Different Ways of Learning MathematicsAuthor(s): Anne ThomasSource: Mathematics in School, Vol. 3, No. 2 (Mar., 1974), pp. 18-19Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211174 .

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Different wa o

learning Mathematics

by Anne Thomas, City of Sheffield College of Education

It is obvious that many different teaching strategies can be used in mathematics lessons in Primary Schools. As a teacher, one must be aware of different methods, so that one can select intelligently the method one considers appropriate to what is being learned, and thus make suitable preparations. I have tried to list different ways of teaching. There is overlap between them and the list is not exhaustive. Some opinions about each are offered and some of the implications concerning preparation for each are drawn. The order is not significant.

1. Direct Teaching Using this method, the teacher dominates the learners, doing a lot of talking and illustrating, perhaps asking questions or encouraging discussion. The teaching may be to the whole class, small groups or individuals.

This method is sometimes appropriate, but like all methods should be used with care. Showing a class how to fill in a Savings Bank withdrawal form might be a good use of the method. This is a factual, right or wrong situation involving specific and circumscribed techniques; the demands on previous experience, under- standing and learning are minimal; the task is not too difficult or too easy for a wide spectrum of ability. On the other hand, to try and teach the technique of sub- tracting in this way would not be effective. To be ready to tackle subtraction sums, a child must have an under- standing of the operation, some knowledge of number bonds, a grasp of the place value system of writing numbers, etc. It is unlikely that many children will reach this point simultaneously. Perhaps more impor- tantly, if children are to understand the technique, they must have the opportunity to handle concrete materials and experiment with them, at their own pace, not at the pace set by the teacher.

If direct teaching is chosen, the teacher will need to make a most careful analysis of the material to be taught, break it up into consecutive teaching points, almost programme it in fact, and give time to the preparation of visual aids.

2. Direct Teaching with Practical Work to Illustrate The teacher still dominates the learners; he has specific objectives, teaches to effect these, but then follows up the direct teaching with work of a practical nature, designed to elucidate the subject matter.

An example would be in the introduction of Metric measures to third or fourth year juniors. The metre and centimetre are introduced and the connection between them explained, using direct teaching. The class are then asked to estimate various lengths, measure them, and find out their errors. In another example, the teacher explains the relationships between variables so that a graph of the relationship is a straight line, and then sets pieces of practical work all of which involve straight line graphs.

This method is frequently used and is often confused with (3) below. It can be a way of introducing new ideas conveniently and efficiently to a number of children, but still allow scope for individual differences. 18

It has the drawbacks of direct teaching, but allows the teacher a chance to remedy these. Its greatest disadvan- tage is that it makes it possible for a child to be able to do the practical work and get the correct results with- out his having attained a conceptual understanding of the material. My own University Physics practical work was entirely of this kind, done correctly but with no understanding at all on my part.

The success of the method depends on the quality and degree of open-endedness of the practical work. For example, the teacher is more likely to discover gaps in children's understanding in the Metric measures example above if he asks them to choose six things of very different lengths, select appropriate units and measuring instruments, estimate, measure and find errors, than if he gives out a carefully typed assignment which has a list of objects with spaces at the side, -m -cm, for the child to fill in.

3. Experimental Work Leading to Pre-determined results The teacher again has specific objectives. He designs a piece of practical work or a range of activities which the children do, from which they are to be led to know- ledge of a particular piece of mathematics. There will usually be need for some discussion and gathering together of results after the practical work has been done.

For example, the aim might be that the children should learn the properties concerning the diagonals of squares and rectangles: (a) the diagonals are equal and bisect each other in both shapes, (b) in the square only, they cut it into four congruent shapes, cut each other at right angles and cut the corners of the square into two equal parts. The children would be given a range of assignments such as: (a) cutting up and rearranging squares and rectangles, (b) making the shapes on nail boards, (c) investigating the symmetries of them, (d) making the shapes with geo-strips and measuring, etc. But for effective and specific learning to take place, there would have to be some discussion, some record- ing of relevant facts. A teacher using this method has a large amount of preparation to do in gathering together suitable assignments, putting these into a form suitable for the children and organizing a class to work in this way. The more specific the aim, the more difficult it becomes to write an assignment card that is open

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enough to allow the learner room to think, yet de- lineated to prevent him from getting side-tracked. The little we know about children's learning at present would suggest that this ought to be a very effective method, and it seems to allow both for the differences in children and the systematization of mathematics.

4. Experimental Work Leading to a Broadening of the Child's Experience

The teacher thinks now in terms of aims rather than objectives, the intention being to give the child experi- ences which will prepare the ground so that later on, the mathematics can be formulated.

Two examples will help to clarify: (a) Five year olds play with balances and a variety of materials such as sand and shells, using spoons, beakers, etc. This is obviously a pre-requisite to any specific work on balance and weighing. (b) Nine year olds make patterns of squares and rect- angles with pegs on board. This will be an experience the children can draw on when thinking about square numbers, primes, factors, the commutative law, sequences and series.

Though the value of this type of work must not be forgotten, it does appear to be carried out to a greater extent than one would.~expect in a subject of such a systematic nature as mathematics. Many children's books and sets of work cards are entirely of this kind. It is easier to make up vague activities than highly specific ones, but teachers must be aware of the need for some schematic development of the subject, even for young children.

5. The Learner does Open-ended Investigations In this learning situation, the teacher allows the child to be a kind of miniature mathematician. The tasks are chosen by the teacher because they might lead to genuinely mathematical activity. The child is allowed to get on with his investigation, following lines that interest him, with intervention from the teacher to suggest further possible developments or to help the child verbalize and symbolize his results. The investiga- tions do not usually involve the handling of materials except at an early stage, and this type of activity is more suitable for older children. Readers of Mathe- matics Teaching will be very familiar with investiga- tions. One example would be to investigate ways of giving instructions as to how to get from A to B on a grid-iron street plan. This can lead to interesting discus- sion of several important mathematical ideas.

6. Topic Work or Projects Work of this kind, where a topic is studied, is under- taken in most Primary Schools. The approach of November brings work on Bonfire Night; a school outing motivates a variety of activities; a pet show starts another line.

Sadly, mathematics is often conspicuous by its absence in this kind of project. There is a wealth of mathematical material in almost any topic and the teacher could use this to interest children in mathe- matics and to show them mathematics in use. It can be very worthwhile to think out beforehand, the mathe- matical opportunities that can arise in a project, so that these are not neglected.

Six ways of learning have been listed without discussion of what might be one of the most important factors in a classroom, the effects that children have on each other's learning. But this is more concerned with how the teacher organizes all the types of learning, which leads to another article!

To find the 0

remainder after

division by 7 without carrying by G. I. Owen, Norwood Technical College, London SE27

Given a positive integer n, it is possible to obtain the remainder when n is divided by 7 as follows: 1. Divide n into 6-digit groups from the right. 2. Treating each of these groups as numbers less than a million,

add them up a column at a time, but write down only the remainder after as many 7s as possible have been subtracted from that column total.

More briefly, use addition modulo 7 in each column. 3. Treating the resulting six digits, in pairs, as three separate

integers (with maximum possible value 66 in each case) write down the remainders after each has been divided by 7.

4. Let the remainders from left to right be x, y, z, then 4x + 2y + z (with maximum value 42) has the same remainder as n.

Examples A. 123 456 789 - 45 67 89

1 23 45 61 35

Remainders x, y, z 3 5 0 4x, 2y, z 12 10 0 -+ (Sum) 22 -+ Remainder 1.

B. 11 223 344 556 677 -* 55 66 77 22 33 44

11 00 22 55

0 1 6

0 2 6 - 8-+ 1.

Those interested in trying out the method could attempt (i) 732 040 964 931 (Ans. 3)

(ii) 1 634 089 211 311 (Ans. 3) (iii) 352 274 908 509 888 (Ans. 0)

Notes (a) It is not necessary to write down the six-digit numbers in columns; the corresponding values can be added in situ. 7, 8, 9 can be treated respectively as 0, 1, 2 during this addition.

Similarly, the values 4x and 2y need not be written down, the value 4x + 2y + z being derived straight from x, y, z.

Abbreviated working for (ii) and (iii) above is shown below; the marks in the top line have been inserted to assist in the location of digits during the addition. (ii) 1'634,089'211,331'

14 53 44 0 4 2- 3

(iii) 352'274,908'509 888' 00 66 64

O 3 1- 0

(b) The method can be used for integers less than a million, as can be verified.

(c) Readers who wonder why the method works may find themselves considering such questions as:

"Does 106k (k = 0, 1, 2, ....) always have remainder 1 when divided by 7?"

"Are there corresponding results for 10104 19

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