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Patterns serve as the cornerstone of algebraic thinking. “Childen watch the sun setting every day; listen to stories, songs, and verses that follow patterns; notice how a puppy plays and sleeps on a schedule; jump rope to patterned chants; and skip over sidewalk bricks laid in patterns” (Copley 2000, 83). “Recognizing, describing, extending, and translating patterns” (NCTM 2000, 90) encourage children to think in terms of algebraic problem solving. Working with patterns invites young children to identify relationships and form generalizations (NCTM 2000). When we consider patterns, many of us focus on a series that repeats. For example, “day, night, day, night, day, night . . .” Indeed, the repeating pattern is one of the distinctive forms of patterning. As the name infers, repeating patterns contain a segment that continuously recurs. The segment can vary in size and level of complexity, but the simplest includes just two items.

With toddlers we can begin building the foundations of algebra through everyday experiences with patterns. For example, a teacher shares with two-year-olds a “clap, tap, clap, tap, clap, tap . . .” pattern. The children are invited to join in as soon as they are ready to do so. For toddlers, simple patterns involving movement and rhythm are quite beneficial.

In a prekindergarten classroom, four-year-old Ethan places blocks in a simple pattern (see Figure 1). Clearly, he knows something about repeating patterns. When Ms. Trep asks him to “read” the pattern, Ethan excitedly explains, “Well, it’s blue, red, blue, red, blue, red, blue, red . . .” Ms. Trep encourages him to point to each block as he “reads” the pattern. Then, to help Ethan move to the next level, Ms. Trep sets up a series of three color blocks in a repeating format (green, yellow, blue, green, yellow, blue, green, yellow, blue . . .). She asks, “What comes next?” When Ethan says he’s not sure, Ms. Trep asks him to read the pattern. In so doing, the child uses many senses; he points to the blocks, looks at them, says and hears the colors in the pattern. Ethan quickly realizes that the green block is next in the pattern. When Ms. Trep asks him to explain why it is a pattern, he replies, “You know it’s a pattern because it goes again and again!”

In this situation, Ms. Trep helps Ethan assimilate information through his senses, allowing him to successfully tackle a more complex pattern. Ethan faces the developmentally appropriate degree of challenge offered to him within a comfortable and nurturing environment. After further similar experiences, he will be ready to try more complex patterns.

Another way to increase the level of pattern complexity is to use attributes other than color. Beginning with a series of two, a teacher can encourage children to focus on shape and size as repeating attributes. Patterns such as “triangle, circle, triangle, circle, triangle . . .” or “big, small, big, small, big, small . . .” invite children to continue repeating patterns using more complex attributes. It is important to remove color as an attribute when working on shape or size patterns. For example, “hexagon, trapezoid, hexagon, trapezoid . . .” may look like a shape pattern, but if all the hexagons are yellow and all the trapezoids are red, a child may simply focus on color. The attribute of focus is quite apparent when the child is asked to read the pattern. With shape or size patterns, teachers can negate color by providing all same-color shapes (for instance, hexagons and trapezoids are blue) or increase the complexity by having a full mixture of colors (such as blue hexagon, green trapezoid, red hexagon, green trapezoid, yellow hexagon, red trapezoid). In this way, children are encouraged to problem solve and focus on the repeating attribute without relying on color.

While young learners can use shape, size, and color as attributes for patterns, they also can use numerous other attributes. Simple position patterns such as “up, down, up, down . . .” or “top, side, front, top, side, front . . .” encourage children to use spatial orientations as the repeating features. Virtually any sorting category—the number of holes in buttons, the texture of shells—can be used as a repeating attribute. Kindergartner Aneil creates a repeating pattern with plastic bugs from the science center (see Figure 2). When I ask Aneil to read his pattern for the class, he proudly exclaims, “Wings, no wings, wings, no wings, wings, no wings . . .” His algebraic thinking and creativity are simply brilliant! He is isolating one attribute but also negating several others, which is higher level thinking in action.

When working with repeating patterns, it is important to present the repeating nature of the pattern. A pattern is not a pattern until it repeats. For example, we should not present children with  and say continue the pattern, because there is not yet a pattern in this series. However, by displaying the repetitive nature, as in  the pattern is apparent and can be extended.

Not only do young children need many experiences with repeating patterns, they also can work with growing patterns that increase (or decrease) by a constant amount. The simplest of all growing patterns increases by one and begins with a small number. For example, “1, 2, 3, 4, 5, 6...” is a growing pattern based on a constant change of plus one. This is best understood by young children through concrete representation. With linking cubes, the plus-one pattern can be visually shared with children. Stack linking cubes in towers and line up the towers in order of size. Children are quick to notice that the result looks like steps or a staircase. They can explain what should come next in the pattern by making the next tower.

To increase the level of difficulty, we can present children with growing patterns that increase (or decrease) by more than one and start with numbers/quantities other than one or zero. Figure 3 shows a growing pattern presented to first-graders. Initially the children share what they discern about the blocks. Ebony and Christopher notice the blocks are lined up. Juan explains, “It looks like steps to a giant’s house—’cause the steps are really big.” Juan’s comment opens the door for the teacher to ask, “How big are the steps?” Maria quickly responds, “Each one is bigger by the same, and that’s 3—so the next one is 14!” The class agrees, and Maria constructs a stack of 14 cubes. The lesson continues and the children construct stacks of 17, 20, 23, and 26. At this point, Khavin exclaims, “This could go on forever!” “You’re right, Khavin,” says the teacher, “and that’s why mathematicians use three dots to show that a series continues!”

Over the next several days the children use their newly acquired algebraic thinking skills
to construct their own growing patterns in their math journals. Lex has a solid understanding of simple growing patterns. She writes, “This is a growing pattern. A growing pattern goes like a stairs—you can add for 2, 3, 4, or 5, and it grows and grows” (see Figure 4). Khavin, who depicts a growing pattern that increases by two, writes, “A growing pattern is like mine. It keeps growing up and up. My secret is 2” (see Figure 5).

The children’s illustrations and words give the teacher a clear understanding of where each child is in her or his construction of knowledge. She sees that Lex is ready to try patterns that increase (or decrease) by more than one; and Khavin, who understands repeated function through the idea of a “secret,” is ready for more elaborate patterns. Each child needs the appropriate challenges to continue to make meaningful connections within these algebraic expressions.

Growing patterns and repeating patterns are often more distinct when displayed in a linear fashion. However, some real-world patterns are nonlinear. Concentric patterns that grow from the inside to the outside are one type of nonlinear pattern and can be found in artwork, fabric designs, and nature. The nautilus shell provides a mathematically brilliant concentric pattern (the ratio of the chambers within the spiral is a perfect 1 to 1.6180). Other nonlinear patterns are common in floor tiles, buildings, even windblown sand. We need to encourage young children to notice and describe the many types of patterns found in their worlds.