Reasoning Skills

The reasoning skills are repeatable across any island. They help provide consistent vocabulary and through explicit highlighting of them, the children are able to develop their reasoning and problem-solving skills across primary maths lessons. They aren’t used in a specific order although there is often similarities across inquiries. They all help with making discoveries and unlocking treasure – the goal in the Isles of What If…? All the skills use the analogy of a team of adventurers exploring an uninhabited island to help the children understand them. They aim to promote the use of the 8 mathematical powers identified by John Mason as being integral to mathematical thinking as well as those that will deepen or advance the inquiry.

Search inquiry reasoning skill in maths

The Search skills define the way that new data and examples are created by the children. In an inquiry context, rather than the children having a list of questions, they are (in a guided manner) creating data of their own choosing. For example, in Distinct Digits to 5555, the children are choosing the digits within their addends that they are using in their addition. Roaming is the creation of examples without a clear strategy – often to get a feel for a situation or early on in the exploration of an island. Combing is where children make small, systematic changes to each one. For example, calculating 2943 + 2612 and then 2945 + 2640 to get a different score. Seeking is where the children create data with a specific goal in mind. They might have a score of 1 to 4 and are trying to get a score of 5 for example. It is possible for the children to be using all 3 Search skills in quick succession. They aren’t mutually exclusive.

The Organise skills are about taking the examples that they have and arranging or comparing them to help in identifying patterns and making discoveries. The Arrange skills involve putting the data into a list, table or diagram. For Distinct Digits to 5555, you might arrange the calculations into a list or table by the scores that they achieve. Rather than just having data randomly across the page, it can really help to have them logically organised to spot patterns. The Compare skill is all about taking one piece or group of data and then comparing it to another. How might the additions that score 8 be different to the additions that score 4? The Classify skill is about naming groups. The naming of groups provides ownership and meaning to them. It highlights similar properties.

The Discover skills are akin to finding the location of treasure – an ‘X’ on the ground. Notice is about spotting something in the data that has already been generated – there is no predictive element although that might come instantaneously with it. For example, they might notice that all the scores of 7 that they have already created involve a 4-digit number and a 3-digit number. The Conjecture skill is about making a prediction about examples or data that hasn’t been created yet. They might conjecture that a score of 9 or 10 is impossible. They haven’t tried every single possible combination but they believe it to be true. It’s important to realise that a conjecture has an element of doubt in the child’s mind. It isn’t something they already know to be true. The Adapt skill is about taking an existing conjecture and altering it somewhat to make a new conjecture. This is important as sometimes a conjecture is proved false with a counterexample but that new evidence causes a new, slightly different, conjecture to be made. For example, they might conjecture that a score of 9 is impossible, they then might conjecture that a score of 9 is possible with decimals when they think of that possibility.

Investigate inquiry reasoning skill in maths

The Investigate skills are about using or creating data to explore or analyse a conjecture. It’s akin to digging the ‘X’ or examining clues. Inspect involves looking at existing data, including that of others, to validate a conjecture. Someone might make a conjecture based on their own examples and then look at what data others have to see if their conjecture is also true in those cases. Dig involves creating new data to try and alter the likelihood of a conjecture being true – perhaps trying to find a counterexample. Expand is about creating unusual data to try and test the validity of a conjecture. For example, a conjecture might be made and children might try decimals as an expansion of data to see whether a conjecture still holds true in that context as well. They hadn’t used decimals previously. Often, the Investigate skills are used in combination with the Search skills. Through these, we reveal the treasure chest – although it is locked until we are convinced of its validity.

The Argue skills are all about trying to prove something in the minds of others or explain something to them. Persuade is about using reasoning to explain an idea and convince others. Show is using diagrams or representations (possibly formed with resources like Numicon) to explain an idea. Prove is the key to unlocking the treasure. Once the class has been convinced of a conjecture being true or false, it is considered proven and the treasure is unlocked. The degree of evidence required is up to the class. It might just be enough data or it could be a more formal proof through reasoning. It’s really important for the children to realise that whether or not a conjecture is proven true or false, it is still treasure. We know more about the situation (and island) as a result and so it is a discovery no matter what. It is important that the children don’t see a conjecture proven false as a failure but instead still treasure. The size and type of the treasure is a good reflection to make about how significant the finding is.

The Explore skills are about deepening the inquiry and exploring more of the island or going to a whole new island completely by asking ‘what if…?’ questions. It is a fundamental aspect of the approach. Reorient thinks about changing the constraints a little to alter our focus. For example, using 3-digit numbers rather than just 4-digit ones. When the children Chart, they are effectively Reorienting in a systematic way. Trying 3-digit, then 2-digit, then 1-digit numbers for example. When the children Depart, they are leaving the current island by asking a ‘what if…?’ question to change the rules and explore a new inquiry. This allows the children to go in a unique direction that you as a teacher perhaps hadn’t considered although it also could come from the teacher. It gives them real ownership of what they are doing and this creates excitement and investment in their learning. For example, they might try and use subtraction rather than addition. The Depart skill enables the children to think creatively and expand the scope of their exploration.