Replace Your Mental Arithmetics Test with a Game!Author(s): Gillian HatchSource: Mathematics in School, Vol. 27, No. 1 (Jan., 1998), pp. 32-34Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211845 .

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RPPLACP YOUR

ME ARithmetic TEST WITH A

Game

by Gillian Hatch

One does not need to look far to find evidence of considerable concern as to the performance of English pupils in the area of numerical fluency. I prefer to use this term to mental arithmetic to avoid all the overtones of being able to recall facts or do simple calculations in one's head at a rapid rate. The front page of a recent Times Educational Supplement (1997) discussed the latest efforts of Professor Sig Preis to convince us that our pupils are lagging behind those in other countries to a significant extent. The research statistics he quotes, for example 'only 36% of nine year olds' and 'only 56% of 13 year olds can subtract 2369from 6000' must be a cause for concern to all those who teach mathematics. One could argue for a long time as to what level of fluency it is worth achieving in an age when complex calculations can always be supported by a calculator or a computer, but it does not seem unreason- able to me that we should expect our pupils to be able to achieve an answer to a calculation such as this which can be done mentally by anyone with a reasonable grasp of place value.

There is evidence that in the UK we teach formal record- ing methods earlier than they are taught in other countries and expect our pupils to do written calculations in a formal and unique format at an early age. It seems to me that this does not in any way support the development of flexible mental strategies. The example in the previous paragraph is especially telling because it is one which shows clearly how inefficient a standard algorithm can be, either as a written or as a mental strategy. While we ask our pupils to do endless calculations in a standard format rather then allowing them to develop their own ways of working things out gradually, there is not likely to be significant improvement in their ability to depart from the standard method when this simplifies things. Pupils need to be led to compare the different ways in which an answer can be obtained so that they gain experi- ence in choosing a good method. In this way the pupil can start to learn to stand back and make choices before getting into the detail of the calculation. This reflective approach is certainly not fostered by an insistence on standard procedures.

However, the main intention of this article is not to discuss how mental skills may be taught either directly or indirectly, but to consider the way in which games can be woven into the curriculum to help pupils to practise their mental skills in a situation quite different from the traditional 'mental' test

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and in a way which reduces the pressure felt by pupils. It is written in the personal conviction that the use of games needs to be given far more attention than it has, perhaps, had up to now. Games can be invented to practise all kinds of numerical fluency including place value, ordering, all four operations, estimation and approximation. Moreover, the numbers with which the games operate can be natural numbers, fractions, decimals, negative numbers or later on even algebraic expres- sions. A girl who had been playing my algebraic games actually expressed the idea that she was doing algebra in her head. A rich source of ideas for numerical games of all kinds is Dave Kirkby's book (1992) Games in the Mathematics Classroom.

For some years now I have been investigating the use of games as a teaching strategy. I have developed a rationale for the use of games in the classroom (Hatch 1997). Some elements of this rationale are particularly relevant when it comes to mental skills :

* There is pressure to work mentally when playing a game " A game does not define the way in which a problem is to

be solved * When playing a game there is usually help available * There is often some choice as to which problem is solved.

(Often a card which is, in some sense a threat, can be discarded or not used at once)

* A game can generate an unreasonable amount of practice

The last of these seems to me to be a crucial reason for using games in the development of mental skills. We need to motivate pupils to work mentally regularly and for extended periods of time without getting bored and wanting to resort to a calculator. In my view, and according to my research observations, when involved in a game pupils scarcely appear to perceive the effort put into computation. Incidentally although I do not intend to discuss this further, eavesdrop- ping on pupils playing games is an excellent way to assess their mental skills.

I shall hope to demonstrate this contribution to mental fluency by discussing a range of arithmetic games. The games will be designed to be ones in which it is clearly possible to vary both the type and range of numbers involved and level of skill which the game requires. There is great advantage in setting up games which are adaptable in this way since pupils will then become familiar with a set of generic game types. This means that they can quickly learn a new version of any of these types. In this way much of the problem of using games in the classroom can be avoided. The games I shall now discuss should thus be seen as only being examples of a whole range of similar games..

The conventional mental arithmetic test has always seemed to me to be an artificial way to attempt to persuade pupils that there is value in being able to work mentally. Unless you are already skilled in mental arithmetic and can enjoy getting a high score, there is little of intrinsic interest in such a test, certainly nothing to motivate you to try your best to extend your skills. On the other hand, when you are playing a game it is a nuisance if you have to spend all your time doing calculations on paper, so there is automatically a premium on managing to achieve the same answers in one's head. Many years ago now I was observing a student in a classroom who was having some difficulty in getting a particular group of low attaining boys in year 4 to work. With his permission, I sat down to see what could be done with these three. I invented a game which seemed to be at about the night level for them, based on a well known investigation:

The first player chooses a number. Each player after the first halves the number if it is even or multiplies it by 3 and then adds 1 if it is odd. The player who obtains 1 wins a point and restarts with a number of his choice. No start- ing number may be repeated in any one game.

Mathematics in School, January 1998

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As I started this game with them they were busy suggesting that they could neither divide by two nor multiply by three, but it was noticeable that once the game started they rapidly discovered a considerable capacity to do both these things. One can interpret this in two ways, either as a change in motivation or as an ability to carry out calculations in their head that they feel they do not know how to do by the standard methods on paper. Discussing this anecdote re- cently with a classroom teacher he also suggested that when we offer pupils an activity which is not tied to formal ques- tions to be answered in an exercise book then something quite different can happen to a pupils' capacity or willingness to make calculations. His anecdote concerned pupils working with a computer program who were quite willing and able to add up angles in their heads when the equivalent calculation done as a textbook example would have seen them reaching for their calculators. I can see the use of a computer program as being psychologically rather like a game - even more so in this age of computer games!

Another factor that seems to be involved when a game is used in the classroom is that however much it may be designed to produce a lot of practice in a particular skill this skill is not explicitly identified in pupils' minds. The players are therefore unaware of what lies behind it and feel free to work things out in any way they choose. I have a strong theory that if pupils were asked to calculate 6000 - 2369 away from the context of a test and the idea that it is a school problem the results might be very different. The idea that a particular kind of 'sum', which they must feel competent to do, is required does not seem to enter their heads when they are playing a game, they approach the situation with an open mind. It is particularly useful if a game allows different levels of response so that pupils can gradually move onto more sophisticated means of solution. Consider for example the following game:

Find the missing number Using the set of cards shown below, deal two or three cards to each player (or pairs playing together.) The re- maining cards are placed face down in a pile. An ordinary die is thrown and the players win any card for which this is the correct missing number. Cards which are won are replaced from the pile until this is exhausted. The players then win any cards for which they can state the missing number.

E + 1 =5 2 x 1 -l=5 3 x 1 -2=13 20 - E = 17

E - 5 = 1 [ - 2 + 3= 4 2( E + 2)=12 25 - E= 20

4x0 = 20 5x[+1=6 4x[-6=6 E+7=13

10 x 0 - 9=1 0 +3-1=1 1-L+2+3=4 3(E +1)--6

(O + 2)+4=1 Ox6-ll=7 xx4+1=17 Ox 3 + 6=9

When this game is played the players can start by just putting the number thrown in the 'box' to see if it works. However, one notices that gradually pairs are saying 'we need a 2, a 5 and a 6.' The final part when the numbers have to be worked out without a die throw also puts pressure on the players to find ways of reversing the process. In developing this they are

Mathematics in School, January 1998

achieving another of the skills of mental arithmetic--the use of inverse operations, (in other situations) invaluable as a check.

Many games also lend themselves to being played in a co-operative fashion, for example these cards can be sorted into piles with the same solution. (A set which has the same number of cards with each solution is a useful check here.) In this particular case the co-operative game is probably harder but in other games it is often useful to allow pupils a chance to work with the game co-operatively before playing competitively. One might ask whether this counts as a game, but in my experience the pupils accept as a game anything which involves a set of playing 'pieces'. Co-operative games also allow the less confident pupils to gain confidence by only attempting to solve problems with which they feel comfort- able. A particularly useful type of co-operative game can be constructed by giving a set of calculations and their answers on separate small cards. Clearly this is very flexible and can be used to practise any level of calculation. It also involves estimation skills. A simple example would be a set containing cards such as:

139 + 273 149 + 253 412 402

This brings into play many skills in estimation, e.g. the use of units digits and relative size of the two answers (here the answer to the second calculation must be 10 less than the first one) Such cards also lend themselves to the pairs game which will be described later.

I referred earlier to the idea that there is help available when a game is being played. When a co-operative game is being played there is time and opportunity for peer tuition as the task is being completed, mental methods can be com- pared and discussed. It is surprising how often a pupil takes on the role of teacher and expounds a method. In a competi- tive game the device of letting two pupils act together as a single player allows each pupil to put forward a point of view and verbalise strategies for the game in a helpful way. Even in competitive games, however, one regularly finds assistance being offered by opponents in order to keep the game going and facilitate the play of the next person.

The type of loop game which was invented by Adrian Pinel (Pinel 1988) is infinitely adaptable to practise any desired skill. The idea is that sets of cards are constructed each containing six cards. The value of the card is given in the top left hand corner, the operation to be carried out to determine the value of the next card to be played is shown in the centre of the card. The particular loop is identified in the top right hand corner. Different loops overlap partially in values so that the game takes a different course in each game. Shown below in its correct order is one of the loops from a game designed to practise tables.

49 A 7 A 35 A 45 A 9 A 63 A

+7 X5 +10 5 X7 -7

The competitive game is played a little like dominoes except that the cards which have been played are kept in a pile so that only the final card is used at the next turn. However one of the strengths of this structure is that it can always be played as a co-operative game when the task is to arrange one or more of the loops into a circle. Pupils seem to find this a compelling activity, being especially intrigued by the self checking nature of the loops. I often invite them to check a loop each to 'make sure that I have made the loops correctly'.

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It is particularly valuable as a preliminary to the competitive game, allowing the pupils to adjust to the new skills involved and to gain a little speed before individual rate of play becomes an issue as it does in a competitive game. If the operations required by a loop game are, at first, right at the extreme of what a group can handle, it is amazing how quickly by using the cards as a co-operative game, they devise strate- gies which make it possible for them to play the game competitively. What is, of course, crucial here, is that no player has to think so long that all the other players get bored!

Another standard game which can be varied to use all sorts of mental arithmetic skills is the familiar pairs game:

Rules for Pairs Games A game for 3 or 4 players Deal all the cards out face down on the table Each player in turn exposes two cards, if they match he/she can keep them and have another go. If they do not match they are turned back and it is the next players turn. The player with the largest number of cards at the end wins.

Pairs could, for example, consist of any of the following types

24 x 5 120

24 x 5 12 x 10

4562 - 2500 2062

4562 - 2500 5662- 3500

124 - 4 31

The questions and answers or the matching pairs of questions can be distinguished to make the game a bit easier if this is felt desirable so that at each turn the players select one card from each set. Such a game involves a very significant amount of mental calculation as differing pairs of cards are turned up. This set of cards can again be used as a co-operative activity in which the task is just to match questions and answers or questions with the same answer. This allows the more tenta- tive pupil to choose a question he feels comfortable with, rather than being forced by the game to accept whatever turns up. If this co-operative activity is used then the pupil has a choice of which calculation to try to match to its answer. To avoid argument a calculator can be offered to be used only when it is needed as a referee as to decide whether a pair really matches.

Spare minute games which can be used for the last few minutes of any lesson as a change of activity are an excellent

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opportunity for pupils to flex their mental arithmetic muscles. A simple one which is a great favourite of mine is:

A Spare Minute Game Ask each player (could be a pair) to write down 6 numbers between I and 100. At each 'turn announce a category of numbers e.g.: greater then 25 less than 64 between 16 and 40 odd number between 25 and 40 even number greater than 80 multiple of 3 multiple of 7 greater than 50 prime number square number triangular number cube number factor of 96 divisible by 13 Each player crosses out one and only one number in that category if they have one. The first player to cross out all his numbers wins. With a smaller number of players, or pupils in 'table' teams, a misere version can be played where the last team with a number left in wins.

Many spare minute games take the form of target games of one kind or another, such as those found in Kirkby (1992). The one that follows was specially invented to address the type of calculation which was discussed in the TES:

Take your pick target Each player draws a box: This will eventually contain a 4 digit number. A target is set e.g. biggest number, smallest number, nearest to 5000, multiple of 6 etc. The teacher throws a dice either an ordinary dice or a 0 to 9 dice. Each player writes each of the dice throws in turn into one of the boxes. Each number must be entered before the next dice throw. Once a number is entered it cannot be changed. After 4 throws the player can choose either to keep the number he has written or 7000-minus this number (replace 7000 with 10000-for a 0 to 9 dice) The winner or winners score a point and a new target is set and the process is repeated. So if the throws are 2 5 1 9 and the order a player writes them in is 5912, then he can score this or 4481 as is more advantageous.

The issue of mental fluency has to be a crucial one in classrooms given the new tests in SATS and the overall emphasis on numeracy. In writing this I have tried to suggest ways in which the need to increase these skills can be met in more creative and enjoyable ways. Since the normal justifi- cation for the use of games in the classroom is one of enjoyment, their use to allow pupils to increase their mental arithmetic powers must have much to recommend it. F*

References Hatch, Gillian (in press) A rationale for the use of Games in the Mathematics

Classroom in Topic Issue 19 (Spring 1998) NFER. Kirkby, Dave (1992) Games in the Mathematics Classroom, CUP Pinel, Adrian (1988) Loop Cards, MAPS Times Educational Supplement Front Page article July 18th 1997

Author Gillian Hatch, Manchester Metropolitan University, Didsbury, Manchester, M20 2RR

Mathematics in School, January 1998

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