I am reading this book at the moment (rather slowly between other things) and am really enjoying it. I am curious if anyone here has read it and has any thoughts to share? As much as I am finding it fascinating reading I am so inexpert on teaching math I am wondering what more knowledgable people might think of it.
I was interested in how differently long division is apparently taught here in Australia.
22b when I was a child we were taught to write the zeros in and why they were there (and I have checked with age peers and they had the same experience). Infact I think I peraonally was first taught to start the problem by writing in the zeros that will be required on each line before starting calculations (highlighting that i was dealing with tens, hundreds, thousands and so on), and it was also later examined why leaving the zeros blank would not change the outcome. I am not sure but I THINK it was eventually allowable to leave blanks but one friend I asked was adamant that this was always frowned upon...
My DD is in 6th and has also been taught to write in zeros and why you need them...
This approach seems to completely cut through a lot of the problems, confusions and angst raised in the book.
Well from the book Val it seems like the USA and china both do not use the zeros, but only the Chinese teachers understood that you COULD use zeros, how, why, etc but would still encourage moving back to blanks. The problem discussed is how to help a child who is right aligning every line of the problem (no blanks OR zeros). Most of the American teachers had a purely procedural knowledge and could do more than tell the kids "do it this way because it works" some thought adding zeros was actively wrong....
I'd be interest in how a whole list of countries deal with this too...
when DD initially started asking about numbers, we did nothing at all before she could expand numbers properly into their ones/tens/hundreds/whatevers - and when it was time to tackle operations, she just organized everything perfectly. she leaves blanks rather than writing them in, but i do hear her muttering about the would-be zeros a lot - maybe that is the secret of her success?
i'm going to get that book - it sounds cool! thanks!
I should add that math teaching in Australia seems completely faddish and often wrongheaded to me, I was just interested in this particular difference. Which I guess ties in with the author's observation that teachers' math teaching ability/process is deeply grounded in how they were taught. This just happens to be how my generation was taught multiplication and it's clearly carrying on.
Faddish example: our schools don't teach long division at all anymore, it got dropped from the curriculum for being irrelevant (or too hard?)
What is the ridiculous method of subtraction referred to above? I can't recall seeing anything that seemed out of the ordinary, but then again I do tend to ignore my kids' math worksheets a lot.
As far as I can tell short division only. And I guess the assumption they'll use calculators instead...
I'm still cracking up that my yr2 child's teacher has just come back from a conference full of the importance of children being proficient at doing all basic math in their heads before learning written algorithms.... Ya think? Like maybe we weren't all crazy to be annoyed at demands to show working for 2+3 when the kid just KNOWS it's 5?
I was interested in how differently long division is apparently taught here in Australia.
What's the difference?
Originally Posted by MumOfThree
22b when I was a child we were taught to write the zeros in and why they were there (and I have checked with age peers and they had the same experience). Infact I think I peraonally was first taught to start the problem by writing in the zeros that will be required on each line before starting calculations (highlighting that i was dealing with tens, hundreds, thousands and so on), and it was also later examined why leaving the zeros blank would not change the outcome. I am not sure but I THINK it was eventually allowable to leave blanks but one friend I asked was adamant that this was always frowned upon...
My DD is in 6th and has also been taught to write in zeros and why you need them...
This approach seems to completely cut through a lot of the problems, confusions and angst raised in the book.
Thanks for the explanation. I'd consider that just a cosmetic difference. The methods are almost identical.
Originally Posted by MumOfThree
Faddish example: our schools don't teach long division at all anymore, it got dropped from the curriculum for being irrelevant (or too hard?)
That's sad. They should at least present it and if just some kids get it, that's okay.
I know that the method for subtraction taught in the USA is ridiculous.
Originally Posted by Val
I'd be interested in learning about how subtraction is taught outside the US.
Originally Posted by Nautigal
What is the ridiculous method of subtraction referred to above? I can't recall seeing anything that seemed out of the ordinary, but then again I do tend to ignore my kids' math worksheets a lot.
There's an explanation here http://en.wikipedia.org/wiki/Subtraction#The_teaching_of_subtraction_in_schools which contrasts the "American method" and the "Austrian (or European) method", the latter being the one which is taught in most countries around the world. The "American method" is unnecessarily more complicated, especially when there is a string of zeroes in the minuend.
Here are some YouTube videos of (various slight variations of) the "Austrian method"
I totally needed the video to make sense on that 22B. That is a much simpler way to write up the same concept. Oddly it seems vaguely familiar to me and yet it's not the primary method I remember from school.
