Follow-up discussions with
Jenny: I am interested that none
of the pupils in the focus group challenged the view that the
circle dots rule did not seem to work for $6$ dots. When I have
used this example to challenge learners I have often had the
response that perhaps, if I arranged the dots differently, I would
find that the rule worked. Of course the only way to test this is
by trying different placement of dots. Did any of your teachers
experience this and how did they tackle it?
Andreas: The statement of the question in the problem is important.
The problem is an adapted version of a problem in John Mason's book
"When there are 15 spots around the circle, is there an easy way
to tell for sure what is the maximum number of non overlapping
Once students engage with the problem they are so determined to
find that for $6$ spots there are $32$ regions that some of them
even count $32$ when there are only $30$ (for a regular hexagon) or
$31$ otherwise. When they do more careful counting and they realise
their cases do not contain $32$ regions (and they can try this as
often as they wish), they realise there is no easy way to be sure
what will happen for $15$ spots. The uncertainly arises from the
fact that no one of them found $32$. This is enough to justify
scepticism about the pattern and it does enable them to say that
the pattern is not trustworthy. The pattern doesn't seem to offer a
secure way for one to be able to say, "I'm confident in the
pattern". What one can say is, "I cannot be $100$% sure that the
pattern will apply for $32$ spots".
Jenny: It was interesting that
not only does the pattern break down with the sixth case but that
pupils tried it. Was this a natural progression through the lesson
or did the teacher have to encourage it. It feels like reaching
this critical point is really important to the whole
Andreas: Many of the students tried the sixth case and several
thought they found $32$ because they did not count them accurately.
This even happened with University students. Originally some
students stopped at the fifth case but then, when someone noted
that it didn't work for the sixth case, it seemed natural for
others to want to try it on their own. But, as I said earlier, it
is unnecessary to show for certain that the pattern doesn't work
for $6$ spots - one only needs to realise and accept that there is
doubt in the continuation of the pattern.
Jenny: In order to come out with
this sequence of activities what kind of teacher action would
support better student progression?
Andreas: If you take out of context one of the problems then it is
very likely that the problem will play out in a way that is
different from that described in the article. Teachers should treat
the sequence of activities as a 'package'. We found that the
sequence of activities plays out consistently in strikingly similar
ways if you are faithful to its structure and particular way of
delivery as explained in the article.
For example, the reflection at the end of the squares problem
helps increase students' awareness of the basis on which they
accepted the pattern. Most students say, "I accepted it because the
pattern seemed to work for the first few cases". This increased
awareness sets the foundation for the surprise that comes to them
later when they encounter the 'failing pattern' in the circle and
Similarly, at the end of the circle and spots problem when we ask
students "How many cases will be enough in order to trust a
pattern?" some students offer different numbers such as twenty,
while others propose to check 'random' or 'spread' cases such as
the 75th case and the 100th case. In trying to develop an argument
for how many cases are enough to check before being in a position
to trust a pattern, students become more aware of the conditions
under which they would accept a pattern. This prepares the ground
for the new surprise that comes to them later with the monstrous
What I'm trying to say with these examples is that not only are the
problems and their sequencing important but also the ways in which
the teacher in the article implemented the problems: the questions
she asked, the reflections she asked the students to make,
Jenny: Have you tried this with
any other group of activities? What can change and what can stay
Andreas: The second problem 'the circle and spots problem' can be
replaced with a problem that has an emerging pattern for a smallish
number of cases but breaks down for a case people can check
relatively easily. So if we had a pattern that broke down for the
30th case, that would probably not be a good choice of a problem
because it would be tedious and time consuming for students to
check so many cases.
The first problem 'the squares problem' can be replaced with a
pattern problem that includes a correct pattern and whose proof is
not obvious to students but, at the same time, is accessible to
them, perhaps after some scaffolding from the teacher. This is a
delicate balance: On the one hand, if the proof were obvious then
perhaps students could come up with it straight away and then there
would be no point in following up with the other two problems in
the sequence. On the other hand, if the proof were not accessible
to students then they would not have an opportunity to see a proof
for the pattern, thus creating a belief that the development of
proofs is difficult and beyond their reach. The squares problem
achieves, I think, a defensible balance between these two
considerations and it may be a bit tricky for teachers to come up
with a good alternative problem that would serve the same
Jenny: I am struck, from my own
experience, that learners do not readily grasp what defines a
logical argument. They will often follow one through but not be
able to create similar arguments because the fundamental
understanding of what constitutes such an argument is not
Andreas: I think there is a distinction to be made between
recognising a logical argument and constructing one of your own.
