Symbolic Representation

Symbolic Representation

Like counting, learning to represent numbers with the corresponding numeral is developmentally sequenced. A child who counts correctly does not necessarily associate the number with its matching symbol. Thus many materials that support learning to countsuch as an abacus, Unifix cubes, or dominoesdo not feature numerals. Conversely, being able to trace or write a number may not mean that the child understands the connection between the numeral and the quantity it represents. In order for that to happen, children need to recognize the symbol, be able to identify and associate it with the correct number of objects in a set, and then represent the number in writing legibly (Charlesworth, 2005, p. 218; Witzel, Ferguson, & Mink, 2012).

Typical examples of materials that focus on one or more of these three tasks are included in Figure 10.2; they should be accessible and used in both preschool and primary classrooms (Witzel, Ferguson, & Mink, 2012). Because of individual variations in the development of this concept, teachers can best help children to acquire symbolic representation through one-on-one and small-group activities such as:

• Making number books with stickers or stamps (Seefeldt & Galper, 2004)
• Writing numerals on sequence picture charts
• Using magnetic letters to represent numbers
• Using calendars with blank spaces and modeling writing in the numbers
• Making numbers out of pipe cleaners, play dough, or wire

Algebra

The foundations for algebra begin with understanding classification, ordering, and patterns (Taylor-Cox, 2003). Classification (sorting and grouping) activities help children begin to distinguish, compare, and categorize concrete objects by characteristics or attributes (such as color, shape, or size) and reinforce the concept of sets. Applying comparison skills leads to identification of simple color, shape, or sound patterns in the environment necessary for algebraic reasoning.

Classification

Classification includes sorting and grouping. When children sort, they separate (subtract) objects into categories; when they group, they combine (add) by noting the characteristics that items have in common, thus making and rearranging sets according to different criteria.

Children begin classifying by a single obvious attribute, such as color or shape (red bears, blue bears, yellow bears; squares, circles, triangles). At first they don’t necessarily know or express what the criteria are, but they do demonstrate observation about how the items are alike or different.

Gradually they begin to classify by less obvious criteria such as material (hard/soft), pattern (striped/checked), texture (rough/smooth), or function (moves/doesn’t move) and more sophisticated characteristics such as animal type (sea/land animals or reptiles/birds). They also move to classifying objects by more than one attribute, such as color and size (big blue cars, small blue cars, big red cars, small red cars). A child’s approach to classification reflects his or her growing sense of logic. It is important to understand, however, that children represent logical thinking in different ways; what seems an obvious attribute to one child might not be apparent to another. You can learn a lot about how children think by asking them to explain the way they sorted a particular group of objects.

Teachers promote classification indirectly with activities such as lining up by shirt color or by boys vs. girls or by listing foods that children like/dislike. They also provide intentional sorting activities with materials such as buttons or beads. Children can also be encouraged to place different toys and propssuch as blocks, play foods, and carsinto appropriately labeled bins during cleanup time.

Teachers facilitate classification skills by introducing and modeling different ways to sort or group and suggesting more challenging criteria. Using effective language will also help. The directions teachers give to children

• Reflect recognition of what they observe, such as “It looks like you are sorting the animals according to where they live. Is that right?”
• Add to the child’s repertoire of criteria, such as “I see you have sorted the buttons into color groups. Can you sort them again by how big they are?”
• Encourage application of multiple criteria as a child’s ability to sort by a variety of single attributes develops, as by saying, for example, “Hmmm we have all these turtles. Some are big, some are small, and they are yellow, brown, and green. Do you think you could sort them into big/green, small/green, big/yellow?”
• Encourage children to describe and label the criteria they are using.

Classification activities provide a natural segue to graphing and representing data, as discussed a bit later in this section.

Ordering

At first children compare pairs of objects and later each additional item to the selected criteria. Ordering, or putting items into seriesalso called seriationrepresents the graduated comparison of more than two things or sets in a larger group. This procedure is more complex than making comparisons in pairs (Charlesworth, 2005).

Seriation activities focus on ordering according to any applicable attribute, by size in ascending or descending order, or temporal ordering of events from first to last (Hendrick, 2007), or ordering by graduated differences such as color shades, sound, or weight.

Any activities that involve paired comparisons can be extended to ordering and seriation simply by addition of materials such as:

• Graduated paper shapes
• Sticks, straws, or rods of different lengths (e.g., Cuisenaire rods)
• Nesting items such as measuring cups and spoons
• Arranging different shades of a single color in order from lightest to darkest
• Storing pots and pans in dramatic play on hooks from smallest to largest
• Arranging pictures of children in order of birthday from January to December
• Lining up by size
• Time lines
• Sequencing picture cards or flannel-board cutouts for familiar stories such as Goldilocks, The Three Little Pigs, or If You Give a Mouse a Cookie
• Playing/singing a favorite song several times, varying the volume from soft to loud
• Using pictures to represent the daily routine and having children put them in order

Note that because of preschoolers’ egocentrism, they often represent people or objects in their drawings and paintings by relative importance rather than by actual size/proportion. Thus, in a picture of mom’s flower garden, mom and the flowers might be bigger than the house! Comparing, seriation, and ordering activities with real objects help children gradually move from psychological to more accurate concrete representations.