Secondary School Heads of MathematicsAuthor(s): J. C. HallSource: Mathematics in School, Vol. 7, No. 3 (May, 1978), pp. 29-30Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214233 .

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SECONDARY SCHOOL HEADS OF MATHEMATICS

by J. C. Hall, Woodway Park Comprehensive School, Coventry

The following discussion of heads of mathematics departments and conceptions of their roles is based upon personal experience, observations, and a questionnaire survey of a sample of 39 heads of mathematics. The questionnaire and full discussion of the sample may be found in Hall'.

In all probability the head of mathematics will be male. Even in single sex girls' schools there is a good chance that the head of mathematics will be a man. This may highlight a shortage of female mathematicians; it may point to a reluctance of females to seek posts of responsibility; or it may indicate a bias against female appointments, although this latter view- point is difficult to reconcile with the incidence of male heads of mathematics in girls' schools. Promotion will have come quickly; about six years to reach Burnham scale 3 and eight years to reach Burnham scale 4. Even though the majority will be graduates, or equivalent, few will have a first degree in mathematics as a single main subject. A not unsubstantial minority will be non-graduates or graduates in disciplines unconnected with mathematics. There is evidence to support the views expressed by teachers in the survey of Grace2 that promotion goes to the movers.

Mathematicians are well aware of the prestige accorded to the subject by their pupils and the "general public." Heads of mathematics believe in the need for close liaison between mathematics and science departments, although there appears to be a strong bias against the practice of appointing a single Head of Science and Mathematics Faculty, unless, as one mathematician commented, the head of mathematics himself was head of this faculty. The desire to establish good relation- ships with staff involved in pastoral work, parents, other departmental heads, and members of their own departments are rather obvious observations. Not so obvious is a particular desire for liaison with lower feeder schools, but a lack of commitment on the part of the heads of mathematics towards the desirability of liaison with local employers.

Heads of mathematics usually believe that they should have teaching experience at a wide variety of ability levels, a wide knowledge of their subject and of curriculum developments in general, and an ability to relate well to people. Considering their general youthfulness and lack of mathematical degree qualifications, it is perhaps not surprising to learn that it is not generally believed that heads of mathematics should necessarily be the best mathematically qualified or the most widely experienced members of the department. A balance of these two attributes is considered most suitable.

Department heads believe that the amount of mathematics teaching carried out by staff without suitable qualifications in mathematics is higher than that reported by the Department of Education and Science Survey of Mathematics in Schools 11-18, 1975. J. C. D. Rainbow (CEO for Lancashire) has

reported that "In Lancashire alone 25% of the children are being taught mathematics by teachers with only an 'O' level or lower qualification in the subject themselves".3 This may explain why heads of mathematics are not always given the important voice that they desire in the appointment of teachers to their department - there are simply not enough qualified mathematics teachers to fill all the vacancies. Many department heads would offer guidance and training to those teachers that they consider to be inadequately qualified, but appear ambivalent and vague over accepting responsibility for the probationer teacher experiencing classroom discipline problems.

The functions of the head of department, in addition to actual teaching responsibilities, may be considered to be representa- tive and managerial. The representative function is a two-way process. The department head interprets the headteacher's philosophy for the school to the department and makes known the needs of the department to the headteacher. Marland4 has described this function as pivotal. Regretfully, regular formal departmental meetings are not considered necessary, although there is a firm commitment to regular full staff and heads of departments meetings. This suggests a desire for a say in the organisation of the school which is not extended to allowing departmental members a similar voice in the organisation of the mathematics department. In the main, department heads appear content to implement policies with which they disagree and many suffer severe restriction over matters of syllabus, etc.

The managerial function is concerned with organisation, direction and control. The choice and use of both personnel and materials would be included in the organisational aspect. Dissatisfaction is often expressed concerning department appointments, as well as the mathematical qualifications and expertise of existing staff. The amount of money allocated to the mathematics department is usually considered adequate, but the size of mathematics classes, which many consider to be larger than the norm, and the lack of mathematics rooms (let alone a suite of rooms) obviously restricts the choice of materials.

The directional aspect involves determination of the aims of the department and guidance to staff on methods of achieving objectives. Many believe that it would be of benefit if the head teacher would set down a clear statement of the responsibilities of heads of department.

Control is exercised in the supervision of the work in the department and in the maintenance of standards. It is the control dimension which produces the greatest difficulty in pinpointing clear viewpoints. Help would be readily available to the weak mathematician, but not so apparent for the weak disciplinarian. Many teachers feel that they ought to be master of their own domain (i.e. classroom) and it may be awkward

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for the head of department to freely invade another teacher's territory. In addition the absence of mathematics rooms and the dispersion of mathematics teachers to all corners of the school make supervision extremely difficult. Formal depart- mental meetings could facilitate control, but on the whole these are not highly regarded. The often expressed desire to teach a broad spectrum of the ability range would certainly aid supervision throughout the school. If, however, objectives were not being realised, it might prove impossible to change direction within the department because of the quality of staff available and the apparent problem of persuading a head- teacher to authorise a major syllabus revision.

