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Abstract

This article outlines teaching ideas appropriate for primary mathematics. It is mainly aimed at primary school teachers and teacher-researchers. James Russo shares his experiences of exploring proof with a group of 8-and 9-year old students in an Australian primary school.
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 difcult 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 innity and
what it represents. I was interested to see whether
or not the introduction of ideas of innity 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 reection and discussion, these
year 3 students concluded:
Because the numbers go forever, there is no way
of knowing. (Divit, age 8)
It [innity] 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 innity 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 innity
should be even… so there is one more even number
than odd number.
Teacher: Why do you think innity 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 innity 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
innity 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 innity 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 sufcient 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 falsied 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 denite
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.
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