Digging Deeper into … Comparing and Ordering (infants to second class)
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Of all the strand units in maths, this topic is one that is very close to the hearts of almost all young children:
- “She’s got more than me! That’s not fair!”
- “I want to be first!”
- “I want to be the biggest!”
This strand unit evolves from the separate strand units of comparing and ordering that, along with the other two strand units of classifying and matching, make up the strand of early mathematical activities. The content objectives for this strand unit are quite similar across the four junior classes, with the main difference being the specific number limits for each class level:
Number > Comparing and Ordering >The child shall be enabled to:
- compare equivalent and non-equivalent sets (to include the symbols <, >, = in second class)
- order sets of objects by number (infants to first only)
- use the language of ordinal number
As mentioned above, even from when they are very young, most children are quite adept at comparing what he/she has with that of another.
As part of the strand early mathematical activities (i.e. pre-number) the children will already have had experience comparing sets by quantity (but without counting) i.e. identifying which of two sets has an obvious amount more (or less) than another. They will also have been identifying two sets/objects as being the same or different.
In Junior Infants, once children are comfortable establishing the cardinality of sets up to five, the next step is comparing and ordering sets of objects up to five. Since the amount of these sets may often only differ by just one or two, then it is not very obvious, from a visual point of view, which one has more or less. Comparing two similar sized sets requires that the child:
- Can identify (and, later, write) the correct numeral for that set
- Understands one-to-one correspondence, and using this can match the items in the two sets, so as to establish which one has more or less
- Understands the conservation of number i.e. that a short line of five objects situated close together still has more than a longer line of four objects further apart.
- Does not assume that the quantity of a set with objects bigger (or smaller) in size must be greater (or less) than the other set.
How many more?
Once a child is able to identify the greater set, the next step is to be able to state the difference between the sets i.e. how many more plates than cups? This can be a very difficult concept, with which children can struggle for many years.
As with the entire Operation Maths programme, a CPA approach is recommended when teaching this concept and, in particular, to use that which is most familiar to the children:
- Use items that typically go together eg knives and forks, cups and saucers/plates, children and chairs/coats/school bags. Take a number of each and ask the children to suggest how we could ascertain the number of each. If not siggested by the children, the teacher should demonstrate how to set out the items in groups toegther eg the first knife with the first fork, the second knife with the second fork etc. If the quantities of each are not equal/the same, ask the children to explain how many more of the lesser quantity is required AND to explain how many extra items there are in the larger amount.
- In a mixed classroom, use girls and boys. Call up a random group of children, ask the boys to line up at the top of the room, and the girls to line up in separate line beside them, so that, where possible, each child is adjacent to one other child in the other line (if you are lucky enough to have square tiles on your floor, ensure that there is a child standing in each square space). Ask the children to identify the children who have a match/partner on the other line and the number of children who do not have a match/partner on the other line. This activity could also be repeated using dolls and teddies, toy farm or zoo animals, attribute bears etc.
- Use concrete manipulatives and pictures. Start with only two sets initially. Impress up on the children that the easiest way to see the comparison is to “line up” the objects, was done with the children previously. Use a grid of squares* to help with this. Once again, ask the children to identify where there is a “partner” fruit on the other line and the number of fruit that do not have a “partner” on the other line. These are the extras. How many more (extra) bananas than apples? How many more (extra) bananas than strawberries? *The 5×5 grid on the Operation Maths Sorting eManipulative is very useful here. The Operation Maths 100 Square eManipulative can also be used; select to show counters only and line up two (or more) rows or columns of different colours.
- Ultimately, it is hoped that the children realise that it is not necessary to establish the exact amount of each set to be able to establish the difference between each set. In the example above, there are two more bananas than strawberries, and it is not necessary to identify the number of each fruit to establish this. This encourages the children to develop efficiency and flexibility in their approaches.
