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multiplication - definition from wolfram

In simple algebra, multiplication is the process of calculating the result when a number is taken times. (simple algebra as opposed to eg groups, matrices, sets, and tensors.)

i can't go back and set up quotes from so many posts - but:

1. i suggested exlpaining 7 x 3 as an array of 3 rows of 7 dots and counting them  7 'lots of'' 3 - wb suggested scaling as an alternative - but if you have a 7-long line and triple it, isn't that just the dots laid end-to-end, witha line drawn through them, and maybe the dots rubbed out 

2. not that i disagree with scaling as a way of understanding multiplication - i believe it works well lfor dyslexics with good spatial skills for example - but  that seems to me a different psychologiocal/neurology understanding of multiplication, not a different definition of multiplication in itself, and drawing or visualizing the line is an algorithm or method , just as grid multiplication is, for example ( i would class 'repeated addition' as an algorithm as well, to be adandoned once you have learnt your tables, and increasingly clunky as your maths gets increasingly complex  - but with scaling, even with small, basic muliplication -)

3. unless you have a ruler or the dots marked in, where do you get an accurate answer?

 (wb herself has suggested 'estimation' as part of how you use 'scaling' to obtain an answer - as with some algebra methods, why estimate when you can calculate - other than the numbers asking for it for ease and/or avoiding spurious accuracy, of course)

i would say wb's trees are a case in point - can't find dm's link, but there is a tree, then a tree 3 times the size, then 3 trees on top of each other, and which picture represents 3 x the original tree - but if i have no ruler or other way of arithemetically scaling up, i would need to draw diagram 3 (tracing paper or copy and paste?) to give me the parameters to draw diagram 2 accurately

4.' 'lots of' etc also leads easily, as someone has already said, onto 10 x ½ = 5  (i could even draw my dots again and cirlce every other one) - and ever more complex fractional multiplication - using scale line for 10½ x 3⅔ is my idea of a nightmare - but if it does work for richard rodgers,say, good luck to him

 and also large numbers

and before wb or anyone else comes back and says for all the maths curriculum, we teach children how to calculate  10½ x 3⅔ or 57 x 62 without having a clue why they're doing it, i wouldn't disagree, but i would again say we are talking methods, not definitions of multiplication

 division: i quite like this definition form the free dictionar:The operation of determining how many times one quantity is contained in another

1. i'm agreeing with someone else again - sorry for plagiarism - , but i don't see why 'sharing' and 'chunking' are seen as separate - i think this comes from children being taught 'sharing' as handing out one by one, akin to dealing cards

but if i have 21 sweets to divide between 7 children, i am actually setting aside the first lot of 7 sweets - that's one each, the second lot of 7  - that's 2 each, the third lot of 7, that's 3 each and i'm done - this is no different arithmetically from if i have to divide the sweets up into bags of 7 (a standard example of chunking rather than sharing) arithmetically, i am doing exactly the samer thing

the difference is my physchological/neurological view of the 2 events, or the method i use to arrive at answer, not the maths (just as i can do 'old fashioned' long division, or chunk my way through the hundereds, then the tens, then the units)

and again, the best method is learn your tables

so i see no imbalance between multiplication and division here 

(oh - an aside - i am surprised you use the word 'partitioning' without comment here, wb - in uk primary schools, 'partitioning' means the operation of splitting numbers by place value eg 364 = 300 + 60 +4 )

2. whereas multiplication as repeated addition, rather than merging 'lots of' is limited, division as repeated subtraction, rather than 'how many lots of' creates at best lazy thinking and quite often confusion - it's a rare child who can create 65 ÷ 10 = 6½ from repeated subtraction,  but if you are looking at 'how many lots of' it follows easily and without much prompting

(mrs f in y5or 6 gat group lesson 'i have 65 metres of ribbon to divide between 10 children - how much do they get each?' almost invariable answer i ask this 'cold'  '6m and you'll have 5m left over' - grrrr)

Flora - your posts are lovely and I will try to give them the attention they deserve as and when I can - it'll probably come in interrupted bits.

Beta, I'm still not really sure what you're after - whatever it is I suspect it's impossible to give.

curlygirly:

We always work on the same topic, but at differentiated levels. Because of the way these schemes were used a teacher could have 33 children doing 33 different things. That was what I found when I took over my class of 4-8 year olds. Totally unworkable ( and boy did I try).

This is what it was like at my school (age4-9).  One of the teachers was a family friend and we talked about it often before she died.  The teachers were very positive about it, because when they considered taking it on, they agreed they would only do it for some lessons and would continue to do what they'd always done well for others.  So we had some class taught, very traditional lesson (I think we used betamaths), and some lessons where we worked on our own taking terms to visit the teacher at her desk to discuss our work.  

They made very sure we were all secure readers before they introduced us to the scheme.  They started it later than they otherwise might have done to ensure this.

I'm not saying it was fab. I think it has an interesting place in history because I think it brought some new visual structures for teaching mathematics into play.  I'd love to see some of the workbooks now to diagnose this more precisely.

But self teaching did - as I said before - smash the glass ceiling on what we could achieve.  When I left the school aged 9 I was already well into the purple book.  It was only a small ordinary state school but I wasn't the only one from my year to go on to do maths at Oxbridge.  Which, given the school I went to next, was really quite a surprise.

Anyway, that was then.  Much better things are possible these days.

 yin not ying    dur

 Reply to florapost on division.

florapost:

(oh - an aside - i am surprised you use the word 'partitioning' without comment here, wb - in uk primary schools, 'partitioning' means the operation of splitting numbers by place value eg 364 = 300 + 60 +4 )

Fair point.  Quotitioning (repeated sutraction/chunking) and partititioning (splitting/how many for one) are the standard vocabulary for this topic too though.

All my vocubularly here is likely to be confusing as I'm reaching into unknown territory so anyone should feel free to challenge it at any time.

florapost:

division as repeated subtraction, rather than 'how many lots of' creates at best lazy thinking and quite often confusion

Okay.  But they are basically two routes through the same image rather than two different images.

florapost:

i don't see why 'sharing' and 'chunking' are seen as separate

I'll tell three stories to explain why I see them as being far apart, one from reception class, one from secondary and a challenge for you.

When my son was in reception class he did an expoloration of odd and even numbers using spot on ladybirds.  The basic idea was that if you can split the dots into two equal groups you have an even number of dots.  If you can't, then you have to put a dot on the middle line.  When he bough home is ladybird we chatted about it and I was impressed with how easily he correctly handled questions like - if there were two dots on the middle line would that be an odd or an even number of spots?  The visual imagery he had been give was robustly supporting his maths.

Then his little friend came to visit, also recently having finished reception class, also having done odd and even numbers.  But when I asked her questions like - would 41 be odd or even, she couldnt do it.  We looked at my son's ladybirds but she was completely lost.  I asked her to show me what she'd done in class and she said they'd had counters and had separated them into groups of two.....  so she could deal easily with saying whether small numbers were odd or even but not with large numbers.  

Of course when I'm talking about odd and even numbers here I'm really talking about dividing by two.  The visual models these children were using were totally incompatible with each other.

In secondary, as I've said before, I'd often set a starter whereby my students had to add, subtract, multiply and divide two quantities. This leads to them having to do two divisions - often difficult ones.  I would not give them full marks unless they could fully draw or explain their solutions (no mathematical tricks allowed).

The explained solutions always fell into one of two camps - either splitting (how many for one, scaling) or chunking.

So here are a couple of problems for you.  

Do them quickly in your head and then work out how you did them.

6090 divided by 3. 

525 divided by 105