You are right that the difference I mentioned in long multiplication is cosmetic, but given the issues raised in the book it's clearly a huge stumbling block for many children and there are many adults who get to adulthood possibly able to follow the procedure but with no comprehension of how/why in works (ie that you move over to the tens column because you are in fact multiplying by 10s now), which putting in the zeros explains (or forces to be explained).
I grew up learning the European method. When my stepson moved in with us in 6th grade and had trouble understanding a lot of math concepts, I tried to learn the methods his teachers were teaching but he still wouldn't get it so I gave up, showed him how I was taught math (not just subtraction but other problems too) and he understood them in no time. And I told him ... if your teacher has a problem with it, tell them to CALL ME! Thankfully they never called, but I was ready to defend our methods
Heh. I've asked 4 adults this afternoon how they to do subtraction with regrouping and got two versions of the Austrian method (a German friend and my DH who learned subtraction in New Zealand), and two versions more like the American system shown in the first video, though both slightly simpler than that one. Now to figure out whether to teach my DD the method our schools use or the Austrian method, which I find far superior.
Obviously she can learn both but I know I have always been resentful of being forced to do things in a less efficient way and I worry she'll be the same and simply refuse the method our schools use if she likes the Austrian. Which may obviously cause problems if she encounters any "My way or the highway" teachers like Val is currently dealing with.
This makes me wonder. If I end up homeschooling my kids. Do I need to follow the American method, living in the US ... or can I teach them the European way that I'm familiar with a lot more? Will that be an issue for them in the future?
If you homeschool you can probably do whatever you like - even if your children had to do standardised testing I'd be astonished if it cared what method they used. Fwiw noone (but me?) ever taught DS subtraction and I'm not sure what you'd call what he does! I do remember him saying with delight "we unwrap this bag of ten, and inside we fiiiiinnndddd...... Ten units!" at a certain stage!
We reached multiple digits addition and substraction with my oldest child through afterschooling in 1st grade. And so (European) taught him the Austrian method.
When they finally reached that last year (3rd grade!) he taught his class the Austrian method. His teacher was blown away.
I found the "standard method", when I was finally introduced to it, incredibly cumbersome, especially in cases like the one 22B shows above. The fact that it forces you to work first left to right then right to left is an issue.
ColinsMum, was your son using Dreambox? That's what they show as a visual aide for this stage.
No, but it's a pretty obvious visual aid!
Does anyone else remember Dienes apparatus? When I was at school we did arithmetic in bases from 2 to 10, pretty much without favouring 10. We had sets of wooden equipment for each base - e.g. in base 7, little unit cubes, then longs that looked like 7 of them stuck together, flats that looked like 7 of those stuck together into a square shape, blocks that were, well, you guess, and in the lower bases long-blocks and flat-blocks. There was lots of "change a block for 7 flats" involved in the arithmetic. Great stuff, IMHO.
I was taught the American traditional method but just realized (by trying to solve 22B's subtraction example) that I do not know how to subtract, no matter which method I use. BTW, I am an engineer that works with numbers all day. The best way to subtract,IMO, is to use excel or a calculator.
ColinsMum and Val, I'm thinking I'm going to hack the multi-base Cuisenaire sets with Popsicle sticks and dry beans. I refuse to pay several hundred dollars for a tool that DS will, in all likelihood, discard within the first hour of use. In the meantime, the gluing can make for some good fine motor work together. I figure I can build bases 2 through 10 on less than $20, then we can have a bonfire once the concept is down.
We had those for base 10; they call them Cuisenaire rods. I'm digging around now for sets like you described that come in bases 2, 3, etc.
But Cuisenaire rods only do the "long"s part, right? What I'm talking about would also take 7 length-7 rods and stick them together side by side to make a 7x7 square "flat", etc.
We had those for base 10; they call them Cuisenaire rods. I'm digging around now for sets like you described that come in bases 2, 3, etc.
But Cuisenaire rods only do the "long"s part, right? What I'm talking about would also take 7 length-7 rods and stick them together side by side to make a 7x7 square "flat", etc.
Yes, exactly. I found them for base 10 quite easily. Did you just use these to make cubes, etc? They seem very nifty.
are these blocks supposed to make math easier? We never did anything like that. Just plain simple numbers written / printed down on a piece of paper. To me all these blocks just complicate things?
