Approach

The Mathematical Isles of What If…? is an inquiry approach to teaching maths in primary schools that has creativity, curiosity, reasoning and problem solving at the very heart of the lessons. It views mathematics as a continuous and collaborative game of asking ‘what if…?’ questions to create rules and then exploring the consequences of following those rules. There is a good explanation of this idea here by Dr. Math.

In the approach, learners are invited to imagine that the exploration of ‘what if…?’ questions is akin to exploring uninhabited islands in search of treasure. An example of this would be asking, ‘What if we created an addition that totalled 5,555 and what if, for each digit that we use in the addends only once, a point is scored?’ This inquiry can be found here.

Asking further ‘what if…?’ questions to deepen inquiries

When exploring an inquiry, we think about it having three phases in terms of depth: the Landing Spot, the Island and the Archipelago.

An individual island represents all the possible examples and directions that we could go in with our inquiry. We start with only a small, confined and accessible area – the Landing Spot. At this point, we are specialising on a small idea or problem. In the case of the above task, we might focus on one set of addends together and determine its score. It is about an accessible start and drawing everyone in.

Then, through further ‘what if…?’ questions, we might reduce constraints, make subtle changes to them or go down different avenues of inquiry and as a result, further data/examples are possible. In our ‘Distinct Digits to 5555’ example, we might ask, ‘what if we changed the addends? What points scores are possible?’ One path might on the island might be to try and reach the highest score possible, one path might be to try and make the lowest score. We could even focus purely on making an individual score like 5. As a class community, we are trying to understand all the possible scores. This is what the whole island represents. We have increased the breadth and depth of our idea or problem beyond the Landing Spot as part of the Island phase. As part of this phase, we might ask questions in aid of our search such as, ‘what if we has decimal addends? Will that help us reach particular scores?’

As we begin to understand the consequences of following the ‘what if…?’ questions that we have asked as part of our island. The children or the teacher might ask further ‘what if…?’ questions to change the rules beyond just thinking about constraints – often in a creative way. Through this, they create new mathematical worlds and we are then on a new island altogether. Our inquiry has stretched to the Archipelago phase of ‘What if…?’ questions. When exploring these related islands of ‘what if…?’ questions, we might generalise across them and use similar strategies. Examples of these might include:

  • What if we used subtraction?
  • What if we had a different target number? Would the range of points that we could score differ?

The ‘what if…?’ questions across the phases drive the exploration and the lesson. They are often made by the teacher but over time come increasingly from the children with the teacher the judge of genuine mathematical investigation. As a result, children see maths as an ever-expanding world full of possibilities. There is never an ‘end’ to an inquiry but instead always new possibilities. These new directions allow for children to challenge themselves in different ways and become experts in their line of inquiry. This is empowering. It promotes maths as being for all in a mixed-ability classroom and one where creativity is valued. After all, to achieve certain scores when adding to 5,555, creative approaches are needed.

Discoveries children make

In our search, our ultimate goal is treasure. The treasure of maths contains discoveries and patterns.

The size of treasure - how bountiful does a conjecture prove to be?

To obtain these prizes, the children make conjectures. This is for the community to then validate. A conjecture, initially, is like an ‘x marks the spot’. To dig up, reveal and unlock the chest, it needs to be accepted as true or false (both are treasure) by the class. For example, the children might conjecture and convince each other of what scores are possible. They might conjecture for example that a score of seven is impossible. Until this is accepted as true by the class or a counterexample is found that shows a score of seven is possible, the treasure remains locked. As such, it is by collaborating and persuading others that treasure hoards are found. Children love proving each other wrong, it can be a real highlight of a lesson, but a conjecture proven false still counts as treasure in the approach – it isn’t a failure. We know more about the island that we have created than we did before.

The size and type of treasure depends on what they find out and whether or not it describes a part of an island, a whole island or even the whole archipelago. Making discoveries is explained in more depth here.

Inquiry skills

The children are supported with the use of a range of skills that focus on reasoning, problem solving or advancing the inquiry. They are clearly defined across six posters and are used across all islands. However, it isn’t the case that they are followed repeatedly in a defined order but instead, in each inquiry, the class and teacher use them as a guide for their paths forwards.

Children’s attitudes towards maths

Each island is an invention, a non-trivial situation that involves multiple possibilities and multiple routes of inquiry. There are lots of different areas to an archipelago that warrant investigation and it is up to the class to decide how we do that – with the assistance of the teacher. We could team up and go somewhere together, we could also follow our own paths and search different areas of the island. The deeper into the unknown we go, the more there is to find out.

What it is not about is providing the ‘right’ answers. Of course, we want children to be fluent in the procedures and concepts that we are teaching them. The Isles of What If…? provides ways of using those concepts and procedures from across the curriculum as contexts to explore. Through this, they deepen their understanding of them and maths itself. They are becoming more fluent in their use of procedures and concepts but doing so in a meaningful way. In the example above, if they find ways to score the entire range of points for 5,555, then great. However, it is about what the community of the classroom uncovers and accepts as collective understanding. They aren’t losing out if they are unable to uncover the need for using 4 digits and 3 digits to score 7 points. It may just be that that class went in a different direction. Regardless, in doing so, they will have thought deeply about column addition and manipulating it fluently to score different numbers of points.

In making discoveries, explaining them and seeking new areas of exploration, children naturally use mathematical language and make connections between mathematical content. They are being encouraged to pursue understanding and patterns. An anecdotal example of this is that a Year 6 girl, who had been exposed to this way of working, did the calculation 4995 ÷ 9 = 555. This wasn’t a deliberately chosen calculation, it came by chance. It would be easy to look at the answer of 555 and consider it surprising but think nothing more of it. In this case though, they were intrigued and came up with 7992 ÷ 9 = 888. This is no longer a coincidence and a direct result of this way of thinking. The ‘wing’ digits (as they named them) of 4 and 5 in the first example are also used in 45 ÷ 9 = 5. In the second one, the ‘wing’ digits are also used in 72 ÷ 9 = 8. A conjecture named after them was formed and this was then the start of an exploration:

  • What if we tried dividing by other numbers?
  • What if we used 5 or 6 digits?
  • What if we tried to replicate it with double-digit divisors?

So many possibilities just from a calculation they did by chance. This is the kind of mindset that I want children to have. Ems Lord (director of NRICH) talked about curiosity and resilience in maths recently and how important it is for children to ‘be’ mathematicians. She mentioned a link the OECD has made between being curious and attaining highly in maths.

So often, to make maths ‘fun’, we feel the need as educators to bring in other contexts and hope that because they like football, parties or fashion designing, they will then be convinced that they also love maths. Practical maths in context is important but I think that we should be promoting maths itself, the patterns that are naturally within and the endless avenues of exploration available as being beautiful, full of creativity and immensely enjoyable to study. I think that even with the curriculum being crammed full of content as it is, we can take those concepts, ask ‘what if…?’ and create whole new worlds. Maths isn’t just a set of procedures – it can be created. There is no reason that children can’t do some of that creating too.

If you are interested to find out more, I recommend that you read through the other pages in the Approach section of the menu and look at a few example inquiries on the Isles page. Alternatively, you can watch the video below that explains the approach. If you have any questions or comments, I’d love to hear from you.