Educational TechnologyAuthor(s): John ClarkeSource: Mathematics in School, Vol. 1, No. 4 (May, 1972), pp. 21-22Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30210774 .

Accessed: 22/04/2014 18:03

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.

http://www.jstor.org

This content downloaded from 185.33.143.13 on Tue, 22 Apr 2014 18:03:58 PMAll use subject to JSTOR Terms and Conditions

Educational Technology by John Clarke

Earlier articles in this series have been concerned with a working definition of educational technology and with the need for making a clear statement of aims and objectives as an integral part of the planning of a courses In this article we begin to show how these notions were employed in the design and construction of a course for primary school childrens

Accent on Place Value The Chinese proverb so often quoted in the context of the Nuffield Science and Mathematics Projects, I hear and I forget - I see and I remember - I do and I understand, provided the rationale for the develop- ment of a mathematical course for primary school children developed in Dundees About four years ago we were asked to investigate the possibility of devising an in-service course for teachers which would give them the security necessary to provide an approach to the learning of mathematics which did not rely for its control on the regular use of text-bookss The accent was to be laid on practices designed to give children a thorough grasp of the principle of place values

In the College of Education we had been working with the ideas promoted by the Nuffield Mathematics team for some time and, prior to this, in my own school, I had been using an approach based on Dienes MAB and AEM materials for some seven yearss From the experiences gained over this long period with both children and teachers I was convinced that children could benefit considerably from the use of concrete materials but that, for such an approach to be really successful, the teachers must appreciate the psycho- logical and mathematical background to the activities they were providings Equally, the importance of each child learning at his own rate as far as was possible and of the teacher stepping into an activity at the appropriate time to help to provide mathematical insight had to be stressed and the teachers prepared for what might be to them radical changes in teaching methods

The limited outline given above of the request made to us provides an idea of the general aim we had before us for the construction of the courses The course content was analysed; a procedure giving us the basic concepts, skills and techniques for which we would need to specify the behavioural objectives, ises what the child would be able to do as a result of mastering each of the areas of learnings In turn, the objectives provided the framework for the construction of assessment procedures to be used to measure progress and to provide diagnostic evidence of a child's difficulties in any aspect of the courses

From the activities described it quickly became evident that, as an understanding of the principle of place value subsumes many concepts and skills, a build-up of these latter elements in a child's repertoire was the first essentials On these and from these an understanding of place value would develop which could be applied to decimal notation used to record any numerical value, measurement or calculation in any of the four basic arithmetical operationss Printed below is a section of the flow chart produced which, to me, clarified the progression of the concepts to which consideration had to be givens

It will be noticed that the chart differs in some respects from the "maps" provided for us by Piaget but this result was based on my experience with

children and served as the model for the structured course to be" produceds

Findings of Piaget Long appreciation of the methods and recommenda- tions of Profs Dienes and of the Nuffield team, both based on the findings of Piaget, provided the justifica- tion for all to depend on the basic notion of a sets For children to use such an idea and to gain from it the concepts of inequalities and equalities, one-to-one correspondence and cardinal and ordinal number, essential experiences had to be built into the courses This was done through all types of activities which might, even remotely, contribute to the background necessary for the development of the ideass From the outset suggestions were made for the use of concrete materials, not structured in any way themselves, to develop an ability to classify, leading in turn through the listed concepts above to an understanding of cardinal value of numbers

The work structure and the notion of structuring may well carry emotional overtoness To many, the idea is an anathema suggesting as it does a rigorous control of work and exercises leading to regimented and convergent thinking unlikely to contribute much to the development of a mathematicians Undoubtedly, programmed learning in its stimulus/response mode used to teach mathematics in a step-by-step fashion, could well attract such criticisms On the other hand, understanding within a scientific subject such as mathematics which relies so heavily on an ability to see and appreciate relationships, to reason logically and to hypothesize, must in my view be built-up by a structured approach which consolidates past experiences and develops techniques, skills and cogni- tive knowledge both horizontally across related aspects of learning and vertically within a single aspects

To be more specific within the case I am presenting, I will describe in some detail my analysis of the FIGURE 1

experience with materials and events

s has iis appreciation of attributes graphica

Js

representation classification

SETS

inequalities t equalities

ordering one-to-one 1 correspondence differece

ss ordinal number

measures

sss jcardinal n number

currency

operations

fractions reversibility

,1i number base

r tio -

proportion place value

[STROsTUR_____s /sss commutative r

_1/ associative lawsroots distributive

closure -

identity elements -/ inverse -

21

This content downloaded from 185.33.143.13 on Tue, 22 Apr 2014 18:03:58 PMAll use subject to JSTOR Terms and Conditions

