The Challenges of Number.
Numbers are all around us; we see them on prices and buses and use them for our PIN codes, but some people struggle to make sense of numbers, they seem alien and incomprehensible.
The 2011 Skills for Life survey in England found that 24% of adults have numeracy levels below those expected in a 9-11 year old child. This means that these adults would not be able to understand pricing on pre-packaged food or household bills, and those with particularly weak numeracy would struggle to withdraw cash from a cash machine and even to select the correct floor number in an elevator.
Although a similar study has not been undertaken in the US, the “National Report Card” – the National Assessment of Educational Progress (NAEP) has found that only 23% of students are proficient at mathematics in Grade 12. So what are the processes that we go through to gain an understanding of number and then onto mathematics?
Acquiring Number Skills
A child learning about mathematics must first make connections between four key components in number and counting:
- their concrete experiences of number,
- the symbols that represent number,
- the language used to describe numbers,
- and pictorial representations of number (such as the number line, an example is given below).
For example, as a child counts out loud whilst pointing to objects in a set, they are making a connection between the concrete experience and the language of numbers and counting.
Numerosity and Magnitude
These kinds of connections allow the child to understand what a number represents. A question such as ‘what is three?’ usually produces two different types of response. The first is that it is a set of three things, specified or not; and the second is that it is the symbol ‘3’. However, the symbol ‘3’ is not the number three, it only allows us to represent the concept of ‘three’. The actual symbol used is arbitrary; indeed III represents the same concept in Roman numerals (note that the latter is a less arbitrary symbol because it does contain three separate segments).
The idea of a set of three does lend itself to understanding number and number is often explained in this way; children are presented with different sets of three things and made aware that ‘three’ is what these sets have in common. To see this equivalence, a child can do one-to-one matching so that three sweets are matched to three teddies and note that they are the same despite physical differences, thus in both sets there are three of them.
Connecting a symbol with a concrete situation such as sets of three things is known as the cardinal aspect of number. An understanding of cardinality is part of the US national curriculum for Kindergarten and is expected to be attained during the first year of formal schooling.
Difficulties with mathematics, particularly arithmetic, can be due to a failure to establish this core concept of cardinality. Sometimes problems can stem from other developmental difficulties that are not necessarily associated with mathematics. For example, children with language impairments may find it difficult to establish the connection between non-verbal properties and the language used for numbers and counting. For others, difficulties with cardinality could be due to deficit of a core number sense which means that numbers lack meaning and may only be learned through rote memory.
Order and Nominal Aspects.
However, cardinality is not the only aspect of number that a child must understand from the outset. Two other properties of number are the ordinal and nominal aspects. The ordinal aspect means that a number can provide information about the order of things. So that if something is labelled ‘3’ such as page 3 in a book then it is the third thing, or page. Moreover this number tells us the relation of the page to other pages, in that it follows page 2 and is followed by page 4. This kind of understanding is important for pictorial representations of number such as the number line (see below), where the symbols for numbers are connected with the ordinal aspect. Number lines are used extensively when teaching the arithmetic operations as they demonstrate the process of adding or subtracting. For example, if the sum 4 + 2 is presented, a child can place their finger or pen over the numeral 4 and make two jumps up the number line. This will lead them to the answer 6.
In contrast, nominal aspect is where a number is just a label used to distinguish between items. Here the number symbol does not represent a property and provides no information on the order of things. Examples of this type of number understanding are from telephone or bus numbers. Thus numbers as symbols can represent very different situations and the child must understand and discriminate between each of these properties very early on to make sense of the everyday use of number around them.
The ordinal aspect of number introduces a spatial element into children’s understanding of number; the magnitude of number increases as we move from left to right. Evidence shows that not only do we learn to represent this visually on a number line but that we have an equivalent mental representation too. Children (and adults) use this spatial representation to work out the magnitude of numbers. However observable problems with the ordinal aspect of number are not limited to those with mathematical difficulties. It has been shown that those with developmental disorders such as dyslexia also show difficulties with sequences that can be represented orderly such as letters and months of the year.
An understanding of time combines many of the key components of number. For example there is an ordinal aspect; 2 o’clock follows 1 o’clock on a pictorial representation of number on an analogue clock face. However teaching young children to tell the time is difficult and they find it much harder than adults anticipate. Despite a good understanding of number, it is not until around 7 years of age that children can tell the time and even then it is not clear that they have a well-developed concept of time. So why is it so difficult?
One reason could be that time on the face of a clock is a linear scale presented as a circle and thus appears very different from the usual linear number line. However clock faces are more complicated than this as each numeral has at least two meanings. For example, the ‘1’ can represent 1 o’clock and also 5 minutes past the hour and because there is information from the two hands on a clock face, integrating this information can be confusing. Furthermore, underlying the representation of time is understanding time as a concept. To understand this aspect of time, children must appreciate that events occur in a temporal order and have a sense of the amount of time between events.
This understanding is often contrasted with lessons on telling the time which present children with many different clock faces and ask them to work out the time depicted. In these lessons, 7 o’clock may be followed by half past three and then quarter past nine. So although the child is able to interpret a clock face, they have no concept of the duration between different time periods.
Difficulties understanding the concept of time are compounded by the language adults use to describe time. Time vocabulary is extensive; not only do we have words to describe the hours and minutes of time, but we can talk about longer durations such as days, weeks and months as well as different seasons, years, decades and millennia. Add to this the use of idioms such as ‘just a minute’ or ‘once in a blue moon’ which are often used by adults in a loose sense (they do not really mean they will take only a minute), and it is not hard to see why understanding time comes sometime after the understanding of number. Nevertheless telling the time is an explicit element of the US curriculum. An understanding of time has been introduced for Grade 1 at a very basic level; tell the hours and half hour time. The level of complexity increases through Grades 2 and 3.