### Numbers

Most kindergarten children understand one-to-one correspondence concepts and are beginning to develop an understanding of additive relationships and cardinality. Because of the development of a system of ones and hierarchical inclusion, most children of this age can not only count, but also understand that with each number they say, the quantity increases by one. When a kindergarten child counts “five, six,” he understands that in essence he is adding “one” to “five.” Because of this additive thinking, children can begin to compose and decompose sets of numbers. For example, they understand that three red chips and two blue chips make five chips (compose). They also understand that a group of five chips can be broken down into smaller groups of four and one or three and two (decompose). While this is not the formal addition that we will see in first grade, this additive thinking makes it possible to introduce some simple addition games and activities to the curriculum. Kindergarten children may not be able to handle addition with numerical notation such as “3 &plus; 5,” but they can count the dots on dominos or other items. Children in kindergarten may count all or, if they are a bit more advanced, they may count on. For example, if playing a path game that requires two number cubes to be thrown and the game piece to be moved based upon the resulting number, a child might count all of the dots on both cubes. This is composition of a new number set because the child is combining two smaller sets of numbers to make a larger number, then moving his game piece according to that new number. Another strategy is for the child to count on from the higher number—he might throw a five and a three and say “five . . .” and then count the dots on the other cube, “six, seven, eight.” This is called counting on. It is still composition of a new set; it is simply a more advanced strategy than counting all (Smith, 1997). Kindergarten teachers can encourage these composition strategies through the use of games. To encourage children to use counting on, teachers can replace one of the cubes that has dots with one that has numerals. Since there are no dots to count, it encourages the child to count on. It is possible that the child will still count to the numeral on his fingers and then count the remaining dots. If this is the case, the child is not ready to advance to counting on. Kindergarten children have sufficient control of their fingers and hand muscles to permit the use of a pencil. In fact, they begin to prefer writing with a pencil rather than a crayon or marker. Their memory level allows them to remember the shapes of numbers and recall them when writing. Number and letter reversals are common in kindergarten children due to factors such as memory or perception. Given their development, this is normal. Children may have no problem aligning numbers on the top-to-bottom axis so they don’t write them upside-down, but coordinating both the top-to-bottom axis and the left-to-right axis may be challenging for children this age.

### Measurement

During this time, many of the cognitive structures of measurement are still developing. Two of the most important of these are transitivity and conservation of length. An understanding of transitivity allows children to compare two lengths even if they are not right next to each other. For example, if shown two towers that are not next to each other and asked if he can use a dowel rod to compare their heights, a child with transitive thought can compare the first tower to the rod and then use the rod to assess the height of tower two, to determine if they are the same. Conservation of length is the understanding that length remains constant regardless of appearance. These two concepts should be the focus of the kindergarten measurement program. In this way, kindergarten children can begin to develop an understanding of iterated units of measurement, such as a ruler broken down into inches (Clements & Stephan, 2004). Iterated units of measurement can be either standard, such as inches or meters or nonstandard, like shoes or paperclips. Both can be used the same way in unit iteration, but a standard unit makes it more efficient for a society to communicate measurements. Some research has shown that the coordination of these two cognitive developments can take until fourth grade to develop to a point where children can use measurement techniques efficiently and flexibly. However, activities and interactions using nonstandard units to find the length of classroom objects support the early development and coordination of these concepts. For example, kindergarteners can use paperclips to measure the length of their books or desks. Children also begin to understand the measurement of area and can cover a picture with a number of squares and then count the squares. They can compare the number of squares it takes to cover different shapes, objects, and pictures.

### Units of Time.

Kindergarten children begin to understand that time can be divided into units such as hours, minutes, and seconds. However, it is often hard for them to mentally gauge the passage of time using these units. For example, a child may be told that the class will be going to the playground in 15 minutes. Five minutes later the child may ask, “Has it been 15 minutes yet?” With kindergarteners, it is often a good idea to use clocks to help them gauge time. The teacher can then say, “It will be 15 minutes when the big hand is on the 12.” Time is most relevant to the kindergarten child when it is linked to his daily schedule and everyday activity. It is difficult for him to link the understanding that time can be segmented into units with how long those units are, because the passage of time is based on perception. The old adage “time flies when you are having fun” is true. Fifteen minutes sitting outside the doctor’s office waiting to get a shot feels a lot longer than 15 minutes playing on the playground.

### Patterns and Algebra

Children’s classification schemes are more complex than they were in preschool. Kindergarteners can now sort objects by one or more attributes. They also use mathematical attributes to sort and seriate objects or groups of objects such as, “This pile has eight this one has four, and this one has two.” They can also sort objects into three distinct classes, such as small, medium, and large. Kindergarten children are also becoming better at using their representational abilities to make mathematical models (Dolk, 2004). Katherine Fosnot describes mathematical modeling as part of “mathematizing,” or the process of using mathematics to structure the world rather than just “learning” it. In her book Young Mathematicians at Work (Fosnot & Dolk, 2001), she shows how children use mathematical models. In one example, a teacher showed children a picture biscuits on a worksheet and asked them if there were enough for everyone in the class to have one. Three students decided to use small cubes to model the problem letting the cubes stand in for biscuits. When asked what they were going to do with the cubes, one child began to put red cubes on each of the biscuits while another gave a brown cube to each child in the class. Once each biscuit had a red cube on it, she counted them and got 22. The other child finished handing out the brown cubes and after ensuring that everyone received one, began collecting the cubes and counted 15. The students then said there were enough, because “22 is more than 15.”

### Geometry and Spatial Sense

Kindergarten children can use positional words in sentences and can give directions to other students. Preschool children learn how to use words like “behind” or “next to” and can understand them, but have difficulty using them correctly. They have no problem saying, “The scissors are on the shelf next to the sink under the paper towels.” They can also locate things in three-dimensional spaces using a two-dimensional representation such as a map. For example, children can follow a treasure map around the classroom to find a bag of candy hidden under the sink. If “X” marks the spot under the sink on the map, they can translate that into the three-dimensional world of the actual classroom. They can also use toys to make a fairly accurate map of their classroom.

### Data Analysis, Probability and Statistics

Kindergarteners can use graphs in more dynamic ways than preschool children, using them to collect, analyze, and represent data. They can then use them to make decisions, such as “What snack are we going to have today?” or “What book are we going to read next?” Kindergarteners are good at making graphs out of things in their environment such as shoes, blocks, or even people. They can also use pictures to create graphs to represent data. While preschoolers can construct similar simple graphs, kindergarteners can use the information in the graph to answer questions such as: “Which group has less? Which group has more? Which group is the smallest? Which group is the biggest?” They can also compare individual factors. For example, if a graph is about a favorite color, each student can put a colored square on the board to make a stacked bar graph. They can then answer questions such as, “Do more people like red or yellow?” Kindergarteners are able to extract useful mathematical information from pictures or other visual media. These can become early versions of “word problems.” For example, children can look at pictures of houses and figure out which one has the most windows. They are able to problem solve and should be involved in what Fosnot called mathematizing, or making sense of the world using mathematics. Kindergarten is the beginning of the formal mathematics that children in the first grade and up will be involved with, such as addition, subtraction, multiplication, fractions, and decimals (Burns & Silbey, 2000).