Maths for 10-Year-Olds. How Is It Organised? What Do Teachers Think about Maths?Author(s): Murray WardSource: Mathematics in School, Vol. 3, No. 3 (May, 1974), pp. 17-19Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211203 .

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Maths for 10-year-olds. How is it organised? What do teachers think about maths? by Murray Ward, Schools Council Primary Mathematics Project

Previous articles (January and March, 1974) have described how a sample of 10 year old pupils' performance on a selection of maths questions related to their teachers' judgement of the importance of each question. In the survey information was also gathered about how mathematics was organised. Further, teachers were asked about their attitudes to the subject.

School organisation Although only two schools of the 16 schools in the survey had streamed classes, in three others children were grouped by ability for maths. Four schools reported some team teaching, and in two schools children were taken from their classes for remedial maths.

Schemes of work All but three of the schools had a written scheme of work. These were based on local courses, local working parties, or one of Miss Biggs' courses. Books quoted as sources included:

Nuffield Guides (Chambers/Murray) Mathematics in Primary Schools (Schools Council Curriculum Bulletin No. 1, HMSO) Alpha and Beta (Schofield and Sims) Flavell and Wakelam: Primary Mathematics (Methuen) Sealey: The Creative Use of Mathematics (Blackwell) Fletcher: "Mathematics for Schools" (Addison- Wesley)

One headteacher reported that his scheme was based finally on "the collective views of the teaching staff". Another added "I find it needs constant revision and adaption and is not meant to be rigid but a source of guidance and ideas"

Tests Some schools gave maths tests to the whole class twice a term, others never; the average was about twice a year. Tests mentioned as being used by more than one school were NFER, Schonell and Bristol ("too expensive"), while half the schools used home-made ones. Tests to individual children are infrequent.

Metrication All the schools seemed to have taken metrication in their stride. "Speeded the whole primary course by at least one year". "In general it has made teaching much easier and has stimulated many children who found maths a difficult subject".

Classroom organisation Do all the children in a class work at maths at the same time? Two schools reported "Always", two "Occasion- ally", the majority "Usually". One headteacher added: "This depends on the classteacher's approach. Teachers are encouraged but not forced to work on integrated lines".

Only one school had a special room for mathemratics; one other had a central pool of maths apparatus. Two headteachers added that the dining area is used as an overflow for practical and group work.

The time spent on mathematics each day varied between 40 minutes and an hour with most schools near the top end of the range. This time was divided between individual, group and class work. Only one teacher reported "no class work at all"; usually the division was fairly even.

Less than About More than No None half half half All data

Inrdividual 12 work rou 20 3 2 work

Class work

The numbers in each diagram refer to classes. In all there were 25 classes from 16 schools.

The main sources of mathematics work were the blackboard, cards (either printed or home-made), and textbooks. There was no teacher who did without text- books; a few did not use work-cards and only one reported not using the blackboard. Over half the teachers said they make their own work-cards. If this is a reflection of what is happening nationally then a staggering number of teacher-hours must be expended on this task. Three teachers mentioned topic work and science as additional sources of maths.

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Less than About More than No None half half half All data

Blackboard 22

Workcards made in 2 17 school

Printed 14 3 workcards

Pupils textbooks4

Other L

Do children help each other with their mathematics in school?

Never Occasionally Usually Always

1 9

Does a child mark his own work? Never Occasionally Usually Always No data

20

Is all the work checked by the teacher? No Yes

6 1

Are mistakes corrected? Never Occasionally Usually Always No data

i 1

Mistakes Teachers described a large number of topics where mistakes commonly occur. (1) Subtraction (the most frequently cited source of

mistakes) borrowing taking the top from the bottom decomposition across more than one column

(2) Zeros (e.g. 8 - 0 = 0; 8 x 0 = 8) (3) Carrying (4) Units, especially square units (5) Place value

not writing figures in columns position of decimal point

(6) Difficulties with 2 pence (7) Not using apparatus correctly (e.g. ruler) (8) Weakness in basic number bonds, including tables.

"Lack of understanding" and "carelessness" were both mentioned in connection with mistakes, as was "a lack of self-critical ability".

Tables All the teachers practised tables using a wide variety of means,: "quick fire practice" clock face with number of table in centre games and puzzles, number bingo, Fizz-Buzz equaliser practice cards "children learn tables for homework" number squares and patterns number graphs "award stars for table knowledge and quickness" counting in 2's, 3's, etc. children test each other practical use missing numbers factors 18

One teacher wrote: "Daily oral work. This is done when clearing up has finished. For example, when waiting for the bell."

Apparatus In general classes seemed well-equipped with apparatus; how much it is used varied from a few times a month to daily.

A few times Once or twice Most Eveiy Never a month a week days (lay

Only one person admitted to having apparatus in the back of the cupboard which was not found useful. This was Dienes algebraic material. "Too much teacher help required and a little too complicated for our children".

Reservations Both heads and classteachers were asked if they had any reservations about aspects of primary mathematics as taught today. Over a quarter were concerned about computation: "Schools go overboard in the acceptance of new ideas and leave basic computation etc. without strength and structure". "The experienced teachers who taught computation are able to ensure that certain processes are learned. Teachers without this experience often miss the point of the activity."

