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bombaysapphire:

DM:

I have a writing contract with a deadline at the beginning of January.

You have a writing contract DM?

You must provide all of the details.  It will give so much more gravitas to your posts.  We will all then accept and agree with paragraphs of long words full of sound and fury and signifying nothing because otherwise it would be a sign of our own ignorance.

I'm so clever that I even spotted the Macbeth quotation there bombay.

DM:

I'm so clever that I even spotted the Macbeth quotation there bombay.

I had a day at a spa yesterday Curlygirl.  That has to be the way to start every holiday, gets you back to feeling human in such a pleasant way.

bombaysapphire:

I always describe multiplication as "lots of" or just "of" with fractional amounts.  When a student makes a mistake with 5 x 0 then reminding them that it is 5 lots of nothing really clarifies what the right answer is and why.

Yep me too.  When students learn to read 'x' as 'of' they access quite a few types of multiplication question with otherwise flaw them. My students were only doing repeated addition for multiplication by they way (as far as I know).

bombaysapphire:

I see repeated addition as a simple case of that, 5 lots of 7 intuitively means the same as adding up 7 five times.  I still don't see an arguement for there being two routes through multiplication.

And you'd be in line with most of the literature here bombaysapphire. 
e.g. Multiplicative Reasoning in the Development of Mathematics (a bit of a bible on these things from 1994) reports that:

" Fischbein, Deri, Nello and Marino (1985) ..... conjectured that the prmitive intuitive model for multiplication was prepeated addition and for division as based on either partitioning or repeated subtraction".

So you're in good company.  

But I'm still not conviced.  Earlier you were arguing that reversing the order of the multiplicaiton gives you the two inverses of partitioning and repeated subtraction.  

But I don't think partitioning and repeating subtraction really cover the territory of divison properly.  With my students I found we often had to restate partitioning as being 'how many for one' for it to make sense.   (e.g. 40 / 1/3)  You can quotition (chunk/repeatedly subtract it) it (there are 120 1/3s in 40) but you can't partition it.  But if you change the vocabulary to make it how many for one it's easy (it's 40 for 1/3 so it's 120 for 1).

So I think the primitives for division are better expressed as being chunking (repeated subtraction) or scaling to one (of which splitting is a sub category).

Then when you reverse it you get repeated addition and scaling (up from one) of which repeated addition the other way is sub category.

Does anyone need a picture?

bombaysapphire:

No I wasn't.

So if "how many in" and "chunking" and "repeated subtraction" are all the same thing, have we got past the misunderstanding?

Or are you disagreeing with Fischbein et al.?  In which case how?