Introduction
Strategies for solving arithmetical problems
In becoming more effective mathematicians, children develop a range of strategies to solve problems. These strategies become more sophisticated as their understanding of mathematics deepens and their numeracy skills are further developed. This growing sophistication and depth of understanding is a crucial element in becoming “confident, creative users...of mathematics” (Mathematics K-10 Syllabus © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2012). The foundations of this development begin as students “emerge” into the realm of mathematics, usually through numbers and counting.
Imagine we asked a student: How many buttons are there?
Children learn the forward sequence of number words initially in the same way as they learn the alphabet, as a continuous string. To find the answer to this question, however, they need to know, and apply, fundamental principles of counting. They need to know:
- we say one number word for each item (one-to-one principle)
- we always say number words in a particular sequence when counting (stable-order principle)
- the last number word we say signifies the total (cardinal principle)
- the order in which you count items does not change the total counted (order-irrelevance principle)
- it does not matter what you count, numbers can be used to count anything and the process for counting is always the same (the abstraction principle)
Now imagine we ask: I have added some more buttons. How many buttons are there altogether now?
Students will vary in their responses to this question, and each response provides information about their individual skills and knowledge about numbers and counting. Each approach shows an increasing knowledge of mathematics. Students who:
- count all of the buttons again, starting back at one, are demonstrating that they are developing confidence in their knowledge of the forward sequence of number words or some of the principles of counting
- count-on from nine know the forward number sequence well enough to continue counting from a given number. They know that the initial collection of nine objects has not been changed and so they can use that as a starting point to combine the two collections into one total sum
- bridge to ten are using a more sophisticated strategy based on their knowledge of known facts and the benefits of using landmark numbers. These students make a collection of ten first and then add the remaining amount
Typically, learners need to develop an understanding of counting principles and fluency in number word sequences (forwards and backwards) and be able to state the number before and after a given number prior to learning how to count-on-and-back to solve number problems. Eventually, students will also use known facts, their understanding of numbers and the multi-unit place value system to solve problems.
One of the challenges with inefficient strategies is that even though they are slower, they often still “work”. A student asked to solve 8 + 3 can first count out 8, then count out 3, and finally, count all of the items again starting from one. Their strategy will give them the correct answer, however, it demands significant mental effort and as the number range increases, their strategy will soon become incredibly problematic.
Teacher: If I have 3 buttons and I add 9 more, how many buttons are there in total?
When approaching number problems with addition, students who use more efficient counting strategies would start with the largest number. To do this, however, they need to know, understand and apply a different principle of mathematics, the commutative property. This means knowing that:
• when using addition to combine 2 amounts, the order in which you combine the amounts does not effect the sum (often referred to as “turn-around facts”)
Students who bridge to ten are employing a deeper level of efficiency by using their knowledge that:
- number benchmarks (landmark numbers) are useful when solving number problems; and
- numbers can be used flexibly to suit the thinking of the mathematician. A student could see nine as a collection of nine objects, but also see it as 8 and 1 more, 3 and 3 and 3, 1 less than 10, 5 and 4, etc. This flexibility in combining and partitioning numbers is referred to as part-whole (or sometimes part-part-whole) partitioning and is a vital skill in students using more efficient strategies beyond counting-on-and-back
The importance of known facts
Fluently applying known number facts to solve problems allows students to attend to other features and work with more challenging problems, often with larger numbers. Basic strategies can persist even after students develop more sophisticated approaches. Competent adults will occasionally revert to using their fingers to count on at times because this strategy achieves the correct answer and doesn’t require as much thinking as using more sophisticated strategies.
Developing multi-unit place value understanding
As well as developing understanding in counting principles, becoming efficient at using ‘counting as a problem solving strategy’ and learning to recall and use known facts, students need knowledge of how our number system works. In other words, they need to understand multi-unit place value.
Crucial to understanding place value is the notion of units within units. As students develop skills in counting, a growing understanding of composite units is required to help them progress. Initially, students may use ten as a count, knowing the counting sequence of multiples of ten. However, to understand place value, students need to develop an understanding of ten as a unit (abstract composite units). This is where students see an amount like 10 as being composed of 10 ones whilst also being composed of 1 unit of ten. For example, students need to see that 24 is composed of 24 ones but also 2 tens and 4 more. This ability to flexibly “convert” between units is at the heart of understanding place value. It is much more important than the ability to the state the positional value of numerals (e.g. in 24, there is a 2 in the tens place and a 4 in the units place). This same idea of forming a unit made up of smaller units is also of fundamental importance in measurement and in multiplication.
Developing part-whole relationships
The notion of units within units is supported by such processes as combining and partitioning. Combining refers to bringing parts together; partitioning refers to separating the parts while maintaining a sense of the original number. It emphasises part-whole (sometimes called part-part-whole) number relations. That is, students see both the parts and the whole.
Can you see the 3, the 2 and the 5?
Interpreting number in terms of part-whole relationships makes it possible for children to think about a number as being composed of other numbers.
We often recognise the number associated with a particular pattern straight away, even before we have had time to “count” the items. This normally applies to small numbers of items.
The process of immediately recognising how many items are in a small group is called subitising. This name comes from the Italian word subito which means “immediately” or “right now”. When playing a game with dice we normally recognise the number of dots without having to count them.
The process of subitising can also be used with seeing parts in the whole. If you look at the dot pattern for five you become aware of seeing it also as four and one or three and two. Developing a rich knowledge of numbers such as five helps children to understand how each number is made up of other numbers.
To assist students in forming a clear understanding of the base ten structure of our number system, we can use organisers such as ten frames.
When these are used in conjunction with counters, students can model combining and partitioning in a structured way. Three counters and four counters can be combined to show the total of seven counters.
The ten frame also provides a visual link between seven and ten. Students can see that three more counters are needed to make ten. That is, the “negative image” is visible as well as the “positive image”. It is also possible to use ten frames to explore partitioning numbers. If you ask students what they see when they look at the seven counters in the ten frame some will notice that seven is made up of five and two.
Indeed all the different number combinations or partitions of seven can be explored from the ten frame. If the ten frame is filled from the left it can be used to build students’ knowledge of doubles and near doubles. For example, seven can be seen as one more than six (double three).
This knowledge of number combinations is very useful with addition and subtraction questions. To add 27 and 5 we often partition the five into 3 and 2 to solve the problem. This process of breaking numbers into parts is important in mental calculations.
Just as ten is essential to our understanding of operations on numbers, five often acts as a base for mental calculation. This is due to students’ early use of finger strategies with arithmetic. The ten frame is organised as five squares and five squares, which replicates the first organised material which students use, namely, two hands with five fingers on each.