We don't have maths wars in UK so can speak freely without being assigned to one side or the other thank goodness. I am going to take issue with you about whether we know enough about how minds develop to ever say we 'understand' except by offering children opportunities to do things and then seeing if they do or not (not if they CAN, but if they DO). In our curriculum and assessment regime teachers are encouraged to believe that certain things have to be learnt and experienced before other things, so what you end up getting is over-simplification of mathematical ideas and fudgy reasoning and then - hey presto - we miss the boat with students who were perfectly capable of understanding more rigorous argument but got into sloppy habits and, worse still, were praised and rewarded with good 'levels' for doing so. By contrast, there are many reports of teachers who work with young children using complex and rigorous argument and find that they DO respond and use these for themselves when in an environment which encourages it. The key in all of these is that children are offered situations where there is a MATHEMATICAL need, not a developmental need, to use more rigorous and precise methods. So, you only need to become more precise if you are aware of non-examples and counter-examples to your current definitions and 'theorems'; you only need to label if you have to keep track of properties; you only need to make statements if you need to generalise or relate; you only need set theory if you need an embracing abstract idea to include several concrete ideas.
-----Original Message----- From: Martin C. Tangora [mailto:email@example.com] Sent: 16 October 2009 21:35 To: MATHEDU@JISCMAIL.AC.UK Cc: Anne Watson Subject: Re: Labels
I was totally with you, right up to your last sentence. There, you talk about the teacher who has not yet been "indoctrinated into stories about what children can or cannot do at certain ages."
That would not be indoctrination, but the understanding of development. As I said before, we do not teach axiomatic set theory to the kindergarten, even though it would be a *logical* subject with which to begin their mathematical education.
I certainly agree that the teacher must be flexible, and not dogmatic. It is a great pleasure to find out, in real time in a real classroom, that the students are ready to move up to the next stage of development, at least on the topic at hand. I agree that it would be a shame if the teacher had rigid ideas about what was too advanced for the class.
The "math wars" live on, because some of us cannot resist taking a dig at others who seem to be "indoctrinating" the wrong side of the argument.
Not to compare myself with Barack Obama, but there are so many polemicists abroad that one despairs of finding common ground.
At 12:51 AM 10/15/2009, Anne Watson wrote: >From Anne again who is very busy but trying to keep up with the argument: There is pedantry on all sides of course there always will be. For me the question is 'is it important to learn about labelling?' My answer is 'yes' because labelling records distinctions and enables us to make and keep track of distinctions. So if 14 year olds (or even 4 year olds) are doing a task which generates distinctions, or which requires distinctions, introduce labelling as the way we handle these. Martin Hughes has done lovely experiments with very young children to show that if they NEED to keep track of specific quantities they will invent symbols for numbers. The same applies to 14 year olds and very recently I observed a lesson in which this happened there was a need to label vertices in corresponding order, so they did! And by the way these were not high fliers, they were students whose prior achievement was below average for their age. Oh yes, and they were being taught by a newis h teacher who has a PhD in Mathematical Physics and has not yet been fully indoctrinated into stories about what children can or cannot do at certain ages her commitment was to their mathematics. >Anne
Martin C. Tangora University of Illinois at Chicago firstname.lastname@example.org