Reasoning, Logic and Proof at Key Stage 2Author(s): Lesley JonesSource: Mathematics in School, Vol. 23, No. 5 (Nov., 1994), pp. 6-8Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215106 .

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Reasoning, Logic and Proof at

Key

Stage

2 by Lesley Jones, Goldsmiths' College, University of London

As a pupil at school in the 1960's my experience of proof was something you learnt and would be asked to reproduce under examination conditions. We were led through other people's reasoning processes as we learnt how you could be sure that the angle subtended by an are at the circumference would be half the size of the angle at the centre. Most of my peers then memorised the proof, ready for the examination. I found it very hard to memorise, but I could work my way through the reasoning process to reconstruct the method. My school experience consisted of learning about proof, rather than learning how to prove conjectures in mathematics.

What is the purpose of teaching children about proof? Is it a part of their culture in the sense that they should know what it is to be a mathematician and proof is an essential element of the discipline of mathematics? I would argue that children do need to know how to reason, explain, justify, and prove their ideas, but it is the process of developing these skills which is important. Associated with proof is the need for rigour, which Freudenthal (1973) does not see as a single static concept. He claims, "in the course of history mathematical rigour has not been the same in the minds of all mathematicians, and unless people are blind they will agree that it is still not the same." He goes on to recognise the value of intuitive thinking and recognises that for a six year old to work out a simple computation on his fingers or using counters, is, at that level, a rigorous proof.

6

Proof has traditionally been an aspect of mathematics which would not be introduced until secondary school stage. However, children learn to reason from an early age and they can be encouraged to extend and develop their reasoning skills in the context of mathematics. Children involved in the PriME project (1991) were encouraged to explain their reasoning. In one example Michael and Richard (aged 9) had indicated that four lots of 0.25 made 1. They were asked to explain what they had done. Richard states that "point two five is a quarter." The teacher asked how he knows it is a quarter, "What if someone didn't believe you. How could you prove it to them?" Michael's justification may not be totally convincing, but he is able to go some way towards justifying his statement, "Because 25 is a quarter of 100, and if you add a nought on the beginning, it would be a quarter of 1 as well." At this stage the children need a lot of number knowledge to be able to put together a reasoned argument and they also need the language skills to be able to express their ideas. Only by providing them with the opportunity to refine these skills will they be able to progress within this strand.

Developing language skills I was interested in exploring children's notions of proof and I worked with a group of four children aged between (8.8) and (9.2). During the work with the children I used different forms of language. Sometimes I asked, "how can you prove that?", sometimes, "how can you be sure?" It became clear that the language the children needed to use to express their ideas in this kind of discussion was problematic to them. My observations led me to think that although the mathematics itself was challenging, the children were capable of reaching a level of personal certainty about it. However, it was sometimes difficult for them to communicate their justification to me or each other.

Odd and even numbers I worked with Ben (9.2) Nikki (8.8) Helena (9.2) and Chris (9.2). After an initial discussion about odd and even numbers in which the children tried to define how they knew a number was odd or even, I asked them what happens when we add an odd number to an odd number.

Ben: "Even, it always does. "

They explored some examples, mostly examples where they added two equal odd numbers. I asked them to think about two odd numbers which were not the same. They decided that the answer would always be even.

LJ: "'How can you be sure" "I think it always does come even. "

LJ: "Can you say why?"

At that point Helena had an idea.

"If I add 11 and 1 that will come to 12 and if I add 11 and 3 that will come to two more than 12, that's 14. If you

Mathematics in School, November 1994

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add 11 and 5 it comes to an equal (sic). It comes to 16. So I just think it will."

Now in a transcript of the conversation it is impossible to give the reader the full flavour of the finality and certainty that were expressed in that statement. Helena has used an inductive process to reach a personal certainty which has all the power of a generalised statement at her level of work.

We moved on.

LJ: "What happens when you add an even number and an even number?" Ben: (very quickly) "It comes to an even number. 'Cos if you have 4 and 4 it's 8, 10 and 10 is 20, 20 and 20 is 40." LJ: "Is that always true? You sound very sure. How could you prove it to me?" Chris: "If you don't use any odd numbers it won't come to an odd number, because even numbers always comes to an even number." Nikki: "Cos like an odd and an odd come to an even, so an even and an even might come to an even. Cos like I said its in the 2 times table. "

Chris: "If 2 evens came to an even, why doesn't 2 odds come to an odd?" LJ: "If I gave you a big number like 248 and asked you to add it to 292, without even working it out, would it be odd or even?a

Ben: "Even" LJ: "Even-you're sure?" Chris: "Well the 100 might not be an even, but the other two would probably be even. ...I think I know the answer, 540"

I confess to being taken aback by this as I had not asked or expected the children to work this out in their heads. There followed a few minutes of confusion as I tried to remember the numbers I had used.