activities which lead to an ability to measure. First, what are the objectives? It is difficult to generalize for all aspects of measurement but an attempt might be made in these terms. Given that the pupil is provided with a suitable measuring instrument (specify in each case), he should be able to measure the... (specify each attribute), calculate if necessary and record the measurement using decimal notation with a given degree of accuracy (specify in each case). A specific objective might be: Given that the pupil is provided with a centimetre tape-measure, he should be able to measure the height of any person and record, using decimal notation, the measurement to the nearest centimetre in centimetres or in metres. This objective makes evident the different skills required. The child must have a knowledge of: (a) the measures - the centimetre and the metre; (b) the nature of a measuring tape; (c) the term height as applied to a person. He must be able to: (d) use decimal notation (this implies a grasp of many concepts and principles including a knowledge of fractional parts, i.e. the relationship between the centimetre and the metre); (e) round off to the nearest centimetre. Clearly, this objective assumes that development for the child will be vertical (within measurement of linear dimensions) and horizontal (relating quantitative measurement to its recording). The steps for this development were devised as follows. Generalizing for all measurement, five basic steps were outlined.: 1. An understanding of the property (attribute) to be measured (e.g. length, area, volume, etc.). 2. An understanding of the nature of a measuring unit (e.g. a square centimetre, a kilometre, a kilogramme, etc.). 3. An ability to select a measure appropriate to the task in hand and an ability to estimate (these are mutually reinforcing - having to estimate established a suitable measure through the logic of using the smallest numbers unless practice decrees otherwise). 4. An understanding of a main measure (again relative to the task in hand) and its fractional parts (a centimetre can be considered as either a part of a metre or a measure in its own right). 5. An ability to use the number system to record and to calculate measurements with a given degree of accuracy.

In a later article these steps will be described in detail. For this contribution I hope that they will serve to illustrate the method of approach used to provide the content for the children's course; the nature of the activities, their presentation and control; the objectives to be met and the forms of assessment; and, finally, the content of the course required for the preparation of the teachers.

In the first year fourteen teachers attended the course, one from each school taking part. This number increased to twenty-two in the second year and evening classes also were organised to encourage a large section of the staff of any school to attend. Authorities other than Dundee joined in and now it is probable that between two and three hundred teachers have been involved with at least some aspects of the approach together with some two to three thousand children. It seems certain that many of these teachers and the children have gained a new appreciation of and insight into mathematics and the fascination it can offer as an intellectual pursuit. An article of this length provides little opportunity to expand in detail the different aspects but an attempt will be made to do so in future issues. 22

Arithmetic:

by Edith E. Biggs

Teacher's Initiative Where have these changes originated? The most interesting feature - distinctive to the British educational system - is that the changes have been initiated by teachers themselves. For many years the emphasis in the teaching of young children has been on the provision of first-hand experience. Of course the materials for this experience have been planned by the teacher who observes what the children do and asks them questions to help their learning. This process is not unstructured and, with few exceptions, it is carefully planned by the teacher, sometimes in consultation with the children themselves. Indeed, discussion between teacher and children - or children and their peers - plays an essential part in learning through first-hand experience and in the development of appropriate vocabulary.

At the infant stage, and often at the junior stage, too, there is frequently little attempt to differentiate "subjects". The teacher's planning includes careful recording for each child to ensure that a balance is achieved between different aspects of the work. This planning and subsequent recording makes stringent demands on teachers but is basic to the success of this method. These changes, at first confined to infant schools (though they are by no means universal at that stage even today) have, particularly during the past twelve years, affected many junior schools - perhaps up to 50 per cent of these.

But it is largely during the past 12 years that the principle of learning through investigation has been applied to mathematics. This has been a period of extensive experiment during which teachers have tried many ideas suggested by courses (run by various providers), by the Nuffield Project or by new textbooks. There has been a great deal of misunderstanding, particularly, perhaps, about the learning of arithmetic. Does a child need to know his multiplication tables as we used to know them? Should we teach him how to subtract, multiply and divide? Should we give him practice in written calculations? If so, how much practice does he need? Is it better that he should enjoy mathematics and be able to reason mathematically (as shown by the way he tackles a problem) even if he cannot do the calculation or that he should be efficient at calculating? No wonder that many teachers, even experienced ones, have been confused about their aims and find it difficult to determine

priorities.

Personal View I want to express a personal point of view about arithmetic, which I hope will help teachers to make some decisions about these important questions. I am convinced that the basic point to keep in mind at all stages is that children should be given opportunities to think for themselves, even in arithmetic. For this reason I should not want to teach them methods of calculation but, through appropriate experience, to help them to devise methods of calculation for themselves. Of course it is necessary to assist them, by questioning, to make these methods as efficient as possible. But it is important that each child should know that he has taken part in the development of the method, that is, in fact, his own. It is also important for his mathematical well-being that he should be aware that mathematics is essentially concerned with pattern - in number as well as in shape and that these patterns occur in nature as well as in man-made forms. Finally, he needs to have confidence in his ability to complete a problem - and since most problems

This content downloaded from 185.33.143.13 on Tue, 22 Apr 2014 18:03:58 PMAll use subject to JSTOR Terms and Conditions