Extract from the Arbitrary and Necessary Wiki
This task is designed to promote further reflection on the issue of what to tell in a
mathematics classroom and how and when we choose to tell it, an issue raised by
several contributors to the problem solving wiki. Hewitt provides a framework for
considering mathematical knowledge by considering mathematics as arbitrary or
necessary. Reflect upon this framework and Hewitt’s view of “educating awareness”
(2002, pp 62) in the light of both your experiences as a learner and as a teacher of
mathematics.
Haggarty, L. (Ed.) (2002) Teaching Mathematics in a Secondary School: A Reader.
Oxon: Routledge Falmer.
PUPIL " What have letters got to do with numbers, what are they for and why is it x and
y"
TEACH " Well, erm cough, it's just a way of showing or including any unknown number,
that we then kind of put in for the letter later on like, kind of. We don't have to use x or y,
we could do the job with n, z, a, or b, it doesn't matter, kind of, do you get me alright ?"
PUPIL "Kind of !
I just sooooo hate algebra, it's so could be this or could be that"
TEACH "I think you mean arbitrary"
PUPIL
" I most certainly do, thankyou for pointing that out sir"
I think Hewitt proposes, it would be reasonable to allow pupils to play around with the
algebra concept and invent something for themselves, as a way of helping them to
understand the nature of using letters to represent unknowns or variables. They can then be
informed of 'givens' to be used and develop the 'necessary' awareness of which operations
to perform when finding an answer. The main focus of learning, being on developing
mathematical thinking, where pupils creatively apply concepts and understanding that they
may be aware of. The teacher fully aware that real learning can easily be side tracked or
bogged down by 'un-necessary' learning convention and 'un-necessary' recall/memory
learning.
Last week I was reminded (as if I'm not everyday), of how important, careful consideration of
pupils understanding and prior knowledge is, when preparing tasks. The tools of their
'awareness' with which to tackle a set task. Problems can be worked out, but only if you are
aware of the concepts or mathematical properties involved and have the ability to
understand and use them in order to move forward. In otherwords have what is 'necessary'
to further develop your mathematical thinking.
I agree with Hewitt's suggestion that real development of mathematical thinking, lies not in
parts of topics that rely on arbitrary learning but in cutting to the chase and focusing learning
on parts of topics that encourage developing real mathematical awareness. The conventions
are helpful in organising and structuring an approach, or to communicate what you are
doing but are ultimately not necessary. As Hewitt puts it " I'm having to remember...so I'm
not learning"
There is however an element of "I'm having to remember other situations and experiences
so I can develop my learning" The idea of received wisdom can be a grey area. Curriculum
pressures and pupil pressures " just tell me", mean we can't keep going back to find an
'appropriate activity' to develop learning, keeping us away from becoming arbitrary teachers
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and developing arbitrary learners. In the real world there has to be a balanced approach,
informed by awareness of deep learning, but also with an eye on what has to be done
with what is in front of me, in a given time frame.
I hope there are not too many " look, it just is .... so get on with it" situations. Careful delivery
of received wisdom can be used to open up pupil awareness and move learning forward.
Participant A
Hello
To me this article is discussing memory of facts versus awareness and understanding. We
all know that pupils who have a greater awareness of why something works and link their
understanding to things they already know will be more likely to succeed. Hewitt talks of
arbitrary: things that just are (little child's question at the beginning) i.e. convention and
necessary: things that can be worked out with the right level of awareness. The choice of
the word necessary seems starnge to me as they are not really necessary if you can work
them out.
I don't think he is saying anything new in the sense that we know as teachers that it is better
if pupils discover things themselves (more likey to remember as their awareness and
understanding will be greater). Hence providing activities to help students educate their own
awareness rather than giving them something to memorise is what we aspire to. It is the
issue of 'readiness' and the fact that the students may not have the necessary awareness to
understand a teacher's explanation which to me is the difficulty in the classroom. Wasn't this
something that was mentioned in the last article on problem solving i.e. the timing of a
teacher's probing questions need to be appropriate.
Hewitt seems to be saying that we use our arbitrary knowledge and conventions of liguistic
(?) proof (if...then) to establish new certainties which may or may not be accepted in the
mathematical community. Again this seems to be stating what most people do when solving
problems. In the modern society we use logic conventions (these are probably arbitrary and I
am sure there is another debate here on which theory of truth should be used). I found this
last bit hard to follow.
In terms of teaching I can see that I am guilty, especially near exam times of 'spoonfeeding.'
It is great when pupils discover things for themselves as my bottom set have been looking at
indices and some of them have realised what happens with negative numbers when
increased to even or odd powers. I have made a conscious decision not to 'teach' this but to
let them discover it by using a calculator and making a comment. It was great to see this
discovery being applied the next lesson and the sense of satisfaction on the girl's face when
she managed to do (-2)^7 and (-2)^6 mentally. If I hadn't given the 'activity' to the class this
would never have happened. Far too often I think the bottom set won't be able to do
something so don't give them the opportunity to discover.
Participant B
Responses to Haggerty chapter 4
The division certainly seems an attractive one and is clearly often true. A great deal of what
we do, know or believe is arbitrary butessential if we are to be able to communicate in any
meaningful way. As a teacher I have certainly spent a great deal of time informing children of
the arbitrary. And still do. I fear that I spend too much time on this and insufficient in
encouraging students to explore the necessary.
