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32 April 2018 www.atm.org.uk

Mathematical proofs in primary schools

James Russo shares his experiences of exploring proof with a group of 8- and 9- year students

in an Australian primary school.

T

he first time I encountered the notion of

mathematical proofs was in a mathematics

class in my nal two years at school. I found

the process of attempting to prove a mathematical

conjecture intriguing, mystifying and intimidating. I was

confounded that mathematical proofs appeared to be

a central aspect of what mathematicians do, yet they

had not appeared in the curriculum until the end of my

formal education

As a primary school teacher, I strive to reveal the

power and elegance of mathematics to young

students. This can be achieved through many different

types of mathematical activity. I think a common thread

running through such activities is that they provide

opportunities for mathematical reasoning. Although

I am a rm believer in using highly contextualised

problem-solving tasks to engage students, I often

wondered whether there might be a role for exploring

mathematical proof in a primary school context. My

feeling is that this might allow students to use the

language of mathematics to reason and build an

argument, whilst deepening their understanding of

important mathematical ideas.

In this article I offer examples of student discussions,

taken from audio recordings, when considering

the properties of even and odd numbers, and

multiplication. I conclude the article by putting forward

three principles of task design that I think might be

useful to other primary educators interested in

developing problems of this type with their students.

Investigating and exploring properties of odd and

even numbers: Proof 1.

Zero is an even number. True or false? Prove it.

This was probably the most accessible of the activities.

All students came up with at least one reason as to

why zero must in fact be an even number. Many of

these explanations focused around an examination

of the pattern of even and odd numbers. For example:

It is an even number because there is a pattern

going even, odd, even, odd. Because 1 is odd, 0

must be even, because they’re next to each other.

(Diane, age 8)

If we count by 2s, there is only even numbers, like

2, 4, 6, 8 ... The number before 2 would be zero,

so zero must be even. (Jake, age 8)

Because if zero was an odd number, it wouldn’t

make any sense. Because if we were counting all

the odd numbers, and it went zero and then 1, 3,

5, 7, 9 such and such. It is not in a pattern and it

doesn’t make sense. (Kiara, age 9)

Negative 1 is an odd number because the only

number in it is 1 and it is an odd number… And

the number ‘actual 1’ is an odd number. So, there

would have to be an even number in between

those two odd numbers to split the odd numbers.

So, zero must be even. (Violet, age 9)

Several students attempted to argue by analogy,

suggesting that all other numbers ending with zero

are even, for example, 10, 20, 30, so zero itself must

be even. As one student stated:

I think zero is an even number because it goes 0,

1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And it keeps going until

100. And if it keeps going until 100, then all of these

(gesturing towards multiples of 10) are even. So all

the numbers counting by 10 are even. (Jeremiah,

age 8)

In some instances, this analogical argument was

linked to the notion that even numbers can be halved.

For example:

If a number has zero at the end, in the units

column, then you can halve this number. Like half

of 30 is 15. If you can halve it, it must be an even

number. So, zero must be even.(Jayesh, age 9)

So, you had 21, you couldn’t halve that, so it is an

odd number. But say you had 20, it is even because

you can halve it into 10 and 10. (Emmy, age 9)

If you halve a number ending with zero, it is either

going to end with a zero or a ve after we halve it.

And if we halve 0 it ends with 0, so 0 must be even.

(Zalika, age 9)

One student explanation focussed on the notion that

even numbers have “partners”, whereas odd numbers

do not. Although this reasoning seems potentially

problematic for proving that zero is an even number,

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April 2018 www.atm.org.uk

Mathematical proofs in primary schools

he attempted to emphasise that the lack of an “odd

one out” proved that zero must be even.

If you have three dots and you put it into two

vertical lines, one side would have two and the

other side would have one. That dot [gesturing

towards the single dot] doesn’t have a friend next

to it, so it is odd. It’s the odd one out. Just like in

seven, because 2, 4, 6 … they all have partners.

But there is one extra dot for seven and that one

doesn’t have a partner. With zero, there is no

par tners needed, so there is no odd one out. (Divit,

age 8)

Indeed, this explanation was challenged by other

students.

Zero doesn’t have a partner, so this doesn’t make

sense. You need a partner to be an even number?

(Jayesh, age 9)

What about negative numbers? They don’t have

partners? (Violet, age 8)

The group reached a near-consensus that given

negative numbers are still odd or even, and that zero

is still odd or even, the idea that even numbers have

partners and odd numbers do not may have outgrown

its usefulness.

Investigating and exploring properties of odd and

even numbers: Proof 2.

There are more even numbers altogether than odd

numbers. True or false? Prove it.

