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bombaysapphire:

I assume that first sentence should end with isn't.

Yes, sorry.  

Right I'll try and write a post on 'grounding maths in the concrete' and another one on 'two routes through multiplication'.

Grounding maths teaching in the concrete

We often assume that because students were one taught the basic operations of mathematics in real contexts, the work they now do is still securely supported by those experiences long ago.

Here's a picture to help us think about that.

Imagine that our knowledge and experiences form nodes in a network.  Our understanding is the links between these nodes, which allow us to use and contextualise our knowledge.  If we have good understanding of something, we can use these links to recreate knowledge if it is lost, or to generate new results or even new concepts. So understanding (the links) is personal because it depends on what we have available in our minds for the knowledge to link to (as well as many other things).

Now, if we were creating a solid and stable network, it woudn't matter if we never revisited stuff we had done before because the links would be robust over time.  But see our minds as being more dynamic and unstable than that.  New ideas come along and supercede old ones. We go through developmental stages and our brains 'reconfigure'.  

I've talked a bit about the teachers who inspired me here in Cumbria and it's time to return to them again.  

They designed their curricula to work in topics by input rather than by expected output.  Each one started at a low level. They starting points would be very easy to understand real contexts for the maths, such as patterns, Dime experiments, early Shell centre stuff etc. Students would spend quite a while exploring these real contexts before beginning to abstract the maths.  Then they took the maths to really high levels if they felt able, or maybe they wouldn't get so far themselves but would benefit from being exposed to the high level work of others which they follow.

It was a very natural and effective way to work.  Like a duck preening its feathers and restoring the links between them so that they work properly. It lent itself very easily to wide spans of ability. It was more natural and engaging for the students because when they got lost, or it got to much, they knew they would be starting with something they would really enjoy and be able to cope with again soon.

When we were all forced to swap to teaching 'level 6, reviewing level 5 and extending into level 7' it was very hard (and of course the teachers who did not swap fast enough were rapidly culled and mixed ability was abolished).

I do actually like having a robust, levelled and tested national curriculum (although I think we could test it better).

But I also try to use some of the ideas of frequently grounding maths in the concrete so that students have firmer foundations which they can use to give them the power to be independently creative.

Otherwise students are relying on abstract algorithms which they do not have ownership of.  What grounding are these for creativity for your average student?

weebecka:

There are quite a few ways multiplication is/has been done which are not explained by repeated addition (or repeated addition the other way round).

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.

Given the amount of time you have spend discussing these matters with your classes I am surprised that you can't communicate your ideas on here more clearly.

Karvol:

 Oh really? One can only attack an argument if one believes that what is being stated is a fact - justified true belief if you will.

It is becoming quite clear that what weebecka has stated is not a fact.

I didn't quote weebecka.

You need a certain amount of context. The exchange goes:

Karvol - mentions ad hominems and personal attacks in the same breath.

bgy1mm - "Hmm, here's probably someone who doesn't know what an ad hominem is. Let's clear that up."

Karvol - bgy1mm shouldn't try to define ad hominem because he's a nobody whilst I've a degree in propositional logic.

bgy1mm "Will anyone see the irony here? Hmm, maybe if I gently point it out?"

Karvol - discussing weebecka's alleged falsification of her experience may be relevant. Yah boo, bgy1mm doesn't know the first thing about logic.

Assembled ng. Hurrah hurrah hurrah!

If the facts can't be established or agreed on you can't proceed to arguing about what they mean, agreed. But only if those facts are relevant. I doubt very much that weebecka is actually fabricating teaching experiences and the class in Cumbria, but I haven't tried to prove that.