Average Life Spans of Animals / Longevidade Média dos Animais

The princess and the... (the Art of SMBC Comics .)

I want one of these :D

from Spice of Life on Facebook

Mathematical Curiosities and Treasures

[and the Professor wrote books on the science of Terry Pratchett's Discworld, no less]
[and this priceless tagline in one of the book's jacket cover:

Forget Sudoku. For keeping your brain limber, nothing can compete with Professor Stewart's tasty assortment of numerical nibbles.

In his quiet way, Ian Stewart may have done more for his subject in these two books than he or his colleagues have done in perhaps the previous 10 or 15 books about mathematics I have read. One has to allow for that warmth towards a book just finished, but I might still feel the same a week or a fortnight from now.

There is no story in these books, no moral, no parable, no implied rebuke for my failure to master the calculus or to remember the difference between a prime and a Mersenne prime. There is only delight and amazement, and of course a tiny bit of entirely self-induced guilt at my own sluggard response to mathematical challenge.

For those who haven't yet looked at them, they are ragbags: almost random jottings of little puzzles, jokes, oddities, anecdotes, commonplaces and calculator curiosities collected over a lifetime. Did I read every word? Probably not. Dippers like me do tend to miss the occasional treasure. And no, I didn't try to solve all the puzzles, but yes, I did get some of them right.

I kept dipping into these books when I was supposed to be reading Jared Diamond's Guns, Germs and Steel. I dare say I shall still be picking them up when I get around to finishing Gibbon's Decline and Fall of the Roman Empire. The entries are short, comprehensible, delightfully distracting and deceptively frivolous.

When I first opened Professor Stewart's Cabinet of Mathematical Curiosities, the first thing I saw was the story about how the Indiana state legislature had passed a law fixing the value of pi. Why not? My school had in effect implemented a law fixing it at 22/7 or perhaps 3.14 (actually I think the first value was at primary school, the other at secondary school, as we went from fractions to decimals). Given that at some point you have to tell an examiner the area of a circle, you need to settle on a value.

But as Stewart points out, firstly it's a myth – also told about Iowa and Idaho – and secondly, the consequences of a "legal truth" (a legal limit on pi) that isn't in fact a "true truth" would be judicially absurd (turn to Cabinet page 25 for the consequences in theorem form).

Stewart can say this with conviction because, as his entertainments confirm, mathematics exposes the reality beneath the semblance of reality that most of us are happy with. There are hundreds of these confections and all of them are presented with an effervescent enthusiasm and good humour missing from the morose maths lessons of my own schooldays.

Some of the charm comes from the telling. I don't know why those recurring postulants the Great Whodunni and Grumpelina are more palatable starting points than A and B; and why Farmer Hogswill and Pigasus, his prize pig on a rope (Cabinet, page 143) seem more easy to manipulate than a blackboard theorem involving an equilateral triangle, but they are.

The other enticing thing about these books is that they are not just an alternative to the cryptic crossword or sudoku. They contain, in snack-sized servings, nourishing bits of intellectual history: Fibonacci series, Fermat's last theorem, chaos theory, the four colour problem, what Byron wrote about Newton, Euler's conjecture, public key cryptography, the inventor of the equals sign, Zeno's paradox, how the Babylonians handled number, the probability theory of monkeys and typewriters, the square root of minus one, celestial resonance and how the Egyptians did fractions with hieroglyphs (not a problem that I'd ever thought about before).

The entries are not all brief: Stewart's discussion of global warming (Hoard, page 164) goes on for pages, just after what Stewart claims is the shortest mathematical joke ever (but you might quarrel with the word "joke").

And how nice to be in a world where e is a Napierian exponent and not a recreational drug, where sliced bread comes in perfectly spherical loaves, and where proverbs become "tautoverbs". Example: If pigs had wings, they'd have wings; they still wouldn't be able to fly, because aerodynamics has laws to stop that sort of thing, but since this is Ian Stewart, the non-flying pig has to become an "unfeasible porcithopter".

My argument (am I the only one to think this?) is that while a little learning may be a dangerous thing, bite-sized ingestion might help some of us chew gratefully on such provocations. Instead of making a three-course meal of one theme in mathematics, Stewart has served up the instructive equivalent of a Michelin-starred tasting menu, or perhaps a smorgasbord of appetisers. And of course, appetisers are designed to give you an appetite for more.

