By the end of the second year of kindergarten, the children should be able to count to at least ten and should be able to use this skill for making reasonable estimates, for ordering, comparing and determining, as well as for manipulating whole number quantities. They should, for example, be able to solve the puzzles set out in the rhyme below by calculation as well as by rhyming, or be able to check the result by using various counting procedures— provided that the sweets mentioned in the rhyme are actually present.
Counting by rhyme
Seven sugary sweets sitting in a jar,
but one was crushed between two bricks. Now there are …
Six sugary sweets. Then came an old man’s wife who took
one away. Then there were just …
Five sugary sweets. Then came my niece Marie, who took
two. Then there were just …
Three sugary sweets. Then came the man from the store,
who brought me a sweet.
Then there were …
Four sugary sweets, and then came Auntie Gwen.
She put six sweets in the jar. And then there were…
Ten sugary sweets. I ate them all myself.
Now the jar is empty sitting on the shelf.
The following classroom observation starts with the line “Four sugary sweets … put six sweets in the jar.” It shows two main strategies.
There is a drawing of a glass jar in the middle of the circle. Some children solve the problem by counting all the sweets and put them in the jar, while others count on from four or six.
This latter strategy is harder to accomplish because it involves doing two things at once: counting-on while remembering how many one has already counted so as to produce a running total. Both strategies can also be practiced with representations rather than real sweets, e.g. with fingers, lines, beads or counters. Representing quantities by various countable images is also one of the objectives for kindergarten.
How many sweets are there? Whether young children in kindergarten can answer a question like this will depend on:
– the number of sweets, whether there are more than six (five)
– the spatial form in which the sweets are arranged—in a row, in a regular pattern as on a die, or in a random heap
– whether or not the sweets are visually or physically present
– the way in which the “How many?” question is put— either directly as “How many sweets are there?” or indirectly as “How many children could you give a sweet to?”
Indirect, context-bound “How many?” questions regarding small quantities of visible objects arranged in a neat pattern are the first to be understood and correctly answered (level 1). Questions about estimating, ordering, comparing, “adding to” and “taking away” that have these characteristics are also meaningful at this level. Straightforward “How many?” questions about quantities of six or more objects that have not been neatly laid out will still initially present many children with insurmountable problems—they count asynchronously, count on past the answer, and so on.
A number of didactical routes are available for easing the transition to the object-bound “bare” numbers at level 2:
– starting with direct but easily grasped “How many?” questions about small quantities (up to four) that can be taken in at a glance, having the children check the answers, and then gradually introducing larger numbers
– using indirect context-bound “How many?” questions about larger quantities and then gradually pushing the context into the background while putting the questions more directly.
The same applies to questions about estimating, ordering, comparing, “adding to” and “taking way.” The transition to calculating operations at level 3 is stimulated by questions about estimation, ordering, comparison, “adding to” and “taking way” without using visible sweets (or other physical objects), which urges the children to work with self-conceived representations.
The task of working out “four sugary sweets plus six sugary sweets” without any visibly present sweets gives the children a lot to do, but this is justified by the results, namely pure numbers that can be conceived concretely in different ways and which can be worked with in several ways. The above attainment targets of counting and elementary calculation (making a reasonable estimate, ordering, comparing, “adding to” and “taking away”) with physical numbers that refer to concrete objects or measurements are put into a teaching framework.
This framework distinguishes five aspects of number and three developmental levels of elementary number sense. The targets are pursued with the help of spontaneous, improvised, and prepared teaching activities in interactive group teaching using a variety of didactical methods, as well as in the more individual settings of the work corners. In this way, every child counts …
Heuvel-Panhuizen, Marja van de. 2008. Children Learn Mathematics: A Learning-Teaching Trajectory with Intermediate Attainment Targets for Calculation with Whole Numbers in Primary School. The Netherlands: Sense Publishers. Open Public version.