Lessons taught; Lessons learnt

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!

(This post is featured in the 188th Carnival of Education. Check it out!)

“In preparing a lecture I find I always have to work hardest on the things I do not say. The things I am sure to say I can easily get up. They are obvious and generally accessible. But they, I find, are not enough. I must have a broad background of knowledge which does not appear in speech. I have to go over my entire subject and see how the things I am to say look in their various relations, tracing out connections which I shall not present to my class.

One might ask what is the use of this? Why prepare more matter than can be used? Every successful teacher knows. I cannot teach right up to the edge of my knowledge without a fear of falling off. My pupils discover this fear, and my words are ineffective. They feel the influence of what I do not say. One cannot precisely explain it; but when I move freely across my subject as if it mattered little on what part of it I rest, they get a sense of assured power which is compulsive and fructifying.”

The Teacher: Essays and Addresses on Education, page 17. Written in 1908. I find this quote deeply relevant and stimulating, as a classroom teacher who is currently preparing for the return of school next week!

I’m a maths teacher, and this next quote could easily have been written about the current trends in the teaching of my subject:

“Among the many changes in mathematical education during the last twenty years, and among the many and often conflicting ideals which have directed these changes, one element at least appears throughout; a desire to relate the subject to reality, to exhibit it as a living body of thought which can and does influence human life at a multitude of points… Our children must learn to think.”

This is from page 35 of Essays on Mathematical Education, written in 1913.

These are just two of more than 8000 results that appear when you search archive.org for texts mentioning ‘education’.

Places like archive.org allow us to correct the notion many people have about the way people were taught in the past. Vocational education; project-based teaching; differentiation; learning styles; curriculum content; the importance of the physical education of youngsters — all of these and more have been considered by teachers for many generations.

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!