A Feasibility Study in Assessing Practical MathsAuthor(s): June MarshSource: Mathematics in School, Vol. 4, No. 6 (Nov., 1975), pp. 21-22Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211456 .

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A feasibility study in

assessing practical maths by June Marsh

Reprinted with permission from the National Foundation for Educational Research's Educational Test Bulletin. A fuller report appears in Tests of Attainments in Mathematics in Schools: Monitoring Feasibility Study, edited by R. Sumner and published by NFER, Slough.

The purpose of the study was to investigate whether reliable estimates of practical mathematics attainment can be obtained when children are tested by a trained assessor in a one-to-one relationship. Also it was hoped to discover which areas of mathematical understanding are not made apparent by traditional examination procedures. To this end, test items were given in both the practical and written modes of presentation. These problems were separately studied for pupils in primary (10-11 year olds) and secondary (14-15 year olds) schools; the account here describes the primary school part of the exercise.

When topics were selected for item-drafting in August 1973 consideration was given to (i) their likelihood of general inclusion in the existing school curriculum; (ii) simplicity of exposition; (iii) reasonable availability of apparatus and (iv) the time and space required for the administration of tests by a visiting assessor. Open-ended questions, then, were temporarily avoided, and the nature of the items ultimately selected bore in some instances, close resemblance to concept- testing; but all involved the handling of clocks or blocks or sticks, shapes, water and scales.

Reception by local schools, approached to assist in item-development, whilst not always of the "So you've come about their sums" calibre encountered in one instance, was nevertheless somewhat guarded, and the request for children of only average ability to test was not generally popular. Space in which to operate is a constant problem in primary schools. Rooms normally used by Heads, Deputies and Staffs were, when they existed, most kindly vacated, but item-development was also conducted in an airless sickroom which certainly merited that title, and in a cupboard behind a stage whereupon an entire class existed behind curtains separating it from an auditorium containing yet another class ...

The October half-term break saw eight broad categories of test activities prepared for experimental use, identified loosely as numeration and structure; area and volume; direction and angles; circles, cylinders and cones; attributes and sets, and time, rate and speed. Each had an established form of words and an accompanying rubric. Marks could be awarded for the employment of a correct strategy, even if an incorrect answer was eventually obtained. Score-sheets gave space to record whether physical or perceptual strategies were used, and other relevant observations. It was now necessary to evaluate the nature of inter-tester reliability, and a small number of teachers was recruited to operate a design which extracted the maximum information from permutations of testers, children, topics, schools, and time allotted. The full ability range of pupils was involved. Apparatus was assembled by the testers, but such fundamental aids as metric blocks

and scales, counting sticks and the three-period abacus were rarely available in participating schools, and had to be specially acquired.

Testers were briefed in approaches and method at an all-day session in November, when video-taped examples of children helping in the item-development stage led to much discussion and valuable modification. Testing occupied two very full days later that month, and completed score-sheets were returned before Christmas.

Notes made on children's reactions and responses were illuminating. There were difficulties with the generally simple mathematical vocabulary; with the reading of number lines such as those on weigh-scales, and, in the later tests, thermometers; with base ten, abacus-reading, and fractional parts other than the most elementary. Metric measures were encountered in some instances for the first time, as were angles, defined by one child as "the things you hold to open the door with". Children were, however, alarmingly ingenious at devising the quick practical remedy to difficult situations. Required to find the weight of the water in a given container, one child, graded by the school as below average, placed the empty container on the scale, adjusted the hand to zero, and poured in the water, to arrive at a correct weight without the trouble of calculation. At least two others gaily poured the large amount of water all over the scale-pan ... and inevitably the table and the tester. 24 hr. clocks, time- tables and written time generally proved unfamiliar. Some children scored no marks at all on this topic. Predictably, ideas of volume and speed, even in this practical context, gave difficulties to most. On the other hand, quite involved processes in logical thinking, concerned with attributes and sets, were well done particularly by the children from the lower ability range.

There were offers of further assistance from this group of testers when their intensive two-day mission was completed. Most had found the experience both rewarding and enlightening, one teacher writing to say that she felt the need to re-think her whole classroom approach as a result. Another remarked on the less- inhibited attitudes of lower ability children to practical work; they were enabled by it, he felt, to score, at times, as highly as those whose thought-processes were more abstract. In this round of testing, some degree of closure was given to every item response, correct or incorrect, so that feelings of inadequacy were minimized. Often, later performance seemed to be enhanced by the confidence this procedure engendered.

Analysis showed that the tester could exercise an influence on the quality of a pupil's responses, but that the measures were reasonably reliable. A shorter test being required for the next phase of the investigation,

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some 28 of the 52 items initially tried-out were selected and adapted according to their statistical reliability, bearing in mind the new requirement for ease of transposition into a comparable written form. This "parallel" pencil and paper test was not entirely satisfactory to its designers owing to the difficulty of matching scoring procedures in the practical mode with the written equivalents.

