Getting the basics right: Counting up, Part 2

This blog post from Nick Hart looks at strategies to develop children's conceptual understanding and help them to think with increasing efficiency.

In the first part of this post, we looked at the prerequisite knowledge necessary to deal with a subtraction calculation by counting up.  Quick and accurate recall of number bonds to one hundred can seriously hinder fluency with a calculation like 534 – 187 but once mastered, the child can get on with the business of thinking mathematically – fluent and efficient manipulation of numbers.

If poor number bond knowledge is a possible hindrance, then so too is the number of steps required in calculations such as 623 – 386.  The premise is simple enough – count from the known part up to the whole.  I would suggest using base ten blocks, starting with ones.  The teacher may need to continue counting in ones to point out how incredibly inefficient this is.   


When that’s evident, switch to counting in tens once the next multiple of ten is reached.  At the next multiple of one hundred, switch to counting in hundreds until the multiple of one hundred before the whole is reached.  Then, switch to counting in tens again until the multiple of ten before the whole is reached.  Finally, count in ones once more until the whole is reached.  This may prove to be an unnecessary stage if, as discussed in detail in part 1, children have fluent recall of number bonds to 10 and 100.  It is however very important to understand what’s happening here.
If the child is at the stage of being able to calculate fairly efficiently with the base ten blocks, they can move on to representing calculations pictorially.  They can simply draw dots, lines and squares for ones, tens and hundreds, counting as they go just as they did when they manipulated the base ten blocks.  


In time and with good practice, the child will soon not need to count from one landmark to the next.  At this point, they can begin to represent what they are doing on a number line in preparation for further improvements in efficiency.  A simple number line starting with the known part and ending with the whole is all that is needed, with three jumps such as below.  The child could combine the previous representation of dots, sticks and squares for each jump but this may not be necessary, instead just writing the digit form of each jump will suffice.


Whilst the child is developing fluency here, I would suggest aiming to sharpen their working memory capacity by slowly increasing the amount of information that they need to keep in their head.  To do this, they could decrease the amount of information recorded on the number line.  We can all remember children who, when faced with multistep calculations such as this, forget key bits of information as they grapple with dysfluency.  This could be very simple to start with: ‘There’s no need to write the known part down, is there? Just keep that in your head,’ or, ‘I’m sure we don’t need to write the whole down at the end of the number line – try to keep that in your head.’  Eventually, this can progress to keeping all the landmarks (the next multiple of one hundred and the multiple of one hundred before the whole) in their head, not writing them at all on the number line and only recording the jumps.  Soon, the number line can be dispensed with completely and the child can simply jot the jumps down roughly on paper before adding them.  This gradual removal of the scaffold of the number line should go a long way to encouraging fluency and improving the child’s working memory for this type of calculation.


Once the child becomes fluent with this slight adaptation to the way of thinking, they can be shown shortcuts.  Perhaps the most obvious is to combine the last two jumps into one, after all, that last jump follows a beautifully simple pattern.  Following the same gradual removal of the number line scaffold above, the child could quite soon be calculating this way by doing two calculations and jotting them down before finding the solution.  

The ultimate step in this sequence is of course not needing to jot down the jumps at all but keeping them in mind and performing the calculation entirely mentally.  If the child becomes capable of doing this, you can be reasonably assured that they have developed sound number sense and fluency of calculation for counting up. 
The beauty of this strategy is its application to subtraction with decimals, with larger numbers, or even with fractions. The key though is planning a sequence of work that will enable a child to first develop their conceptual understanding and then invest the time to get them to think increasingly efficiently.
Looking for resources to help support the development of number fluency? Take a look at our Arithmetic Practice Tests and Mental Maths Tests.



Mathematics, Maths

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