Follow-up discussions with Andreas

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 'Thinking Mathematically'.

"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 regions?"

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 process.

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 spots problem.

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 counterexample illustration.

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, etc.

Jenny: Have you tried this with any other group of activities? What can change and what can stay the same?

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 purposes.

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 developed.

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 proofs.

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 following criteria:

- 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 the criteria.

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 criteria.

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!