Should We Say Times?Author(s): Julie AnghileriSource: Mathematics in School, Vol. 14, No. 3 (May, 1985), pp. 24-26Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214000 .

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should should

I,

times?3

by Julie

Anghileri Froebel College

(Roehampton Institute) London

The symbolic expression "3 x 4" is interpreted by adults and children in a variety of ways. A recent, small-scale survey revealed that very few people use the words "3 multiplied by 4". Most interpret the symbols "3 x 4" as "3 times 4" or simply "3 fours" and among children the expression "3 lots of 4" is also found.

Yet, a survey of currently popular primary school math- ematics texts will show that some consideration has been given to the interpretation of the symbol "x" and the majority specify that since the operation in question is multiplication, the correct interpretation is "multiplied by"

School Texts In Mathematics for Schools, Level I, (1979),' multipli- cation is introduced through the idea of "sets of" objects and associates the idea of combining equivalent sets with re- peated addition.

"2(3) can be written as 3)+ 3 > 6" (Level I, Teachers' Resource Book, p. 199).

In Level II when the multiplication symbol is introduced we find:

"3 sets of 6 cakes can be written as 3(6) or 6 x 3" (Level II Teachers' Resource Book, 182, p. 33).2

The pupil's work book introduced the symbol as follows:

3(4)=+ 4+4+4

"We say a set of 4 buttons 3 times can be written as 4 multiplied by 3. We write 4 x 3 = 12" (Work Book I, Section 5, p. 38).

In the Teachers' Resource Book, it is noted that "the numbers are reversed in this notation, so it is useful to use an alternative phrase, 'a set of four sweets three times'." (Level II Teachers' Resource Book 1 & 2, p. 29).

In Nuffield Maths 5-11 (1980),3 multiplication is again introduced as repeated addition through the "intermediary form of notation: 3(4) meaning '3 sets of 4'." (Nuffield Maths 3, Teachers' Handbook, p. 58.)

In Pupil's Book 3 we find the multiplication symbol introduced without use of the word "times", as follows:

5 sets of 2 5(2) 10.

5 and 2 are the factors of 10.

10 is the product of 2 and 5 and we write 2 x 5 = 10"

(Nuffield Maths 3, Pupil's Book 3, p. 43).

In the Teachers' Handbook, we are told, "when the child fully understands the significance of 5(2)= 10 he can then be introduced to the notation 2 x 5 = 10, "two multiplied by five equals ten". (Nuffield Maths 3, Teachers Handbook, p. 58). In Ginn Maths., (1983)4, the Teachers' Resource Book warns that "multiplied by" is often replaced by "times". However, this term is not strictly correct and it is inconsis- tent with the use of + and -." (Ginn Mathematics, Teachers' Resource Book, Level 3, p. 43).

When introducing the children to multiplication they are required to set out three groups of two counters and add to find the answer. "They should read 2x3=6 as 'two multiplied by three equals six'." (Teachers' Resource Book, Level 3, p. 44).

In the childrens' workbooks the word times is not used, instead we find:

"2+2+2+2+2= 10. 5 twos= 10.

2 x 5= 10."

(Level 3, Workbook 2, p. 15).

In S.P.M.G., Primary mathematics (1975)s however, the Teachers' Manual states that "The multiplication sign is read as 'times' and practice is given in terms of this sign".

24 Mathematics in School, May 1985

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(S.P.M.G., Primary Mathematics, Stage 1, Teachers' Notes, p. 40).

In contrast to the texts above, in S.P.M.G.:

"2 x 5 = 5 + 5"

(Stage I, Workbook 4, p. 3).

Indeed the Teachers' Notes specify:

"Fives threes is seen to be five sets each set containing 3 objects:

This is described as 5 x 3 (read as 'five times three').

3 x4=4+4+4= 12"

(Stage I, Teacher's Notes, p. 40).

With these different interpretations of the multiplication symbol, teachers should exert caution when working with several texts and careful consideration should be given when children change teachers or change texts. There also appears to be a discrepancy between the way the expression "3 x 4" ought to be read and the way it actually is read.

The Meaning of Multiplication What then is meant by "3 x 4" and how should it be introduced to children?

Hervey (1966)6 notes that "in the seventh book of Euclid's Elements multiplication is defined as follows:

'One number is said to multiply another when the number multiplied is so often added to itself, as there are units in the number multiplying, and another number is produced'." Thus multiplication is defined using equal addends and "3 x 4" means 3 + + 3 + 3+3.

This definition satisfied mathematicians for many cen- turies but after the work of Georg Cantor, a definition based on sets was formulated which is independent of addition.

If A= {a,, a2} and B=

{b,, b2, b3} are two finite sets,

the cartesian product

Ax B= {(aj, bl), (a, b2), (a,, b3). (a2, bi), (a2, b2), (a2, b3)} is the set of all ordered pairs that may be formed whose first entry is an element of A and whose second entry is an element of B. Now the number of elements in A x B is the product of 2 and 3, the numbers of elements in A and B respectively. This may be illustrated by the following physical situation:

Anne has two skirts, a brown one and a black one. She has 3 different coloured blouses, a white one, a yellow one and a pink one. How many different ways could she choose to dress in the morning, i.e., how many different colour combinations of skirts and blouses could she make?

Neither of these definitions suggests the use of the word "times".

There is no doubt, however, that the word "times" has been used for generations and "times tables" are still learned by many children. The word "times" itself appears to be interpreted in more than one way. The phrase "3 times 4" is understood by some to mean "3 times ... 4", that is, 3 lots of 4. By others it is understood to mean "3 ... times 4" as in the four times table. This second interpretation would be consistent with the verb "times" where 3 is

"timesed by" 4. Indeed, the words "timesed by" are used by some children.

Does it matter if "3 x 4" is variously read as "3 multiplied by 4", "3 times 4", "3 fours", or "3 lots of 4"?, Does it matter that the phrase "3 times 4" has different interpreta- tions put on it? Will it affect children if teachers and texts are inconsistent in the interpretation made?

There is an important distinction between these different interpretations. On one hand "3 times 4" and "3 fours" usually relate to 3 sets of 4 objects and are consistent with "3 lots of 4". On the other hand,,'3 multiplied by 4" relates to 4 sets of 3 objects and is consistent with "4 lots of 3". For young children, 3 lots of 4 and 4 lots of 3 are fundamentally different. They think in concrete terms where 3 children each having 4 sweets are luckier than 4 children each having 3 sweets although the total number of sweets is the same. In an adult world, 3 pieces of material of 2 metres length are not the same as 2 pieces of material of 3 metres length.

In the Classroom It is well accepted that young children learn best through experience. By actively working with materials, the lan- guage for discussion is acquired and an understanding of the mathematical concept will eventually grow.

Activity

Language]

Symbolism

So what are the activities that embody the concept of multiplication? There are several different aspects of the operation, each associated with different situations and each involving particular language.

Some of the various aspects are the following:

(i) equal steps on a number line: exemplified by stepping across stepping stones "2 at a time" or climbing steps "3 at a time". This aspect concerns the ability to move an object along a calibrated scale so that each displacement has the same magnitude,

(ii) the array: in which multiplication, appears in a visual form as a pattern of "rows" and "columns". This aspect concerns the ability to perceive an array as made up of discrete rows or as discrete columns,

(iii) equal groupings: in which several sets appear with the same number of objects "in each" set. This aspect concerns the ability to construct a sequence of sets where the structure of each set is clearly seen to be the same as each previous set,

(iv) scale factor (multiplying factor): in which one object may be referred to as "3 times as big" as another, or where for example, one child has four sweets and another has "3 times as many". This aspect does not concern repeated sets but an en- largement which may apply to either discrete objects or a continuous medium,

(v) allocation (rate): perhaps, even the "sharing" aspect of multiplication exemplified when 4 children are giving 3 sweets "each". This aspect concerns the ability to construct a many to one matching in which equal sets (portions) are matched using a tally of objects (owners),

Mathematics in School, May 1985 25

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(vi) cartesian product: sometimes called the "partner- ing" aspect as illustrated earlier by the girl Anne and her different coloured skirts and blouses. This as- pect requires the ability to make a many-to-many matching between two distinct sets in which "each" element of one set corresponds to "every" element of the other set.

As may be seen, some aspects require a one-to-many matching (e.g. scale factor and allocation) whilst others require a many-to-many matching (e.g. cartesian product).

Some of these aspects are easier for children to under- stand, Brown (1981)7 concluded that the rate aspect was easier than the cartesian product aspect perhaps because it is more easily identified as "repeated addition" whereas with the cartesian product model there is more difficulty in seeing how to co-ordinate the two numbers.

Hervey (1966) compared two types of multiplication word problems and concluded that equal addends word problems "were significantly less difficult than the cartesian product problems for children".

Tests based on the different aspects have been devised and trials with 4-8 years olds suggest that children find a problem requiring a one-to-many matching easier than one based on the scale factor aspect of multiplication. Both of these aspects appear easier than the cartesian product which requires a many-to-many matching. Difficulties arose for many children in these tests with the language used for multiplication. When asked to make a "pattern stick" of unifix cubes using five different colours and three cubes of each colour, a number of children were unable to use both the "five" and the "three" correctly. Some produced "pattern sticks" with 3 cubes of each colour but only 3 different colours whilst others produced a stick of 5 differ- ent coloured cubes to which they added 3 more cubes of a single colour. They certainly found this activity more difficult than allocating 3 counters to each of 5 conkers.

Further test items have been used to analyse the different strategies children use to solve a variety of multiplication problems based on the above aspects. The strategies used vary, not only according to the age of the child but also within each age band according to the ability of the child. Among the strategies used were the following:

(i) the use of concrete materials to model a given situation;

(ii) the use of counting in groups, often using fingers to represent these groups and sometimes using fingers as a tally of the groups already counted;

(iii) the reciting of number patterns e.g., "3, 6, 9, 12, 15 "

(iv) the direct application of a known multiplication fact.

Most of these aspects are fundamental to mathematics and science work that will be developed in later school years. For example, a child who is unable to recognise an array of squares as a multiplication situation may well have difficulty calculating areas and volumes and if the scale factor aspect is not well understood, magnifications and reductions may present difficulties in later school work.

Children do appear to lack an adequate understanding of multiplication. Reporting on the secondary school children's understanding of arithmetic, Margaret Brown (1977)" states:

"In general we feel that the results show that teachers of first year secondary school children should not, except in the case of very bright children, take their understanding of multiplica- tion and division for granted".

Perhaps their lack of understanding is due, in part, to inconsistencies in the interpretation of the symbol " x ". As we have seen, in many currently popular primary school texts, young children are expected to interpret "3 x 4" as "3 multiplied by 4" or "3+3+3+3". However, when algebraic notation is used in secondary schools, the symbols "2n" are used as a shorthand for "2 x n" and are interpreted as "2 lots of n" or "n+n" and not as "n lots of 2", or "2 + 2 + 2 ...+ 2" as we might expect.

Perhaps many more different activities in multiplication are necessary to give children sufficient experience of the language and concepts before the symbolic representation is introduced. Through these activities a "natural" language would surely lead to the use of "times" as appropriate for most occurrences of the operation of multiplication.

The mathematical language would well be left until the children understand the operation in the context of mean- ingful problems.

References 1. (1979) Mathematics in School, Level 1, Teachers' Resource Book,

Addison-Wesley. 2. (1980) Mathematics in School, Level II, Teachers' Resource Book 1 and

2, Addison-Wesley. 3. (1980) Nuffield Maths 3, Teachers' Handbook & Pupils' Book.

Longman. 4. (1983) Ginn Mathematics, Level 3, Teachers' Resource Book & Work-

book 2. Ginn & Co. 5. (1975) Scottish Primary Maths Group, Primary Mathematics, Stage 1.

Teachers' Notes, Heinemann. 6. Hervey, M. (1966) Childrens' responses to two types of multiplication

problems. The Arithmetic Teacher, April. 7. Brown, M. (1981) Number Operations. In Children's Understanding of

Mathematics: 11-16 (Ed.) K. M. Hart, John Murray. 8. Brown, M. & Kuchemann, D. (1977) Is it an add, Miss? Part 2.

Mathematics in School. 6, 1.

Games

in the Learning of Mathematics

The Editors are planning a special edition of Mathematics in School which will focus on the single theme of games in the learning of mathe- matics. It will contain a wide range of articles related to the rationale, planning and organisation of games in the classroom; the mathematical basis of some popular games; games to make and games to buy; a select bibliography plus a range of games for use in the classroom and at clubs, which will include calculator and computer games.

If you have managed to initiate a successful learning experience through the use of a game in the classroom, then please write to the Editors, and so make this experience available to more children. Articles could include details of the game, tips on how to produce it, if it is home made, an account of the classroom activity together or hints on how to adapt the game to special groups of children. The Editors will be pleased to advise in detail.

26 Mathematics in School, May 1985

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