Assessment Workshop: Assessment in MathematicsAuthor(s): E. D. TaggSource: Mathematics in School, Vol. 1, No. 4 (May, 1972), pp. 28-29Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30210777 .

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Assessment Workshop

Assessment in Mathematics

by E. D. Tagg

(Dr. Tagg was Director of the Accelerated Teaching of Higher Mathematics Project at the University of Lancaster)

For the last three years Miss Amy Wootten and I have been running a first year mathematics course for students of social and life sciences at the University of Lancaster. These students had, in general, done no mathematics beyond O-level previously; many of them had been glad to give up mathematics at that stage and were rather dismayed to find that a modern study of their chosen subject required a greater degree of numeracy than they possessed. With assessment required to qualify for the second year courses we had to decide on methods of testing which would not hinder the efforts we were making to solve the students' problems of lack of motivation and lack of confidence.

In the past mathematics has largely been taught at universities for the future mathematician or physical scientist and methods of assessment have been such as would show which students had real mathematical promise. This certainly is not the objective in testing students of the type attending our course; we therefore had to think out again what we were trying to assess. Some of our findings may be of interest to teachers in schools - between us Miss Wootten and I have over 50 years experience of teaching mathematics in schools. All but a very small proportion of schoolchildren have no intention of becoming either mathematicians or physical scientists.

Five specific objectives 1. Removal - or reduction - of fear of examinations. 2. Giving opportunities to students to show their knowledge as well as to reveal their ignorance. 3. Suiting the methods of expression required in the work for assessment to those which the students use with facility. 4. Giving opportunities for students to show divergent as well as convergent thinking. 5. Encouraging students to discuss mathematics with each other.

The overall aim of the course is to enable the

students to form and use mathematical models in the analysis of situations in their own fields of study. The emphasis is therefore largely on the putting of a problem into mathematical form, and the knowledge of what methods are available for solution, rather than the detailed knowledge of these methods in' order to be able to use them under examination pressure. There is also the need for fluency in the use of mathematical notation, and for some facility in basic mathematical skills in the handling of routine formulae.

We have therefore introduced regular fortnightly tests which consist of straightforward questions set in a multiple-choice or structured response form. These tests count as 40% of the final assessment, and students gain in confidence through regular testing, so that they can have confirmation that they do understand the new work. They are also saved from the fear that going to pieces in any one test or examination will completely spoil their chances of passing. These questions, in general, relate to one specific item of skill or knowledge at a time, so that they will not score zero in a question through ignorance of any one of a number of possible items. It might be thought that students might rest on their laurels if they did well in these tests and not bother about revision for the final examination, but this has not been found to be the case. The main disadvantage at Lancaster has been that 28

students in the first year with three subjects to take have tended to spend more time in the third term on other subjects which will be their major subjects and in which the final examination is much more important relatively.

Final Examination The final examination consists of two papers. The first is to test retention of knowledge and understanding. The second consists of two types of question, multi-facet and essay. In the first a situation is examined from several aspects; in the second the student is allowed to show his knowledge of a topic in his own way. Students of subjects in which description in the English Language is the usual form of expression find themselves more at home with such questions. They are able to use their imagination and give their own choice of example. Examples of such questions for possible use at a school level are given below.

In addition regular weekly assignments are given. These include practice in routine operations with some simple problems which involve going from verbal descriptions to mathematical statements and vice-versa. Such questions can often be done in more than one way and so stimulate discussion of their relative merits. These assignments are intended first of all as part of the process of learning but they are assessed. (20% of the total.) So far as the learning process is concerned, it is probably best that students should discuss these with each other so the assessment is not emphasised.

Integrated Thinking The number of students involved (about 120) makes it difficult to introduce project work so far, but the students will be able to think naturally in mathe- matical terms when they have projects in their own subjects at a later stage. There is a great deal to be said for students at all levels being able to integrate their

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thinking in different fields, and also for their being able to see their teachers co-operating in discussing a situation from different points of view.

In business, industry and research it is common for different skills and interests to be blended together to form a team for developing or investigating fields of activity and solving the problems which arise. It is desirable that such co-operation should start at an early stage in school and that it should take place without any pressures on individuals to "play to the camera" for individual assessment.

The techniques of multiple-choice testing will no doubt be considered in other articles. Some suggested topics for divergent writing are now given.

Topics for Divergent Thinking 1. Compare vulgar and decimal fractions as media for working with quantities, illustrating their advantages and disadvantages in calculation and measurement. 2. A 20-year-old man weighs 80 kg. What will a 30-year-old man weigh? Why can this question not be answered? Give examples of questions which properly involve proportion and others which do not. 3. What are the disadvantages of considering the "average" or "mean" age of a class as the "standard" value. Give examples of situations where the mean is clearly the best "standard" value to take and also examples of situations where other "standard" values would be better. 4. On a car journey a man goes the same distance at 30m.p.h. on ordinary roads as he does along the motorway (at 70 m.p.h.). Explain why the average speed is not the same as if he had gone for equal times at these speeds. 5. Describe three different situations which might lead to the equations

x+y= 7 3x + 4y = 25

In each case make clear whether your equations are strictly true or are approximations. 6. What are the advantages and disadvantages of the following methods of expressing the relationship between 2 variables: (i) A table of figures; (ii) A graph; (iii) An algebraic formula? Use some practical situation to illustrate your points. 7. Why is it necessary to consider negative numbers? Give practical illustrations of situations where they may arise including some which will involve: (i) subtraction of a negative number from a positive

one,; (ii) the multiplication of a negative number by a positive number;

(iii) the division of a negative number by a positive number;

(iv) the multiplication of two negative numbers. 8. Tables are given for the numbers of cows, pigs and sheep slaughtered in England and Wales, Scotland and Northern Ireland for the years 1969, 1970, 1971.

1969 EW S NI

C a b c P d e f S g h i

1970 EW S NI j h I m n o p q r

1971 EW S NI

s t u v w x

y z

Write down six different tables to illustrate different ways of processing these data. Include addition (3 types) subtraction and transposition in your operations.

(The letters may be replaced by actual figures.)

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