Once a basic number sense has developed for numbers up to ten (see Developing Early Number Sense) a strong ‘sense of ten’ needs to be developed as a foundation for both place value and mental calculations. (This is not to say that young children do not have an awareness of much larger numbers. Indeed, there is no reason why children should not explore larger numbers while working in depth on ‘tenness’).
Ten-Frames are two-by-five rectangular frames into which counters are placed to illustrate numbers less than or equal to ten, and are therefore very useful devices for developing number sense within the context of ten. The use of ten-frames was developed by researchers such as Van de Walle (1988) and Bobis (1988).
Various arrangements of counters on the ten frames can be used to prompt different mental images of numbers and different mental strategies for manipulating these numbers, all in association with the numbers’ relationship to ten.
For example, examine the three ten-frames below. What numbers are illustrated? What does the particular arrangement of the counters prompt you to think about the numbers? What can you say about each number’s relationship to ten?
Frame A: There are five counters; perhaps seen as a sub-groups of three and two, either by looking at the clusters at either end of the frame, or by looking at the number in the top and bottom rows. Frame B: Again there are five counters; perhaps seen as three in top row and three in the bottom, or as four and one, or two and two and one. It is also noticeable that there are five empty boxes remaining, in a similar shape to the full boxes. This prompts the awareness that ‘five and five make ten’.
In Edward Hopper's painting," Nighthawks", there is a sense of lonliness in the picture. There are four people in a late night coffee shop that is in the middle of a deserted downtown business area. The coffee chop is brightly lit and there are no decorations of the walls or counters. There is a curved wall of windows where all four of the patrons can be seen.There is a man and a woman together at ...
Frame C: This arrangement strongly illustrates the idea that ‘five and five make ten’. It also suggests the idea that half of ten is five. This type of thinking would not occur if the five counters were presented without the context of the ten-frame.
Plenty of activities with ten-frames will enable children to automatically think of numbers less than ten in terms of their relationship to ten, and to build a sound knowledge of the basic addition/subtraction facts for ten which are an integral part of mental calculation. For example, a six year old child, when shown the following ten-frame, immediately said, “There’s eight because two are missing.”
This child had a strong sense of ten and its subgroups and was assisted by the frame of reference provided by the ten-frame. Once this type of thinking is established, the ten-frame is no longer needed. Although dealing with whole numbers initially, the ‘part-part-whole’ thinking about numbers supports the understanding of fractions, in particular tenths.
‘Ten’ is of course the building block of our Base 10 numeration system. Young children can usually ‘read’ two-digit numbers long before they understand the effect the placement of each digit has on its numerical value. For example, a 5 year-old might be able to correctly read 62 as sixty-two and 26 as twenty-six, and even know which number is larger, without understanding why the numbers are of differing values.
Ten-frames can provide a first step into understanding two-digit numbers simply by the introduction of a second frame. Placing the second frame to the right of the first frame, and later introducing numeral cards, will further assist the development of place-value understanding.
Ten-Frame Flash (5-7 years) 4 players
Materials: A dozen ten-frames with dot arrangements on them, a blank ten-frame for each child, counters. Rules: One child shows a ten-frame for a count of three, then hides it while the other children place counters in the same positions on their frames from memory. The ‘flasher’ shows the card again and helps each child check his/her display. After three cards the next child becomes the ‘flasher’ and so on, until everyone has had a turn.
In this task I am going to describe Jean Piaget’s and Tina Bruce’s theories about how children’s understandings of mathematical develop. Piaget Jean Piaget’s research led him to believe that we develop by taking in information, which is then processed by the brain and as a result of this our behaviour changes. He started that there are staged of development that children move through. The ages are ...