The post of head of mathematics is seen by many incumbents as being a stepping-stone towards a headship or deputy head- ship. Perhaps the experience in curriculum organisation and

administration (e.g. timetabling) which many heads of mathematics are able to gain is considered to be suitable training for these posts. Much more effort, however, must be put into consideration of the role of the head of department in order to provide suitable preparation for these later posts and to aid selection and training of future heads of mathematics.

References 1. J. C. Hall (1975) The Role of the head of Mathematics Department in the

Secondary School. Unpublished MSc dissertation, Loughborough University of Technology.

2. G. R. Grace (1972) Role Conflict and the Teacher. Routledge and Kegan Paul. 3. J. C. D. Rainbow (1977) "In-Service Training - A Lancashire

Perspective" British Journal ofj In-Service Education. Vol. 3, No. 2, 87-92.

4. M. Marland (1971) Head of Department. Heinemann.

Solutions to Puzzles, Pastimes and Problems on p. 17

1. Black and White Patterns The greatest number of black squares is 15, with 3 in each of the five rows and five columns. Figure 6 shows a pattern with

Fig. 6 Fig. 7

one line of symmetry (a diagonal); Figure 7 is symmetrical about both diagonals. There are other patterns with rotational symmetry only; can you find one?

2. Fun With Numbers (a) 19, 28, 37, 46, 55, 64, 73, 82, 91. Three of these are prime numbers. The nine numbers form an A.P. with common dif- ference 9; their sum is 9 x 55.

(b) 5+7=12, 5x7=36-1. 5+19=24, 5x19=96-1. 7+17=24, 7x17=120-1. 11+13=24, 11x13=144-1. All prime numbers (greater than 3) are 1 more or 1 less than a multiple of 6. So (6n- 1)+(6m + 1) is a multiple of 12 when n + m is even (= 2k). Then the difference between m and n is also even as m- n = 2(k-n). The product (6n-1X6m+ 1) = 36mn + 6(n - m)- 1 is one less than a multiple of 12 for all values of m and n.

3. A Rectangle Within a Rectangle The converse is also true. One solution is shown in Figure 8, where QS is the line joining the mid-points of the rectangle's two shorter sides, and P and R lie on the circle on QS as diameter.

If PQRS is a rectangle, then the triangles APQ and BQR are similar, and their corresponding sides are proportional. Hence

Fig. 8 Fig. 9 P

cl

UR Z

X

Y

x/a = cl(b- x), which leads to the quadratic equation x2- bx + ac = 0, that has equal roots when b2= 4ac, giving x = 12b, and Q and S are the mid-points of AB and CD. If b2 > 4ac, there are two positive solutions for x, lying between 0 and b.

(x = 12(b h /b2- 4ac).)

4. Number Scales (i) 32+3+1=13; 42+4+1=21; 52+5+1=31; 62+6+1 = 43; 72 + 7 + I = 57; 82 + 8 + 1= 73; 92 + 9 + I = 91; 102- 10+ 1= 111; 112+ 11+1=133; 122+12+ 1= 157. Only five are prime numbers - in the scales of 2, 3, 5, 6 and 8. (ii) In the scale of r, 121 represents the number r2+2r+1 = (r+ 1)2), where r > 2, and 144 represent r2+ 4r+ 4 = (r+ 2)2, where r > 4. So both numbers represent square numbers in every scale greater than 4. (iii) (a) (r + 1Xr + 2)= r2+ 3r + 2 for every value of r; True when r > 3. (b) (r + 2)(r + 3)= 2r2 + 2r + 1 only when r2- 3r- 4 = 0, i.e. when r= 1 or 4. True only when r= 4. (c) (r+2X2r+ 1)=3r2+2, only when r'=5r, i.e. when r= 0 or 5. True only when r= 5. (d) (r + 2)(2r + 1)= 3r2+ r + 2 only when r2= 4r, i.e. when r= 0 or 4. True only when r= 4. (e) (rXr+ 1)=r2+1 only when r=1, but this scale does not exist. Never true. (f) (r + 1X2r + 2)= r+ r + 2 when ra- 2r2- 3r= 0, i.e. when r= 0, - 1 or 3. True only when r= 3. (g) (r + 1)(2r + 2)= r + when 2r2 + r- 1 = 0, i.e. when r= - 1 or 1/2. Never true. (h) (2r + 3)(3r + 2)= 2r3 + r2 + 2r + 2when 2r3 - 5r2- llr - 4 = 0 when r= - 1, - 1/2 or 4. True only when r= 4. (j) (3r+2)2=r3+2r+4 when r3-9r2-10Or=0, i.e. when r= 0, - 1 or 10. True only in the decimal scale.

5. A Dissection Puzzle (i) Figures 9, 10 and 11 show the solutions. (ii) In Figure 12, M is the mid-point of AB, MK is per- pendicular to AC, and MN is parallel to AC.

Fig. 10

X Z Y

Fig. 11

Y Z

A

M

K

Z

c ,/"e/ /N /X/ j/ // //

"y/' Fig. 12

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