Children’s ability to understand mathematics takes a big leap forward in the preschool years. Their strengthening ability to represent using manipulatives, symbols, and signs opens up many new possibilities. As early as age three, children can hold up fingers to indicate a quantity. The child is sometimes incorrect, but it is a mathematics behavior. By age four, children are learning to count. They can usually count to 5 or 10, and can tell you what number comes next in a series. For example, if you counted 1, 2, 3, 4, 5 and asked what comes next, the child could say, “6!” However, at this stage, saying the words and understanding the quantity linked to them is not a certainty (D. Clements & Sarama, 2004; Sarama & Clements, 2004). In geometric thinking, children can recognize and name shapes that have different sizes and orientations. They are also learning and using directional words such as “up,” “down,” “over,” “under,” and many others. By age four or five, they can use these words in a sentence to describe the orientation of an object. Measurement also takes a leap forward as children begin to compare objects by length. Three-year-olds can put two pencils next to each other and tell you which one is longer. Four-year-olds can begin to use nonstandard units to measure things; for example, they can tell you how many shoes wide the teacher’s desk is. They will need a lot of shoes to do this, because they cannot yet use one shoe repeatedly (D. Clements & Stephan, 2004; D. H. Clements, 2001; D. H. Clements & Bright, 2003). Children also make strides in algebra and patterning. As mentioned earlier, they can sequence events in time by age three. By age four, they can re-create patterns or make their own repeating pattern. During the preschool years, the child’s ability to problem solve takes on a new zeal. Children’s ability to classify objects is more developed; they can sort and organize objects into different categories and tell which pile has more.

##### Developmental Milestones in Preschool Mathematics

 3–4 Years 4–41⁄2 –Years 4 1⁄2–5 Years Numbers and Operations Child can count a collection of 1–4 items. He understands that the last word tells “how many” even if his counting lacks numerical understanding.Counting is more likesaying “ABCs”. There is sequence but not quantity. Can add by “counting all.” For example, children roll two dice that come up three and four. To determine how many spaces to move, the child counts “One, two, three” on one die and “Four, five, six, seven” on the second die. Can “count on” when using dice to play a math game or in other addition situations. The child rolls two dice. One displays three dots and the other displays four dots. He looks at the first die and says “Three” then counts on with the second die “Four, five, six, seven.” Geometry and Spatial Sense Child begins to associate names of two-dimensional and three-dimensional shapes with the same size and orientation. Can identify two-dimensional and three-dimensional shapes regardless of size differences and orientations. A triangle is still a triangle even if it is upside-down. Can use tangrams to combine shapes into new shapes. Measurement Recognizes and labels measurable attributes of objects such as length and weight. For example, they may ask “is this long enough?” Children can use a stick that is longer than the object to be measures to reproduce a length, but is unable to use something shorter. Begins to use nonstandard suits for measurement. They can use an item that is shorter then the item to be measured, but they need a number of them. For example, the child can measure the teacher’s desk with five shoes. Children begin to use standard units such as rulers to measure objects. They can use a 1-foot ruler to measure a 3-foot desk using “unit iteration.” Displaying and Analyzing Data Sorts objects and is able to count and compare the categories formed. Creates categories based on complex characteristics of objects such as “things that cut,” which could include knives, scissors, and saws. Child organizes and displays data through simple numerical representations such as bar graphs. Children can count the number in each group.