Inchworms and grasshoppers: thinking styles in maths

|3 min read

Editor’s Note:

This is an updated version of a blog post published on Oct 2 2019

Every child has a different thinking style, and when we teach maths, we need to factor this in. Teaching with the diversity of thinking in mind makes maths experiences much more positive.

The way that a learner processes information is called their cognitive or thinking style. Knowing a child’s cognitive style gives you an insight into how they approach their maths work. If children don’t learn the way we teach, then we have to teach them the way they learn.

Sticking to one style of teaching can frustrate children. It’s at this point they can switch off and opt-out.

Inchworms and grasshoppers: two maths thinking styles

There are two distinct maths thinking styles for children learning maths. You might be thinking that this sounds very binary, but this isn’t about being in one camp or the other. These are extremes at the ends of a continuum. The most capable maths thinkers are skilled at moving between the two styles when solving problems.

In his work on identifying thinking styles in maths, influential author Steve Chinn has landed on two distinct styles: inchworms and grasshoppers.

Still with us? It may sound odd, but knowing how inchworms and grasshoppers think helps us consider our own approaches. Your style of teaching is likely to use a bit of both, but at the same time, it could lean more towards one more than another.



First approach to a problem

Focuses on the parts and the details and sees topics in isolation. Fails to have an overview.

Overviews, puts together and tends to link topics. Tends to have a holistic view.

Looks at the numbers and facts to select a suitable formula or procedure. Prefers to follow a rule.

Looks at the numbers and facts to estimate an answer, or narrow down the range of answers. Prefers controlled exploration.

May have a poor grasp of concepts.

More likely to grasp concepts.

Solving the problem

Formula, process-oriented. Prefers to follow a procedure.

Answer-oriented. May redesign or simplify the problem.

Constrained focus. Uses one method.

Flexible focus. Uses a range of methods.

Tends to have inflexible focus, identifies the question only not in relation to previous learning.

Flexible in focus — may use a range of methods based on previous learning.

Works in serially ordered steps, usually forward.

Often works back from a trial answer.

Uses numbers exactly as given.

Adjusts, breaks down or builds up numbers to make an easier calculation.

Prefers pencil and paper methods. Documents methods.

Performs calculations mentally and intuitively. Rarely documents methods.

Checking and evaluating answers

Finds checking solutions difficult and unlikely to evaluate an answer. If a check is done it will be by the same procedure or method.

Likely to appraise and evaluate an answer against the original estimate. May use an alternative method to check an answer.

Often does not understand procedures or values of numbers. Tends to reproduce procedures by rote (often inaccurately). Works mechanically.

Good understanding of numbers, methods and relationships.

Encourage flexibility in learning maths

In working out a word problem, inchworm thinkers might work things out step by step.

A grasshopper thinker might provide the right answer without any working out. Frustratingly for a teacher, and when asked how they solved the problem, the learner might respond, “I just knew”.

“I just knew” can be an issue for questions that require pupils to show their working out. Grasshopper thinkers can struggle at higher levels because this evidence is often needed. The flexibility in learning is essential. Try to encourage your grasshoppers to try inchworm methods and vice-versa.

Grasshopper may excel at mental arithmetic, whereas algebra is more inchworm friendly.

Many children use both cognitive styles. But keep in mind, it might be harder to reach extreme inchworms and extreme grasshoppers. Teaching to their style may be more beneficial than trying to force flexibility.

The importance of knowing your own thinking style

As teachers, our own maths thinking plays a huge role. Teachers who are uncertain with their personal maths knowledge may regress to the inchworm ‘formulaic’ methods.

As teachers, we need to reappraise our cognitive style. Ask yourself, whether you are a flexible thinker and whether flexibility permeates your lessons. Look at the maths lessons you teach. Were they more inchworm or grasshopper, or a mixture of both?

One of the ongoing challenges of teaching is managing learner individuality in a classroom. When it comes to maths thinking styles, encouraging flexibility is key to your classroom’s success. Because diversity is a strength and it is important to have an inclusive approach to maths teaching.

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Bath, J.B., Chinn, S.J. and Know, D.E. (1986) The Test of Cognitive Style in Mathematics. East Aurora, NY. Slosson

Chinn, S. J. (2004) The trouble with maths: A practical guide to helping learners with numeracy difficulties. RoutledgeFalmer: London