Perhaps more troubling is the amount of teaching of the necessary I do, rather than
encouraging students to explore and find things out for themselves. I am justifying this to
myself because many of my students, certainly most, are fearful of failure because of their
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histories and lack the confidence to think about things and come to conclusions. I do try to
encourage this but it is both very time consuming and, at least for my SEBD students, also
requires a great deal of emotional support.
It is less time consuming to teach than to encourage finding out and exploring and where
there is pressure on a timetable the explorati9on is often what is most easily sacrificed.
However there is no doubt in my mind that when students do come to correct determinations
of ‘necessary, truths their understanding is both deeper and more capable of being applied
than when taught.
As a learner I find that I am also somewhat ambiguous as I much prefer to see patterns and
make sense of them , but when in a hurry or tired it is easier to accept givens than to derive.
However this makes my knowledge much less flexible, I can do more with what I have seen.
As a child I can still remember feeling in Maths that quite a lot of what I did was like being
reminded of something that I already knew, something that I have always just assumed was
there. This chapter suggests to me that that may have been a subconscious awareness of
the necessary. I have not however found many other people I have talked to who have this
memory so I wonder if it is simply a peculiarity of a combination of the way I was taught at
primary school (very well I believe) and my own way of thinking.
Secondary school in my memory did not allow the time for exploration (except with one
individual teacher) and we were taught a lot of isolated processes. I did OK because I had a
good memory but my 1st year in secondary school nearly put me off for life. Being informed
of everything then endless book exercises then sitting waiting for the rest to catch up.
I think the following is a very important comment from the passage.
‘The necessary is about properties and one possibility is for students to 'receive' properties
through a teacher informing them just as for the arbitrary. However, this turns the necessary
into received wisdom and students may well treat this as something else to be memorised.
Indeed, they will have no other choice unless they are able or willing to do the work
necessary to become aware of the necessity of this received wisdom. Some students may
be able to do this work, in which case the received wisdom will become a derived certainty
and be known through awareness rather than memory’
If we are restricted by time and other constraints let us at least offer enough sense of joy in
the maths to encourage at least some of our students to take that step.
Participant C
Response to Hewitt Chapter 4
I can clearly see how this article relates to the previous article by Polyna. It provides further
evidence that pupils need to be allowed to discover maths themselves, in order to be able to
own the information and make it semantic. Thinking back to when I was a pupil and even
now I need to often know the reason behind methods and theories and will often fail to fully
understand the topic if I do not find the meaning behind the work I am doing. I guess in a
similar way to what Karen writes I was the ‘but why?’ child, which to some of my teachers
was perhaps a breath of fresh air, whilst to others merely an inconvenience. I rediscovered
this feeling recently when we were working on indices and being able to actually understand
why anything to the power of 0 =1. I have come to terms with this through the work that we
completed and am realising how difficult it must sometimes be for my pupils.
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I agree with the writing of Hewitt particularly the comment “so much time is spent on what is
necessary and so much time is spent on memorising and practising conventions”. I feel that
due to the way our schools are set up and the emphasis on success and results leads
teachers to rote teach pupils and lessons do become a cycle of practise and yet again that
phrase of ‘spoon feeding’. Teachers can become stuck in the rut of teaching for results and
not for understanding. I feel that when a teacher takes over a class of pupils who are nearing
the end of their time in Maths education and have experienced being taught by a number of
teachers throughout their life it can be difficult to undo the ingrained ‘facts’ that previous
teachers may have given and develop pupils skills to actually discover information
themselves.
Hewitt suggests “awareness can be adopted from the arbitrary”. I think the skill as a maths
teacher is to know how much to tell and when. I believe that it is important that pupils are
armed with some arbitrary information to enable them to access necessary maths concepts.
However this is dependent on the topic and it is up to us as mathematicians to decipher what
a child is capable of discovering for themselves in each area through our experience and
discussion with pupils.
From a teachers point of view it can be difficult to know what to tell pupils and when to allow
them to use their ‘realm of awareness’ to discover what is necessary. I myself as a teacher
have been guilty of “providing explanations based on my awareness”, as opposed to the
pupils. It can be difficult to see things from their perspective, but I believe the more
experience that I gain as a learner myself and also as a teacher, the more this will allow me
to develop my awareness of the way that pupils think and approach maths. Through the use
of problem solving, as Polyna suggested and through self discovery activities I think I would
be able to encourage pupils to take more control of their own learning, in order to gain a
more developed and concrete understanding for themselves that will last them a lifetime
rather than a few months in a run up to an exam. I also believe that taking this approach to
teaching will encourage pupils to become more motivated about maths rather than viewing
the subject as an upward struggle on a mountain of facts.
I am in agreement with Janet in her comment that Hewitt is not actually proposing ‘anything
new’, but I do feel that if this topic arises so frequently in education and we are all so aware,
why in fact to we constantly need to readdress it? Is it due to the way education is geared
towards achieving results that prevents teachers being able to allow pupils discovery and
essentially the enjoyment?
Participant D
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