It's all about understanding place value. This paper has some of the history:http://www.ianthompson.pi.dsl.pipex.com/index_files/teaching%20place%20value%20in%20the%20uk%20-%20time%20for%20a%20reappraisal.pdf
And concludes that the multibase stuff I remember stopped being used because it was too hard and in fact place value is too hard for children until year 4 (age 8)! Sad, particularly if true. The multibase stuff seems to have gone completely.
My kids loved Cuisenaire rods. I used them to help them see the concepts behind addition and subtraction. For subtraction, we'd remove a rod of ten and replace it with ten little ones.
ColinsMum, I meant to ask, "did you just use the shorter rods to make flats and cubes in other bases?"
it's still not making any sense to me. I need my numbers on a paper! I guess it's good I never had issues with place value. Can't imagine having to explain these rods to my kids in the future! lol
It's not that I don't grasp the concept. I just feel like it's adding too many extra steps to what's as simple as math? ... probably the same reason why I (and DS5) have serious issues with Kindergarten homework. It's too much hands on and not much math that would move us forward.
My kids loved Cuisenaire rods. I used them to help them see the concepts behind addition and subtraction. For subtraction, we'd remove a rod of ten and replace it with ten little ones.
ColinsMum, I meant to ask, "did you just use the shorter rods to make flats and cubes in other bases?"
Sorry, I'm having trouble being clear! Friday night... There was a completely different set of equipment for each base; only the units were in common. Just as a rod looks like (say) 7 unit cubes stuck together but is actually solid, a flat looked like 7 longs stuck together but was actually solid. But you can see that in the picture, so maybe I'm still not understanding the question? Sorry!
It's not that I don't grasp the concept. I just feel like it's adding too many extra steps to what's as simple as math? ... probably the same reason why I (and DS5) have serious issues with Kindergarten homework. It's too much hands on and not much math that would move us forward.
By kindergarten, I agree it could well be too hands-on. But, I find it perfect for DS22mo now. He's loved those sorts of manipulables for doing simple addition and subtraction--especially math cubes--for quite a while now. But, obviously, don't fix what isn't broken!
It's not that I don't grasp the concept. I just feel like it's adding too many extra steps to what's as simple as math? ... probably the same reason why I (and DS5) have serious issues with Kindergarten homework. It's too much hands on and not much math that would move us forward.
Can your DS5 do arithmetic in arbitrary base? For example if you tell him that on Planet Zog the inhabitants have three digits* on each hand rather than five, so they do arithmetic with six playing the role ten plays for us, and then you show him a couple of two-digit example sums, can he immediately do multidigit addition and subtraction as reliably in base 6 as in base 10? For my DS there was a stage when he was quite reliable in base 10 but would get completely confused in any other base.
But I never bought manipulatives for him unless you count the linking elephants or the jar of counters, both bought when he was a baby and used occasionally for maths. For child-me, the Dienes stuff was tactically appealing (it used to be natural wood in those days!) and the multi-base stuff was a fascinating start for enrichment (I remember being asked to design a machine that would do sums taking input using blocks - remember this was before even calculators were familiar objects, never mind computers!). My teachers never, that I remember, made me use the blocks if I didn't want to. So I remember them with affection.
*if he's anything like mine this is necessary to let you opt out of a finger/thumb discussion
I was not taught any other base but 10, I'm not sure I even encountered the concept before adulthood! DH was. But I suspect that was in NZ.
These days children in the UK don't meet it either (until they meet base 2 in the context of computer science if they're lucky); the Dienes MAB stuff I was talking about was briefly fashionable in the 70s, and even then not many schools used it. Still, it's great stuff to do with children who'd enjoy it!
It's not that I don't grasp the concept. I just feel like it's adding too many extra steps to what's as simple as math? ... probably the same reason why I (and DS5) have serious issues with Kindergarten homework. It's too much hands on and not much math that would move us forward.
This was the exact problem that DD had with math during preK through about 5th grade. She has (maybe always??) thought abstractly/symbolically without needing manipulatives, so to her, the entire concept of a physical, three-dimensional representation felt weird and unnecessary-- and it freaked her out and made her feel like "oh, well, what the heck is THIS for??"
I still remember how fascinating it was to my 13-yo self to learn binary and base-8 math during my first forays into computer programming. It was AWESOME.
My DD thought I was hilarious when I sang Tom Lehrer's New Math...
(He discusses regrouping in another 'base' system towards the end of the song. It's very charming/funny.)
