Kevin Lees has been supporting home education for a long time. He has now been contributing to Otherways for twenty years! We thank him for his continued and enthusiastic support of home education.

by Kevin Lees

Many parents have told me they were put off maths at school, and probably most people regard it as hard, or dull, or generally both. And yet not everybody will enjoy maths anyway, just as not everybody enjoys ballet or footy. But it can be fun � at least mentally exciting. As well as the approaches proposed by Susan Wight and Thomas Armstrong in Otherways 105, I believe there is value (for those children who enjoy it) in exploring mathematical ideas themselves. And, with one or two exceptions, my suggestions are EASY. It’s just a slightly different approach to maths.

NUMBER is splendid for seeing patterns, which can help maths to make sense. Young learners using beads can “?nd” that some numbers can be laid out in box shapes, e.g. 6 is 2 rows of 3, and others (the so-called “primes”, like 5 and 11) can’t. And a few
numbers like 4, 9 and 16 can be set out as squares.
A bit later on, when learning “tables”, children can be encouraged to notice things about the numbers. For instance, all the numbers in the 5 times table end in 5 or 0. The 3s and 9s are less obvious � try adding up the digits of numbers in the 3x table: as an example, 27 � the digits are 2 and 7, and 2 + 7 = 9, which is in the 3x table. (So would 49380437 divide by 3? � Easy if you’ve found the key!) You might photocopy Diagram A (below) and use the copies to shade in (e.g.) all the ’6x’ numbers, 6, 12, 18, and so on � remember you’ll ?nd some ’6x’ numbers outside the actual 6x table lines, e.g. 2 x 3 is 6, 8 x 9 is 72. Most ‘x’ numbers will give a pattern. One or two won’t, and that itself is a minor enquiry.

Children with fairly good mental arithmetic skills can look at some well-known series of numbers such as the square numbers, 1, 4,
9, 16, 25 and so on. (You can make them by setting out beads in squares, e.g. 2×2, 3×3 etc.) Also try the triangular numbers 1, 3,
6, 10, 15, 21 (yes, beads in triangles) and the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13 (see encyclopedia or Internet for how they
come about). For each set, ask your children What can you notice? To start off Well, the square numbers go up by 3 (from 1 to 4), then 5, then 7 Anything else? Try ADDING consecutive square numbers: 1 + 4 = 5, thats prime. Then 4 + 9 = 13, thats prime. 9 + 16 is 25, thats not prime has the adding pattern broken down completely, or is it just a bit more complicated than it seemed at first?

It’s worth remembering that not every investigation will turn up discoveries. And, sometimes there may be a pattern, only not as simple as it ?rst looked, as in the adding-pattern. Yet again, sometimes there seems to be no pattern at all for a bit. For instance, looking at the last (“ones” or “units”) digit of the square numbers: 1, 4, 9, 6, 5 don�t seem to make much sense, so to speak, but then the next square number is 36, its last digit is
6 � another 6 � so is something happening after all?

The primes themselves contain one or two curiosities.
The prime numbers are those which won’t divide by anything [except by 1 of course, and themselves]
i.e. (1), 2, 3, 5, 7, 11, 13, 17� (Mathematicians argue about 1 itself. It doesn�t matter here).

The history of number is only hard if you try to calculate using some of the old systems – for example, the cost of IV bags of chook food at $VII.XCV per bag. Most early societies counted in tens as we do, but not all: Mayan arithmetic had a base of 20 and the Babylonians used 60. The idea of placing numerals in Hundreds, Tens, Units positions did not develop for quite a time: thus the Roman system used different letters (I, V, X. L, C etc for 1, 5, 10, 50, 100 etc). In fact the Greeks used a more elaborate version of that idea, letting their letters ?, ?, ?, etc (i.e. Their a, b, c…) represent 1, 2, 3, up to 10, then using their next alphabet letters for 20, 30, 40 and so on. Even our ‘place value’ system took time to develop: until the idea of 0 came to us from the Hindus through the Arabs around the 1100s (Fibonacci had a lot
to do with that), you couldn’t tell whether “15″ meant “15″, “105″, or “150″. There are library books on the history of maths, also again encyclopedias and Internet.

While on this topic, the mathematicians themselves were a varied lot: Blaise Pascal became a monk. Evariste Galois was a revolutionary who got himself killed in a duel. Pierre de Fermat stopped somewhat short of this, but irritated his contemporaries by presenting “proofs” full of comments like “so it is instantly obvious that…” which gave them days of hard work checking his results.

Sophie Germain’s parents tried to stop her studying maths because they believed maths was too hard for a girl’s brain. (An attitude which was pretty widespread at least until recent times � some would argue about “recent times” � and which has a lot to do with why there aren’t very many female mathematicians in history.)

Fractions are sometimes hard, but they behave so strangely that I think they’re worth examining. One strange behaviour is that, when you multiply fractions, the answer is smaller instead of bigger: this makes sense if you think of if as taking part of a part (i.e., instead of say 3 lots or helpings or 5, answer 15, you take 1/3 of a helping of 1/5). But the conversion of simple fractions to
decimals can be even weirder. For instance, to make 1/3 a decimal, you divide 3) 1.00000, 3 into 1 won’t go so you count the 1 as 10 tenths, it goes 3 with 1 (tenth) left over, you count this as ten hundredths�and at the end of the day you get .333333� or point 3 recurring. Now if you multiply that by 3, your answer has to be .999999… and we call it �1�, after all it does round up.

But it raises a question: might there be some quantities which we cannot express exactly? ? and ?2 are some other instances,
which I�ll look at next time. (?�s fairly easy ?2isn�t).
Finally, though a bit off standard maths, “Martian fractions” are interesting too. Not easy). On Earth, we write this amount:

As 1/2, i.e. 1 part taken (shaded) out of 2 parts of the candy bar. On Mars, they write it as 1/1 meaning 1 part taken and 1 left. (How do I know? I didn’t hear that, sorry). Using the Martian system, you can still identify large and small fractions, equivalent fractions, fractions equal to 1. In fact, a lot of it isn’t much different from our system. And you can add and multiply Martian fractions however, if you really think our calculating system is bad, don’t try the Martian one. It’s complicated. It’s possible to explore other ways of writing fractions, eg (2-1) for our 1/2, as on Uranus. They’re complicated too.

If anybody actually reads this and wants to contact me to comment, complain, let off steam, or whatever, Im happy to respond (if I can) to emails: Kevinlees@dodo.com.au