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The final big idea in algebraic thinking is change. As noted in the math standards, “The understanding that most things change over time, that such changes can be described mathematically, and that changes are predictable helps lay a foundation for applying mathematics” (NCTM 2000, 95). We must encourage young children to notice and describe many different changes.

According to the standards, there are two algebraically oriented types of change: qualitative and quantitative. Qualitative change entails many familiar experiences that are part of children’s lives: A pair of shoes feels smaller as the child’s feet grow; the sunflower is taller than it was last week; a bucket fills with water as the rain continues all day. All of these changes are qualitative. They are described with relative mathematical labels such as smaller, taller, and fuller. The changes occur over time and are fairly predictable. Quantitative changes are also part of children’s lives: The child’s shoe size changes from 10 to 11; the sunflower grows 3 centimeters in one week; the amount of water in the bucket increases by 50 milliliters every 30 minutes of a three-hour rainfall. The mathematic language to describe changes incorporates more precise numeric language. Essentially, the use of exact amounts differentiates quantitative change from qualitative change. Young children need to notice and think about both types of change.

Play offers young children many experiences with change. While filling a bag with blocks, prekindergartners notice that the bag gets heavier with each additional block. Using a cardboard ramp and a tennis ball, kindergartners discover that changing the height of the ramp changes the distance the ball rolls. With chain links and weights, first-graders discover and discuss how the length of the chain influences the number of pendulumtype swings. In each of these situations, the children analyze changes that occur as they experiment while playing.