An algorithm is a formula or set of rules or steps for solving a particular problem or accomplishing a particular task. This set of rules or steps must be unambiguous, and have a clear finishing point.

Algorithms can be in any language, from English, Spanish etc to mathematical or computer programming languages.

We use algorithms every day. For example, a cake recipe is an algorithm. A set of instructions for checking the oil level in your car is an algorithm.

Algorithms are designed to be helpful and show you an efficient way of doing something. Are they dangerous? It depends. Certainly not intentionally. But the ability to use an algorithm does not imply that a person understands the underpinning of that algorithm… in other words, they may not understand why they’re doing what they’re being told to do.

In Maths, algorithms are frequently used in calculating the answers to addition, subtraction, multiplication and division problems. (They have many other applications too, of course.) However, they do not help children to understand the mathematical concepts behind addition, subtraction, multiplication and division. And when they are used as the sole method for teaching children how to calculate, many children do not gain the understanding of what is actually going on mathematically. In this way, algorithms CAN be dangerous.

Let’s look at an addition problem and use a standard algorithm to calculate the answer:

The usual procedure is to add the right hand column first (the ‘ones’), and if the result is 10 or above, to ‘carry’ the ‘one’ by writing it somewhere in the ‘tens’ column and write down the second digit (the ‘ones’ number) below in the ‘ones’ column. Then we add all the single digits in the ‘tens’ column.

On the face of it, this looks very easy because you only have to add small numbers at any point in the process, and most children find it quite easy to add small numbers. So where’s the danger?

Algorithms generally don’t correspond to the way people think about numbers. Studies have shown that most children stop thinking logically about the numbers they’re working with when they’re using algorithms. They just think about the numbers column by column. If they make an error, they’re unlikely to notice it because they don’t think about the ‘reasonableness’ of their answer.

Algorithms also tend to ‘unteach’ place value. For example, a 7-8 year old child can usually show that 16 is actually 10 + 6 (and that the 1 in 16 actually means 1 group of 10 ones) if they have materials and are focusing just on this sort of partitioning task. When this child is taught to add using an algorithm, however, they are told to add the numbers in the ‘ones’ column (in this case arriving at 16), and then put the 1 from the 16 into the ‘tens’ column. This is where their fledgling understanding of place value is instantly challenged. Often, in this context, the child will not know that the 1 actually means 10 ones, or understand why they’re putting it in the next column to add to the other numbers.

Check your child’s current understanding of place value! Draw 23 circles quickly on a piece of paper as balloons. Ask your child first to count them and then write down the number of balloons on the piece of paper. Do not say the number ‘twenty-three’ aloud at any point, even to confirm they’re right. Instead, underline the 2 in the 23 they’ve written. Then point to the 2 and say, “Use a pen to show me how many balloons this underlined digit represents.”

So should we teach algorithms to young children if they can actually hinder the development of their mathematical understanding?

Current Maths educational theory tells us that it is not a good idea to introduce addition, subtraction, multiplication and division using algorithms. It is fine to teach algorithms as an alternative method when you are sure that a child fully understands these mathematical concepts and can partition numbers flexibly.

So how should we teach addition, subtraction, multiplication and division then?

Just looking at addition, there are many different strategies that are meaningful AND build the flexibility in thinking about numbers that children should be developing.

Here are a couple to think about:

*Near doubles*

For example, 6+5 is almost 5+5 or ‘double 5’. You just have to add 1. 6+5 is also nearly 6+6 or ‘double 6’. If you look at it like this, you just have to subtract 1. Your child should be thinking about doubles and halves, and learning doubles facts, if not already doing so. Doubles facts (eg ‘double 1’ or ‘1+1’ is 2) should be taught using materials if it’s a new thing.

*Bridging to 10* (or 20, 30 etc)

We can work out what 9 plus ‘something’ is very easily. For example, 9+3 can be looked at as 10+2 if we think about it flexibly. By taking 1 away from the 3 and adding it to the 9, 9+3 becomes 10+2. Adding 10+2 (or 10+ anything) is a breeze! This idea is best introduced using two grids with ten squares in each (easy to draw), and counters or small objects. Ask the child to put 9 counters on the first grid (one per square) and 3 counters on the second grid. So 9 is then represented on one grid and 3 on the other. Then show your child how to bridge to 10. Get them to move 1 of the counters away from the 3 and across to fill up the 9 in the other grid to make 10. Tell your child they’re looking at the answer, ie 10+2 (12)!

We could use this strategy to solve our 19+7 problem above. We’d be bridging to 20 in this case, and would end up seeing 19+7 as 20+6. (Children need to be familiar with bridging to 10 before attempting to bridge to 20.)

Aha! You have just given me an insight, for which many thanks. 🙂

I had not appreciated the importance of differentiated delivery of learning at such a young age. My own success stories (or “aha” moments) have been when delivering customised delivery to adult learners.

We lose many children early on in their maths journey because their experiences became too abstract too quickly. As adults, they’ll often say, “I was always hopeless at maths”. When we try to ensure that young children understand what’s going on mathematically, more of them engage with Maths, enjoy it and are likely to become numerate. Algorithms are abstract processes, and so the use of more meaningful calculation strategies involving both standard and non-standard partitioning is recommended initially. It’s wonderful to see a child who has been confused for a long time suddenly have an ‘aha’ moment. Frequently, this does not happen without the use of actual hands-on materials.

I first encountered this material when I was six years old. Of course, I did not understand it fully at the time. What six year-old would? However, as a tool it served me well in the days of pounds, shillings and pence with its variety of numerical bases. My knowledge became fully consolidated when I encountered polynomials at age 16, when the links between my original learning, polynomials, and integer and irrational bases became immediately obvious. My work in computers, where some of my work is in binary, ternary, decimal, hexadecimal and base-64, uses this knowledge base at the level of being an unconscious competent.

The same pattern repeated itself when I learned about vectors, matrices and constrained (linear) optimisation, also at age 16, and it was not until I came to use that knowledge 20 years later that I really understood what it was all about. Again, I now use that knowledge as an unconscious competent.

Was it dangerous for me? Obviously not. Perhaps we are looking at the need for applicability of such learning. Any thoughts?