Only after listening to the tape did I pick up Chris' observation about the hundreds digit being odd. He appears to be considering the digits individually in terms of odd and even.

Immediately after this comment was Ben's next remark. "At least I got it right that it was even."

LJ: "1 still want to know how you are so sure." Ben: '"It's like the numbers are people and its only fair that if the odds come to an even and the odds get to see the evens and come to an odd and then if you add 2 evens you can come to an even number." Helen: "Don't get it" Chris: "Nor do I" LJ: "It's putting it into words that's difficult. Have another go Ben. "" Ben: "If you were going to be fair to each other, if the numbers were people you could have an odd and even to come to an odd and 2 odds that come to an even and 2 evens that come to an even."

Maybe it was the body language that made this statement so convincing, but it also referred back to a definition of even numbers that Nikki had given at the beginning of the session, where the definition had depended on sharing a number equally between two people, without having halves left over. Ben seemed to have a clear idea of his "proof", but was finding it difficult to find a way of expressing his ideas. Nikki took up the explanation.

"Its like if you have a packet of cards and you was going to play a game of pairs and there was 4 cards all the same. If you put the cards on the table, if you put 2 cards together you get a pair and that'd be two and if you counted the cards together they might come to an even numbers .. cos they're all fours and if there was an odd number then you won't get two pairs will you?" Chris: "If you had some pairs and if when you've got two

Mathematics in School, November 1994

pairs and when you add those pairs up it mzght not come to another even, cos it might be an odd. Cos though the cards got two when it comes to adding them up it might be an odd or an even. 9

Several of Chris' statements made me realise the potential for confusion in the language we use. By now the children had made clear links with "Pairs" a game with which they all seemed familiar. Chris seemed to be clear in his mind that when the game had finished and all the pairs were made, the players might end up with an odd number of pairs. When children check to see if a number is odd or even by making pairs there may be confusion about numbers which contain an odd number of pairs. Perhaps "even" is seen as a relative term, where 8 is more even than 6.

Helen recounted an example of a game which she had played,

"Cos at home my little sister lost 2 cards that were both different. When we added them up it came to an odd number because we put them all together in twos but one of the cards didn't have a pair and the other didn't have a pair so I had to put those together to make a pair."

The next activity I initiated by setting out a line of counters in a simple pattern, alternating green and red.

I asked Helena to count them for me and meanwhile heard Chris thinking out loud,

"It'll probably be....it should be an even number." "The even numbers are green and the red ones are odd, 'cos one is an odd number and 2 is an even number. If you added another one it would come to a red. " LJ: "You said without counting that it was an even number, Chris. If we got to number 100 what colour would it be?" Chris: "Even...a green one." LJ: "What about 99?" "Red because it's the last number." LJ: "125 would be....?" "CRed"J

LJ: "'How do you know?'" Nikki: "Because if you know your two times table it would be 2, 4, 6, 8, 10. Helena: "There's another way you know...if you count the last number, if that is odd, the number is odd."

This seemed to refer to the method of testing the number by considering the units digit.

Patterns I rearranged the counters into 6 rows of three, with alternate colours as shown below

RGR GRG RGR GRG RGR GRG

LJ: "If I add on two more rows, how many green counters will be in the last row? (indicating that the rows will be added to the top of the array) The children used the counters to demonstrate what colour the counter would be.

LJ: "How can you prove it to me?" Chris: "Because like if you have three counters and put that there, you've added these here, red, green, red, green, red, green, red, because there's four greens and five reds there. (This refers to the numbers in the columns) Ben: "Cos there's an odd number there and the reds are odd and there's more odds than evens. 'cos there's 5 reds and only four greens. "

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This justification seemed to consist mainly of a demon- stration of the continuation of the pattern, but the children themselves seemed satisfied that the existence of the pattern provided assurance that the result would be as predicted. They were using their reasoning powers and their knowl- edge of generalisation to form hypotheses and justify their conclusions.

Consecutive numbers For the next activity I asked the children each to give me a small number, less than 10. For each number I wrote down the number, and two consecutive numbers.

