of national celebration would be fitting!

“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:

• Pascal’s Triangle PDF [Zipped SVG]
• List of Questions on Pascal’s Triangle PDF [SVG]

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: