TRENDS IN MATHEMATICS

Press, Volume CIII, Issue 30536, 3 September 1964, Page 15

TRENDS IN MATHEMATICS

During the last few years much has been said of changes In mathematics teaching and the content of mathematics programmes in schools. The Canterbury Mathematical Association, along with other similar bodies throughout the country, has been aware of these changes and, wherever possible, have given assistance in formulating curricula. The bulk of the changes envisaged are in the two regions: approach and content. It is this latter which becomes of increasing importance in the future as, with the change to decimal coinage, much more time will become available to the genuine study of mathematics.

The changes in approach are many and varied but in this article space does not permit me to give a sufficiently wide range of examples to give a clear appreciaition of their full significance. Most of us learnt what mathematics we know as a series of operations, each of which was developed independently and in almost total isolation. This is in no way criticising our own teachers or the many first class “traditional” mathematics teachers still in the forefront of the teaching profession. To suggest that these teachers do not teach for understanding would be fatuous. Without their able guidance we could not have reached our present position. But many teachers are not availing themselves of many unifying concepts that have only recently become available at the school level.

Depending on the extent to which we studied mathematics, many of us have never stopped to think that the ordinary method of long multiplication is precisely the method of carrying out a calculation by “practice" or of expanding an algebraic expression of Die type (a+b)(c+d>. Cooildtr three MleuUtUMe !1) 47xk2 11) «7 Kite •« 11/lt- eerb. al) Ux ♦ ?)(V ♦ a) (i) *7 Mk . S 7«» M • t? iu> Sit > tt «(»♦»> (11) <7 eeke et £2 - £l%- 0-0 *7 .o». et 2/- - 67 «!• at £Z/V(ui)(fca7X«r«4, . (Xu . 1) . . 1) - (a» 2 ♦ UyHCMy ♦ i») . a, 2 . kOy .1k V H tan Mlralatla J • 10, y 7 . 100 ee («y.7X*r«a>7w 2 . utr ♦ ik.. 2k I too a w a 10 » Ik . atk Each of these calculations is merely an example of a very important generalisation known to mathematicians for quite a long time as the distributive principle, but how many of us who learnt how to carry out even the first two of these calculations, or much more difficult calculations of the same type, ever learnt that the distributive principle ax(b+c) “ aXb + aXc where a, b, and c represent numbers existed as a universally true statement. One of the alms then of the present trend in mathematics teaching is to bring such principles to the explicit level so that the child, through examples, discovers these principles, learns to use and generalise them, and hence understands the many operations based on them. In carrying out a simple multiplication in fractions where simplification, or cancellation, is involved how many of us realise that we are using the very important idea ;that multiplication by 1 leaves the value of the number unchanged. For example: lhat is | of 7r « 2 x 15 3x2 5x6 “ ixT 5 6 ’l X Z «£xl «5 1

(There are many alternatives here.) We are well aware that to ask for such a method of calculation on every occasion would certainly discourage and bore even the most interested pupil but the principle of multiplication by 1 is so fundamental, and at a more advanced level so important, that as a principle it should be taught and understood, and when understood become a genuine aid to more accurate calculation.

Another topic, which like the development of understanding of basic mathematical principles cannot be considered modern, is the use of numbers to bases other than 10, and the resulting “modulo” arithmetics. When we count in the ordinary way we group numbers in powers erf ten as units, tens, hundreds, thousands etc. Mathematically, it is just as convenient to group in powers of any number and of course early computers being based on electric circuits depended on two positions, on and off, and hence used base two. If we recall that

2 2 -4 2 5 = 8 3 16 etc* then the number 23 may be written as IXI6 + OXB + IX4 + IX2 + 1 and so in base 2 as 10111. The same number for base 3 could be arranged as 2X9+ IX3+ 2 in base 3 as 212. By asking children to write numbers in other bases we are not only testing their tables and manipulative skill but ensuring a better understanding of place value in the decimal system. In “modulo” arithmetics we are interested only in the remainder after division by the base number. So that in modulo 7 the decimal number 16 would be represented by 2 just as would the decimal number 44 because 44= 6X7 + 2. Hence in answer to the question “What day of the week will it be in 44days time?’ we can say immediately Saturday i.e. 2 days after today, Thursday. Modulo arithmetics and other bases occur far more frequently in our daily lives than we realise. The relating of numbers to points on a numberline has been introduced very early in the education of young children with considerable success.