4,5,6 2,3,4 7,8,9

I asked the children to find the total of each row, then asked them to comment on anything they noticed. They discussed which "times tables" they appeared in; "The 2 times, the 5 times and the 9 times."

LJ: "Do they all come into one of the tables?" (A leading question!) "Yes, they are all in the 3 times.

They tried some more examples to see if the total was always a multiple of 3. For each example they worked out how many threes gave the total and wrote this alongside:

4,5,6 3 x 5 2,3,4 3 x 3 7,8,9 3 x 8 5,6,7 3 x 6

LJ: "They are all in the three times table. Why is that then?" Nikki: "I think it's because they're all next to each other. 'Cos they're next to each other they come in the three times table. "'

This statement was one of those which leave you wondering whether it was a "shot in the dark", an amazing piece of insight or simply the child bringing together the significant pieces of information about the numbers we were considering.

I tried to explore this idea further by setting out groups of counters to represent the numbers we were considering.

Nikki: "They're all 3 numbers so they must be in the 3 times table.

LJ: "Nikki, I can see you have got an idea. Can you try to explain again?" Nikki: "There are 3 numbers; 1,2,3; 4,5,6; 7,8,9. Is it because of the three numbers. Is it because they're next to each other?" LJ: "We know that 3,4,5 makes the same as 3 x 4. Can you use the counters to show me that?" "Take away one from the three and add it to the four and that makes 5. Then add the five and the five together and then add the two and that makes an even number that is in an odd and even times table"

It is interesting that this method relies on the construction of 12 in its most familiar form i.e. as 5 + 5 + 2, rather than shifting directly from 3,4,5, to 3 x 4. The child has focused on the number 12, rather than the sets of 4. I tried again, laying out piles of 4, 5 and 6 counters.

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"How can we make that into 3 fives?"

Nikki moved a counter from the pile of 6 to the pile of 4.

"I took one away from the four (sic), so that's 5. Put it here so that's a 5 and then that's a 5. LJ: "'Let's try it with a different number. " Nikki: "You could probably do it with any number." Helen: "You could probably do it with any number. It could be because there's an even in it.

Helena and Nikki had the counters and were talking through the moves.

Nikki was able to demonstrate how you could make three equal piles from any three consecutive numbers. I offered her large numbers to think about and she could describe how they could be shown to be multiples of three. During this session Nikki developed a very clear idea of this number pattern, supported by visual images of the numbers. She was able to demonstrate a way of proving that this pattern would hold for a general case. The language she used for this; "You could probably do it with any number," is tentative, but she was well on the way to personal confidence in the stability of this conjecture.

Conclusion Prior to the Dearing proposals, the National Curriculum had no requirement to ask children to justify their statements until level four. It certainly appears to be a higher order skill, which requires reasoning ability and good communication skills. I have referred in this article to the difficulty children have in expressing their ideas clearly when they are working at a relatively high intellectual level. Teachers pick up cues from body language and from their detailed knowledge of the children in their care. These provide information about children's levels of understanding, which inform the teacher's planning, but which may be difficult to justify in the formal assessment of children's work. Any assessment of the child's ability to reason and justify his/her ideas will be limited by the capacity for communicating and expressing them clearly.

During the process of this work the children used different ways of justifying their ideas. They explained, they demonstrated using apparatus, they drew on previous experience of similar situations, they reasoned. Balacheff (1988) distinguishes between pragmatic and conceptual proof. Pragmatic proof is "proof by showing", whereas conceptual proof depends on abstraction from the specific examples and provides a reasoned argument generalising the case. The children with whom I worked were clearly working at the level of pragmatic proof. For the mathemat- ical pedant we would have to say that this is not a proof in the strict understanding of the term. Yet, as Balacheff states, "We talk of proof because they are recognised as such by their producers." Children in the primary school are certainly capable of reasoning and we owe it to them to provide them with situations which challenge their thinking and allow them to experience the creative and satisfying aspects of mathematics. Clearly work of this kind is teacher intensive if the children are to be challenged to "stretch" their thinking. However, children can be encour- aged to reason and justify their ideas to each other and can be helped to develop the linguistic skills which are essential to communicate their ideas effectively. M

References Balacheff, N. (1988) Aspects of Proof in Pupils' Practice of school

Mathematics, in Pimm, D. (Ed.) Mathematics Teachers and Children, O.U.P.

Freudenthal, H. (1973) Mathematics as an Educational Task, Reidel. Shuard, H. et al. (1991) Prime: Calculators, Children and Mathematics,

Simon and Schuster.

Mathematics in School, November 1994

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