# Maths: the big ideas

This month, Nick Hart takes a look at the big ideas of mathematics and how they can help children to connect different areas of mathematical understanding.

The beauty of maths lies in the interconnectedness of ideas and concepts yet this concept of relationships is often lacking in children who struggle with maths.  Mike Askew, Professor of Education at Monash University, Melbourne, has written about what he calls the big ideas of maths. These help children to connect different areas of mathematical understanding, yet are small enough to understand in their own right.

His big ideas are:

Position on a number line - when considering numbers as discrete quantities, they can be assigned a place on a number line.

Place value - the fact that 10 of something can be worth one of another is the root of our base 10 number system.

Equivalence - the idea that two or more things can have the same value.

Symbols - the idea that complex mathematical thinking can be represented by simple symbols.

Estimation - instead of a precise calculation, approximate the solution to a problem.

Classification - sorting numbers, shapes or calculations based on their properties.

Patterns - identifying relationships between two or more numbers, shapes, objects or calculations.

Numerical reasoning - the relationship and differences between additive and multiplicative reasoning, rather than seeing the four operations as separate entities.

These big ideas transcend national curriculum objectives.  All of them can be developed at any stage of the primary curriculum and with almost all topics.  For example, classification can be used in the Early Years for sorting numbers which are more or less than 3, or it can be used in Year 6 for sorting percentage-of-a-number questions that have an answer of more or less than 20.  These are not objectives in their own right, but broad areas of understanding that underpin great mathematical thinking.

The most obvious way of developing these areas of thinking is to design tasks that require children to pay attention to and think hard about these ideas.  Let’s take the simple example of columnar subtraction.  We’d teach the standard algorithm but there are many further possibilities using these big ideas:

35,041 – 19,553

Position on a number line

On an empty number line, from 0 to the whole (35,041), represent the part to be taken away and the position of the remaining part.  Children need to make judgements about where on the number line they would mark and in doing so, develop their number sense alongside the calculation.

Place value

If you multiply the whole by 10 and then complete the calculation, can you predict the new solution? By doing this, children will be thinking about scaling which can help with trickier calculations later on.

Equivalence

So many children misunderstand the equals sign as ‘the answer is coming’.  By giving children regular opportunities to find equivalence in mathematical ideas, this misconception can be eradicated.

Here’s an example:

Symbols

As a way in to algebraic thinking, we can simply provide some questions where symbols are used instead of digits where the task is to figure out the value of each symbol, like this:

Estimation

The obvious task is to estimate the answer to the calculation first.  However, children could also estimate the number of solutions to this problem, for example:

Classification

Classifying is not just for shapes.  Children could sort calculations based on observable characteristics such as:

• Those that require exchanging and those that do not

• Those that have the solution the same as what is being subtracted and those that do not

• Those that will give an answer of 3000 or less and those that will give an answer of 3001 or more.

Patterns

Patterns are abundant in maths and finding them out is rather satisfying.  Children could generalise – what happens when you add 10 to the whole?  Does the known part change?  What about the unknown part?

Numerical reasoning

Here, children should be given the chance to manipulate calculations.  Take the question given and re-write it as an addition statement or with the ‘unknown’ in different positions in the calculation:

Askew’s big ideas are fantastic starting points for task design and if children habitually think about these ideas, their mathematical thinking can be finely attuned.  There are so many opportunities to get children to make connections by designing tasks from the big ideas, whilst also exposing possible misconceptions.  Here are some of my favourites:

Position on a number line

• When adding, subtracting, multiplying or dividing fractions, get children to represent the calculations on a number line.

• When looking at squared numbers, plot them on an empty number line.  Do children see the exponential gap between each number?  Can they space the points appropriately?

• Use number lines to represent the time so that children can see that a clock is a number line.  Can children represent the time accurately with just one arrow?

• Represent the perimeter of a shape on a number line:

Equivalence

• One fifth of a 30 is the same as a quarter of what?

• Draw a square with the same perimeter as a given triangle.

• Draw a rectangle with the same area as a given one but with different dimensions.

Classification

• Sort shapes into those where the area and perimeter are the same value and those where they are not.

• Sort these word problems based on whether the whole is unknown or a part is unknown.

Working on these big ideas absolutely complements a mastery curriculum.  Planned thoroughly, children will revisit key ideas many, many times over their primary years.  If used as tasks, teachers can develop units of work to deepen children’s understanding whilst giving children that struggle more time to work on the basics, ensuring that all children have the opportunity to master national curriculum objectives whilst also catering for the needs of children who grasp concepts quickly.

Interested in resources to support and embed a mastery curriculum? Take a look at Rising Stars Mathematics,  a whole-school mastery textbook programme that uses a CPA approach to help all children master the new curriculum.

Mathematics

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