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weebecka:

DM if you could lift the whole thing and put it in this thread, together with the sentence underneath, that would be great. 

I can't unless the picture is uploaded to photobucket or some other webhosting site.   Here is the direct link if that is any help.

http://www.tes.co.uk/Download.aspx?storycode=6066431&type=X&id=6106230

DM:

Here is the direct link

Ta.  How's the writing going?  Swap you for 49 essays to mark......

DM:

Actually I have been quite productive in the last couple of hours.   I don't want your essays but you wouldn't want my 62 Year 11 reports either.

That's almost a whole Year 11 cohort in some schools

bombaysapphire:

How do we get the numerical value for scaling one number by another without thinking of "lots of"?

Very good question.  Perhpaps a combination of estimation and know facts?

Working with students in the way I did brougth to light the huge variety of and subtleties in variations in strategies they use, most of which had long ago been abstracted from on or both the primitive pictures (either by them or by someone else and passed to them as an algorithm).  The strategy employed depended on the numbers presented.  It was also interesting to see that when two people use the same alorithm, they can be using different primitives to support their thinking.

So I'm not sure we do use scaling alone. But I'm not considerering interventions which may enhance students' overall ability to do multiplication by working first on their scaling skills and then on how they integrate them into their overall strategies for multiplication.

bombaysapphire:

If you are arguing for scaling as something distinct from that (repeated addition) then surely division can also be classified as "de-scaling" or multiplying by the reciprocal.

That gives us 2 approaches to multiplication and 3 now to division.

Repeated addition (multiplication),

Chunking/repeated subtraction (division)                (adding, removing a dimension)

Scaling 'up' (which includes multipling the other way) (multiplication)

Scaling 'down' (which includes splitting) (division)   (scaling in the same dimension).

But I'm very happy for you to keep challenging it.  I'm certainly not sure.

Someone on this thread tried to ram home the importance of known facts in multiplication.
I think here we could have some useful insight into why it is useful to practice tables.  Being put on the spot to multiply very rapidly encourages the development and use of scaling.  Maybe.  There are other ways it could be done I think.

seren_dipity:

When EXACTLY did Scottish Maths introduce 'teach yourself'?  Only I'm quite interested in Scottish Maths 

What, PRECISELY, do you mean by 'year 3' in terms of Scottish education?

Which book is THE foundation stage book?  Could you give me title and publisher so that I can drag it out of our maths cupboard and have a look.

What do you mean by 'foundation stage' in terms of Scottish education?

The SPMG Scottish Maths program was introduced in the late 70s.

Here in front of me I'm looking at:

"Infant Mathematics, A development through activity, First Stage: Teacher's Notes" which was for age 4-7.  This book is a teacher led course based around pratical acitivites with short workbooks.  It was published by Heinemann. 

After that came the 7-12 teach yourself scheme.  There were (I think) four red workbooks, four green workbooks, a blue text book, an orange text book and a purple text book.  I don't think these books were published by Heinemann (although they were later taken over by Heinemann - but elements of the way the maths was taught changed significantly at that point).

I just use the current english stage references and years for the ease of most reader.  We started the red workbooks in the year when were 6/7 (so year 2 not year 3!) - scottish schools probably started them in P3 or P4 I suppose. 

Dragging them out is probably not much use to you - John says it's a struggle to make them work as stand alone books as they were written as a self contained scheme.    https://www.ncetm.org.uk/community/thread/74409

I can't find anyone with any of the workbooks, so if you've got any I'll happy come and kowtow to you in person for the chance of a nosey.  

Meanwhile we were developing the SMP booklets here in Englands.

The most beneficial aspect of teach yourself schemes was that they shattered the glass ceiling on attainment which is much harder to break with whole class teaching.  They were also quite good as a short term intervention in teaching because they rolled out a lot of clearer explanations about topics which some teachers benefitted from.  And of course they were useful where there was a lack of specialist maths teachers.

But they had serious endemic problems for many students too.  Never having the whole class working on the same topic wiped out so many opportunities for and ways of learning.

Their main consequence for me was that they left me with a nagging awareness of that glass ceiling and a sense that I wanted to remove it as far as possible and as often as possible.