The number 0
Back over here, where much less blogging takes place (! How do you manage it, Clare?! Taking my hat off to you..) Lyddie picked up a maths book when we were out shopping yesterday. I'd noticed them in the past and given them a wide berth. I mean, gold stars?? They'll be handing out chocolate drops next. Or.. erm.. EMAs. Perish the thought that anyone might learn something because they were interested in it.
"I need to be doing this for my school work," said Lyddie, and put it in the trolley. Hmmm. School work? This hasn't come from me. I can only imagine TV to be the culprit - that and the way they're presented to look oh-so-shinily attractive. It reminds me of the Child Catcher from Chitty Chitty Bang Bang:
- who wouldn't look out of place on our government's front benches, don't you think..? (Was Ian Fleming home educated?) I would just rather she learned maths in a more organic way, which she is doing all the time, of course, but I'm a bit worried that this kind of prescribed process might thwart her natural learning. Anyway, it's only a book, I hear you saying, and we've done schooly stuff before, without any apparent ill-effects. So let's get to the point.
She raced through the first pages, which involved "adding to 12" - until she got to this sum, which required her to fill in the gap:
- and she could not get it! It didn't make sense to her at all. We had the split peas out, and the abacus, and I tried to help, by saying: "What do you have to add to 2, to get 2?" But she was completely flummoxed.
It was then that I started thinking about the number 0, and what a crazy, contrived question "What do you have to add to 2, to get 2?" actually is. When in life, outside a school environment, are we ever expected to apply such convoluted logic?
We went to look up the history of the number 0 and read that "the ancient Greeks seemed unsure about the status of zero as a number. They asked themselves, 'How can nothing be something?', leading to philosophical and, by the Medieval period, religious arguments about the nature and existence of zero and the vacuum."
"How can nothing be something?" seems to me to be a far more sensible question than "What do you have to add to 2, to get 2?" No wonder children in the UK struggle so much with maths. If Lyddie had been faced with that question in a classroom environment, I think she'd have given up there and then.
posted by Gill at 6:45 am
15 Comments:
Ha, Bizarre. Yesterday A demanded she "sit down and do some sums". I had bought these books from Amazon (http://tinyurl.com/6uuqc8) without exactly realising they were *workbooks*, but, nevertheless, A sat down and worked her way through a couple of sheets. She needed a bit of help with the reading, but she managed to do the sums, but, like L, did get completely confused, not so much about the 0, but about the way those sums were written. For example, she got really stuck on: _ - 3 = 2.
She loved the word problems, like: "On a string there are 9 beads, 3 are red, 3 are blue and the rest are white. How many white beads are there?"
I bought a Terry Jones dvd about numbers, and we watched it the other day. It was quite interesting.
http://tinyurl.com/7e5g7v
Like a borrow?
I think the number 0 is a wonderful invention. I am astonished that ancient civilisations could achieve so much in terms of engineering and astronomy without it.
I think it illustrates very pointedly that mathematics is an *art form* (not a science), aiming at its core for things like beauty and completeness.
I read a very interesting article recently about this, which made me worry about the dangers of using maths workbooks.
On the other hand, the problem of mathematical poverty in our culture is not something we can solve on our own, and we know that there are equally great dangers in refusing to use any resources our children choose. Both our kids like to work through workbooks or practice books sometimes.
I think using them in the context of an autonomous home education is probably very different from in a classroom. Lyddie's confusion about zero is entirely logical and reasonable, and looking at the history is the only way to find an answer, imo.
Dani thank you for that wonderful article. I am only half way through (off to get another mug of coffee!) but it's great, and I am planning to send it to all our relatives who ask me (worriedly): "but what about maths?" The article exactly (so far) sums up my intuitive thinking about what's wrong with maths in schools. I have been trying to collect up some dvds which might inspire this kind of creative thinking about maths (something like David Attenborough for Maths). The Story of 1 (link in my comment above) was a good start, but have you found anything else that might keep a creative spark for maths alive? A seems to be very fascinated in numbers, she surprised me the other day when I said I'd be back in around half an hour. "How many minutes in an hour?" she asked, and after I'd replied: "so you'll be back in 30 minutes?" Fractions + division all at once, and we've only 'done' fractions as slices of pizza, etc. Also she has always counted to high numbers, up to several hundreds to put herself to sleep at night from around age 4. And now she's interested in working out new names for high numbers herself, so before she found out there was such a thing as 'trillions', she was making up her own name for it. This is in complete contrast to M, who doesn't show the slightest interest in numbers.
