Unrecorded, Unobserved and Suppressed Attainment: Can Our Pupils Do More than We Know?Author(s): Ann MacNamara and Tom RoperSource: Mathematics in School, Vol. 21, No. 5 (Nov., 1992), pp. 12-13Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214926 .

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UNRECORDED

UNOBSERVED AND

UPPRESSED

TTAINMENT CAN OUR PUPILS DO MORE THAN WE KNOW?

by Ann MacNamara and Tom Roper, Centre for Studies in Science & Mathematics Education, Leeds University

Recent pronouncements by Government Ministers have consistently stressed that teachers under-estimate what their pupils are capable of doing. As Michael Fallon said:

"... in many cases, the tests (the SATs) showed teachers much that was new about their pupils, usually that they could do more than teachers thought." (Michael Fallon's speech to the NAHT Primary Conference.)

Ministers are echoing the views of HMI who, in their recent report, claimed that:

"much of the work for Year 7 pupils was initially based on Levels 2 or 3 which represented serious underexpectation by teachers and lack of attention to previous attainment." (HMI report on Mathematics at KS 1 and 3.)

The Secretary of State for Education repeated a similar view of the reliability of assessment by teachers when he said:

"I share the wide-spread public concern - supported by HMI - that certain types of coursework, particularly where internally assessed and conducted under open-ended conditions, may not reliably assess pupils' performance." (Kenneth Clarke, Secretary of State for Education.)

New Attainment Target 1 is given considerable impor- tance by HMI and NCC. The problem of under-estimation of pupils' achievements in Mathematics is of most con- cern in this attainment target - "Using and Applying Mathematics".

We suggest that teachers may be under-estimating the attainment of pupils for three reasons: firstly, because they do not - or are not able to - observe pupils' attainment; secondly, because pupils are not able to record attainment efficiently; and thirdly, because pupils are suppressing evidence of their attainment.

In order to trial some materials developed for a KS 3 Inset course on NAT 1, we taught in two comprehensive schools, one in West Yorkshire and the other in North Yorkshire. In the first school, two lessons, one with Year 8 pupils, the other with Year 9, were videoed and the participation of certain groups was audio taped with port- able recorders. The two classes were reported by the teachers to be of average to below average in mathematical ability. In the second school, audio recordings were made of two groups within a class of Year 9 pupils of low ability.

The pupils had been asked to find how many different solid shapes could be made with four Multilink* cubes. The main focus of the two lessons in the first school was to encourage the pupils to write down, in their own words, a definition about "same" and "different". They worked in pairs, having more than sufficient cubes to make the eight different solids. The written evidence provided by the pupils showed attainment in NAT 1 at Levels 5 and 6. The following is what one pupil wrote:

"Sometimes if you make one solid it can look the same as some of the others. But if it does not fit into the other shape it is different. We have two shapes that look the same but it doesn't fit into the other. They can be a mirror image of each other, but if you put it to a mirror it would look the same but it isn't." (Year 9 pupil.)

12 Mathematics in School, November 1992

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We would postulate that this is evidence that the pupil has achieved NAT 1 Level 5(c) "make a generalisation and test it".

Other pupils were unable to bring that kind of clarity to their written descriptions, even after being encouraged to by the teacher. The video and tape recordings later demonstrated that pupils seemed to be showing by their manipulations of the cubes, and by pupil-pupil comment, the distinctions that they were unable to verbalise or to record.

There was evidence of attainment, clearly observed and observable, but with no written evidence for it. It was unrecorded by the pupil - and possibly not within the capabilities of the pupil to record. Is it acceptable just to say attainment has been observed or must we have it recorded by the pupil in some form? Must the pupil always provide written evidence?

As the lesson progressed, pupils were invited to consider the surface area of the shapes which they had made. Later, they were considering the maximum surface area that solid shapes made with different cubes might have. The short transcript below is from the work of two boys: (T is the teacher.)

