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Maths Café


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Understanding versus doing.

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Graham1970 24 August 2011 15:07
Posts83
Joined20/07/09

  Thinking about the expression "rote learning has made me come to realise it is a bit oxy-moronic. For example, how many times did you have to learn 5 x 5 is 25? I would probably only once. When you were sat there in primary school with your abacus, block or what ever and you gathered in 5 lots of 5 or whatever you did to find it was 25 that was when you "learnt" it. Everything you after that is reinforcing the memory you made such as repetition of the times table or flicking the beads again.
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Christine_Jones 24 August 2011 15:18
Posts735
Joined09/10/09
Jane, that link contains a very interesting document which I haven't read. There is so much on this site that I haven't managed to access yet. I'm sure there are ways of finding all this information but I find it very difficult and confusing to sort out where items of interest may be. That probably is a problem unique to myself but I do find it frustrating to see documents that I would have liked to have read but I had no idea they were on this site.

With regards to issues to do with textbooks I'm very much in favour of their use. I see them as a way of helping students develop independent learning. I also see them as a way of insuring that the student has a least one coherent copy of a maths syllabus.

I have thought about the following remark you made:

I think that the intentions of exercises vary (as, indeed, does the quality): sometimes (often?) they aim only to develop fluency,

and I've been trying to think about what you may mean by fluency and why you would have said 'only fluency'. Fluency, to me, implies being able to use something easily and accurately and I think that is a very good outcome. If you can use something accurately that must indicate some level of understanding so, again, this seems a good outcome. This is where semantics come in to the word choice we make and the possible implications we intend to result from that word choice. Also, others can read our words and take a completely different meaning from them which can cause problems in communication, especially within forums etc.
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petegriffin 24 August 2011 15:55 - Last edited by petegriffin on 24 August 2011 15:59
Assistant Director (Maths Hubs)
Posts1319
Joined29/06/06
Graham says:
"Thinking about the expression "rote learning has made me come to realise it is a bit oxy-moronic. For example, how many times did you have to learn 5 x 5 is 25? I would probably only once. When you were sat there in primary school with your abacus, block or what ever and you gathered in 5 lots of 5 or whatever you did to find it was 25 that was when you "learnt" it. Everything you after that is reinforcing the memory you made such as repetition of the times table or flicking the beads again".


I am thinking that someone who has learnt 5 x 5 = 25 might be stumped when asked to calculate 25 divided by 5. "Learning" something like this could mean anything from being able to repeat it faultlessly to being able to know it in lots of different ways and different contexts.

I know I have quoted these two things from John Holt before in other threads, so apologies to those for whom this is unnecessary repetition, but I have found these very helpful in thinking about doing and understanding.

But pieces of information like 7 x 8 = 56 are not isolated facts. They are parts of the landscape, the territory of numbers, and that person knows them best who sees most clearly how they fit into the landscape and all the other parts of it.
The mathematician knows, among many other things, that 7 x 8 = 56 is an illustration of the fact that products of odd and even integers are even, that 7 x 8 is the same as 14 x 4 or 28 x 2 or 56 x 1; that only these pairs of positive integers will give 56 as a product; that 7 x 8 is (8 x 8) – 8, or (7 x 7) + 7, or (15 x 4) – 4; and so on.
He also knows that 7 x 8 = 56 is a way of expressing in symbols a relationship that may take many forms in the world of real objects; thus he knows that a rectangle 8 units long and 7 units wide will have an area of 56 square units.
But the child who has learned to say like a parrot, “Seven times eight is fifty-six” knows nothing of its relation either to the real world or to the world of numbers. He has nothing but blind memory to help him. When memory fails, he is perfectly capable of saying that 7 x 5 = 23, or that 7 x 8 is smaller than 7 x 5, or larger than 7 x 10. Even when he knows 7 x 8, he may not know 8 x 7, he may say it is something quite different.
And when he remembers 7 x 8, he cannot use it. Given a rectangle of 7cm. by 8cm., and asked how many 1sq. cm. pieces he would need to cover it, he will over and over again cover the rectangle with square pieces and laboriously count them up, never seeing any connexion between his answer and the multiplication tables that he has memorized.
(From “How Children Fail” by John Holt, page 110.)

and

“I feel I understand something if and when I can do some, at least, of the following:
1. State it in my own words; 
2. Give examples of it;
3. Recognise it in various guises and circumstances;
4. See connections between it and other facts or ideas; 
5. Make use of it in various ways;
6. Foresee some of its consequences;
7. State its opposite or converse.”
(From “How Children Fail” by John Holt.)