Sometimes the most arcane dish is spiced with even more arcane flavours: a preposterous anecdote from Snorri Sturluson's Heimskringla (Hoard, page 223) is accompanied by two footnotes on the identities of Olaf, Olof the Treasurer, and Sigrid the Haughty.

I had, of course, come across Fibonacci and Fermat and quite a few other mathematical stars before, often in Stewart's earlier books, but these bits of semi-detached instruction seem a lot more reader-friendly when surrounded by unexpected titbits and not-so-silly jokes. For instance, in Hoard page 139 – between a short history of the square root symbol and a description of the ham sandwich theorem – is a tiny little squib headed "Please bear with me.

Q. What's a polar bear?

A. A Cartesian bear after a change of co-ordinates."

Yes, I'm still thinking about that one.

The Guardian

A Tabuada / Times Tables

Click to compute ;)

Why do times tables matter - and how should they be taught?

The question above is half asked by me, and half by my 8-year-old daughter. She is currently learning her tables, and wondering why they are important. I've told her they're vital, but she remains unconvinced (er, the cheery phrase, "times tables suck" was written on a piece of paper left on the breakfast table one morning last week.)

I, on the other hand, remain convinced that tables really do matter, both for extending maths ability and for life outside the classroom. They can help, as I've tried to explain to her, from working out how much money you need to buy something, to measuring a room for a carpet (not that appealing to a child, I do realise). They're also important for recipes - what if you need to double or triple the quantities? - and for saving time. If you know your tables, then you're ahead of the game.

Clive Portman teaches Year 5 at primary school (that's 9 and 10 year olds). He says that tables are vital, because if you don't know them, you can't do the maths that follows. It sounds obvious, but it's so important. If you don't know your tables, you risk being lost when maths gets a bit more difficult. Clearly, you need to know what these tables mean, and how to apply them, but I don't think that's really so hard (groups of numbers, anyone? Or why not use the plastic cups method shown on Monday night's Dispatches programme?)

"Learning their tables is also good for a child's self-esteem," adds Mr Portman. "We find that parents often see them as an indicator of how good their child's maths is."

Adam Creen, head of maths at a secondary school, Salesian School, in Surrey, agrees that tables are crucially important. "They're used all the time," he says. "Half of the GCSE still needs to be done without a calculator, and knowing your tables speeds everything up. It's important for squares, square roots and powers. And, of course, you use them out of school, in all sorts of jobs too."

Peter Watt, a fellow secondary school maths teacher backs this up. "Remember that numbers and algebra are connected, so from a secondary teacher's viewpoint, this basic understanding of numeracy allows the pupils to engage in much deeper, more abstract maths beyond just counting," he says. "So, if you are confident with your numbers, then the fun stuff like algebra becomes instantly more accessible. Take for example the simple problem, I buy 8 albums for £24, if they are all the same price how much is each one? Which is just a wordy way of saying 8a = 24, what is 'a' worth?

"Confidence with numbers breeds confidence in maths, if you are stressing about 6 times 3 then you will struggle to access the other topics."

So, multiplication tables are important, and not just in an abstract sense, but for educational achievement and life in general. My daughter appears to be slowly coming around to this, as we have begun pointing out when we use tables in everyday life (usually concerning money!). She has also realised that division is so much easier if you can do multiplication, something she didn't seem to have picked up before.

The Conservatives are thinking of making all children take a tables test in primary school (possibly instead of KS1 Sats), so I wonder why our children often don't seem realise how important they are. Is this partly because they're taught in so many "fun" ways these days, they don't want to buckle down and actually learn something by heart (I do realise that makes me sound very old-fashioned)?

So, how should tables be taught? There seems to be much less of an emphasis on learning by rote now, although I actually think this is a good way to learn some things. As I mentioned above, I'm unconvinced  that everything at school needs to be desperately imaginative, or fun. 

Adam Creen, however, says that he's not convinced that rote learning is best for everyone, and at a recent workshop at my daughter's school, we were told that as learning by rote only worked for around 80 percent (!) of children, it wasn't particularly encouraged. The problem, as I see it, is that you simply need to know your tables. If you are having to work them out by adding up each answer (2,4,6,8,10 etc), you will find it a very hard and slow process.

I'm joined in my views by Clive Portman, who says that he "strongly believes in the rote learning" and by Dina Strasser, an American teacher (and blogger) who tells me that the same issues are talked about in the States. "I learned mine (late 70s early 80s) by rote, with much angst,  Mom tells the story of coming upstairs to hear me reciting them worriedly in my sleep," she says. "My daughter isn't old enough to have hit them yet, but I can say from the math she's bringing home and the math my 7th graders are learning, there's a tension between rote learning and more conceptual, investigative learning that continues in the US-- and not just in math, but in all subjects."

