Making Maths Stick: The Power of Spatial Thinking

Encouraging more spatial thinking in maths can have a big impact on children’s confidence and overall ability. Gill Cochrane explains the close link between spatial thinking and reasoning skills and outlines some practical things teachers can do to help. [Don’t miss our free resource at the end].

Whilst teaching maths as a primary school teacher, I became fascinated with the idea of ‘semantic glue’ – or what makes information ‘stick’ in our memory. The pursuit of answers to ‘the glue question’ led me to undertake a psychology degree. During my studies, I came across these words from the Nobel prize-winning scientist Eric Kandel: 
 

“Memory is everything.
Without it we are nothing.“
– Eric Kandel

Kandel studied Californian sea slugs to reveal how memories are formed. This research helped demonstrate how memory supports the process of monitoring for safety when we are moving through space. Human functioning and reasoning essentially evolved to serve this need. Grasping this powerful idea allows us to appreciate how inextricably linked spatial processing and reasoning are.

There are three essential steps in both processes:

• adding new information to what you already know and hold in memory
• relating and connecting this information to what’s already known
• using this new information to achieve a purpose or find solutions to challenging situations.

Essentially, both boil down to extracting meaningful patterns from the environment. Understanding this helps to explain why the latest research into how children learn maths has demonstrated a causal relationship between spatial thinking and academic outcomes, prompting increasing calls to spatialise the maths curriculum. It could also explain why weak verbal and spatial working memory are commonly associated with maths difficulties and maths anxiety. 

So how can we, as teachers, exploit the connection between reasoning and spatial skills to help re-address difficulties with maths learning?

Pattern detection and relational reasoning skills

Relational reasoning makes a unique and positive contribution to maths outcomes. Giving pupils early and explicit experience in rule-following will improve their ability to monitor sets of information for patterns.

Attribute blocks vary in shape, thickness, colour and size; early sorting work with these blocks encourages learners to observe and describe relative differences and similarities and boosts maths vocabulary. Venn diagrams give a great framework for further explicit categorisation work using the blocks. Children can then build on the knowledge gained sorting shapes to generate patterns. This is an important skill, as it involves the application and creation of rules.

Activity 1 – Sort it out: Present children with an assortment of attribute blocks and a simple rule card that says, for example, ‘circle’. Ask them to sort shapes according to the rule. Gradually make this more complex by adding more than one rule for example, ‘circle’ and ‘thick’. You can see this in Figure 1.
 

Activity 2 – One at a time: Give pupils a set of attribute blocks each and ask them to play a domino game where the rule ‘only one change at a time’ governs the flow of play. They will need to take turns to put down blocks between them, but they won’t always be able to go. An example sequence from a game is shown in Figure 2 below.
 

For progression, you could also ask pupils to work together or independently to determine the steps they must go through in order to get from an initial state to an end state in the shortest number of moves by changing only one attribute at a time.  For example, ‘How many steps does it take to get from a large, red, thick circle, to a small, blue, thin triangle?’  

Activity 3 – Patterns: You can encourage children to discuss their reasoning by providing prompts to structure their analysis of a pattern. In Figure 3 you can see how pupils are asked to fill in information about every stage of a pattern. For example, has the number of objects changed? Are they the same shape? The grey ‘difference diamonds’ featured in the speech bubbles prompt the recording of this crucial bit of between-item information. Providing the numerical term positions on the roofs helps to structure discussion of the pattern and predictions of the next item in the series. Developing pattern detection skills like this will help children move on to more complex non-verbal reasoning.

Analogical reasoning

This is perhaps the most familiar type of reasoning, and it features in many popular matrix tasks, such as Figure 4. In activities like this, the learner needs to choose the right shape to complete a pattern. Information can be deduced by looking at the relationships between the two other given sequences; potential answers are displayed immediately below each question.

It’s important for us, as teachers, to understand the steps in the exploration that lead to the solution. 

Reasoning by anomaly 

This involves working out the rule behind various patterns and then spotting the sequence that isn’t following it. Look at Figure 5. By studying each of the patterns and the interrelations within them you will start to appreciate how practice in this type of task promotes structured scrutiny of the relationships within and across the set items.

Reasoning by antithesis 

This requires learners to identify when a parallel or opposite rule has been applied. This is more complex, as the process must be identified so the sub- item featuring the opposite process can be chosen. Practice with all these types of non-verbal reasoning tasks is done without processing number systems, allowing analytical skills to be strengthened separately from numeric work.

There is evidence that this type of reasoning practice can help when it comes to work with numbers.

Purposeful scanning

A rich body of research demonstrates that purposeful scanning of information can significantly aid the ability to solve problems. There are indications that providing practice in explicitly noticing relationships within written equations can boost performance on similar maths problems. For example, building meaningful gesturing into the appraisal of sums can lead to better performance on tasks involving the equivalence concept. 

Ask pupils to use their index and middle fingers to touch each number before the equals sign (and any after it) and then to use their index finger to point to the missing information. (See Figure 7). The gesturing enhances the processing of the equation as a whole, working from left to right.

Changing how we present sums can alter the way pupils monitor the equations set and boost performance. Re-organising the presentation of a problem (see Figure 8), promotes scanning of the whole sum, and examination of the interrelationships within it.

In time, with supporting, structured discussion, learners can develop flexibility in strategy use when calculating, which can reduce the need to calculate in some cases. The children will learn to exploit the inter-relations to reduce the cognitive load of computation.

Read also: Understanding maths difficulties

Organising numbers and fractions spatially

Recent research has shown that lower levels of maths anxiety are found in people who ‘spatialise’, organising sequences from left to right in verbal working memory.

Here are some ways we can help children spatialise their mathematical understanding:

• Number lines and bar models should be modelled as important organisers of computational thought.
• Work on fractions should include their ordering on number lines to make relative value explicit.
• Cross-sectional drawing can provide an additional way to map space to solve word problems.

There is a causal link between spatial thinking and achievement and confidence in maths. Spatialising maths builds an appreciation of the relations between and within objects and ultimately contributes to a more secure mathematical understanding in the longer term. 

Gill Cochrane is a former primary school teacher. She is now the lead developer on the specialist literacy and maths courses run by Real Training in partnership with Dyslexia Action.

Find out more by visiting realtraining.co.uk/maths-courses

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