Perhaps these two things are not well distinguished even in the
literature. In a recent study we did with University students we
asked them to construct a proof and then criticise their own
arguments and answer the question whether they had actually
constructed a proof. It turned out that many of them constructed
empirical arguments but were able to criticise them, realising the
limitations of such arguments. We, teachers, might be tempted to
say, "Ah well they don't know what a proof is". But then many
students said in their reflections and critical evaluations things
like, "I know this is not really a proof but it was the best
argument I could come up with at that time". So students'
constructions are not always indicative of their level of
understanding - constructing a proof is, I think, much more
challenging than recognising the limitations of empirical arguments
or a proof when you see one.
It seems to me that a necessary and foundational step in teaching
proof to students is to help them understand that empirical
arguments are not proofs. This empirical conception is so dominant
amongst students of all levels. If students think an empirical
argument is good enough to justify a generalisation, then they
don't see a need to learn about different kinds of argument,
notably proofs. Once students realise the limitations of empirical
arguments, then they are ready to engage in thinking about what
might count as a proof. The next stage for us as teachers is to
help students learn how to construct their own arguments including
Jenny: So, in school it might be
worth considering not only pupils critically examining their own
proofs but, perhaps as a next stage, looking at other people's
proofs and identifying gaps or flaws in other people's arguments?
Are there any other major barriers to proof apart from the
empirical argument problem?
Andreas: This seems to be the most striking barrier to proof and
perhaps the most well documented in the literature, but there are
also other barriers that we need to help students overcome. For
example students tend to construct circular arguments - this
creates a need for us to help them make a distinction between what
is given and what they need to show. Also students tend to believe
that proof always needs to use algebra - this creates another need
for us, namely, to offer to students opportunities to encounter
proof in multiple contexts, not only algebraic.
Jenny: How did the teacher
continue with the proof once the three problems and material had
been worked through?
Andreas: Next the teacher addressed the students' emerging need to
respond to the questions: "What can we do that will prove the
pattern in the squares problem? Even if you try millions of cases
you cannot be sure, but what then?"
If you leave students with these questions then there might be a
danger of them developing misconceptions such as that "in
mathematics one can never be sure of anything". So it is important
at that stage for teachers to help students develop an
understanding of what criteria they might need to fulfil in order
to ensure an argument qualifies as a proof. A class can then use
those criteria to work on other proof problems to negotiate and
agree on what the criteria mean - having criteria does not imply a
shared understanding of their meaning.
After some brainstorming asking the students what they thought
might be such criteria, we prepared a PowerPoint slide with five
criteria that the teacher showed to the class and asked the
students to read and see whether they made sense to them.
An argument that counts as proof in our class should satisfy the
- It can be used to convince not only myself or a friend but also
a sceptic. It should not require someone to make a leap of faith
(e.g., "This is how it is" or "You need to believe me that this
[pattern] will go on forever.")
- It should help someone understand why a statement is true
(e.g., why a pattern works the way it does).
- It should use ideas that our class knows already or is able to
understand (e.g., equations, pictures, diagrams).
- It should contain no errors (e.g., in calculations).
- It should be clearly presented.
Following the presentation of the criteria, the teacher asked the
students which criterion they thought was the most challenging to
fulfil, which one they thought would be the most important, whether
any of them were unnecessary, and whether they wanted to add
anything to the criteria to make them clearer. The students were
asked further to mention some examples to show how they understood
After they had the discussion about the criteria the teacher led
the class back to the squares problem and said: "Now, with these
criteria in mind, let's try to decide what might be a proof for the
$(n - 2)^2$ pattern." The students worked on the development of a
proof with some scaffolding, and they succeeded. We really wanted
them to get an image of what a proof looks like. With this, the
students saw that a proof is actually within their reach and that
an argument doesn't have to use algebra in order to fulfil the
In the following couple of lessons the students worked on other
pattern problems trying to prove the emerging results. The list of
criteria was something the class kept coming back to. There is no
canonical definition of what a proof is, but nevertheless it is
important for students to have a list of criteria that can guide
their work. And when the teacher says to them, "No this argument is
not a proof'", it is fair for the class to have a basis on which to
explain why the argument is not a proof and how it might be
developed in order to become one.
There is much more that can be said about the teaching of proof - I
hope that the article will inspire teachers to work on this
important issue with their students!