- As the children move into first and second class, they should still be encouraged to “line up” the sets. If comparing the number of items in two static sets that cannot be lined up, eg an image in their books, the children can represent the number of items in each set using cubes and these cubes can then be lined up to make it easier to identify the difference between each set. This would link very well with their experiences of comparing quantities in pictograms and block graphs from the strand of Data.
It is important that teachers are aware that establishing the extra number in the larger/greater set and establishing how many less/fewer in the smaller/lesser set requires the children comparing the amounts in two different ways. In the example above, to identify how many more bananas there are than strawberries, requires identifying the number of bananas for which there are no corresponding strawberries. However, to identify how many fewer strawberries there are than bananas, requires identifying the number of empty spaces in the strawberries that there are, opposite the extra bananas. While the answer is the same both time, the route to the answer is different, and the latter approach requires the children to count empty spaces, which is more challenging due to its abstractness.
In second class, the children will begin to use the inequalities symbols (<, >). Many children will struggle with selecting the correct symbol to use, even if they can identify the larger or smaller quantity. Thus flashcards or reference cards such as the ones at this link can be very useful to connect both the language and the symbol. Interactive quizzes like this one from That Quiz or this one from ixl.ie can provide opportunities for extra practice. However, as emphasised previously, it may still be necessary to use a visual representation of both numbers being compared, for example using stacks of cubes, base ten blocks, straws or base ten money (10c and 1c coins). In this way, the children are now beginning to use their place value understanding also to compare quantities. As well as using the actual concrete materials, the Sorting eManipulative can be used to demonstrate how to do this using images of base ten materials; see Ready to go activities 2.3 and 2.4 as examples (screenshots below).
As part of a early mathematical activities, the children will already have experienced ordering objects by length, size etc. Now, they are extending this understanding to order by quantity.
In Junior Infants, once the children are able to count individual sets of up to five objects, this enables them to start ordering the sets of objects.
Counting and numeration are both very important when it comes to ordering:
- The children are beginning to understand how higher numbers correlate with greater numbers of objects and vice versa.
- When ordering sets we must also consider the number word sequence i.e. number five comes after the number four so five must be a greater amount than four.
The nature of the English words for the ordinal numbers (first, second, third, fourth etc) and the nature of their abbreviated forms (1st, 2nd, 3rd, 4th etc) can pose significant difficulties for children as, at first glance, there appears to be little correspondence between the forms, and the abbreviations may not appear to follow any rule or pattern. Another difficulty lies in the apparent contradiction between ordinal numbers and cardinal numbers; it is typically better to have 10 rather than 1 of anything, but it is typically better to be 1st rather than 10th in any competitive activity.
- Initially the focus should be on the spoken words only and the activities used should reflect this eg lining up children at the classroom door, asking the rest of the class to identify who is first, who is second, third, last etc.
- When ready, flashcards of the ordinal words should be introduced and these can be incorporated into the familiar activities eg the flashcard with “first” can be given to a child who must give it to the child in that position in the line.
- It is better to avoid using the abbreviations until first class and it is also better to start with the words, fourth, sixth seventh and tenth. Write the word fourth on the board and establish that the children can read and understand the word. Explain that for speed we want to find a quicker way to write/indicate this position and ask them to suggest what might be written to replace the underlined part of the word (ie 4th). Repeat this with the ordinal words sixth, seventh and tenth. Ask the children to suggest how fifth, eighth and ninth might be abbreviated and then finally ask for suggestions for the words first, second and third; ultimately, tell them the correct answers if they do not arrive at them themselves. In this way, the children are being prompted to discover the system of abbreviations that we use, as opposed to being just told.
- The Happy Number Blog has an excellent, must-read post, dedicated to comparing, entitled How to Teach Kinders to Compare? Start with their Intuition! and a separate, but equally informative, post focusing on number word sequence and ordering, entitled 4 Steps toward Number Sequence Automaticity
- The Operation Maths Digital Resources have specific resources designed to support this strand unit, and have suggestions of online games in the Weblinks section. Full details of these can be found in your TRB. Click here for the Quick Start Guide to the Digital Resources.