“Abstraction is only a sort of trickery and deflection of the mind if it doesn’t constitute the crowning stage of a series of previously concrete actions. The real cause of failure in formal education is therefore essentially the fact that one begins with language instead of beginning with real and material action.” -Jean Piaget, 1976
After reading this quote from Jean Piaget, I am reminded of babies learning to walk, babies don’t just walk, there are steps to get there, crawling, balance, and more. When you think of it, you might not think of all these steps. I think the same can be said with learning math, without learning the smaller steps grasping abstract concepts can be futile. Looking back at my own experience with math, I can remember teachers helping me to grasp concrete actions before moving on to more abstract ideas. Making sure I was understanding the concepts before moving on helped to create the pieces that I would then use to grasp abstract concepts in older grades. Word problems are a good example of progression into using abstraction. At a certain point, students, don’t need the concrete or representational objects in front of them, they can take the data from the word problem to find their answer. Lessons that start with concrete actions might be lessons that utilize blocks to represent number sentences, this can progress into students knowing what the numbers and symbols represent and will eventually complete automatically without attaching concrete example to them.