Number bonds - the foundations for future success in maths

As a Kumon Instructor, I am sometimes asked why we require repeated practice of basic mental addition without allowing our students to count on their fingers? If they get the right answer using their fingers, why do they need to keep practising 3+2, 8+2, 6+2, mentally?  

Why do many of our students, even those who are already dealing with multiplication or fractions at school, often start the Kumon programme on basic addition (plus 1, plus 2, plus 3) before progressing to mixed addition and subtraction, multiplication and division, fractions and higher maths? 

The answer is that we are laying the foundations for future success in more advanced maths. Most parents accept that it is important to know the multiplication tables off by heart - we take the view that it is just as important to have automatic recall of the number bonds.

Basic Number Bonds
Think about it; if you know instantly that 8+2 =10, then you also know instantly that 2+8 =10, that 10−8 =2, and that 10−2 =8. That is, you can successfully and immediately answer four questions based on immediate recall of one number bond. Compare the time likely to be taken and the errors likely to be made by a student counting on (or counting backwards) for each of these sums separately.

This is why primary schools focus in the early years on the number bonds up to 10 (1+9, 2+8, etc.) and up to 20 (16+4, 12+8 etc.). However, on its own, knowing number bonds up to 20 is not sufficient to help a student dealing with column addition or subtraction.

Column Addition and Subtraction
Take the example of 198+477. If a student is dependent on finger counting or mental “counting on” to find out the answer to 8+7, by the time he finds the answer, he may well have forgotten what he is doing! The working memory can only hold information for a short period of time. A student who recognises immediately that 8+7=15, can write down the 5, carry the 1, and move to the next column (9+7). Again, immediate recall that 9+7=16 will enable the student to move to the next stage, adding back the carried number to make 17 and then carrying the 1 to the third column.

A student who relies on counting on his fingers, 'counting on' in his head or 'counting on' making pencil marks, may lose track, take far longer, may be prone to make errors, or forget the purpose of his counting. This is why the Kumon programme emphasises fluency and accuracy in addition of all single digit numbers. Students often practice +6, +7, +8, etc. until they can accurately answer, for example, one hundred addition questions involving +8 within a target time of less than ten minutes.

Long Multiplication and Division
Fluency becomes all the more important when a student moves on to long multiplication and long division. To multiply 348 by 37, a student needs not only to have quick recall of his 3, 4, and 8 times tables, he also needs to be able to add together the resulting 4-digit and 5-digit numbers quickly and accurately. When carrying out my initial assessment of a student’s abilities in maths I often find that he or she will make basic errors in the addition/subtraction element of long multiplication/division whilst successfully managing the more complex task of multiplying/dividing - an indicator that the student has spent more time memorising his times tables than was committed to memorising the number bonds.

Factorisation and Quadratic Equations

Leaping forward to factorisation and quadratic equations, again, in order to factorise a quadratic equation quickly and accurately (e.g. x² + 6x + 8 = 0, which factorises into (x + 2)(x + 4) = 0), a student needs not only immediate recall of his multiplication tables (in this case the factors of 8), but also the ability to see quickly which of the factors of 8 also add up to 6. Picture how long it will take a student who is still dependent on finger counting to hold these pieces of information in his head whilst calculating the correct solution.

The moral is that it is not always helpful to push a student on too quickly when they are learning the basics of addition, even where it appears that the student is capable of 100% accuracy on the basis of his or her methods. The good news is that the young student’s brain is very effective at memorising the information gained from practice and praise for their achievements. Once memorised, that information will remain there for the rest of the student’s life - and certainly for the duration of his or her academic life—providing the solid foundations needed to ensure future academic success in maths.

Anna Dalglish is the Instructor of the Kumon Guildford North Study Centre in Surrey. She has nine years’ experience of working with Kumon students and has a long-held interest in education.

“I was motivated to move into the field of education because I wanted to inspire others with the same feeling. I am myself educated to post-graduate level and have a great love of learning, including learning into adulthood.”

"I was particularly drawn to the Kumon Method of Learning because of its ethos of encouraging students to become independent learners. I believe that this is where a student finds real fulfilment - not depending on others for their achievements but experiencing the self-esteem that comes from seeing the rewards for their own efforts.”