T. " ... make a note, for four, the biggest is eighteen. What's the biggest for five? ... twenty-two ... what do you think it is for six? Make a little table ... biggest area is eighteen, for the five shape, twenty-two. Make one and see what you can do."

P, "(sub voce counting ...) ... has to be twenty-six, I'm sure." P2 "Write it down ..." P, "Golly, it goes up in fours, I bet you ..."

Finishes counting. "Twenty-six."

T "Twenty-six - OK."

The teacher appears to be present all the time and, therefore, might well have observed the glorious "Eureka!" moment when one of the boys makes a prediction that "it goes up in fours, I bet you" and finds that he is correct. The crucial point, in this instance, is that the teacher was distracted by something else prior to the moment in question, and on return saw only a table with yet another entry. The boys themselves offered no information to the teacher as to how the pattern went on, nor about the conversation they had just had. Indeed, as the teacher was called away again, the opportunity to follow up this particu- lar issue was lost. The pupils had shown evidence of attainment of NAT 1, Level 5(c), but this had gone unnoticed, unremarked and unrecorded by the teacher, appearing eventually only as an entry in a table of results, providing evidence for attainment at Level 3(c).

Teachers cannot be everywhere in classrooms, cannot observe everything, nor have they the time to listen to or transcribe all that pupils say. The teacher's dilemma is how to know what the pupils have done. It is possible that one solution is to encourage the pupils to write down exactly what happened, especially in terms of prediction and confirmation. Perhaps pupils need to be trained in what might be termed the "construction of an investigative diary" in which they record all these thoughts and actions as they occur. This training should begin as early as possible -preferably in the Primary School - but cer- tainly could begin in the Year 7 when many pupils arrive in the Secondary sector. However, even this may be insufficient, particularly with children for whom English is not a mother tongue, or for children with language problems.

In the second school, the activity was the making of loops based on octagons*, looking at the number of octa-

gons from which a complete loop could be made when one side touched another, and investigating free edges. In reading the following transcript, the reader should bear in mind that the girls were being recorded but then over- heard, as the tape did, the two boys on an adjacent table. As before, the teacher was totally unaware of this conversation.

P1 "So, maybe it's the odd numbers that don't work and the even numbers that do." P2 "See if the pattern goes on."

P1 "That's what I think anyway."

P2 "I'm making a table on a separate piece of paper."

A nearby group of boys are talking excitedly about what they had found- the following is overheard by the girls:

P3 "Can't make this one with seven ... all the even numbers you can do and all the odd numbers you can't."

The girls continue:

P, "Someone else knows it as well now ... you don't need to write it down."

P2 "But three isn't an even number is it?" P, "Three, can't make it with three."

The question of how to assess this is a very real one. The girls had made a discovery, but immediately on hearing that it would seem to be common knowledge consider that the value of the discovery is worthless -"someone else knows it as well now ..." - resulting in "you don't need to write it down".

The girls made, for them, an original discovery. Unfortu- nately they chose not to record their findings so no account of it appears in the record of their work. It is as though it had never happened, it has been actively suppressed. In this case, it is not obvious that simply keeping some form of diary about an investigation would have encouraged pupils to make this kind of record of evidence of attainment.

In any form of recording, pupils make value judgements as to what is worth recording and what is not. Pupils' attitudes to recording will depend on a number of factors: the value which they themselves attribute to their own work; that which they think that their peers will attribute to it and that which they think the teacher will attribute to it. If they consider that in any of these areas their work will not be sufficiently valued, then they will not record it and possibly not attempt to develop or extend it.

There do not seem to be any clear-cut answers to any of these issues, but in the light of the evidence available, it would seem that there is danger of under-estimating the attainment of pupils

(i) because they are incapable of adequately verbalising their deductions and thought processes in permanent form, or

(ii) it is not always possible to observe vital moments when evidence is apparent, or

(iii) vital moments are deliberately suppressed by the pupils themselves.

*Multilink: linking cubes marketed by NES Arnold, Nottingham. *from the trial SAT at KS 3.

Mathematics in School, November 1992 13

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