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This gives the child a far better understanding of positive and negative numbers, or numbers and their opposites, and when associated with fundamental principles such as the distributive principle leads to a clear understanding of why (-3) X (-4)=12 and not just the blind obedience to a rule. Certainly the child must still have plenty of drill in number work, but understanding and development skill must go hand in ment of skill must go hand in hand.

Traditionally in school we have always been concerned only with conditions of equality. In fact, of course, relatively few things in this world are “equal” much as we seek this nebulous state. With the present trend comes the study of things which are unequal, or inequalities. The cost of one brand of tea is “greater than” the cost of another; one man’s salary “exceeds” another’s by at least £2OO, to quote but two examples. One number is of course greater than another if to the second number we must add a positive number to get the first: e.g., 5 is greater than 3 because we must add 2 to 3 to obtain 5 and 2 is a positive number. The study of inequalities in the numerical, algebraic and graphical situations is now receiving much attention and opens to the student a wealth of genuine practical problems which may lead him to a study of linear programming, a fundamental process in modern industrial economic planning. Development of number with the aid of the number line, and the many structured materials which may be used at all levels with it, gives an excellent introduction to graph work which at last has become an integral part of any mathematics course and not something just tagged on the end.

Probably the most far reaching concept being introduced in school programmes is the idea of a “set,” and the language, implications and understanding that go with this

standing that go with this idea. This concept is one that even the youngest child has, e.g. the set which consists of members of his family; and set of toys he owns, or shares with his sister; the set of food he eats, the set of foods he rejects and so on. The language associated with sets is of such immense use that the experienced teacher finds it difficult not to use these ideas in almost every mathematical situation. The set of whole numbers, the set of prime numbers, the set of multiples of two, the set of multiples of three, the set of multiples of both 2 and 3. Let us write these down: The set of multiples of 2: A=(2, 4,6, 8, 10, 12, 14, 16, 18, 20 ...) The set of multiples of 3: B=(3, 6,9,12,15,18, 21...) The set of multiples of 2 and 3: C=(6, 12, 18 . . .) This set C, which contains those numbers which are multiples of both 2 and 3, is known as the intersection set of sets A and B and contains only those numbers common to both these sets. If all the numbers in either A or B or in both were required we would describe this as the union of sets A and B. These ideas of union and intersection have far reaching consequences whether we be thinking of sets of numbers, of points on a line, of regions of space, of probability, of logic, or electrical circuits. Associated with sets we normally think of Venn diagrams which generally consists of two or more geometrical objects used to represent sets. Our sets of multiples of two and three could be represented thus: SaltlylM Of Tw

The region where the two rectangles intersect containing multiples of 6. As an illustration of a problem which may be solved with the aid of a Venn diagram let us take as an example: A soap manufacturer producing two brands of soap “Aromatic,” and “Beauty” decided to take a survey of 1000 homes to find out which soap had the greatest appeal. From the 1000 visited he obtained the following information: 640 used "Aromatic,” 570 used “Beauty,” and 280 used neither of these brands. He asked the canvasser to explain these figures as he really wanted to know how many used “Aromatic” only, and how many used “Beauty” only. The diagram his agent drew was:

The rectangle represents the 1000 homes visited and the two circles the users of soap “Aromatic” and “Beauty.” The region outside the circles but within the rectangle represents those who use neither soap and so represents 280 homes. This means that A and B combined can only represent 720 homes. If 640 must be in A then 80 must be in the part of B not in A. If 570 are in B, then 150 must be in that part of A not also in B. This leaves 480 homes using both soaps. The final diagram diagram could appear as:

Here, then, are just a few examples of how mathematics teaching is changing. If you are one of the many parents who is finding that you can no longer help young hopeful with his mathematics homework, please do not push it to one side and say it’s all daft just because it’s different; remember his teacher is endeavouring to make use of some firmly based mathematical concepts in order to not only get your child calculating but thinking at the same time, and if you need help then his teacher, the Canterbury Mathematical Association, or W.E.A., who have a mathematics course in their annual programme, will be only too willing to assist.