One-to-One Correspondence

One-to-One Correspondence

The most fundamental concept of number is one-to-one correspondence, which is basic to understanding equivalence and conservation and necessary for counting. Children demonstrate one-to-one correspondence when they distribute items saying, “one for me, one for you,” or match materials to silhouette or picture labels on shelves and baskets during cleanup time. When you label each child’s cubby with a photograph, they associate these spaces with their belongings. Likewise, providing a sign-in sheet where each child has a preprinted line with his or her name and an adjacent blank space for signing in reinforces one-to-one correspondence.

Matching and one-to-one correspondence activities and materials include:

• Lotto boards and matching games (see Figure 10.2)
• Tossing rings of different colors onto matching-colored posts
• Musical chairs
• Repeating hand-clap patterns or clapping once for each word in a rhyme
• Solving puzzles that have one space for each matching piece (see Figure 10.2)
• Place mats with outlines for plate and utensils that children use to set their table
• Using tweezers or tongs to remove one item at a time from a full bowl and transfer it to an empty bowl
• Shadowing gameschildren repeat/mimic the motions of a leader
• Making puzzle cards (such as the ones Ms. Phyllis made for her insect unit in Chapter 6 (Figure 6.8)

Counting

A child’s ability to recite numbers, or count by rote, doesn’t necessarily mean that he or she associates the name of a number with quantity or the name of the number with its numerical symbol. To rote count, children memorize number sequences, and it is not unusual for them to skip a number or group of numbers, as in “one, two, three, six, eight, nine, ten.”

Once past ten, they also sometimes have difficulty mastering the number names as they are expressed in English and may say “eleventeen” or “twelvety.” Children gradually move from rote counting to rational counting, correctly associating the name of a number with objects in a group (Charlesworth, 2005).

It is important to work with children at their level of understanding, so a teacher would not, for example, lead rote-counting practice or use finger plays that count in descending (backwards) order before children had mastered ascending order. Rote counting is reinforced through the use of rhymes/finger plays such as, “one, two buckle my shoe, three four, shut the door. . . ,” or counting songs, like “one little, two little, three little monkeys, four little, five little . . . .” These are reinforced by regular practice and routines like “Let’s count to ten before I open the doorrepeat after me: one, two . . . .”

You promote rational counting to associate number with discrete quantities by pointing to each object as children count; also by asking children to count groups of objects and then saying, “How many did we count in this pile?” Daily opportunities abound for rational counting, including:

• Counting different numbers of sticks or straws and putting them in a can with a corresponding number of dots.
• Counting the number of children in a group seated at a table and then counting the correct number of red crayons needed so that each child has one.
• Counting the number of steps to “4” as they step on each number of a number line taped to the floor.
• Counting the number of fish in the aquarium.
• Counting off while standing in line waiting to go outside.
• Counting the number of stacking blocks needed for a construction in groups of one, two, three, etc.

Operations

Understanding numbers as parts of other numbers is the basis for the operations of addition and subtraction. Once children have achieved rational counting, whether they can represent numbers in writing or not, they can begin to perform operations. The child who takes three bears from a bin and puts them on the table, counting “one, two, three, I have three bears,” and then takes two more bears from the bin and counts, “four, fivefirst I had three and now I have five,” is beginning to perform the simple operation of combining or adding sets of objects. The child with five bears who announces separating them into two separate piles of 3 and 2, is demonstrating understanding of the concept of assigning referent numbers to subsets (Campbell, 1999). There is some evidence to suggest that in early education, emphasizing parts and wholes over direct teaching of computation steps and base 10 operations promotes a more flexible understanding of algorithms (multiple-step problem solving) (Campbell, 1999; Witzel et al, 2012). Thus if you use an open-ended question to ask a child to partition 12 Unifix cubes in as many ways as possible, the child might construct sets of 1 + 11, 2 + 10, 3 + 9, etc., but subsets of 1 + 4 + 5 + 2 would also be correct. Later on, that approach may lead to computation strategies that do not necessarily have to match the “one right way” you might remember from your own experiences with math instruction.

Understanding the relationship between parts and wholes is also the beginning of fractions. Children learn that parts may be of either equivalent or nonequivalent size. Eventually they learn that fractions represent equally divided subparts that can be combined and expressed in different ways (Charlesworth, 2005). Children should be encouraged to combine and divide whole objects and groups of objects in different ways, such as cutting or tearing paper, separating piles of objects into multiple containers, putting interlocking puzzles together, or counting the number of slices in a pizza.

Children need manipulatives to work out operations and the symbols for addition, subtraction, and equivalence before representing them abstractly with their number symbols. You help them make this transition by modeling with objects and gradually moving to two-dimensional representations (flannel board, overhead projector, paper and pencil).