Where’s the Maths in That? – Maths across the curriculum
Conjecturing and Convincing
This month, instead of looking at where maths appears in other curriculum subjects, we look at a concept  conjecture  that has a place in every lesson on the timetable (English, history, art, for example), but which we feel can play an especially powerful part in mathematics learning. You can find previous Where's the Maths in That? features here.
The character Captain Conjecture appears throughout the new assessment materials. He voices a conjecture as in the example below, and children are invited to discuss whether or not they agree, seeking to convince others that their explanation is correct:

Captain Conjecture says, 'I can double any number,
but I can only halve some numbers.'
Do you agree?
Explain your reasoning.

The process of conjecturing and convincing are important in learning mathematics and are promoted in the middle aim of the National Curriculum to reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language (National Curriculum p3). Sometimes a conjecture may need a qualification, such as in the one above where the conjecture is correct when we are working with whole numbers only, or objects that cannot be dissected. However the statement is incorrect where we allow for fractions or decimal numbers to be involved, since any number can be halved, although this will result in a decimal or fraction for odd numbers. This question appears in the Year 1 materials, where either explanation is acceptable as long as it is accompanied by a valid argument. It is the argument that is important; it demonstrates the child’s ability to give an explanation based on mathematical reasoning. Teachers often comment that children find these types of questions the most difficult In SATs tests. However they provide good opportunities for demonstration of mastery. Not only does the child have to be familiar with the mathematics involved, but they also have to reason about it and communicate their understanding.
The process of conjecturing and convincing is promoted in the book Thinking Mathematically (Mason et al 2010). It is identified as one of a set of natural powers that young children arrive at school with, that can be very useful when learning mathematics. Others that are included are, specialising and generalising; imagining and expressing; and sorting and classifying. These are all things that children do quite naturally in an attempt to make sense of the world around them. They do however need to be nurtured and developed within mathematics lessons if they are to be effectively applied to learning mathematics.
Another example from the Y1 assessment materials is:




Captain Conjecture says, 'All of the glasses contain the same quantity of lemonade.'
Do you agree?
Explain your reasoning.

In this example the purpose of the question is to assess whether a child is able to reason about the capacity of a container. Conservation of liquid is an important concept that children need to grasp. Some children may ignore the width of the container and only focus on the level of the liquid, hence claiming that the three containers contain the same quantity. A child however that has mastered the concept will not only take into account the level of the liquid but also the width of the containers. They will disagree with Captain Conjecture and be able to explain why they disagree.
A further example is taken from Y5:

Captain Conjecture says, 'Using the digits 0 to 9 we can write any number, no matter how large or small.'

This might seem an obvious generalisation; however it may be something that many children have not explicitly thought about before. Although presented in Year 5, it is a question that might be valuable as a discussion point in any year group, where children can think about the numbers they are currently familiar with. In Y5 children should make reference to place value in their explanations: These digits can take on different values, depending on where we place them, so the digit one might have a value of one thousand as in the number 1249 or one tenth as in the number 45.1, and so by placing them in different positions we can construct different numbers, in fact any number. Our place value system is a base ten system and so only uses the digits nought to nine. Once ten is reached, we move into the next column or position to express the value of the number, and ten ones becomes one ten, and ten tens becomes one hundred etc, so we only need the digits nought to nine to express any number.
Most of the conjectures posed by Captain Conjecture lead to a mathematical generalisation. Generalisation lies at the heart of mathematics and is mentioned in the middle aim of the curriculum, as referenced above. Generalisation involves making a statement that is true for all examples. In making a generalisation we cut down on the amount of mathematics to learn, and we deepen our conceptual understanding. Generalisation is a key process in the development of mastery of mathematics. If we understand an idea well enough to generalise it for the range of examples it applies to, then we have probably mastered the concept.
Captain Conjecture appears many times across the assessment materials and is a valuable tool in providing opportunities for children to reason and generalise, and for the teacher to assess their understanding. You can of course make up your own Captain Conjecture statements, as can your pupils. This will support the journey to mastering mathematics.
Reference
MASON, J., BURTON, L., & STACEY, K. (2010), Thinking mathematically, 2nd Edition Pearson