A lack of progression and continuity worried teachers too: "A tendency to drift around in circles if not careful." "The child can meander aimlessly through a lot of work and draw few conclusions, learning little."

Several teachers were doubtful about some of the new topics in mathematics: "I believe some topics such as set theory, vectors, matrices need approaching with caution". "Teachers sometimes suffer from an excess of enthusiasm ".

Other reservations expressed more than once were: (1) bright children not fully stretched, and (2) lack of liaison between schools.

Strong points Fun and enjoyment were the keynote of the teachers' comments on the strong points of primary mathematics as taught today. "Maths is now lively, interesting and understood better than it has ever been". "Maths can be so colourful, so full of physical activity, amusing, fascinating and stimulating". "Making children think for themselves" was another frequent comment, linked with the emphasis on under- standing instead of rote learning. "No longer are children merely memory-boxes".

In a similar vein many of the teachers wrote of discovery and practical work, with concepts developing from the child's own experience. "The awakening to the fact that maths is all round us and used every day by everybody and not only in computation". "Children learning by experience backed up by practice, e.g. multiplication, know what they are doing and why ".

Help needed The help that teachers stated they would find most useful is more in-service courses, particularly those dealing with new apparatus and specific topics. "In-service training is usually a voluntary affair attended only by the keen, the converted or the

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ambitious. Therefore any enlightenment must come from a 'school-based' situation."

Opportunities to meet teachers from other schools would be welcome; help is also looked for from more experienced staff within the school and from local mathematical advisers.

Need for guidance was felt in choosing the contents of the maths curriculum. "A positive line to follow which nevertheless gives freedom to broaden and develop". "A scheme of work that is flexible but enables the teacher to know what the average child in the class would be capable of attaining during the year".

As mentioned in the previous article, this was not a random sample so we cannot generalise from these results. However I think it likely that, for instance, the sorts of mistakes listed are those which most teachers of this age group encounter. More than one person has commented that if the same enquiry had been made

10 or even 20 years ago, the list of common mistakes would have been much the same.

Some ten years ago a big study of primary mathe- matics in England and Wales divided schools up accord- ing to the type of structural apparatus they used. Thus there were Cuisenaire schools, Dienes schools and so on. Clearly this would not be appropriate today, because schools quite rightly use a variety of methods and apparatus to suit the needs of each boy and girl.

The content of the mathematics lesson is finally decided at teacher level and it is hoped that this pro- ject will provide information about some of the options available and about the standards which might be expected, so that in the light of local conditions and by discussion within a school each teacher is helped to come to his own decision.

Acknowledgement I am very grateful to the teachers and pupils who took part in the survey, and especially to the teachers who added many useful comments to their questionnaires.

Numbers out of Logi-blocs by Michael Holt

Select a set of logic-blocks in, say, 2 colours and 3 shapes. For example, like this:

Select your own shape order. Mine, as you see, is

as shown by the one-way arrows. Each arrow stands for a single change of shape in this pattern. The double-headed arrows are for two-way changes of colour.

A teacher once asked me to set this pattern to numbers. (Number tunes have, I suspect, wider appeal than logical harmonies.) Three changes and two changes coming back on themselves. I pondered; like the snake in the ancient symbol biting its own tail. This suggested 2x3 = 6 on a 6-hour clock, what mathematicians call modulo 6 arithmetic. For instance, on this clock, 5+3 = 2.

o

Note: On this 6-hour 5

clock, 6 o'clock is shown as zero-like zero hours on a real 24-hour clock. 2

3

Which is no more puzzling than you or I looking at our watch at 11 a.m. and noting that in 3 hours time it will be 2 p.m.

A sheer guess led me to call the one-way arrows +4 (on the 6-hour clock, of course). The two-way arrows must be +3 (clock 6) since +3 then +3 must bring you back to where you started on a 6-hour clock.

Now call the round block zero. This is at your choice. After all, you've got to start somewhere! Then via the one-way arrow O becomes 4 because

+4

And 0 becomes 3 because

Now follow round the inner triangle. The inner square becomes 1 because +4 . (You can check this on the 6-hour clock: 3+4 = 1.) Now for the "lock-up" test. Does the two-way arrow change 1 into 4 and back? Well, 1+3 = 4 alright-this works in ordinary addition- and, wonder of wonders, going the other way 4+3 = 1 on a 6-hour clock.

The completed network is:

0

2 4

A better known numerical*version (or isomorphism, as they say) is that of the Klein group. In it each and every change (arrow) is double-headed or two-way; so two in a row bring you back to where you started from.

x3 1 < >

3

x5 "I

,

5 < 7

All the arrows stand for multiplying on an 8-hour clock (modulo 8). Multiply 1 by 3 and you get 3 via -. Multiply 3 by 3 (clock 8) and you get back to 1 again. The same happens for starting at 1 and multiplying by 5 (clock 8) via >.

What about multiplying by 3 then 5 (clock 8). 3x5 = 15 which is 7 on an 8-hour clock. Multiply 7 by 7 on this clock and you get via ---- to 49 or 1 (clock 8). Back again to where you started. The three changes lock up into the square pattern. Of course, you need another changer-the "times 1" which leaves you where you are. For the interested reader this is all part of ths topic called groups

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