It would be an appalling shame if I squashed this burgeoning interest inadvertantly. And watching dvds has been a failsafe method so far of keeping passion sparked, and enabling her to keep asking herself interesting questions about a subject. (eg last night while lying in bed after watching "Earth": "why do you think we feel so at home here when our whole planet is a collection of rocks lost in space?")
Anyone have any ideas?
I think I'm about to blog another response to Gill...perhaps I should make it a category!
Have you noticed that the Romans had nothing for zero? And they were pretty smart people :-)...
Mine adore workbooks! Flopsy said the other day, when I said I'd found and printed off a sums bingo game for her, "Yes!!! I love sums!!!". I think they actually learn very little from them compared to what they learn from life, but they see them more as fun puzzle books. Off to read stuff about the number zero, although I do remember learning that it had to be invented from a tv documentary about maths once.
Thanks for the link - now I'm under pressure to continue it! ;-)
Yes please, Lucy!
I can't even think how to phrase a question like: _ - 3 = 2. Um... What number, when you take away three, changes into two? No, that's not right.. I give up, really!
I've come across Lockhart's Lament before Dani, and the first few paragraphs actually illustrate very well the way I was made to learn music - there could be no surer way of putting a person off a subject than that and indeed I now have very little interest in it.
"..and we know that there are equally great dangers in refusing to use any resources our children choose.."
Yes I agree with this, hence the purchase of said maths book, in the end. I think she just wanted to explore, and she does seem to be enjoying it.
Lucy, "why do you think we feel so at home here when our whole planet is a collection of rocks lost in space?" has to be one of the best questions I've ever heard! Mindblowing. I think the answer lies in the protective structures of our minds and the way we protect ourselves from fear, and therefore the risk of breakdown.
Jax, I'm having a trying day today but will make my way over there to read it ASAP :-) (And much looking forward to it too.)
Good point from you, as usual Augustin!
Clare I seem to remember learning that it had to be invented too, and comparitavely recently as well. (The invention, not the learning!)
Finally getting around to re-reading the rest of Lockhart, and this bit's absolutely classic, isn't it?
"Obviously the current
practical training program isn’t working, and for good reason: it is
excruciatingly boring, and nobody ever uses it anyway. So why do
people think it’s so important? I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them. It might do some good, though, to show them
something beautiful and give them an opportunity to enjoy being
creative, flexible, open-minded thinkers— the kind of thing a real
mathematical education might provide."
Such a long article - not sure I'll get to the end this evening, there's so much going on here - but soooo good! Thanks for the reminder, Dani.
how to phrase _-3=2
how about... what number would you have, if you took away 3 from it and the answer was 2?
hmm, still a bit confused. much easier just to play with counters (or peas or whatever) on the table - that uncovers numbersense and how numbers fit together more organically than that type of question...
loved the article dani. read it all instead of doing stuff for work. i do think some people seem to see and think more intuitively in maths than others [boy portico on the blogring for example] maybe more of us could see in maths better if we were given the right glasses to look through at the beginning
I got some of the same series while in the UK but the gold stars are usually peeled off by my 2 year old before they get anywhere near the workbook pages! my 2nd son sometimes makes workbooks for me!But the maths ones from school they have here in Sweden seem to assume colouring in is a good reward for getting the answer right (ie colour the sums adding to 8 red etc) there's only so much colouring a child can take and I actually finished the colouring off for him once as he obviously understood the maths.
On the other end of spectrum, my first son was asking me when he was about 5, what's the biggest number.Explaining infinity, noy easy at any age me thinks...
I think Lockhart would approve of that question :-) He mentions a similar one: "Is infinity a number?" (I suppose that if zero is, then logically infinity must be?)
just wanted to say thankyou Dani, that's an AWESOME article, I LOVE IT and we're quoting it all over the house at the mo!
and it doesn't just apply to the american curiculum, or just to maths... that general attitude is behind much of the school system at times and completely baffles me.
How to phrase the 'missing addend' question...
You start with a bowl of smarties, with an unknown number of smarties in it. Then you eat three of them (so there's three in your tummy now) and notice that now there are only 2 left in the bowl, how many smarties must you have started with (bowl plus tummy)?
That's how we do it here :)
Now of course this piece of actual information is not necessarily useful in and of it's own right, but the application of the logic required to reach the answer IS useful.
e.g. - If a bus takes 20 minutes to make a journey and I need to arrive at 2.30, what roughly what time do I need to catch the bus?
It's the same logic applied to a different set of figures and setting.
It really annoys me that the government has changed the name for maths to numeracy. This implies that maths at primary school level is confined to number crunching and doesn't include anything else. There are all sorts of other topics such a geometry that are used in applications such as tile patterns and molecular structures. Should these be pushed aside in favour of subjects like long division or fractions and decimals?
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