How do children come to understand a concept? More specifically, how do they develop a concept of number? This is the underlying question to the work we do at the AIMS Center for Math and Science Education. In seeking the answer to this question, we have been reading research around cognition. Needless to say, we spend a lot of time thinking about thinking. One idea that I have come to understand is what an abstract concept number is. As adults, we take this for granted because we are so far removed from our experiences of constructing our own concept of number. Dr. Leslie Steffe visited the center in January and gave us a great example of this. He had us look at a picture of the woods outside his office and count eight oak trees. In the process of counting eight oak trees and reflecting on that activity, we had a conversation around the realization the “eight-ness” doesn’t reside in nature. Each of us brought forth the concept of eight within us from the construction of our many experiences of counting eight. It is through these collected experiences over many years that we have constructed a concept of eight. If we did not have these experiences and a symbol to it connect with we might scratch our heads befuddled when someone asks us to count eight oak trees.
Our ability to construct concepts relies on our physical experiences and interactions with our world. For example, over the winter break, my husband and I spent some time with my parents who live in Shaver Lake. During our time there we took a walk to an area where my father’s family had built two cabins in the 1920s. Growing up, my family spent the whole summer up there until I was about five years old and then my father’s family sold the land that the cabins were on. The cabins are long gone. The only structures that are left are the stone chimneys of both cabins, and some of the water pipes. Even without the physical structures being there, my parents and I could still vividly describe the layout of the cabins and decor to my husband. With our detailed descriptions my husband constructed his own concept of the cabins, but his concept would not be the same as my concept, because I constructed my concept based on my interaction with the physical environment of the real structures. I had so many interactions that even though it was years later I could still recall the concept of the cabins I constructed from those memories.
Now think of putting the symbol for “eight” in front of a kindergarten student who hasn’t had many experiences counting to eight and expecting them to perform mathematical operations such as adding or subtracting. How can we expect them to add on to eight or take away from eight if they barely have a concept of eight? They may simply know “eight” as the word following “seven,” and not even have any concept about “eight-ness.” For us adults it seems easy, but children are just at the beginning of a journey toward developing their concept of number.