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Christine_Jones 24 August 2011 16:05
Posts735
Joined09/10/09
Pete, there is a lot in your post to think about. 

It all seems to boil down to what exactly we mean by understand and Bruner appears to mean something different to John Holt.
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maggieh 24 August 2011 16:43
Posts67
Joined26/09/08
Pete I absolutely love that list of 7 points - have not seen this before so a big thank you.
I know this is a bit off topic but I plan on simplifying the language a bit and making a poster of the seven points for my Entry Functional Skills classroom (as a focus at the end of lessons).
Thanks again for the inspiration :) and thanks to all for such an interesting thread.
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petegriffin 24 August 2011 17:19
Assistant Director (Maths Hubs)
Posts1319
Joined29/06/06
Christine says:
"It all seems to boil down to what exactly we mean by understand and Bruner appears to mean something different to John Holt."

I do take Mary's point that if we go along with Bruner, one of the key ways we "check" for understanding (both within ourselves as well as in others) is to see what we (or they) can do and so doing has a part to play in any definition of understanding.
It is interesting in the Holt quote about understanding that all the things in his list (with the possible exception of 4 and 6) are "doing" things, behaviours.

The now (in)famous Gattegno quote "Only Awareness is Educable" has been developed by John Mason into the trio:
Only awareness is educable;
Only behaviour is trainable;
Only emotion is harnessable.

... and this prompts me to think about the importance of keeping all three of these strands present (the "emotion" is meant to indicate the importance of motivation or "emotional buy-in" in the learning process).
Mason has developed this into a framework for preparing to teach a topic and writes about it in his article "Asking Mathematical Questions Mathematically" (Versailles, March 1998) on pages 7 and 8.

(BTW - A suite of materials called "Preparing to Teach ....." was produced by the Centre for Mathematics Education, the Open University in the late 1980's based on this framework. If you are interested in these, information about them is on the CME web-site).
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petegriffin 24 August 2011 17:27
Assistant Director (Maths Hubs)
Posts1319
Joined29/06/06
Maggie - I have always thought the Holt check list was ideal material for a classroom poster but not quite the right language. I would love to see what you come up with.
Would you mind sharing here?
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maggieh 24 August 2011 17:48
Posts67
Joined26/09/08
I'll be happy to Pete. Am not teaching the class in question until after 4 weeks of induction (ie October 1st - I work in FE) but will rustle up some drafts which I will probably have to adapt after 'trialling'.

By the way, thanks to you, I just downloaded a PDF version of the Holt book http://iwcenglish1.typepad.com/Documents/Holt_How_Children_Fail.pdf in fact I'm trying to figure out how to email it to myself so I can read at leisure on my Kindle.

I searched the PDF for the list of 7 and note he follows the list with:

"This list is only a beginning; but it may help us in the future to find out what our students really know as opposed to what they can give the appearance of knowing, their real learning as opposed to their apparent learning."

So you never know I might end up with more than 7 points! (Any further suggestions welcomed).  Although, as my class is Entry Level, I might also end up with fewer.... 
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mary_pardoe 24 August 2011 18:06
Posts156
Joined29/01/08
In Thinking Mathematically John Mason, Leone Burton and Kaye Stacey list 5 assumptions that are relevant to ideas about ways of using textbook exercises to help understanding:

Assumption 1   You can think mathematically

Assumption 2   Mathematical thinking can be improved by practice with reflection

Assumption 3   Mathematical thinking is provoked by contradiction, tension and surprise

Assumption 4   Mathematical thinking is supported by an atmosphere of questioning, challenging and reflecting

Assumption 5   Mathematical thinking helps in understanding yourself and the world 


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Jane_Imrie 24 August 2011 18:17
Deputy Director
Posts636
Joined26/06/06
Apologies for not keeping up with the thread and so responding to Christine's 15:18 post when several other fascinating ones have come between!

First - you are right about the wealth of material on the portal now. We are gradually finding ways of packaging it so that people can find everything. The search facility is excellent - but I agree would not have helped in this case.

I didn't mean to suggest that fluency wasn't a good outcome. It's a very important outcome. There are many techniques in which it helps to be fluent in order to progress in mathematics and I think that often such fluency removes barriers to understanding. What I was trying to say is that I think we need a balance of mathematical outcomes across our teaching  of which fluency is one, but evidence suggests (and there is some of this in the Mathematics Matters report) that we don't achieve that balance, even when we think it's desirable, and that the focus is on fluency. The report goes on to discuss possible contributory factors.

Perhaps there's also a question about how we develop fluency which a number of people have alluded to here.
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