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Patterns and Algebraic Thinking

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Can young children do algebra? This resource illustrates the links between algebraic thinking and patterns.

by Herbert P. Ginsburg

In what sense do children engage in algebraic thinking when they solve pattern problems? Consider a six-year-old boy in Grade 1. Nicole began by showing Brian three towers of stacked blocks and then asked him to make the fourth. After he did this, Nicole asked him to explain his work. 

Brian begins by saying that he put two cubes first, and after that, four, six, and eight, suggesting that counting by twos is involved; and he argues that the bottom-most cube has to be blue because of the alternating color pattern. Nicole then asks, pointing to the towers, “What is happening to the towers each time, like from here to here?” (an excellent question). Brian says, and indicates with his fingers, that they increase by “two more” each time. Clearly, he has the idea of a pattern growing by twos.Next Nicole asks Brian to think about what a fifth tower would entail.

He correctly says that there would be 10 cubes, and explains his answer by referring to increases of two. When pressed to think about how many blue cubes he would need for the fifth tower, Brian counts the blues in the fourth tower and says that he would need to add one more to make the fifth.

Nicole then poses a challenging question: “How many blocks do you think we’re going to need for the eighth tower?” Note that solving the problem requires some fairly abstract thinking. Brian cannot simply add two more to the number of the previous tower to get the answer. 

Instead, he uses his count on by twos method to determine the numbers of the imaginary sixth, seventh, and then eighth towers. 

We can say that Brian’s thinking was algebraic in several essential ways:

  • Brian recognized and analyzed the patterns of growing by twos and of alternating blue and white cubes. 
  • He studied the relationship between the numbers beneath the towers and the number of cubes, although he did not have to represent the relationships abstractly, as in this statement: If n is the number of the tower, and if c is the number of cubes in the tower, then c = 2n.
  • He certainly generalized when he figured out the number of cubes in the eighth tower.
  • And he analyzed change when he talked about increases by two.

Many pre-kindergarten and kindergarten teachers will not (and should not) employ tasks as difficult as those Brian was asked to solve. Nevertheless, it is important to keep in mind what children are capable of learning in the future. In other words, your teaching should help to place children on a path to employ meaningful algebraic thinking like Brian’s. 

Patterns pervade children’s everyday lives (and ours too). Children see mathematical patterns in art, music, language, space, shape, and number. They frequently engage in patterned activities like singing, dancing or making designs and block constructions.  Teachers can help children to extend their relatively unsophisticated knowledge of mathematical patterns using art, music, and action. They can also help children explore important patterns in mathematics itself. Engaging in meaningful thinking about pattern is a first step on the road towards algebraic thinking. The study of pattern can be a joy for both teachers and children. If only older children could find similar thoughtfulness and delight in their more “advanced” (but often meaningless) study of algebra! Algebra is a subject in which techniques used in early math education, particularly exploring pattern through the use of manipulatives, returns benefits once children are elementary school students.

Additional Exercises

For additional exercises to accompany this thinking story, visit Using Unedited Videos in Your Courses.

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