Although intended to build on the previous discussion,

this problem seemed more difcult for students to

make meaningful progress with. However, part of the

reason for me choosing to go in this direction was

that, in addition to exploring the properties of odd

and even numbers, this problem gave students an

opportunity to grapple with the notion of innity and

what it represents. I was interested to see whether

or not the introduction of ideas of innity might be

overwhelming for students, limiting their capacity to

attempt to reason mathematically. For some students,

this might have been the case. For example, after

several minutes of reection and discussion, these

year 3 students concluded:

Because the numbers go forever, there is no way

of knowing. (Divit, age 8)

It [innity] goes up to like a quadrillion, billion, so

you can’t tell. (Jake, age 8)

Other students proposed that there are more even

numbers because innity itself was likely to be even.

However, in students’ minds, this appeared contingent

on where you started counting from. Below is an

exchange I had with two students who proposed that

if we began counting at zero, there would be more

even numbers, whereas if we began counting at one,

there would be more odd numbers. The logic appears

to be that beginning at one rather than zero shifts the

entire set of possible numbers by one, meaning that

instead of beginning and ending our counting on an

even number, we will begin and end our counting on

an odd number.

Jayesh: If the rst number zero is even, then innity

should be even… so there is one more even number

than odd number.

Teacher: Why do you think innity might be an even

number?

Zalika: Because the rst number you started counting,

you would probably end on what you started on, but

higher. But if people started counting from one, that

would mean there would be more odd numbers. It

kind of depends on how you count them.

Teacher: So, it depends on where you start?

Zalika: Yep, yep.

Jayesh: So, if you start from an odd number, there

will be one more odd number, but if you start from an

even number, there will be one more even number.

Teacher: But if you start from one instead of zero

Jayesh, how does the number you end on change?

Jayesh: Because innity keeps on going, it goes odd,

even, odd, even. But if you start from 1 instead of

zero, it will go one back, and it will go one back from

innity from the even.

Zalika: So, if you start on zero you will have one more

even number, and if you start on one, you will have

one more odd number.

Teacher: So, changing where you begin shifts

everything over? Is that what you are saying?

Jayesh Yeah, yeah.

Jayesh and Zalika were clearly genuinely engaged with

the notion of innity and how it might be manipulated

to address this particular problem. However, the

explanation that most closely bore semblance to a

formal proof was provided by Violet and Emmy, who

argued that there were more even numbers than odd

numbers:

Teacher: So, why do you think there are more even

numbers than odd numbers altogether?

34 April 2018 www.atm.org.uk

Mathematical proofs in primary schools

Emmy: If we count up to 10 with the even numbers, it

goes 0, 2, 4, 6, 8, 10 – there are six numbers. But if

we count by 2s in the odd numbers,1, 3, 5, 7, 9 – there

are only ve.

Teacher: So, up to 10 there are more even numbers

than odd numbers.

Emmy: And then if you count every group of 10 in the

number alphabet …

Teacher: But, in the next set of 10, say up to 20, you

wouldn’t start on 10, you would start on 11. So, there

wouldn’t be more even numbers in this set. There

would be ve odd numbers and ve even numbers.

Emmy: But that wouldn’t matter because there would

still be more even numbers, because there would be

one more here [gesturing towards the zero to 10 set].

Teacher: So, can you summarise your argument?

Violet: If we count to 10 in the 2 times tables in the

even numbers, it goes 0, 2, 4, 6, 8, 10. And there are

six even numbers. But we count in the 2 times tables

for the odd numbers, it goes 1, 3, 5, 7, 9. And then

there are only ve odd numbers. And if we do that to

every (set of) 10 that there is, there will always be one

more even number. Even though for the next group

of 10 we start at 11, and then 21… For these groups,

there’s the same (amount of even and odd numbers).

But it doesn’t matter, because there is still always one

more even number, because of the rst group.

Although it is possible to offer an alternative proof

which would contradict Emmy and Violet’s conclusion,

for example, ending our sets on odd numbers, like

9, 19, 29 and so on, and similar reasoning could be

used to prove there are an equal number of odd and

even numbers, it seems that these students have

harnessed the language and logic of mathematics to

form their argument.

Investigating and exploring the properties of

multiplication: Proof 1.

If I know all my other times-tables, l don’t need to

learn my 7 times-tables because I will know them

anyway. True or false? Prove it.

This problem was intended to engage students

in reflecting on the commutative property of

multiplication. However, two students struggled to

make progress with the problem, even after some

teacher prompting. For example, after following the

prompt to write out his 7 times-tables and look for

patterns, one student concluded,

“It’s false, because if you don’t learn your 7’s, you

won’t know much about your 7 times-tables.” However,

all other students in the group cited the commutative

property of multiplication as the reason why learning

the 7 times-tables independently was unnecessary.

Because you know your 1s, 2s, 3s, 4s and other

times-tables, you don’t need to know your 7s. And

it is the same with any other [times-table]. Because

you know one 1 x 7, 2 x 7, 3 x 7. (Violet, age 9)

Because if you learn like 1 x 7 and 2 x 7 it would be

the same as 7 x 1 and 7 x 2. It is the same answer,

even though it is the other way. (Jayesh, age 9)

If you wrote out all your times-tables, and then just

found the one’s with 7s it would be the same. You

swap it around and it would be the same thing.

