There is an analogy in maths teaching which asks ‘are you an inchworm or a grasshopper?’

‘What is an inchworm?’

Now for us Brits, the first question is actually ‘what is an inchworm?’ So just to fill that in straight away for you they are one of those caterpillars that move forward by arching and straightening its body ‘an inch at a time.’

So what have these two insects got to do with maths? Well, they are analogies for two much more complex words and theories that are not in adult-friendly and certainly not child-friendly language — quantitative and qualitative learning styles. Put as simply as I can quantitative – means can be measured by quantity or is measurable. Qualitative – means measured by a looking at the whole and with a description. 

How these two concepts words are difficult to understand or to apply to real situations but the inchworm and grasshopper are easier to understand and explain to children. 

This graphic shows the way inchworms and grasshoppers think and react to learning. Which one do you think you are? What about your child? 

So which insect is it best to be? Successful mathematicians are generally those who are skilled at applying and using both approaches as success in maths tends to require flexibility in thinking.

The most able children within a class will be the ones that can adapt and flip between both styles dependent on what the task is. For them, they are able to be a grasshopper jumping between concepts to notice that a question can be solved more efficiently and with less working out by linking the idea to another and then adjusting the answer. But equally, if a question is a harder or more complex one, they can work carefully and methodically using a set method. 

Looking at this modelled example, the inchworm has learnt how to do long multiplication in columns or even by another pencil and paper method. They see that the question is multiplying so they use that standard method. The grasshopper, however, will look at the specific numbers given and notice how close to twenty 21 is and realise they can do this particular example in their head. Multiplying by 20 is same as doubling then times by 10. They would then add on the extra 32 to make 672. 

Now look at the second example, this time isn’t so clear cut for our more able child. They could do the same double 43 is 86, times by 10 is 860. But this time when they go to add on the extra 43 the numbers are a bit more tricky and they might at this point chose to use a pencil method to add the 860 and 43 to make 903. 

Our third example shows that the more flexible mathemetician will look at the specific numbers involved this time and because they have good number sense, they will realise that there are no easy, quick ways around this. They will have to do it by a standard written method for long multiplication.

Younger children might show their learning styles by linking number bonds to 10 with the specific numbers in the question. The inchworm will work from left to right each time, adding the next number on to their total. They will have to make five additions and cross the tens boundary, which also requires number bond to 10 knowledge. A grasshopper will have good number sense and can link between topics easily. For them, this question because of the specific numbers is very simple. They will notice straight away that there are three pairs of number bonds to ten. 8 + 2, 3 + 7 and 4 + 6. So quickly 3 x 10 = 30. 

Another example is where the inchworm will do the whole standard layout, including the row of zeros. But the grasshopper will spot straight away that the numbers both end in zeros and they can use known facts and then adjust the place value all in their heads. 

So you might be thinking from these examples I have given, why would you want any children to be aiming to be an inchworm rather than this seemingly very smart grasshopper? 

Looking back at the original lists, there are negatives from being a grasshopper. They find it hard to be methodical; they prefer to calculate mentally not on paper and find learning set methods tricky. Lack of these three strategies can be the downfall of a child who only works like a grasshopper.

At certain stages in a child’s education, tests and exams require them to do their working out in a set way. These children will find it almost impossible to do this and will often come away with a low test score as they haven’t been able to conform to the standard methods expected. This is very frustrating for both child and teacher as they are more than capable and given different requirements they would probably excel. Equally, another reason for the downfall of a strict grasshopper is that they will go very fast and miss a crucial step or trip themselves up somehow. 

It would seem that it is better to be a grasshopper, but sometimes grasshoppers just leap in, and there is no thought trail to the reasoning. They miss a crucial clue in the question or misread it and go down a blind alley. They are less logical, and sometimes this means they come unstuck in a way that the inchworm will not as they will be methodical even if this is slower. As the child becomes older and preparing for GCSE getting reliably to the correct final answer every time becomes critical. 

So on things that we can quickly and efficiently do differently, it is good to be a grasshopper, but if something is really tricky, the inchworm is the best approach.

How can we encourage children to be both? 

Questions like ‘is there another way to get to the answer?’ and ‘what other topic uses …?’ can help give prompts and scaffolding, so you encourage a child to think around a question and not just follow the same procedure each time. Also changing the numbers in the question in the way I did. Each time asking ‘let’s look at the numbers this time, is there a quicker way to do it this time?’