However, Dina adds that "for all its dreadly drill-and-kill feel, I feel that some thing are just worth memorising - as well as understanding. I'd put the times tables amongst the very few pieces of knowledge that are."

So I'm all for learning tables and there are many ways to do this. Peter Watt recommends playing Brain Training to practice, and there are loads of times tables games on the internet, not to mention CDs (we liked times tables disco and also learnyourtimestables, though some of it is a bit strange...). 

But I do realise (as mentioned in that 80 percent comment above) that some children do find times tables extremely hard and need more help with them. This is what prompted Penny Topsom, who's severely dyslexic, to come up with a new way to teach them to her children (who are also dyslexic).

"I had a fear of maths, a real problem with it," says Penny. "So when I had to help my sons with their tables, and one of them pointed out that the list of tables I'd produced didn't look like a 'table.'. I decided to change the way they were written and come up with my own version."

Penny's grid produced patterns to the tables that she didn't realise existed before. Suddenly maths began to make sense.

She has now written a book about her method, called "Multiplication rules" and she adds: "whether a child should be answer them instantaneously, with a teacher and an entire class staring at them, I'm not sure. Most of us when put under pressure, panic and go completely blank  and this is where I think most peoples' dread of times tables comes from. I'm sure it was this that made me few like a complete 'maths numptie' never once being able to answer these question!"

However, despite this, she agrees that kids simply need to learn to multiply. "I could never get a grip on the times tables when I was growing up," she says, "but now I realise that if you understand them, it makes fractions a lot simpler, and division. In fact, every part of maths seems to come back to them. I never understood why I needed to learn them at school, but now I do."

~

 

How did you learn to tell the time? Here's a new method...

We all have to learn to tell the time, and many of us probably can't remember how we ever managed to do so. What you might recall is how difficult it was, and this is something I've been reminded of in recent years as my children have started learning about clocks.

After all, it's hard to do things in 60s. It would all be so much easier if minutes were 100 seconds and hours were a hundred minutes. And all that big hand/little hand stuff is very complicated too. One minute we're saying that a hand on the one means one o'clock. The next we're saying it means five past the hour. How confusing....

Jamie Rugge-Price first thought about this when his children - who are now grown-up and have children of their own - were small. He also spoke to various teachers about it and realised that they found telling the time frustrating to teach. It took a while, but he finally decided to do something about it.

"Telling the time is a basic life skill," he says. "It should be easy and fun, but it isn't. Then I had a Eureka moment. I thought 'what shape is an hour?' If it could be visualised, telling the time would be so much easier."

Jamie decided to come up with a concept for telling the time, and he called it a nonsense name (and acronym of his four daughters' names), Aramazu.

I shall briefly stop the story for a moment. I get sent a lot of things - books, teaching aids etc - and look at them all. Some impress me more than others. I have to admit that I was very impressed by Aramazu, and especially when my son, who's four, started to understand the concept of telling the time. He understood the hours and half pasts (seen in this method as time climbing a mountain) very quickly. The minutes were a bit more complicated, but he was still keen to learn more.

Jamie wrote about his method in a series of storybooks. The key to them is how visual they are - an hour is the shape of a mountain, and it takes 30 minutes to walk to the top of the hour or down to the half past. The hour hand is a finger, and the minute hand a foot.

Jamie tested the books and refined them. Then he tested them again. The results have been good; unsurprisingly he just wants more people to know about it.

Cheryl Hossle is a Year 1/2 teacher at a state school in the Forest of Dean. She has used the Aramazu method for teaching children to tell the time for the last two years and is very impressed. She also acts as an educational consultant to Jamie.

"Aramazu is not abstract," she says. "From the books the children can see why we need time, and how everything can go wrong if we don't have it. They can also work out how to use the feet and finger method. They make the connections."

Hossle says that this method works well for dyslexic children too, because it is so visual. "It addresses thinking skills and is humorous," she says. "I've been very pleased with the results we've had".

The Aramazu method (you can see a clock in the illustration) comes in different forms, for children of different ages. As I say, I am quite convinced by it, and would be interested to know what other people think. Or if anyone has any other brilliant ways to teach children how to tell the time.

TimesOnline School Gate