(Zalika, age 9)

I was thinking it wasn’t true, but then I got convinced

because if you know all the others, you’ll probably

know your 7s. [… ] Because if you count by 2’s all

the way to 14 it would be 7 times, and 14 is 2 times

7. (Diane, age 8)

Towards the end of the discussion, several students

realized that one multiplication fact would still remain

missing, that is 7 × 7. The group decided that this

fact would need to be learnt independently. I thought

that this realisation would lead to a discussion about

how the distributive property could be used as an

alternative means of showing that learning the 7

times-tables independently is unnecessary (that is 7

groups of 7 is the same as 7 groups of 5 and 7 groups

of 2), however, this was not the case. Instead, several

students proceeded to discuss whether learning

your 7 times-tables would be sufcient for knowing

all other times-tables, that is, exploring whether the

commutative property could be used to prove the

inverse of the original problem. The group reached

a consensus that this would in fact be impossible, as

many multiplication facts would be “left out” such as,

“5 times 6 and 6 times 5.”

Investigating and exploring the properties of

multiplication: Proof 2

If I know all my times-tables up to 10 × 10, l don’t

need to learn my 12 times-tables because I can

easily work them out. True or false? Prove it.

This problem was aimed at students developing

an appreciation for the distributive property of

multiplication, through them demonstrating that 12

times-tables are equivalent to either ‘adding together’

our 10 times-tables and our 2 times-tables, or doubling

our 6 times-tables. Most students in the group did in

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April 2018 www.atm.org.uk

Mathematical proofs in primary schools

fact use the distributive property to ‘prove’ that the

statement was true. The most notable exchange was

again with Violet and Emmy.

Emmy: We reckon it’s true.

Teacher: Why?

Emmy: Say you have 12 × 5. You would add the 10s,

so 10 × 5 = 50, and then you would add the 2s, so 2

× 5 = 10. And 50 and 10 equals 60.

Teacher: Wow, that seems like a fairly powerful idea.

I wonder whether you could you use the same logic

for the 13 times-tables?

Emmy: Yeah, say you had 13 × 7. You would do 10 × 7

which is 70 and 3 × 7 which is 21. And you could add

it together, and it would be 91.

Teacher: Wow. And, if that is the case, I wouldn’t need

to know my 14s or my 15s either. I wouldn’t need to

know any of my times-tables up until my 20s.

Violet: But, 20s would be easy. Say we had to do 24

× 5. You would times the 5 by 10, and then you would

do it again. And then you would add 4 more groups of

5. So it would be 120.

Task design principles for developing ‘proof-type’

problems.

Hopefully the above discussion has demonstrated

that asking students to engage in problems requiring

them to prove or falsify a particular conjecture is

worth pursuing, even in a primary school context. I

will conclude by proposing three task design principles

which teachers interested in developing similar

problems may wish to consider.

Principle 1: The problem should be worded as a

statement, with an attached follow-up question

“True or false? Prove it”.

Putting forward the problem as a conjecture that

needs to be proven or falsied is very different from

conventional question-type problems we normally

ask of students. I feel that presenting problems in this

form both encourages students to rst take a denite

position. They have to decide whether they believe

the statement to be true or false which gets them into

the mindset of needing to nd evidence to support

their position. This shifts the focus of the lesson to

the quality of students’ mathematical reasoning and

their capacity to adopt the language of mathematics

to develop a persuasive argument.

Principle 2: The mathematical knowledge required

to engage productively with the problem is

accessible to most students beforehand.

It seemed to me that asking students to engage in

these problems is cognitively demanding, particularly

when students are unfamiliar with problems of this

structure. Consequently, I feel that such problems

are probably better introduced towards the end of a

particular unit of work, as they offer a great opportunity

for students to apply recently acquired mathematical

knowledge and skills.

Principle 3: An important mathematical idea

should lie at the heart of the problem.

I think that these types of activities are an excellent

means of exposing students to important mathematical

ideas. In some instances, it may be that such problems

allow the discovery of such ideas, as appeared

to be the case with Violet and Emmy realising the

power of organising numbers into discrete sets whilst

attempting to prove that there are more even numbers

than odd numbers altogether. In other instances, it

may be that such problems allow already known ideas

to be brought to life and given new relevance, as was

the case when the same two students demonstrated

that applying the distributive property of multiplication

meant that learning our 12 times-tables, and even our

24 times-tables, was redundant.

Finally, I would love to hear from other primary school

teachers and teacher-educators who have developed

similar problems which they have used successfully

in their own classrooms or with pre-service teachers.

James Russo teaches at Belgrave South Primary

School and Monash University, Melbourne,

Australia. Email: james.russo@monash.edu.