From the 61 Lancashire schools invited to partici- pate in this third phase by releasing a member of staff for two separate days' involvement, and providing six children and some accommodation for testing on one day, 28 agreed to help. Instead of an anticipated sample size of about 250, the number secured was barely 170.

As before, the nominated testers attended a briefing session, where they obtained apparatus kits and detailed instruction; topic sheets were examined and discussed, video-taped interviews observed, and a water-tight scoring system established. Five testers from the previous round, (nobly bridging gaps where small schools, willing to participate, could not release a teacher), were able to contribute informed opinion as a result of their previous experiences. It should be noted that, throughout the study, a high degree of involve- ment was met in all who volunteered their assistance.

Practical tests were administered in the days that followed this meeting, each tester visiting another's school on one day to test six previously-selected children drawn from the full ability range. The sample of schools had been halved for printed-test administra- tion, one section completing these papers not less than three weeks before the practicals, and the other not less than three weeks after the practicals. It was hoped in this way to minimize the effects of one mode upon the other. In addition, a Verbal Reasoning paper con- firmed gradings of pupils. Reactions from testers, on completion of this phase of the study, employed such phrases as "well-devised", "ability-revealing", "thought- provoking", in connection with the practical items, which, it was remarked, were relevant to work in schools for children of this age. The printed items were less well received; either they had suffered in their

transposition or were simply just not as attractive. It was a frequent observation that some children

were extremely nervous in this one-to-one situ~ation. Reluctance to handle the materials was often encountered, and encouragement and persuasion were needed before a response could be obtained. Such children often gained in confidence as a test progressed, and most testers remarked on the enjoyment expressed by many children at this kind of testing. Those of lower ability in particular much preferred to use equipment and talk, than to think and write. There were those also who rushed at a solution without sufficient thought, and here, "Are you sure?" was often enough to elicit a correction. Otherwise, the only closure offered in any event was "Thank you"-or, all else failing, "Fine". Any more could have influenced later pencil and paper performance. The same insecure concepts as those exposed in the earlier rounds were again in evidence. The mathematical equipment avail- able in some schools had to be supplemented; for example only six metric weigh-scales and six litre cubes could be raised from the 28 schools involved.

Above all, this practical testing has emphasized the necessity for clear communication of ideas in both directions between child and assessor. Item developers must give this requirement absolute priority. Other more precise findings await the completion of data processing at Slough. In the meantime, however, a scattergram, presenting a relationship between results obtained in the two modes, reveals a distinct bias towards better performance in the practical tests. It is interesting to note that those who experienced ("practicals") before the written papers have scored noticeably higher marks in the latter than those who did their written work first.

There's an educational slogan here somewhere. M

(Mrs. Marsh, who teaches in a Blackpool school, was seconded on a half-time appointment to carry out this research. The NFER wishes to thank the Lancashire and Blackpool authorities for permission to carry out this development work in schools. The assistance of teachers and pupils is gratefully acknowledged.)

XVI1 International Olyathematical

Olympiad papers

Burgas, Bulgaria 7, 8 July 1975.

Questions 1-3, 4 hours. Questions 4-6, 4 hours.

1. Let xi,yi, (i = 1 to n) be real numbers such that

x, > x2 >3

*. Xn and y, > y2 >3 *

s >

A A Yn

Prove that if z,, zz, . .., zn is any permutation of n n

Y , Y2, . . ., Y n, then (xi-yi)2 <.

(xi-zi)2. (CZ i =1 i =1

6 pts)

2. Let a,, a2, az,... be any infinite sequence of strictly positive integers such that ak < ak+, for 1 < k. Prove that infinitely many an can be written in the form

an = xap+yaq, with x,y strictly positive integersand p

4 q. (GB 7 pts)

22

3. On the sides of an arbitrary triangle ABC triangles ABR, BCP and CA Q are constructed externally with LPBC = LCAQ = 450, LBCP = LQCA = 300 and LABR = LRAB = 150.

Prove that LQRP = 900, and QR = RP. (NL 7 pts)

4. When 44444444 is written in decimal notation the sum of its digits is A. Let B be the sum of the digits of A. Find the sum of the digits of B. (A and B are written in decimal notation.) (SU 6 pts)

5. Determine, with proof, whether one can find 1975 points on the circumference of a circle of unit radius such that the distance (along a chord) between any two of them is a rational number. (SU 6 pts)

6. Find all polynomials P, in two variables, with the properties

(i) for a positive n, and all real t,x,y, P(tx, ty) = tnp(x,y). (i.e. P is homogeneous and of degree n)

(ii) for all real a,b,c, P(a+b,c)+P(b+c,a)+P(c+a,b) = 0.

and (iii) P(1,0) = 1. (GB 8 pts)

The country of origin and the maximum mark is given in brackets. W

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